## Envision Math Common Core 4th Grade Answers Key Topic 10 Extend Multiplication Concepts to Fractions

Essential Questions:
How can you describe a fraction using a unit fraction? How can you multiply a fraction by a whole number?
A fraction can be converted into a unit fraction as explained below:
Put the whole number over 1 to turn it into a fraction.
Ex:
The whole number is: 5
The conversion of 5 into a unit fraction is: $$\frac{1}{5}$$
A fraction can be multiplied by a whole number as explained below:
Step 1:
Multiply across the top number by top number (numerator) and bottom number by bottom number (denominator).
Step 2:
Simplify your answer by dividing both numerator and denominator by the same number, when applicable.

enVision STEM Project: Light and Multiplication
Do Research
Use the Internet or other sources to research the words transparent, translucent, and opaque? Write a definition for each word.
Transparent:
If any material allows all light to pass through it, then it is called “Transparent”
Translucent:
If any material allows some light to pass through it, then it is called “Translucent”
Opaque:
When any material allows no light to pass through it, then it is called “Opaque”

Journal: Write a Report
Include what you found. Also in your report:
PART A:
List 1 example of each of the items that are transparent, translucent, or opaque.
Examples of Transparent items are:
a. Clean glass
b. Water
c. Air
Examples of Translucent items are:
a. Frosted glass
b. Wax paper
c. Smoke
d Butter paper
Examples of Opaque items are:
a. Stone
b. Metal
c. Wood

PART B:
Suppose one-third of each of 5 same-sized posters is covered with opaque paper. What fraction of the posters are not covered by opaque paper? Explain how to use multiplication to find what parts of the posters are not covered by opaque paper?
It is given that
One-third of each of 5 same-sized posters is covered with opaque paper
Now,
The representation of the 5 same-sized posters with opaque paper is:

Now,
From the given figure,
We can observe that
The total number of parts are: 5
The number of parts covered with opaque paper is: 3
The number of parts that are not covered with opaque paper is: 2
Now,
The fraction of the posters that are covered by the opaque paper = $$\frac{The number of parts that are covered with opaque paper}{The total number of parts}$$
= $$\frac{3}{5}$$
So,
The fraction of the posters that are not covered by the opaque paper = $$\frac{The number of parts that are not covered with opaque paper}{The total number of parts}$$
= $$\frac{2}{5}$$
Hence, from the above,
We can conclude that
The fraction of the posters that are not covered by the opaque paper is: $$\frac{2}{5}$$

Review What You Know

Vocabulary

• equivalent fractions
• mixed number
• fraction
• whole number

Choose the best term from the box. Write it on the blank.
Question 1.
A _________ has a whole number and a fraction.
We know that,
A “Mixed number” has a whole number and a fraction.
Hence, from the above,
We can conclude that
The best term that is suitable for the given sentence is: Mixed number

Question 2.
Fractions that name the same region, part of a set or part of a segment are called __________
We know that,
Fractions that name the same region, part of a set, or part of a segment are called “Equivalent fractions”
Hence, from the above,
We can conclude that
The best term that is suitable for the given sentence is: Equivalent fractions

Question 3.
A _________ has a numerator and a denominator.
We know that,
A “Fraction” has a numerator and a denominator
Hence, from the above,
We can conclude that
The best term that is suitable for the given sentence is: Fraction

Identifying Fractions

Write the fraction shown by each model.
Question 4.

The given model is:

Now,
From the given model,
The total number of parts are: 4
So,
The fraction represented by the given model = $$\frac{The shaded part}{The total number of parts}$$
= $$\frac{1}{4}$$
Hence, from the above,
We can conclude that
The fraction represented by the given model is: $$\frac{1}{4}$$

Question 5.

The given model is:

Now,
From the given model,
The total number of parts are: 3
So,
The fraction represented by the given model = $$\frac{The shaded part}{The total number of parts}$$
= $$\frac{2}{3}$$
Hence, from the above,
We can conclude that
The fraction represented by the given model is: $$\frac{2}{3}$$

Question 6.

The given model is:

Now,
From the given model,
The total number of parts are: 8
So,
The fraction represented by the given model = $$\frac{The shaded part}{The total number of parts}$$
= $$\frac{5}{8}$$
Hence, from the above,
We can conclude that
The fraction represented by the given model is: $$\frac{5}{8}$$

Question 7.

The given model is:

Now,
From the given model,
The total number of parts are: 5
So,
The fraction represented by the given model = $$\frac{The shaded part}{The total number of parts}$$
= $$\frac{1}{5}$$
Hence, from the above,
We can conclude that
The fraction represented by the given model is: $$\frac{1}{5}$$

Question 8.

The given model is:

Now,
From the given model,
The total number of parts are: 10
So,
The fraction represented by the given model = $$\frac{The shaded part}{The total number of parts}$$
= $$\frac{5}{10}$$
Hence, from the above,
We can conclude that
The fraction represented by the given model is: $$\frac{5}{10}$$

Question 9.

The given model is:

Now,
From the given model,
The total number of parts are: 2
So,
The fraction represented by the given model = $$\frac{The shaded part}{The total number of parts}$$
= $$\frac{1}{2}$$
Hence, from the above,
We can conclude that
The fraction represented by the given model is: $$\frac{1}{2}$$

Unit Fractions

Write a fraction for each statement.
Question 10.
3 copies of $$\frac{1}{6}$$ is _______.
The given statement is:
3 copies of $$\frac{1}{6}$$
So,
The representation of the given statement is: 3 × $$\frac{1}{6}$$
So,
3 × $$\frac{1}{6}$$
= $$\frac{3}{6}$$
Hence, from the above,
We can conclude that the fraction for the given statement is: $$\frac{3}{6}$$

Question 11.
9 copies of $$\frac{1}{12}$$ is _________
The given statement is:
9 copies of $$\frac{1}{12}$$
So,
The representation of the given statement is: 9 × $$\frac{1}{12}$$
So,
9 × $$\frac{1}{12}$$
= $$\frac{9}{12}$$
Hence, from the above,
We can conclude that the fraction for the given statement is: $$\frac{9}{12}$$

Question 12.
5 copies of $$\frac{1}{5}$$ is ___________
The given statement is:
5 copies of $$\frac{1}{5}$$
So,
The representation of the given statement is: 5 × $$\frac{1}{5}$$
So,
5 × $$\frac{1}{5}$$
= $$\frac{5}{5}$$
Hence, from the above,
We can conclude that the fraction for the given statement is: $$\frac{5}{5}$$

Question 13.
3 copies of $$\frac{1}{10}$$ is _________.
The given statement is:
3 copies of $$\frac{1}{10}$$
So,
The representation of the given statement is: 3 × $$\frac{1}{10}$$
So,
3 × $$\frac{1}{10}$$
= $$\frac{3}{10}$$
Hence, from the above,
We can conclude that the fraction for the given statement is: $$\frac{3}{10}$$

Question 14.
6 copies of $$\frac{1}{8}$$ is _________.
The given statement is:
6 copies of $$\frac{1}{8}$$
So,
The representation of the given statement is: 6 × $$\frac{1}{8}$$
So,
6 × $$\frac{1}{8}$$
= $$\frac{6}{8}$$
Hence, from the above,
We can conclude that the fraction for the given statement is: $$\frac{6}{8}$$

Question 15.
7 copies of $$\frac{1}{10}$$ is _________.
The given statement is:
7 copies of $$\frac{1}{10}$$
So,
The representation of the given statement is: 7 × $$\frac{1}{10}$$
So,
7 × $$\frac{1}{10}$$
= $$\frac{7}{10}$$
Hence, from the above,
We can conclude that the fraction for the given statement is: $$\frac{7}{10}$$

Equivalent Fractions

Question 16.
Draw a rectangle that shows 8 equal parts. Shade more than $$\frac{3}{8}$$ of the rectangle but less than $$\frac{5}{8}$$. What fraction did you model? Use multiplication or division to write two equivalent fractions for your model.
It is given that
Draw a rectangle that shows 8 equal parts. Shade more than $$\frac{3}{8}$$ of the rectangle but less than $$\frac{5}{8}$$
Now,
The representation of the rectangle with the total number of parts and shaded parts is:

Now,
From the given figure,
We can observe that
The total number of parts are: 8
The number of shaded parts is: 4
The number of non-shaded parts is: 4
Now,
The fraction of the shaded part from the model = $$\frac{The number of shaded parts}{The total number of parts}$$
= $$\frac{4}{8}$$
= $$\frac{1}{2}$$
Hence, from the above,
We can conclude that
a. $$\frac{4}{8}$$
b. $$\frac{1}{2}$$

Pick a Project

PROJECT 10A
Would you like to work with tiles?
Project: Design with Tiles

PROJECT 10B
What cause would you donate your time or money to?
Project: Set Up a Charity Event

PROJECT 10C
How fast can a jet aircraft travel?
Project: Write and Perform a Skit

PROJECT 10D
How would you like to run a marathon?
Project: Make a Game about Marathon Winners

### Lesson 10.1 Fractions as Multiples of Unit Fractions

Solve & Share
Kalil and Mara were working on their math homework. Mara wrote $$\frac{4}{5}$$ as $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$. Kalil looked at Mara’s work and said, “I think you could use multiplication to rewrite your equation.” Is Kalil’s observation correct? Explain.
I can … use fraction strips or number lines to understand a fraction as a multiple of a unit fraction.

It is given that
Kalil and Mara were working on their math homework. Mara wrote $$\frac{4}{5}$$ as $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$. Kalil looked at Mara’s work and said, “I think you could use multiplication to rewrite your equation.”
Now,
We know that,
“Repeated addition” is also known as “Multiplication”. If the same number is repeated then in short we can write that in the form of multiplication
Ex:
2 is repeated 5 times so in short, we can write this addition as 2 x 5.
So,
According to Kalil, Mara’s work can also be represented in the form of multiplication
So,
$$\frac{4}{5}$$ can also be written as:
$$\frac{4}{5}$$ = 4 × $$\frac{1}{5}$$
Hence, from the above,
We can conclude that Kalil’s observation is correct

Look Back! Model with Math
Write an equation to show the relationship between Mara’s work and Kalil’s observation.
According to Mara,
$$\frac{4}{5}$$ = $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$ + $$\frac{1}{5}$$
Now,
We know that,
“Repeated addition” is also known as “Multiplication”. If the same number is repeated then in short we can write that in the form of multiplication
So,
According to Kalil,
Mara’s work can also be represented as:
$$\frac{4}{5}$$ = 4 × $$\frac{1}{5}$$
Now,
The relationship between Mara’s work and Kalil’s observation is:
$$\frac{4}{5}$$ = 4 × $$\frac{1}{5}$$
So,
$$\frac{1}{5}$$  is repeated 4 times
So,
In short, we can write this addition as 4 x $$\frac{1}{5}$$
Hence, from the above,
We can conclude that
The relationship between Mara’s work and Kalil’s observation is:
$$\frac{1}{5}$$  is repeated 4 times
So,
In short, we can write this addition as 4 x $$\frac{1}{5}$$

Essential Question
How Can You Describe a Fraction Using a Unit Fraction?
A fraction can be converted into a unit fraction as explained below:
Put the whole number over 1 to turn it into a fraction.
Ex:
The whole number is: 5
The conversion of 5 into a unit fraction is: $$\frac{1}{5}$$

Visual Learning Bridge
Courtney ran $$\frac{3}{4}$$ of the way to school. Describe $$\frac{3}{4}$$ using unit fractions.

A unit fraction is a fraction that describes one part of the whole. Unit fractions always contain the numerator 1.

When a whole is divided into four equal parts, each part is described as the unit fraction $$\frac{1}{4}$$.
Decompose $$\frac{3}{4}$$ into unit fractions.

Repeated addition can be represented as multiplication.

Convince Me! Reasoning
The number $$\frac{5}{8}$$ is a multiple of what unit fraction? Explain.
The given fraction is: $$\frac{5}{8}$$
In the given fraction,
The numerator represents the shaded parts
The denominator represents the total number of parts
Now,
The representation of $$\frac{5}{8}$$ is:

Now,
From the given model,
We will add $$\frac{1}{8}$$ 5 times
So,
5 × $$\frac{1}{8}$$ = $$\frac{5}{8}$$
Hence, from the above,
We can conclude that
$$\frac{5}{8}$$ is a multiple of $$\frac{1}{8}$$

Another Example!
Describe $$\frac{5}{4}$$ as a multiple of a unit fraction.

Guided Practice

Do You Understand?
Question 1.
Draw a picture to explain why $$\frac{3}{5}$$ = 3 × $$\frac{1}{5}$$.
The given fraction is: $$\frac{3}{5}$$
In the given fraction,
The numerator represents the shaded parts
The denominator represents the total number of parts
Now,
The representation of the given fraction in the form of unit fractions is:

Now,
From the given model,
We will add $$\frac{1}{5}$$ 3 times
So,
3 × $$\frac{1}{5}$$ = $$\frac{3}{5}$$
Hence, from the above,
We can conclude that
$$\frac{3}{5}$$ = 3 × $$\frac{1}{5}$$

Question 2.
Write a multiplication equation to show each part of the following story. Mark’s family ate $$\frac{7}{4}$$ chicken pot pies for dinner. There are 7 people in Mark’s family. Each family member ate $$\frac{1}{4}$$ of a pie.
It is given that
Mark’s family ate $$\frac{7}{4}$$ chicken pot pies for dinner. There are 7 people in Mark’s family. Each family member ate $$\frac{1}{4}$$ of a pie.
So,
The representation of the given information in the form of a bar diagram is:

So,
The equation that shows the given information is:
(The total number of people present in Mark’s family) × (The fraction of pie eaten by each family member of Mark) = (The total amount of chicken pot pies eaten by Mark’s family for dinner)
7 × $$\frac{1}{4}$$ = $$\frac{7}{4}$$
Hence, from the above,
We can conclude that
The multiplication equation to show the given information is:
(The total number of people present in Mark’s family) × (The fraction of pie eaten by each family member of Mark) = (The total amount of chicken pot pies eaten by Mark’s family for dinner)

Do You Know How?
For 3-6, write each fraction as a multiple of a unit fraction. Use a tool as needed.
Question 3.
$$\frac{2}{3}$$ = ______ × $$\frac{1}{3}$$
The given fraction is: $$\frac{2}{3}$$
Now,
In the given fraction,
The numerator represents the shaded part
The denominator represents the total number of parts
Now,
The representation of the given fraction as a multiple of a unit fraction is:

Hence, from the above,
We can conclude that
$$\frac{2}{3}$$ = 2 × $$\frac{1}{3}$$

Question 4.
$$\frac{5}{6}$$ = 5 × $$\frac{1}{}$$
The given fraction is: $$\frac{5}{6}$$
Now,
In the given fraction,
The numerator represents the shaded part
The denominator represents the total number of parts
Now,
The representation of the given fraction as a multiple of a unit fraction is:

Hence, from the above,
We can conclude that
$$\frac{5}{6}$$ = 5 × $$\frac{1}{6}$$

Question 5.
$$\frac{4}{2}$$ = 4 × $$\frac{1}{}$$
The given fraction is: $$\frac{4}{2}$$
Now,
In the given fraction,
The numerator represents the shaded part
The denominator represents the total number of parts
Now,
The representation of the given fraction as a multiple of a unit fraction is:

Hence, from the above,
We can conclude that
$$\frac{4}{2}$$ = 4 × $$\frac{1}{2}$$

Question 6.
$$\frac{6}{5}$$ = 6 × $$\frac{1}{}$$
The given fraction is: $$\frac{6}{5}$$
Now,
In the given fraction,
The numerator represents the shaded part
The denominator represents the total number of parts
Now,
The representation of the given fraction as a multiple of a unit fraction is:

Hence, from the above,
We can conclude that
$$\frac{6}{5}$$ = 6 × $$\frac{1}{5}$$

Independent Practice

Leveled Practice For 7-12, write each fraction as a multiple of a unit fraction. Use a tool as needed.
Question 7.

The given fraction is: $$\frac{7}{8}$$
So,
The representation of the given fraction as a unit fraction is:

Hence, from the above model,
We can conclude that
$$\frac{7}{8}$$ = 7 × $$\frac{1}{8}$$

Question 8.

The given fraction is: $$\frac{3}{6}$$
So,
The representation of the given fraction as a unit fraction is:

Hence, from the above model,
We can conclude that
$$\frac{3}{6}$$ = 3 × $$\frac{1}{6}$$

Question 9.

The given fraction is: $$\frac{2}{5}$$
So,
The representation of the given fraction as a unit fraction is:

Hence, from the above model,
We can conclude that
$$\frac{2}{5}$$ = 2 × $$\frac{1}{5}$$

Question 10.
$$\frac{6}{4}$$
The given fraction is: $$\frac{6}{4}$$
So,
The representation of the given fraction as a unit fraction is:

Hence, from the above model,
We can conclude that
$$\frac{6}{4}$$ = 6 × $$\frac{1}{4}$$

Question 11.
$$\frac{9}{6}$$
The given fraction is: $$\frac{9}{6}$$
So,
The representation of the given fraction as a unit fraction is:

Hence, from the above model,
We can conclude that
$$\frac{9}{6}$$ = 9 × $$\frac{1}{6}$$

Question 12.
$$\frac{8}{5}$$
The given fraction is: $$\frac{8}{5}$$
So,
The representation of the given fraction as a unit fraction is:

Hence, from the above model,
We can conclude that
$$\frac{8}{5}$$ = 8 × $$\frac{1}{5}$$

Problem Solving

Question 13.
Mark slices $$\frac{4}{6}$$ of a tomato. Each slice is of the tomato. How many slices does Mark have? Explain by writing $$\frac{4}{6}$$ as a multiple of $$\frac{1}{6}$$.
It is given that
Mark slices $$\frac{4}{6}$$ of a tomato. Each slice is of the tomato
Now,
From the given fraction,
We can observe that
The number of shaded slices of tomato is: 4
The total number of slices of tomato are: 6
So,
The representation of the given fraction as a unit fraction is:

So,
The representation of the given fraction as a multiple of $$\frac{1}{6}$$ is:
$$\frac{4}{6}$$ = 4 × $$\frac{1}{6}$$
Hence, from the above,
We can conclude that
The number of slices does Mark have is: 6 slices
The representation of the given fraction as a multiple of $$\frac{1}{6}$$ is:
$$\frac{4}{6}$$ = 4 × $$\frac{1}{6}$$

Question 14.
Delia flew 2,416 miles the first year on the job. She flew 3,719 miles the second year. Delia flew 2,076 more miles the third year than the first and second years combined. How many miles did Delia fly the third year?
It is given that
Delia flew 2,416 miles the first year on the job. She flew 3,719 miles the second year. Delia flew 2,076 more miles the third year than the first and second years combined
So,
The number of miles Deli flew the third year = 2,076 + (The number of miles Delia flew the first and the second years combined)
= 2,076 + (2,416 + 3,719)
= 2,076 + 6,135
= 8,211 miles
Hence, from the above,
We can conclude that
The number of miles did Delia fly the third year is: 8,211 miles

Question 15.
The model with Math
The picture below shows $$\frac{6}{2}$$ pears. Write $$\frac{6}{2}$$ as repeated addition and as a multiple of a unit fraction.

It is given that
The picture below shows $$\frac{6}{2}$$ pears
Now,
The given picture is:

So,
The representation of the given fraction as a unit fraction is:

Now,
The representation of the given fraction as a repeated addition is:
$$\frac{6}{2}$$ = $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$
Now,
The representation of the given fraction as a multiple of the unit fraction is:
$$\frac{6}{2}$$ = 6 × $$\frac{1}{2}$$
Hence, from the above,
We can conclude that
The representation of $$\frac{6}{2}$$ as a repeated addition and as a multiple of unit fraction respectively is:
a. $$\frac{6}{2}$$ = $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$
b. $$\frac{6}{2}$$ = 6 × $$\frac{1}{2}$$

Question 16.
The picture below shows $$\frac{7}{2}$$ apples. Write $$\frac{7}{2}$$ as repeated addition and as a multiple of a unit fraction.

It is given that
The picture below shows $$\frac{7}{2}$$ apples
Now,
The given picture is:

So,
The representation of the given fraction as a unit fraction is:

Now,
The representation of the given fraction as a repeated addition is:
$$\frac{7}{2}$$ = $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$
Now,
The representation of the given fraction as a multiple of the unit fraction is:
$$\frac{7}{2}$$ = 7 × $$\frac{1}{2}$$
Hence, from the above,
We can conclude that
The representation of $$\frac{7}{2}$$ as repeated addition and as a multiple of unit fraction respectively is:
a. $$\frac{6}{2}$$ = $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$
b. $$\frac{7}{2}$$ = 7 × $$\frac{1}{2}$$

Question 17.
enVision® STEM Light travels at a speed of about 186,000 miles per second. How far does light travel in 5 seconds?
It is given that
Light travels at a speed of about 186,000 miles per second
So,
The distance traveled by light in 5 seconds = 5 × (The distance traveled by light in 1 second)
= 5 × 186,000
= 930,000 miles
Hence, from the above,
We can conclude that
The distance traveled by light in 5 seconds is: 930,000 miles

Question 18.
Higher-Order Thinking Kobe drinks $$\frac{1}{3}$$ cup of juice each day. He has 2$$\frac{1}{3}$$ cups of juice left. For how many days will it last? Explain by writing 2$$\frac{1}{3}$$ as a fraction and then writing the fraction as a multiple of $$\frac{1}{3}$$.
It is given that
Kobe drinks $$\frac{1}{3}$$ cup of juice each day. He has 2$$\frac{1}{3}$$ cups of juice left.
So,
The representation of 2$$\frac{1}{3}$$ as a fraction is:
2$$\frac{1}{3}$$ = $$\frac{8}{3}$$
But,
It is given that
Kobe drinks $$\frac{1}{3}$$ cup of juice each day
So,
The representation of $$\frac{8}{3}$$ as a multiple of the unit fraction is:
$$\frac{8}{3}$$ = 8 × $$\frac{1}{3}$$
So,
The total number of cups of juice left = (The total number of days) × (The number of cups Kobe drinks each day)
Hence, from the above,
We can conclude that
The number of days that $$\frac{8}{3}$$ cups of juice will last is: 8 days

Assessment Practice

Question 19.
Which multiplication equation describes the fraction plotted on the number line?

The given number line is:

Now,
We know that,
We will write any fraction in terms of a multiple of the unit fraction as:
$$\frac{x}{y}$$ = x × $$\frac{1}{y}$$
Now,
From the given number line,
We can observe that,
The value of y is: 8
So,
From the given options,
We have to check whether the given fraction is in the form of
$$\frac{x}{8}$$ = x × $$\frac{1}{8}$$
Hence, from the above,
We can conclude that
The multiplication equation that describes the fraction plotted on the number line is:

Question 20.
Which multiplication equation describes the picture below?

The given number line is:

Now,
We know that,
We will write any fraction in terms of a multiple of the unit fraction as:
$$\frac{x}{y}$$ = x × $$\frac{1}{y}$$
Now,
From the given number line,
We can observe that,
The value of y is: 2
So,
From the given options,
We have to check whether the given fraction is in the form of
$$\frac{x}{2}$$ = x × $$\frac{1}{2}$$
Hence, from the above,
We can conclude that
The multiplication equation that describes the picture is:

### Lesson 10.2 Multiply a Fraction by a Whole Number: Use Models

Solve & Share
How much tomato juice is needed for a group of 4 people if each person gets $$\frac{1}{3}$$ cup of juice? How much tomato juice is needed if they each get $$\frac{2}{3}$$ cup of juice? Solve these problems any way you choose.
I can… use drawings, area models, or number lines to multiply fractions by whole numbers.

It is given that
There are 4 people and each person gets $$\frac{1}{3}$$ cup of juice and after some time gets $$\frac{2}{3}$$ cup of juice
So,
The amount of tomato juice needed for 4 people when each person gets $$\frac{1}{3}$$ of juice
= (The number of people) × (The amount of juice each person gets)
= 4 × $$\frac{1}{3}$$
= $$\frac{4}{3}$$ cups of tomato juice
So,
The amount of tomato juice needed for 4 people when each person gets $$\frac{2}{3}$$ cups of juice
= (The number of people) × (The amount of juice each person gets)
= 4 × $$\frac{2}{3}$$
= $$\frac{4 × 2}{3}$$
= $$\frac{8}{3}$$ cups of tomato juice
Hence, from the above,
We can conclude that
a. The amount of tomato juice needed for 4 people when each person gets $$\frac{1}{3}$$ cup of juice is: $$\frac{4}{3}$$ cups of tomato juice
b. The amount of tomato juice needed for 4 people when each person gets $$\frac{2}{3}$$ cups of juice is: $$\frac{8}{3}$$ cups of tomato juice

Look Back! Use Structure How does finding the total juice for 4 people with $$\frac{2}{3}$$ cup servings compare to finding it for $$\frac{1}{3}$$ cup servings? Why?
From the above problem,
We can observe that
The amount of tomato juice needed for 4 people when each person gets $$\frac{1}{3}$$ cup of juice is: $$\frac{4}{3}$$ cups of tomato juice
The amount of tomato juice needed for 4 people when each person gets $$\frac{2}{3}$$ cups of juice is: $$\frac{8}{3}$$ cups of tomato juice
So,
The amount of tomato juice needed for 4 people when each person gets $$\frac{2}{3}$$ cups of juice = $$\frac{8}{3}$$ cups of tomato juice
= 4 × $$\frac{2}{3}$$
= 4 × 2 × $$\frac{1}{3}$$
= 2 × (The amount of tomato juice needed for 4 people when each person gets $$\frac{1}{3}$$ cups of juice)
Hence, from the above,
We can conclude that
The amount of tomato juice needed for 4 people when each person gets $$\frac{2}{3}$$ of juice is “2 Times” of the amount of tomato juice needed for 4 people when each person gets $$\frac{1}{3}$$ of juice

Essential Question
How Can You Multiply a Fraction by a Whole Number?
A fraction can be multiplied by a whole number as explained below:
Step 1:
Multiply across the top number by top number (numerator) and bottom number by bottom number (denominator).
Step 2:
Simplify your answer by dividing both numerator and denominator by the same number, when applicable.

Visual Learning Bridge
Dori lives $$\frac{1}{4}$$ mile from school. If she walks to and from school each day, how far does Dori walk during a school week?

Remember, multiplication is repeated addition. So, you can use addition or multiplication to solve this problem.

One Way
Draw a picture to show Dori walks $$\frac{1}{4}$$ mile, 10 times.

Since $$\frac{2}{4}$$ is equivalent to $$\frac{1}{2}$$, 2$$\frac{2}{4}$$ is equivalent to 2$$\frac{1}{2}$$. Dori walks 2$$\frac{1}{2}$$miles to and from school each week.

Another Way
Draw a number line to show Dori walks $$\frac{1}{4}$$ mile, 10 times

Convince Me! Generalize Why can both addition and multiplication be used to represent the problem above? Write an equation to explain.
We know that,
The addition is the process of combining a number of individual items together to form a new total. This means that in multiplication, groups are created to represent the numbers being multiplied, and then the groups are added together to produce a total. Relating addition to multiplication is relatively simple.
Example:
We can write $$\frac{10}{4}$$ as the repeated addition of $$\frac{2}{4}$$ and as a multiplication as a multiple of $$\frac{2}{4}$$
So,
$$\frac{10}{4}$$ = $$\frac{2}{4}$$ + $$\frac{2}{4}$$ + $$\frac{2}{4}$$ + $$\frac{2}{4}$$ + $$\frac{2}{4}$$
In the form of multiplication,
$$\frac{10}{4}$$ = 5 × $$\frac{2}{4}$$

Another Example!
How far did Jess bike to practice if he biked mile each day for 4 days?

$$\frac{3}{5}+\frac{3}{5}+\frac{3}{5}+\frac{3}{5}=\frac{12}{5}=\frac{5}{5}+\frac{5}{5}+\frac{2}{5}=2 \frac{2}{5}$$
Jess bikes 2$$\frac{2}{5}$$ miles.

Use multiplication.
$$4 \times \frac{3}{5}=\frac{12}{5}=\frac{5}{5}+\frac{5}{5}+\frac{2}{5}=2 \frac{2}{5}$$
Jess bikes 2$$\frac{2}{5}$$ miles.

Guided Practice

Do You Understand?
Question 1.
Draw a picture to explain how to find 3 × $$\frac{2}{5}$$.
The given fraction is:
3 × $$\frac{2}{5}$$
Now,
In the given fraction,
The numerator represents the shaded parts
The denominator represents the total number of parts
So,
The representation of the given fraction in the form of a bar diagram is:

Now,
From the given figure,
We can observe that
3 × $$\frac{2}{5}$$ = $$\frac{2}{5}$$ + $$\frac{2}{5}$$ + $$\frac{2}{5}$$
Hence, from the above,
We can conclude that
The representation of the given fraction is:
3 × $$\frac{2}{5}$$ = $$\frac{2}{5}$$ + $$\frac{2}{5}$$ + $$\frac{2}{5}$$

Do You Know How?
For 2-3, write and solve a multiplication equation.
Question 2.

The given figures are:

Now,
From the first figure,
We can observe that
The number of shaded parts is: 2
The total number of parts are: 6
From the second figure,
We can observe that
The number of shaded parts is: 2
The total number of parts are: 6
So,
The representation of the given figures in the form of fractions are:
For the first figure,
The fraction of the shaded part = $$\frac{The number of shaded parts}{The total number of parts}$$
= $$\frac{2}{6}$$
For the second figure,
The fraction of the shaded part = $$\frac{The number of shaded parts}{The total number of parts}$$
= $$\frac{2}{6}$$
So,
The multiplication equation for the given figures is:
$$\frac{2}{6}$$ + $$\frac{2}{6}$$ = 2 × $$\frac{2}{6}$$
Hence, from the above,
We can conclude that
The multiplication equation for the given figures is:
$$\frac{2}{6}$$ + $$\frac{2}{6}$$ = 2 × $$\frac{2}{6}$$

Question 3.

The given number line is:

Now,
From the given number line,
We can observe that
The number line is showing the gap of 2 numbers
Now,
The representation of the multiplication equation for the given number line is:
n × $$\frac{1}{3}$$
Where,
n = 2, 4, 6, 8
Hence, from the above,
We can conclude that
The representation of the multiplication equation for the given number line is:
n × $$\frac{1}{3}$$
Where,
n = 2, 4, 6, 8

Independent Practice

For 4-7, write and solve a multiplication equation. Use drawings or number lines as needed.
Question 4.

The given model is:

Now,
From the given model,
We can observe that
$$\frac{1}{8}$$ is repeated 5 times
So,
The multiplication equation for the given model is:
$$\frac{1}{8}$$ + $$\frac{1}{8}$$+ $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ (or) 5 × $$\frac{1}{8}$$
Hence, from the above,
We can conclude that
The multiplication equation for the given model is:
$$\frac{1}{8}$$ + $$\frac{1}{8}$$+ $$\frac{1}{8}$$ + $$\frac{1}{8}$$ + $$\frac{1}{8}$$ (or) 5 × $$\frac{1}{8}$$

Question 5.

The given model is:

Now,
From the given model,
We can observe that
$$\frac{2}{10}$$ is repeated 3 times
So,
The multiplication equation for the given model is:
$$\frac{2}{10}$$ + $$\frac{2}{10}$$ + $$\frac{2}{10}$$ (or) 3 × $$\frac{2}{10}$$
Hence, from the above,
We can conclude that
The multiplication equation for the given model is:
$$\frac{2}{10}$$ + $$\frac{2}{10}$$ + $$\frac{2}{10}$$ (or) 3 × $$\frac{2}{10}$$

Question 6.
Calculate the distance Margo rides her bike if she rides $$\frac{7}{8}$$ mile each day for 4 days.
It is given that
Margo rides her bike $$\frac{7}{8}$$ mile each day for 4 days
So,
The distance traveled by Margo for 4 days = 4 × (The number of miles Margo rides each day)
= 4 × $$\frac{7}{8}$$
= $$\frac{4 × 7}{8}$$
= $$\frac{28}{8}$$ miles
Hence, from the above,
We can conclude that
The distance Margo rides her bike for 4 days is: $$\frac{28}{8}$$ miles

Question 7.
Calculate the distance Tom rides his bike if he rides $$\frac{5}{6}$$ mile each day for 5 days.
It is given that
Tom rides his bike $$\frac{5}{6}$$ miles each day
So,
The distance Tom rides his bike for 5 days = 5 × (The number of miles Tom rides his bike each day)
= 5 × $$\frac{5}{6}$$
= $$\frac{5 × 5}{6}$$
= $$\frac{25}{6}$$ miles
Hence, from the above,
We can conclude that
The distance Tom rides his bike for 5 days is: $$\frac{25}{6}$$ miles

Problem Solving

Question 8.
Kiona fills a measuring cup with $$\frac{3}{4}$$ cup of juice 3 times to make punch. Write and solve a multiplication equation with a whole number and a fraction to show the total amount of juice Kiona uses.

It is given that
Kiona fills a measuring cup with $$\frac{3}{4}$$ cup of juice 3 times to make a punch
So,
The total amount of juice Kiona uses = (The number of times Kiona fills a measuring cup) × (Each cup of juice that Kiona fills)
= 3 × $$\frac{3}{4}$$
= $$\frac{3 × 3}{4}$$
= $$\frac{9}{4}$$ cups of juice
Hence, from the above,
We can conclude that
The total amount of juice Kiona uses is: $$\frac{9}{4}$$ cups of juice

Question 9.
Each lap around a track is $$\frac{3}{10}$$ kilometer. Eliot walked around the track 4 times. How far did Eliot walk?

It is given that
Each lap around a track is $$\frac{3}{10}$$ kilometer. Eliot walked around the track 4 times.
So,
The distance that Eliot walked = (The number of times Eliot walked around the track) × (Each lap Eliot walked around a track)
= 4 × $$\frac{3}{10}$$
= $$\frac{4 × 3}{10}$$
= $$\frac{12}{10}$$ Kilometers
Hence, from the above,
We can conclude that
The distance that Eliot walked is: $$\frac{12}{10}$$ Kilometers

Question 10.
A chef serves $$\frac{5}{6}$$ of a pan of lasagna. Each piece is $$\frac{1}{6}$$ of the pan. How many pieces did the chef serve? Solve by writing $$\frac{5}{6}$$ as a multiple of $$\frac{1}{6}$$.
It is given that
A chef serves $$\frac{5}{6}$$ of a pan of lasagna. Each piece is $$\frac{1}{6}$$ of the pan.
Now,
Let the number of pieces did the chef serve be: x
So,
The amount of a pan of lasagna the chef serves = (The number of pieces did the chef serve) × (Each piece of lasagna)
$$\frac{5}{6}$$ = x × $$\frac{1}{6}$$
5 × $$\frac{1}{6}$$ = x × $$\frac{1}{6}$$
x = 5 pieces
Hence, from the above,
We can conclude that
The number of pieces did the chef serves is: 5 pieces

Question 11.
Model with Math
Wendy uses $$\frac{2}{12}$$ of a loaf of bread to make one sandwich. Write and solve an equation to find b, how much of the loaf of bread she uses to make 4 sandwiches. Use a drawing, as needed.
It is given that
Wendy uses $$\frac{2}{12}$$ of a loaf of bread to make one sandwich
So,
The amount of the loaf of bread Wendy uses to make 4 sandwiches = 4 × (The loaf of a bread used by Wendy to make one sandwich)
= 4 × $$\frac{2}{12}$$
= $$\frac{4 × 2}{12}$$
= $$\frac{8}{12}$$ loaves of bread
Hence, from the above,
We can conclude that
The amount of loaf of bread used by Wendy to make 4 sandwiches is: $$\frac{8}{12}$$ loaves of bread

Question 12.
Higher-Order Thinking A baker uses $$\frac{2}{3}$$ cup of rye flour in each loaf of bread. How many cups of rye flour will the baker use in 3 loaves? in 7 loaves? in 10 loaves?
It is given that
A baker uses $$\frac{2}{3}$$ cup of rye flour in each loaf of bread
So,
The number of cups of rye flour will the baker use in 3 loaves = 3 × (The cup of rye flour used by the baker in each loaf of bread)
= 3 × $$\frac{2}{3}$$
= $$\frac{3 × 2}{3}$$
= $$\frac{6}{3}$$ cups of rye flour
So,
The number of cups of rye flour will the baker use in 7 loaves = 7 × (The cup of rye flour used by the baker in each loaf of bread)
= 7 × $$\frac{2}{3}$$
= $$\frac{7 × 2}{3}$$
= $$\frac{14}{3}$$ cups of rye flour
So,
The number of cups of rye flour will the baker use in 10 loaves = 10 × (The cup of rye flour used by the baker in each loaf of bread)
= 10 × $$\frac{2}{3}$$
= $$\frac{10 × 2}{3}$$
= $$\frac{20}{3}$$ cups of rye flour
Hence, from the above,
We can conclude that
The number of cups of rye flour will the baker use in 3 loaves is: $$\frac{6}{3}$$ cups
The number of cups of rye flour will the baker use in 7 loaves is: $$\frac{14}{3}$$ cups
The number of cups of rye flour will the baker use in 10 loaves is: $$\frac{20}{3}$$ cups

Assessment Practice

Question 13.
Elaine jogged $$\frac{4}{5}$$ mile each day for 4 days. Select all the expressions that tell how far Elaine jogged in all. Use drawings or number lines as needed.
☐ 4 × $$\frac{4}{5}$$
☐ $$\frac{16}{5}$$
☐ 3 $$\frac{1}{5}$$
☐ 4 × $$\frac{1}{5}$$
☐ 2$$\frac{1}{5}$$
It is given that
Elaine jogged $$\frac{4}{5}$$ mile each day for 4 days
So,
The number of miles Elaine jogged for 4 days = 4 × (The number of miles Elaine jogged each day)
= 4 × $$\frac{4}{5}$$
= $$\frac{4 × 4}{5}$$
= $$\frac{16}{5}$$ miles
= 3$$\frac{1}{5}$$ miles
Hence, from the above,
We can conclude that all the expressions that tell how far Elaine jogged are:

Question 14.
Freddie skated $$\frac{1}{2}$$ mile each day for 6 days. Select all the equations that can be used to find s, the total distance Freddie skated.
☐ s = $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$
☐ s = 6 × $$\frac{1}{2}$$
☐ s = 6 + $$\frac{1}{2}$$
☐ s = 6 + 2 × $$\frac{1}{2}$$
☐ s = 6 × 2
It is given that
Freddie skated $$\frac{1}{2}$$ mile each day for 6 days
So,
The number of miles (s) Freddie skated for 6 days = 6 × (The number of miles Freddie skated each day)
= 6 × $$\frac{1}{2}$$
= $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$ + $$\frac{1}{2}$$
= $$\frac{6 × 1}{2}$$
= $$\frac{6}{2}$$ miles
Hence, from the above,
We can conclude that all the equations that can be used to find s, the total distance Freddie skated are:

### Lesson 10.3 Multiply a Fraction by a Whole Number: Use Symbols

Solve & Share
A recipe for 1 gallon of fruit punch calls for $$\frac{3}{4}$$ cup of orange juice. How many cups of orange juice are needed to make 8 gallons of fruit punch? Solve this problem any way you choose.
I can… use properties and equations to multiply a fraction by a whole number.

It is given that
A recipe for 1 gallon of fruit punch calls for $$\frac{3}{4}$$ cup of orange juice
So,
The number of cups of orange juice needed to make 8 gallons of fruit punch = 8 × (The number of cups of orange juice needed to make 1 gallon of fruit punch)
= 8 × $$\frac{3}{4}$$
= $$\frac{8 × 3}{4}$$
= $$\frac{24}{4}$$
= 6 cups of orange juice
Hence, from the above,
We can conclude that
The number of cups of orange juice needed to make 8 gallons of fruit punch is: 6 cups of orange juice

Look Back! Be Precise Look back at your solution. What units should you use to label your answer?
From the above problem,
We can observe that
The solution for the above problem is: 6 cups of orange juice (Which is the number of cups)
Hence, from the above,
We can conclude that

Essential Question
How Can You Use Symbols to Multiply a Fraction by a Whole Number?
A fraction can be multiplied by a whole number as explained below:
Step 1:
Multiply across the top number by top number (numerator) and bottom number by bottom number (denominator).
Step 2:
Simplify your answer by dividing both numerator and denominator by the same number, when applicable.

Visual Learning Bridge
Stanley makes ice cream sundaes. Today Stanley made 2 ice cream sundaes. How much ice cream did Stanley use? Find 2 × $$\frac{3}{4}$$.

You can use structure when multiplying a fraction and a whole number.

Convince Me! Use Structure Use properties of operations to calculate 3 × $$\frac{3}{6}$$. Show your work.
The given fraction is: 3 × $$\frac{3}{6}$$
Now,
By using the unit fractions,
3 × $$\frac{3}{6}$$
= 3 × 3 × $$\frac{1}{6}$$
By using he Associative Property of Multiplication,
= (3 × 3) × $$\frac{1}{6}$$
= 9 × $$\frac{1}{6}$$
= $$\frac{9 × 1}{6}$$
= $$\frac{9}{6}$$
= $$\frac{3}{2}$$
Hence, from the above,
We can conclude that
The value of 3 × $$\frac{3}{6}$$ is: $$\frac{3}{2}$$

Guided Practice

Do You Understand?
Question 1.
Sarah has $$\frac{1}{2}$$ of a granola bar. Her friend has 5 times as many granola bars. How many granola bars does Sarah’s friend have?
It is given that
Sarah has $$\frac{1}{2}$$ of a granola bar. Her friend has 5 times as many granola bars
So,
The number of granola bars does Sarah’s friend have = (The number of times Sarah’s friend has as many granola bars) × (The number of pieces of each granola bar)
= 5 × $$\frac{1}{2}$$
= $$\frac{5}{2}$$ granola bars
Hence, from the above,
We can conclude that
The number of granola bars does Sarah’s friend have is: $$\frac{5}{2}$$ granola bars

Question 2.
Sue needs a $$\frac{5}{6}$$ cup of cocoa to make one batch of chocolate pudding. She wants to make 4 batches of pudding to take to a party. Write and solve an equation to find how many cups of cocoa, c, Sue will need for all 4 batches of pudding.
It is given that
Sue needs a $$\frac{5}{6}$$ cup of cocoa to make one batch of chocolate pudding. She wants to make 4 batches of pudding to take to a party
So,
The number of cups of cocoa Sue will need for 4 batches of pudding (c) = 4 × (The amount of cocoa needed to make one batch of chocolate pudding)
= 4 × $$\frac{5}{6}$$
= $$\frac{4 × 5}{6}$$
= $$\frac{20}{6}$$
= $$\frac{10}{3}$$ cups of cocoa
Hence, from the above,
We can conclude that
The number of cups of cocoa Sue will need for 4 batches of pudding is: $$\frac{10}{3}$$ cups of pudding

Do You Know How?
For 3-4, multiply.
Question 3.
8 × $$\frac{1}{2}$$
The given fraction is: 8 × $$\frac{1}{2}$$
So,
8 × $$\frac{1}{2}$$
= $$\frac{8 × 1}{2}$$
= $$\frac{8}{2}$$
= 4
Hence, from the above,
We can conclude that
The value of the given fraction is: 4

Question 4.
3 × $$\frac{3}{4}$$
The given fraction is: 3 × $$\frac{3}{4}$$
So,
3 × $$\frac{3}{4}$$
= $$\frac{3 × 3}{4}$$
= $$\frac{9}{4}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{9}{4}$$

For 5-6, write and solve a multiplication equation.
Question 5.
Calculate the amount of medicine taken in 5 days if the dose is $$\frac{3}{4}$$ fluid ounce per day.
It is given that
The dose is $$\frac{3}{4}$$ fluid ounce per day
So,
The amount of medicine taken in 5 days = 5 × (The amount of dose taken per day)
= 5 × $$\frac{3}{4}$$
= $$\frac{5 × 3}{4}$$
= $$\frac{15}{4}$$ fluid ounces
Hence, from the above,
We can conclude that
The amount of medicine taken in 5 days is: $$\frac{15}{4}$$ fluid ounces

Question 6.
Calculate the total length needed to decorate 9 boxes if each box uses $$\frac{2}{3}$$ yard of ribbon.
It is given that
Each box uses $$\frac{2}{3}$$ yard of ribbon
So,
The total length of ribbon needed to decorate 9 boxes = 9 × (The length of ribbon used for each box)
= 9 × $$\frac{2}{3}$$
= $$\frac{9 × 2}{3}$$
= $$\frac{18}{3}$$
= 6 yards of ribbon
Hence, from the above,
We can conclude that
The total length of the ribbon needed to decorate 9 boxes is: 6 yards of ribbon

Independent Practice

For 7-15, multiply.
Question 7.
4 × $$\frac{1}{3}$$
The given fraction is: 4 × $$\frac{1}{3}$$
So,
4 × $$\frac{1}{3}$$
= $$\frac{4 × 1}{3}$$
= $$\frac{4}{3}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{4}{3}$$

Question 8.
6 × $$\frac{3}{8}$$
The given fraction is: 6 × $$\frac{3}{8}$$
So,
6 × $$\frac{3}{8}$$
= $$\frac{3 × 6}{8}$$
= $$\frac{18}{8}$$
= $$\frac{9}{4}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{9}{4}$$

Question 9.
8 × $$\frac{2}{5}$$
The given fraction is: 8 × $$\frac{2}{5}$$
So,
8 × $$\frac{2}{5}$$
= $$\frac{8 × 2}{5}$$
= $$\frac{16}{5}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{16}{5}$$

Question 10.
2 × $$\frac{5}{6}$$
The given fraction is: 2 × $$\frac{5}{6}$$
So,
2 × $$\frac{5}{6}$$
= $$\frac{2 × 5}{6}$$
= $$\frac{10}{6}$$
= $$\frac{5}{3}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{5}{3}$$

Question 11.
4 × $$\frac{2}{3}$$
The given fraction is: 4 × $$\frac{2}{3}$$
So,
4 × $$\frac{2}{3}$$
= $$\frac{4 × 2}{3}$$
= $$\frac{8}{3}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{8}{3}$$

Question 12.
5 × $$\frac{7}{8}$$
The given fraction is: 5 × $$\frac{7}{8}$$
So,
5 × $$\frac{7}{8}$$
= $$\frac{5 × 7}{8}$$
= $$\frac{35}{8}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{35}{8}$$

Question 13.
7 × $$\frac{3}{4}$$
The given fraction is: 7 × $$\frac{3}{4}$$
So,
7 × $$\frac{3}{4}$$
= $$\frac{7 × 3}{4}$$
= $$\frac{21}{4}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{21}{4}$$

Question 14.
9 × $$\frac{3}{4}$$
The given fraction is: 9 × $$\frac{3}{4}$$
So,
9 × $$\frac{3}{4}$$
= $$\frac{9 × 3}{4}$$
= $$\frac{27}{4}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{27}{4}$$

Question 15.
4 × $$\frac{5}{8}$$
The given fraction is: 4 × $$\frac{5}{8}$$
So,
4 × $$\frac{5}{8}$$
= $$\frac{4 × 5}{8}$$
= $$\frac{20}{8}$$
= $$\frac{5}{2}$$
Hence, from the above,
We can conclude that
The value of the given fraction is: $$\frac{5}{2}$$

For 16-17, write and solve a multiplication equation.
Question 16.
Calculate the total distance Mary runs in one week if she runs a $$\frac{7}{8}$$ mile each day.
It is given that
Mary runs $$\frac{7}{8}$$ miles each day
Now,
We know that,
1 week = 7 days
So,
The total distance Mary runs in 1 week = 7 × (The number of miles Mary runs each day)
= 7 × $$\frac{7}{8}$$
= $$\frac{7 × 7}{8}$$
= $$\frac{49}{8}$$ miles
Hence, from the above,
We can conclude that
The total distance Mary runs in 1 week is: $$\frac{49}{8}$$ miles

Question 17.
Calculate the length of 5 pieces of ribbon laid end to end if each piece is $$\frac{2}{3}$$ yard long.
It is given that
Each piece is $$\frac{2}{3}$$ yard long
So,
The length of 5 pieces of ribbon laid end to end = 5 × (The length of each piece)
= 5 × $$\frac{2}{3}$$
= $$\frac{5 × 2}{3}$$
= $$\frac{10}{3}$$ yards
Hence, from the above,
We can conclude that
The length of 5 pieces of ribbon laid end to end is: $$\frac{10}{3}$$ yards

Problem Solving

Question 18.
A baseball team bought 8 boxes of baseballs. If the team spent a total of $1,696, what was the cost of 1 box of baseballs? Answer: A baseball team bought 8 boxes of baseballs and the team spent a total of$1,696
Now,
Let the cost of 1 box of baseballs be $x So, The total cost team spent = (The number of baseballs) × (The cost of 1 box of baseballs)$1,696 = 8 × x
x = $$\frac{1,696}{8}$$
x = $212 Hence, from the above, We can conclude that The cost of 1 box of baseballs is:$212

Question 19.
Oscar wants to make 4 chicken pot pies. The recipe requires $$\frac{2}{3}$$ pound of potatoes for each pot pie. How many pounds of potatoes will Oscar need?
It is given that
Oscar wants to make 4 chicken pot pies. The recipe requires $$\frac{2}{3}$$ pound of potatoes for each pot pie
So,
The number of pounds of potatoes oscar will need = (The number of chicken pot pies) × (The number of pounds of potatoes for each pot pie)
= 4 × $$\frac{2}{3}$$
= $$\frac{4 × 2}{3}$$
= $$\frac{8}{3}$$ pounds of potatoes
Hence, from the above,
We can conclude that
The number of pounds of potatoes Oscar will need is: $$\frac{8}{3}$$ pounds of potatoes

Question 20.
It takes Mario $$\frac{1}{4}$$ hour to mow Mr. Harris’s lawn. It takes him 3 times as long to mow Mrs. Carter’s lawn. How long does it take Mario to mow Mrs. Carter’s lawn? Write your answer as a fraction of an hour, then as minutes.

It is given that
It takes Mario $$\frac{1}{4}$$ hour to mow Mr. Harris’s lawn. It takes him 3 times as long to mow Mrs. Carter’s lawn
So,
The time taken for Mario to mow Mrs. harris’s lawn = 3 × (The time taken for Mario to mow Mr.Harris’s lawn)
= 3 × $$\frac{1}{4}$$
= $$\frac{3 × 1}{4}$$
= $$\frac{3}{4}$$ hours
Now,
It is given that
$$\frac{1}{4}$$ hour = 15 minutes
So,
The time is taken for Mario to mow Mrsharris’s lawn = 3 × 15
= 45 minutes
Hence, from the above,
We can conclude that
The time taken for Mario to mow MrsHarris’s lawn is: $$\frac{3}{4}$$ hours (or) 45 minutes

Question 21.
Vocabulary Use numerator, denominator, and whole number.
When you multiply a fraction by a whole number, the __________ in the product is the same as the denominator of the fraction. The ___________ in the product is the product of the and the numerator of the fraction.
When you multiply a fraction by a whole number, the “Denominator” in the product is the same as the denominator of the fraction. The “Numerator” in the product is the product of the “Whole number” and the numerator of the fraction.

Question 22.
Model with Math
Malik swims $$\frac{9}{10}$$ mile each day. Write and solve an equation to find n, how many miles Malik swims in 4 days.
It is given that
Malik swims $$\frac{9}{10}$$ mile each day
So,
The number of miles Malik swims in 4 days = 4 × (The number of miles Malik swims each day)
= 4 × $$\frac{9}{10}$$
= $$\frac{4 × 9}{10}$$
= $$\frac{36}{10}$$
= $$\frac{18}{5}$$ miles
Hence, from the above,
We can conclude that
The number of miles Malik swims in 4 days is: $$\frac{18}{5}$$ miles

Question 23.
Higher-Order Thinking Sam is making 7 fruit tarts. Each tart needs $$\frac{3}{4}$$ cup of strawberries and $$\frac{1}{4}$$ cup of blueberries. What is the total amount of fruit that Sam needs for his tarts? Use properties of operations to solve.
It is given that
Sam is making 7 fruit tarts. Each tart needs $$\frac{3}{4}$$ cup of strawberries and $$\frac{1}{4}$$ cup of blueberries
So,
The total cups of fruit Sam needs for each tart = (The number of cups of strawberries needed for 1 fruit tart) + (The number of cups of blueberries needed for 1 fruit tart)
= $$\frac{3}{4}$$ + $$\frac{1}{4}$$
= $$\frac{3 + 1}{4}$$
= $$\frac{4}{4}$$
= 1 cup
So,
The total amount of fruit Sam needs for his tarts = (The total number of fruit tarts) × (The numebr of cups of fruit Sam needs for each tart)
= 7 × 1
= 7 cups
Hence, from the above,
We can conclude that
The total amount of fruit Sam needs from his tarts is: 7 cups of strawberry and blueberries

Assessment Practice

Question 24.
Sean is making picture frames. Each frame uses $$\frac{4}{5}$$ yard of wood. What is the total length of wood Sean will need to make 2 frames? Complete the equation.

It is given that
Sean is making picture frames. Each frame uses $$\frac{4}{5}$$ yard of wood.
So,
The total length of the wood Sean will need to make 2 frames = (The total number of frames) × (The length of wood needed to make each frame)
=
Hence, from the above,
We can conclude that
The total length of the wood Sean will need to make 2 frames is: $$\frac{8}{5}$$ yards

Question 25.
Ellen is making plant boxes. Each box uses $$\frac{3}{6}$$ yard of wood. What is the total length of wood Ellen will need to make 7 plant boxes? Complete the equation.

It is given that
Ellen is making plant boxes. Each box uses $$\frac{3}{6}$$ yard of wood
So,
The total length of wood Ellen need to make 7 plant boxes = 7 × (The length of wood used by Ellen for each box)
=
Hence, from the above,
We can conclude that
The total length of wood Ellen need to make 7 plant boxes is: $$\frac{21}{6}$$ yards

### Lesson 10.4 Solve Time Problems

Solve & Share
The Big Sur International Marathon is run on the California coast each spring. Sean’s mother was the women’s overall winner. How much faster was Sean’s mother than the women’s winner in the Ages 65-69 group? Tell me how you decided. Solve this problem any way you choose.
I can… use addition, subtraction, multiplication, or division to solve problems involving time.

It is given that
The Big Sur International Marathon is run on the California coast each spring. Sean’s mother was the women’s overall winner
Now,
The given table is:

Now,
From the given table,
We can observe that
The time is taken by Sean’s mother to complete the marathon = 2 hours 50 minutes
The time is taken by Women’s Age group 65 – 69 to complete the marathon = 3 hours 58 minutes
So,
The time difference between Sean’s mother and women’s Age group 65 – 69 to complete the marathon = 3 hours 58 minutes – 2 hours 50 minutes
= 1 hour 8 minutes
Hence, from the above,
We can conclude that
Sean’s mother was faster by 1 hour 8 minutes than the women’s winner in the Ages 65-69 group

Look Back! The men’s winner in the Ages 70-74 group took 4$$\frac{1}{3}$$ hours. Sean’s grandfather, who is only 68, took 3$$\frac{2}{3}$$ hours. How can you find the difference in these times?
It is given that
The men’s winner in the Ages 70-74 group took 4$$\frac{1}{3}$$ hours. Sean’s grandfather, who is only 68, took 3$$\frac{2}{3}$$ hours.
Now,
The given table is:

Now,
From the given table,
We can observe that
The time taken by Men’s winner Ages 70 – 74 group = 4 hours 20 minutes
Now,
We know that,
1 hour = 60 minutes
So,
4 hours 20 minutes = 4 + $$\frac{20}{60}$$
= 4$$\frac{1}{3}$$ hours
Now,
The time is taken by Sean’s grandfather who is only 68 = 3 hours 34 minutes
≈ 3 hours 40 minutes
= 3 + $$\frac{40}{60}$$
= 3$$\frac{2}{3}$$ hours
So,
The difference of time taken between men’s winner Ages 70 – 74 group and Sean’s grandfather
= 4$$\frac{1}{3}$$ – 3$$\frac{2}{3}$$
= $$\frac{13}{3}$$ – $$\frac{11}{3}$$
= $$\frac{13 – 11}{3}$$
= $$\frac{2}{3}$$ hours
Hence, from the above,
We can conclude that
The difference of time taken between men’s winner Ages 70 – 74 group and Sean’s grandfather is about $$\frac{2}{3}$$ hours

Essential Question
How Can You Solve Problems Involving Time?
We know that,
When we want to find the total time,
We will add the given times
When we want to find the difference between time,
We will subtract the given times
Now,
We know that,
1 hour = 60 minutes
1 minute = $$\frac{1}{60}$$ hours

Visual Learning Bridge
Krystal is training for a race. She trains every day for 8 days. How many hours does Krystal train?

Krystal spends an equal amount of time sprinting, walking, and jogging. How many minutes does Krystal spend on each activity during her 8 days of training?

You can use what you know about time to help solve these problems.

Find how many hours Krystal trains.
Find 8 × $$\frac{3}{4}$$

Find how many minutes Krystal spends on each activity during her training.
1 hour = 60 minutes
6 × 60 minutes = 360 minutes of training
In 8 days, Krystal spends 360 minutes sprinting, walking, and jogging.
Divide to find how many minutes Krystal spends on each activity. Find 360 ÷ 3.

In 8 days, Krystal spends 120 minutes, or 2 hours, training on each activity.

Convince Me! Construct Arguments Why do you multiply to convert 6 hours to minutes?
From the previous example,
We can observe that
The amount of time taken by Krystal to train for 8 days is: 6 hours
But,
It is given that we have to find the amount of time in minutes
Now,
We know that,
1 hour = 60 minutes
So,
6 hours = 6 × 60 minutes
= 360 minutes
Hence, from the above,
We can conclude that
We will convert 6 hours into minutes because it has been asked to find the amount of time taken in minutes

Another Example!

Guided Practice

Do You Understand?
Question 1.
How are adding and subtracting measures of time like adding and subtracting whole numbers?
The adding and subtracting measures of time is the same as adding and subtracting whole numbers when the measures of time are also whole numbers
But,
When the measures of time are the decimal numbers, we will borrow 60 minutes when the borrowing number is less than the second number

Do You Know How?
For 2-3, solve. Remember there are 60 minutes in 1 hour and 7 days in 1 week.
Question 2.
How many minutes are in a school day of 7 hours 25 minutes?
It is given that
1 hour = 60 minutes
1 week = 7 days
So,
7 hours 25 minutes = (7 × 60) + 25
= 420 + 25
= 445 minutes
Hence, from the above,
We can conclude that
The number of minutes present in a school day of 7 hours 25 minutes is: 445 minutes

Question 3.
How much is 3$$\frac{2}{4}$$ weeks + 2$$\frac{3}{4}$$ weeks?
The given fraction is:
3$$\frac{2}{4}$$ weeks + 2$$\frac{3}{4}$$ weeks
So,
3$$\frac{2}{4}$$ weeks + 2$$\frac{3}{4}$$ weeks
= $$\frac{14}{4}$$ weeks + $$\frac{11}{4}$$ weeks
= $$\frac{14 + 11}{4}$$ weeks
= $$\frac{25}{4}$$ weeks
Hence, from the above,
We can conclude that the value of 3$$\frac{2}{4}$$ weeks + 2$$\frac{3}{4}$$ weeks is:
$$\frac{25}{4}$$ weeks

Independent Practice

For 4-7, add, subtract, multiply, or divide.

Question 4.

We know that,

Now,
We know that,
When the number of minutes is greater than 60,
Convert 60 minutes into 1 hour and add that 1 hour to the number of hours we obtained in the result
Hence,

Question 5.

We know that,

Now,
We know that,
When the number of months is less than the second number of months,
We have to borrow and we will borrow 12 months and add that 12 months into the first number of months
Hence,

Question 6.

We know that,

Hence,

Question 7.
How long must each person work for 4 people to evenly share 48 hours of work?
48 ÷ 4 = ☐ hours
We know that,

So,
The amount of time each person work for 4 people to evenly share 48 hours of work is:

Hence, from the above,
We can conclude that
The amount of time each person work for 4 people to evenly share 48 hours of work is: 12 hours

Problem Solving

For 8-9, use the table at the right.

Question 8.
How long do all of the activities at the reunion last?
The given data is:

So,
The time that all of the activities at the reunion last = The total time is taken for all the activities to end
So,
The time that all of the activities at the reunion last = 4 hours 15 minutes + 55 minutes + 1 hour 30 minutes + 1 hour 35 minutes
= 6 hours 135 minutes
= 6 hours + 2 hours + 15 minutes
= 8 hours 15 minutes
Hence, from the above,
We can conclude that
The time that all of the activities at the reunion last is: 8 hours 15 minutes

Question 9.
There are 55 minutes between the time dinner ends and the campfire begins. What is the elapsed time from the beginning of dinner to the beginning of the campfire?
It is given that
There are 55 minutes between the time dinner ends and the campfire begins
Now,
The given data is:

So,
The elapsed time from the beginning of dinner to the beginning of the campfire = (The end time of the slide show) – (The end time of Dinner)
= 1 hour 30 minutes – 55 minutes [1 hour = 60 minutes]
= 90 – 55
= 35 minutes
Hence, from the above,
We can conclude that
The elapsed time from the beginning of dinner to the beginning of the campfire is: 35 minutes

Question 10.
Make Sense and Persevere The band boosters spent $4,520 on airline tickets and$1,280 on hotel costs for the 8 members of the color guard. How much was spent for each member of the color guard?
It is given that
The band boosters spent $4,520 on airline tickets and$1,280 on hotel costs for the 8 members of the color guard
So,
The total amount spent by the band boosters = (The amount spent by the band boosters on airline tickets) + (The amount spent by the band boosters on hotel costs)
= $4,520 +$1,280
= $5,800 So, The amount spend for each member of the color guard = $$\frac{The total amount spent by the band boosters}{The total members present in the color guard}$$ = $$\frac{5,800}{8}$$ =$725
Hence, from the above,
We can conclude that
The amount spend for each member of the color guard is: $725 Question 11. Higher-Order Thinking A boat ride at the lake lasts 2$$\frac{2}{4}$$ hours. A canoe trip down the river lasts 3$$\frac{1}{4}$$ hours. Show each time on the number line. How much longer is the canoe trip than the boat ride in hours? in minutes? Answer: It is given that A boat ride at the lake lasts 2$$\frac{2}{4}$$ hours. A canoe trip down the river lasts 3$$\frac{1}{4}$$ hours So, The difference of time between the canoe trip and the boat ride in hours = 3$$\frac{1}{4}$$ – 2$$\frac{2}{4}$$ = $$\frac{13}{4}$$ – $$\frac{10}{4}$$ = $$\frac{13 – 10}{4}$$ = $$\frac{3}{4}$$ hours Now, The difference of time between the canoe trip and the boat ride in minutes = $$\frac{3}{4}$$ × 60 minutes = $$\frac{3 × 60}{4}$$ = $$\frac{180}{4}$$ = 45 minutes Now, The representation of the canoe time and the time taken for a boat ride in the number line is: Hence, from the above, We can conclude that The difference of time between the canoe trip and the boat ride in hours is: $$\frac{3}{4}$$ hours The difference of time between the canoe trip and the boat ride in minutes is: 45 minutes Assessment Practice Question 12. It takes Krys and Glen $$\frac{1}{4}$$ hour to walk a mile. This week Krys walked 9 miles and Glen walked 3 miles. How much longer did Krys walk than Glen? ________ hours Answer: It is given that It takes Krys and Glen $$\frac{1}{4}$$ hours to walk a mile. This week Krys walked 9 miles and Glen walked 3 miles So, The difference between the number of miles walked by Krys ad Glen = 9 – 3 = 6 miles So, The difference of time taken for Krys to walk faster than Glen = (The difference between the number of miles walked by Krys and Glen) × (The time taken by Krys and Glan to walk a mile) = 6 × $$\frac{1}{4}$$ = $$\frac{6 × 1}{4}$$ = $$\frac{6}{4}$$ = $$\frac{3}{2}$$ = 1.5 hours Hence, from the above, We can conclude that The difference of time taken for Krys to walk faster than Glen is: 1.5 hours Question 13. Henry’s first flight lasts 1 hour 12 minutes. The second flight lasts 2 hours 41 minutes. How much time did Henry spend on the flights? _______ hours _______ minutes Answer: It is given that Henry’s first flight lasts 1 hour 12 minutes. The second flight lasts 2 hours 41 minutes So, The total time spent by Henry on the flights = 1 hour 12 minutes + 2 hours 41 minutes = 3 hours 53 minutes Hence, from the above, We can conclude that The total time spent by Henry on the flights is: 3 hours 53 minutes ### Lesson 10.5 Problem Solving Model with Math Solve & Share Pierre’s mother owns an ice cream shop. She puts $$\frac{3}{12}$$ cup of vanilla extract and $$\frac{1}{12}$$ cup of almond extract in each 10-gallon batch of ice cream. How much total extract is used to make 5 batches of ice cream? Use the bar diagrams to represent and solve this problem. I can …use various representations to solve problems. Thinking Habits Be a good thinker! These questions can help you. • How can I use the math I know to help solve this problem? • How can I use pictures, objects, or an equation to represent the problem? • How can I use numbers, words, and symbols to solve the problem? Answer: It is given that Pierre’s mother owns an ice cream shop. She puts $$\frac{3}{12}$$ cup of vanilla extract and $$\frac{1}{12}$$ cup of almond extract in each 10-gallon batch of ice cream. So, The total amount of extract that is used to make 1 batch of icecream = (The amount of vanilla extract) + (The amount of almond extract) = $$\frac{3}{12}$$ + $$\frac{1}{12}$$ = $$\frac{3 + 1}{12}$$ = $$\frac{4}{12}$$ = $$\frac{1}{3}$$ cups So, The total extract present in 5 batches of icecream = 5 × (The total extract present in 1 batch of icecream) = 5 × $$\frac{1}{3}$$ = $$\frac{5 × 1}{3}$$ = $$\frac{5}{3}$$ cups Now, The representation of the given information in the form of a bar diagram is: Hence, from the above, We can conclude that The total amount of extract present in 5 batches of ice cream is: $$\frac{5}{3}$$ cups Look Back! Model with Math What number sentences can you write to model the problem? Answer: The number of sentences that you can write to model the above problem is: a. The total amount of extract that is used to make 1 batch of icecream = (The amount of vanilla extract) + (The amount of almond extract) b. The total extract present in 5 batches of icecream = 5 × (The total extract present in 1 batch of icecream) Essential Question How Can You Represent a Situation with a Math Model? Answer: A “Math model” usually describes a system by a set of variables and a set of equations that establish relationships between the variables Visual Learning Bridge Mr. Finn gives the number of snacks shown to the baseball team’s coach every time the team wins a game. How many total pounds of snacks does Mr. Finn give the coach after the baseball team wins 3 games? What hidden question do you need to find and solve first? How many total pounds of snacks does Mr. Finn give the coach when the baseball team wins one game? How can I model with math? I can • use previously learned concepts and skills. • find and answer any hidden questions. • use bar diagrams and equations to represent and solve this problem. Here’s my thinking. Let p = the pounds of snacks after one game. Mr. Finn gives the coach 2$$\frac{5}{8}$$ pounds of snacks after the team wins 3 games. Convince Me! Reasoning Explain how to solve this problem another way. Answer: It is given that Mr. Finn gives the number of snacks shown to the baseball team’s coach every time the team wins a game. Now, The given figure is: Now, From the given figure, We can observe that The total number of pounds of snacks Mr.Finn give the coach after the baseball team wins 1 game = (The number of pounds of red licorice) + (The number of pounds of peanuts) = $$\frac{3}{8}$$ + $$\frac{4}{8}$$ = $$\frac{3 + 4}{8}$$ = $$\frac{7}{8}$$ pounds of snacks Now, The total number of pounds of snacks Mr.Finn give the coach after the baseball team wins 3 games = 3 × (The total number of pounds of snacks Mr.Finn give the coach after the baseball team wins 1 game) = 3 × $$\frac{7}{8}$$ = $$\frac{3 × 7}{8}[/altex] = $\frac{21}{8}$$ pounds of snacks Hence, from the above, We can conclude that The total number of pounds of snacks Mr.Finn give the coach after the baseball team wins 3 games is: $$\frac{21}{8}$$ pounds Guided Practice Model with Math Colton and his classmates are making maps of the streets where they live. How much green and black felt does his teacher need to buy so 5 groups of students can each make a map? Question 1. Draw the bar diagrams and write equations to find g, the amount of green, and b, the amount of black felt. Answer: It is given that Colton and his classmates are making maps of the streets where they live Now, The given information is: Now, From the given information, The amount of green felt needed for each map (g) = $$\frac{4}{6}$$ sheet of green The amount of black felt needed for each map (b) = $$\frac{5}{6}$$ sheet of black Hence, The representation of the given information in the form of a bar diagram is: Question 2. Write and solve an equation to find t, the amount of green and black felt the class will use. Answer: From Question 1, We can observe that The amount of green felt needed for each map (g) = $$\frac{4}{6}$$ sheet of green The amount of black felt needed for each map (b) = $$\frac{5}{6}$$ sheet of black So, The total amount of green and black felts needed for each map (t) = (The amount of green felt needed fro each map) + (The amount of black felt needed for each map) = $$\frac{4}{6}$$ + $$\frac{5}{6}$$ = $$\frac{4 + 5}{6}$$ = $$\frac{9}{6}$$ = $$\frac{3}{2}$$ So, the amount of green and black felt the class will use = 5 × $$\frac{3}{2}$$ = [altex]\frac{5 × 3}{2}$ = $$\frac{15}{2}$$ Hence, from the above, We can conclude that The total amount of green and black felt the class will use is: $$\frac{15}{2}$$ Independent Practice Model with Math Moira swims $$\frac{3}{6}$$ hour before school 5 days a week and $$\frac{5}{6}$$ hour after school 4 days a week. For how long does she swim each week? Use Exercises 3-5 to answer the question. Question 3. Draw a bar diagram and write an equation to find b, how many hours Moira swims before school each week. Answer: It is given that Moira swims $$\frac{3}{6}$$ hour before school 5 days a week and $$\frac{5}{6}$$ hour after school 4 days a week. So, The number of hours Moira swims before school a week (b) = (The number of days Moira swims before school in a week) × (The number of hours Moira swims each day before school in a week) = 5 × $$\frac{3}{6}$$ = $$\frac{5 × 3}{6}$$ = $$\frac{15}{6}$$ = $$\frac{5}{2}$$ hours Hence, The representation of the information about the number of hours Moira swims before school each week is: The number of hours Moira swims before school a week is: $$\frac{5}{2}$$ hours Question 4. Draw a bar diagram and write an equation to find a, how many hours she swims after school each week. Answer: It is given that Moira swims $$\frac{3}{6}$$ hour before school 5 days a week and $$\frac{5}{6}$$ hour after school 4 days a week. So, The number of hours Moira swims after school a week (a) = (The number of days Moira swims after school in a week) × (The number of hours Moira swims each day after school in a week) = 4 × $$\frac{5}{6}$$ = $$\frac{5 × 4}{6}$$ = $$\frac{20}{6}$$ = $$\frac{10}{3}$$ hours Hence, The representation of the information about the number of hours Moira swims after school each week is: The number of hours Moira swims after school a week is: $$\frac{10}{3}$$ hours Question 5. Draw a bar diagram and write an equation to find h, how many hours Moira swims each week. Answer: From Questions 3 and 4, We can observe that The number of hours Moira swims before school a week (b) is: $$\frac{5}{2}$$ hours The number of hours Moira swims after school a week (a) is: $$\frac{10}{3}$$ hours So, The total number of hours Moira swims each week (h) = b + a = $$\frac{5}{2}$$ + $$\frac{10}{3}$$ = $$\frac{15}{6}$$ + $$\frac{20}{6}$$ = $$\frac{15 + 20}{6}$$ = $$\frac{35}{6}$$ hours Hence, The representation of the bar diagram for the number of hours Moira swims each week is: The number of hours Moira swims each week is: $$\frac{35}{6}$$ hours Problem Solving Performance Task Seeing Orange Perry mixed $$\frac{5}{8}$$ gallon of red paint and $$\frac{3}{8}$$ gallon of yellow paint to make the right shade of orange paint. He needs 2 gallons of orange paint to paint the basement floor. How many gallons of red and yellow paint should Perry use to make enough orange paint to cover the floor? Question 6. Reasoning What do you need to know to find how many gallons of each color Perry should use? Answer: It is given that Perry mixed $$\frac{5}{8}$$ gallon of red paint and $$\frac{3}{8}$$ gallon of yellow paint to make the right shade of orange paint. He needs 2 gallons of orange paint to paint the basement floor Hence, To find how many gallons of each color Perry should use, we have to answer: How many gallons of red paint and yellow paint are needed to make the right side of orange paint? Question 7. Model with Math Draw the bar diagrams and write equations to find g, how many gallons of paint are in a batch, and b, how many batches Perry needs to make. Answer: It is given that Perry mixed $$\frac{5}{8}$$ gallon of red paint and $$\frac{3}{8}$$ gallon of yellow paint to make the right shade of orange paint. He needs 2 gallons of orange paint to paint the basement floor So, The total amount of paint needed to make the right side of orange paint = (The number of gallons of red paint) + (The number of gallons of yellow paint) = $$\frac{5}{8}$$ + $$\frac{3}{8}$$ = $$\frac{5 + 3}{8}$$ = $$\frac{8}{8}$$ = 1 gallon Now, The number of gallons needed for Perry to paint the basement floor = (The number of gallons of orange paint needed to paint the basement floor) × (The total amount of paint needed to make the right side of orange paint) = 2 × 1 = 2 gallons Hence, The representation of the given information in the form of a bar diagram is: The number of gallons needed for Perry to paint the basement floor is: 2 gallons Question 8. Model with Math Draw the bar diagrams and write and solve equations to show how to find how many gallons of each color Perry should use. Tell what your variables represent. Answer: It is given that Perry mixed $$\frac{5}{8}$$ gallon of red paint and $$\frac{3}{8}$$ gallon of yellow paint to make the right shade of orange paint. He needs 2 gallons of orange paint to paint the basement floor So, The number of gallons of each color Perry should use = 2 × (The number of gallons of red paint) + 2 × (The number of gallons of yellow paint) = 2 × $$\frac{5}{8}$$ + 2 × $$\frac{3}{8}$$ = $$\frac{2 × 5}{8}$$ + $$\frac{2 × 3}{8}$$ = $$\frac{10}{8}$$ + $$\frac{6}{8}$$ = $$\frac{10 + 6}{8}$$ = $$\frac{16}{8}$$ = 2 gallons Hence, The representation of the given information in the form of a bar diagram is: The number of gallons of red paint Perry should use is: $$\frac{10]{8}$$ gallons The number of gallons of yellow paint Perry should use is: $$\frac{6]{8}$$ gallons ### Topic 10 Fluency Practice Activity Pointe Tall Find a partner. Get paper and a pencil. Each partner chooses a different color: light blue or dark blue. Partner 1 and Partner 2 each point to a black number at the same time. Each partner adds the two numbers. If the answer is on your color, you get a tally mark. Work until one partner has twelve tally marks. I can … add multi-digit whole numbers. ### Topic 10 Vocabulary Review Understand Vocabulary Word List • denominator • equivalent fractions • fraction • mixed number • multiple • numerator • unit fraction Write T for true and F for false. Question 1. _______ The fraction $$\frac{3}{4}$$ is a multiple of $$\frac{1}{4}$$. Answer: The given statement is: The fraction $$\frac{3}{4}$$ is a multiple of $$\frac{1}{4}$$ Now, We know that, n × $$\frac{1}{y}$$ is always a multiple of $$\frac{1}{y}$$ Hence, from the above, We can conclude that The fraction $$\frac{3}{4}$$ is a multiple of $$\frac{1}{4}$$ is: True Question 2. __________ Equivalent fractions are fractions where the numerator and the denominator have the same value. Answer: The given statement is: Equivalent fractions are fractions where the numerator and the denominator have the same value Now, We know that, The “Equivalent fractions” are the fractions that have different values of numerator and denominator Hence, from the above, We can conclude that Equivalent fractions are fractions where the numerator and the denominator have the same value is: False Question 3. __________ The denominator of a fraction tells the number of equal parts in the whole. Answer: The given statement is: The denominator of a fraction tells the number of equal parts in the whole. Now, We know that, In the fraction $$\frac{x}{y}$$, x represents the number of shaded parts y represents the total number of equal parts Hence, from the above, We can conclude that The denominator of a fraction tells the number of equal parts in the whole is: True Question 4. ___________ A fraction names part of a whole, part of a set, or a location on a number line. Answer: The given statement is: A fraction names part of a whole, part of a set, or a location on a number line. Now, We know that, A symbol used to name a part of a whole, a part of a set, or a location on a number line is called a “Fraction” Hence, from the above, We can conclude that A fraction names part of a whole, part of a set, or a location on a number line is: True Question 5. ________ The numerator is the number below the fraction bar in a fraction. Answer: The given statement is: The numerator is the number below the fraction bar in a fraction. Now, We know that, The “Denominator” is the number below the fraction bar in a fraction Hence, from the above, We can conclude that The numerator is the number below the fraction bar in a fraction is: False Write always, sometimes, or never. Question 6. A unit fraction __________ has a numerator of 1. Answer: The given statement is: A unit fraction __________ has a numerator of 1. Hence, from the above, We can conclude that A unit fraction “always” has a numerator of 1. Question 7. A numerator is _________ greater than its denominator. Answer: The given statement is: A numerator is _________ greater than its denominator. Hence, from the above, We can conclude that A numerator is “sometimes” greater than its denominator. Question 8. A mixed number _________ has just a fraction part. Answer: The given statement is: A mixed number _________ has just a fraction part. Hence, from the above, We can conclude that A mixed number “never” has just a fraction part. Use Vocabulary in Writing Question 9. Samatha wrote $$\frac{1}{2}$$. Use at least 3 terms from the Word List to describe Samantha’s fraction. Answer: It is given that Samatha wrote $$\frac{1}{2}$$. Hence, from the above, We can conclude that The 3 terms from the word list to describe Samantha’s fraction are: a. Fraction b Unit fraction c Equivalent fraction ### Topic 10 Reteaching Set A pages 385-388 Talia used $$\frac{5}{8}$$ yard of ribbon. Remember a unit fraction will always have a numerator of 1 Write each fraction as a multiple of a unit fraction. Question 1. $$\frac{5}{5}$$ Answer: The given fraction is: $$\frac{5}{5}$$ So, The representation of the given fraction in the form of a unit fraction is: Hence, from the above, We can conclude that The given fraction as a multiple of a unit fraction is: $$\frac{5}{5}$$ = 5 × $$\frac{1}{5}$$ Question 2. $$\frac{3}{8}$$ Answer: The given fraction is: $$\frac{3}{8}$$ So, The representation of the given fraction in the form of a unit fraction is: Hence, from the above, We can conclude that The given fraction as a multiple of a unit fraction is: $$\frac{3}{8}$$ = 3 × $$\frac{1}{8}$$ Question 3. $$\frac{4}{3}$$ Answer: The given fraction is: $$\frac{4}{3}$$ So, The representation of the given fraction in the form of a unit fraction is: Hence, from the above, We can conclude that The given fraction as a multiple of a unit fraction is: $$\frac{4}{3}$$ = 4 × $$\frac{1}{3}$$ Question 4. $$\frac{6}{5}$$ Answer: The given fraction is: $$\frac{6}{5}$$ So, The representation of the given fraction in the form of a unit fraction is: Hence, from the above, We can conclude that The given fraction as a multiple of a unit fraction is: $$\frac{6}{5}$$ = 6 × $$\frac{1}{5}$$ Question 5. $$\frac{15}{8}$$ Answer: The given fraction is: $$\frac{15}{8}$$ So, The representation of the given fraction in the form of a unit fraction is: Hence, from the above, We can conclude that The given fraction as a multiple of a unit fraction is: $$\frac{15}{8}$$ = 15 × $$\frac{1}{8}$$ Question 6. $$\frac{7}{4}$$ Answer: The given fraction is: $$\frac{7}{4}$$ So, The representation of the given fraction in the form of a unit fraction is: Hence, from the above, We can conclude that The given fraction as a multiple of a unit fraction is: $$\frac{7}{4}$$ = 7 × $$\frac{1}{4}$$ Set B pages 389-392 James runs $$\frac{3}{5}$$ mile each week. How far does James run after 2 weeks? Use multiplication to find the product. $$2 \times \frac{3}{5}=\frac{3}{5}+\frac{3}{5}=\frac{6}{5}=\frac{5}{5}+\frac{1}{5}=1 \frac{1}{5}$$ James ran $$\frac{6}{5}$$ or 1$$\frac{1}{5}$$ miles. Remember you can record answers as fractions or mixed numbers. Write and solve an equation. Question 1. Answer: The given model is: Now, From the given model, We can observe that 3 × $$\frac{1}{10}$$ + 3 × $$\frac{1}{10}$$ = $$\frac{3}{10}$$ + $$\frac{3}{10}$$ = $$\frac{3 + 3}{10}$$ = $$\frac{6}{10}$$ = $$\frac{3}{5}$$ Hence, from the above, We can conclude that The value for the given model is: $$\frac{3}{5}$$ Question 2. Answer: The given model is: Now, From the given model, We can observe that The total number of parts are: 6 The number of shaded parts is: 4 So, The fraction of the shaded part in the given model = $$\frac{The number of shaded parts}{The total number of parts}$$ = $$\frac{4}{6}$$ = $$\frac{2}{3}$$ Hence, from the above, We can conclude that The value for the given model is: $$\frac{2}{3}$$ Set C pages 393-396 Alisa has 7 puppies. Each puppy eats $$\frac{2}{3}$$ cup of food each day. How many cups of food does Alisa need to feed the puppies each day? Multiply 7 × $$\frac{2}{3}$$ Multiply the whole number and the numerator. $$7 \times \frac{2}{3}=\frac{7 \times 2}{3}$$ = $$\frac{14}{3}$$ = $$\frac{3}{3}+\frac{3}{3}+\frac{3}{3}+\frac{3}{3}+\frac{2}{3}$$ = 4$$\frac{2}{3}$$ cups Alisa needs 4$$\frac{2}{3}$$ cups of food to feed the puppies each day. Remember you multiply the whole number and the numerator and write the product above the denominator of the fraction. Question 1. Milo makes 5 batches of muffins. In each batch he uses $$\frac{2}{3}$$ bag of walnuts. How many bags of walnuts does Milo use? Answer: It is given that Milo makes 5 batches of muffins. In each batch he uses $$\frac{2}{3}$$ bag of walnuts. So, The number of bags of walnuts does Milo use = (The number of batches of muffins) × (The amount of bag of walnuts present in each batch) = 5 × $$\frac{2}{3}$$ = $$\frac{5 × 2}{3}$$ = $$\frac{10}{3}$$ bags of walnuts Hence, from the above, We can conclude that The number of bags of walnuts does Milo used is: $$\frac{10}{3}$$ bags of walnuts Question 2. A bird feeder can hold $$\frac{7}{8}$$ pound of seeds. How many pounds of seeds can 4 bird feeders hold? Answer: It is given that A bird feeder can hold $$\frac{7}{8}$$ pound of seeds So, The number of pounds of seeds can 4 bird feeders hold = 4 × (The number of pounds of seeds a bird feeder can hold) = 4 × $$\frac{7}{8}$$ = $$\frac{4 × 7}{8}$$ = $$\frac{28}{8}$$ = $$\frac{7}{2}$$ pounds of seeds Hence, from the above, We can conclude that The number of pounds of seeds can 4 bird feeders hold is: $$\frac{7}{2}$$ pounds of seeds Set D pages 397-400 Remember you may need to regroup when solving problems with time. Question 1. Answer: We know that, If the number of minutes is greater than 60 and we know that 1 hour = 60 minutes Add that hour to the hours present in the result and keep the extra minutes intact Hence, Question 2. 7 × $$\frac{3}{4}$$ hour Answer: The given fraction is: 7 × $$\frac{3}{4}$$ hours So, 7 × $$\frac{3}{4}$$ = $$\frac{7 × 3}{4}$$ = $$\frac{21]{4}$$ hours Hence, from the above, We can conclude that The value of the given fraction is: $$\frac{21}{4}$$ hours Question 3. Answer: We know that, 1 week = 7 days So, When the number of days is less than 7, borrow that 7 days from the number of weeks and add it to the number that is less than 7 and do the subtraction Hence, Question 4. Divide 560 days into groups of 8. Answer: It is given that Divide 560 days into groups of 8. So, Using the method of Long division, Hence, from the above, We can conclude that 560 days is divided into 70 equal parts when divided by 8 Question 5. Li Marie practices piano 1$$\frac{2}{3}$$ hours during the week and 2$$\frac{1}{3}$$ hours on the weekend. Show each time on the number line. How many more hours does she practice on the weekend than on the weekdays? Answer: It is given that Li Marie practices piano 1$$\frac{2}{3}$$ hours during the week and 2$$\frac{1}{3}$$ hours on the weekend Now, We know that, The number of weekdays is: 5 days The number of days in a weekend are: 2 days So, The number of hours Li Marie practices piano during the week = 5 × 1$$\frac{2}{3}$$ = 5 × $$\frac{5}{3}$$ = $$\frac{5 × 5}{3}$$ = $$\frac{25}{3}$$ hours So, The number of hours Li Marie practices piano on the weekend = 2 × 2$$\frac{1}{3}$$ = 2 × $$\frac{7}{3}$$ = $$\frac{2 × 7}{3}$$ = $$\frac{14}{3}$$ hours So, The difference between the number of hours Li Marie practices on the weekend and during the week = $$\frac{25}{3}$$ – $$\frac{14}{3}$$ = $$\frac{25 – 14}{3}$$ = $$\frac{11}{3}$$ hours So, The representation of the number f hours Li Mario practices piano during the week and on the weekend on a number line is: Hence, from the above, We can conclude that Li Mario practices $$\frac{11}{3}$$ hours more on the weekend than the weekdays Set E pages 401-404 Think about these questions to help you model math. Thinking Habits • How can I use the math I know to help solve this problem? • How can I use pictures, objects, or an equation to represent the problem? • How can I use numbers, words, and symbols to solve the problem? Julie makes chili with 2$$\frac{3}{8}$$ cups of red beans, 4$$\frac{1}{8}$$ cups of chili beans, and $$\frac{7}{8}$$ cup of onions. How many more cups of chili beans did Julie use than red beans and onions combined? Question 1. Write and solve an equation to find r, how many cups of red beans and onions Julie uses. Answer: It is given that Julie makes chili with 2$$\frac{3}{8}$$ cups of red beans, 4$$\frac{1}{8}$$ cups of chili beans, and $$\frac{7}{8}$$ cup of onions So, The number of cups of red beans and onions Julie uses to make chili (r) = (The number of cups of red beans used to make chili) + (The number of cups of onions used to make chili) = 2$$\frac{3}{8}$$ + $$\frac{7}{8}$$ = $$\frac{19}{8}$$ + $$\frac{7}{8}$$ = $$\frac{19 + 7}{8}$$ = $$\frac{26}{8}$$ = $$\frac{13}{4}$$ cups Hence, from the above, We can conclude that The number of cups of red beans and onions Julie uses to make chili is: $$\frac{13}{4}$$ cups Question 2. Write and solve an equation to find c, how many more cups of chili beans Julie used than red beans and onions. Answer: It is given that Julie makes chili with 2$$\frac{3}{8}$$ cups of red beans, 4$$\frac{1}{8}$$ cups of chili beans, and $$\frac{7}{8}$$ cup of onions Now, From Question 1, The number of cups of red beans and onions Julie uses to make chili is: $$\frac{13}{4}$$ cups Now, The number of more cups of chili beans Julie used than red beans and onions (c) = (The number of cups of chili beans Julie used to amke chili) – (The number of cups of red beans and onions Julie uses to make chili) = 4$$\frac{1}{8}$$ – $$\frac{13}{4}$$ = $$\frac{33}{8}$$ – $$\frac{26}{8}$$ = $$\frac{33 – 26}{8}$$ = $$\frac{7}{8}$$ cups Hence, from the above, We can conclude that The number of more cups of chili beans Julie used than red beans and onions (c) is: $$\frac{7}{8}$$ cups ### Topic 10 Assessment Practice Question 1. Margo practices her flute $$\frac{1}{4}$$ hour each day. A. Write and solve an equation to find how many hours Margo practices her flute in 1 week. Answer: It is given that Margo practices her flute $$\frac{1}{4}$$ hour each day. Now, The given information is: So, The number of hours Margo practices her flute in 1 week = (The number of days in 1 week) × (The number of hours Margo practices her flute each day) = 7 × $$\frac{1}{4}$$ = $$\frac{7 × 1}{4}$$ = $$\frac{7}{4}$$ hours Hence, from the above, We can conclude that The number of hours Margo practices her flute in 1 week is: $$\frac{7}{4}$$ hours B. Write and solve an equation to find how many minutes Margo practices her flute in 1 day. Then use that to find the number of minutes she practices in 1 week. Answer: It is given that Margo practices her flute $$\frac{1}{4}$$ hour each day. Now, The given information is: Now, We know that, 1 day = 24 hours So, The number of minutes Margo practices her flute in 1 day = (The number of hours Margo practices her flute in 1 day) × 60 = $$\frac{1}{4}$$ × 60 = $$\frac{60}{4}$$ = 15 minutes So, The number of minutes Margo practices her flute in 1 week = (The number of days in 1 week) × (The number of minutes Margo practices her flute in 1 day) = 7 × 15 = 7 × (10 + 5) = (7 × 10) + (7 × 5) = 70 + 35 = 105 minutes Hence, from the above, We can conclude that The number of minutes Margo practices her flute in 1 day is: 15 minutes The number of minutes Margo practices her flute in 1 week is: 105 minutes Question 2. Which of the following represents the fraction $$\frac{8}{9}$$ as a multiple of a unit fraction? A. $$\frac{8}{8}$$ = 1 × $$\frac{8}{9}$$ B. $$\frac{8}{9}$$ = 8 × 9 C. $$\frac{8}{9}$$ = 8 × $$\frac{1}{9}$$ D. $$\frac{8}{9}$$ = 4 × $$\frac{2}{9}$$ Answer: The given fraction is: $$\frac{8}{9}$$ So, The representation of the given fraction as a multiple of a unit fraction is: $$\frac{8}{9}$$ = 8 × $$\frac{1}{9}$$ Hence, from the above, We can conclude that The following equations represent the fraction $$\frac{8}{9}$$ as a multiple of a unit fraction are: Question 3. Ben played at a friend’s house for 2 hours 35 minutes. Later he played at a park for 1 hour 10 minutes. He played for another 1 hour 20 minutes in his backyard. How long did Ben play in all? A. 6 hours 27 minutes B. 5 hours 15 minutes C. 5 hours 5 minutes D. 5 hours Answer: It is given that Ben played at a friend’s house for 2 hours 35 minutes. Later he played at a park for 1 hour 10 minutes. He played for another 1 hour 20 minutes in his backyard So, The total amount of time taken by Ben to play in all = 2 hours 35 minutes + 1 hour 10 minutes + 1 hour 20 minutes = 4 hours 65 minutes = 5 hours 05 minutes Hence, from the above, We can conclude that The time taken by Ben to play in all is: Question 4. Choose numbers from the list to fill in the missing values in the multiplication equations. Use each number once. Answer: The given number list is: Hence, from the above, We can conclude that The numbers from the list that filled in the missing values in the multiplication equations are: Question 5. Chris found the products of whole numbers and fractions. Match each expression with its product. Answer: The given expressions with their matched products are: Question 6. What is the product of 4 and $$\frac{4}{8}$$? Write another expression that is equal to the product of 4 and $$\frac{4}{8}$$. Answer: The given numbers are: 4 and $$\frac{4}{8}$$ So, 4 × $$\frac{4}{8}$$ = 4 × 4 × $$\frac{1}{8}$$ = $$\frac{4 × 4}{8}$$ = $$\frac{16}{8}$$ = 2 Hence, from the above, We can conclude that The product of 4 × $$\frac{4}{8}$$ is: 2 The another expression that is equal to 4 × $$\frac{4}{8}$$ is: 4 × 4 × $$\frac{1}{8}$$ Question 7. Complete the multiplication equation that describes that is shown by the model. Answer: The given model is: Now, From the given model, We can observe that $$\frac{1}{6}$$ is repeated 8 times and $$\frac{2}{6}$$ is repeated 4 times So, Hence, from the above, We can conclude that The completed multiplication equation that is shown by the model is: Question 8. Use a unit fraction and a whole number to write a multiplication equation equal to a $$\frac{7}{8}$$. Answer: The given fraction is: $$\frac{7}{8}$$ So, The representation of the given fraction in the form of a unit fraction and a whole number is: $$\frac{7}{8}$$ = 7 × $$\frac{1}{8}$$ Hence, from the above, We can conclude that The multiplication equation for the given fraction in the form of a unit fraction and a whole number is: $$\frac{7}{8}$$ = 7 × $$\frac{1}{8}$$ Question 9. Juan is making cookies. He makes 2 batches on Monday and 4 batches on Tuesday. He uses $$\frac{3}{4}$$ cup of flour in each batch. How much flour does Juan use? Explain. Answer: It is given that Juan is making cookies. He makes 2 batches on Monday and 4 batches on Tuesday. He uses $$\frac{3}{4}$$ cup of flour in each batch So, The total number of batches = (The number of batches made by Juan on Monday) + (The number of batches made by Juan on Tuesday) = 2 + 4 = 6 batches So, The total amount of flour Juan used = (The total number of batches) × (The amount of flour used in each batch) = 6 × $$\frac{3}{4}$$ = $$\frac{6 × 3}{4}$$ = $$\frac{18}{4}$$ = $$\frac{9}{2}$$ cups of flour Hence, from the above, We can conclude that The total amount of flour Juan used is: $$\frac{9}{2}$$ cups of flour Question 10. Lee uses $$\frac{1}{5}$$ yard of wire for each ornament he makes. He makes 3 ornaments for his grandmother and 2 ornaments for his mother. How many yards of wire did Lee use? A. $$\frac{3}{5}$$ B. 1$$\frac{2}{5}$$ C. $$\frac{2}{5}$$ D. 1 Answer: It is given that Lee uses $$\frac{1}{5}$$ yard of wire for each ornament he makes. He makes 3 ornaments for his grandmother and 2 ornaments for his mother So, The total number of ornaments Lee made = (The number of ornaments he made for his grandmother) + (The number of ornaments made for his mother) = 3 + 2 = 5 ornaments So, The total number of yards of wire Lee used = (The total number of ornaments Lee made) × (The number of yards of wire he used for each ornament) = 5 × $$\frac{1}{5}$$ = $$\frac{5 × 1}{5}$$ = $$\frac{5}{5}$$ = 1 yard Hence, from the above, We can conclude that The total number of yards of wire Lee used is: Question 11. Lucas is making one dozen snacks for his team. He uses $$\frac{1}{4}$$ cup of dried cherries and $$\frac{2}{4}$$ cup of dried apricots for each snack. How many cups of dried fruit does Lucas need for his one dozen snacks? Remember, there are 12 snacks in one dozen. Write and solve equations to show how you found the answer. Answer: It is given that Lucas is making one dozen snacks for his team. He uses $$\frac{1}{4}$$ cup of dried cherries and $$\frac{2}{4}$$ cup of dried apricots for each snack. So, The total number of cups of dried cherries and dried apricots for each snack = $$\frac{1}{4}$$ + $$\frac{2}{4}$$ = $$\frac{1 + 2}{4}$$ = $$\frac{3}{4}$$ So, The total number of cups of dried cherries and dried apricot for his 1 dozen snacks = 12 × (The total number of cups of dried cherries and dried apricots for each snack) = 12 × $$\frac{3}{4}$$ = $$\frac{12 × 3}{4}$$ = $$\frac{36}{4}$$ = 9 cups Hence, from the above, We can conclude that The total number of cups of dried cherries and dried apricot for his 1 dozen snacks are: 9 cups ### Topic 10 Performance Task School Mural Paul has permission to paint a 20-panel mural for his school. Part of the mural is shown in the Painting a Mural figure. Paul decides he needs help. The Helpers table shows how much several of his friends can paint each day and how many days a week they can paint. Question 1. The students want to find how long it will take to paint the mural if each works on a different part of the panels a different number of days a week. Part A How many panels can Leeza paint in a week? Use fraction strips to explain. Answer: It is given that Paul has permission to paint a 20-panel mural for his school. Part of the mural is shown in the Painting a Mural figure. Paul decides he needs help. The Helpers table shows how much several of his friends can paint each day and how many days a week they can paint. Now, The given table is: Now, From the given table, We can observe that Leeza can paint $$\frac{3}{4}$$ panels a day for 3 days a week So, The number of panels Leeza can paint in a week = (The amount Leeza can paint a day) × (The number of days she can paint a week) = $$\frac{3}{4}$$ × 3 = $$\frac{3 × 3}{4}$$ = $$\frac{9}{4}$$ panels Hence, The representation of the number of panels Leeza can paint in a week in the form of a bar diagram is: The number of panels Leeza can paint in a week is: $$\frac{9}{4}$$ panels Part B How many panels can Kelsey paint in a week? Use equations to explain. Answer: The given table is: Now, From the given table, We can observe that Kelsey paints $$\frac{7}{8}$$ panels a day for 4 days a week So, The number of panels Kelsey can paint in a week = (The number of days can Kelsey paint in a week) × (The number of panels Kelsey can paint in a day) = 4 × $$\frac{7}{8}$$ = $$\frac{4 × 7}{8}$$ = $$\frac{28}{8}$$ = $$\frac{7}{2}$$ panels Hence, from the above, We can conclude that The number of panels Kelsey can paint in a week is: $$\frac{7}{2}$$ panels Part C Paul can work 5 days a week. How many panels can Paul paint in a week? Explain. Answer: It is given that Paul works $$\frac{9}{10}$$ panels a day So, The number of panels Paul can paint in a week = (The number of days Paul can paint in a week) × (The number of panels Paul can paint in a day) = 5 × $$\frac{9}{10}$$ = $$\frac{5 × 9}{10}$$ = $$\frac{45}{10}$$ = $$\frac{9}{2}$$ panels Hence, from the above, We can conclude that The number of panels Paul can paint in a week is: $$\frac{9}{2}$$ panels Part D How many panels can Tony paint in a week? Draw a bar diagram. Write and solve an equation. Answer: The given table is: Now, From the given table, We can observe that Tony paints $$\frac{5}{6}$$ panels a day for 3 days a week So, The number of panels Tony can paint in a week = (The number of days can Tony paint in a week) × (The number of panels Tony can paint in a day) = 3 × $$\frac{5}{6}$$ = $$\frac{3 × 5}{6}$$ = $$\frac{15}{6}$$ = $$\frac{5}{2}$$ panels Hence, The representation of the number of panels Tony can make in a week in the form of a bar diagram is: The number of panels Tony can paint in a week is: $$\frac{5}{2}$$ panels Question 2. The Time Spent Painting Each Day table shows how much time each of Paul’s friends helped with the mural each day that they worked on it. How much more time did Kelsey spend each day than Tony and Leeza combined? Explain. Answer: It is given that The Time Spent Painting Each Day table shows how much time each of Paul’s friends helped with the mural each day that they worked on it. Now, The given table is: Now, From the given table, We can observe that The time spent painting each day by Tony and Leeza combined = 1 hour 45 minutes + 30 minutes = 1 hour 75 minutes = 2 hours 15 minutes So, The difference between the time spent each day by Kelsey and Tony and Leeza combined = 2 hours 30 minutes – 2 hours 15 minutes = 15 minutes Hence, from the above, We can conclude that Kelsey spent each day 15 minutes more than Tony and Leeza combined #### enVision Math Common Core Grade 4 Answer Key ## Envision Math Common Core Grade 4 Answer Key Topic 11 Represent and Interpret Data on Line Plots ## Envision Math Common Core 4th Grade Answers Key Topic 11 Represent and Interpret Data on Line Plots Essential Questions: How can you solve problems using data on a line plot? How can you make a line plot? enVision STEM Project: Safety and Data Do Research Use the Internet or other sources to find what causes an earthquake and how the power of an earthquake is measured. Tell how people can stay safe during earthquakes. Journal: Write a Report Include what you found. Also in your report: • The size, or magnitude, of an earthquake is measured with the Richter scale. Explain how the scale is used. • Research the magnitudes of at least 6 earthquakes that have occurred in your lifetime. Make a table showing when they occurred and their magnitudes, and then show their magnitudes on a line plot. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • compare • data • line plot • scale Question 1. A _________ is a way to organize data on a number line. Answer: A __line plot_______ is a way to organize data on a number line. Question 2. Numbers that show the units used on a graph are called a __________ Answer: Numbers that show the units used on a graph are called a __scale________. Question 3. _________ are pieces of information. Answer: ___Data______ are pieces of information. Comparing Fractions Write >, <, or = in the . Question 4. $$\frac{7}{8}$$ $$\frac{3}{4}$$ Answer: $$\frac{7}{8}$$ > $$\frac{3}{4}$$. Explanation: $$\frac{7}{8}$$ $$\frac{3}{4}$$ => 0.875 __>__ 0.75. Question 5. $$\frac{1}{2}$$ $$\frac{5}{8}$$ Answer: $$\frac{1}{2}$$ < $$\frac{5}{8}$$. Explanation: $$\frac{1}{2}$$ $$\frac{5}{8}$$ => 0.5 __<___ 0.625. Question 6. $$\frac{1}{4}$$ $$\frac{2}{8}$$ Answer: $$\frac{1}{4}$$ = $$\frac{2}{8}$$ Explanation: $$\frac{1}{4}$$ $$\frac{2}{8}$$ => 0.25 __=___ 0.25. Fraction Subtraction Find the difference. Question 7. 10$$\frac{3}{8}$$ – 4$$\frac{1}{8}$$ = _______ Answer: 10$$\frac{3}{8}$$ – 4$$\frac{1}{8}$$ = 6$$\frac{1}{4}$$. Explanation: 10$$\frac{3}{8}$$ – 4$$\frac{1}{8}$$ = ??? => 6$$\frac{1}{4}$$ . Question 8. 5$$\frac{1}{4}$$ – 3$$\frac{3}{4}$$ = _______ Answer: 5$$\frac{1}{4}$$ – 3$$\frac{3}{4}$$ = 1$$\frac{1}{2}$$. Explanation: 5$$\frac{1}{4}$$ – 3$$\frac{3}{4}$$ = ??? => 1$$\frac{1}{2}$$. Question 9. 7$$\frac{4}{8}$$ – 2$$\frac{4}{8}$$ = __________ Answer: 7$$\frac{4}{8}$$ – 2$$\frac{4}{8}$$ = 5. Explanation: 7$$\frac{4}{8}$$ – 2$$\frac{4}{8}$$ = ??? => 5. Interpreting Data Use the data in the chart to answer each exercise. Question 10. What is the greatest snake length? What is the least snake length? Answer: The greatest snake length is 24 inches. The least snake length is 12 $$\frac{1}{2}$$ inches. Question 11. Which of the snake lengths are recorded more than once? Which length was recorded the most? Answer: The snake lengths are recorded more than once are 12 $$\frac{1}{2}$$ inches, 16 inches and 17 inches. Length that was recorded the most is 16 inches. Question 12. What is the difference between the greatest length and the shortest length recorded? Answer: The difference between the greatest length and the shortest length recorded is 11 $$\frac{1}{2}$$ inches. Explanation: The difference between the greatest length and the shortest length recorded is: The greatest snake length – The least snake length = 24 inches – 12 $$\frac{1}{2}$$ inches. = 11 $$\frac{1}{2}$$ inches. Pick a Project PROJECT 11A What are fun ways to get up off the couch and move? Project: Design a Park Answer: PROJECT 11B What are the most commonly chosen state insects? Project: Write a Poem and Make a Graph about a State Insect Answer: PROJECT 11C Have you ever baked a pie? Project: Make a Pamphlet of Pie Recipes Answer: 3-ACT MATH PREVIEW Math Modeling It’s a Fine Line I can… model with math to solve a problem that involves analyzing and interpreting data on line plots. ### Lesson 11.1 Read Line Plots Solve & Share Emily went fishing. She plotted the lengths of 12 fish caught on the line plot shown below. What was the length of the longest fish caught? What was the length of the shortest fish caught? I can … interpret data using line plots. Look Back! What other observations can you make from the line plot about the lengths of fish caught? Answer: Other observations can be made from the line plot about the lengths of fish caught is that every quarterly he caught new fish added one more than it. Essential Question How Can You Read Data in a Line Plot? Answer: It is read easily because the line plot is clear stating its refers to lengths of different fishes caught and represented on the line its catching points of intervals, by this way I can read given Data in a Line Plot. Visual Learning Bridge A line plot shows data along a number line. Each dot above a point on the line represents one number in the data set. The table below shows the distance Eli walked his dog each day for seven days. Here is how the data look on a line plot. The numbers along the bottom of the line plot are the scale of the graph. Interpret the data on the line plot. The most dots are above 1 on the line plot. The most common distance walked is 1 mile. The longest distance walked is 3 miles. The shortest distance walked is $$\frac{1}{2}$$ mile. What is the difference between the longest distance and the shortest distance Eli walked his dog? 3 – $$\frac{1}{2}$$ = $$\frac{6}{2}$$ – $$\frac{1}{2}$$ = $$\frac{5}{2}$$ or 2$$\frac{1}{2}$$ miles Convince Me! Model with Math Write and solve an equation to find how many miles m, Eli walked his dog in all for the 7 days. Answer: Total distance walked by Eli his dog in all for the 7 days = 9 $$\frac{1}{2}$$ miles. Explanation: Distance walked by Eli on Monday = $$\frac{1}{2}$$ mile. Distance walked by Eli on Tuesday = 1 $$\frac{1}{2}$$ miles. Distance walked by Eli on Wednesday = 1 mile. Distance walked by Eli on Thursday = 1 $$\frac{1}{2}$$ miles. Distance walked by Eli on Friday = 3 miles. Distance walked by Eli on Saturday = 1mile. Distance walked by Eli on Sunday = 1mile. Total distance walked by Eli his dog in all for the 7 days = Distance walked by Eli on Monday + Distance walked by Eli on Tuesday + Distance walked by Eli on Wednesday + Distance walked by Eli on Thursday + Distance walked by Eli on Friday + Distance walked by Eli on Saturday + Distance walked by Eli on Sunday = 1 + $$\frac{1}{2}$$ + 1 $$\frac{1}{2}$$ + 1 + 1 $$\frac{1}{2}$$ + 3 + 1 = 1 $$\frac{1}{2}$$+ 1 $$\frac{1}{2}$$ + 1 + 1 $$\frac{1}{2}$$ + 3 + 1 = 3 + 1 + 1 $$\frac{1}{2}$$ + 3 + 1 = 4 + 1 $$\frac{1}{2}$$ + 3 + 1 = 5 $$\frac{1}{2}$$ + 3 + 1 = 8 $$\frac{1}{2}$$ + 1 = 9 $$\frac{1}{2}$$ miles. Guided Practice Do You Understand? Question 1. How can you tell the longest distance Eli walked his dog from the line plot? Answer: It can be said the longest distance Eli walked his dog from the line plot by checking and comparing the numerical values given in the given data and finding out the highest among them in all. Question 2. If a line plot represented 10 pieces of data, how many dots would it have? Explain. Answer: If a line plot represented 10 pieces of data, dots would be of 10 because the data is of 10 pieces. Do You Know How? For 3-5, use the line plot below. Question 3. How many giraffes are 14 feet tall? Answer: Two or 2 giraffes are 14 feet tall. Question 4. What is the most common height? Answer: The most common height is 15 feet. Question 5. How tall is the tallest giraffe? Answer: 16 feet is the tallest giraffe. Independent Practice For 6-10, use the line plot at the right. Question 6. How many people ran the 100-meter sprint? Answer: 20 people ran the 100-meter sprint. Question 7. Which time was the most common? Answer: 11 hours is the most common time. Question 8. What is the difference between the fastest sprint and the slowest sprint? Answer: The difference between the fastest sprint and the slowest sprint is 3 $$\frac{3}{4}$$ seconds. Explanation: The fastest sprint = 10 $$\frac{1}{4}$$. The slowest sprint = 12 $$\frac{2}{4}$$. The difference between the fastest sprint and the slowest sprint: = 12 $$\frac{2}{4}$$ – 10 $$\frac{1}{4}$$ = 6 – 2 $$\frac{1}{4}$$ = 3 $$\frac{3}{4}$$ seconds. Question 9. How many more people ran 100 meters in 11$$\frac{2}{4}$$ seconds than in 10$$\frac{1}{4}$$ seconds? Answer: 2 more people ran 100 meters in 11$$\frac{2}{4}$$ seconds than in 10$$\frac{1}{4}$$ seconds. Explanation: More people ran 100 meters in 11 $$\frac{2}{4}$$ seconds than in 10 $$\frac{1}{4}$$ seconds = 2. Question 10. Curtis said more than half the people ran 100 meters in less than 11 seconds. Do you agree? Explain. Answer: Yes, I agree with what Curtis said because its 17 people who ran 100 meters in less than 11 seconds. Problem Solving For 11-12, use the line plot at the right. Question 11. Reasoning Mr. Dixon recorded the times it took students in his class to complete a project. Which time was most often needed to complete the project? Answer: 3 Hours was most often needed to complete the project. Question 12. How much longer was the greatest amount of time spent completing the project than the least amount of time? Answer: 1 hour longer was the greatest amount of time spent completing the project than the least amount of time. Explanation: The greatest amount of time spent completing the project = 3 $$\frac{2}{4}$$ hours. The least amount of time spent completing the project = 2 $$\frac{2}{4}$$ hours. Difference: The greatest amount of time spent completing the project – The least amount of time spent completing the project = 3 $$\frac{2}{4}$$ – 2 $$\frac{2}{4}$$ = 1 hour. Question 13. Number Sense Jorge collects sports cards. He displays his cards in an album. There are 72 pages in the album. Each page holds 9 cards. Explain how to decide whether or not the album holds more than 600 cards. Answer: Yes, the album holds more than 600 cards because by the calculation of total number of cards and result is 648. Explanation: Number of pages in the album = 72. Number of cards in each page = 9. Total number of cards the album holds = Number of pages in the album × Number of cards in each page = 72 × 9 = 648. Question 14. Higher Order Thinking Bob and 2 friends each were able to juggle with bean bags for $$\frac{3}{4}$$ minute. How long did they juggle altogether? Answer: Total time taken by them = $$\frac{9}{4}$$ minutes. Explanation: Number of people were juggling with bean bags = 3. Time taken for each to juggle with bean bags = $$\frac{3}{4}$$ minute = 3/4 minute. Total time taken by them = Number of people were juggling with bean bags × Time taken for each to juggle with bean bags = 3 × $$\frac{3}{4}$$ minute = $$\frac{9}{4}$$ minutes. Assessment Practice For 15-16, use the line plot at the right. Question 15. How much longer is the longest nail than the shortest nail? A. 1$$\frac{1}{4}$$ inches B. 1$$\frac{2}{4}$$ inches C. 1$$\frac{3}{4}$$ inches D. 2$$\frac{1}{4}$$ inches Answer: 1$$\frac{2}{4}$$ inches is the longest nail than the shortest nail. B. 1$$\frac{2}{4}$$ Explanation: The longest nail = 2 $$\frac{1}{4}$$ inches. The shortest nail = $$\frac{3}{4}$$ inches. Difference: The longest nail – The shortest nail = 2 $$\frac{1}{4}$$ – $$\frac{3}{4}$$ = 1$$\frac{2}{4}$$ inches. Question 16. Ed measured the nails that were 2$$\frac{1}{4}$$ inches long incorrectly. They were each actually $$\frac{3}{4}$$ inch longer. What was the length of the nails? A. $$\frac{3}{4}$$ inch B. 1$$\frac{2}{4}$$ inches C. 3 inches D. 3$$\frac{1}{4}$$ inches Answer: 1$$\frac{2}{4}$$ inches is the the length of the nails. B. 1$$\frac{2}{4}$$ Explanation: Wrong measurement of nails = 2$$\frac{1}{4}$$ inches. Correct measurement of nails = $$\frac{3}{4}$$ inch. the length of the nails = Wrong measurement of nails – Correct measurement of nails = 2$$\frac{1}{4}$$ – $$\frac{3}{4}$$ = 1$$\frac{2}{4}$$ inches. ### Lesson 11.2 Make Line Plots Solve & Share The manager of a shoe store kept track of the lengths of the shoes sold in a day. Complete the line plot using the data from the shoe store. What length shoe was sold the most? I can … make a line plot to represent data. Look Back! Generalize How can you use a line plot to find the data that occur most often? Answer: We can use a line plot to find the data that occur most often by counting the number of times which occurred more in the given data. Explanation: Essential Question How Can You Make Line Plots? Answer: We can make line plots by using the data given representing it on line. To create a line plot, ​first create a number line that includes all the values in the data set. Visual Learning Bridge Serena measured the lengths of her colored pencils. How can Serena make a line plot to show these lengths? Making a Line Plot Step 1 Draw a number line and choose a scale based on the lengths of Serena’s pencils. Mark halves, fourths, and eighths. The scale should show data values from the least to the greatest. Step 2 Write a title for the line plot. Label the line plot to tell what the numbers represent. Step 3 Draw a dot for each pencil length. Convince Me! Model with Math Write and solve an equation to find the difference , in length between Serena’s two shortest colored pencils. Answer: Difference , in length between Serena’s two shortest colored pencils = $$\frac{1}{4}$$ inches. Explanation: Length of First shortest colored pencil = 4 $$\frac{1}{2}$$ inches. Length of Second shortest colored pencil = 4 $$\frac{3}{4}$$ inches. Difference: Length of Second shortest colored pencil – Length of First shortest colored pencil = 4 $$\frac{3}{4}$$ – 4 $$\frac{1}{2}$$ = $$\frac{1}{4}$$ inches. Guided Practice Do You Understand? Question 1. The scale of the line plot, Lengths of Serena’s Pencils, goes from 4 to 5 by eighths. Why is this a good scale to use? Answer: This is a good scale to use because its easy to plot the measurements on the line and easy to understand the values. Explanation: The scale of the line plot, Lengths of Serena’s Pencils, goes from 4 to 5 by eighths. This is a good scale to use because its easy to plot and to understand too. Question 2. Use the table shown at the right to compare the lengths of Sandy’s pencils with the lengths of Serena’s pencils shown on the previous page. Who has more pencils that are the same length, Serena or Sandy? Which set of data was easier to compare? Why? Answer: Serena has more more pencils that are the same length of 4 $$\frac{3}{4}$$ inches. Both set of data are easier to compare because the values are clear to understand. Explanation: Do You Know How? Question 3. Complete the line plot. Answer: Explanation: Line plotting for the following data: Independent Practice Leveled Practice For 4-5, use the table at the right. Question 4. Use the data in the table to make a line plot. Answer: Explanation: Line plotting of data: Question 5. What is the length of the longest bracelet? What is the shortest length? What is the difference? Answer: Difference = 2 inches. Explanation: Length of the longest bracelet = 8 $$\frac{1}{2}$$ inches. Length of the shortest bracelet = 6 $$\frac{1}{2}$$ inches. Difference: Length of the longest bracelet – Length of the shortest bracelet = 8 $$\frac{1}{2}$$ – 6 $$\frac{1}{2}$$ = 2 inches. Problem Solving Question 6. Nora weighed each of the 7 beefsteak tomatoes she picked from her garden. The total weight of the 7 tomatoes was 10$$\frac{3}{4}$$ pounds. Her line plot shows only 6 dots. What was the weight of the missing tomato? Answer: Weight of the missing tomato = 1$$\frac{3}{4}$$ pounds. Explanation: Total weight of the 7 tomatoes =10 $$\frac{3}{4}$$ pounds. Number of dots shown on line plot = 6. Total weight of the 7 tomatoes = dots value plotted on line + Weight of the missing tomato 10 $$\frac{3}{4}$$ = 1 + 1 + 1$$\frac{1}{4}$$ + 1$$\frac{2}{4}$$ + 2 + 2$$\frac{1}{4}$$ + Weight of the missing tomato => 10 $$\frac{3}{4}$$ = 9 + Weight of the missing tomato => 10 $$\frac{3}{4}$$ – 9 = Weight of the missing tomato => 1$$\frac{3}{4}$$ pounds = Weight of the missing tomato. Question 7. Make Sense and Persevere Alyssa made a pink-and-white-striped blanket for her bed. There are 7 pink stripes and 6 white stripes. Each stripe is 8 inches wide. How wide is Alyssa’s blanket? Explain. Answer: Total length of Alyssa’s blanket = 104 inches. Explanation: Number of blanket with pink stripes = 7. Number of blanket with pink stripes = 6. Length of each stripe = 8 inches. Total length of Alyssa’s blanket = (Number of blanket with pink stripes + Number of blanket with pink stripes) × Length of each stripe = (7 + 6) × 8 = 13 × 8 = 104 inches. For 8-9, use the table at the right. Question 8. Trisha measured how far her snail moved each day for 5 days. Make a line plot of Trisha’s data. Answer: Explanation: Question 9. Higher Order Thinking Write a question that would require addition or subtraction to solve using Trisha’s data. What is the answer? Answer: Total distance Trisha’s snail moved in 5 days = 7 $$\frac{2}{8}$$ inches. Explanation: Distance Trisha’s snail moved on Monday = 1$$\frac{4}{8}$$ Distance Trisha’s snail moved on Tuesday =1$$\frac{3}{8}$$ Distance Trisha’s snail moved on Wednesday =1$$\frac{1}{8}$$ Distance Trisha’s snail moved on Thursday = 2$$\frac{1}{8}$$ Distance Trisha’s snail moved on Friday = 1$$\frac{1}{8}$$ Total distance Trisha’s snail moved in 5 days = Distance Trisha’s snail moved on Monday + Distance Trisha’s snail moved on Tuesday + Distance Trisha’s snail moved on Wednesday + Distance Trisha’s snail moved on Thursday + Distance Trisha’s snail moved on Friday = 1$$\frac{4}{8}$$ + 1$$\frac{3}{8}$$ + 1$$\frac{1}{8}$$ + 2$$\frac{1}{8}$$ + 1$$\frac{1}{8}$$ = 2 $$\frac{7}{8}$$ + 1$$\frac{1}{8}$$ + 2$$\frac{1}{8}$$ + 1$$\frac{1}{8}$$ = 4 + 2$$\frac{1}{8}$$ + 1$$\frac{1}{8}$$ = 6 $$\frac{1}{8}$$ + 1$$\frac{1}{8}$$ = 7 $$\frac{2}{8}$$ inches. Assessment Practice Question 10. Brianna is making bracelets for her friends and family members. The bracelets have the following lengths in inches: 6, 6$$\frac{3}{4}$$, 6$$\frac{1}{4}$$, 5$$\frac{3}{4}$$, 5, 6, 6$$\frac{2}{4}$$, 6$$\frac{1}{4}$$, 6, 5$$\frac{3}{4}$$ Use the data set to complete the line plot. Answer: Explanation: The bracelets lengths in inches: 6, 6$$\frac{3}{4}$$, 6$$\frac{1}{4}$$, 5$$\frac{3}{4}$$, 5, 6, 6$$\frac{2}{4}$$, 6$$\frac{1}{4}$$, 6, 5$$\frac{3}{4}$$ ### Lesson 11.3 Use Line Plots to Solve Problems Solve & Share Ms. Earl’s class measured the lengths of 10 caterpillars in the school garden. The caterpillars had the following lengths in inches: $$\frac{3}{4}$$, 1$$\frac{1}{4}$$, 1$$\frac{3}{4}$$, 1$$\frac{1}{2}$$, 1, 1, $$\frac{3}{4}$$, 1$$\frac{1}{4}$$, 1$$\frac{3}{4}$$, 1$$\frac{1}{2}$$ Plot the lengths on the line plot. Write and solve an equation to find the difference in length between the longest and shortest caterpillars. I can … use line plots to solve problems involving fractions. Look Back! How can a line plot be used to find the difference between the greatest and least values? Answer: A line plot can be used to find the difference between the greatest and least values by doing the subtraction function between the two numbers. Essential Question How Can You Use Line Plots to Solve Problems involving Fractions? Answer: We can use line plots to Solve Problems involving Fractions by dividing the fractions by fractions to find the required solution to the problem. Visual Learning Bridge Alma and Ben are filling water balloons. The line plots show the weights of their water balloons. Who filled more water balloons? How many more? How much heavier was Alma’s heaviest water balloon than Ben’s heaviest water balloon? Who filled more water balloons? How many more? Each dot in the line plots represents 1 water balloon. Alma filled 20 water balloons. Ben filled 15 water balloons. 20 – 15 = 5 Alma filled 5 more water balloons than Ben. How much heavier was Alma’s heaviest water balloon than Ben’s heaviest water balloon? The dot farthest to the right in each line plot represents the heaviest water balloon. Alma’s heaviest water balloon was 2$$\frac{2}{8}$$ pounds. Ben’s heaviest water balloon was 2$$\frac{1}{8}$$ pounds. Subtract. 2$$\frac{2}{8}$$ – 2$$\frac{1}{8}$$ = $$\frac{1}{8}$$ Alma’s heaviest water balloon was pound heavier than Ben’s heaviest water balloon. Convince Me! Make Sense and Persevere How much heavier was Alma’s heaviest water balloon than her lightest water balloon? How much heavier was Ben’s heaviest water balloon than his lightest water balloon? Write and solve equations. Answer: 1$$\frac{1}{8}$$ pounds heavier was Alma’s heaviest water balloon than her lightest water balloon. $$\frac{6}{8}$$ pounds heavier was Ben’s heaviest water balloon than her lightest water balloon. Explanation: Weight of Alma’s heaviest water balloon = 2$$\frac{2}{8}$$ pounds Weight of Alma’s lightest water balloon = 1$$\frac{1}{8}$$ pounds Difference: Weight of Alma’s heaviest water balloon – Weight of Alma’s lightest water balloon = 2$$\frac{2}{8}$$ – 1$$\frac{1}{8}$$ = 1$$\frac{1}{8}$$ pounds. Weight of Ben’s heaviest water balloon = 2$$\frac{1}{8}$$pounds Weight of Ben’s lightest water balloon = 1$$\frac{4}{8}$$ pounds Difference: Weight of Ben’s heaviest water balloon – Weight of Ben’s lightest water balloon = 2$$\frac{1}{8}$$ – 1$$\frac{4}{8}$$ = $$\frac{6}{8}$$ pounds. Another Example! Rowan’s class measured the snowfall for 5 days. The line plot shows the heights of snowfall they recorded. How many inches of snow were recorded? What amount of snowfall occurred most often? Find the total number of inches of snowfall recorded. $$\frac{1}{4}$$ + $$\frac{2}{4}$$ + $$\frac{2}{4}$$ + $$\frac{2}{4}$$ + $$\frac{3}{4}$$ = 2$$\frac{2}{4}$$ inches The amount of snowfall that occurred most often was $$\frac{2}{4}$$ inch. Guided Practice Do You Understand? Question 1. Use Structure How could you use the Commutative and Associative Properties of Addition to make the addition in the Another Example easier? Answer: We can use the Commutative and Associative Properties of Addition to make the addition in the Another Example easier by rewriting the values and doing addition because the result is same. Do You Know How? For 2-3, use the example on the previous page. Question 2. Who filled more water balloons over 2 pounds? Answer: Ben filled more water balloons over 2 pounds. Explanation: Ben’s heaviest water balloon was 2$$\frac{1}{8}$$ pounds. Question 3. How much heavier were Alma’s two heaviest water balloons than Ben’s two heaviest? Answer: $$\frac{1}{8}$$ pounds heavier were Alma’s two heaviest water balloons than Ben’s two heaviest. Explanation: Weight of Alma’s two heaviest water balloons = 2 × 2$$\frac{2}{8}$$ Weight of Ben’s two heaviest water balloon = 2 × 2$$\frac{1}{8}$$ pounds. Difference: Weight of Alma’s two heaviest water balloons – Weight of Ben’s two heaviest water balloon = 2 × 2$$\frac{2}{8}$$ – 2 × 2$$\frac{1}{8}$$ = 4 $$\frac{2}{8}$$ – 4 $$\frac{1}{8}$$ = $$\frac{1}{8}$$ pounds. Independent Practice For 4-5, use the line plot at the right. Question 4. What is the difference in height between the tallest and shortest patients? Answer: 1 $$\frac{3}{4}$$ feet is the difference in height between the tallest and shortest patients. Explanation: Height of the tallest patient = 6 feet. Height of the shortest patient = 4$$\frac{1}{4}$$ feet. Difference: Height of the tallest patient – Height of the shortest patient = 6 – 4$$\frac{1}{4}$$ = 6 – 4.25 = 1 $$\frac{3}{4}$$ feet. Question 5. Oscar says 5 feet is the most common height Dr. Chen measured. Do you agree? Explain. Answer: No, 5 feet is not the most common height Dr. Chen measured as Oscar says because 5 $$\frac{2}{4}$$ feet is most common height Dr. Chen measured. Explanation: Most common height Dr. Chen measured = 5$$\frac{2}{4}$$ feet in the given data. Problem Solving Question 6. Make Sense and Persevere Marcia measured her dolls and showed the heights using a line plot. How much taller are Marcia’s two tallest dolls combined than her two shortest dolls? Explain. Answer: 2$$\frac{3}{4}$$ inches taller are Marcia’s two tallest dolls combined than her two shortest dolls. Explanation: Marcia’s two tallest dolls heights = 7 inches and 6$$\frac{3}{4}$$ inches. Marcia’s two tallest dolls combined = 7 inches + 6$$\frac{3}{4}$$ inches. = 13 $$\frac{3}{4}$$ inches. Marcia’s two shortest dolls heights = 5$$\frac{1}{4}$$ inches and 5 $$\frac{3}{4}$$ inches. Marcia’s two shortest dolls combined = 5$$\frac{1}{4}$$ inches + 5 $$\frac{3}{4}$$ inches. = 5.25 + 5.75 = 11 inches. Difference: Marcia’s two tallest dolls combined – Marcia’s two tallest dolls combined = 13 $$\frac{3}{4}$$ inches – 11 inches = 2$$\frac{3}{4}$$ inches. Question 7. Higher Order Thinking Marlee is knitting a scarf. The line plot shows the length she knits each day. How many more inches does Marlee need to knit so the scarf is 30 inches long? Answer: 2 inches more inches Marlee needs to knit so the scarf is 30 inches long. Explanation: The length she knits each day = 2, 2$$\frac{2}{8}$$ inches, 2$$\frac{2}{8}$$ inches, 2$$\frac{4}{8}$$ inches, 2$$\frac{4}{8}$$ inches, 2$$\frac{4}{8}$$ inches, 2$$\frac{5}{8}$$ inches, 2$$\frac{6}{8}$$ inches, 2$$\frac{6}{8}$$ inches, 2$$\frac{7}{8}$$ inches, 3 inches. Total of the lengths she knits all days = 2 + 2$$\frac{2}{8}$$ + 2$$\frac{2}{8}$$ + 2$$\frac{4}{8}$$ + 2$$\frac{4}{8}$$ + 2$$\frac{4}{8}$$ + 2$$\frac{5}{8}$$ + 2$$\frac{6}{8}$$ + 2$$\frac{6}{8}$$ + 2$$\frac{7}{8}$$ + 3 inches = 28 inches. More inches Marlee needs to knit so the scarf is 30 inches long = 30 – Total of the lengths she knits all days = 30 – 28 inches = 2 inches. Assessment Practice For 8-9, use the line plot. Question 8. Which of the following statements are true? Select all that apply. ☐ Most of the players are 6 feet or taller. ☐ Five players are 6 feet tall. ☐ The combined height of two of the shortest players is 1$$\frac{1}{2}$$ feet. ☐ The difference between the tallest and the shortest players is $$\frac{3}{4}$$ foot. ☐ All of the players are taller than 5$$\frac{3}{4}$$ feet. Answer: Statements which are true: Most of the players are 6 feet or taller. All of the players are taller than 5$$\frac{3}{4}$$ feet. Explanation: Statements which are true: ☐ Most of the players are 6 feet or taller. ☐ Five players are 6 feet tall. ☐ The combined height of two of the shortest players is 1$$\frac{1}{2}$$ feet. Combined height of two of the shortest players = 5$$\frac{3}{4}$$ + 5$$\frac{3}{4}$$ = 11$$\frac{2}{4}$$. ☐ The difference between the tallest and the shortest players is $$\frac{3}{4}$$ foot. Tallest player height = 6$$\frac{3}{4}$$ feet. Shortest player height = 5$$\frac{3}{4}$$ feet. Difference: Tallest player height – Shortest player height = 6$$\frac{3}{4}$$ – 5$$\frac{3}{4}$$ = 1 feet. ☐ All of the players are taller than 5$$\frac{3}{4}$$ feet. Question 9. If one of the shortest players grew $$\frac{3}{4}$$ foot before the next season started, how tall would the player be? A. $$\frac{6}{4}$$ feet B. 5$$\frac{3}{4}$$ feet C. 6 feet D. 6$$\frac{2}{4}$$ feet Answer: 6$$\frac{2}{4}$$ feet taller the player would be. Explanation: Shortest player height = 5$$\frac{3}{4}$$ feet. If one of the shortest players grew $$\frac{3}{4}$$ foot. => 5$$\frac{3}{4}$$ + $$\frac{3}{4}$$ => 6$$\frac{2}{4}$$ feet. ### Lesson 11.4 Problem Solving Critique Reasoning Solve & Share A class made a line plot showing the amount of snowfall for 10 days. Nathan analyzed the line plot and said, “The difference between the greatest amount of snowfall recorded and the least amount of snowfall recorded is 3 because the first measurement has one dot and the last measurement has 4 dots.” How do you respond to Nathan’s reasoning? I can … use what I know about line plots to critique the reasoning of others. Thinking Habits Be a good thinker! These questions can help you. • What questions can ask to understand other people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Look Back! Critique Reasoning Millie said that the total amount of snowfall for the 5 days above was 10 inches. Is Millie correct? Answer: No, Millie is incorrect because Total amount of snowfall for the 5 days is 5 $$\frac{2}{4}$$ inches. not 10 inches. Explanation: Amount of snowfall for the 5 days = $$\frac{3}{4}$$, 1, 1, 1$$\frac{1}{4}$$, 1$$\frac{2}{4}$$ Total amount of snowfall for the 5 days = $$\frac{3}{4}$$ + 1 + 1 + 1$$\frac{1}{4}$$ + 1$$\frac{2}{4}$$ = 5 $$\frac{2}{4}$$ inches. Essential Question How Can You Critique the Reasoning of Others? Answer: We can Critique the Reasoning of Others by checking the process how the problem is solved and finally tallying their solution. Visual Learning Bridge The line plots show the amount of rainfall for two months. Val said, “The total rainfall for February was greater than the total rainfall for January because $$\frac{7}{8}$$ + $$\frac{7}{8}$$ equals $$\frac{14}{8}$$, and the highest rainfall in January was $$\frac{5}{8}$$“. What is Val’s reasoning? Val compared the two highest amounts of rainfall for each month. How can I critique the reasoning of others? I can • ask questions for clarification. • decide if the strategy used makes sense. • look for flaws in estimates or calculations. Here’s my thinking. Val’s reasoning is not correct. She compared the days with the greatest amount of rainfall for the two months. The days with the greatest amounts of rainfall are not the total for the months. Val should have added the amounts for each month. Then she could compare the amounts. January $$\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{3}{8}+\frac{3}{8}+\frac{3}{8}+\frac{5}{8}=\frac{17}{8}$$ inches February: $$\frac{1}{8}+\frac{1}{8}+\frac{5}{8}+\frac{7}{8}+\frac{7}{8}=\frac{21}{8}$$ inches During February, there was $$\frac{21}{8}-\frac{17}{8}=\frac{4}{8}$$ inch more rain than January Convince Me! Critique Reasoning Bev thought January had more rainfall because it rained on 7 days and February only had rain on 5 days. How do you respond to Bev’s reasoning? Answer: Well, Bev’s reasoning is not correct because to find the more rainfall in between January and February months, you should calculate the rainfall not the number of days. Guided Practice Critique Reasoning At a dog show, a judge wrote down the heights of 12 dogs. Cole made a line plot of the heights, shown to the right. He concluded, “The height with the most dots is 1$$\frac{1}{4}$$ feet, so that is the greatest height of the dogs at the dog show.” Question 1. What is Cole’s conclusion? How did he reach this conclusion? Answer: Cole concluded, “The height with the most dots is 1$$\frac{1}{4}$$ feet, so that is the greatest height of the dogs at the dog show.” He reached this conclusion by marking most dots on 1$$\frac{1}{4}$$ feet in the line plot chart. Question 2. Is Cole’s conclusion correct? Explain. Answer: No, Cole’s conclusion incorrect because what he marked as the highest height of dog is the most common height of dogs and highest height of dog is 3 feet not 1$$\frac{1}{4}$$ feet. Independent Practice Critique Reasoning Natasha keeps a log of the total amount of time her students practiced on their violins outside of their weekly lesson. She creates the line plot shown. Each dot represents one student who practices a specific amount of time in one week. Natasha says that 5 of her students’ practice times combined is 1$$\frac{1}{4}$$ hours because there are 5 dots above 1$$\frac{1}{4}$$. Question 3. What is Natasha’s argument? How does she support it? Answer: Natasha’s argument is that 5 of her students’ practice times combined is 1$$\frac{1}{4}$$ hours. She supports it because there are 5 dots above 1$$\frac{1}{4}$$. Question 4. Critique Natasha’s reasoning. Answer: Natasha’s reasoning says that 5 of her students’ practice times combined is 1$$\frac{1}{4}$$ hours because there are 5 dots above 1$$\frac{1}{4}$$. She thinks its correct but she is saying about more common hours the students who practiced a specific amount of time in one week. Problem Solving Performance Task Taking Inventory Mr. Pally is building a desk using screws of different lengths. The instructions show how many screws of each length he will need to use. Mr. Pally concludes he will use more of the shortest screws than the longest screws. Question 5. Model with Math Draw a line plot to show the screw lengths Mr. Pally will use to build the desk. Answer: Explanation: Question 6. Reasoning How can you use the line plot to find which length of screw Mr. Pally will need the most? Answer: We use the line plot to find which length of screw Mr. Pally will needs the most by counting the dots potted on the line, which occurred many times. Question 7. Critique Reasoning is Mr. Pally’s conclusion correct? How did you decide? If not, what can you do to improve his reasoning? Answer: Yes, Mr. Pally conclusion he will use more of the shortest screws than the longest screws is correct because in the line plotted shows many dots on shorter screws than the longest screws. Explanation: Mr. Pally concludes he will use more of the shortest screws than the longest screws. ### Topic 11 Fluency Practice Activity Find a Match Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match. Find a match for every clue. I can …add and subtract multi-digit whole numbers. Clues A. The sum is between 3,510 and 3,520. B. The difference is exactly 3,515. C. The sum is between 3,560 and 3,570. D. The difference is between 3,530 and 3,540. E. The sum is exactly 3,584. F. The difference is between 3,590 and 3,600. G. The sum is exactly 3,987. H. The difference is between 1,000 and 2,000. Answer: Explanation: Clues A. The sum is between 3,510 and 3,520. B. The difference is exactly 3,515. C. The sum is between 3,560 and 3,570. D. The difference is between 3,530 and 3,540. E. The sum is exactly 3,584. F. The difference is between 3,590 and 3,600. G. The sum is exactly 3,987. H. The difference is between 1,000 and 2,000. ### Topic 11 Vocabulary Review Understand Vocabulary Word List • data set • graph • line plot • number line • scale • table Write T for true and F for false. Question 1. ________ Graphs are used to display and represent data. Answer: __True______ Graphs are used to display and represent data. Question 2. _________ A data set is a collection of pieces of information. Answer: ____True_____ A data set is a collection of pieces of information. Question 3. __________ A table is never used to display data. Answer: ___False_______ A table is never used to display data. Question 4. __________ A line plot shows data along a number line. Answer: ___True_______ A line plot shows data along a number line. Question 5. _________ A line plot may have more points than there are numbers in the data set. Answer: ___False______ A line plot may have more points than there are numbers in the data set. Write always, sometimes, or never. Question 6. A line plot _________ displays data. Answer: A line plot ___always______ displays data. Question 7. The scale on a line plot is ________ numbered using fractions. Answer: The scale on a line plot is ___sometimes_____ numbered using fractions. Question 8. A number line is __________ numbered out of order. Answer: A number line is ___never_______ numbered out of order. Use Vocabulary in Writing Question 9. Use at least 3 terms from the Word List to describe another way Patrick can display his data. Answer: Graph, Line plot and Number line are the 3 terms from the Word List to describe another way Patrick can display his data. Explanation: Word List given: • data set • graph • line plot • number line • scale • table ### Topic 11 Reteaching Set A pages 417-420 The line plot shows the number of hours Mrs. Mack was at the gym each day, during a two week period. Remember each dot above the line plot represents one value in the data set. Question 1. How many days did Mrs. Mack go to the gym? Answer: Mrs. Mack went to the gym for 11 days. Question 2. What is the least amount of time Mrs. Mack spent at the gym? Answer: 1$$\frac{1}{4}$$ hours is the least amount of time Mrs. Mack spent at the gym. Question 3. How many hours was Mrs. Mack at the the gym during the two weeks? Answer: 25$$\frac{3}{4}$$ hours Mrs. Mack was at the the gym during the two weeks. Explanation: Number of hours Mrs. Mack was at the the gym during the two weeks: 1, 1$$\frac{1}{4}$$ , 1$$\frac{2}{4}$$ ,2 , 2 , 2$$\frac{1}{4}$$ , 2 $$\frac{1}{4}$$ , 2$$\frac{1}{4}$$ ,2 $$\frac{2}{4}$$ , 2$$\frac{3}{4}$$ ,3 , 3. Total number of hours Mrs. Mack was at the the gym during the two weeks = 1 + 1$$\frac{1}{4}$$ + 1$$\frac{2}{4}$$ + 2 + 2 + 2$$\frac{1}{4}$$ + 2 $$\frac{1}{4}$$ + 2$$\frac{1}{4}$$ + 2 $$\frac{2}{4}$$ + 2$$\frac{3}{4}$$ + 3 + 3 = 25$$\frac{3}{4}$$ hours. Set B pages 421-424 Lilly measured the lengths of the ribbons in her craft kit and drew a line plot. The number line shows the lengths from least to greatest. The labels show what the dots represent. Remember to choose a reasonable scale for your number line. A zoo in Australia studied platypuses. Their masses are recorded below. Question 1. Draw a line plot for the data set. Answer: Explanation: Question 2. What is the difference in mass of the platypus with the greatest mass and the platypus with the least mass? Answer: 1$$\frac{1}{8}$$ kg is the difference in mass of the platypus with the greatest mass and the platypus with the least mass. Explanation: The greatest mass of the platypus = 2$$\frac{6}{8}$$ kg. The least mass of the platypus = 1$$\frac{5}{8}$$ kg. Difference: The greatest mass of the platypus – The least mass of the platypus = 2$$\frac{6}{8}$$ – 1$$\frac{5}{8}$$ = 1$$\frac{1}{8}$$ kg. Set C pages 425-428 Carly and Freddie pick up trash. The line plots show how much they picked up each day for 14 days. What is the difference between the greatest and least amounts Carly picked up? The greatest amount of trash Carly picked up was 3 pounds. The least amount was $$\frac{1}{2}$$ pound. Subtract. 3 – $$\frac{1}{2}$$ = 2$$\frac{1}{2}$$ pounds Remember you can use equations to help solve problems with data from line plots. For 1-3, use the line plots at the left. Question 1. Explain how to find the total weight of the trash Freddie picked up. Answer: We can find the total weight of the trash Freddie picked up in 14 days by adding the every day trash he collected. Explanation: Question 2. Write and solve an equation to find , the difference between the greatest amount Freddie collected and the least amount he collected. Answer: 2 pounds is the difference between the greatest amount Freddie collected and the least amount he collected. Explanation: The greatest amount Freddie collected = 2$$\frac{1}{2}$$ pounds. The least amount he collected = $$\frac{1}{2}$$ pounds. Difference: The greatest amount Freddie collected – The least amount he collected = 2$$\frac{1}{2}$$ pounds – $$\frac{1}{2}$$ pounds = 2 pounds. Question 3. What is the sum of Carly’s most frequent weight and Freddie’s most frequent weight? Explain. Answer: 5 pounds is the sum of Carly’s most frequent weight and Freddie’s most frequent weight. Explanation: Carly’s most frequent weight = 2$$\frac{1}{2}$$ pounds. Freddie’s most frequent weight = 1$$\frac{1}{2}$$ pounds. Sum: Carly’s most frequent weight + Freddie’s most frequent weight = 2$$\frac{1}{2}$$ pounds + 1$$\frac{1}{2}$$ pounds = 5 pounds. Set D pages 429-432 Think about these questions to help you critique the reasoning of others. Thinking Habits! • What questions can ask to understand other people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Remember you can use math to identify mistakes in people’s thinking. Question 1. Spencer says 2$$\frac{3}{8}$$ miles is the most common delivery distance. Do you agree? Explain. Answer: No, I disagree with what Spencer says 2$$\frac{3}{8}$$ miles is the most common delivery distance because the most common delivery distance is $$\frac{6}{8}[/Latex] miles. ### Topic 11 Assessment Practice Question 1. What is the difference between the heaviest and lightest weights? Answer: 1 pounds is the difference between the heaviest and lightest weights. Explanation: The heaviest weight = 3[latex]\frac{2}{4}$$ pounds. The lightest weight = 2$$\frac{2}{4}$$ pounds. Difference: The heaviest weight – The lightest weight = 3$$\frac{2}{4}$$ – 2$$\frac{2}{4}$$ = 1 pounds. Question 2. How many dots would be placed above 1$$\frac{3}{4}$$ in a line plot of these data? A. 3 dots B. 2 dots C. 1 dot D. 0 dots Answer: 3 dots dots would be placed above 1$$\frac{3}{4}$$ in a line plot of these data. A. 3 dots. Explanation: Dots would be placed above 1$$\frac{3}{4}$$ in a line plot of these data: 3 dots. Question 3. Which is the most common length of snail Fred has in his backyard? Answer: 3 inches is the most common length of snail Fred has in his backyard. Explanation: 3 inches is the most common length of snail Fred has in his backyard. Question 4. During a sleep study, the number of hours 15 people slept was recorded in the table below. A. Use the data in the table to draw a line plot. Answer: Explanation: B. How many more hours did the person who slept the greatest number of hours sleep than the person who slept the least number of hours? Explain. Answer: 3$$\frac{1}{2}$$ more hours the person who slept the greatest number of hours sleep than the person who slept the least number of hours. Explanation: The person who slept the greatest number of hours = 9 . The person who slept the least number of hours = 5$$\frac{1}{2}$$. Difference: The person who slept the greatest number of hours – The person who slept the least number of hours = 9 – 5$$\frac{1}{2}$$ = 3$$\frac{1}{2}$$. Question 5. Use the line plot below. Select all the true statements. ☐ The greatest height is 2$$\frac{1}{2}$$ inches. ☐ More plants have a height of 2 inches than 1$$\frac{1}{2}$$ inches. ☐ There are 3 plants with a height of 1 inch. ☐ There are 3 plants with a height of 2 inches and 3 plants with a height of 2$$\frac{1}{2}$$ inches. ☐ The tallest plant is 1$$\frac{1}{2}$$ inches taller than the shortest plant. Answer: All the true statements: The greatest height is 2$$\frac{1}{2}$$ inches. There are 3 plants with a height of 2 inches and 3 plants with a height of 2$$\frac{1}{2}$$ inches. The tallest plant is 1$$\frac{1}{2}$$ inches taller than the shortest plant. Explanation: Select all the true statements: ☐ The greatest height is 2$$\frac{1}{2}$$ inches. (True) ☐ More plants have a height of 2 inches than 1$$\frac{1}{2}$$ inches. (False) ☐ There are 3 plants with a height of 1 inch. (False) No, There are 2 plants with a height of 1 inch. ☐ There are 3 plants with a height of 2 inches and 3 plants with a height of 2$$\frac{1}{2}$$ inches. (True) ☐ The tallest plant is 1$$\frac{1}{2}$$ inches taller than the shortest plant. (True) => Tallest plant height = 2$$\frac{1}{2}$$ inches. Shortest plant height = 1inches. Difference: Tallest plant height – Shortest plant height = 2$$\frac{1}{2}$$ – 1 = 1$$\frac{1}{2}$$. Question 6. Mr. Tricorn’s class measured the lengths of crayons. How many crayons did they measure? Use the line plot. Answer: Total number of crayons they measured = 10. Explanation: Number of crayons of length 1$$\frac{1}{2}$$ inches = 1. Number of crayons of length 2$$\frac{1}{2}$$ inches = 4. Number of crayons of length 3$$\frac{1}{2}$$ inches = 2. Number of crayons of length 4$$\frac{1}{2}$$ inches = 3. Total number of crayons measured = Number of crayons of length 1$$\frac{1}{2}$$ inches + Number of crayons of length 2$$\frac{1}{2}$$ inches + Number of crayons of length 3$$\frac{1}{2}$$ inches + Number of crayons of length 4$$\frac{1}{2}$$ inches = 1 + 4 + 2 + 3 = 10. Question 7. Use the line plot from Exercise 6. How many crayons were greater than 3 inches long? A. 9 B. 5 C. 6 D. 3 Answer: Number of crayons greater than 3 inches long = 5. B. 5. Explanation: Number of crayons of length 1$$\frac{1}{2}$$ inches = 1. Number of crayons of length 2$$\frac{1}{2}$$ inches = 4. Number of crayons of length 3$$\frac{1}{2}$$ inches = 2. Number of crayons of length 4$$\frac{1}{2}$$ inches = 3. Number of crayons greater than 3 inches long = Number of crayons of length 3$$\frac{1}{2}$$ inches + Number of crayons of length 4$$\frac{1}{2}$$ inches = 2 + 3 = 5. Question 8. Ms. Garcia measured the heights of her students. A. Use the data in the table to draw a line plot. Answer: Explanation: B. Use the data in Exercise 8. Select all of the statements that are true. ☐ The tallest student is 4 feet tall. ☐ The tallest student is 4$$\frac{2}{4}$$ feet tall. ☐ The shortest student is 3$$\frac{3}{4}$$ feet tall. ☐ The tallest student is 1 foot taller than the shortest student. ☐ The most common height of the students was 4 feet tall. Answer: All of the statements that are true: The tallest student is 4$$\frac{2}{4}$$ feet tall. The tallest student is 1 foot taller than the shortest student. The most common height of the students was 4 feet tall. Explanation: Select all of the statements that are true. ☐ The tallest student is 4 feet tall. (False) ☐ The tallest student is 4$$\frac{2}{4}$$ feet tall. (True) ☐ The shortest student is 3$$\frac{3}{4}$$ feet tall. (False) ☐ The tallest student is 1 foot taller than the shortest student. (True) => Tallest student = 4$$\frac{2}{4}$$ feet. Shortest student = 3$$\frac{2}{4}$$ feet. Difference: Tallest student – Shortest student = 4$$\frac{2}{4}$$ feet – 3$$\frac{2}{4}$$ feet = 1 feet. ☐ The most common height of the students was 4 feet tall. (True) ### Topic 11 Performance Task Measuring Pumpkins Mr. Chan’s class picked small pumpkins from the pumpkin patch and then weighed their pumpkins. Question 1. The class made the Pumpkin Weights line plot of the data. Part A What is the most common weight of the pumpkins? Answer: The most common weight of the pumpkins = 4$$\frac{1}{4}$$ pounds. Part B Write and solve an equation to find, how much more the heaviest pumpkin weighs than the lightest pumpkin. Answer: 1$$\frac{3}{4}$$ pounds more the heaviest pumpkin weighs than the lightest pumpkin. Explanation: The heaviest pumpkin weight = 5pounds. The lightest pumpkin weight = 3$$\frac{1}{4}$$ pounds. Difference: The heaviest pumpkin weight – The lightest pumpkin weight = 5 – 3$$\frac{1}{4}$$ = 1$$\frac{3}{4}$$ pounds. Part C Ayana said 3 pumpkins weigh 4$$\frac{2}{4}$$ pounds. Critique Ayana’s reasoning. Is she correct? Answer: Yes, she is correct because there are three pumpkins weigh 4$$\frac{2}{4}$$ pounds. Question 2. The class also measures the distance around their pumpkins to the nearest half-inch. They recorded their data in the Pumpkin Size list. Pumpkin Size: 19$$\frac{1}{2}$$, 20$$\frac{1}{2}$$, 19$$\frac{1}{2}$$, 20, 20$$\frac{1}{2}$$, 21$$\frac{1}{2}$$, 20, 21, 22, 19$$\frac{1}{2}$$, 20$$\frac{1}{2}$$, 21$$\frac{1}{2}$$, 21, 21, 21$$\frac{1}{2}$$, 20$$\frac{1}{2}$$ Part A Draw a line plot of Pumpkin Size data. Answer: Explanation: Pumpkin Size: 19$$\frac{1}{2}$$, 20$$\frac{1}{2}$$, 19$$\frac{1}{2}$$, 20, 20$$\frac{1}{2}$$, 21$$\frac{1}{2}$$, 20, 21, 22, 19$$\frac{1}{2}$$, 20$$\frac{1}{2}$$, 21$$\frac{1}{2}$$, 21, 21, 21$$\frac{1}{2}$$, 20$$\frac{1}{2}$$ Part B Drew says 1 more pumpkin was 20$$\frac{1}{2}$$ inches around than was 19$$\frac{1}{2}$$ inches because 20$$\frac{1}{2}$$ – 19$$\frac{1}{2}$$ = 1. Critique Drew’s reasoning. Answer: No, he is incorrect because pumpkin size of 20$$\frac{1}{2}$$ – pumpkin size of 19$$\frac{1}{2}$$ = 1 is the difference between two different sized pumpkins but not 1 more pumpkins added to the count. Explanation: 20$$\frac{1}{2}$$ – 19$$\frac{1}{2}$$ = 1. Difference between different pumpkins. Part C What is the difference between the longest distance and the shortest distance ? Write and solve an equation. Answer: 3 inches is the difference between the longest distance and the shortest distance Explanation: The longest pumpkin sized = 22 inches. The shortest pumpkin sized = 19 inches. Difference: The longest pumpkin sized – The shortest pumpkin sized = 22 – 19 = 3 inches. #### enVision Math Common Core Grade 4 Answer Key ## Envision Math Common Core Grade 4 Answer Key Topic 12 Understand and Compare Decimals ## Envision Math Common Core 4th Grade Answers Key Topic 12 Understand and Compare Decimals Essential Questions: How can you write a fraction as a decimal? How can you locate points on a number line? How do you compare decimals? enVision STEM Project: Energy and Decimals Do Research Use the Internet or other sources to research other sports or games where players transfer energy to cause collisions in order to score points and win. Journal: Write a Report Include what you found. Also in your report: • Explain how the transfer of energy helps the player or team score. • A game of curling is broken into ten rounds called ends. Suppose a team wins 6 of the 10 ends. Write a fraction with a denominator of 10 and an equivalent fraction with a denominator of 100. Then, write an equivalent decimal that represents the same value. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • hundredth • tens • place value • tenth Question 1. A ________ is one of 10 equal parts of a whole, written as $$\frac{1}{10}$$. Answer: A tenth is one of 10 equal parts of a whole, written as $$\frac{1}{10}$$. Question 2. __________ is the position of a digit in a number that is used to determine the value of the digit. Answer: tens is the position of a digit in a number that is used to determine the value of the digit. Question 3. A __________ is one of 100 equal parts of a whole, written as $$\frac{1}{100}$$. Answer: A hundredth is one of 100 equal parts of a whole, written as $$\frac{1}{100}$$. Comparing Fractions Write >, <, or = in the . Question 4. $$\frac{5}{100}$$ $$\frac{5}{10}$$ Answer: $$\frac{5}{100}$$ < $$\frac{5}{10}$$ Question 5. $$\frac{1}{10}$$ $$\frac{1}{100}$$ Answer: $$\frac{1}{10}$$ > $$\frac{1}{100}$$ Question 6. $$\frac{2}{10}$$ $$\frac{20}{100}$$ Answer: $$\frac{2}{10}$$ = $$\frac{20}{100}$$ Parts of a Whole Complete each fraction to represent the shaded part of the whole. Question 7. Answer: There are 10 blocks of which 3 are shaded. So, the fraction of the shaded part of the whole is $$\frac{3}{10}$$ Question 8. Answer: There are 10 blocks of which 0 are shaded. So, the fraction of the shaded part of the whole is 0. Question 9. Answer: There are 10 blocks of which 7 are shaded. So, the fraction of the shaded part of the whole is $$\frac{7}{10}$$ Shade the part of the whole that represents the fraction. Question 10. Answer: Question 11. Answer: Question 12. Answer: Problem Solving Question 13. Reasoning Rob walked $$\frac{2}{10}$$ block. Drew walked $$\frac{5}{10}$$ block. Write a comparison for the distance Rob and Drew each walked. Answer: Given, Rob walked $$\frac{2}{10}$$ block. Drew walked $$\frac{5}{10}$$ block. Drew walked more than Rob. Pick a Project PROJECT 12A How much will it cost to visit a national park? Project: Write a Travel Journal PROJECT 12B How do you know who won the event? Project: Compare Olympic Racing Times PROJECT 12C Would you like to win an award for a presentation? Project: Make a Presentation about Adding Fractions PROJECT 12D How did railroads help build Florida? Project: Build a Miniature Railroad ### Lesson 12.1 Fractions and Decimals Solve & Share According to a survey, 7 out of 10 pet owners have a dog. Represent this in a drawing. I can … relate fractions and decimals. Look Back! How many pet owners do NOT have a dog? Write your answer as a fraction. Essential Question How Can You Write a Fraction as a Decimal? Visual Learning Bridge On Kelsey Street, 6 out of 10 houses have swing sets. Write $$\frac{6}{10}$$ as a decimal. A decimal is another representation for a fraction and also names parts of wholes. A decimal is a number with one or more digits to the right of the decimal point. Fractions with denominators of 10 and 100 may be written as decimals. Sixth tenths or $$\frac{6}{10}$$ of the houses have swing sets. You can write $$\frac{6}{10}$$ as a decimal by putting a 6 in tenths place. The tenths place is to the right of the decimal point. $$\frac{6}{10}$$ and $$\frac{60}{100}$$ are equivalent. You can write $$\frac{60}{100}$$ as a decimal by using tenths and hundredths places The hundredths place is to the right of tenths place. So, 0.6 or 0.60 of the houses have swing sets. Convince Me! Reasoning in the Kelsey Street neighborhood, 75 out of 100 houses are two-story homes. Write $$\frac{75}{100}$$ as a decimal. Shade the grid to show the equivalent fraction and decimal. Another Example! You can use grids to show how money relates to fractions and decimals. Guided Practice Do You Understand? Question 1. How can you use grids to represent$4.71?

Do You Know How?
Question 2.
Write a decimal and a fraction for the part of the grid that is shaded.

So, the decimal of the shaded part is 0.5
The fraction of the shaded part is $$\frac{5}{10}$$

Independent Practice

For 3-6, write a decimal and fraction for each diagram.
Question 3.

So, the decimal of the shaded part is 0.04
The fraction of the shaded part is $$\frac{4}{100}$$

Question 4.

So, the decimal of the shaded part is 0.77
The fraction of the shaded part is $$\frac{154}{200}$$

Question 5.

So, the decimal of the shaded part is 0.39
The fraction of the shaded part is $$\frac{39}{100}$$

Question 6.

Problem Solving

Question 7.
The arena of the Colosseum in Rome was about $$\frac{15}{100}$$ of the entire Colosseum. Write this amount as a decimal.

Given,
The arena of the Colosseum in Rome was about $$\frac{15}{100}$$ of the entire Colosseum.
The fraction $$\frac{15}{100}$$ can be written as 0.15.

Question 8.
What fraction of the Colosseum was NOT the arena? Write and solve an equation.
$$\frac{100}{100}$$ – $$\frac{15}{100}$$ = $$\frac{85}{100}$$

Question 9.
Vocabulary Write the vocabulary word that best completes the sentence:
Jelena says, “One dime is one ________ of a dollar.”
Jelena says, “One dime is one tenth of a dollar.”

Question 10.
Number Sense About how much of the rectangle is shaded green? Write this amount as a fraction and as a decimal.

Answer: $$\frac{1}{10}$$ or 0.1

Question 11.
Critique Reasoning Cher adds up the money in her piggy bank. She has a one-dollar bill and 3 dimes. Did Cher write the amount of money correctly? If not, what mistake did Cher make? $1.3 Answer: Given, Cher adds up the money in her piggy bank. She has a one-dollar bill and 3 dimes. The amount of money is always written in hundredths. Cher should have written$1.30

Question 12.
Higher Order Thinking The diagram models the plants in a vegetable garden. Write a fraction and a decimal for each vegetable in the garden.

Radishes: $$\frac{23}{100}$$, 0.23
Carrots: $$\frac{35}{100}$$, 0.35
Corn: $$\frac{15}{100}$$, 0.15
Lettuce: $$\frac{27}{100}$$, 0.27

Assessment Practice

Question 13.
Which decimal represents $$\frac{5}{100}$$?
A. 0.05
B. 0.5
C. 0.50
D. 0.95
Given the fraction $$\frac{5}{100}$$
It can be written in the decimal form as 0.05.
Thus the correct answer is option A.

Question 14.
Which fraction and decimal represent twenty-nine hundredths?
A. 0.29 and $$\frac{29}{10}$$
B. 0.29 and $$\frac{100}{29}$$
C. 2.9 and $$\frac{29}{100}$$
D. 0.29 and $$\frac{29}{100}$$
twenty-nine hundredths mean 29 of hundred.
The decimal form will be 0.29
The fraction form is $$\frac{29}{100}$$.
Thus the correct answer is option D.

### Lesson 12.2 Fractions and Decimals on the Number Line

Solve & Share
Name the fractions and/or decimals of each lettered point on the number lines. Tell how you decided.
I can … locate and describe fractions and decimals on number lines.

Look Back! Is the name for point B above different from the name for point B on the number line below? Explain.

Essential Question
How Can You Locate Points on a Number Line?

Visual Learning Bridge
In long-track speed skating, each lap is $$\frac{4}{10}$$ kilometer. During practice, Elizabeth skated 3.75 kilometers. Draw a number line to show $$\frac{4}{10}$$ and 3.75.
You can use a number line to locate and describe fractions and decimals.

Locate $$\frac{4}{10}$$ on a number line.
Draw a number line and divide the distance from 0 to 1 into 10 equal parts to show tenths.
The distance from 0 to 0.4 is four tenths the distance from 0 to 1.
Draw a point at $$\frac{4}{10}$$

Locate 3.75 on a number line.
You can show 3.75 on a number line divided into tenths by plotting a point halfway between 3.7 and 3.8.

You can use a second number line to show the interval between 3.7 and 3.8. The points on both number lines are at 3.75.

Convince Me! Be Precise Which decimal shown on the number line is not placed in the correct location? Explain.

Another Example!
Fractions and decimals can name the same points on a number line.

Mixed numbers and decimals can name the the same points on a number line.

Guided Practice

Do You Understand?
Question 1.
Locate$$\frac{45}{100}$$ on the number line.

Question 2.
Draw a number line to represent both the decimal and fraction for eight tenths.

Do You Know How?
For 3-6, name the decimal and fraction for each point on the number line.

Question 3.
E
The decimal at E point is 0.6.
The fraction at E point is 6/100.

Question 4.
H.
The decimal at H point is 1.4.
The fraction at H point is 7/5.

Question 5.
F
The decimal at F point is 1.33
The fraction at F point is 133/100.

Question 6.
G
The decimal at G point is 1.39
The fraction at G point is 139/100

Independent Practice

For 7-8, label the number lines with the given fractions and decimals.
Question 7.
Represent the decimals and fractions from 3.08 to 3.13.

Question 8.
Represent the fractions and decimals from $$\frac{4}{10}$$ to 1.

For 9-16, name the decimal and fraction for each point on the number line.

Question 9.
J
The decimal at the point J is 4.66
The fraction at the point J is 233/50

Question 10.
K
The decimal at the point K is 4.62
The fraction at the point K is 231/50.

Question 11.
L
The decimal at the point L is 4.8
The fraction at the point L is 48/100.

Question 12.
M
The decimal at the point M is 4.53
The fraction at the point M is 453/100.

Question 13.
N
The decimal at the point N is 4.69
The fraction at the point N is 469/100.

Question 14.
O
The decimal at the point 0 is 4.76
The fraction at the point 0 is 119/25.

Question 15.
P
The decimal at the point P is 4.6
The fraction at the point P is 23/5.

Question 16.
Q
The decimal at the point Q is 3.59
The fraction at the point Q is 359/100

Problem Solving

Question 17.
Write the five missing decimals on the number line.

The five missing decimals on the number line are 0.4, 0.6, 0.8, 1.2, and 1.4.
The difference of the numbers in the number line is 0.2

Question 18.
Write the five missing fractions on the number line.

The five missing fractions on the number line are 41/100, 43/100, 46/100, 47/100, and 50/100.
The difference of the fractions on the number line is 1/100.

Question 19.
Draw a number line to show 60 cents. Use the number line to write 60 cents as a fraction and as a decimal.

Question 20.
Make Sense and Persevere Neil is learning about unusual units of volume. There are 2 pecks in 1 kenning. There are 2 kennings in 1 bushel. There are 8 bushels in 1 quarter. There are 5 quarters in 1 load. Write a number sentence to show how many pecks are in 1 load.
Given that,
2 pecks in 1 kenning.
2 kinnings in 1 bushel.
8 bushels in 1 quarter.
Therefore 2 pecks in 1 load.

Question 21.
Draw a number line and plot a point at each number shown.
2$$\frac{71}{100}$$ 2.6 2$$\frac{82}{100}$$

Question 22.
Higher Order Thinking Use a number line to name two numbers that are the same distance apart as 3.2 and 3.8.

Assessment Practice

Question 23.
What decimals or fractions do the points on the number lines show? Choose the decimals and fractions from the box to label the number lines.

Question 24.
What decimals or fractions do the points on the number lines show? Choose the decimals and fractions from the box to label the number lines.

### Lesson 12.3 Compare Decimals

Solve & Share
A penny made in 1982 weighs about 0.11 ounce. A penny made in 2013 weighs about 0.09 ounce. Which penny weighs more? Solve this problem any way you choose.
I can … compare decimals by reasoning about their size.

Look Back! Construct Arguments Simon and Danielle are eating oranges. Danielle says, “Because we each have 0.75 of an orange left, we have the same amount left to eat.” Do you agree with Danielle? Explain.

Essential Question
How Do You Compare Decimals?

Visual Learning Bridge
Donovan ran the 100-meter race in 10.11 seconds. Sal ran the same race in 10.09 seconds. Who had the faster time?
There is more than one way to compare decimals.

One Way
Use hundredths grids.
The whole numbers are the same. Compare the digits in the tenths place.

Another Way
Use place value.
The whole number parts are the same.
The decimal parts are both to the hundredths.
11 hundredths is greater than 9 hundredths.
10.11 >10.09

Another Way
Start at the left.
Compare each place value. Look for the first place where the digits are different.
10.11 10.09
1 tenth > 0 tenths
10.11 > 10.09

Convince Me! Reasoning Write four different digits in the blank spaces to make each comparison true. Explain your reasoning.
0. ______ 8 < 0.______ 0.5 _____ > 0.______ 9

Another Example!
You can also use place-value blocks or number lines to compare.
Grids, place-value blocks, and number lines are all appropriate tools to use for comparing decimals. When using place-value blocks, let
the flat equal one whole.

Guided Practice

Do You Understand?
Question 1.
Cy says, “0.20 is greater than 0.2 because 20 is greater than 2.” Do you agree? Explain.
Answer: No 0.20 and 0.2 are equal in the decimal. Because 0 after 2 will remain the same and 0 before 2 will make a lot of change as it becomes hundredths.

Do You Know How?
For 2-5, write >, <, or = in each . Use an appropriate tool as needed to compare.
Question 2.
0.70 0.57
0.70 > 0.57

Question 3.
0.41 0.14
0.41 > 0.14

Question 4.
6.28 7.31
6.28 < 7.31

Question 5.
1.1 1.10
1.1 = 1.10

Independent Practice

Leveled Practice For 6-14, write >,<, or = in each . Use an appropriate tool as needed to compare.
Question 6.

0.17 < 0.2

Question 7.

0.31 > 0.29

Question 8.

0.44 > 0.22

Question 9.
0.1 0.1 0
0.1 = 0.10

Question 10.
$2.98$2.56
$2.98 >$2.56

Question 11.
7.01 7.1
7.01 > 7.1

Question 12.
0.08 0.7
0.08 < 0.7

Question 13.
3.40 3.4
3.40 = 3.4

Question 14.
$21.50$20.99
$21.50 >$20.99

For 15-20, write a decimal to make each comparison true.
Question 15.
______ < 0.23
0.20 < 0.23

Question 16.
8.60 = _______
8.60 = 8.60

Question 17.
______ > 4.42
3.43 > 4.42

Question 18.
13.2 > ______
13.2 > 12.4

Question 19.
5.2 < ______
5.2 < 4.3

Question 20.
6.2 = ______
6.2 = 6.2

Problem Solving

Question 21.
Use Appropriate Tools Maria timed how long it took her Venus Fly Trap to close. The first time it took 0.43 second to close. The second time took 0.6 second to close. Which was the faster time? Draw place-value blocks to show your comparison.
Given that,
Maria took a time to her venus fly trap to close for first time = 0.43
Maria took a time to her venus fly trap to close for second time = 0.6
Second time was the faster time.

Question 22.
Fishing lures have different weights. Which lure weighs more?

Given that,
The weight of the yellow minnow fishing lures = 0.63
The weight of the green minnow fishing lures = 0.5
Yellow minnow fishing lures is more than green minnow fishing lures.
So, 0.63 > 0.5

Question 23.
Number Sense Ellen wants to give 100 toys to each of 9 charities. In one week, she collects 387 toys. The next week, she collects 515 toys. Has Ellen reached her goal? Use an estimate to explain.
Given that,
Ellen wants to give 100 toys to each of 9 charities.
Total number of toys she collected in one week = 387
Total number of toys she collected in next week = 515
Total number of toys she collected = 387 + 515 = 902
100 toys to each of 9 charities = 9 × 100 = 900
So, she reached her goal.

Question 24.
Higher Order Thinking Tori has two different-sized water bottles. In the larger bottle, she has 0.81 liter of water. In the smaller bottle, she has 1.1 liters of water. Can you tell whether one bottle has more water? Explain.
Given that,
In the longer bottle she has a water = 0.81 liter.
In the smaller bottle she has a water  = 1.1 liter.
0.81 > 1.1
So, the smaller has more water.

Assessment Practice

Question 25.
Stanley found the weights of two minerals, quartz and garnet. The quartz weighed 3.76 ounces and the garnet weighed 3.68 ounces.

Explain how Stanley can use a tool to find which mineral weighed more.

Explain how Stanley can use place value to find which mineral weighed less.
Given that,
The weight of a quartz = 3.76 ounces
The weight of a garnet = 3.68 ounces
Therefore 3.68 > 3.75
The weight of a quartz is more than the weight of a garnet
Using the place value
3.68 and 3.75
Ten’s place of a numbers are
6 < 7
The weight of a quartz is more.

### Lesson 12.4 Add Fractions with Denominators of 10 and 100

Solve & Share
The mural is divided into 100 equal parts. Marilyn’s class painted io of the mural, and Cal’s class painted 20 of the mural. How much of the mural have the two classes painted? Solve this problem any way you choose.
I can … use equivalence to add fractions with denominators of 10 and 100.

You can use appropriate tools. Think about how you can use the grid to find how much of the mural the two classes painted. Show your work in the space above!

Look Back! How much of the mural remains to be painted? Write the amount as a decimal.

Essential Question
Hall How Can You Add Fractions with Denominators of 10 and 100?

Visual Learning Bridge
Steve and Jana collected money for an animal shelter. Steve collected $$\frac{4}{10}$$ of their goal while Jana collected $$\frac{5}{100}$$. How much of their goal did Jana and Steve collect?

Use like denominators to add fractions.

The red shows $$\frac{4}{10}$$ of the goal, and the blue shows $$\frac{5}{100}$$ of the goal.
The amount they collected can be written as to $$\frac{4}{10}$$ + $$\frac{5}{100}$$
You can use equivalent fractions to write tenths as hundredths.

Rename $$\frac{4}{10}$$ as an equivalent fraction with a denominator of 100.
Multiply the numerator and denominator by 10.
$$\frac{4 \times 10}{10 \times 10}=\frac{40}{100}$$

Add the numerators and write the sum over the like denominator.
$$\frac{40}{100}$$ + $$\frac{5}{100}$$ = $$\frac{45}{100}$$
Jana and Steve collected $$\frac{45}{100}$$ of their goal.

Convince Me! Construct Arguments in the problem above, why is the denominator of the total 100 and not 200?

Guided Practice

Do You Understand?
Question 1.
Suppose Jana collected another $$\frac{25}{100}$$ of their goal. What fraction of the goal have they now collected?
Given that,
Jana collected money = $$\frac{5}{100}$$.
Jana collected another $$\frac{25}{100}$$.
Fraction of goal have they collected =
$$\frac{5}{100}$$ + $$\frac{25}{100}$$ =
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
$$\frac{5}{100}$$ + $$\frac{25}{100}$$ =$$\frac{30}{100}$$

Question 2.
Write a problem that represents the addition shown below, then solve.

Do You Know How?
Question 3.
$$\frac{3}{10}+\frac{4}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
LCM of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{3}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{4}{100}$$ × $$\frac{1}{1}$$)
$$\frac{30}{100}$$ + $$\frac{4}{100}$$ = $$\frac{34}{100}$$

Question 4.
$$\frac{71}{100}+\frac{5}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{71}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{5}{10}$$ × $$\frac{10}{10}$$)
$$\frac{71}{100}$$ + $$\frac{50}{100}$$ = $$\frac{121}{100}$$

Question 5.
$$\frac{4}{100}+\frac{38}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{4}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{38}{10}$$ × $$\frac{10}{10}$$)
$$\frac{4}{100}$$ + $$\frac{380}{100}$$ = $$\frac{384}{100}$$

Question 6.
$$\frac{90}{100}+\frac{1}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{90}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{1}{10}$$ × $$\frac{10}{10}$$)
$$\frac{90}{100}$$ + $$\frac{10}{100}$$ = $$\frac{100}{100}$$ = 1

Question 7.
$$\frac{8}{10}+\frac{1}{10}+\frac{7}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{9}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{7}{100}$$ × $$\frac{1}{1}$$)
$$\frac{90}{100}$$ + $$\frac{7}{100}$$ = $$\frac{97}{100}$$

Question 8.
$$\frac{38}{100}+\frac{4}{10}+\frac{2}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{38}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{6}{10}$$ × $$\frac{10}{10}$$)
$$\frac{38}{100}$$ + $$\frac{60}{100}$$ = $$\frac{98}{100}$$

Independent Practice

Leveled Practice For 9-23, add the fractions.
Question 9.
$$\frac{21}{100}+\frac{2}{10}=\frac{21}{100}+\frac{ }{100}$$
Given,
$$\frac{21}{100}+\frac{2}{10}=\frac{21}{100}+\frac{ }{100}$$
We have to find the missing number.
$$\frac{21}{100}+\frac{2}{10}$$
LCM of 10 and 100 is 100.
= $$\frac{21}{100}+\frac{20}{100}$$ = $$\frac{41}{100}$$
So, the missing fraction is $$\frac{20}{100}$$
$$\frac{21}{100}+\frac{2}{10}=\frac{21}{100}+\frac{20}{100}$$

Question 10.
$$\frac{ }{10}+\frac{68}{100}=\frac{30}{100}+\frac{68}{100}$$
Given,
$$\frac{ }{10}+\frac{68}{100}=\frac{30}{100}+\frac{68}{100}$$
We have to find the missing number.
LCM of 10 and 100 is 100.
$$\frac{30}{100}$$ = $$\frac{3}{10}$$
So, the missing fraction is $$\frac{3}{10}$$
$$\frac{3}{10}+\frac{68}{100}=\frac{30}{100}+\frac{68}{100}$$

Question 11.
$$\frac{4}{10}+\frac{60}{100}=\frac{ }{10}+\frac{ }{10}$$
Given,
$$\frac{4}{10}+\frac{60}{100}=\frac{ }{10}+\frac{ }{10}$$
We have to find the missing number.
LCM of 10 and 100 is 100.
$$\frac{60}{100}$$ = $$\frac{6}{10}$$
$$\frac{4}{10}$$ = $$\frac{4}{10}$$
So, the missing fraction is $$\frac{6}{10}$$ and $$\frac{4}{10}$$
$$\frac{4}{10}+\frac{60}{100}=\frac{4}{10}+\frac{6}{10}$$

Question 12.
$$\frac{32}{100}+\frac{28}{100}+\frac{6}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{60}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{6}{10}$$ × $$\frac{10}{10}$$)
$$\frac{60}{100}$$ + $$\frac{60}{100}$$ = $$\frac{120}{100}$$

Question 13.
$$\frac{11}{10}+\frac{41}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{11}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{41}{100}$$ × $$\frac{1}{1}$$)
$$\frac{110}{100}$$ + $$\frac{41}{100}$$ = $$\frac{151}{100}$$

Question 14.
$$\frac{72}{100}+\frac{6}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{72}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{6}{10}$$ × $$\frac{100}{100}$$)
$$\frac{72}{100}$$ + $$\frac{600}{100}$$ = $$\frac{672}{100}$$

Question 15.
$$\frac{5}{10}+\frac{3}{10}+\frac{18}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{8}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{18}{100}$$ × $$\frac{1}{1}$$)
$$\frac{80}{100}$$ + $$\frac{18}{100}$$ = $$\frac{98}{100}$$

Question 16.
$$\frac{7}{100}+\frac{6}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{7}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{6}{10}$$ × $$\frac{10}{10}$$)
$$\frac{7}{100}$$ + $$\frac{60}{100}$$ = $$\frac{67}{100}$$

Question 17.
$$\frac{9}{10}+\frac{4}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{9}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{4}{100}$$ × $$\frac{10}{10}$$)
$$\frac{90}{100}$$ + $$\frac{40}{100}$$ = $$\frac{130}{100}$$

Question 18.
$$\frac{30}{100}+\frac{5}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{30}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{5}{10}$$ × $$\frac{10}{10}$$)
$$\frac{30}{100}$$ + $$\frac{50}{100}$$ = $$\frac{80}{100}$$

Question 19.
$$\frac{39}{100}+\frac{2}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{39}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{2}{10}$$ × $$\frac{10}{10}$$)
$$\frac{39}{100}$$ + $$\frac{20}{100}$$ = $$\frac{59}{100}$$

Question 20.
$$\frac{8}{10}+\frac{9}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{8}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{9}{100}$$ × $$\frac{1}{1}$$)
$$\frac{80}{100}$$ + $$\frac{9}{100}$$ = $$\frac{89}{100}$$

Question 21.
$$\frac{44}{100}+\frac{34}{100}+\frac{9}{10}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{78}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{9}{10}$$ × $$\frac{10}{10}$$)
$$\frac{78}{100}$$ + $$\frac{90}{100}$$ = $$\frac{168}{100}$$

Question 22.
$$\frac{70}{10}+\frac{33}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{70}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{33}{100}$$ × $$\frac{1}{1}$$)
$$\frac{700}{100}$$ + $$\frac{33}{100}$$ = $$\frac{733}{100}$$

Question 23.
$$\frac{28}{10}+\frac{72}{10}+\frac{84}{100}$$
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{100}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{84}{100}$$ × $$\frac{1}{1}$$)
$$\frac{1000}{100}$$ + $$\frac{84}{100}$$ = $$\frac{1084}{100}$$

Problem Solving

Question 24.
Algebra A mail carrier made a total of 100 deliveries in a day. $$\frac{76}{100}$$ of the deliveries were letters, $$\frac{2}{10}$$ were packages, and the rest were postcards. Write and solve an equation to find the fraction that represents how many of the deliveries were letters and packages.
Given,
A mail carrier made a total of 100 deliveries in a day.
$$\frac{76}{100}$$ of the deliveries were letters, $$\frac{2}{10}$$ were packages, and the rest were postcards.
$$\frac{76}{100}$$ + $$\frac{2}{10}$$
$$\frac{76}{100}$$ + $$\frac{20}{100}$$ = $$\frac{96}{100}$$

Question 25.
Make Sense and Persevere Balloons are sold in bags of 30. There are 5 giant balloons in each bag. How many giant balloons will you get if you buy 120 balloons? Explain.

Given that,
Total number of balloons solids in bags = 30
Balloons in each bag = 5
If you buy 120 balloons how many bags do you have = 120/5 = 24
For 120 balloons 24 bags are filled.

Question 26.
Higher Order Thinking of the first 100 elements on the periodic table, $$\frac{13}{100}$$ were discovered in ancient times, and $$\frac{21}{100}$$ were discovered in the Middle Ages. Another $$\frac{5}{10}$$ were discovered in the 1800s. What fraction of the first 100 elements was discovered after the 1800s? Explain.

$$\frac{13}{100}$$ were discovered in ancient times.$$\frac{21}{100}$$ were discovered in the Middle Ages.
Another $$\frac{5}{10}$$ discovered after 1800.
Find how many discovered after 1800 is
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{34}{100}$$ × $$\frac{1}{1}$$) + ($$\frac{5}{10}$$ × $$\frac{10}{10}$$)
$$\frac{35}{100}$$ + $$\frac{50}{100}$$ = $$\frac{85}{100}$$
$$\frac{85}{100}$$ are discovered after 1800.

Assessment Practice

Question 27.
Delia hiked $$\frac{7}{10}$$ mile one day and $$\frac{67}{10}$$ mile the next. She wanted to know how far she hiked in all. Her work is shown below.

Is Delia’s work correct? Explain.
Delia hiked $$\frac{7}{10}$$ mile one day
$$\frac{67}{10}$$ mile the next.
She hiked in all is
The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
L.C.M of 10 and 100 is 100.
Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1.
($$\frac{7}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{67}{10}$$ × $$\frac{10}{10}$$)
$$\frac{70}{100}$$ + $$\frac{670}{100}$$ = $$\frac{737}{100}$$
Delia hiked in all =$$\frac{737}{100}$$

### Lesson 12.5 Solve Word Problems involving Money

Solve & Share
A flash drive costs $24, including tax. A customer purchases 3 flash drives and pays the cashier$80. How much change should the cashier give back to the customer? Solve this problem any way you choose.
I can … use fractions or decimals to solve word problems involving money.

Look Back! Generalize How can you estimate and check if your solution is reasonable?

Essential Question
How Can You Solve Word Problems Involving Money?

Visual Learning Bridge
Marcus buys a toy airplane and a toy car. How much does Marcus spend? How much more does the toy airplane cost than the toy car?

You can draw or use bills and coins to solve problems involving money.

Find $3.32 +$1.12.

Add the bills, then count on to add each type of coin.
$4.00 +$0.40 + $0.04 =$4.44
Marcus spent $4.44. Find$3.32 – $1.12. Start with the cost of the airplane, then subtract the cost of the car. Count the remaining bills and coins.$2.00 + $0.20 =$2.20
The toy airplane costs $2.20 more than the toy car. Convince Me! Use Structure in the examples above, how can you use place value to help add or subtract? Another Example! Find$6.33 ÷ 3. Draw or use bills and coins.
You can use multiplication or division to solve problems involving money.

Guided Practice

Do You Understand?
Question 1.
Write a fraction and a decimal to describe how the quantities are related.

One coin = 1/10 of a dollar = $0.10 Three coins = 1/10 + 1/10 + 1/10 =$0.3

Question 2.
Write a fraction and a decimal to describe how the quantities are related.

One coin = 1/100 of a dollar = $0.01 Three coins = 1/100 + 1/100 + 1/100 =$0.01

Do You Know How?

For 3, use the bills and coins to solve.
Question 3.
Marcus has $15.00. He buys a movie ticket for$11.25. How much money does Marcus have left?

Given that,
Marcus has a total amount = $15.00 Marcus buy a movie ticket for$11.25
Marcus left amount = $15.00 –$11.25 = 3.75

Independent Practice

For 4-5, you may draw or use bills and coins to solve.
Question 4.
Sarah bought 3 wool scarves. The price of each scarf was $23.21. How much did 3 scarves cost? Answer: Given that, Sarah bought wool scarves = 3 The price for each scarf=$23.21
Total amount for 3 scarves = $23.21 × 3 = 69.63 Question 5. Carlos spends$14.38 on equipment. How much change should Carlos receive if he gives the clerk $20.00? Answer: Given that, Carlos spends on equipment =$14.38
He give the clerk = $20.00 Carlos received tha change =$20.00 – $14.38 =$6.62

Problem Solving

Question 6.
Use Structure Leo went to lunch with his parents. The bill was $17.85. Complete the table to show two different combinations of coins and bills that can represent$17.85.

Question 7.
Kenya purchases a new tablet for $109.78. She pays with six$20 bills. Show how you would estimate how much change Kenya should receive.
Given that,
Kenya purchases a new tablet for $109.78 She pay with six$20 bills = 6 × 20 = 120
Kenya received change = $120 –$109.78 =$11.78 Question 8. Algebra Marco paid$12 for 3 jump ropes. If each jump rope costs the same amount, how much does 1 jump rope cost? Write and solve an equation.
Given that,
Algebra Marco paid $12 for 3 jump ropes. For each jump = 12/3 =$4
For each jump cost = $4 Question 9. Number Sense Jiang has a collection of 3,788 toy building bricks. He used 1,229 bricks to build a city. About how many bricks does Jiang have left? Explain how you estimated. Answer: Given that, Jiang has a collection of toy building bricks = 3788 Jiang used bricks to build a city = 1229 Jiang has left = 3788 – 1229 = 2559 Question 10. Higher Order Thinking Edward bought 7 concert tickets for himself and six friends for a total of$168. Each friend paid Edward back for his or her ticket. If one of Edward’s friends gave him a $50 bill, how much change should Edward return? Explain. Answer: Edward bought a concert tickets for himself and six friends for total =$168
Money for each ticket = 168/7 = $24 Assessment Practice Question 11. Rajeev bought a skateboard for$37.74. How much change should Rajeev receive if he gave the cashier $40.00? You may draw or use bills and coins to solve. A.$2.26
B. $2.74 C.$3.26
D. $3.74 Answer:$2.26

Explanation:
Given,
Rajeev bought a skateboard for $37.74. Rajeev receive if he gave the cashier$40.00
40.00 – 37.74 = 2.26
Thus the correct answer is option A.

Question 12.
Genevieve bought a catcher’s mitt for $30.73 and a bat for$19.17. How much did Genevieve spend? You may draw or use bills and coins to solve.
A. $11.56 B.$49.17
C. $49.90 D.$50.73
Given,
Genevieve bought a catcher’s mitt for $30.73 and a bat for$19.17.
30.73 + 19.17 = $49.90 Thus the correct answer is option C. ### Lesson 12.6 Problem Solving Look For and Use Structure Solve & Share Three people hiked the same 1-mile trail. The distance for each hiker is represented in the drawings. Show about where the 1-mile mark should be on each drawing. Explain. I can … use the structure of the place-value system to solve problems. Thinking Habits Be a good thinker! These questions can help you. • What patterns can I see and describe? • How can I use the patterns to solve the problem? • Can I see expressions and objects in different ways? Look Back! Look For Relationships The three drawings represent 0.5, 0.25, and 0.75 mile with equivalent lengths. How does this affect where 1-mile is located on each drawing? Essential Question How Can You Look for and Make Use Question of Structure to Solve Problems? Visual Learning Bridge Maps from two different ski resorts show a 1-mile cross-country ski trail for beginners. Show about where to mark 0.25, 0.5, and 0.75 mile on each trail. How can you determine where to mark the points on each drawing? I need to analyze each drawing and decide about where the given decimals should be located on each. How can I make use of structure to solve this problem? I can • break the problem into simpler parts. • use what I know about decimal meanings to locate the points. • use equivalent forms of numbers. Here’s my thinking. The size of a decimal depends on the size of the whole. The size of the whole is not the same for each drawing. Divide each whole in half to show 0.5 on each whole. Convince Me! Use Structure Use the drawing of the trail shown. Where is the 1.5-mile mark on the trail? How did you decide? Guided Practice Use Structure Margie painted 0.4 of her banner blue. Helena painted 0.5 of her banner blue. Question 1. Complete the drawings to show the whole, or 1, for each banner. Answer: Question 2. Explain how you determined where to draw 1 whole for each banner. Answer: Question 3. Do the drawings show 0.4 < 0.5? Explain. Answer: Independent Practice Use Structure Kaitlin is making a map for the walk/run race. She wants the water stops to be at 0.5 mile, 0.3 mile, and 0.85 mile from the start. Question 4. Label 0.25, 0.5, 0.75 on the number line as a scale reference. Explain how you decided where to mark the number line. Answer: Question 5. Estimate where 0.3 and 0.85 are located compared to the other points. Mark the points 0.3 and 0.85. Explain how you estimated. Answer: Problem Solving Performance Task Watching Savings Grow Tomas deposits money in his savings account every month. If he continues to save$3.50 each month, how much money will he have at the end of 6 months? 12 months? Use the table and Exercises 6-11 to help solve.

Question 6.
Reasoning What quantities are given in the problem and what do the numbers mean?

Question 7.
Make Sense and Persevere What do you need to find?

To find the total money in the savings account.

Question 8.
Use Structure What is the relationship between the amount of money Tomas will have in his savings account in the fourth month and the amount in the third month?

Given that,
The amount of money Tomas will have in his savings account in the third month = $20.50 For each month he saves =$3.50
Total amount in the forth month = $20.50 +$3.50 = $24 Question 9. Model with Math Write an expression that can be used to find the amount saved at the end of 6 months. Answer: For each month he save$3.50
For 6 months = $3.50 × 6 =$21.25
From the given table at 0 month he have $10 in her savings account. So, for 6 months =$21.25 + $10 =$31.25

Question 10.
Model with Math Complete the table to find how much Tomas will have saved in 6 months.

Question 11.
Be Precise Use the answers from the table to find how much money Tomas will have at the end of 12 months. Show your work.
For each month he save $3.50 For 6 months =$3.50 × 12 = $42 From the given table at 0 month he have$10 in her savings account.
So, for 6 months = $42 +$10 = $52 ### Topic 12 Fluency Practice Activity Follow the path Shade a path from Start to finish. Follow the sums or differences that round to 2,000 when rounded to the nearest thousand. You can only move up, down, right, or left. I can … add and subtract multi-digit whole numbers. ### Topic 12 Vocabulary Review Understand Vocabulary Word List • decimal • decimal point • equivalent • fraction • greater than symbol (>) • hundredth • less than symbol (<) • tenth Choose the best term from the box. Write it on the blank. Question 1. A dot used to separate dollars from cents or ones from tenths in a number is called a _________ Answer: A dot used to separate dollars from cents or one’s from tenths in a number is called a one decimal point. Question 2. One part of 100 equal parts of a whole is called a ___________ Answer: One part of 100 equal parts of a whole is called a one hundredth. Question 3. Numbers that name the same amount are __________ Answer: Numbers that name the same amount are equivalent. Question 4. A symbol, such as $$\frac{2}{3}$$, $$\frac{5}{1}$$, or $$\frac{8}{5}$$, used to name part of a whole, part of a set, or a location on a number line is called a __________ Answer: fraction Question 5. One out of ten equal parts of a whole is called a _________ Answer: One out of ten equal parts of a whole is called a one hundredth. For each of these terms, give an example and a non-example. Answer: Use Vocabulary in Writing Question 9. Krista wrote $$\frac{75}{100}$$ and 0.75. Use at least 3 terms from the Word List to describe Krista’s work. Answer: ### Topic 12 Reteaching Set A pages 445-448 The essay question on a 100-point test was worth 40 points. Write this part as a fraction and a decimal. There are 100 points, so each point is $$\frac{1}{100}$$. $$\frac{40}{100}$$ is 0.40. Remember that the name of a fraction can help you write it as a decimal. Write a decimal and a fraction for each model. Question 1. Answer: 57/100 Question 2. Answer: 7/10 Question 3. Donnie has 4 dollars, 6 pennies, and 9 dimes. Write a decimal for the amount of money Donnie has. Answer: Set B pages 449-452 Locate 0.8 and 0.62 on a number line. The distance from 0 to 0.8 is eight-tenths the distance from 0 to 1. Draw a number line showing hundredths. 0.62 is between 0.6 and 0.7. Name the decimal and fraction at each point. Question 1. K Answer: The decimal at the point K is 5.47 The fraction at the point K is 547/100. Question 2. M Answer: The decimal at the point M is 5.55 The fraction at the point M is 111/20 Question 3. O Answer: The decimal at the point O is 5.68 The fraction at the point O is 144/25 Question 4. N Answer: The decimal at the point N is 5.60 The fraction at the point N is 560/100. Question 5. L Answer: The decimal at the point L is 5.50 The fraction at the point L is 11/2 Question 6. J Answer: The decimal at the point J is 5.42 The fraction at the point J is 542/100. Set C pages 453-456 Compare 1.74 and 1.08. The digits in the ones place are the same, so look at the digits after the decimal point to compare. 1.74 1.08 7 tenths > 0 tenths 1.74 > 1.08 Remember you can use tools such as place value blocks, number lines, or grids to compare decimal amounts. Write >, <, or = in each . Question 1.$4.13 $4.32 Answer: The digits in the ones place are the same, so look at the digits after the decimal point to compare.$4.13 and $4.32 1 tenths < 3 tenths$4. 13 < $4.32 Question 2. 0.6 0.60 Answer: The digits in the ones place are the same, so look at the digits after the decimal point to compare. 0.6 and 0.60 0. 6 < 0.60 Question 3. 5.29 52.9 Answer: The digits in the ones place are the same, so look at the digits after the decimal point to compare. 5.29 and 52.9 5.29 < 52.9 Question 4. 12.91 12.19 Answer: The digits in the ones place are the same, so look at the digits after the decimal point to compare. 12.91and 12.19 12.91 > 12.19 Set D pages 457-460 Find $$\frac{9}{10}+\frac{49}{100}$$ Rewrite $$\frac{9}{10}$$ as an equivalent fraction with a denominator of 100. $$\frac{9 \times 10}{10 \times 10}=\frac{90}{100}$$ $$\frac{90}{100}+\frac{49}{100}=\frac{139}{100}$$ or 1$$\frac{39}{100}$$ Remember to find equivalent fractions with like denominators to add. Add. Use grids or place-value blocks as needed to help. Question 1. $$\frac{8}{10}+\frac{40}{100}$$ Answer: Rewrite $$\frac{8}{10}$$ as an equivalent fraction with a denominator of 100. $$\frac{8 \times 10}{10 \times 10}=\frac{80}{100}$$ $$\frac{80}{100}+\frac{40}{100}=\frac{120}{100}$$ or 1 $$\frac{1}{5}$$ Question 2. $$\frac{24}{100}+\frac{6}{10}$$ Answer: Rewrite $$\frac{6}{10}$$ as an equivalent fraction with a denominator of 100. $$\frac{6 \times 10}{10 \times 10}=\frac{60}{100}$$ $$\frac{24}{100}+\frac{60}{100}=\frac{84}{100}$$ Set E pages 461-464 Find$5.21 + $1.52. Add the bills, then count on to add each type of coin.$6.00 + $0.50 +$0.20 + $0.03 =$6.73

Remember to take away each type of bill and coin when subtracting money.

Question 1.
Chelsea had $71.18. She bought a new pair of glasses for$59.95. Can she buy a case that costs $12.95? Explain. Answer: Given that, Chelsea had =$71.18
She bought a new pair of glasses = $59.95 Remaining money with Chelsea =$71.18 – $59. 95 = 12. 77 She buy a case for =$12.95
She can’t buy a case because she has only $12.77 but the cost of a case is$12.95.

Question 2.
Eddie bought 3 train tickets for $17.00 each. If he paid with three$20 bills, how much change did Eddie receive?
Given
Eddie bought 1 train tickets = $17.00 For three tickets =$17.00 × 3 = $51 He paid bills = 3 ×$20 = $60 Thus Eddie received money =$60 – $51 =$9

Set F pages 465-468

Thinking Habits

• What patterns can I see and describe?
• How can I use the patterns to solve the problem?
• Can I see expressions and objects in different ways?

Remember you can use structure to break a problem into simpler parts.

Raven joined a walk-a-thon. The red dot shows how far Raven walked in one hour.
Question 1.
Complete the number line below.

Question 2.
Estimate how far Raven walked in the first hour. Explain.
Answer: Raven walked 2 miles in one hour.

### Topic 12 Assessment Practice

Question 1.
Which represent the decimal 0.7? Select all that apply.
☐ 0.07
☐ 7.00
☐ $$\frac{7}{10}$$
☐ $$\frac{70}{10}$$
☐ $$\frac{70}{100}$$
☐ 0.07
☐ 7.00
$$\frac{7}{10}$$ can be written in the decimal as 0.7
☐ $$\frac{70}{10}$$
$$\frac{70}{100}$$ can be written in the decimal as 0.7

Question 2.
Select all the statements that correctly compare two numbers.
☐ 29.48 > 29.69
☐ 29.48 < 29.69 ☐ 15.36 > 15.39
☐ 16.99 < 17.99
☐ 21.30 = 21.03
☐ 29.48 > 29.69
29.48 < 29.69 ☐ 15.36 > 15.39
16.99 < 17.99
☐ 21.30 = 21.03

Question 3.
Lucy buys a puzzle for $3.89, a model airplane for$12.75, and a stuffed animal for $2.50. How much money did she spend in all? Draw or use bills and coins to solve. A.$19.14
B. $19.00 C.$16.64
D. $16.00 Answer: Given, Lucy buys a puzzle for$3.89, a model airplane for $12.75, and a stuffed animal for$2.50.
$3.89 +$12.75 + $2.50 =$19.14
Therefore she spent $19.14 in all. Thus the correct answer is option A. Question 4. Which point is incorrectly labeled? Explain. Answer: C (4.9) is incorrectly labeled. Question 5. Catalina takes the money shown to the bookstore. A. Does Catalina have enough for all three books? If not, how much more money does Catalina need? Explain. Draw or use bills and coins to solve. Answer: B. Catalina chooses to buy only 2 of the books. Choose two books for Catalina to buy, and then find how much money she will have left. Draw or use bills and coins to solve. Answer: Question 6. Write a fraction and a decimal that represent the part of the grid that is green. Answer: 63/100 There are 63 shaded blocks out of 100. So, the fraction is $$\frac{63}{100}$$ Question 7. Match each number on the left to its equivalent fraction. Answer: 20 can be written as 20/100 2 can be written as 200/100 = 2 0.02 can be written in the fraction form as 2/100 0.20 can be written in the fraction form as 20/100 or 2/10. Question 8. Select all the statements that correctly compare two numbers. ☐ 7.27 > 74.7 ☐ 1.24 < 1.42 ☐ 58.64 > 48.64 ☐ 138.5 < 13.85 ☐ 12.56 > 12.65 Answer: ☐ 7.27 > 74.7 1.24 < 1.42 58.64 > 48.64 ☐ 138.5 < 13.85 ☐ 12.56 > 12.65 Question 9. What fraction is equivalent to 0.4? Answer: The fraction is equivalent to 0.4 is $$\frac{4}{10}$$ Question 10. Explain how to find the sum of $$\frac{3}{10}+\frac{4}{100}$$ Answer: The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator. LCM of 10 and 100 is 100. Multiply both the numerator and denominator of each fraction by the number that makes its denominator equal to the LCD. This is basically multiplying each fraction by 1. ($$\frac{3}{10}$$ × $$\frac{10}{10}$$) + ($$\frac{4}{100}$$ × $$\frac{1}{1}$$) $$\frac{30}{100}$$ + $$\frac{4}{100}$$ = $$\frac{34}{100}$$ Question 11. Use the table below. Create a number line and plot the value of each letter. Answer: Question 12. What decimal represents $$\frac{44}{100}$$? Answer: The decimal that represents $$\frac{44}{100}$$ is 0.44 ### Topic 12 Performance Task Nature Club The nature club at the school devoted a month to learning about different local birds. The Bird Traits photos show information about several birds they observed. Question 1. The club leader asked students to analyze and compare the measures from the Bird Traits photos. Part A Randall was asked to write the mass of a red-tailed hawk as a fraction. Label the mass on the number line and write the equivalent fraction. Answer: Part B Melanie was assigned to compare the wingspans of the blue jay and the sandpiper. Which bird had a greater wingspan? Show the decimals on the grids, and write the comparison using symbols. Answer: Part C Mila compared the wingspans of the red-tailed hawk and the great horned owl. Explain how to use place value to find the greater wingspan. Show the comparison using symbols. Answer: Question 2. Gerald found the mass of a great horned owl and a sandpiper combined. Show how to write each mass as a fraction and then write and solve an addition equation. Answer: Given that mass of the great horned owl = 1.8 Mass of the sandpiper = 0.06 1.8 fraction is 18/10 0.06 fraction is 3/10 Mass of sandpiper and great horned owl is = 18/10 + 3/10 = 21/10. Question 3. The Blue Jay photo shows the wingspan of a blue jay Susannah observed. Susannah said the wingspan of the blue jay was greater than the wingspan of the great horned owl since 1.4 > 1.3. Do you agree? Explain. Answer: Yes I will agree because the wingspan of the blue jay is 1.4 Wingspan of the great horned owl is 1.3 So the wingspan of the blue jay is greater than the wingspan of the great horned. #### enVision Math Common Core Grade 4 Answer Key ## Envision Math Common Core Grade 4 Answer Key Topic 13 Measurement: Find Equivalence in Units of Measure ## Envision Math Common Core 4th Grade Answers Key Topic 13 Measurement: Find Equivalence in Units of Measure Essential Questions: How can you convert from one unit to another? How can you be precise when solving math problems? enVision STEM Project: Erosion and Measurement Do Research The Colorado River has played a large part in shaping North America. Use the Internet and other resources to research the states through which the river travels. Journal: Write a Report Include what you found. Also in your report: • Look up geology and geometry in the dictionary. Write the definitions and explain how these words are related. What does the prefix “geo” mean in both words? • A.J. takes a 4-mile tour of the Grand Canyon. Explain how to convert the length of A.J.’s tour from miles to feet. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • capacity • gram • liter • mass Question 1. The amount of liquid a container can hold is called its ________. Answer: The amount of liquid a container can hold is called its __capacity______. Question 2. ________ is the amount of matter that something contains. Answer: ___Mass_____ is the amount of matter that something contains. Question 3. One metric unit of capacity is a ________. Answer: One metric unit of capacity is a __Liter______. Perimeter Find the perimeter of each shape. Question 4. Answer: Perimeter of the given rectangle shape = 134 centimeters. Explanation: Length of the given rectangle shape = 42 centimeters. Width of the given rectangle shape = 25 centimeters. Perimeter of the given rectangle shape = 2 × (Length of the given rectangle shape × Width of the given rectangle shape) = 2 × (42 + 25) = 2 × 67 = 134 centimeters. Question 5. Answer: Perimeter of the given square shape = 28 feet. Explanation: Length of the given square shape = 7 feet. Perimeter of the given square shape = 4 × Length of the given square shape = 4 × 7 = 28 feet. Question 6. Answer: Perimeter of the given Triangular shape = 9 yards. Explanation: Length of the given Triangular shape = 3 yards. Perimeter of the given Triangular shape = Length of the given Triangular shape + Length of the given Triangular shape + Length of the given Triangular shape = 3 + 3 + 3 = 9 yards. Question 7. Answer: Perimeter of the given Quadrilateral shape = 62 inches. Explanation: Length of the given Quadrilateral shape: Side 1 length = 17 inches. Side 2 length = 12 inches. Side 3 length = 21 inches. Side 4 length = 12 inches. Perimeter of the given Quadrilateral shape = Side 1 length + Side 2 length + Side 3 length + Side 4 length = 17 + 12 + 21 + 12 = 29 + 21 + 12 = 50 + 12 = 62 inches. Question 8. Answer: Perimeter of the given hexagon shape = 90 centimeters. Explanation: Length of the given hexagon shape = 15 centimeters. Perimeter of the given hexagon shape = 6 × Length of the given hexagon shape = 6 × 15 = 90 centimeters. Question 9. Answer: Perimeter of the given rectangle shape = 40.67 feet. Explanation: Length of the given rectangle shape = 19 (11/12) feet. Width of the given rectangle shape = 7 (5/12) feet. Perimeter of the given rectangle shape = 2 × (Length of the given rectangle shape × Width of the given rectangle shape) = 2 × (19 11/12 + 7 5/12) = 2 × (209/12 + 35/12) = 2 × 244/12 = 488/12 = 40.67 feet. Area Find the area of each shape. Question 10. Answer: Area of the given rectangle shape = 10 square yards. Explanation: Length of the given rectangle shape = 5 yards. Width of the given rectangle shape = 2 yards. Area of the given rectangle shape = Length of the given rectangle shape × Width of the given rectangle shape = 5 × 2 = 10 square yards. Question 11. Answer: Area of the given rectangle shape = 1/2 or 0.5 inches. Explanation: Length of the given rectangle shape = 2 inches. Width of the given rectangle shape = 1/4 inches. Area of the given rectangle shape = Length of the given rectangle shape × Width of the given rectangle shape = 2 × 1/4 = 1/2 or 0.5 inches. Question 12. Answer: Area of the given rectangle shape = 21 square centimeters. Explanation: Length of the given rectangle shape = 7 centimeters. Width of the given rectangle shape = 3 centimeters. Area of the given rectangle shape = Length of the given rectangle shape × Width of the given rectangle shape = 7 × 3 = 21 square centimeters. Problem Solving Question 13. Make Sense and Persevere A league is a nautical measurement equal to about 3 miles. If a ship travels 2,000 leagues, about how many miles does the ship travel? Answer: Number of miles the ship travelled = 6000 miles. Explanation: 1 league = 3 miles. Number of leagues the ship travelled = 2000. Number of miles the ship travelled = 3 × Number of leagues the ship travelled => 3 × 2000 => 6000 miles. Pick a Project PROJECT 13A What makes the St. Johns River special? Project: Make a Travel Brochure About Rivers in Your Home State Answer: PROJECT 13B How are tin cans useful? Project: Cooking on a Budget Answer: Ways to Cook on a Budget 1. Make meal plans. 2. Shop for groceries at a discount grocer. 3. Start with the grocery store flyer. 4. Pair sale items with low-cost staple foods. 5. Use a slow cooker. 6. Make “pantry meals” a habit. 7. Make extras and freeze them. 8. Eat your leftovers. PROJECT 13C Who invented the jigsaw puzzle? Project: Make Your Own Jigsaw Puzzle Answer: 3-ACT MATH PREVIEW Math Modeling A Pint’s a Pound I can … model with math to solve a problem that involves estimating and computing with units of weight and capacity. ### Lesson 13.1 Equivalence with Customary Units of Length Solve & Share Jeremy jogged 75 yards from his house to school. How many feet did Jeremy jog? Solve this problem any way you choose. I can … convert customary units of length from one unit to another and recognize the relative size of different units. Look Back! Look for Relationships What do you notice about the relationship between the number of yards and the number of feet Jeremy jogged? Answer: The relationship between the number of yards and the number of feet Jeremy jogged is 3 times more than the number of yards Jeremy jogged. Explanation: Number of yards Jeremy jogged = 75. Number of feet Jeremy jog = 3 × Number of yards Jeremy jogged = 3 × 75 = 225 feet. Essential Question How Can You Convert from One Unit of Length to Another? Answer: We can Convert from One Unit of Length to Another, by following steps: 1. Write the conversion as a fraction (that equals one) 2. Multiply it out (leaving all units in the answer) 3. Cancel any units that are both top and bottom. Visual Learning Bridge Maggie has a tree swing. How many inches long is each rope from the bottom of the branch to the swing? Step 1 Find the length of the rope in feet. r = 10 – 2$$\frac{1}{4}$$ Each rope is 7$$\frac{3}{4}$$ feet long. Step 2 Convert the length of the rope to inches. Each rope is 93 inches long. Convince Me! Generalize How do you know the answer is reasonable when converting a larger unit to a smaller unit? Answer: When converting a larger unit to a smaller one, you multiply; when you convert a smaller unit to a larger one, you divide. Hence, the answer is reasonable when converting a larger unit to a smaller unit. Another Example! Mark moved forward yard when doing a back flip. Daisy moved forward 3$$\frac{1}{6}$$ feet. How much more did Daisy move forward than Mark? One yard is 3 times as long as a foot. Guided Practice Do You Understand? Question 1. Does it take more inches or feet to equal a given length? Explain. Answer: Converting Between Units of Length: There are many more inches for a measurement than there are feet for the same measurement, as feet is a longer unit of measurement. You could use the conversion factor. Question 2. Which is a greater distance, 9 yards or 9 miles? Answer: 9 miles is a greater distance between 9 yards or 9 miles. Explanation: 1 miles = 1760 yards. 9 miles = ??? yards. => 9 × 1760 => 15840 yards. Do You Know How? For 3-4, convert each unit. Question 3. 2 miles = _________ yards Answer: 2 miles = __3520 _______ yards. Explanation: 1 miles = 1760 yards. 2 miles = 2 × 1760 => 3520 yards. Question 4. $$\frac{2}{3}$$ yard = _____feet Answer: $$\frac{2}{3}$$ yard = __2___feet. Explanation: 1 yard = 3 feet. $$\frac{2}{3}$$ yard = $$\frac{2}{3}$$ × 3 => 2 feet. Independent Practice In 5-7, write > or < in each to compare the measures. Question 5. 6 inches 6 feet Answer: 6 inches < 6 feet. Explanation: 1 feet = 12 inches. 6 feet = 6 × 12 => 72 inches. Question 6. 2 yards 7 feet Answer: 2 yards < 7 feet. Explanation: 1 yard = 3 feet. 2 yards = 2 × 3 => 6 feet. Question 7. 4 yards 100 inches Answer: 4 yards > 100 inches. Explanation: 1 yard = 36 inches. 4 yards = 4 × 36 => 144 inches. For 8-11, convert each unit. Question 8. 8 yards = _______ inches Answer: 8 yards = __288_____ inches. Explanation: 1 yard = 36 inches. 8 yards = 8 × 36 => 288 inches. Question 9. 28 yards = ______ feet Answer: 28 yards = __84____ feet. Explanation: 1 yard = 3 feet. 28 yards = 3 × 28 => 84 feet. Question 10. 18 feet = ________ inches Answer: 18 feet = ___216_____ inches. Explanation: 1 feet = 12 inches. 18 feet = 12 × 18 => 216 inches. Question 11. 7 miles = ______ yards Answer: 7 miles = __12320____ yards. Explanation: 1 mile = 1760 yards. 7 miles = 1760 × 7 => 12320 yards. Problem Solving Question 12. Be Precise Lou cuts 3 yards from a 9-yard roll of fabric. Then he cuts 4 feet from the roll. How many feet of fabric are left on the roll? Answer: Remaining feet of fabric are left on the roll = 14. Explanation: Length of fabric Lou cuts = 3 yards. Length of roll of fabric = 9 yards. Remaining length of roll of fabric = Length of roll of fabric – Length of fabric Lou cuts = 9 – 3 = 6 yards. Conversion: 1 yard = 3 feet. = 6 yards = ??? feet => 6 × 3 feet = 18 feet. Length of fabric later cut = 4 feet. Remaining feet of fabric are left on the roll = Remaining length of roll of fabric – Length of fabric later cut = 18 – 4 = 14 . Question 13. What customary units would you use to measure the length of a praying mantis? Explain. Answer: The customary system of measurement is inches used to measure the length of a praying mantis. Explanation: The customary system of measurement is defined as a set of weights and measures used for measuring length, weight, capacity, and temperature. Most praying mantis are no more than 5 inches in length. The smaller species measure around 2 to 3 inches while the bigger specimens reach the overall length of 6 to 7 inches. Question 14. Algebra On the field trip, Toni collected 4 times as many bugs as Kaylie. Kaylie collected 14 bugs. Draw a bar diagram, and write and solve an equation to find b, how many bugs Toni collected. Answer: Number of bugs Toni collected = 56. Explanation: Number of bugs Kaylie collected = 14. Toni collected 4 times as many bugs as Kaylie. Number of bugs Toni collected = 4 × Number of bugs Kaylie collected = 4 × 14 => 56. Question 15. Which is greater, 3 miles or 5,000 yards? How much greater? Explain. Answer: 3 miles is greater by 280 yards than 5,000 yards. Explanation: 3 miles,5000 yards. Conversion: 1 mile = 1760 yards. 3 miles = ??? yards => 3 × 1760 => 5280 yards. Difference: 5280 yards – 5000 yards => 280 yards. Question 16. Higher Order Thinking Jenna uses $$\frac{1}{2}$$ yard of ribbon for each box she wraps. How many inches of ribbon does she need to wrap 4 boxes? Use the linear model to help solve. Answer: Number of inches of ribbon she needs to wrap 4 boxes = 72 inches. Explanation: Length of ribbon Jenna uses for each box she wraps = $$\frac{1}{2}$$ yard. Total Length of ribbon she needs to wrap 4 boxes = 4 × Length of ribbon Jenna uses for each box she wraps = 4 × $$\frac{1}{2}$$ yard = 2 yards. Conversion: 1 yard = 36 inches. Number of inches of ribbon she needs to wrap 4 boxes = Total Length of ribbon she needs to wrap 4 boxes × 36 inches = 2 × 36 => 72 inches. Assessment Practice Question 17. Connor has 3$$\frac{3}{4}$$ feet of brown fabric and $$\frac{3}{4}$$ yard of green to make a costume for the school play. How many more feet of brown than green fabric does Connor have? Show both measures with points on the number line. Answer: 1 $$\frac{2}{4}$$ feet more feet of brown than green fabric Connor have. Explanation: Length of brown fabric Connor has = 3$$\frac{3}{4}$$ feet. Length of green fabric Connor has = $$\frac{3}{4}$$ yard Conversion: 1 yard = 3 feet. $$\frac{3}{4}$$ yard = ?? feet => 3 × $$\frac{3}{4}$$ => 2 $$\frac{1}{4}$$ feet. Difference: Length of brown fabric Connor has – Length of green fabric Connor has = 3$$\frac{3}{4}$$ feet – 2 $$\frac{1}{4}$$ feet = 1 $$\frac{2}{4}$$ feet. Question 18. Charlotte made $$\frac{11}{2}$$ yard of a paper chain for the school dance. Josh made 4$$\frac{1}{12}$$ feet, and Mika made 3$$\frac{4}{12}$$ feet. How many feet of chain did they make in all? A. 8$$\frac{4}{12}$$ feet B. 9$$\frac{4}{12}$$ feet C. 10$$\frac{2}{12}$$ feet D. 10$$\frac{4}{12}$$ feet Answer: Total length of feet of chain they make in all = 9$$\frac{4}{12}$$ feet. B. 9$$\frac{4}{12}$$ feet. Explanation: Length of a paper chain Charlotte made = $$\frac{11}{2}$$ yard. Conversion: 1 yard = 3 feet. $$\frac{11}{2}$$ yard = $$\frac{11}{2}$$ × 3 = 16 $$\frac{1}{2}$$ feet. Length of a paper chain Josh made = 4$$\frac{1}{12}$$ feet. Length of a paper chain Mika made = 3$$\frac{4}{12}$$ feet. Total length of feet of chain they make in all = Length of a paper chain Charlotte made + Length of a paper chain Josh made + Length of a paper chain Mika made = 16 $$\frac{1}{2}$$ feet + 4$$\frac{1}{12}$$ feet + 3$$\frac{4}{12}$$ feet = 9$$\frac{4}{12}$$ feet. ### Lesson 13.2 Equivalence with Customary Units of Capacity Solve & Share Casey has $$\frac{1}{2}$$ gallon of juice. How many 1-pint containers can he fill? Solve this problem any way you choose. I can … convert customary units of capacity from one unit to another and recognize the relative size of different units. Look Back! Generalize How did you convert from a larger unit of capacity to a smaller unit of capacity? Did you use the same process you used to convert from a larger unit of length to a smaller unit of length? Explain. Answer: When converting a larger unit of capacity to a smaller one of capacity, you multiply; when you convert a smaller unit of capacity to a larger one of capacity, you divide. Yes, I used the same process you used to convert from a larger unit of length to a smaller unit of length. Essential Question How Can You Convert from One Unit of Capacity to Another? Answer: We can convert from One Unit of Capacity to Another by converting a larger unit of capacity to a smaller one of capacity, you multiply; when you convert a smaller unit of capacity to a larger one of capacity, you divide. Visual Learning Bridge Ms. Nealy’s class needs 5 gallons of punch for family math night. How much of each ingredient is needed to make enough punch with the recipe shown? Units of capacity include gallons, quarts, pints, cups, and fluid Ounces. Capacity is how much liquid a container can hold. This diagram shows the relative sizes of customary units of capacity. 1 gallon is 4 times as much as 1 quart Step 1 Convert 5 gallons to pints. 5 gallons = 40 pints Step 2 Add the number of pints in the recipe to find how many batches the class needs to make. 5 + 4 +1 = 10 10 × n = 40 n = 4 The class needs to make 4 batches of the recipe. Step 3 Find how much of each ingredient is in 4 batches. 4 × 5 = 20 pints 4 × 4 = 16 pints 4 × 1 = 4 pints 20 pints of apple juice, 16 pints of lemon/lime soda, and 4 pints of frozen orange juice are needed. Convince Me! Reasoning Complete the sentence below. One gallon equals _______ quarts, _______ pints, or _______ cups. Answer: One gallon equals ___4____ quarts, ____8___ pints, or ___16____ cups. Explanation: 1 gallon = 8 pints. 1 gallon = 4 quarts. 1 gallon = 16 cups. Guided Practice Do You Understand? Question 1. How many cups of punch does 5 gallons of the punch from the previous page make? Answer: Number of cups of punch 5 gallons of the punch from the previous page makes = 80 cups. Explanation: Number of cups of punch 5 gallons of the punch from the previous page makes = 5 gallons = ??? cups Conversion: 1 gallon = 16 cups. 5 gallons = 5 × 16 = 80 cups. Question 2. Which size container holds more, 3 pints or 3 quarts? Answer: 3 quarts size container holds more. Explanation: 3 pints and 3 quarts. 1 quarts = 2 pints. 3 quarts = 3 × 2 = 6 pints. Do You Know How? For 3-5, convert each unit. Question 3. 2 cups = _______ fluid ounces Answer: 2 cups = ___16____ fluid ounces. Explanation: Conversion: 1 cup = 8 fluid ounces. 2 cups = 8 × 2 = 16 fluid ounces. Question 4. $$\frac{1}{2}$$ gallon = _______ pints Answer: $$\frac{1}{2}$$ gallon = ___4____ pints. Explanation: Conversion: 1 gallon = 8 pints. $$\frac{1}{2}$$ gallon = 8 × $$\frac{1}{2}$$ = 4 pints. Question 5. 5 pints = ______ cups Answer: 5 pints = __10____ cups. Explanation: Conversion: 1 pint = 2 cups. 5 pints = 2 × 5 = 10 cups. Independent Practice In 6-8, write > or < in each to compare the measures. Question 16. 2 pints 2 gallons Answer: 2 pints < 2 gallons. Explanation: 2 pints and 2 gallons. Conversion: 1 gallon = 8 pints. 2 gallons = 8 × 2 = 16 pints. Question 7. 5 quarts 8 pints Answer: 5 quarts > 8 pints. Explanation: 5 quarts and 8 pints. Conversion: 1 quart = 2 pints. 5 quarts = 2 × 5 = 10 pints. Question 8. 10 cups 2 quarts Answer: 10 cups > 2 quarts. Explanation: 10 cups and 2 quarts. Conversion: 1 quart = 4 cups. 2 quarts = 4 × 2 = 8 cups. For 9-12, convert each unit. Question 9. 7 quarts = ________ cups Answer: 7 quarts = ____28____ cups. Explanation: Conversion: 1 quart = 4 cups. 7 quarts = 4 × 7 = 28 cups. Question 10. 12 gallons = ______ quarts Answer: 12 gallons = ___48___ quarts. Explanation: Conversion: 1 gallon = 4 quarts. 12 gallons = 4 × 12 = 48 quarts. Question 11. 7 pints = _______ fluid ounces Answer: 7 pints = __112_____ fluid ounces. Explanation: Conversion: 1 pint = 16 fluid ounces. 7 pints = 16 × 7 = 112 fluid ounces. Question 12. $$\frac{3}{4}$$ gallon = ______ pints Answer: $$\frac{3}{4}$$ gallon = ___6___ pints. Explanation: Conversion: 1 gallon = 8 pints. $$\frac{3}{4}$$ gallon = 8 × $$\frac{3}{4}$$ = 6 pints. For 13-14, convert each unit. Question 13. Answer: Explanation: Conversion: 1 pint = 16 fluid ounces. $$\frac{2}{4}$$ = 16 × $$\frac{2}{4}$$ = 8 fluid ounces. 1 pint = 16 fluid ounces. 2 pints = 16 × 2 = 32 fluid ounces. 4 pints = 16 × 4 = 64 fluid ounces. Question 14. Answer: Explanation: Conversion: 1 gallon = 16 cups. 2 gallon = 16 × 2 = 32 cups. 3 gallon = 16 × 3 = 48 cups. 4 gallon = 16 × 4 = 64 cups. Problem Solving Question 15. en Vision® STEM Scientists measure how much water and debris flow past a river station at different times of the year. The water and debris are called discharge. The table shows the average discharge at the Camp Verde station on the Verde River in two months. How many more quarts of discharge per second are there in December than November? Answer: 2664 more quarts of discharge per second are there in December than November . Explanation: Average discharge at the Camp Verde station on the Verde River for November month = 1619 gallons per second. Average discharge at the Camp Verde station on the Verde River for December month = 2285 gallons per second. Conversion: 1 gallon = 4 quarts. Number of quarts of discharge per second in November = Average discharge at the Camp Verde station on the Verde River for November month × 4 = 1619 × 4 = 6476 quarts. Number of quarts of discharge per second in December = Average discharge at the Camp Verde station on the Verde River for December month × 4 = 2285 × 4 = 9140 quarts. Difference: Number of quarts of discharge per second in December – Number of quarts of discharge per second in November = 9140 – 6476 = 2664 . Question 16. How many quarts of discharge per second were recorded in November and December? Answer: Number of quarts of discharge per second in November = 6476 quarts. Number of quarts of discharge per second in December = 9140 quarts. Explanation: Conversion: 1 gallon = 4 quarts. Number of quarts of discharge per second in November = Average discharge at the Camp Verde station on the Verde River for November month × 4 = 1619 × 4 = 6476 quarts. Number of quarts of discharge per second in December = Average discharge at the Camp Verde station on the Verde River for December month × 4 = 2285 × 4 = 9140 quarts. Question 17. Make Sense and Persevere Annabelle had the following containers of paint left over: $$\frac{1}{2}$$ gallon, $$\frac{3}{4}$$ quart, and $$\frac{1}{4}$$ gallon. How many quarts of paint does Annabelle have left over? Explain. Answer: Total quarts of paint does Annabelle have left over = 3 $$\frac{3}{4}$$ quarts. Explanation: Quantity of containers of paint left over: First container = $$\frac{1}{2}$$ gallon. Second container = $$\frac{3}{4}$$ quart. Third container = $$\frac{1}{4}$$ gallon. Conversion: 1 gallon = 4 quarts. First container = $$\frac{1}{2}$$ gallon. = $$\frac{1}{2}$$ × 4 = 2 quarts. Third container = $$\frac{1}{4}$$ gallon. = $$\frac{1}{4}$$ × 4 = 1 quarts. Total quarts of paint does Annabelle have left over = First container + Second container + Third container = 2 quarts + $$\frac{3}{4}$$ + 1 quarts = 3 $$\frac{3}{4}$$ quarts. Question 18. Higher Order Thinking A caterer combines 3 quarts of orange juice, 5 pints of milk, and 5 cups of pineapple juice to make smoothies. How many cups can be filled with smoothies? Explain. Answer: Number of cups can be filled with smoothies = 27 cups. Explanation: Quantity of orange juice = 3 quarts. Quantity of pints of milk = 5 pints. Quantity of cups of pineapple juice = 5 cups. Conversion: 1 pint = 2 cups. Quantity of pints of milk = 5 pints. => 5 pints = 2 × 5 = 10 cups. Conversion: 1 quarts = 4 cups. Quantity of orange juice = 3 quarts. => 3 quarts = 4 × 3 = 12 cups. Number of cups can be filled with smoothies = Quantity of orange juice + Quantity of pints of milk + Quantity of cups of pineapple juice = 12 cups + 10 cups + 5 cups = 27 cups. Assessment Practice Question 19. Which equals 3 quarts? A. 6 cups B. 12 cups C. 12 pints D. 3 gallons Answer: 3 quarts = 12 cups. B. 12 cups Explanation: 1 quarts = 4 cups. 3 quarts = 4 × 3 = 12 cups. 1 quart = 2 pint. 3 quarts = 2 × 3 = 6 pints. 1 quart = 0.25 gallon. 3 quarts = 3 × 0.25 = 0.75 gallon. Question 20. Select all the comparisons that are true. ☐ 4 cups < 4 pints ☐ 7 pints > 7 fluid ounces ☐ 2 gallons > 9 quarts ☐ 3 quarts < 14 cups ☐ 3 gallons < 18 pints Answer: All the comparisons that are true: 4 cups < 4 pints (TRUE) 7 pints > 7 fluid ounces (TRUE) 3 quarts < 14 cups (TRUE) 3 gallons < 18 pints (TRUE) Explanation: All the comparisons that are true: ☐ 4 cups < 4 pints (TRUE) ☐ 7 pints > 7 fluid ounces (TRUE) ☐ 2 gallons > 9 quarts (FALSE) ☐ 3 quarts < 14 cups (TRUE) ☐ 3 gallons < 18 pints (TRUE) ☐ 4 cups < 4 pints 1 pint = 2 cups. 4 cups < 4 pints => 4 × 2 = 8 cups < 4 pints. ☐ 7 pints > 7 fluid ounces 1 pint = 16 fluid ounces => 7 × 16 = 112 > 7 fluid ounces. 1 gallon = 4 quarts. ☐ 2 gallons > 9 quarts => 2 × 4 = 8 quarts > 9 quarts. 1 quarts = 4 cups. ☐ 3 quarts < 14 cups => 3 × 4 = 12 < 14 cups. 1 gallon = 8 pints. 3 gallons < 18 pints => 3 × 8 = 24 < 18 pints. ### Lesson 13.3 Equivalence with Customary Units of Weight Solve & Share When Lori’s puppy, Bay, was born she weighed $$\frac{3}{8}$$ pound. What was Bay’s weight in ounces? Solve this problem any way you choose. I can … convert customary units of weight from one unit to another and recognize the relative size of different units. Look Back! Be Precise How did you know you needed to convert units to solve the problem above? Answer: Well, we need to know the units given in the problem to convert. Next we should find the units value into which, the value should be converted into and later convert into the required units to solve the problem asked above. Essential Question How Can You Convert from One Unit of Weight to Another? Answer: We have also learnt conversion of one unit to another. We can convert from One Unit of Weight to Another by knowing the value of the conversion unit and multiplying it. For example: We can convert kilogram into grams by multiplying the number of kilograms by 1000. To convert grams into kilograms, we divide the number of grams by 1000. Visual Learning Bridge Mark made dinner for his family using the ingredients shown. How many 6-ounce servings did Mark make? Weight is how heavy an object is. Units of weight include ounces, pounds, and tons. This table shows the relative sizes of customary units of weight. 1 pound is 16 times as heavy as 1 ounce. To convert the weight of the pasta and the meatballs to ounces, multiply each weight by 16. Pasta: 2 × 16 = 32 = 6$$\frac{2}{5}$$ ounces Meatballs: $$\frac{3}{5}$$ × 16 = $$\frac{48}{5}$$ = 9$$\frac{3}{5}$$ ounces Add the weights of all the ingredients to find the total ounces. The weight of all the ingredients is 24 ounces. Divide to find s, the number of servings. 24 = 6 = s s = 4 Mark made four 6-ounce servings. Convince Me! Generalize How do you convert a larger unit of weight to a smaller unit of weight? Answer: We can convert a larger unit of weight to a smaller unit of weight by multiplying. To convert from a smaller unit to a larger one, divide. Guided Practice Do You Understand? Question 1. Would it make sense to describe the total weight of Mark’s dinner in tons? Why or why not? Answer: No, it does not makes sense to describe the total weight of Mark’s dinner in tons because it is not asked in the problem to find. Do You Know How? For 2-4, convert each unit. Question 2. 9 tons = ________ pounds Answer: 9 tons = ________ pounds. Explanation: Conversion: 1 ton = 2000 pounds. 9 tons = 2000 × 9 = 18000 pounds Question 3. $$\frac{3}{4}$$ pound = _______ ounces Answer: $$\frac{3}{4}$$ pound = ___12____ ounces Explanation: Conversion: 1 pound = 16 ounces. $$\frac{3}{4}$$ pound = 16 × $$\frac{3}{4}$$ = 12 ounces. Question 4. 17 pounds = _________ ounces Answer: 17 pounds = ___272______ ounces. Explanation: Conversion: 1 pound = 16 ounces. 17 pounds = 16 × 17 = 272 ounces. Independent Practice In 5-7, write > or < in each to compare the measures. Question 5. 6 ounces 6 pounds Answer: 6 ounces < 6 pounds. Explanation: 6 ounces and 6 pounds. Conversion: 1 pound = 16 ounces. 6 pounds = 16 × 6 = 96 ounces. Question 6. 3 pounds 40 ounces Answer: 3 pounds > 40 ounces. Explanation: 3 pounds and 40 ounces. Conversion: 1 pound = 16 ounces. 3 pounds = 16 × 3 = 48 ounces. Question 7. 5,000 pounds 2 tons Answer: 5,000 pounds > 2 tons. Explanation: 5,000 pounds and 2 tons. Conversion: 1 ton = 2000 pounds. 2 tons = 2000 × 2 = 4000 pounds. For 8-13, convert each unit. Question 8. 15 pounds = ________ ounces Answer: 15 pounds = ___240_____ ounces. Explanation: Conversion: 1 pound = 16 ounces. 15 pounds = 16 × 15 = 240 ounces. Question 9. 7 tons = _______ pounds Answer: 7 tons = _14000______ pounds. Explanation: Conversion: 1 ton = 2000 pounds. 7 tons = 2000 × 7 = 14000 pounds. Question 10. 46 pounds = ________ounces Answer: 46 pounds = __736______ounces. Explanation: Conversion: 1 pound = 16 ounces. 46 pounds = 16 × 46 = 736 ounces. Question 11. $$\frac{1}{8}$$ pound = ________ ounces Answer: $$\frac{1}{8}$$ pound = ____2____ ounces. Explanation: Conversion: 1 pound = 16 ounces. $$\frac{1}{8}$$ pound = 16 × $$\frac{1}{8}$$ = 2 ounces. Question 12. 6 tons = _______ pounds Answer: 6 tons = ___12000______ pounds. Explanation: Conversion: 1 ton = 2000 pounds. 6 tons = 2000 × 6 = 12000 pounds. Question 13. 3 pounds = _______ ounces Answer: 3 pounds = __48_____ ounces. Explanation: Conversion: 1 pound = 16 ounces. 3 pounds = 16 × 3 = 48 ounces. For 14-15, complete each table. Question 14. Answer: Explanation: Conversion: 1 ton = 2000 pounds. 2 tons = 2000 × 2 = 4000 pounds. 3 tons = 2000 × 3 = 6000 pounds. Question 15. Answer: Explanation: Conversion: 1 pound = 16 ounces. $$\frac{2}{4}$$ pound = 16 × $$\frac{2}{4}$$ = 8 ounces. 1 pound = 16 ounces. 2 pounds = 16 × 2 = 32 ounces. Problem Solving For 16-19, use the line plot at the right. Question 16. Be Precise What is the total weight in ounces of the three kittens that weight the least? Answer: Total weight in ounces of the three kittens that weight the least = 40 ounces. Explanation: Weight of Three kittens that weight the least = $$\frac{3}{4}$$ pound + $$\frac{3}{4}$$ pound + 1 pound. = 2$$\frac{2}{4}$$ pound. Conversion: 1 pound = 16 ounces. Total weight in ounces of the three kittens that weight the least = 2$$\frac{2}{4}$$ pound = 16 × 2$$\frac{2}{4}$$ = 40 ounces. Question 17. Higher Order Thinking Two kittens had a total weight of 3$$\frac{1}{4}$$ pounds. What could their individual weights have been? Answer: Their individual weights could be = 1$$\frac{2}{4}$$ pounds + 1$$\frac{3}{4}$$ pounds. Explanation: Weight of two kitten had a total weight of 3$$\frac{1}{4}$$ pounds: Their individual weights could be = 1$$\frac{2}{4}$$ pounds + 1$$\frac{3}{4}$$ pounds. = 3$$\frac{1}{4}$$ pounds. Question 18. Algebra How many more pounds did the heaviest kitten weigh than the lightest kitten? Answer: 1$$\frac{2}{4}$$ more pounds the heaviest kitten weights than the lightest kitten, Explanation: Weight of the heaviest kitten = 2$$\frac{1}{4}$$ pounds. Weight of the lightest kitten = $$\frac{3}{4}$$ pounds. Difference: Weight of the heaviest kitten – Weight of the lightest kitten = 2$$\frac{1}{4}$$ pounds – $$\frac{3}{4}$$ pounds = 1$$\frac{2}{4}$$ pounds. Question 19. Each of the greatest number of kittens weighed how many pounds? Answer: 1$$\frac{1}{4}$$ pounds is the greatest number of kittens weighed of 3$$\frac{3}{4}$$ pounds. Explanation: Greatest number of kittens weighed = 3 × 1$$\frac{1}{4}$$ pounds = 3$$\frac{3}{4}$$ pounds. Question 20. About how many pounds does this African elephant weigh? Complete the table to solve. Answer: This African elephant weights = 10000 pounds. Explanation: This African elephant weights = 5 tons. Conversion: 1 ton = 2000 pounds. => 5 tons = 2000 × 5 = 10000 pounds. Table : $$\frac{2}{4}$$ ton = 2000 × $$\frac{2}{4}$$ = 1000 pounds. 1 ton = 2000 pounds. 2 ton = 2000 × 2 = 4000 pounds. 3 ton = 2000 × 3 = 6000 pounds. 4 ton = 2000 × 4 = 8000 pounds. 5 ton = 2000 × 5 = 10000 pounds. Assessment Practice Question 21. Which is most likely to weigh 3 ounces? A. A shoe B. A large spider C. A box of cerea! D. A loaded pick-up truck Answer: A shoe is most likely to weigh 3 ounces. A. A shoe Explanation: A. A shoe – 9.5 ounces. B. A large spider – 5300 ounces. C. A box of cereal – 12 ounces to 18 ounces. D. A loaded pick-up truck – 64000 ounces. Question 22. Which comparison is true? A. 7,000 pounds < 3 tons B. 5 pounds > 85 ounces C. 50 ounces > 3 pounds D. 4 pounds < 60 ounces Answer: The following below comparison is true: C. 50 ounces > 3 pounds. Explanation: A. 7,000 pounds < 3 tons (False) Conversion: 1 ton = 2000 pounds. => 3 tons = 2000 × 3 = 6000 pounds. B. 5 pounds > 85 ounces (False) Conversion: 1 pound = 16 ounces. => 5 pounds = 16 × 5 = 80 ounces. C. 50 ounces > 3 pounds (True) => 3 pounds = 16 × 3 = 48 ounces. D. 4 pounds < 60 ounces (False) => 4 pounds = 16 × 4 = 64 ounces. ### Lesson 13.4 Equivalence with Metric Units of Length Solve & Share Find the length of the marker shown in both centimeters and millimeters. Describe the relationship between the two units. I can …convert metric units of length from one unit to another and recognize the relative size of different units. Look Back! The length of Toby’s giant pencil is 25 centimeters. How could you find the length of his pencil in millimeters? Answer: We can find the length of his pencil in millimeter by using conversion of units into another required unit. Length of his pencil in millimeters = 250 millimeters. Explanation: Length of Toby’s giant pencil = 25 centimeters. Conversion: 1 centimeter = 10 millimeters. Length of his pencil in millimeters = 25 centimeters = 10 × 25 = 250 millimeters. Essential Question How Can You Convert from One Metric Unit of Length to Another? Answer: To convert from one unit to another within the metric system usually means moving a decimal point. If you can remember what the prefixes mean, you can convert within the metric system relatively easily by simply multiplying or dividing the number by the value of the prefix Visual Learning Bridge For the standing long jump, Corey jumped 2 meters and Gary jumped 175 centimeters. Who jumped farther? Step 1 Convert 2 meters to centimeters. 1 meter = 100 centimeters Corey jumped 200 centimeters. Step 2 Compare the lengths jumped. 200 centimeters is greater than 175 centimeters. Corey jumped farther than Gary Convince Me! Critique Reasoning Shayla says 5 kilometers are equal to 500 meters. Do you agree? Explain. Answer: Yes, I agree with Shayla saying ” 5 kilometers are equal to 500 meters.” Explanation: Conversion: 1 kilometer = 1000 meters. 5 kilometers = 1000 × 5 = 5000 meters. Another Example! Kendra and Lili measured the length of several rosebuds. The longest one Kendra measured was 2.4 centimeters long. The longest one Lili measured was 1.8 centimeters. Who measured the longest rosebud? Use a linear model to help explain. Since 2.4 cm > 1.8 cm, Kendra measured the longest rosebud. Guided Practice Do You Understand? Question 1. What metric unit would you use to measure the length of a field? Answer: The most common units that we use to measure length of a field in the metric system are the millimeter, centimeter, meter, and kilometer. Do You Know How? For 2-3, convert each unit. Question 2. 5 kilometers = _______ meters Answer: 5 kilometers = __5000_____ meters. Explanation: Conversion: 1 kilometer = 1000 meters. 5 kilometers = 1000 × 5 = 5000 meters. Question 3. 75 centimeters = ________ millimeters Answer: 75 centimeters = _____750_______ millimeters. Explanation: Conversion: 1 centimeters = 10 millimeters. 75 centimeters = 10 × 75 = 750 millimeters. Independent Practice In 4-6, tell what metric unit you would use to measure each. Question 4. The length of your math book Answer: Metric unit used to measure the length of your math book centimeter (cm). Question 5. The distance between cities Answer: Metric unit used to measure the distance between cities is Kilometers per hour. Question 6. The length of a fly Answer: Metric unit used to measure the length of a fly is meter. For 7-8, complete each table. Question 7. Answer: Explanation: Conversion: 1 meter = 1000 millimeter. 2 meter = 2 × 1000 = 2000 millimeter. 3 meter = 3 × 1000 = 3000 millimeter. Question 8. Answer: Explanation: Conversion: 1 centimeter = 10 millimeter. 2 centimeter = 10 × 2 = 20 millimeter. 3 centimeter = 10 × 3 = 30 millimeter. Problem Solving For 9-10, use the table at the right. Question 9. Be Precise The table shows the amount of rainfall students measured for a week. What was the total rainfall for the week, in millimeters? Answer: Total rainfall for the week, in millimeters = 110 millimeters. Explanation: Amount of rainfall on Monday = 3 cm. Amount of rainfall on Tuesday = 0 cm. Amount of rainfall on Wednesday = 1 cm. Amount of rainfall on Thursday = 5 cm. Amount of rainfall on Friday = 2 cm. Total rainfall for the week = Amount of rainfall on Monday + Amount of rainfall on Tuesday + Amount of rainfall on Wednesday + Amount of rainfall on Thursday + Amount of rainfall on Friday = 3 cm + 0 cm + 1 cm + 5 cm + 2 cm = 11 cm. Conversion: 1 cm = 10 mm. Total rainfall for the week, in millimeters = Total rainfall for the week × 10 => 11 × 10 = 110 millimeters. Question 10. How many more millimeters of rain fell on Thursday than on Monday and Wednesday combined? Answer: 10 more millimeters of rain fell on Thursday than on Monday and Wednesday combined. Explanation: Amount of rainfall on Monday = 3 cm. Amount of rainfall on Wednesday = 1 cm. Amount of rainfall on Thursday = 5 cm. Combine Rain fell on Monday and Wednesday = Amount of rainfall on Monday + Amount of rainfall on Wednesday = 3 cm + 1 cm = 4 cm. Difference: Amount of rainfall on Thursday – Combine Rain fell on Monday and Wednesday = 5 cm – 4 cm = 1 cm. Conversion: 1 centimeter = 10 millimeter. Difference in millimeter of rain fell on Thursday than on Monday and Wednesday combined = 1 centimeter = 10 millimeter. Question 11. Which is greater, 2,670 meters or 2 kilometers? Explain. Answer: 2,670 meters is greater. Explanation: 2,670 meters or 2 kilometers. Conversion: 1 kilometer = 1000 meters. 2 kilometers = 1000 × 2 = 2000 meters. Question 12. Critique Reasoning Milo thinks 8 hours is greater than 520 minutes. Is Milo correct? Remember 1 hour is equal to 60 minutes. Answer: No, Milo is incorrect because 8 hours = 480 minutes less than 520 minutes. Explanation: Conversion: 1 Hour = 60 minutes. 8 hours = 60 × 8 = 480 minutes. Question 13. Algebra Leah ran around the track 8 times. She ran a total of 2,000 meters. How many meters equal 1 lap? Use the bar diagram to write an equation which can be used to find m, the meters in one lap. Answer: 1 Lap = 400 meters. Explanation: Number of meters She ran = 2000m. Number of times Algebra Leah ran around the track = 8. Number of meters equal 1 lap = Number of meters She ran ÷ Number of times Algebra Leah ran around the track = 2000 ÷ 8 = 400 meters. 1 Lap = 400 meters Question 14. Higher Order Thinking Signs are placed at the beginning and at the end of a 3-kilometer hiking trail. Signs are also placed every 500 meters along the trail. How many signs are along the trail? Explain. Answer: Total signs along the trail = 7. Explanation: Distance of Signs are placed at the beginning and at the end of a hiking trail = 3 kilometer. Signs placed along the trail = 500 meters. Conversion: 1 km = 1000 meters. 3 km = 1000 × 3 = 3000 meters. Total signs along the trail = Distance of Signs are placed at the beginning and at the end of a hiking trail ÷ Signs placed along the trail + 1 = 3000 ÷ 500 + 1 = 6 + 1 = 7. Assessment Practice Question 15. Select all the true statements. ☐ 14 meters = 1,400 centimeters ☐ 10 centimeters = 1,000 millimeters ☐ 55 kilometers = 5,500 meters ☐ 3 meters = 3,000 millimeters ☐ 5 meters = 500 centimeters Answer: All the true statements: 14 meters = 1,400 centimeters 55 kilometers = 5,500 meters 3 meters = 3,000 millimeters 5 meters = 500 centimeters Explanation: ☐ 14 meters = 1,400 centimeters. (True) Conversion: 1 meters = 100 centimeters. 14 meters = 10 × 14 = 1400 centimeters. ☐ 10 centimeters = 1,000 millimeters. (False) Conversion: 1 centimeters = 10 millimeters. 10 centimeters = 10 × 10 = 100 millimeters. ☐ 55 kilometers = 5,500 meters. (True) Conversion: 1 kilometers = 1000 meters. 55 kilometers = 1000 × 55 = 5500 meters. ☐ 3 meters = 3,000 millimeters. (True) Conversion: 1 meter = 1000 millimeters. 3 meters = 1000 × 3 = 3000 millimeters. ☐ 5 meters = 500 centimeters. (True) Conversion: 1 meter = 100 centimeters. 5 meters = 100 × 5 = 500 centimeters. Question 16. Select all the true statements. ☐ 3 meters > 3,000 centimeters ☐ 2 kilometers < 2,500 meters ☐ 4 centimeters > 38 millimeters ☐ 3.5 meters < 3.2 meters ☐ 5 kilometers < 5,200 meters Answer: All the true statements: 2 kilometers < 2,500 meters. 4 centimeters > 38 millimeters. 5 kilometers < 5,200 meters. Explanation: ☐ 3 meters > 3,000 centimeters. (False) Conversion: 1 meter = 100 centimeters. 3 meters = 100 × 3 = 300 centimeters. ☐ 2 kilometers < 2,500 meters . (True) Conversion: 1 kilometers = 1000 meters. 2 kilometers = 1000 × 2 = 2000 meters. ☐ 4 centimeters > 38 millimeters. (True) Conversion: 1 centimeters = 10 millimeters. 4 centimeters = 10 × 4 = 40 millimeters. ☐ 3.5 meters < 3.2 meters. ( False) ☐ 5 kilometers < 5,200 meters. (True) Conversion: 1 kilometers = 1000 meters. 5 kilometers = 1000 × 5 = 5000 meters. ### Lesson 13.5 Equivalence with Metric Units of Capacity and Mass Solve & Share Jenny has 3 liters of water. How many milliliters of water does she have, and what is the mass of the water in grams? Solve this problem any way you choose. I can… convert metric units of capacity and mass from one unit to another and recognize the relative size of different units. Look Back! Why did you need to convert units to solve the problem above? Answer: Well, it is the conversion between an amount in one unit to the corresponding amount in a desired unit using various conversion factors. This is valuable because certain measurements are more accurate or easier to find than others. Essential Question How Can You Convert from One Metric Unit of Capacity or Mass to Another? Answer: We convert from One Metric Unit of Capacity or Mass to Another, larger unit multiply, smaller unit divide. Visual Learning Bridge Louis needs 8 liters of apple juice. He has 5,000 milliliters of juice. Does Louis have enough apple juice? Step 1 Find how many milliliters of apple juice Louis needs. 1 liter = 1,000 milliliters Louis needs 8,000 milliliters of juice. Step 2 Compare to find if Louis has enough apple juice. 8,000 milliliters > 5,000 milliliters Louis does not have enough apple juice. How much more does he need? 8,000 – 5,000 = 3,000 Louis needs 3,000 milliliters more. Convince Me! Be Precise Why did you need to convert liters to milliliters? Answer: We need to convert liters to milliliters because Louis needs milliliters of juice not liters. Another Example! How many grams of apples are needed to make 1 liter of apple juice? 1 kilogram = 1,000 grams 2 kilograms = 2 × 1,000 grams = 2,000 grams 2,000 grams of apples make 1 liter of apple juice. Guided Practice Do You Understand? Question 1. What metric unit would you use to measure your mass? the amount of blood in your body? Explain. Answer: The gram and kilogram are two units used to measure mass in the metric system. the amount of blood in your body is measured in units of millimeters of mercury (mmHg). Do You Know How? For 2-3, convert each unit. Question 2. 6 grams = _______ milligrams Answer: 6 grams = ___6000____ milligrams. Explanation: Conversion: 1 gram = 1000 milligrams. 6 grams = 1000 × 6 = 6000 milligrams. Question 3. 9 liters = _________ milliliters Answer: 9 liters = ___9000______ milliliters. Explanation: Conversion: 1 gram = 1000 milligrams. 9 liters = 1000 × 9 = 9000 milliliters. Independent Practice In 4-6, tell what metric unit you would use to measure each. Question 4. Medicine in a pill Answer: Orally administered liquid medications should be dosed exclusively by using metric-based dosing with milliliters (ie, mL) to avoid confusion and dosing errors associated with common kitchen spoons Question 5. Ink in a pen Answer: Metric unit used to measure Ink in a pen is grams. Question 6. The mass of a pencil Answer: Metric unit used to measure the mass of a pencil is grams. For 7-10, convert each unit. Question 7. 5 kilograms = ________ grams Answer: 5 kilograms = ___5000_____ grams. Explanation: Conversion: 1 kilograms = 1000 grams. 5 kilograms = 1000 × 5 = 5000 grams. Question 8. 2 liters = _______ milliliters Answer: 2 liters = ___2000______ milliliters. Explanation: Conversion: 1 liter = 1000 milliliters. 2 liters = 1000 × 2 = 2000 milliliters. Question 9. 4 grams = _______ milligrams Answer: 4 grams = _____4000______ milligrams. Explanation: Conversion: 1 gram = 1000 milligrams. 4 grams = 1000 × 4 = 4000 milligrams. Question 10. 9 kilograms = ________ grams Answer: 9 kilograms = _____9000______ grams. Explanation: Conversion: 1 kilogram = 1000 grams. 9 kilograms = 1000 × 9 = 9000 grams. Problem Solving Question 11. Reasoning A cardboard box has a mass of 800 grams. When 4 books of equal mass are put into the box, the filled box has a mass of 8 kilograms. What is the mass of each book in grams? Explain. Answer: The mass of each book in grams = 1800 grams. Explanation: Mass of a cardboard box has = 800 grams. 4 books of equal mass are put into the box, Mass of the filled box has = 8 kilograms. Conversion: 1 kilogram = 1000 grams. => 8 kilograms = 1000 × 8 = 8000 grams. The mass of each book in grams = Mass of the filled box has – Mass of a cardboard box has ÷ 4 = 8000 grams – 800 grams ÷ 4 = 7200 grams ÷ 4 = 1800 grams. Question 12. en Vision® STEM The Cape Hatteras Lighthouse was a kilometer from the shore in 1870. How far was the lighthouse from the shore in 1970? Explain. Answer: Distance of the lighthouse away from the shore in 1970 = 920 meters. Explanation: Distance of the Cape Hatteras Lighthouse from the shore in 1870 = 1 kilometer. Conversion: 1 kilometer = 1000 meters. Distance of the beach near the Cape Hatteras Lighthouse eroded = 8 meters each year. Distance of the lighthouse away from the shore in 1970 = 1 kilometer – (Distance of the beach near the Cape Hatteras Lighthouse eroded × number of years from 1870 – 1970) =1000 – (8 × 10) = 1000 – 80 = 920 meters. Question 13. The mass of 4 large zucchini is about 2 kilograms. About how many grams will 1 large zucchini have? Answer: Number of grams will 1 large zucchini = 500 grams. Explanation: Mass of 4 large zucchini = 2 kilograms. Conversion: 1 kilogram = 1000 grams. => 2 kilograms = 1000 × 2 = 2000 grams. Number of zucchini = 4. Number of grams will 1 large zucchini = Mass of 4 large zucchini ÷ Number of zucchini = 2000 ÷ 4 = 500 grams. Question 14. Higher Order Thinking A small sofa has a mass of 30 kilograms. A pillow on the sofa has a mass of 300 grams. How many pillows would it take to equal the mass of the sofa? Answer: Number of pillows it take to equal the mass of the sofa = 100. Explanation: Mass of small sofa = 30 kilograms. Conversion: 1 kilograms = 1000 grams. => 30 kilograms = 1000 × 30 = 30000 grams. Mass of a pillow on the sofa = 300 grams. Number of pillows it take to equal the mass of the sofa = Mass of small sofa ÷ Mass of a pillow on the sofa = 30000 ÷ 300 = 100 . Assessment Practice Question 15. Which shows a correct comparison? A. 5 milliliters > 50 liters B. 2 liters < 200 milliliters C. 100 liters < 1,000 milliliters D. 3,200 milliliters > 3 liters Answer: A correct comparison: D. 3,200 milliliters > 3 liters Explanation: A. 5 milliliters > 50 liters. (False) Conversion: 1 liter = 1000 milliliters. => 50 liters = 1000 × 50 = 50000 liters. B. 2 liters < 200 milliliters. (False) Conversion: 1 liter = 1000 milliliters. => 20 liters = 1000 × 2 = 2000 milliliters. C. 100 liters < 1,000 milliliters. (False) Conversion: 1 liter = 1000 milliliters. => 100 liters = 1000 × 100 = 100000 milliliters. D. 3,200 milliliters > 3 liters. (True) Conversion: 1 liter = 1000 milliliters. => 3 liters = 1000 × 3 = 3000 milliliters. Question 16. Write the missing numbers in the table. Answer: Conversion: 1 kilograms = 1000 grams. 2 kilograms = 1000 × 2 = 2000 grams. 3 kilograms = 1000 × ?? = 3000 grams. => 3000 ÷ 1000 = 3. 4 kilograms = 1000 × 4 = 4000 grams. ### Lesson 13.6 Solve Perimeter and Area Problems Solve & Share A can of paint is used to cover all 168 square feet of a wall. The wall is 8 feet high. Tape is placed along the top, bottom, and sides of the wall. What is the width of the wall? How much tape is needed? Solve this problem any way you choose. I can … find the unknown length or width of a rectangle using a known area or perimeter. Look Back! Describe the steps you would use to solve the problem. Answer: The steps used to solve the problem: 1. Collect the given complete information of the problem. 2. Find the width of the wall. 3. Find out the perimeter of the wall to find the how much tape to be needed. . Essential Question How Can You Use Perimeter and Area to Solve Problems? Answer: The perimeter is the distance around the outside of an object. The perimeter is found by adding the lengths of all of the sides of the object together. Area is the size of the surface of the object or the total amount of space that the object covers. Visual Learning Bridge The state park shown has a perimeter of 36 miles. What is the area of the state park? Step 1 Find the length of the state park. Use the perimeter, 36 miles, and the width, 7 miles, to find the length. Opposite sides of a rectangle are the same length, so multiply the width by 2. 7 × 2 = 14 Subtract 14 from the perimeter. 36 – 14 = 22 22 miles is the length of two sides of the park. Divide 22 by 2 to find the length of one side. 22 =2= 11 The length of the park is 11 miles. Step 2 Find the area of the state park. w = 7 miles l = 11 miles A = l × w = 11 × 7 = 77 The area of the state park is 77 square miles, Convince Me! Make Sense and Persevere If the area of another state park is 216 square miles, and the park has a width of 8 miles, what is the park’s length? What is the perimeter of this state park? Answer: Length of another state park = 27 miles. Perimeter of this state park = 70 miles. Explanation: Area of the another state park = 216 square miles. Width of another state park = 8 miles. Length of another state park = L miles. Area of the another state park = L × W = 216 square miles. => L × 8 = 216 => L = 216 ÷ 8 => L = 27 miles. Perimeter of this state park = 2(L + W) => 2(27 + 8) => 2 × 35 = 70 miles. Guided Practice Do You Understand? Question 1. A sandbox is shaped like a rectangle. The area is 16 square feet. The side lengths are whole numbers. What are the possible dimensions of the sandbox? Do all possible dimensions make sense? Answer: The possible dimensions of the sandbox = (8miles and 2 miles) or (4miles and 4 miles). Explanation: Area of sandbox = 16 square feet. The side lengths are whole numbers. Let the Length of sandbox be L. Width of sandbox be W. Area of sandbox = L × W => 16 = L × W => 16 ÷ L = W. or 16 ÷ W = L. Length of sandbox and Width of sandbox dimensions can be 8 miles and 2 miles or 4 miles or 4 miles. Question 2. Write and solve an equation to find the width of a room if the length of the floor is 8 feet and the area of the room is 96 square feet. Answer: Width of the room = 12 feet. Explanation: Area of the room = 96 square feet. Length of the floor = 8 feet. Let Width of the room = W feet. Area of the room = L × W = 96 square feet. => 8 × W = 96 => W = 96 ÷ 8 => W = 12 feet. Do You Know How? For 3-5, complete each calculation. Question 3. Find n. Perimeter = 46 in. Answer: Width of the rectangle = 15 inches. Explanation: Perimeter of the rectangle = 46 inches. Length of the rectangle = 8 inches. Width of the rectangle = n inches. Perimeter of the rectangle = 2(L + W) = 46 inches. =>2(8 + n) = 46 => 16 + 2n = 46 => 2n = 46 – 16 => n = 30 ÷ 2 => n = 15 inches. Question 4. Find n and A. Perimeter = 26 cm Answer: Width of the rectangle = 4 cm. Area of the rectangle = 36 sq cm. Explanation: Perimeter of the rectangle = 26 cm. Length of the rectangle = 9 cm. Width of the rectangle = n cm. Perimeter of the rectangle = 2(L + W) = 26 cm. => 2(9 + n) = 26 => 18 + 2n = 26 => 2n = 26 – 18 => n = 8 ÷ 2 => n = 4 cm. Area of the rectangle = L × w = 9 × 4 = 36 square cm. Question 5. Find the perimeter. Answer: Perimeter of the square = 22 yards. Explanation: Side of the square = 5$$\frac{2}{4}$$ yards. Perimeter of the square = 4 × Side of square = 4 × 5$$\frac{2}{4}$$ = 22 yards. Independent Practice For 6-9, find the missing dimension. Question 6. Find n. Answer: Width of the rectangle = 10 ft. Explanation: Area of the rectangle = 60 sq ft. Length of the rectangle = 6 ft. Width of the rectangle = n ft. Area of the rectangle = L × W = 60 sq ft => 6 × n = 60 => n = 60 ÷ 6 => n = 10 ft. Question 7. Find n. Perimeter = 65 in. Answer: Width of the rectangle = 21 inches. Explanation: Perimeter of the rectangle = 65 inches. Length of the rectangle = 11$$\frac{2}{4}$$ inches. Width of the rectangle = n inches. Perimeter of the rectangle = 2(L + W) = 65 inches. => 2 (11$$\frac{2}{4}$$ + n) = 65 => 23 + 2n = 65 => 2n = 65 – 23 => n = 42 ÷ 2 => n = 21 inches. Question 8. Find n. Perimeter = 84 yd Answer: Width of the rectangle = 20 yd. Explanation: Perimeter of the rectangle = 84 yd. Length of the rectangle = 22 yd. Width of the rectangle = n yd. Perimeter of the rectangle = 2(L + W) = 84 yd. => 2 (22 + n) = 84 => 44 + 2n = 84 => 2n = 84- 44 => n = 40 ÷ 2 => n = 20 yd. Question 9. A rectangle has a length of 9 millimeters and an area of 270 square millimeters. What is the width? What is the perimeter? Answer: Width of the rectangle = 30 millimeters. Perimeter of the rectangle = 78 millimeters. Explanation: Length of the rectangle = 9 millimeters. Area of the rectangle = 270 square millimeters. Width of the rectangle = w millimeters. Area of the rectangle = L × W = 270 square millimeters. => 9 × w = 270 => w = 270 ÷ 9 => w = 30 millimeters. Perimeter of the rectangle = 2(L+ W) = 2(9 + 30) = 2 × 39 = 78 millimeters. Problem Solving Question 10. Greg built the picture frame shown to the right. It has a perimeter of 50$$\frac{2}{4}$$ inches. How wide is the picture frame? Answer: Width of the picture frame = 10 inches. Explanation: Length of the picture frame = 15$$\frac{1}{4}$$ inches. Perimeter of the picture frame = 50$$\frac{2}{4}$$ inches. Let Width of the picture frame be w inches. Perimeter of the picture frame = 2(L + W) = 50$$\frac{2}{4}$$ inches. => 2(15$$\frac{1}{4}$$ + w) = 50$$\frac{2}{4}$$ => 30$$\frac{2}{4}$$ + 2w = 50$$\frac{2}{4}$$ => 2w = 50$$\frac{2}{4}$$ – 30$$\frac{2}{4}$$ => w = 20 ÷ 2 => w = 10 inches. Question 11. Greg covered the back of the picture with a piece of felt. The picture is 17 inches shorter than the frame and 1$$\frac{1}{4}$$ inch less in width. What is the area of the felt? Answer: Area of the felt = 70 inches. Explanation: Length of the picture frame = 15$$\frac{1}{4}$$ inches. Width of the picture frame = 10 inches. Perimeter of the picture frame = 50$$\frac{2}{4}$$ inches. The picture is 17 inches shorter than the frame and 1$$\frac{1}{4}$$ inch less in width. => Width of the felt = Width of the picture frame – 1$$\frac{1}{4}$$ inch => 10 – 1$$\frac{1}{4}$$ => 8$$\frac{3}{4}$$ inches. => Perimeter of the picture frame – 17 inches = Perimeter of the felt. => 50$$\frac{2}{4}$$ – 17 = Perimeter of the picture. => 33$$\frac{2}{4}$$ inches = Perimeter of the picture. Length of the felt = L inches. Perimeter of the felt = 2(L + W) 33$$\frac{2}{4}$$ = 2(L + 8$$\frac{3}{4}$$ ) 33$$\frac{2}{4}$$ = 2L + 17$$\frac{2}{4}$$ 33$$\frac{2}{4}$$ – 17$$\frac{2}{4}$$ = 2L 16 ÷ 2 = L 8 inches = L. Area of the felt = L × W = 8 × 8$$\frac{3}{4}$$ = 70 inches. Question 12. Al has a goal to read 2,000 pages over summer break. He has read 1,248 pages. How many more pages does Al need to read to reach his goal? Answer: Number of more pages Al needs to read to reach his goal = 752. Explanation: Number of pages Al has a goal to read = 2000. Number of pages he has read = 1248. Number of more pages Al needs to read to reach his goal = Number of pages Al has a goal to read – Number of pages he has read = 2000 – 1248 = 752. Question 13. The area of a tabletop is 18 square feet. The perimeter of the same table is 18 feet. What are the dimensions of the tabletop? Answer: Dimensions of the tabletop (L and W) are: (9 feet and 2 feet) or (6 feet and 3 feet) or (2 feet and 9 feet) or (3 feet and 6 feet) Explanation: Area of a tabletop = 18 square feet. Perimeter of a tabletop = 18 feet. Area of a tabletop = L × W = 18 sq feet. => L = 18 ÷ W or W = 18 ÷ L. Dimensions of the tabletop are: (9 feet and 2 feet) or (6 feet and 3 feet) or (2 feet and 9 feet) or (3 feet and 6 feet) Question 14. Construct Arguments Amy and Zach each have 24 feet of fencing for their rectangular gardens. Amy makes her fence 6 feet long. Zach makes his fence 8 feet long. Whose garden has the greater area? How much greater? Explain. Answer: Amy garden has the greater area than Zach garden by 4 sq ft. Explanation: Perimeter of the rectangular garden fencing Amy and Zach each have = 24 feet. Length of rectangular garden fence Zach makes = 8 feet. Length of rectangular garden fence Amy makes = 6 feet. Width of rectangular garden fence Amy makes = W feet. Perimeter of the rectangular garden Amy makes = 2(L + W) = 24 feet. => 2(6 + W) = 24 => 12 + 2W = 24 => 2W = 24 – 12 => W = 12 ÷ 2 => W = 6 feet. Area of the rectangular garden Amy makes = Length of rectangular garden fence Amy makes × Width of rectangular garden fence Amy makes = 6 × 6 = 36 sq ft. Length of rectangular garden fence Zach makes = 8 feet. Width of rectangular garden fence Zach makes = w feet. Perimeter of the rectangular garden Zach makes = 2(L + W) = 24 feet. => 2(8 + w) = 24 => 16 + 2w = 24 => 2w = 24 – 16 => w = 8 ÷ 2 => w = 4 feet. Area of the rectangular garden Zach makes = Length of rectangular garden fence Zach makes × Width of rectangular garden fence Zach makes = 8 × 4 = 32 sq ft. Difference in areas of Zach and Amy rectangular gardens : Area of the rectangular garden Amy makes – Area of the rectangular garden Zach makes = 36 sq ft – 32 sq ft = 4 sq ft. Question 15. Higher Order Thinking Nancy made a table runner that has an area of 80 square inches. The length and width of the table runner are whole numbers. The length is 5 times greater than the width. What are the dimensions of the table runner? Answer: The dimensions of the table runner: Length of the table = 40 inches. Width of the table = 8 inches. Explanation: Area of the table runner which Nancy made = 80 square inches. The length is 5 times greater than the width. Let the Length of the table be L inches. =>L = 5W Width of the table be W inches. Area of the table runner = Length of the table × Width of the table = 80 square inches. => 5W × W = 80 => 2W = 80 ÷ 5 => 2W = 16 => W = 16 ÷ 2 => W = 8 inches. Length of the table = 5W = 5 × 8 = 40 inches. Assessment Practice Question 16. The rectangle has an area of 144 square centimeters. Which is its perimeter? A. 26 cm B. 48 cm C. 52 cm D. 72 cm Answer: Perimeter of the rectangle = 52 cm. C. 52 cm Explanation: Area of the rectangle = 144 square centimeters. Length of the rectangle = 8 cm. Width of the rectangle = W cm Area of the rectangle = L × W = 144 square centimeters. => 8 × W = 144 => W = 144 ÷ 8 => W = 18 cm. Perimeter of the rectangle = 2(L + W ) = 2 (8 + 18) = 2 × 26 = 52 cm. ### Lesson 13.7 Problem Solving Precision Solve & Share Mr. Beasley’s science class wants to decorate one wall in the classroom like an underwater scene. They use sheets of blue poster board that are 2 feet long and 2 feet wide. How many sheets of blue poster board are used to cover the entire area of the wall? Use math words and symbols to explain how you solve. I can … be precise when solving math problems. Thinking Habits Be a good thinker! These questions can help you. • Am I using numbers, units, and symbols appropriately? • Am I using the correct definitions? • Am I calculating accurately? • Is my answer clear? Look Back! Be Precise How can calculating the area of the whole wall and the area of one sheet of poster board help you determine the total number of sheets of poster board needed to cover the entire area of the wall? Answer: The wall is to be covered by the sheets so, it is required for calculating the area of the whole wall and the area of one sheet of poster board help you determine the total number of sheets of poster board needed to cover the entire area of the wall. Essential Question How Can You Be Precise When Solving Math Problems? Answer: We can Precise When Solving Math Problems by the following the below steps: 1. Teach with Intention. 2. Use Practice Sheets Appropriately. 3. Use Best Practice Math Strategies and Instruction. 4. Use Problems with Multiple Solutions. 5. Differentiation is Critical. Visual Learning Bridge Piper has a fish tank and wants to cover all four sides $$\frac{6}{10}$$ of the way to the top with clear plastic for insulation. She measures and finds the dimensions shown. How much plastic does Piper need? Use math words and symbols to explain how to solve. What do you need to know so you can solve the problem? I need to find how much plastic is needed for the fish tank. I need to be precise in my calculations and explanation. How can I be precise in solving this problem? I can • correctly use the information given. • calculate accurately. • decide if my answer is clear and appropriate. • use the correct units. Here’s my thinking. The height of the plastic is $$\frac{6}{10}$$ times 15 inches. $$\frac{6}{10}$$ × 15 = $$\frac{90}{10}$$ or 9 The plastic is 9 inches high. Front and back: A = 9 × 24 A = 216 square inches Each side: A = 9 × 12 A = 108 square inches Add: 216 + 216 + 108 + 108 = 648 square inches Piper needs 648 square inches of plastic. Convince Me! Be Precise How did you use math words and numbers to make your explanation clear? Answer: To justify a solution, we need to be able to use appropriate mathematical language to give reasons for the particular approach used to solve a problem. Any time we produce a ‘solution’ in an attempt to solve a problem, that ‘solution’ needs to be justified. Guided Practice Be Precise Jeremy uses $$\frac{2}{3}$$ yard of tape for each box he packs for shipping. How many inches of tape does Jeremy need to pack 3 boxes? Question 1. How can you use the information given to solve the problem? Answer: We can use the information given to solve the problem for : • Defining the problem. • Generating alternatives. • Evaluating and selecting alternatives. • Implementing solutions. Question 2. How many inches of tape does Jeremy need to pack 3 boxes? Explain. Answer: Number of inches of tape Jeremy needs to pack 3 boxes = 72 inches. Explanation: Jeremy uses $$\frac{2}{3}$$ yard of tape for each box he packs for shipping. How many inches of tape does Jeremy need to pack 3 boxes? Length of the tape Jeremy uses to tape each box he packs for shipping = $$\frac{2}{3}$$ yard. Number of tapes Jeremy needs to pack 3 boxes = Length of the tape Jeremy uses to tape each box he packs for shipping × 3 = $$\frac{2}{3}$$ × 3 = 2 yards. Conversion: 1 yard = 36 inches. Number of inches of tape Jeremy needs to pack 3 boxes = Number of tapes Jeremy needs to pack 3 boxes × 36 inches = 2 × 36 = 72 inches. Question 3. Explain why you used the units you did in your answer. Answer: Units are important because without proper measurement and units to express them, we can never express physical laws precisely just from qualitative reasoning. Independent Practice Be Precise Mrs. Reed collects shells. Each shell in her collection weighs about 4 ounces. Her collection weighs about 12 pounds in all. About how many shells are in Mrs. Reed’s collection? Use Exercises 4-6 to solve. Question 4. How can you use the information given to solve the problem? Answer: We can use the information given to solve the problem for : • Defining the problem. • Generating alternatives. • Evaluating and selecting alternatives. • Implementing solutions. Question 5. What is the total weight of Mrs. Reed’s shell collection, in ounces? Answer: Total weight of her collection in ounces = 192 ounces. Explanation: Weight of each shell in Mrs. Reed’s collection = 4 ounces. Total weight of her collection = 12 pounds. Conversion : 1 pound = 16 ounces. Total weight of her collection in ounces = 12 pounds = 16 × 12 = 192 ounces. Question 6. How many shells are in Mrs. Reed’s shell collection? Answer: 48 shells are in Mrs. Reed’s shell collection. Explanation: Weight of each shell in Mrs. Reed’s collection = 4 ounces. Total weight of her collection = 12 pounds. Conversion : 1 pound = 16 ounces. Total weight of her collection in ounces = 12 pounds = 16 × 12 = 192 ounces. Number of shells are in Mrs. Reed’s collection = Total weight of her collection ÷ Weight of each shell in Mrs. Reed’s collection = 192 ÷ 4 = 48. Problem Solving Performance Task Making Thank You Cards Tanesha is making cards by gluing 1 ounce of glitter on the front of the card and then making a border out of ribbon. She makes each card the dimensions shown. How much ribbon does Tanesha need? Question 7. Reasoning What quantities are given in the problem and what do the numbers mean? Answer: Quantities are given in the problem: Quantity of glitter used for making Thank You Cards by Tanesha = 1 ounce. Length of the card = 9 cm. Width of the card = 85 mm. Number mean the amount of quantity measured and measurements of the cards. Question 8. Reasoning What do you need to find? Answer: We need to find how much quantity of ribbon Tanesha needs. Question 9. Model with Math What are the hidden questions that must be answered to solve the problem? Write equations to show how to solve the hidden questions. Answer: The hidden questions that must be answered to solve the problem is that we need the length of the ribbon required for which we should find out the perimeter of the card and convert the units. Quantity of glitter used for making Thank You Cards by Tanesha = 1 ounce. Length of the card = 9 cm. Conversion: 1 cm = 10 mm. Width of the card = 85 mm. Perimeter of the card = 2(Length of the card + Width of the card) Question 10. Be Precise How much ribbon does Tanesha need? Use math language and symbols to explain how you solved the problem and computed accurately. Answer: Length of the ribbon required = 350 mm. Explanation: Quantity of glitter used for making Thank You Cards by Tanesha = 1 ounce. Length of the card = 9 cm. Conversion: 1 cm = 10 mm. => 9 cm = 10 × 9 = 90 mm. Width of the card = 85 mm. Perimeter of the card = 2(Length of the card + Width of the card) = 2(90 + 85) = 2 × 175 = 350 mm. Question 11. Reasoning What information was not needed in the problem? Answer: Quantity of glitter used for making Thank You Cards by Tanesha = 1 ounce. This information was not needed in the problem. ### Topic 13 Fluency Practice Activity Find a Match Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match. Find a match for every clue. Clues A. The sum is between 2,000 and 2,500. B. The difference is exactly 10,000. C. The sum is exactly 6,000. D. The difference is exactly 4,500. E. The sum is exactly 16,477. F. The sum is between 5,500 and 5,600. G. The difference is between 1,000 and 2,000. H. The difference is between 8,000 and 9,000. Answer: Explanation: 10005+6472 = 16477. 1050+1200=2250. 7513-5676=1837. 3778+2222 = 6000. 35000-25000=10000. 10650-2150=8500. 1234+4321=5555. 9000-4500=4500. ### Topic 13 Vocabulary Review Understand Vocabulary Word List • area • capacity • centimeter (cm) • cup (c). • formula • gallon (gal) • gram (g) • kilogram (kg) • kilometer (km) • liter (L) • mass • meter (m) • milligram (mg) • milliliter (mL) • millimeter (mm) • ounce (oz) • perimeter • pint (pt) • pound (lb) • quart (qt) • ton (T) • weight • fluid ounce (fl oz) Question 1. Cross out the units that are NOT used to measure length. centimeter (cm) pint (pt) pound (lb) kilogram (kg) Answer: The units that are NOT used to measure length: centimeter (cm) pint (pt) pound (lb) kilogram (kg) X Question 2. Cross out the units that are NOT used to measure capacity. millimeter (mm) ounce (oz) gallon (gal) milliliter (mL) Answer: The units that are NOT used to measure capacity: millimeter (mm) X ounce (oz) gallon (gal) milliliter (mL) Question 3. Cross out the units that are NOT used to measure weight. cup (c) liter (L) meter (m) ton (T) Answer: The units that are NOT used to measure weight: cup (c) liter (L) meter (m) X ton (T) Question 4. Cross out the units that are NOT used to measure mass. liter (L) kilometer (km) milligram (mg) quart (qt) Answer: The units that are NOT used to measure mass: liter (L) kilometer (km) X milligram (mg) quart (qt) Label each example with a term from the Word List. Question 5. 2 × 4 = 8 square units _________ Answer: 2 × 4 = 8 square units __Area_______. Question 6. 3 + 7 + 3 + 7 = 20 units __________ Answer: 3 + 7 + 3 + 7 = 20 units ___perimeter_______. Question 7. Area = l × w _________ Answer: Area = l × w __Formula_______. Use Vocabulary in Writing Question 8. Mike uses 24 meters of fence to enclose a rectangular garden. The length of the garden is 10 meters. What is the width? Use at least 3 terms from the Word List to explain. Answer: Width of the garden = 2 meters. Explanation: Length of fence to enclose a rectangular garden Mike uses = 24 meters. Perimeter of the rectangular garden Mike uses = 24 meters. Length of the garden =10 meters. Width of the garden = W meters. Formula: Perimeter of the rectangular garden = 2(L + W) = 24 meters. => 2(10 + W) = 24 => 20 + 2W = 24 => 2W = 24- 20 =>W = 4 ÷ 2 => W = 2 meters. ### Topic 13 Reteaching Set A pages 481-492 Customary units can be used when measuring length, capacity, and weight. Remember when converting from a larger unit to a smaller unit, multiply. Use the conversion charts to help solve. Question 1. 9 yards = _________ inches Answer: 9 yards = ____324_____ inches. Explanation: 1 yard = 36 inches. 9 yards = 36 × 9 = 324 inches. Question 2. 5 miles = _________ yards Answer: 5 miles = ____8800_______ yards. Explanation: 1 mile = 1760 yards. 5 miles = 1760 × 5 = 8800 yards. Question 3. 215 yards = _________ feet Answer: 215 yards = ____645_____ feet. Explanation: 1 yard = 3 feet. 215 yards = 3 × 215 = 645 feet. Question 4. 9 pints = _________ fluid ounces Answer: 9 pints = ___144______ fluid ounces. Explanation: 1 pint = 16 fluid ounces. 9 pints = 16 × 9 = 144 fluid ounces. Question 5. 372 quarts = _________ cups Answer: 372 quarts = ___1488______ cups. Explanation: 1 quart = 4 cups. 372 quarts = 4 × 372 = 1488 cups. Question 6. 1,620 gallons = _________ pints Answer: 1,620 gallons = _____12960_______ pints. Explanation: 1 gallon = 8 pints. 1,620 gallons = 8 × 1620 = 12960 pints. Question 7. 9 pounds = _________ ounces Answer: 9 pounds = ___144______ ounces. Explanation: 1 pound = 16 ounces. 9 pounds = 16 × 9 = 144 ounces. Question 8. 5 tons = _________ pounds Answer: 5 tons = ____10000_____ pounds. Explanation: 1 ton = 2000 pounds. 5 tons = 2000 × 5 = 10000 pounds. Question 9. 12 feet = _________ inches Answer: 12 feet = ___144______ inches. Explanation: 1 feet = 12 inches. 12 feet = 12 × 12 = 144 inches. Set B pages 493-500 Metric units can be used to measure length, capacity, and mass. Remember metric units can be converted using multiples of 10. Use the conversion charts to help. Question 1. 9 kilometers = _________ meters Answer: 9 kilometers = ___9000______ meters. Explanation: 1 kilometer = 1000 meters. 9 kilometers = 1000 × 9 = 9000 meters. Question 2. 55 centimeters = _________ millimeters Answer: 55 centimeters = ____550_____ millimeters. Explanation: 1 centimeter = 10 millimeters. 55 centimeters = 10 × 55 = 550 millimeters Question 3. 2 meters = _________ centimeters Answer: 2 meters = ___200______ centimeters. Explanation: 1 meter = 100 centimeters. 2 meters = 100 × 2 = 200 centimeters. Question 4. 9 liters = _________ milliliters Answer: 9 liters = ___9000______ milliliters. Explanation: 1 liter = 1000 milliliters. 9 liters = 1000 × 9 = 9000 milliliters. Question 5. 4 grams = _________ milligrams Answer: 4 grams = ___4000______ milligrams. Explanation: 1 gram = 1000 milligrams. 4 grams = 1000 × 4 = 4000 milligrams. Question 6. 5 kilograms = _________ grams Answer: 5 kilograms = ___5000______ grams. Explanation: 1 kilogram = 1000 grams. 5 kilograms = 1000 × 5 = 5000 grams. Question 7. 8 kilograms = _________ grams Answer: 8 kilograms = _____8000______ grams. Explanation: 1 kilograms = 1000 grams. 8 kilograms = 1000 × 8 = 8000 grams. Question 8. 5 grams = _________ milligrams Answer: 5 grams = ___5000______ milligrams. Explanation: 1 gram = 1000 milligrams. 5 grams = 1000 × 5 = 5000 milligrams. Set C pages 501-504 The perimeter of Ted’s pool is 16 yards. The pool is 3 yards wide. He has 150-square feet of plastic. Does Ted have enough plastic to cover the pool? Use the formula for perimeter to find the length. Substitute the numbers you know. Perimeter = (2 × l) + (2 × w) 16 = (2 × l) + (2 × 3) l = 5 The length of the pool is 5 yards. 3 yards wide × 3 = 9 feet wide 5 yards long × 3 = 15 feet long Find the area of the pool. A = 15 × 9 A = 135 The area of the pool is 135 square feet. 135 < 150, so Ted has enough plastic to cover the pool. Remember to label your answer with the appropriate unit. Question 1. Find n. P = 108 inches Answer: Width of rectangle = 36 inches. Explanation: Perimeter of rectangle = 108 inches. Length of rectangle = 18 inches. Width of rectangle = n inches. Perimeter of rectangle = 2(L + W) = 108 inches. => 2 (18 + n) = 108 => 36 + 2n = 108 => 2n = 108 -36 =>n = 72 ÷ 2 => n = 36 inches. Question 2. Find the area. P = 26 m Answer: Area of rectangle = 36 sq m. Explanation: Perimeter of rectangle = 26 m. Length of rectangle = 4m. Width of rectangle = Wm. Perimeter of rectangle = 2(L + W) =26m. =>2(4 + W) = 26 => 8 + 2W = 26 => 2W = 26 – 8 =>W = 18 ÷ 2 => W = 9m. Area of rectangle = L × W = 4 × 9 = 36 sq m. Question 3. Find the perimeter of the square. Answer: Perimeter of square = 10 yards. Explanation: Side of square = 2$$\frac{2}{4}$$ yards. Perimeter of square = 4 × Side = 4 × 2$$\frac{2}{4}$$ = 10 yards. Set D pages 505-508 Think about these questions to help you be precise. Thinking Habits • Am I using numbers, units, and symbols appropriately? • Am I using the correct definitions? • Am I calculating accurately? • Is my answer clear? Remember to give an explanation that is clear and appropriate. A puppy pen is 4 feet wide and 5 feet long. Question 1. Is 21 square feet of fabric large enough to make a mat for the pen? Explain. Answer: Yes, it is enough to make a mat for the pen because it requires 20 sq feet for the mat and its 1 sq feet more than required. Explanation: Width of the puppy pen = 4 feet. Length of the puppy pen = 5 feet. Area of the puppy pen = L × W = 5 × 4 = 20 sq feet. Question 2. Puppy fencing comes in sizes that are 12 feet, 24 feet, and 30 feet in length. Which length would be the best for the pen? How much, if any, will have to be left over? Explain. Answer: 24 feet length would be the best for the pen and it leaves with 6 feet of more. Explanation: Puppy fencing comes in sizes that are 12 feet, 24 feet, and 30 feet in length. Width of the puppy pen = 4 feet. Length of the puppy pen = 5 feet. => Perimeter of fencing required for puppy pen = 2(L + W) => 2(5 + 4) => 2 × 9 => 18 feet. Difference: 18 feet – 12 feet = 6 feet. 24 feet – 18 feet = 6 feet. 30 feet – 18 feet = 12 feet. ### Topic 13 Assessment Practice Question 1. A window is 5 feet long. What is the length of the window in inches? Answer: Length of the window = 60 inches. Explanation: Length of the window = 5 feet. Conversion: 1 feet = 12 inches. 5 feet = 12 × 5 = 60 inches. Question 2. Mrs. Warren bought 6 liters of lemonade for a party. How many milliliters of lemonade did she buy? A. 9,000 milliliters B. 6,000 milliliters C. 3,000 milliliters D. 1,200 milliliters Answer: 6000 milliliters of lemonade did she buy. B. 6,000 milliliters. Explanation: Quantity of lemonade Mrs. Warren bought for a party = 6 liters. Conversion: 1 liter = 1000 milliliters. = 1000 × 6 = 6000 milliliters. Question 3. Select the equivalent measurement for each measurement on the left. Answer: Explanation: Conversion: 1 feet = 12 inches. 9 feet = 12 × 9 = 108 inches. 1 cup = 8 fluid ounces. 9 cups = 8 × 9 = 72 fluid ounces. 1 ton = 2000 lb. 4 tons = 2000 × 4 = 8000 lb. 1 yard = 36 inches. 4 yards = 36 × 4 = 144 inches. Question 4. A picnic table is 9 feet long and 3 feet wide. Write and solve an equation to find the area of the rectangular surface of the table. Answer: Area of the rectangular surface of the table = 27 square feet. Explanation: Length of a picnic table = 9 feet. Width of a picnic table = 3 feet. Area of the rectangular surface of the table = L × W = 9 × 3 = 27 square feet. Question 5. The Girl’s Club is making muffins. Mindy’s recipe calls for 3 cups of buttermilk. Josie’s recipe calls for 20 fluid Ounces of buttermilk. Georgia’s recipe calls for 1 pint of buttermilk. Whose recipe calls for the most buttermilk? Explain. Answer: Georgia’s recipe calls for the most buttermilk because its most between 0.05 cup and 3 cups. Explanation: Quantity of buttermilk Mindy’s recipe calls for = 3 cups. Quantity of buttermilk Josie’s recipe calls for = 20 fluid Ounces. Quantity of buttermilk Georgia’s recipe calls for = 1 pint . Conversion: 1 cup = 8 fluid ounces. => 20 fluid Ounces = 1 ÷ 20 = 0.05 cups. 1 pint = 2 cups. Question 6. Andrea ran 4 kilometers over the weekend. How many meters did Andrea run? Answer: Number of meters Andrea run =4000 meters. Explanation: Distance Andrea ran over the weekend = 4 kilometers. Conversion: 1 kilometer = 1000 meters. Number of meters Andrea run = 1000 × 4 = 4000 meters. Question 7. Choose numbers from the box to complete the table. Some numbers will not be used. Answer: Explanation: Conversion: 1 pound = 16 ounces. 1$$\frac{2}{4}$$ pounds = 16 × 1$$\frac{2}{4}$$ = 24 ounces. 2 pounds = 16 × 2 = 32 ounces. 2$$\frac{2}{4}$$ pounds = 16 × 2$$\frac{2}{4}$$ = 40 ounces. 3 pounds = 16 × 3 = 48 ounces. 3$$\frac{2}{4}$$ pounds = 16 × 3$$\frac{2}{4}$$ pounds = 56 ounces. Question 8. Select each correct equation. ☐ 1l = 100 mL ☐ 1 kg = 1,000 g ☐ 4 yd = 14 ft ☐ 15 cm = 150 mm ☐ 1 gal = 13 cups Answer: Correct equation: 1 kg = 1,000 g. Explanation: ☐ 1l = 100mL (False) ☐ 1 kg = 1,000 g (True) ☐ 4 yd = 14 ft (False) ☐ 15 cm = 150 mm (False) ☐ 1 gal = 13 cups (False) Conversion: 1 l = 100 mL. (False) 1 kg = 1000 g. 1 yd = 3 ft => 4 yd = 3 × 4 = 12 ft. 1 cm = 100 mm. => 15 cm = 100 × 15 = 1500 mm. 1 gal = 16 cups. Question 9. Morgan rode her bike 2 kilometers from her house to her friend’s house. From her friend’s house, she rode 600 meters in all going to and from the library. Then she rode back home. How many meters did Morgan bike in all? Answer: Distance in meters she rode in all = 2600 meters. Explanation: Distance Morgan rode her bike from her house to her friend’s house = 2 kilometers. Distance From her friend’s house, she rode in all going to and from the library = 600 meters. Distance in meters she rode in all = Distance Morgan rode her bike from her house to her friend’s house + Distance From her friend’s house, she rode in all going to and from the library = 2 kilometers + 600 meters Conversion: 1 km = 1000 m. => 2 kilometers = 1000 × 2 = 2000 meters. => 2000 + 600 = 2600 meters. Question 10. Which statement is true about the bedrooms in the drawings below? A. Erin’s room has a greater area than Steve’s room. B. Steve’s room has a greater perimeter than Erin’s room. C. They both have the same perimeter. D. None of the above Answer: Statement is true about the bedrooms in the drawings below: C. They both have the same perimeter. Explanation: Length of Steve’s room = 9 ft. Width of Steve’s room = 8 ft. Area of Steve’s room = L × W = 9 × 8 = 72 square feet. Perimeter of Steve’s room = 2(L + W) = 2(9 + 8) = 2 × 17 = 34 feet. Length of Erin’s room = 10 ft. Width of Erin’s room = 7 ft. Area of Erin’s room = L × W = 10 × 7 = 70 square feet. Perimeter of Erin’s room = 2(L + W) = 2(10 + 7) = 2 × 17 = 34 feet. A. Erin’s room has a greater area than Steve’s room. (False) B. Steve’s room has a greater perimeter than Erin’s room. (False) C. They both have the same perimeter. (True) D. None of the above. (False) Question 11. Tim has 3 meters of yarn. How many centimeters of yarn does Tim have? Answer: 33.34 centimeters of yarn Tim has. Explanation: Length of yarn Tim has = 3 meters. Conversion: 1 m = 100 cm. 3 meters = 100 ÷ 3 = 33.34 centimeters. Question 12. Mrs. Li’s classroom is 34 feet wide and 42 feet long. A. What is the area of the classroom? Answer: Area of Mrs. Li’s classroom = 1428 square feet. Explanation: Length of Mrs. Li’s classroom = 34 feet. Width of Mrs. Li’s classroom = 42 feet. Area of Mrs. Li’s classroom = L × W = 34 × 42 = 1428 square feet. B. How much area is taken up by the objects in the classroom? How much area is left for the students’ desks? Write and solve equations to find the area. Answer: Area left for the students’ desks = 1194 square feet. Explanation: Area of Mrs.Li’s desk = 8 square feet. Area of Fish tank = 6 square feet. Area of Math center = 100 square feet. Area of Reading center = 120 square feet. Area taken up by the objects in the classroom = Area of Mrs.Li’s desk + Area of Fish tank + Area of Math center + Area of Reading center = 8 + 6 + 100 + 120 = 234 square feet. Area left for the students’ desks = Area of Mrs. Li’s classroom – Area taken up by the objects in the classroom = 1428 square feet – 234 square feet = 1194 square feet. Question 13. Select the equivalent measurement for each measurement on the left. Answer: Explanation: Conversion: 1g = 1000 mg. => 3 g = 1000 × 3 = 3000 mg. 1kg = 1000 g. => 3 kg = 1000 × 3 = 3000 g. 1L = 1000 mL. => 3 L = 1000 × 3 = 3000 mL. 1m = 1000 mm. => 3 m = 1000 × 3 = 3000 mm. ### Topic 13 Performance Task Watermelons Kasia grows watermelons. Question 1. Kasia plants her watermelons in rows. Kasia’s watermelon field has a perimeter of 71$$\frac{1}{3}$$ yards and is 14$$\frac{2}{3}$$ yards wide. Each row is yard wide and the rows will be planted 2 yards apart. Part A What is the length of Kasia’s field? Explain. Answer: The length of Kasia’s field = 14$$\frac{2}{3}$$ yards because its given in the problem. Explanation: The length of Kasia’s field = 14$$\frac{2}{3}$$ yards. Part B What is the area of Kasia’s field? Complete the table to convert the length to feet. Be sure to use the correct units on your answer. Explain. Answer: Area of Kasia’s field = 308 square yards. Explanation: Length of Kasia’s field = 14$$\frac{2}{3}$$ yards. Width of Kasia’s field = W yards. Perimeter of Kasia’s field = 71$$\frac{1}{3}$$ yards. Perimeter of Kasia’s field = 2(L + W) = 71$$\frac{1}{3}$$ yards. => 2 (14$$\frac{2}{3}$$ + W) = 71$$\frac{1}{3}$$ => 28$$\frac{1}{3}$$ + 2W = 71$$\frac{1}{3}$$ => 2W = 71$$\frac{1}{3}$$ – 29$$\frac{1}{3}$$ => W = 42 ÷ 2 => W = 21 yards. Area of Kasia’s field = L × W = 14$$\frac{2}{3}$$ × 21 = 308 square yards. Conversion: 1 yard = 3 feet. 10 yards = 3 × 10 = 30 feet. 20 yards = 3 × 20 = 60 feet. 21 yards = 3 × 21 = 63 feet. Part C How many rows can Kasia plant? Explain. Answer: Number of rows Kasia plant = 154. Explanation: Number of rows Kasia plant = 2. Length of each row = 1 yard. Width of the rows will be planted = 2 yards. Area of Kasia’s field = 308 square yards. Number of rows Kasia plant = Area of Kasia’s field ÷ Width of the rows will be planted = 308 ÷ 2 = 154. Question 2. Use the information in the Watermelon table. Part A If there are twenty-eight 8-ounce servings in a 20-pound watermelon, how many pounds does the rind weigh? Explain. Answer: Quantity of rind in pounds = 2 pounds. Explanation: Quantity of watermelon = 20 pounds. Quantity of watermelon servings = 288 ounces. Conversion: 1 pound = 16 ounces. => 20 pounds = 16 × 20 = 320 ounces. Quantity of rind = Quantity of watermelon – Quantity of watermelon servings = 320 – 288 = 32 ounces. Conversion: 1 pound = 16 ounces. => 32 ounces = 1 × 32 ÷ 16 = 32 ÷ 16 = 2 pounds. Part B How many cups of fruit does Kasia get from a 20-pound watermelon? Explain. Show your computations. Do not include the weight of the rind. Answer: Number of cups of fruit does Kasia get from a 20-pound watermelon = 40$$\frac{2}{4}$$ cups. Explanation: Quantity of watermelon = 20 pounds. Quantity of rind in pounds = 2 pounds. Quantity of watermelon Kasia gets in pounds = Quantity of watermelon – Quantity of rind in pounds = 20 – 2 = 18 pounds. Conversion: 1 pound = 2$$\frac{1}{4}$$ cups Number of cups of fruit does Kasia get from a 20-pound watermelon = => 18 pounds = 2$$\frac{1}{4}$$ × 18 => 40$$\frac{2}{4}$$ cups. Question 3. Use the information from the Watermelon and Nutrition picture to answer the question. How many more milligrams of fiber than potassium are in a serving of watermelon? Explain. Answer: 730 mg more milligrams of fiber than potassium are in a serving of watermelon. Explanation: Quantity of serving of fiber = 1 gram. Conversion: 1g = 1000 mg Quantity of serving of potassium = 270 milligram. Difference: Quantity of serving of fiber – Quantity of serving of potassium = 1000 – 270 = 730 mg. #### enVision Math Common Core Grade 4 Answer Key ## Envision Math Common Core Grade 4 Answer Key Topic 14 Algebra Generate and Analyze Patterns ## Envision Math Common Core 4th Grade Answers Key Topic 14 Algebra Generate and Analyze Patterns Essential Questions: How can you use a rule to continue a pattern? How can you use a table to extend a pattern? How can you use a repeating pattern to predict a shape? enVision STEM Project: Patterns and Waves Journal: Write a Report Include what you found. Also in your report: Do Research Use the Internet or other sources to learn about 2 industries where oscilloscopes can be used. Name the industry and what can be observed using the oscilloscope. Oscilloscopes are used to observe patterns in waves. Suppose a scientist created a pattern with three levels of sounds: quiet, loud, medium. If the scientist repeats the pattern of sounds, what would be the 41st sound in the pattern? Explain. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • even number • odd number • inverse operations • variable Question 1. A(n) ________ can be divided into groups of 2 without a remainder. Answer: Question 2. A symbol or letter that stands for a number is called a(n) ___________. Answer: Question 3. Operations that undo each other are called _________ Answer: Addition and Subtraction Patterns Add or subtract to find the missing number in each pattern. Question 4. 3, 6, 9, 12, ____,18 Answer: Question 5. 4,8, 12, ____, 20, 24 Answer: Question 6. 8, 7, 6, ____,4, 3 Answer: Question 7. 30, 25, 20, 15, ____,5 Answer: Question 8. 1, 5, 9, ____, 17, 21 Answer: Question 9. 12, 10, 8, 6, ___, 2 Answer: Multiplication and Division Patterns Multiply or divide to find the missing number in each pattern. Question 10. 1, 3, 9, 27, ___, 243 Answer: Question 11. 64, 32, 16, ____, 4, 2 Answer: Question 12. 1,5, 25, ____, 625 Answer: Question 13. 1, 2, 4, 8, ____, 32 Answer: Question 14. 1, 4, 16, ____, 256 Answer: Question 15. 729, 243, 81,27, 9, ____ Answer: Problem Solving Question 16. Look for Relationships James places 1 counter in the first box. He places 2 counters in the second box, 4 counters in the third box, 8 counters in the fourth box, and continues the pattern until he gets to the tenth box. How many counters did James place in the tenth box? Answer: Pick a Project PROJECT 14A How have roller coasters changed through the years? Project: Make a Model Roller-Coaster Car PROJECT 14B How can you use currency from different countries? Project: Make Your Own Currency PROJECT 14C How can patterns be used in sidewalks? Project: Design Your Own Sidewalk PROJECT 14D How many stadiums in the United States have retractable roofs? Project: Make a Seating Diagram ### Lesson 14.1 Number Sequences Solve & Share Look at the rules and starting numbers below. What are the next 6 numbers in each pattern? Tell how you decided. Describe features of the patterns. Solve these problems any way you choose. I can … use a rule to create and extend a number pattern and identify features of the number pattern not described by the rule. Look Back! Look for Relationships Create two patterns that use the same rule but start with different numbers. Identify a feature of each pattern. For example, identify whether the numbers are all even, all odd, or alternate between even and odd. Essential Question How Can You Use a Rule to Continue a Pattern? Visual Learning Bridge The house numbers on a street follow the rule “Add 4.” If the pattern continues, what are the next three house numbers? Describe a feature of the pattern. You can use a number line to help make sense of the problem and find the next three house numbers. Use a number line to continue the pattern. A rule is a mathematical phrase that tells how numbers or shapes in a pattern are related. The rule for the house numbers is “Add 4.” The next three house numbers are 20, 24, and 28. Describe features of the pattern. Some patterns have features that are not given in the rule. 16, 20, 24, 28 One of the features of this pattern is all of the house numbers are even numbers. Another feature is all of the house numbers are multiples of 4. Convince Me! Generalize Can you use the rule “Add 4” to create a different pattern with all odd numbers? Explain. Another Example! On another street, the house numbers follow the rule “Subtract 5.” What are the next three house numbers after 825? Describe a feature of the pattern. The next three house numbers are 820, 815, and 810. All of the house numbers are multiples of 5. Guided Practice Do You Understand? Question 1. Rudy’s rule is “Add 2.” He started with 4 and wrote the numbers below. Which number does NOT belong to Rudy’s pattern? Explain. 4,6, 8, 9, 10, 12 Answer: Do You Know How? Continue the pattern. Describe a feature of the pattern. Question 2. Subtract 6 48, 42, 36, 30, 24, _____, _____, ______ Answer: Independent Practice For 3-6, continue each pattern. Describe a feature of each pattern. Question 3. Subtract 3: 63, 60, 57, _____, _____ Answer: Question 4. Add 7: 444, 451, 458, _____, _____ Answer: Question 5. Add 25: 85, 110, 135, _____, _____ Answer: Question 6. Subtract 4: 75, 71, 67, _____, _____ Answer: For 7-12, use the rule to generate each pattern. Question 7. Rule: Subtract 10 90, _____, _____ Answer: Question 8. Rule: Add 51 16, _____, _____ Answer: Question 9. Rule: Add 5 96, _____, _____ Answer: Question 10. Rule: Add 107 43, _____, _____ Answer: Question 11. Rule: Subtract 15 120, _____, _____ Answer: Question 12. Rule: Subtract 19 99, _____, _____ Answer: Problem Solving Question 13. Reasoning Orlando delivers mail. He sees one mailbox that does not have a number. If the numbers are in a pattern, what is the missing number? Answer: Question 14. A bus tour runs 9 times a day, 6 days a week. The bus can carry 30 passengers. Find the greatest number of passengers who can ride the tour bus each week. Answer: Question 15. The year 2017 was the year of the Rooster on the Chinese calendar. The next year of the Rooster will be 2029. The rule is “Add 12.” What are the next five years of the Rooster? Answer: Question 16. Describe a feature of the year of the Rooster pattern. Answer: Question 17. Vocabulary Define rule. Create a number pattern using the rule “Subtract 7.” Answer: Question 18. Higher Order Thinking Some patterns use both addition and subtraction in their rules. The rule is “Add 3, Subtract 2.” Find the next three numbers in the pattern. 1, 4, 2, 5, 3, 6, 4, 7, _____, _____, _____ Answer: Assessment Practice Question 19. Rima used “Subtract 3” as the rule to make a pattern. She started with 60, and wrote the next six numbers in her pattern. Which number does NOT belong in Rima’s pattern? A. 57 B. 54 C. 45 D. 26 Answer: Question 20. Ivan counted all the beans in a jar. If he counted the beans in groups of 7, which list shows the numbers Ivan could have named? A. 77, 84, 91, 99 B. 301, 308, 324, 331 C. 574, 581, 588, 595 D. 14, 24, 34, 44 Answer: ### Lesson 14.2 Patterns: Number Rules Solve & Share There are 6 juice boxes in 1 pack, 12 in 2 packs, and 18 in 3 packs. How many juice boxes are in 4 packs? in 5 packs? in 6 packs? Use the rule to complete the table. Describe features of the pattern. Then find how many juice boxes are in 10 packs and 100 packs. I can … use a rule to extend a number pattern, identify features of the number pattern, and use the number pattern to solve a problem. Look Back! Reasoning Create a table showing the relationship between the number of bicycles and the number of bicycle wheels. Start with 1 bicycle. Complete 5 rows of the table using the rule “Multiply by 2.” Describe features of the pattern. Essential Question What is the Pattern? Visual Learning Bridge There are 3 leaflets on 1 cloverleaf. There are 6 leaflets on 2 cloverleaves. There are 9 leaflets on 3 cloverleaves. How many leaflets are on 4 cloverleaves? How many cloverleaves will have 12 leaflets? You can use a table to create, extend, and identify features of a pattern. How many leaflets are on 4 cloverleaves? There are 12 leaflets on 4 cloverleaves. The number of leaflets is a multiple of the number of cloverleaves. How many cloverleaves for 12 leaflets? There are 4 cloverleaves for 12 leaflets. The number of cloverleaves is a factor of the number of leaflets. Convince Me! Model with Math If you know the number of leaflets, l, what expression can you use to find the number of cloverleaves, c? If you know the number of cloverleaves, what expression can you use to find the number of leaflets? Guided Practice Do You Understand? Question 1. The rule for this table is “Multiply by 4.” What number does not belong? Answer: Do You Know How? Complete the table. Describe a feature of the pattern. Question 2. Rule: Divide by 4 Answer: Independent Practice For 3-6, use the rule to complete each table. Describe a feature of each pattern. You can multiply or divide to find the patterns in these tables. Question 3. Rule: Multiply by 8 Answer: Question 4. Rule: Divide by 5 Answer: Question 5. Rule: Multiply by 16 Answer: Question 6. Rule: Divide by 2 Answer: Problem Solving Question 7. The table shows how much money Joe makes painting. How much money will Joe make when he paints for 6 hours? Rule: Multiply by 45 Answer: Question 8. The table shows the total number of pounds of potatoes for different numbers of bags. How many bags does it take to hold 96 pounds of potatoes? Rule: Divide by 8 Answer: Question 9. Number Sense What is the greatest number you can make using each of the digits 1,7,0, and 6 once? Answer: Question 10. Algebra A penguin can swim 11 miles per hour. At this speed, how far can it swim in 13 hours? Use s as a variable. Write and solve an equation. Answer: For 11-12, the rule is “Multiply by 3.” Question 11. Reasoning Using the rule, how many batteries do 8 flashlights need? Answer: Question 12. Higher Order Thinking How many more batteries do 20 flashlights need than 15 flashlights? Explain. Answer: Assessment Practice Question 13. There are 6 rolls in each package. Use the rule “Divide by 6” to show the relationship between the number of rolls and the number of packages. Use each digit from the box once to complete the table. Answer: ### Lesson 14.3 Patterns: Repeating Shapes Solve & Share The rule for the repeating pattern below is “Square, Triangle.” What will be the 37th shape in the pattern? Explain. Solve this problem any way you choose. I can … use a rule to predict a number or shape in a pattern. Look Back! When the pattern has 37 shapes, how many are triangles? Essential Question How Can You Use a Repeating Pattern to Predict a Shape? Visual Learning Bridge Rashad is making a repeating pattern for the rule “Triangle, Square, Trapezoid.” What will be the 49th shape in the pattern? A repeating pattern is made up of shapes or numbers that form a part that repeats. Look for Features of the Repeating Pattern The trapezoid is the 3rd, 6th, and 9th shape in the pattern. The positions of the trapezoids are multiples of 3. The triangle is the 15th, 4th, and 7th shape in the pattern. The positions of the triangles are 1 more than a multiple of 3. The square is the 2nd, 5th, and 8th shape in the pattern. The positions of the squares are 1 less than a multiple Use the Repeating Pattern to Solve When you divide 49 by 3, the quotient is 16 R1. The pattern repeats 16 times. The 1st shape in the repeating pattern, a triangle, then appears. 49 is one more than a multiple of 3. The 49th shape is a triangle. Convince Me! Be Precise Suppose the rule is “Square, Triangle, Square, Trapezoid” in a repeating pattern. What is the 26th shape in the pattern? Describe features of the repeating pattern. Be precise in your description. Another Example! Write the next three numbers in the repeating pattern. Then name the 100th number in the pattern. Rule: 1, 3, 5, 7 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5, 7, 1, 3, 5 … There are 4 items in the repeating pattern. To find the 100th number, divide by 4. The pattern repeats 25 times. The 100th number is 7. Guided Practice Do You Understand? Question 1. In the “Triangle, Square, Trapezoid” example on the previous page, what will be the 48th shape? the 50th shape? Explain. Answer: Do You Know How? Question 2. What is the 20th shape? The rule is “Triangle, Circle, Circle.” Answer: Question 3. Write the next three numbers. The rule is “9, 2, 7, 6.” 9, 2, 7, 6, 9, 2, 7, 6, ____, _____, ______ Answer: Independent Practice For 4-7, draw or write the next three items to continue each repeating pattern. Question 4. The rule is “Square, Triangle, Square.” Question 5. The rule is “Up, Down, Left, Right.” Answer: Question 6. The rule is “1, 1, 2.” 1, 1, 2, 1, 1, 2, ____ , ____, ____ …… Answer: Question 7. The rule is “5, 7, 4, 8.” 5, 7, 4, 8, 5, 7, 4, 8, 5, 7, ____, ____, ____ …… Answer: For 8-9, determine the given shape or number in each repeating pattern. Question 8. The rule is “Tree, Apple, Apple.” What is the 19th shape? Answer: Question 9. The rule is “1, 2.” What is the 42nd number? 1, 2, 1, 2, 1, 2, … Answer: Problem Solving Question 10. Create a repeating pattern using the rule “Triangle, Square, Square.” Answer: Question 11. enVision® STEM Margot measured the distance for 6 wavelengths of visible light as 2,400 nanometers. What is the distance for 1 wavelength? Answer: Question 12. Look for Relationships Hilda is making a repeating pattern with the shapes below. The rule is “Heart, Square, Triangle.” If Hilda continues the pattern, what will be the 11th shape? Answer: Question 13. Look for Relationships Josie puts beads on a string in a repeating pattern. The rule is “Blue, Green, Yellow, Orange.” There are 88 beads on her string. How many times did Josie repeat her pattern? Answer: Question 14. How many more years passed between the first steam locomotive and the gasoline-powered automobile than between the gasoline-powered automobile and the first diesel locomotive? Answer: Question 15. Louisa used the rule “Blue, Green, Green, Green” to make a bracelet with a repeating pattern. She used 18 green beads. How many beads did Louisa use to make the bracelet? How many beads were NOT green? Answer: Question 16. Higher Order Thinking Marcus is using shapes to make a repeating pattern. He has twice as many circles as squares. Make a repeating pattern that follows this rule. Answer: Assessment Practice Question 17. Which rules give a repeating pattern that has a square as the 15th shape? Select all that apply. ☐ Square, Circle ☐Circle, Square, Triangle ☐ Square, Circle, Triangle ☐ Circle, Triangle, Square ☐ Trapezoid, Circle, Square Answer: Question 18. Which rules give a repeating pattern that has a 7 as the 15th number? Select all that apply. ☐ 1, 7 ☐ 1, 7, 9 ☐ 1, 9, 7 ☐ 1, 7, 7 ☐ 7, 1, 9 Answer: ### Lesson 14.4 Problem Solving Look For and Use Structure Solve & Share Evan’s baby brother is stacking blocks. Using the rule “Add 1 block to the number of blocks in the previous stack,” how many blocks will be in the 6th stack? Explain. Justify your answer. I can … use patterns to help solve problems. Thinking Habits • What patterns can I see and describe? • How can I use the patterns to solve the problem? · Can I see expressions and objects in different ways? Look Back! Look For Relationships How many blocks are in the 10th stack? Explain. Essential Question How Can I Look For and Make Use of Structure? Visual Learning Bridge Alisa made three walls with cubes. She recorded her pattern. If she continues the pattern, how many cubes will be in a 10-layer wall? a 100-layer wall? What do you need to do to find the number of cubes in a 10-layer and 100-layer wall? I need to continue the pattern using the rule and analyze the pattern to find features not stated in the rule itself. How can I make use of structure to solve this problem? I can • look for and describe patterns in three dimensional shapes. • use the rule that describes how objects or values in a pattern are related. • use features of the pattern not stated in the rule to generate or extend the pattern. Here’s my thinking. Make a table and look for patterns. There are 4 cubes in each layer. Multiply the number of layers by 4 to calculate the number of cubes. A 10-layer wall contains 10 × 4 = 40 cubes. A 100-layer wall contains 100 × 4 = 400 cubes. Convince Me! Look for Relationships How could you use multiples to describe Alisa’s pattern? Guided Practice Use Structure Leah arranged triangular tiles in a pattern like 1 row the one shown. She used the rule “Multiply the number of rows by itself to get the number of small triangles.” How many small triangles would be in the pattern if there were 10 rows? Question 1. Complete the table to help describe the pattern. Answer: Question 2. Describe the pattern another way. Answer: Question 3. How many triangles would be in 10 rows? Answer: Independent Practice Look for Relationships Alan built the towers shown using the rule “Each story has 2 blocks.” How many blocks will a 10-story tower have? Use Exercises 4-6 to answer the question. Question 4. Complete the table to help describe the pattern. Answer: Question 5. What is another way to describe the pattern that is not described by the rule? Answer: Question 6. How many blocks are in a 10-story tower? Explain. Answer: Problem Solving Performance Task Glass Stairs An art gallery staircase is built using glass cubes. The diagram below shows 4 steps are 4 cubes high and 4 cubes across. Five steps are 5 cubes high and 5 cubes across. How many glass cubes are used to make 7 steps? Use Exercises 7-10 to answer the question. Question 7. Make Sense and Persevere What do you know, and what do you need to find? Answer: Question 8. Reasoning Complete the table. Answer: Question 9. Look For Relationships What pattern can you determine from the table? Answer: Question 10. Reasoning How many cubes are needed for 7 steps? Write and solve an equation. Answer: ### Topic 14 Fluency Practice Activity Point & Tally Find a partner. Get paper and a pencil. Each partner chooses a different color: light blue or dark blue. Partner 1 and Partner 2 each point to a black number at the same time. Each partner adds the two numbers. If the answer is on your color, you get a tally mark. Work until one partner has twelve tally marks. I can … add multi-digit whole numbers. ### Topic 14 Vocabulary Review Understand Vocabulary Word List • equation • even number • factor • multiple • odd number • repeating pattern • rule • unknown Question 1. Circle the term that best describes 28. even odd equation unknown Answer: Question 2. Circle the term that best completes this sentence: 4 is a _________ of 16. even odd factor multiple Answer: Question 3. Circle the term that best describes 17. even odd equation unknown Answer: Question 4. Circle the term that best completes this sentence: 9 is a _________ of 3. even odd factor multiple Answer: Question 5. Draw a line from each term to its example. Answer: Use Vocabulary in Writing Question 6. Use at least 3 terms from the Word List to describe the pattern. 50, 48, 46, 44, 42 … Answer: ### Topic 14 Reteaching Set A pages 521-524 You can use the rule “Subtract 3” to continue the pattern. The next three numbers in the pattern are 9,6, and 3 A feature of the pattern is all the numbers are multiples of 3. Another feature is all the numbers in the pattern alternate even, odd. Remember to check that the numbers in your pattern follow the rule. Use the rule to continue each pattern. Describe a feature of the pattern. Question 1. Rule: Add 20 771, 791, 811, _____, _____, _____ Answer: Question 2. Rule: Subtract 12 122, 110, 98, _____, _____, ______ Answer: Set B pages 525-528 The regular price is twice the sale price. You can use the rule “Divide by 2” to continue the pattern. Remember to look for features of the pattern not described by the rule. Use the rule to continue each pattern. Describe a feature of the pattern. Question 1. Rule: Multiply by 18 Answer: Question 2. Rule: Divide by 9 Answer: Question 3. Rule: Multiply by 24 Answer: Set C pages 529-532 You can use the rule “Circle, Triangle, Square” to continue the repeating pattern. You can use the rule to find the 25th shape in the pattern. 25 ÷ 3 = 8 R1. The pattern will repeat 8 times, then the 1st shape will appear. The circle is the 25th shape in the pattern. Remember to use the rule to continue the pattern. Question 1. a. Draw the next three shapes in the repeating pattern. The rule is “Right, Up, Up.” Answer: b. Draw the 50th shape in the pattern. Answer: Question 2. a. Write the next three numbers in the repeating pattern. The rule is “3, 5, 7, 9.” 3, 5, 7, 9, 3, 5, 7, ____, _____, ______ Answer: b. What will be the 100th number in the pattern? Answer: Set D pages 533-536 Think about these questions to help you Look For and Use Structure. Thinking Habits • What patterns can I see and describe? • How can I use the patterns to solve the problem? • Can I see expressions and objects in different ways? Remember to use the rule that describes how objects or values in a pattern are related. Sam creates a pattern using the rule “Each story has 3 blocks.” Question 1. Draw the next shape in Sam’s pattern. Answer: Question 2. Use the rule to continue Sam’s pattern. Answer: Question 3. How many blocks are in the 10th shape in Sam’s pattern? Answer: ### Topic 14 Assessment Practice Question 1. Football players come out of the tunnel, and their jerseys have the number pattern shown below. They follow the rule “Add 4.” A. What number belongs on the front of the blank jersey? Explain. Answer: B. Describe two features of the pattern. Answer: Question 2. One dozen eggs is 12 eggs. Two dozen eggs is 24 eggs. Match the number of dozens to the number of eggs. The rule is “Multiply by 12.” Answer: Question 3. Use the rule “Multiply by 6” to continue the pattern. Then describe a feature of the pattern. Answer: Question 4. Use the rule “Divide by 3” to continue the pattern. Then write 4 terms of a different pattern that follows the same rule. 729, 243, 81, ____, _____ Answer: Question 5. Nicole arranges her shopping purchases by price. Each item costs$6 more than the last. The first item costs $13. The last costs$61. Her brother John says that the price of each item is an odd number. Is John correct? Find the cost of each item to explain.

Question 6.
The rule for the repeating pattern is “5, 7, 2, 8.” Write the next three numbers in the pattern. Then tell what will be the 25th number in the pattern. Explain.
5, 7, 2, 8, 5, 7, 2, 8, 5, _____, _____, _____

Question 7.
Jackson wrote different patterns for the rule “Subtract 5.” Select all of the patterns that he could have written. Then write 4 terms of a different pattern that follows the same rule.
☐ 27, 22, 17, 12,7
☐ 5, 10, 15, 20, 25
☐ 55, 50, 35, 30, 25
☐ 100, 95, 90, 85, 80
☐ 75, 65, 55, 45, 35

Question 8.
The rule is “Subtract 7.” What are the next 3 numbers in the pattern? Describe two features of the pattern.
70,63, 56, 49, 42,35

Question 9.
The table shows the different number of teams formed by different numbers of players. The rule is “Divide by 8.”

A. How many teams can be formed with 40 players?
_______ teams

B. How many players are there on 13 teams? How do you know?

Question 10.
A. Select all the true statements for the repeating pattern. The rule is “Circle, Heart, Triangle.”

☐ The next shape is the circle.
☐ The circle only repeats twice.
☐ The 10th shape is the heart.
☐ The 12th shape is the triangle.
☐ The circle is the 1st, 4th, 7th, etc. shape.

B. How many triangles are there among the first 22 shapes?

Wall Hangings Michael uses knots to make wall hangings to sell.

Question 1.
The Michael’s Basic Wall Hanging figure shows a simple wall hanging Michael makes by repeating the shapes shown. What is the 16th shape in the repeating pattern? The rule is “Circle, Triangle, Square.” Explain.

Question 2.
The Snowflake Design figure shows a knot Michael likes to use.

Part A
List the number of knots that Michael uses to form 1 to 6 snowflake designs. The rule is “Add 11.”

Part B
Describe a feature of the pattern you listed in Part A that is not part of the rule. Explain why it works.

Question 3.
The Michael’s Wall Hanging figure shows the design of a wall hanging Michael makes using the Snowflake Design. Answer the following to find how many knots Michael ties to make a wall hanging with 28 snowflakes.

Part A
Each column of 4 snowflakes has 4 connectors. There are also 4 connectors between columns. Complete the Connectors table using the rule “Add 8 connectors for each column.” Describe a feature of the pattern.

Part B
Complete the Total Knots table using the following rules.
Snowflake Knots rule: Multiply the number of snowflakes by 11.
Connector Knots rule: Multiply the number of connectors from the Connectors table by 3.
Total Knots rule: Add the number of snowflake knots and the number of connector knots.

## Envision Math Common Core 4th Grade Answers Key Topic 15 Geometric Measurement: Understand Concepts of Angles and Angle Measurement

Essential Questions:
What are some common geometric terms? How can you measure angles?

enVision STEM Project: Lines and Angles
Do Research Use the Internet or other sources to research the area of the world’s largest bumper car floor. Find where it is located and when it was built.
Journal: Write a Report Include what you found. Also in your report:

• Draw a diagram of a bumper car collision. Use an angle to show how a car might change direction after it collides with something. Measure and label the angle you drew.
• Describe your angle using some of the vocabulary terms in this topic.

Review What You Know

Vocabulary

Choose the best term from the box. Write it on the blank.

• angle
• right angle
• line
• sixth

Question 1.
A(n) _________ is one of 6 equal parts of a whole, written as $$\frac{1}{6}$$.
Answer: A sixth is one of 6 equal parts of a whole, written as $$\frac{1}{6}$$.

Question 2.
A(n) _________ is a figure formed by two rays that share the same endpoint.
Answer: An angle is a figure formed by two rays that share the same endpoint.

Question 3.
A(n) ___________ is an angle that forms a square corner.
Answer: A right angle is an angle that forms a square corner.

Find the sum or difference.
Question 4.
45 + 90
Given the two numbers 45 and 90
We have to find the sum of 45, 90.
The sum of 45 + 90 = 135.

Question 5.
120 – 45
Given the two numbers 120 and 45
We have to find the difference of 120, 45.
The difference of 120 – 45 = 75.

Question 6.
30 + 150
Given the two numbers 30 and 150
We have to find the sum of 30, 150.
The sum of 30 + 150 = 180.

Question 7.
180 – 135
Given the two numbers 180 and 135
We have to find the difference of 180, 135.
The difference of 180 – 135 = 45.

Question 8.
60 + 120
Given the two numbers 60 and 120
We have to find the sum of 60, 120.
The sum of 60 + 120 = 180.

Question 9.
90 – 45
Given the two numbers 90 and 45
We have to find the difference of 90, 45.
The difference of 90 – 45 = 45.

Parts of a Whole

Tell the fraction that represents the shaded part of the whole.
Question 10.

The fraction that represents the shaded part of the whole number is 1/6.

Question 11.

The fraction that represents the shaded part of the whole number is 1/4.

Question 12.

The fraction that represents the shaded part of the whole number is 1/10.

Dividing

Find the quotient.
Question 13.
360 ÷ 6
The quotient of 360 ÷ 6 = 60
60 is the quotient.

Question 14.
180 ÷ 9
The quotient of 180 ÷ 9 = 20
20 is the quotient.

Question 15.
360 ÷ 4
The quotient of 360 ÷ 4 = 90
90 is the quotient.

Problem Solving

Question 16.
Make Sense and Persevere Gary has $4. Mary has twice as many dollars as Gary. Larry has 4 fewer dollars than Mary. How much money do Gary, Mary, and Larry have in all? Answer: Given that, Total number of dollars at Gary =$4
Mary has twise as many dollars sa Gray = 2 × $4 =$8.
Larry has 4 fewer dollars than Mary = $8 – 4 =$4.
Total money with Gary, Marry, and Larry = $8 +$4 + $4 =$16.

Pick a Project

PROJECT 15A
Can you find angles in stringed instruments?
Project: Make a Stringed Instrument

PROJECT 15B
How are angles important in origami?
Project: Present an Origami Animal

PROJECT 15C
How are angles formed by airplane paths?
Project: Trace a Flight Plan

3-ACT MATH PREVIEW

Math Modeling
Game of Angles

I can … model with math to solve a problem that involves measuring angles and computing with angle measures.

### Lesson 15.1 Lines, Rays, and Angles

Solve & Share
A right angle forms a square corner, like the one shown below. Find examples of right angles in your classroom. Draw two angles that are open less than the right angle. Solve this problem any way you choose.
I can … recognize and draw lines, rays, and different types of angles.

Look Back! Reasoning Draw an angle that is open more than a right angle.

Essential Question
What Are Some Common Geometric Terms?

Visual Learning Bridge
Point, line, line segment, ray, right angle, acute angle, obtuse angle, and straight angle are common geometric terms.

An angle is formed by two rays that have the same endpoint.

Convince Me! Look for Relationships Complete each figure to show the given angle.

Guided Practice

Do You Understand?
Question 1.
What geometric term describes a part of a line that has one endpoint? Draw an example.
A ray is a part of a line that has one endpoint and continues on forever in one direction.

Question 2.
What geometric term describes a part of a line that has two endpoints? Draw an example.
A line segment is a part of a line with two endpoints.

Question 3.
What geometric term describes an angle that forms a square corner? Draw an example.
A right angle forms a square corner.

Do You Know How?
For 4-7, use geometric terms to describe what is shown.
Question 4.

∠PX represents the line segment with two end points.

Question 5.

∠PQR is a right angle triangle. A right angle forms a forms a square corner.

Question 6.

∠BY is a straight angle.

Question 7.

∠LMN is an acute angle. An acute angle is open less than a right angle.

Independent Practice

For 8-11, use geometric terms to describe what is shown.
Question 8.

∠HOS is an obtuse angle. An obtuse angle is open more than a right angle but less than a straight angle.

Question 9.

∠BD represents the line segment with two endpoints.

Question 10.

Answer: ∠BY is a straight angle.

Question 11.

∠STP is a right angle.

For 12-15, draw the geometric figure for each term.
Question 12.
Line segment

Question 13.
Point

Question 14.
Ray

Question 15.
Line

Problem Solving

For 16-18, use the map of Nevada. Write the geometric term that best fits each description. Draw an example.

Question 16.
Be Precise The route between 2 cities.

Question 17.
The cities

Question 18.
The corner formed by the north and west borders

Question 19.
Vocabulary Write a definition for right angle. Draw a right angle. Give 3 examples of right angles in the classroom.

Question 20.
Higher Order Thinking Nina says she can make a right angle with an acute angle and an obtuse angle that have a common ray. Is Nina correct? Draw a picture and explain.
Answer: If an angle is acute, it can’t also be a right angle.

Assessment Practice

Question 21.
Which geometric term describes ∠HJK?

A. Acute angle
B. Obtuse angle
C. Right angle
D. Straight angle
The geometric term ∠HJK represents a straight angle.
Thus the answer is option D.

Question 22.
Lisa drew 2 rays that share an endpoint. Which of the following is Lisa’s drawing?

### Lesson 15.2 Understand Angles and Unit Angles

Solve & Share
If a clock shows it is 3 o’clock, how could you describe the smaller angle made by the two hands of the clock? Solve this problem any way you choose.
I can … use what I know about fractions to measure angles.

Look Back! At 3 o’clock, what two fractions do the hands divide the clock into?

Essential Question
What is the Unit Used to Measure Angles?

Visual Learning Bridge
An angle is measured with units called degrees. An angle that turns through $$\frac{1}{360}$$ of a circle is called a unit angle. How can you determine the angle measure of a right angle and the angles that turn through $$\frac{1}{6}$$ and $$\frac{2}{6}$$ of a circle?
You measure length in inches, area in square centimeters, and capacity in ounces. You measure an angle in degrees, o. A full circle has an
angle measure of 360°.

Divide to find the angle measure of a right angle.

Right angles divide a circle into 4 equal parts.
360° ÷ 4 = 90°
The angle measure of a right angle is 90°.

Divide to find the measure of an angle that turns through $$\frac{1}{6}$$ of a circle.

$$\frac{1}{6}$$ of a circle is one part of the circle that is divided into 6 equal parts.
360° ÷ 6 = 60°
The angle measure is 60°.

Add to find the measure of an angle that turns through $$\frac{2}{6}$$ of a circle.

Remember $$\frac{2}{6}$$ = $$\frac{1}{6}$$ + $$\frac{1}{6}$$
Add to calculate the measure of $$\frac{2}{6}$$ of a circle.
60° + 60° = 120° The angle measure of of a circle is 120°.

Convince Me! Critique Reasoning Susan thinks the measure of angle B is greater than the measure of angle A. Do you agree? Explain.

Another Example!
Find the fraction of a circle that an angle with a measure of 45° turns through

A 45° angle turns through $$\frac{45}{360}$$ of a circle.
45° × 8 = 360°, so 45° is $$\frac{1}{8}$$ of 360°.
One 45° angle is $$\frac{1}{8}$$ of a circle.

Guided Practice

Do You Understand?
Question 1.
What fraction of the circle does a 120° angle turn through?
A circle contains 360°
To find the fraction of the circle’s 120° angle.
Divide 120° by 360°
120°/360° = 1/3
Thus the fraction of the circle that does a 120° angle turn through is 1/3.

Question 2.
Mike cuts a pie into 4 equal pieces. What is the angle measure of each piece? Write and solve an equation.
Given,
Mike cuts a pie into 4 equal pieces.
A circle contains 360°
360/90 = 4
90 degrees because when you cut each of them they will each measure 90 degrees.

Do You Know How?
Question 3.
A circle is divided into 9 equal parts. What is the angle measure of one of those parts?

Given that,
A circle is divided into 9 equal parts.
There are 360 degrees in a circle.
That means that we first divide 360 by 9.
We have the measure for one of the nine parts of the circle, but we need two.
All we do is multiply 40 by 2 to get 80 degrees.

Question 4.
An angle turns through of the circle. What is the measure of this angle?

An angle turns through the circle.
There are 360 degrees in a circle.
That means that we first divide 360 by 4.
360 ÷ 4 = 90
Thus the measure of the angle of the above circle is 90 degrees.

Independent Practice

For 5-8, find the measure of each angle.
Question 5.
The angle turns through $$\frac{1}{5}$$ of the circle.

An angle turns through the circle.
There are 360 degrees in a circle.
That means that we first divide 360 by 5.
360 ÷ 5 = 72
Thus the measure of the angle of the above circle is 72 degrees.

Question 6.
The angle turns through $$\frac{3}{8}$$ of the circle.

An angle turns through the circle.
There are 360 degrees in a circle.
That means that we first divide 360 by $$\frac{3}{8}$$.
360 × $$\frac{3}{8}$$ = 135
Thus the measure of the angle of the above circle is 135 degrees.

Question 7.
The angle turns through $$\frac{2}{5}$$ of the circle.

The angle turns through $$\frac{2}{5}$$ of the circle.
There are 360 degrees in a circle.
That means that we first divide 360 by $$\frac{2}{5}$$.
360 × $$\frac{2}{5}$$ = 144
Thus the measure of the angle of the above circle is 144 degrees.

Question 8.
The angle turns through $$\frac{2}{6}$$ of the circle.

The angle turns through $$\frac{2}{6}$$ of the circle.
There are 360 degrees in a circle.
That means that we first divide 360 by $$\frac{2}{6}$$.
360 × $$\frac{2}{6}$$ = 120
Thus the measure of the angle of the above circle is 120 degrees.

Problem Solving

Question 9.
Use the clock to find the measure of the smaller angle formed by the hands at each time.

a. 3:00
b. 11:00
c. 2:00

11:00
Thus the correct answer is option B.

Question 10.
Algebra Jacey wrote an equation to find an angle measure. What do variables a and b represent in Jacey’s equation? 360° ÷ a = b
We know that Jacey wrote an equation to find an angle measure.
The equation is as follows:
360 ÷ a = b
So, if the angle is 360 degrees is a central angle, i.e, an angle between 2 radii in a circle, then
a = the number of equal parts the circle is divided.
b = the measure of the central angle of each part.

Question 11.
enVision® STEM A mirror can be used to reflect a beam of light at an angle. What fraction of a circle would the angle shown turn through?

Given,
A mirror can be used to reflect a beam of light at an angle.
120° ÷ 360° = 1/3
The fraction of a circle would the angle turn is 1/3.

Question 12.
Malik paid $32.37 for three books. One book cost$16.59. The second book cost $4.27. How much did the third book cost? Use bills and coins to solve. Answer: Given, Malik paid$32.37 for three books. One book cost $16.59. The second book cost$4.27.
$16.59 +$4.27 = $20.86$32.37 – $20.86 =$11.51
Thus the cost of third book is $11.51 Question 13. Make Sense and Persevere A pie was cut into equal parts. Four pieces of the pie were eaten. The 5 pieces that remained created an angle that measured 200°. What was the angle measure of one piece of pie? Answer: Given that, A pie was cut into equal parts. Four pieces of the pie were eaten. The 5 pieces that remained created an angle that measured 200°. 200/5 = 40 Thus the angle measure of one piece of pie is 40 degrees. Question 14. Higher Order Thinking Jake cut a round gelatin dessert into 8 equal pieces. Five of the pieces were eaten. What is the angle measure of the dessert that was left? Answer: Given, Jake cut a round gelatin dessert into 8 equal pieces. Five of the pieces were eaten. Fraction is the number of pieces eaten by the total number of pieces. 5/8 360 × 5/8 = 225 Thus the angle measure of the desert that was left. Assessment Practice Question 15. Select all choices that show an angle measure of 120°. Use the clock to help. ☐ 10 o’clock ☐ $$\frac{2}{6}$$ of a pie ☐ $$\frac{2}{3}$$ of a circle ☐ 4 o’clock ☐ 8 o’clock Answer: $$\frac{2}{6}$$ of a pie ### Lesson 15.3 Measure with Unit Angles Solve & Share The smaller angles on the tan pattern block shown each measure 30°. How can you use the angles on the pattern block to determine the measure of the angle below? Solve this problem any way you choose. I can … use angles I know to measure angles I do not know. Look Back! Two right angles make a straight angle. How many 45° angles form a straight angle? Explain. Essential Question How Can You Measure Angles? Visual Learning Bridge Holly traced around a trapezoid pattern block. She wants to find the measure of the angle formed shown to the right. What can Holly use to measure the angle? The measure of a unit angle is 1 degree. Just like adding inches + inches, you can add degrees + degrees. So 5° = 1° + 1° + 1° + 1° + 1° or 5 × 1° Use an angle you know to find the measure of another angle. The smaller angle of the tan pattern block measures 30°. A 30-degree angle turns through 30 one-degree angles. The angle of the trapezoid pattern block is equal to 2 of the smaller angles of the tan pattern block. Each smaller angle is 30°. The measure of the trapezoid angle is 60°. A 60-degree angle turns through 60 one-degree angles. Convince Me! Generalize What do you notice about the number of one-degree angles in an angle measure? Guided Practice Do You Understand? Question 1. How many 30° angles are in a 180° angle? Explain. Answer: 180 ÷ 30 = 6 Thus there are 6 30° angles are in a 180° angle. Question 2. How many 15° angles are in a 180° angle? Use your answer to Exercise 1 to explain. Answer: 180 ÷ 15 = 12 Thus there are 12 15° angles are in a 180° angle. Do You Know How? For 3-4, use angles you know to find the measure of each angle. Explain how the angles in the square can help. Question 3. Answer: By seeing the above figure we can say that it is a right angle. 90 ÷ 2 = 45 degrees Question 4. Answer: By seeing the above figure we can say that it is an obtuse angle. 90° + 45° = 135° Independent Practice For 5-13, find the measure of each angle. Use pattern blocks to help. Question 5. Answer: The measure of the angle is 70 degrees. Question 6. Answer: The measure of the angle is 90 degrees. Question 7. Answer: The measure of the angle is 60 degrees. Question 8. Answer: The measure of the angle is 110 degrees. Question 9. Answer: The measure of the angle is 120 degrees. Question 10. Answer: The measure of the angle is 90 degrees. Question 11. Answer: The measure of the angle is 90 degrees. Question 12. Answer: The measure of the angle is 180 degrees. Question 13. Answer: The measure of the angle is 90 degrees. Problem Solving Question 14. Use Appropriate Tools What is the measure of the angle of the yellow hexagon pattern block? Answer: Question 15. What is the measure of the smaller angle formed by the clock hands when it is 5:00? Answer: 120 degrees Question 16. How many 30° angles are in a circle? Write and solve a multiplication equation to explain. Answer: The total angle in a circle is 360 degrees 360°/30° = 12 Thus there are 120 30° angles in a circle. Question 17. How many unit angles make up the smaller angle formed by the hands of a clock when it is 3:00? Explain. Answer: 3/12 = x°/360° 1/4 = x°/360° x° = 90° Question 18. Veronica purchases a rug with a length of 16 feet and a width of 4 feet. One fourth of the rug is purple and the rest is blue. What is the area of the blue part of the rug? Answer: Veronica purchases a rug with a length of 16 feet and a width of 4 feet. One-fourth of the rug is purple and the rest is blue. A = base × height Area = 16 × 4 = 64 sq. ft Determine the blue area: 64 sq. ft — 1 x —- 3/4 x = 3/4 × 64 x = 192/4 x = 48 sq.ft Question 19. Higher Order Thinking The hands of a clock form a 120° angle. Name two different times it could be. Answer: The 120° angles make up 1/3 of one full rotation of the hands-on clock, with a full rotation divided up into 12 equal sections. 1/3 of those 12 sections would be 4 sections, and since we are looking for exact hours that represent this angle, we need four sections forming each of the 120° angles with the minute hand on the clock being at 12. Therefore, the hour hand can be at the 4 for 4 o’clock and it could also be at the 8 for 8 o’clock. Assessment Practice Question 20. The clock reads 9:00. What is the angle measure? A. 90° B. 180° C. 270° D. 360° Answer: A. 90° The angle measure is 90 degrees. Thus the correct answer is option A. Question 21. How many 60° angles are in 360° angle? A. 3 B. 6 C. 10 D. 12 Answer: 6 Explanation: 360/60 = 6 There are 6 60° angles are in 360° angle. Thus the correct answer is option B. ### Lesson 15.4 Measure and Draw Angles Solve & Share Find the measure of ZABC. Solve this problem any way you choose. I can … use a protractor to measure and draw angles. Look Back! Use Appropriate Tools Use the protractor to draw an angle that measures 110°. Essential Question How Do You Use a Protractor? Visual Learning Bridges A protractor is a tool that is used to measure and draw angles. A partially folded crane is shown at the right. Measure ∠PQR. The angle, ∠PQR can also be written as ∠RQP. Measure Angles Measure ∠PQR. Place the protractor’s center on the angle’s vertex, Q. Place one of the 0° marks on $$\overrightarrow{Q R}$$. Read the measure where $$\overrightarrow{Q P}$$ crosses the protractor. If the angle is acute, use the lesser number. If the angle is obtuse, use the greater number. The measure of ∠PQR is 45°. Draw Angles Draw an angle that measures 130°. Draw $$\overrightarrow{T U}$$. Place the protractor so the center is over point T, and one of the 0° marks is on $$\overrightarrow{T U}$$. Place a point at 130°. Label it W. Draw $$\overrightarrow{T W}$$. The measure of ∠WTU is 130°. Convince Me! Be Precise How do you know the measure of ∠UTS is 60° and not 120°? Guided Practice Do You Understand? Question 1. What is the angle measure of a straight line? Answer: The angle measure of a straight line is 0° or 180° Question 2. What are the vertex and rays of ∠ABC? Explain. Answer: The vertex is the common point at which the two lines or rays are joined. Point B in the figure above is the vertex of the angle ∠ABC. Do You Know How? For 3-4, use a protractor to measure each angle. Question 3. Answer: 50 degrees Question 4. Answer: 120 degrees For 5-6, use a protractor to draw each angle. Question 5. 110° Answer: Question 6. 50° Answer: Independent Practice For 7-14, measure each angle. Tell if each angle is acute, right, or obtuse. Remember an acute angle is less than 90° and an obtuse angle is greater than 90° but less than 180°. Question 7. Answer: The above angle is greater than 90° so it is an obtuse angle. Question 8. Answer: The above angle is less than 90° so it is an acute angle. Question 9. Answer: The above angle is less than 90° so it is an acute angle. Question 10. Answer: The above angle is greater than 90° so it is an obtuse angle. Question 11. Answer: The above angle is equal to 90° so it is a right angle. Question 12. Answer: The above angle is less than 90° so it is an acute angle. Question 13. Answer: The above angle is greater than 90° so it is an obtuse angle. Question 14. Answer: The above angle is less than 90° so it is an acute angle. For 15-18, use a protractor to draw an angle for each measure. Question 15. 140° Answer: Question 16. 180° Answer: Question 17. 65° Answer: Question 18. 25° Answer: Problem Solving Question 19. Measure all the angles created by the intersection of Main Street and Pleasant Street. Explain how you measured. Answer: 25 degrees Question 20. Use a protractor to find the measure of the angle, then use one of the angle’s rays to draw a right angle. Find the measure of the the angle that is NOT a right angle. Answer: 110 degrees Question 21. Critique Reasoning Gail and 3 friends share half a pie. Each piece of pie is the same size. Gail believes each piece of pie has an angle measure of 25°. Is Gail correct? Explain. Answer: Given, Gail and 3 friends share half a pie. Each piece of pie is the same size. Gail believes each piece of the pie has an angle measure of 25°. Half of a pie is 50%, divide that by three and that equals about 16.67 if you round it, but if each piece has an angle of 25, those pieces aren’t equal. Question 22. Janet made 5 three-point shots in her first game and 3 in her second game. She also made 4 two-point shots in each game. How many total points did Janet score in the two games? Answer: Janet score 35 points Question 23. Higher Order Thinking Maya designed two intersecting roads. She drew the roads so one of the angles at the intersection was 35o. What are the three other angle measurements formed by the intersection? Answer: To find the other three angles in the intersection, we need to know that in an intersection of two lines, one angle is the supplement of the other, and the other two are equal to the first and the second one, that is, if the first angle is x, the values of the four angles are x, 180 – x, x and 180 – x so if one angle is 35 degrees, we have that the three other angles are 145 degrees, 35 degrees, and 145 degrees. Assessment Practice Question 24. Find the measure of the angle shown. Answer: 35 degrees Question 25. Find the measure of the angle shown. Answer: 110 degrees ### Lesson 15.5 Add and Subtract Angle Measures Solve & Share Draw $$\overrightarrow{B C}$$ that divides ∠ABD into two smaller angles. Measure each angle. Solve this problem any way you choose. I can … use addition and subtraction to solve problems with unknown angle measures. Look Back! How can you relate the measures of the two smaller angles to the measure of the larger angle above using an equation? Essential Question How Can You Add and Subtract to Find Unknown Angle Measures? Visual Learning Bridge Elinor designs wings for biplanes. First she draws a right angle, ∠ABC. Then she draws $$\overrightarrow{B E}$$. She finds ∠EBC measures 30°. How can Elinor find the measure of ∠ABE without using a protractor? ∠ABC is decomposed into two non-overlapping parts. ∠EBC and ∠ABE do not overlap, so the measure of right ∠ABC is equal to the sum of the measures of its parts. The measure of ∠ABC equals the measure of ∠ABE plus the measure of ∠EBC. All right angles measure 90° Write an equation to determine the missing angle measure. n + 30° = 90° Solve the equation. n = 90° – 30° n = 60° The measure of ∠ABE is 60°. Convince Me! Make Sense and Persevere ∠ABD is a straight angle. What is the measure of ∠ABE if the measure of ∠DBC is 115° and the measure of ∠CBE is 20°? How did you decide? Write and solve an equation. Guided Practice Do You Understand? Question 1. Use the information below to draw and label a diagram. ∠PQR measures 45°. ∠RQS measures 40°. ∠PQR and ∠RQS do not overlap. Write and solve an equation to find the measure of ∠PQS. Answer: ∠PQR + ∠RQS + n = 90 Do You Know How? For 2-3, use the diagram to the right of each exercise. Write and solve an equation to find the missing angle measure. Question 2. What is the measure of ∠EBC if ∠ABE measures 20°? Answer: ∠EBC + ∠ABE = 90° ∠EBC + 20° = 90° ∠EBC = 90° – 20° ∠EBC = 70° Question 3. What is the measure of ∠AEB if ∠CEB measures 68°? Answer: ∠AEB + ∠CEB = 180° ∠AEB + 68° = 180° ∠AEB = 180° – 68° ∠AEB = 112° Independent Practice For 4-7, use the diagram to the right. Write and solve an addition or subtraction equation to find the missing angle measure. Question 4. What is the measure of ∠FGJ if ∠JGH measures 22°? Answer: ∠FGJ + ∠JGH = 90° ∠FGJ + 22° = 90° ∠FGJ = 90° – 22° ∠FGJ = 68° Question 5. What is the measure of ∠KGF if ∠EGK measures 59°? Answer: ∠KGF + ∠EGK = 90° ∠KGF + 59°= 90° ∠KGF = 90° – 59° ∠KGF = 31° Question 6. Use the angle measures you know to write an equation to find the angle measure of ∠EGH. What kind of angle is ∠EGH? Answer: ∠EGK + ∠KGF + ∠FGJ + ∠JGH = 180° Question 7. Which two non-overlapping angles that share a ray make an obtuse angle? Use addition to explain. Answer: ∠KGH and ∠EGJ are the two non-overlapping angles that share a ray that makes an obtuse angle. Problem Solving Question 8. Shane says a straight angle always has 180° degrees. Is Shane correct? Explain. Answer: Yes Shane is correct. Question 9. Model with Math Talla earns 85¢ for cans she recycles. If she gets a nickel for each can, how many cans does Talla recycle? Draw a bar diagram to represent how to solve the problem. Answer: Question 10. Alex draws an angle that measures 110°. He then draws a ray that divides the angle into 2 equal parts. What is the measure of each smaller angle? Answer: To solve for this use the expression 110/2 since you are dividing an angle into two equal parts. When you simplify, you should get 55 degrees as the measure of each angle. Question 11. Six angles share a vertex. Each of the angles has the same measure. The sum of the measures of the angles is 330°. What is the measure of one angle? Answer: Given, Six angles share a vertex. Each of the angles has the same measure. The sum of the measures of the angles is 330°. The measure of one angle is 55 because 330 divided by six is 55 Question 12. Higher Order Thinking Li uses pattern blocks to make a design. He puts 5 pattern blocks together, as shown in the diagram. The measure of ∠LJK is 30°. Name all the 60° angles shown that have point J as a vertex. Answer: Li uses pattern blocks to make a design. He puts 5 pattern blocks together, as shown in the diagram. The measure of ∠LJK is 30°. divide 120 by 4 you get 30 then multiply 30 by 2. Assessment Practice Question 13. Carla drew two acute non-overlapping angles that share a ray and labeled them ∠JLK and ∠KLM. The two angles have different measures. Carla says ∠JLM is greater than a right angle. Part A Is it possible for Carla to be correct? Write to explain. Answer: No, because the acute angle is less than the right angle. Part B Write an equation showing one possible sum for Carla’s angles. Answer: ∠JLK + ∠KLM = 90 degrees ### Lesson 15.6 Problem Solving Use Appropriate Tools Solve & Share Caleb is standing next to the tallest building in a city. Determine the measure of the 3 angles with the vertex at the tallest building and rays on the music hall, the live theater, and the art museum. Tell what tool you used and explain why the measures make sense relative to each other. I can … use appropriate tools strategically to solve problems. Thinking Habits Be a good thinker! These questions can help you. • Which tools can I use? • Why should I use this tool to help me solve the problem? • Is there a different tool I could use? • Am I using the tool appropriately? Look Back! Use Appropriate Tools Could you use a ruler to find the angle measures? Explain. Essential Question How Can You Select and Use Appropriate Tools to Solve Problems? Visual Learning Bridge Trevor and Holly are drawing trapezoids to make a design. They need to find the measures of the angles formed by the sides of the trapezoid and the length of each side of the trapezoid. What tools are needed to find the measures of the angles and the lengths of the sides? What do you need to do to copy the trapezoid? I need to measure the angles, then measure the sides. Which tool can I use to help me solve this problem? I can • decide which tool is appropriate. • explain why it is the best tool to use. • use the tool correctly. Here’s my thinking. Use a protractor to measure the angles. The angles measure 120° and 60°. Then, use a ruler to measure the length of each side. The lengths are $$\frac{3}{4}$$ inch, $$\frac{3}{4}$$ inch, $$\frac{3}{4}$$ inch, and 1$$\frac{1}{2}$$ inches. Convince Me! Use Appropriate Tools What other tools could be used to solve this problem? Why are a protractor and a ruler more appropriate than other tools? Guided Practice Lee brought 1$$\frac{3}{5}$$ pounds of apples to the picnic. Hannah brought $$\frac{4}{5}$$ pound of oranges. Lee said they brought 2$$\frac{2}{5}$$ pounds of fruit in all. Lee needs to justify that 1$$\frac{3}{5}$$ + $$\frac{4}{5}$$ = 2$$\frac{2}{5}$$ Question 1. What tool could Lee use to justify the sum? Answer: Fraction strips Question 2. How can Lee use a tool to justify the sum? Draw pictures of the tool you used to explain. Answer: Independent Practice Use Appropriate Tools What are the measures of the sides and angles of the parallelogram shown? Use Exercises 3-5 to help solve. Question 3. What tools can you use to solve this problem? Answer: Question 4. Explain how to use the tool you chose to find the measures of the angles. Label the figure with the measures you find. Answer: Question 5. Explain how to use the tool you chose to find the lengths of the sides. Label the figure with the measures you find. Answer: Problem Solving Performance Task Mural Before Nadia paints a mural, she plans what she is going to paint. She sketches the diagram shown and wants to know the measures of ∠WVX, ∠WVY, ∠XWY, and ∠YVZ. Question 6. Reasoning What quantities are given in the problem and what do the numbers mean? What do you know from the diagram? Answer: Question 7. Make Sense and Persevere What do you need to find? Answer: Question 8. Use Appropriate Tools Measure ∠WVX, ∠WVY, and ∠YVZ. What is the best tool to use? Answer: Question 9. Model with Math Write and solve an equation which could be used to find the measure of ∠XVY. What is the measure of the angle? Answer: ### Topic 15 Fluency Practice Activity Follow the path Shade a path from START to FINISH. Follow sums and differences that are between 20,000 and 25,000. You can only move up, down, right, or left. I can … add and subtract multi-digit whole numbers. Answer: ### Topic 15 Vocabulary Review Word List • acute angle • angle measure • degree (0) • line • line segment • obtuse angle • point • protractor • ray • right angle • straight angle • unit angle • vertex Understand Vocabulary Question 1. Cross out the terms that do NOT describe an angle with a square corner. acute angle right angle obtuse angle straight angle Answer: Straight angle. Explanation: We can describe acute angle, obtuse angle, right angle with a square corner. A straight angle cannot be described with a square corner. Thus the answer is option D. Question 2. Cross out the terms that do NOT describe an angle open less than a right angle. acute angle right angle obtuse angle straight angle Answer: obtuse angle, straight angle are the terms that do NOT describe an angle open less than a right angle. Question 3. Cross out the terms that do NOT describe an angle that forms a straight line. acute angle right angle obtuse angle straight angle Answer: acute angle, right angle, obtuse angle are the terms that do NOT describe an angle that forms a straight line. Question 4. Cross out the terms that do NOT describe an angle open more than a right angle, but less than a straight angle. acute angle right angle obtuse angle straight angle Answer: acute angle, right angle and straight angle are the terms that do NOT describe an angle open more than a right angle, but less than a straight angle. Label each example with a term from the Word List. Question 5. ___________ Answer: straight line, because it has does not have endpoints. Question 6. ___________ Answer: ray, because it has one endpoint. Question 7. ___________ Answer: line segment, because it has two endpoints. Question 8. ____________ Answer: 2 rays with a single endpoint Use Vocabulary in Writing Question 9. Describe how to measure an angle. Use at least 3 terms from the Word List in your explanation. Answer: We can measure an angle using a protractor, compass, and blocks. ### Topic 15 Reteaching Set A pages 549-552 A ray has one endpoint and continues on forever in one direction. A line segment is a part of a line with two endpoints. An angle is formed by two rays with a common endpoint. Remember that a line segment is a part of a line. Use geometric terms to describe what is shown. Question 1. Answer: line segment, because it has two endpoints. Question 2. Answer: right angle Question 3. Answer: obtuse angle Question 4. Answer: ray, because it has one endpoint Set B pages 553-556 The angle below is $$\frac{1}{3}$$ of the circle. $$\frac{1}{3}$$ means 1 of 3 equal parts. 360° ÷ 3 = 120° The measure of this angle is 120°. Remember there are 360° in a circle. A circle is cut into eighths. What is the angle measure of each piece? Question 1. Use division to solve. Answer: Given, A circle is cut into eighths. 360/8 = 45 degrees 90/2 = 45 degrees Question 2. Use multiplication to solve. Answer: Set C pages 557-560 You can use an angle you know to find the measure of other angles. The smaller angle of the tan pattern block has a measure of 30° Three of the 30° angles will fit into the angle. Add: 30° + 30° + 30o = 90° The measure of this angle is 90°. Remember you can use any angle that you know the measure of to find the measure of other angles. Find the measure of each angle. Use pattern blocks. Question 1. Answer: 55 degrees Question 2. Answer: 110 degrees Set D pages 561-564 The measure of this angle is 60°. Remember that a straight angle has a measure of 180°. Measure the angles. Question 1. Answer: 90 degrees Question 2. Answer: 120 degrees Set E pages 565-568 When an angle is decomposed into non-overlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Remember you can subtract to find angle measures. ∠ABD is decomposed into two non-overlapping angles, ∠ABC and ∠CBD. Complete the table. Answer: Set F pages 569-572 Think about these questions to help you use appropriate tools strategically. Thinking Habits • Which tools can I use? • Why should I use this tool to help me solve the problem? • Is there a different tool could use? • Am I using the tool appropriately? Remember there may be more than one appropriate tool to use to solve a problem. One-eighth of the pie is missing from the tin. Question 1. What tools can Delia use to measure the angle of the missing piece? Answer: protractor can be used to measure the angle of the missing piece. Question 2. How can you calculate the measure? Answer: We can calculate the measure of an angle using protractor. ### Topic 15 Assessment Practice Question 1. What is the measure of the angle shown below? Name a type of angle that has an angle measure greater than the angle shown. Answer: The measure of the angle is 35 degrees. The name of the angle is an acute angle. Question 2. Megan needs to find the measures of the angles on a bridge. A. Find the measure of ∠YXW if ∠YXZ is 85° and ∠ZXW is 40°. Write and solve an addition equation. Answer: B. Find the measure of ∠CAD if ∠CAB is a right angle and ∠DAB is 45°. Write and solve a subtraction equation. Answer: Question 3. If you divide a circle into 360 equal angles, what is the angle measure of each angle? Answer: you divide a circle into 360 equal angles 360 ÷ 6 = 60 degree 60 + 60 + 60 + 60 + 60 + 60 Thus the angle measure of each angle is 60 degrees. Question 4. Choose the correct term from the box to complete each statement. Line Segment Ray A __________ has one endpoint. A __________ has two endpoints. Answer: A Ray has one endpoint. A Line segment has two endpoints. Question 5. Draw an example of a line $$\overleftrightarrow{R S}$$. Label a point T between points Rand S. Using point T, draw ray $$\overrightarrow{T V}$$. Answer: Question 6. JKL is a straight angle decomposed into 2 non-overlapping angles, ∠JKM and ∠MKL. If ∠MKL measures 104°, what type of angle is ∠JKM? What is the measure of ∠JKM? Answer: Question 7. ∠ABC has a measure of 40°and ∠CBD has a measure of 23°. The angles share a ray and form ZABD. Write and solve an equation to find the measure of ∠ABD. Answer: ∠ABC + ∠CBD + ∠ABD = 90° Question 8. Emma cuts slices from pies. Match each fraction with the equal angle measure. Answer: Question 9. Select all the true statements. ☐ An acute angle is open less than a right angle. ☐ An obtuse angle makes a square corner. ☐ A right angle is open less than an obtuse angle. ☐ A straight angle forms a straight line. ☐ All obtuse angles have the same measure. Answer: ☐ An acute angle is open less than a right angle. ☐ A right angle is open less than an obtuse angle. ☐ A straight angle forms a straight line. Thus A, C, D are the true statements. Question 10. Two wooden roof beams meet at a 60° angle. Draw an angle to represent how the beams meet. Answer: Question 11. Which geometric term best describes the light that shines from a flashlight? A. Point B. Ray C. Line segment D. Line Answer: Ray Question 12. Terry is measuring ∠RST using pattern blocks. The smaller angle of each of the tan pattern blocks shown below measures 30°. What is the measure of ∠RST? Explain. Answer: Given, Terry is measuring ∠RST using pattern blocks. The smaller angle of each of the tan pattern blocks shown below measures 30°. Thus the measure of ∠RST is 30° + 30° + 30° + 30° = 120 Question 13. Identify an acute angle, a right angle, and an obtuse angle in the figure below. Answer: ∠AEB is an acute angle ∠DEC is a right angle ∠BED is an obtuse angle ### Topic 15 Performance Task Ancient Roads The ancient Romans built roads throughout their empire. Many roads were paved with stones that fit together. The spaces between the stones were filled with sand and gravel. Many of these roads still exist today, over 2,000 years after they were built. Question 1. As seen in the Roman Road figure, the stones formed angles and geometric figures. Part A What geometric figure has one endpoint at F and goes on forever through point G? Answer: E Part B Is ∠EDA right, acute, or obtuse? Explain. Answer: ∠EDA is an obtuse angle. Part C ∠EDG turns through $$\frac{1}{8}$$ of a circle. What is its angle measure? Explain. Answer: ∠EDG turns through $$\frac{1}{8}$$ of a circle. 80 Question 2. Answer the following to find the measure of ∠HJK and ∠HJL in the Measuring a Roman Road figure. Part A Name two tools you could use to measure the angles. Answer: Protractor and blocks pattern could be used to measure the angles. Part B The smaller angle of the tan pattern block measures 30°, as shown in the Tan Pattern Block figure. A 30-degree angle is 30 one-degree angles. What is the measure of ∠HJK in the Measuring a Roman Road figure? Explain. Answer: Part C What is the measure of ∠HJL in the Measuring a Roman Road figure? Write and solve an equation to find the measure of the angle. Answer: #### enVision Math Common Core Grade 4 Answer Key ## Envision Math Common Core Grade 4 Answer Key Topic 16 Lines, Angles, and Shapes ## Envision Math Common Core 4th Grade Answers Key Topic 16 Lines, Angles, and Shapes Essential Questions: How can you classify triangles and quadrilaterals? What is line symmetry? enVision STEM Project: Senses and Symmetry Do Research The location of an animal’s eyes helps it to survive in the wild. Use the Internet or other sources to find why some animals have eyes on the sides of their head and others have eyes on the front. Journal: Write a Report Include what you found. Also in your report: Most animals are the same on both sides of their body. Use a line of symmetry to help make a simple drawing of your favorite animal’s face. Draw both sides of the animal’s face the same. Explain how you know that both sides of your drawing are the same. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • angle • polygon • quadrilateral • triangle Question 1. A ___________ is a closed figure made up of straight line segments. Answer: A polygon is a closed figure made up of straight line segments Question 2. A polygon with three sides is a(n) _________. Answer: A polygon with three sides is a(n) Triangle Question 3. A(n) _________ is formed by two rays with the same endpoint. Answer: A(n) Angle is formed by two rays with the same endpoint. Shapes Choose the best term to describe each shape. Use each term once. Rectangle Rhombus Trapezoid Question 4. Answer: Rhombus Explanation : 2 pairs of parallel sides 4 equal sides Question 5. Answer: Trapezoid – only 1 pair of parallel sides Question 6. Answer: Rectangle Explanation : 2 pairs of parallel sides 4 right angles Lines Use geometric terms to describe what is shown. Question 7. Answer: It is said as line AB and represented a $$\overleftrightarrow{\mathrm{AB}}$$ Explanation : A line is a straight path of points that goes without end in both directions . Question 8. Answer: It is called as Ray $$\overrightarrow{C D}$$. The symbol of a ray is → Explanation : A ray is a part of line that has one end point and goes on without end in one direction . Question 9. Answer: Line segment – It is represented as $$\overline{E F}$$ Explanation : It is a line segment EF. Line segments are represented by a single overbar with no arrowheads over the letters representing the two endpoints. Problem Solving Question 10. Generalize Which generalization about these figures is NOT true? A. Each figure is a quadrilateral. B. Each figure has two pairs of parallel sides. C. Each figure has at least two sides of equal length. D. Each figure has 4 angles. Answer: Option B is not true Explanation : Figure 2 that is trapezoid EFGH has only one pair of parallel lines . Pick a Project PROJECT 16A How are dictionaries useful? Project: Create a Picture Dictionary PROJECT 16B How can shapes be used in art at the Dali Museum? Project: Create Cubist Art PROJECT 16C Do snowflakes have lines of symmetry? Project: Make Snowflakes PROJECT 16D Can animals have symmetry? Project: Draw a Line-Symmetric Animal ### Lesson 16-1 Lines Solve & Share The number line below is an example of a line. A line goes on forever in a straight path in two directions. Draw the following pairs of lines: two lines that will never cross, two lines that cross at one point, two lines that cross at two points. If you cannot draw the lines, tell why. I can … draw and identify perpendicular, parallel, and intersecting lines. Look Back! Terry said, “The lines shown intersect at three points.” Is Terry correct? Explain. Answer : No Terry is not correct . Explanation : A line can be defined as a straight one- dimensional figure that has no thickness and extends endlessly in both directions. It is often described as the shortest distance between any two points. Essential Question How Can You Describe Pairs of Lines? Answer : A pair of lines, line segments or rays are intersecting if they have a common point. This common point is their point of intersection. For example, two adjacent sides of a sheet of paper, a ruler, a door, a window and letters. Visual Learning Bridge A line is a straight path of points that goes on and on in opposite directions. A pair of lines can be described as parallel, perpendicular, or intersecting. Pairs of lines are given special names depending on their relationship. Convince Me! Be Precise Find examples in your classroom where you can identify parallel lines, intersecting lines, and perpendicular lines. Explain. Answer : (i) Intersecting lines – Edges of my Textbook and Notebook through any corner . (ii) Parallel lines – Opposite Edges of my textbook and notebook (iii) Perpendicular lines – Adjacent edges of my textbook and black board in my classroom. Guided Practice Do You Understand? Question 1. What geometric term could you use to describe the top and bottom edges of a book? Why? Answer: The top and bottom edges of a book are parallel lines. Explanation : Both the lines are equidistant from each other and they never intersect each other lines . Question 2. The blades of an open pair of scissors look like what pair of lines? Why? Answer: perpendicular lines . Explanation : When the scissor is open it look like perpendicular lines intersecting at right angle . Do You Know How? For 3-6, use the diagram. Question 3. Name four points. Answer: The Four points are W, X, Y and Z . Explanation : A point is a location represented by a dot. A point does not have any length, width, shape or size, it only has a position. When two distinct points are connected they form a line. Question 4. Name four lines. Answer: The four lines are WX, XZ, ZY and YW . Question 5. Name two pairs of parallel lines. Answer: The two pairs of parallel lines are WX ll YZ . Question 6. Name two pairs of perpendicular lines. Answer: The two pairs of perpendicular lines are WY ⊥ YZ and ZX ⊥ WX . Independent Practice For 7-12, use geometric terms to describe what is shown. Be as specific as possible. Question 7. Answer: LM ll to HI . Explanation : Both the lines are equidistant and doesn’t intersect so the given lines are parallel lines . Question 8. Answer: GH ⊥ EF Explanation : Both the lines are intersecting at a point forming Right angle so, the lines are perpendicular lines . Question 9. Answer: It is a point A . Explanation : A point is a location represented by a dot. A point does not have any length, width, shape or size, it only has a position. When two distinct points are connected they form a line. Question 10. Answer: Perpendicular paths Explanation : Perpendicular lines are lines that intersect at a right (90 degrees) angle. Both the Road paths are intersecting at a point forming Right angle so, the Road paths are perpendicular to each other . Question 11. Answer: Parallel paths . Explanation : The two straight lines in a plane that do not intersect at any point are said to be parallel. Question 12. Answer: Intersecting Lines Explanation : A pair of lines, line segments or rays are intersecting if they have a common point. This common point is their point of intersection . For 13-15, draw what is described by the geometric terms. Question 13. Perpendicular lines Answer: Perpendicular lines – SW UV and TX UV . Question 14. Intersecting lines Answer: Intersecting lines – Line SW is interescted at point U and Line TX is interescted at V . Question 15. Parallel lines Answer: Parallel lines – SW ll TX . Problem Solving Question 16. Critique Reasoning Bella names this line $$\overleftrightarrow{\mathrm{LM}}$$. Miguel names the line $$\overleftrightarrow{L N}$$. Who is correct? Explain. Answer: Both are correct Explanation : These three points L, M and N all lie on the same line. This line could be called ‘Line LM’, ‘Line MN’, ‘Line LN’, ‘Line NL’, ‘Line NM’, or ‘Line ML’ Question 17. Construct Arguments if all perpendicular lines are also intersecting lines, are all intersecting lines also perpendicular lines? Explain. Answer: Perpendicular lines always intersect each other, however, all intersecting lines are not always perpendicular to each other. The two main properties of perpendicular lines are: Perpendicular lines always meet or intersect each other. The angle between any two perpendicular lines is always equal to 90. Perpendicular lines intersect at a right angle. and do not intersect in this image, but if you imagine extending both lines, they will intersect soon. So, they are neither parallel nor perpendicular. Question 18. Draw three lines so two of the lines are perpendicular and the third line intersects the perpendicular lines at exactly one point. Label the lines with points. Answer: Explanation : line m and n are perpendicular lines and line l intersects the perpendicular line at D point as shown in above figure . Question 19. Higher Order Thinking $$\overleftrightarrow{A B}$$ is parallel to $$\overleftrightarrow{C D}$$, and $$\overleftrightarrow{C D}$$ is perpendicular to $$\overleftrightarrow{E F}$$. If a line through B and D is perpendicular to $$\overleftrightarrow{A B}$$, what is the relationship between $$\overleftrightarrow{B D}$$ and $$\overleftrightarrow{E F}$$? Answer: Explanation : $$\overleftrightarrow{A B}$$ ll $$\overleftrightarrow{C D}$$ . $$\overleftrightarrow{C D}$$ $$\overleftrightarrow{E F}$$ if $$\overleftrightarrow{A B}$$ $$\overleftrightarrow{B D}$$ then , $$\overleftrightarrow{B D}$$ and $$\overleftrightarrow{E F}$$ are parallel lines asshown in above figure . Assessment Practice Question 20. Which geometric term would you use to describe the power cables shown at the right? A. Perpendicular lines B. Parallel lines C. Intersecting lines D. Points Answer: Option B – Parallel lines . Explanation : Both the power cables are parallel to each other as there are equidistant from each other . ### Lesson 16.2 Classify Triangles Solve & Share Sort the triangles shown below into two or more groups. Explain how you sorted them. Solve this problem any way you choose. I can … reason about line segments and angles to classify triangles. Look Back! Generalize What is true about all 7 triangles you sorted? Answer : Essential Question How Can You Classify Triangles? Visual Learning Bridge Triangles can be classified by the line segments that make their sides. Triangles can be classified by their angle measures. Convince Me! Be Precise Can a triangle have more than one obtuse angle? Explain. Answer : No Explanation : Because sum of angles in the triangle = 180 degree . if one angle is obtuse angle that is 120 degrees . then , other two angles will be 180 – 120 = 60 degrees . that means a triangle can have only one obtuse angle . Another Example! The pattern follows the rule: right triangle, acute triangle, right triangle, acute triangle…. It also follows the rule: isosceles, scalene, scalene, isosceles, scalene…. Draw a triangle that could be next in the pattern and explain. For the first rule, the next triangle is acute. For the second rule, it is scalene. So, the next triangle is an acute, scalene triangle. It can be the same as the second triangle in the pattern or it can be a different acute, scalene triangle. Guided Practice Do You Understand? Question 1. Is it possible to have an obtuse acute triangle? Explain. Answer: No, it is not possible Explanation : Because sum of angles in the triangle = 180 degree . if one angle is obtuse angle that is 120 degrees . then , other two angles will be 180 – 120 = 60 degrees . that means a triangle can have only one obtuse angle . Question 2. Can a triangle have more than one right angle? If so, draw an example. Answer: No a triangle cannot have more than one right angle Explanation : As if there are 2 right angles the their sum will be 180° and the third angle will exceed the sum. … As the sum of all three angles is , the third angle would have to be zero resulting in a degenerate shape which is a line rather than a triangle. Do You Know How? For 3-4, classify each triangle by its sides, and then by its angles. Question 3. Answer: Isosceles triangle Acute Triangle Explanation : An Isosceles triangle has at least 2 sides of same length . An Acute triangle has three acute angles. all angles measure less than a right angle . Question 4. Answer: Equilateral Triangle Acute Triangle Explanation : An Equilateral Triangle has 3 sides equal . An Acute triangle has three acute angles. all angles measure less than a right angle . Independent Practice For 5-10, classify each triangle by its sides, and then by its angles. Question 5. Answer: Scalene Triangle Right angle Triangle Explanation : Scalene Triangle has no sides of same length . A Right Triangle has one Right Triangle . Question 6. Answer: Obtuse Triangle Scalene Triangle Explanation : An Obtuse Triangle has one obtuse angle . One angle has a measure greater than a right angle . Scalene Triangle has no sides of same length . Question 7. Answer: Isosceles Triangle Right Triangle Explanation : A Right Triangle has one Right Triangle . An Isosceles triangle has at least 2 sides of same length . Question 8. Answer: Isosceles Triangle Right Triangle Explanation : A Right Triangle has one Right Triangle . An Isosceles triangle has at least 2 sides of same length . Question 9. Answer: Scalene Triangle Obtuse Triangle Explanation : An Obtuse Triangle has one obtuse angle . One angle has a measure greater than a right angle . Scalene Triangle has no sides of same length . Question 10. Answer: Isosceles Triangle Acute triangle Explanation : An Acute triangle has three acute angles. all angles measure less than a right angle . An Isosceles triangle has at least 2 sides of same length . Problem Solving Question 11. Reasoning The backyard shown at the right is an equilateral triangle. What do you know about the lengths of the other two sides that are not labeled? Explain. Answer: the length of one side = 45 feet . The lengths of the other two sides = 45 feet Explanation : An Equilateral Triangle has 3 sides equal . Question 12. en Vision® STEM A rabbit’s field of vision is so wide that it can see predators that approach from behind. The diagram shows the field of vision of one rabbit and the field where the rabbit cannot see. Classify the triangle by its sides and its angles. Answer: The Triangle formed is a Isosceles Triangle and Acute Triangle . Explanation : An Acute triangle has three acute angles. all angles measure less than a right angle . An Isosceles triangle has at least 2 sides of same length . Question 13. A pattern follows the rule: obtuse triangle, obtuse triangle, right triangle, obtuse triangle…. It also follows the rule: isosceles, scalene, isosceles, scalene… Draw a triangle that could be the fifth shape in the pattern and explain. Answer: Obtuse triangle , Isosceles Triangle is the fifth shape Question 14. Higher Order Thinking Mitch draws a triangle with one obtuse angle. What are all the possible ways to classify the triangle by its angle measures and side lengths? Explain. Answer: Explanation : Obtuse angle angle is formed with all different sides and with two sides equal . Assessment Practice Question 15. Draw each triangle in its correct angle classification. Answer: ### Lesson 16.3 Classify Quadrilaterals Solve & Share Draw three different four-sided shapes that have opposite sides parallel. Explain how your shapes are alike and how they are different. Solve this problem any way you choose. I can … reason about line segments and angles to classify quadrilaterals. Look Back! What attributes do your shapes have in common? Answer : A Parallelogram is a quadrilateral in which both pairs of opposite sides are parallel . A Rectangle is a parallelogram with four right angles, so all rectangles are also parallelograms and quadrilaterals. On the other hand, not all quadrilaterals and parallelograms are rectangles. A Trapezoid is a quadrilateral with exactly one pair of parallel sides. Essential Question How Can You Classify Quadrilaterals? Visual Learning Bridge Quadrilaterals can be classified by their angles or the line segments that make their sides. Which of the quadrilaterals shown have only one pair of parallel sides? Which have two pairs of parallel sides? A rhombus is a quadrilateral that has opposite sides that are parallel and all of its sides are the same length. It is also a parallelogram. A trapezoid is a quadrilateral with only one pair of parallel sides. Trapezoids have only one pair of parallel sides. Parallelograms, rectangles, squares, and rhombuses all have two pairs of parallel sides. Convince Me! Use Structure How are a parallelogram and a rectangle the same? How are they different? Another Example! Perpendicular sides form right angles. Can a trapezoid have perpendicular sides? A trapezoid can have two right angles that form perpendicular sides. A trapezoid with two right angles is called a right trapezoid. Guided Practice Do You Understand? Question 1. What is true about all quadrilaterals? Answer: All Quadrilaterals have four sides, are coplanar, have two diagonals, and the sum of their four interior angles equals 360 degrees. Question 2. What is the difference between a square and a rhombus? Answer: The sides of a square are perpendicular to each other and its diagonals are of equal length. A rhombus is a quadrilateral in which the opposite sides are parallel and the opposite angles are equal. Question 3. Shane drew a quadrilateral with at least 2 right angles and at least 1 pair of parallel sides. Name three quadrilaterals Shane could have drawn. Answer: The Three Quadrilaterals with at least 2 right angles and at least 1 pair of parallel sides are Trapezoid , Rectangle and Square . Explanation : A trapezoid can have two right angles that form perpendicular sides. A trapezoid with two right angles is called a right trapezoid. A Rectangle is a parallelogram with four right angles, so all rectangles are also parallelograms and quadrilaterals. On the other hand, not all quadrilaterals and parallelograms are rectangles. A Square can be defined as a rhombus which is also a rectangle – in other words, a parallelogram with four congruent sides and four right angles. Do You Know How? For 4-7, write all the names possible for each quadrilateral. Question 4. Answer: Rectangle Explanation : Opposite sides are equal and parallel Four Right Angles . Question 5. Answer: Rhombus Explanation : All Four sides are equal and parallel Question 6. Answer: Square Explanation : All four sides are equal and parallel Four Right Angles . Question 7. Answer: Scalene Quadrilateral Explanation : A scalene quadrilateral is a four-sided polygon that has no congruent sides. Independent Practice For 8-11, write all the names possible for each quadrilateral. Question 8. Answer: Parallelogram Explanation : A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel Question 9. Answer: Rhombus Explanation : A Rhombus is a parallelogram with four equal sides. Question 10. Answer: Square Explanation : All four sides are equal and parallel Four Right Angles . Question 11. Answer: Rectangle Explanation : A Rectangle is a parallelogram with four right angles . Problem Solving Question 12. The pattern follows the rule: quadrilateral with no parallel sides, quadrilateral with two pairs of parallel sides, quadrilateral with two pairs of parallel sides, quadrilateral with no parallel sides, quadrilateral with two pairs of parallel sides…. Draw quadrilaterals that could be the next three in the pattern. Answer: Question 13. Critique Reasoning Tia says every square is a rectangle, and every square is a rhombus, so every rectangle must be a rhombus. Do you agree? Explain. Answer: No, Every Rectangle is not a Rhombus . Explanation : A square is a quadrilateral with all 4 angles right angles and all 4 sides of same length. So a square is a special kind of rectangle, it is one where all the sides have the same length. Thus every square is a rectangle because it is a quadrilateral with all 4 angles right angles. Yes, a rhombus is a quadrilateral with 4 equal sides. Every square has 4 equal length sides, so every square is a rhombus. No, every rectangle is not a rhombus because Rectangle have only opposite sides , whereas in Rhombus all four sides are equal . Question 14. Number Sense What number comes next in the pattern? The rule is “Multiply the position number by itself.” Describe a feature of the pattern. 1, 4, 9, 16, ☐ Answer: 1, 4, 9, 16, 25 . Explanation : Multiply the position to itself means squaring the position of the number Question 15. Higher Order Thinking Could you use the formula for finding the perimeter of a square to find the perimeter of another quadrilateral? Explain. The formula for the perimeter of a square is P = 4 × s. Answer: Assessment Practice Question 16. Select all the possible names for the shape below. ☐ Quadrilateral ☐ Rhombus ☐ Trapezoid ☐ Parallelogram ☐ Rectangle Answer: Quadrilateral and Trapezoid Question 17. Which shape has only 1 pair of parallel sides? A. Rhombus B. Square C. Right trapezoid D. Parallelogram Answer: Right Trapezoid ### Lesson 16.4 Line Symmetry Solve & Share Question : How many ways can you fold the square so one half fits exactly on top of the other half? How many ways can you fold the letter so one half fits exactly on top of the other half? Solve this problem any way you choose. I can …recognize and draw lines of symmetry and identify line-symmetric figures. Look Back! Reasoning What figures can you form when you fold a square in half? How are the different figures related to symmetry? Answer : The G have zero lines of symmetry. The rest of the letters, A, B, C, D, and E all have only 1 line of symmetry. Notice that the A has a vertical line of symmetry, while the B, C, D, and E have a horizontal line of symmetry. Essential Question What is Line Symmetry Visual Learning Bridge A figure is line symmetric if it can be folded on a line to form two matching parts that fit exactly on top of each other. The fold line is called a line of symmetry. There is one line of symmetry drawn on the picture of the truck. How many lines of symmetry do the figures below have? Count the lines of symmetry drawn on each figure below. A figure can have more than one line of symmetry. This figure is line symmetric. It has 2 lines of symmetry. It can be folded on each line of symmetry into matching parts. A figure can have many lines of symmetry. This figure is line symmetric. It has 6 lines of symmetry. It can be folded on each line of symmetry into matching parts. A figure can have no lines of symmetry. This figure is NOT line symmetric. It has O lines of symmetry. It cannot be folded to have matching parts. Convince Me! Look for Relationships Find two capital letters that have exactly one line of symmetry. Find two capital letters that have exactly two lines of symmetry. Guided Practices Do You Understand? Question 1. How many lines of symmetry does the letter R have? Answer: Letter R have Zero lines of symmetry as it cannot be divided into two identical shapes . Question 2. How many lines of symmetry does the figure below have? Answer: The given figure have One line of symmetry . Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 3. How many lines of symmetry can you find for a circle? Do you think you can count them? Answer: A circle has infinite lines of symmetry. A circle is symmetric about all its diagonals. Do You Know How? For 4-5, tell if each line is a line of symmetry. Question 4. Answer: Line of symmetry Question 5. Answer: Yes , it is a line of symmetry as two identical shapes are formed . For 6-7, tell how many lines of symmetry each figure has. Question 6. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 7. Answer: This figure is NOT line symmetric. It has O lines of symmetry. It cannot be folded to have matching parts. Independent Practice For 8-11, tell if each line is a line of symmetry. Question 8. Answer: This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 9. Answer: No Explanation : This figure is NOT line symmetric. It has O lines of symmetry. It cannot be folded to have matching parts. Question 10. Answer: Question 11. Answer: This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. For 12-19, decide if each figure is line symmetric. Draw and tell how many lines of symmetry each figure has. Question 12. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 13. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 14. Answer: Explanation : A line of symmetry is defined as the line that a figure can be divided into half, with the end result of the two halves matching up exactly. Question 15. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 16. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 17. Answer: Explanation : This figure is line symmetric. It has 3 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 18. Answer: zero . Explanation : This figure is NOT line symmetric. It has O lines of symmetry. It cannot be folded to have matching parts. Question 19. Answer: Explanation : This figure is line symmetric. It has 4 lines of symmetry. It can be folded on each line of symmetry into matching parts. Problem Solving Question 20. The Thomas Jefferson Memorial is located in Washington, D.C. Use the picture of the memorial at the right to decide whether the building is line symmetric. If so, describe where the line of symmetry is. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 21. Name the type of triangle outlined in green on the picture of the memorial. Answer: Acute Triangle . Question 22. Construct Arguments How can you tell when a line is NOT a line of symmetry? Answer: Folding Test : You can find if a shape has a Line of Symmetry by folding it. When the folded part sits perfectly on top (all edges matching), then the fold line is a Line of Symmetry. It cannot be folded to have matching parts. then it have zero lines of symmetry . Question 23. Higher Order Thinking How many lines of symmetry can a parallelogram have? Explain. Answer: zero Explanation : • A parallelogram has no lines of symmetry. • It has rotational symmetry of order two. Assessment Practice Question 24. Which figure has six lines of symmetry? Draw lines as needed. Answer: Option C – has 6 lines of symmetry . Explanation : Question 25. Which figure is NOT line symmetric? Answer: Option A . ### Lesson 16.5 Draw Shapes with Line Symmetry Solve & Share Craig and Julia are designing kites. A kite will fly well if the kite has line symmetry. Does Craig’s or Julia’s kite have line symmetry? Explain. Then, design your own kites. Design one kite with 2 lines of symmetry and another kite with 3 lines of symmetry. Solve this problem any way you choose. I can…draw a figure that has line symmetry. Question : Look Back! Can both Craig’s and Julia’s kites be folded into matching parts? If one of the kites is not line symmetric, can it be changed so that it is? Explain. Answer : Essential Question How Can You Draw Figures one with Line Symmetry? Visual Learning Bridge Sarah wants to design a line-symmetric tabletop. She sketched half of the tabletop. What are two ways Sarah can complete her design? The tabletop is line symmetric if the design can be folded along a line of symmetry, into matching parts. One Way Draw a line of symmetry. Complete Sarah’s design on the opposite side of the line of symmetry. The design for the tabletop is now line symmetric. Another Way Draw a different line of symmetry. Complete Sarah’s design on the opposite side of the line of symmetry. The design for the tabletop is now line symmetric. Convince Me! Question : Model with Math Sarah sketched different designs for a smaller tabletop. Use the lines of symmetry to draw ways Sarah can complete each design. Answer : Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Guided Practice Do You Understand? Question 1. Chandler tried to complete Sarah’s design from the previous page. Describe the error Chandler made. Answer: the image formed is not the exact reflection . Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 2. How can folding a piece of paper help to determine if a line in a figure is a line of symmetry? Answer: If the figure can be folded along a straight line so that one half of the figure exactly matches the other half, the figure has line symmetry. The crease is the line of symmetry. Do You Know How? For 3-4, use the line of symmetry to draw a line-symmetric figure. Question 3. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 4. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Independent Practice For 5-10, use the line of symmetry to draw a line-symmetric figure. Question 5. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 6. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 7. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 8. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 9. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 10. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Problem Solving Question 11. Draw a figure that has no lines of symmetry. Answer: A scalene triangle, have no lines of symmetry – it is not possible to fold the shape about a line so that the two halves fit exactly on top of one another. Question 12. Vanessa drew a figure that has an infinite number of lines of symmetry. What figure could Vanessa have drawn? Answer: Circle have infinite number of lines of symmetry . Question 13. enVision® STEM Dogs can smell odors that humans cannot. Dogs can be trained to alert their owners when they smell odors associated with illness. If a dog trains 2 hours every day for 1 year, how many hours has the dog trained? Answer: Number of hours dog train for a day = 2 hours Number of hours dog train for 1 year = 2 hours × 365 days = 730 hours . Question 14. Make Sense and Persevere Clare trained for a long-distance marathon. She ran a total of 225 miles in 3 months. The first month she ran 50 miles. If she ran 25 more miles each month, how many miles did she run in her third month of training? Answer: Total Distance ran = 225 miles Distance covered in First month = 50 miles . Distance covered in Second month = 50 + 25 = 75 miles . Distance covered in third month = 75 + 25 = 100 miles . Question 15. Higher Order Thinking Can you draw a line that divides a figure in half but is NOT a line of symmetry? Use the figures below to explain. Answer: No, because the given line drawn are lines of symmetry it divides the figure into identical halves . Assessment Practice Question 16. Which of the following figures is line symmetric about the dashed line? Answer: Option B – as two identical halves are formed by line of symmetry . Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. ### Lesson 16.6 Problem Solving Critique Reasoning Solve & Share Nathan gave the answer shown to the following question. True or False? All right triangles have two sides the same length. How do you respond to Nathan’s reasoning? I can … critique the reasoning of others by using what I know about two-dimensional shapes. Nathan That’s true. Here are three different sizes of right triangles. In each, two sides are the same length. Thinking Habits Be a good thinker! These questions can help you. • What questions can I ask to understand other people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Look Back! Critique Reasoning Nathan answered another question. True or false: A triangle can have two right angles. Nathan says this is not possible. Do you agree or disagree? Explain. Answer : Because of the fact that the sum of the three interior angles of a triangle must be 180 degrees, a triangle could not have two right angles Essential Question How Can You Critique the Reasoning of Others? Visual Learning Bridge Abby gave the answer shown to the following question. True or False? Every quadrilateral has at least one right angle. Abby True. Here are different quadrilaterals. They all have four sides and four right angles. What is Abby’s reasoning to support her statement? Abby drew quadrilaterals that have right angles. How can I critique the reasoning of others? I can • ask questions about Abby’s reasoning • look for flaws in her reasoning. • decide whether all cases have been considered. Here’s my thinking. Abby’s reasoning has flaws. She used only special kinds of quadrilaterals in her argument. For these special cases, the statement is true. Here is a quadrilateral that has no right angles. It shows the statement is not true about every quadrilateral. The statement is false. Convince Me! Be Precise Would Abby’s reasoning be correct if the question was changed to: True or False? Some quadrilaterals have at least one right angle. Explain. Guided Practice Critique Reasoning Anthony said all multiples of 4 end in 2, 4, or 8. He gave 4, 8, 12, 24, and 28 as examples. Question 1. What is Anthony’s argument? How does he support it? Answer: No – Anthony is wrong . Explanation : all multiples of 4 end in 0, 2, 4, 6 and 8 . Examples – 4 × 4 = 16 4 × 5 = 20 and e t c . Question 2. Describe at least one thing you could do to critique Anthony’s reasoning. Answer: Look flaws in the statement and checking for all numbers . Question 3. Does Anthony’s reasoning make sense? Explain. Answer: Yes – As he is calculated the values of 4 of squares of 2 and their respective square numbers . Independent Practice Critique Reasoning Marista said the polygons shown all have the same number of angles as they have sides. Question 4. Describe at least one thing you could do to critique Marista’s reasoning. Answer: Number of interior angles of a polygon is equal number of Number of sides irrespective of exterior angles . Question 5. Does Marista’s reasoning make sense? Explain. Answer: Question 6. Can you think of any examples that prove all polygons don’t have the same number of sides as angles? Explain. Answer: No, A polygon has the same number of sides and angles. Problem Solving Performance Task Dog Pen Caleb is designing a dog pen for the animal shelter. He has 16 feet of fence, including the gate. His designs and explanation are shown. Critique Caleb’s reasoning. Dog pens usually have right angles, so I just used rectangles. Both my pens used 16 feet of fence. I think the square one is better, because it has more area. Question 7. Reasoning What quantities are given in the problem and what do the numbers mean? Answer: The Quantities given are the length and Breadth of the Rectangle pen and the side of the square . the numbers help in calculating the Area . Question 8. Critique Reasoning What can you do to critique Caleb’s thinking? Answer: The two shapes are given Rectangle and Square with their respective Quantities . The Area of Rectangle is smaller than the Rectangle of the Square . The area of the Square is preferred because it has more area than Rectangle . Question 9. Be Precise Did Caleb correctly calculate the perimeter of each fence? Explain. Answer: No, Explanation : Length of Rectangle = 5 feet Breadth of Rectangle = 2 feet Perimeter of Rectangle = 2 ( Length + Breadth ) = 2 ( 5 + 2 ) = 2 ( 7 ) = 14 feets . Side of Square = 4 feet . Perimeter of Square = 4 ( Side ) = 4 ( 4 ) = 16 feets . Question 10. Critique Reasoning Does Caleb’s reasoning make sense? Explain. Answer: No, Explanation : As, perimeter and area are not calculated correctly . Length of Rectangle = 5 feet Breadth of Rectangle = 2 feet Perimeter of Rectangle = 2 ( Length + Breadth ) = 2 ( 5 + 2 ) = 2 ( 7 ) = 14 feet . Side of Square = 4 feet . Perimeter of Square = 4 ( Side ) = 4 ( 4 ) = 16 feet . Question 11. Be Precise Explain how you know what units to use in your explanation. Answer: Perimeter measured in feet . Area measured in square feet . ### Topic 16 Fluency Practice Activity Find a Match Work with a partner. Point to a clue. Read the clue. Look below the clues to find a match. Write the clue letter in the box next to the match. Find a match for every clue. I can … add multi-digit whole numbers. Clues A. The sum is between 650 and 750. B. The sum is between 1,470 and 1,480. C. The sum is exactly 1,550. D. The sum is between 1,350 and 1,450. E. The sum is exactly 790. F. The sum is exactly 1,068. G. The sum is between 1,100 and 1,225. H. The sum is exactly 1,300. Answer : ### Topic 16 Vocabulary Review Understand Vocabulary Word List • acute triangle • equilateral triangle • intersecting lines • isosceles triangle • line of symmetry • line symmetric • obtuse triangle • parallel lines • parallelogram • perpendicular lines • rectangle • rhombus • right triangle • scalene triangle • square • trapezoid Write T for true and F for false. Question 1. ________ An acute triangle is a triangle with one acute angle. Answer: True Question 2. _________ An isosceles triangle has at least two equal sides. Answer: True Question 3. ______ A figure is line symmetric if it has at least one line of symmetry Answer: True Question 4. __________ Perpendicular lines form obtuse angles where they intersect. Answer: False Question 6. __________ A trapezoid has two pairs of parallel sides. Answer: False Write always, sometimes, or never. Question 6. An equilateral triangle __________ has three equal sides. Answer: An equilateral triangle always has three equal sides. Question 7. Parallel lines _________ intersect. Answer: Parallel lines Never intersect. Question 8. A scalene triangle __________ has equal sides. Answer: A scalene triangle never has equal sides. Question 9. A rectangle is _________ a square. Answer: A rectangle is sometimes a square. Question 10. A rhombus __________ has opposite sides that are parallel. Answer: A rhombus always has opposite sides that are parallel. Use Vocabulary in Writing Question 11. Rebecca drew a figure. Describe Rebecca’s figure. Use at least 3 terms from the Word List in your description. Answer: Quadrilateral Parallelogram ### Topic 16 Reteaching Set A pages 585-588 Pairs of lines are given special names: parallel, intersecting, or perpendicular. $$\overleftrightarrow{D E}$$ and $$\overleftrightarrow{F G}$$ are parallel lines. Remember to use geometric terms when describing what is shown. Question 1. Answer: $$\overleftrightarrow{P Q}$$ and $$\overleftrightarrow{R S}$$ are perpendicular lines. Question 2. Answer: Both lines are perpendicular intersect at point Y . Set B pages 589-592 Triangles can be classified by their sides and angles. Two sides are the same length, and each angle measures less than a right angle. It is an isosceles, acute triangle. Remember to classify each triangle by its sides and then by its angles. Question 1. Answer: Two sides are the same length – Isosceles All angles are less than 90 degrees so, isosceles acute triangle . Question 2. Answer: all sides are different Right angle so, Scalene right triangle Set C pages 593-596 Name the quadrilateral. Opposite sides are parallel. There are no right angles. All sides are not the same length. It is a parallelogram, but not a rectangle, rhombus, or square. Remember that a quadrilateral can be a rectangle, square, trapezoid, parallelogram, or rhombus. Write all the names possible for each quadrilateral. Question 1. Answer: Trapezoid – only 1 pair of parallel sides Question 2. Answer: Rhombus Explanation : 2 pairs of parallel sides 4 equal sides Set D pages 597-600 How many lines of symmetry does the figure have? Fold the figure along the dashed line. The two halves are equal and fit one on top of the other. The figure is line symmetric. It cannot be folded on another line, so it has 1 line of symmetry. Remember that figures can have many lines of symmetry. Draw and tell how many lines of symmetry for each figure. Question 1. Answer: Explanation : This figure is line symmetric. It has 4 lines of symmetry. It can be folded on each line of symmetry into matching parts. Question 2. Answer: Explanation : This figure is line symmetric. It has 1 lines of symmetry. It can be folded on each line of symmetry into matching parts. Set E pages 601-604 Complete a design with line symmetry. Draw a line of symmetry for the shape. Complete the design on the opposite side of the line of symmetry. Remember, for a figure to be line symmetric, it must have a line of symmetry. Complete the designs. Question 1. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Question 2. Answer: Explanation : We say there is symmetry when the exact reflection or mirror image of a line, shape or object gets created. The line of symmetry can be defined as the axis or imaginary line that passes through the center of the shape or object and divides it into identical halves. Set F pages 605-608 Think about these questions to help you critique the reasoning of others. Thinking Habits Be a good thinker! These questions can help you. • What questions can I ask to understand other people’s thinking? • Are there mistakes in other people’s thinking? • Can I improve other people’s thinking? Remember that it only takes one counterexample to show the statement is false. Derek says, “All triangles have 1 right angle.” Question 1. Use the figures above to critique Derek’s statement. Answer: The First two triangles ( from left to Right )in the figure are Right triangle but the third triangle is a Equilateral Triangle . Question 2. What kinds of triangles NEVER have right angles? Answer: Since a right-angled triangle has one right angle, the other two angles are acute. Therefore, an obtuse-angled triangle can never have a right angle; and vice versa. The side opposite the obtuse angle in the triangle is the longest. Any triangle that is not a right triangle is an oblique triangle. Solving an oblique triangle means finding the measurements of all three angles and all three sides. ### Topic 16 Assessment Practice Question 1. Of a parallelogram, rectangle, rhombus, and trapezoid, which cannot describe a square? Explain. Answer: A square can be a rhombus, parallelogram, a rectangle do not have to be a Trapezoid . Explanation: A rhombus is a quadrilateral with all four sides being of equal length. A parallelogram is a quadrilateral with two pairs of parallel sides. A rectangle is a quadrilateral with four right angles and 2 parallel sides . A square is a quadrilateral with four right angles and equal side lengths. Where as a A trapezoid is a quadrilateral with at least one pair of parallel sides. it sides are not of different lengths . Looking at the definitions, a square is already guaranteed to be a rhombus. Here are examples of each of the other shapes which are not rhombuses. Question 2. How many acute angles are there in an equilateral triangle? Answer: An equilateral triangle has three acute angles . Question 3. Gavin drew different-colored lines. Draw a line that is parallel to $$\overleftrightarrow{S R}$$. Answer: Explanation : A line is drawn from M which is parallel to $$\overleftrightarrow{S R}$$ . Question 4. Marci described the light from the sun as a line that starts at the sun and continues on forever. Which geometric term best describes Marci’s description of the sun’s light? Answer: It is a Ray Explanation : A ray can be defined as a part of a line that has a fixed starting point but no end point. It can extend infinitely in one direction. Question 5. Four of Mrs. Cromwell’s students decorated a bulletin board with the shapes shown below. Order the students’ shapes in order from fewest lines of symmetry to most lines of symmetry. Answer: Question 6. Are all intersecting lines perpendicular? Draw a picture to help explain your answer. Answer: Perpendicular lines are a particular case of intersecting lines when an angle between them is 90o. Explanation: Perpendicular lines always intersect each other, however, all intersecting lines are not always perpendicular to each other. The two main properties of perpendicular lines are: Perpendicular lines always meet or intersect each other. The angle between any two perpendicular lines is always equal to 90. Question 7. A four-sided figure with two pairs of parallel sides cannot be what type of quadrilateral? Explain. Answer: Trapezoids have only one pair of parallel sides; parallelograms have two pairs of parallel sides. A trapezoid can never be a parallelogram. The correct answer is that all trapezoids are quadrilaterals. . Trapezoids are four-sided polygons, so they are all quadrilaterals. Question 8. Equilateral triangle ABC has one side with a length of 4 inches. What are the lengths of each of the other two sides of the triangle? Explain. Answer: An equilateral triangle is a triangle with all three sides of equal length , corresponding to what could also be known as a “regular” triangle . if one side is 4 inches then other two ides will also be 4 inches . Question 9. Which set of angles could form a triangle? A. Two right angles, one acute angle B. One obtuse angle, one right angle, one acute angle C. Two obtuse angles, one acute angle D. One right angle, two acute angles Answer: Option D . Question 10. A figure has one angle formed from a pair of perpendicular lines, one pair of parallel sides, and no sides with equal lengths. What geometric term can be used to name this figure? Answer: Question 11. Dina’s teacher asks her to describe the top and bottom edges of her ruler using a geometric term. What term could Dina use? Answer: The top and bottom edge of the ruler is like a straight line . Question 12. Shapes are divided into two groups. These are the shapes in the first group. The following shapes do not belong in the group above. These are the shapes in the second group. What generalization can be made about the shapes in the first group? Answer: In the above group we have shapes like Parallelogram , Trapezoid and Equilateral Triangle In the above group, We have shapes like Trapezoid , right Triangle and Square . Generalization made is square can be a parallelogram in few statements . Question 13. Complete the drawing so the figure is line symmetric. Answer: Explanation : Fold the figure along the dashed line. The two halves are equal and fit one on top of the other. The figure is line symmetric. It cannot be folded on another line, so it has 1 line of symmetry. ### Topic 16 Performance Task Ottoman Art The Ottoman Empire lasted from 1299 until 1922. Much of the art from this period contained geometric shapes. Question 1. Use the Ottoman Empire figure to answer the following. Part A – Question Name a pair of parallel lines and explain why the lines are parallel. Answer: AB ll CD . Explanation : Because the points A and B are equi distant from points C and D . and they don’t intersect each other Part B – Question The enlarged part of the figure shows 4 triangles that are all the same type. Classify these triangles by their sides and by their angles. Explain. Answer: all the sides are equal then it is Equilateral Triangle . all angles will be 60 degrees . Part C Olivia said the 4 triangles were inside a square. When asked other possible names for the square, she said it was a quadrilateral, a parallelogram, and a rectangle. Critique Olivia’s reasoning. Answer: A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. A quadrilateral is a polygon having only four sides. A rectangle is a parallelogram in which one angle is of 90 degrees . Yes, a square is a special type of rectangle because it possesses all the properties of a rectangle. Similar to a rectangle, a square has: interior angles which measure 90 each. opposite sides that are parallel and equal. Question 2 The basic shape used in the Ottoman Scarf is a quadrilateral. Answer the following about this shape. Part A – Question What are all the names you can use for this quadrilateral? Explain. Answer: Rhombus and parallelogram A parallelogram is a quadrilateral in which each pair of opposite sides is parallel. A rhombus is a parallelogram in which adjacent sides are equal. Part B – Question Corbin drew a triangle by connecting the points W, X, and Y. He said the triangle is acute because it has acute angles. Critique Corbin’s reasoning. Answer: Yes the Triangle formed is WXY . The triangle is acute Triangle because the triangles have acute angles . Part C – Question Draw all lines of symmetry on the Decorative Plate. How many lines of symmetry does the plate have? Explain. Answer: Explanation : Fold the figure along the dashed line. The two halves are equal and fit one on top of the other. The figure is line symmetric. It cannot be folded on another line, so it has 1 line of symmetry. #### enVision Math Common Core Grade 4 Answer Key ## Envision Math Common Core Grade 5 Answer Key Topic 4 Use Models and Strategies to Multiply Decimals ## Envision Math Common Core 5th Grade Answers Key Topic 4 Use Models and Strategies to Multiply Decimals Essential Question: What are some common procedures for estimating and finding products involving decimals? enVision STEM Project: Solar Energy Do Research Use the Internet or other sources to learn about solar energy. Find at least five ways that we use the Sun’s energy today. Answer: To dry your clothes. To grow your food. To heat your water. To power your car. To generate your electricity. Explanation: In the above-given question, given that, The sun has gone to a lot of trouble to send us its energy. we can use solar energy in many ways. they are: to dry your clothes. to grow your food. to heat your water. to power your car. to generate your electricity. Journal: Write a Report Include what you found. Also in your report: • Describe at least one way that you could use solar energy. Could it save you money? • Estimate how much your family pays for energy costs such as lights, gasoline, heating, and cooling. • Make up and solve problems by multiplying whole numbers and decimals. Review What You Know Vocabulary Choose the best term from the box. Write it on the blank. • exponent • hundredths • overestimate • partial products • power • round • tenths • thousandths • underestimate Question 1. One way to estimate a number is to ____ the number. Answer: One way to estimate a number is to round the number. Explanation: In the above-given question, given that, one way to estimate a number is to round the number. for example: 2.456 round to tenths. 2.556. Question 2. Using 50 for the number of weeks in a year is a(n) _____. Answer: Using 50 for the number of weeks in a year is a(n) is exponents. Explanation: In the above-given question, given that, using 50 for the number of weeks in a year is a(n) is exponents. for example: a(n) = 50. a = 5. 50/5 = 10. Question 3. In the number 3.072, the digit 7 is in the ___ place and the digit 2 is in the ____ place. Answer: In the number 3.072, the digit 7 is in the hundredths place and the digit 2 is in the thousandths place. Explanation: In the above-given question, given that, In the number 3.072, the digit 7 is in the hundredths place and the digit 2 is in the thousandths place. for example: 3.072. 7 is in the hundredths place. 2 is in the thousands place. Question 4. 10,000 is a(n) ____ of 10 because 10 × 10 × 10 × 10 = 10,000. Answer: 10,000 is a(n) power of 10 because 10 x 10 x 10 x 10 = 10,000. Explanation: In the above-given question, given that, for example: 10 x 10 x 10 x 10. 100 x 100 = 10,000. Whole Number Multiplication Find each product. Question 5. 64 × 100 Answer: The product is 6400. Explanation: In the above-given question, given that, the two numbers are 64 and 100. multiply the two numbers. 64 x 100 = 6400. so the product is 6400. Question 6. 7,823 × 103 Answer: The product is 7823000. Explanation: In the above-given question, given that, the two numbers are 7823 and 1000. multiply the two numbers. 7823 x 1000 = 7823000. so the product is 7823000. Question 7. 10 × 1,405 Answer: The product is 14050. Explanation: In the above-given question, given that, the two numbers are 10 and 1405. multiply the two numbers. 10 x 1405 = 14050. so the product is 14050. Question 8. 53 × 413 Answer: The product is 21889. Explanation: In the above-given question, given that, the two numbers are 53 and 413. multiply the two numbers. 53 x 413 = 21889. so the product is 21889. Question 9. 906 × 57 Answer: The product is 51,642. Explanation: In the above-given question, given that, the two numbers are 906 and 57. multiply the two numbers. 906 x 57 = 51,642. so the product is 51,642. Question 10. 1,037 × 80 Answer: The product is 82,960. Explanation: In the above-given question, given that, the two numbers are 1037 and 80. multiply the two numbers. 1037 x 80 = 82,960. so the product is 82,960. Round Decimals Round each number to the nearest tenth. Question 11. 842.121 Answer: The number to the nearest tenth = 842.10. Explanation: In the above-given question, given that, the number is 842.121. to round a number to the nearest tenth look at the number of ones. if this is 5 or more round up. if it is 4 or less round down. 842.10. so the number to the nearest tenth = 842.10. Question 12. 10,386.145 Answer: The number to the nearest tenth = 10386.10. Explanation: In the above-given question, given that, the number is 10,386.145. to round a number to the nearest tenth look at the number of ones. if this is 5 or more round up. if it is 4 or less round down. 10386.10. so the number to the nearest tenth = 10386.10. Question 13. 585.055 Answer: The number to the nearest tenth = 585.155. Explanation: In the above-given question, given that, the number is 585.055. to round a number to the nearest tenth look at the number of ones. if this is 5 or more round up. if it is 4 or less round down. 585.055. so the number to the nearest tenth = 585.155. Properties of Multiplication Use the Commutative and Associative Properties of Multiplication to complete each multiplication. Question 14. 96 × 42 = 4,032 so 42 × 96 = ___ Answer: 42 x 96 = 4032. Explanation: In the above-given question, given that, the two numbers are 96 and 42. multiply the two numbers. 96 x 42 = 4032. 42 x 96 = 4032. Question 15. 4 (58 × 25) = 4 × (25 × ___) = (___ × ___) × 58 = ___ Answer: 4(58 x 25) = 4 x (25 x 58) = (25 x 4) x 58 = 5800. Explanation: In the above-given question, given that, the two numbers are 58, 4, and 25. multiply the two numbers. 4 x 25 x 58. 25 x 4 x 8 = 5800. Question 16. (293 × 50) × 20 = 293 × (50 × ___) = ___ Answer: 293 x 50 x 20 = 293 x 50 x 20 = 2,93,000. Explanation: In the above-given question, given that, the two numbers are 293, 50, and 20. multiply the two numbers. 293 x 50 x 20. 50 x 293 x 20 = 2,93,000. pick a Project PROJECT 4A How can you set up an exercise plan? Project: Plan an exercise Program PROJECT 4B How much does it cost to dress a team? Project: Budget a Team PROJECT 4C How far can a rocket go in 100 seconds? Project: Make a Poster Answer: The rocket can go in 100 sec = 790 km. Explanation: The above-given question, given that, the rocket can go in 100 sec is: 1 minute = 60 sec. 1 sec = 7.9 km. 100sec = 7.9 x 100. 790 km. PROJECT 4D How much extra do you have to pay? Project: Make a Data Display ### Lesson 4.1 Multiply Decimals by Powers of 10 Activity Solve & Share Javier is helping his parents put up posters in their movie theater. Each poster has a thickness of 0.012 inch. How thick is a stack of 10 posters? 100 posters? 1,000 posters? Solve this problem any way you choose. Answer: The thick is a stack of 10 posters = 0.12. the thick is a stack of 100 posters = 1.2. the thick is a stack of 1000 posters = 12. Explanation: In the above-given question, given that, Javier is helping his parents put up posters in their movie theater. Each poster has a thickness of 0.012. 0.012 x 10 = 0.12. 0.012 x 100 = 1.2. 0.012 x 1000 = 12. You can use the structure of our number system and mental math to help you. Look Back! Use Structure How is your answer for 1,000 posters similar to 0.012? How is it different? Visual Learning Bridge Essential Question What Patterns Can Help You Multiply Decimals by Powers of 10? A. You can use place value and what you know about whole numbers to multiply decimals by powers of 10. What patterns can you find? We already know what happens when a whole number is multiplied by powers of 10. B. In a place-value chart, the same pattern appears when a decimal is multiplied by powers of 10. A digit in one place is worth 10 times more when moved to the place on its left. Every time a number is multiplied by 10, the digits of the number shift to the left. C. Holding the numbers still, another pattern appears. 3.63 × 1 = 3.63 3.63 × 101 = 36.3 3.63 × 102 = 363.0 3.63 × 103 = 3630.0 It looks like the decimal point moves to the right each time. Convince Me! Use Structure Complete the chart. What patterns can you use to place the decimal point? Answer: 1.275 x 10 = 12.75, 1.275 x 100 = 127.5, 1.275 x 1000 = 1275. 26.014 x 10 = 260.14, 26.014 x 100 = 2601.4, 26.014x 1000 = 26014. 0.4 x 10 = 4, 0.4 x 100 = 40, 0.4 x 1000 = 400. Explanation: In the above-given question, given that, the numbers are 1.275, 26.014, and 0.4. multiply the numbers by 10, 100, and 1000. 1.275 x 10 = 12.75, 1.275 x 100 = 127.5, 1.275 x 1000 = 1275. 26.014 x 10 = 260.14, 26.014 x 100 = 2601.4, 26.014x 1000 = 26014. 0.4 x 10 = 4, 0.4 x 100 = 40, 0.4 x 1000 = 400. Guided Practice Do You Understand? Question 1. When multiplying by a power of 10, like 4.58 × 103, how do you know you are moving the decimal in the correct direction? Answer: The number is 4580. Explanation: In the above-given question, given that, 4.58 × 103. 4.58 x 10 x 10 x 10. 4.58 x 1000 = 4580. so the number is 4580. Do You Know How? In 2-5, find each product. Question 2. 0.009 × 10 Answer: The product is 0.09. Explanation: In the above-given question, given that, the numbers are 0.009 and 10. multiply the numbers. 0.009 x 10 = 0.09. so the product is 0.09. Question 3. 3.1 × 103 Answer: The product is 3100. Explanation: In the above-given question, given that, the numbers are 3.1 and 1000. multiply the numbers. 3.1 x 1000 = 3100. so the product is 3100. Question 4. 0.062 × 102 Answer: The product is 6.2. Explanation: In the above-given question, given that, the numbers are 0.062 and 100. multiply the numbers. 0.062 x 100 = 6.2. so the product is 6.2. Question 5. 1.24 × 104 Answer: The product is 1240. Explanation: In the above-given question, given that, the numbers are 1.24 and 10000. multiply the numbers. 1.24 x 10000 = 1240. so the product is 1240. Independent Practice Leveled Practice in 6 and 7, find each product. Place-value patterns can help you solve these problems. Question 6. 42.3 ×1 = ___ 42.3 × 10 = ___ 42.3 × 102 = ___ Answer: 42.3 x 1 = 42.3. 42.3 x 10 = 423. 42.3 x 100 = 4230. Explanation: In the above-given question, given that, the numbers are 42.3, 42.2 x 10, and 42.3 x 100. 42.3 x 1 = 42.3. 42.3 x 10 = 423. 42.3 x 100 = 4230. Question 7. ____ = 0.086 × 101 ___ = 0.086 × 100 ____ = 0.086 × 1,000 Answer: 0.086 x 10 = 0.86. 0.086 x 100 = 8.6. 0.086 x 1000 = 86. Explanation: In the above-given question, given that, the numbers are 0.086 x 10, 0.086 x 100, and 0.086 x 1000. 0.086 x 10 = 0.86. 0.086 x 100 = 8.6. 0.086 x 1000 = 86. In 8-15, find each product. Question 8. 63.7 × 10 Answer: The product is 637. Explanation: In the above-given question, given that, the numbers are 63.7 and 10. multiply the numbers. 63.7 x 10 = 637. so the product is 637. Question 9. 563.7 × 102 Answer: The product is 56370. Explanation: In the above-given question, given that, the numbers are 563.7 and 100. multiply the numbers. 563.7 x 100 = 56370. so the product is 56370. Question 10. 0.365 × 104 Answer: The product is 3650. Explanation: In the above-given question, given that, the numbers are 0.365 and 10000. multiply the numbers. 0.365 x 10000 = 3650. so the product is 3650. Question 11. 5.02 × 100 Answer: The product is 502. Explanation: In the above-given question, given that, the numbers are 5.02 and 100. multiply the numbers. 5.02 x 100 = 502. so the product is 502. Question 12. 94.6 × 103 Answer: The product is 94600. Explanation: In the above-given question, given that, the numbers are 94.6 and 1000. multiply the numbers. 94.6 x 1000 = 94600. so the product is 94600. Question 13. 0.9463 × 102 Answer: The product is 94.63. Explanation: In the above-given question, given that, the numbers are 0.9463 and 100. multiply the numbers. 0.9463 x 100 = 94.63. so the product is 94.63. Question 14. 0.678 × 100 Answer: The product is 67.8. Explanation: In the above-given question, given that, the numbers are 0.678 and 100. multiply the numbers. 0.678 x 100 = 67.8. so the product is 67.8. Question 15. 681.7 × 104 Answer: The product is 6817000. Explanation: In the above-given question, given that, the numbers are 681.7 and 10000. multiply the numbers. 681.7 x 10000 = 6817000. so the product is 6817000. In 16-18, find the missing exponent. Question 16. 0.629 × = 62.9 Answer: The missing exponent is 2. Explanation: In the above-given question, given that, 0.629 x 10 x 10. 0.629 x 100 = 62.9. 0.629 Question 17. × 0.056 = 560 Answer: The missing exponent is 4. Explanation: In the above-given question, given that, 10 x 10 x 10 x 10 x 0.056. 100 x 100 x 0.056. 10000 x 0.056. 560. Question 18. 1.23 = × 0.123 Answer: The missing exponent is 0. Explanation: In the above-given question, given that, 1.23 x 10. 1.23. Problem Solving In 19-21, use the table to find the answers. Question 19. Monroe uses a microscope to observe specimens in science class. The microscope enlarges objects to 100 times their actual size. Find the size of each specimen as seen in the microscope. Answer: The size of each specimen as seen in the microscope = 0.8, 11, 0.25, and 0.4. Explanation: In the above-given question, given that, Monroe uses a microscope to observe specimens in science class. The microscope enlarges objects to 100 times their actual size. 0.008 x 100 = 0.8. 0.011 x 100 = 11. 0.0025 x 100 = 0.25. 0.004 x 100 = 0.4. Question 20. Monroe’s teacher wants each student to draw a sketch of the longest specimen. Which specimen is the longest? Answer: Specimen B is the longest. Explanation: In the above-given question, given that, Monroe’s teacher wants each student to draw a sketch of the longest specimen. 0.008 x 100 = 0.8. 0.011 x 100 = 11. 0.0025 x 100 = 0.25. 0.004 x 100 = 0.4. so specimen B is the longest. Question 21. Seen through the microscope, a specimen is 0.75 cm long. What is its actual length? Answer: The actual length = 75 cm. Explanation: In the above-given question, given that, a specimen is 0.75 cm long. 0.75 x 100 = 75. so the actual length of the specimen is 75 cm. Question 22. Jon’s binoculars enlarge objects to 10 times their actual size. If the length of an ant is 0.43 inch, what is the length as seen up close through his binoculars? Answer: The length as seen up close through his binoculars = 4.3 inches. Explanation: In the above-given question, given that, Jon’s binoculars enlarge objects to 10 times their actual size. If the length of an ant is 0.43 inches. 0.43 x 10 = 4.3 inches. so the length as seen up close through his binoculars = 4.3 inches. Question 23. Higher Order Thinking Jefferson drew a line 9.5 inches long. Brittany drew a line 10 times as long. What is the difference in length between the two lines? Answer: The difference in length between the two lines = 85.5 inches. Explanation: In the above-given question, given that, Jefferson drew a line 9.5 inches long. Brittany drew a line 10 times as long. 9.5 x 10 = 95. 95 – 9.5 = 85.5. so the difference in length between the two lines = 85.5 inches. Question 24. Construct Arguments José ran 2.6 miles. Pavel ran 2.60 miles. Who ran farther? Explain your reasoning. Answer: Jose and Pavel ran the same. Explanation: In the above-given question, given that, José ran 2.6 miles. Pavel ran 2.60 miles. 2.6 is equal to 2.60. so Jose and Pavel ran the same. Assessment Practice Question 25. Choose all equations that are true. 4.82 × 1,000 = 482,000 4.82 × 102 = 482 482 × 101 = 48.2 482 × 103 = 482 48.2 × 104 = 4,820 Answer: 4.82 x 100 = 482. Explanation: In the above-given question, given that, The equations are: 4.82 x 1000 = 482000. 4.82 x 10 = 482. 482 x 1000 = 482. 48.2 x 10000 = 4820. so the equation that is true is 4.82 x 100 = 482. Question 26. Choose all equations that are true when 102 is placed in the box. o 37 = × 0.37 0 0.37 = × 0.037 0370 = × 3.7 0.37 = × 3.7 3.7 = × 0.037 Answer: None of the equations are true. Explanation: In the above-given question, given that, 37 x 102 = 0.370. 0.37 x 102 = 0.037. 0370 x 102 = 3.7. 3.7 x 102 = 0.037. so none of the equations are true. ## Lesson 4.2 Estimate the Product of a Decimal and a Whole Number Activity Solve & Share Renee needs 32 strands of twine for an art project. Each strand must be 1.25 centimeters long. About how many centimeters of twine does she need? Solve this problem any way you choose! Answer: The centimeters of twine does she need = 40 cm. Explanation: In the above-given question, given that, Renee needs 32 strands of twine for an art project. Each strand must be 1.25 centimeters long. 32 x 1.25 = 40. so the centimeters of twine does she need = 40 cm. Generalize How can you relate what you know about estimating with whole numbers to estimating with decimals? Show your work! Look Back! Is your estimate an overestimate or an underestimate? How can you tell? Visual Learning Bridge Essential Question What Are Some Ways to Estimate Products of Decimals and Whole Numbers? A. A wedding planner needs to buy 16 pounds of sliced cheddar cheese. About how much will the cheese cost? B. One Way Round each number to the nearest dollar and nearest ten.$2 × 20 = $40 The cheese will cost about$40.

C.
Another Way
Use compatible numbers that you can multiply mentally.

$2 × 15 =$30
The cheese will cost about $30. Convince Me! Reasoning About how much money would 18 pounds of cheese cost if the price is$3.95 per pound? Use two different ways to estimate the product. Are your estimates overestimates or underestimates? Explain.

Another Example
Manuel walks a total of 0.75 mile to and from school each day. If there have been 105 school days so far this year, about how many miles has he walked in all?
Round to the nearest whole number.

Use compatible numbers.

Be sure to place the decimal point correctly.

Both methods provide reasonable estimates of how far Manuel has walked.

Guided Practice

Do You Understand?

Question 1.
Number Sense There are about 20 school days in a month. In the problem above, about how many miles does Manuel walk each month? Write an equation to show your work.

The number of miles does Manuel walks each month = 16.

Explanation:
In the above-given question,
given that,
There are about 20 school days in a month.
0.75 x 20.
0.8 x 20 = 16.
so the number of miles does Manuel walks each month = 16.

Question 2.
Without multiplying, which estimate in the Another example do you think is closer to the exact answer? Explain your reasoning

The number of miles does Manuel walks each month = 16.

Explanation:
In the above-given question,
given that,
There are about 20 school days in a month.
0.75 x 20.
0.8 x 20 = 16.
so the number of miles does Manuel walks each month = 16.

Do You Know How?

In 3-8, estimate each product using rounding or compatible numbers.

Question 3.
0.87 × 112

The product is 112.

Explanation:
In the above-given question,
given that,
the numbers are 0.87 and 112.
0.87 x 112.
0.8 is equal to 1.
112 x 1 = 112.
so the product is 112.

Question 4.
104 × 0.33

The product is 52.

Explanation:
In the above-given question,
given that,
the numbers are 104 and 0.33.
104 x 0.33.
0.33 is equal to 0.5.
104 x 0.5 = 52.
so the product is 52.

Question 5.
9.02 × 80

The product is 720.

Explanation:
In the above-given question,
given that,
the numbers are 9.02 and 80.
9.02 x 80.
9.02 is equal to 9.
9 x 80 = 720.
so the product is 720.

Question 6.
0.54 × 24

The product is 12.

Explanation:
In the above-given question,
given that,
the numbers are 0.54 and 24.
0.54 x 24.
0.54 is equal to 0.5.
0.5 x 24 = 12.
so the product is 12.

Question 7.
33.05 × 200

The product is 6600.

Explanation:
In the above-given question,
given that,
the numbers are 33.05 and 200.
33.05 x 200.
33.05 is equal to 33.
33 x 200 = 6600.
so the product is 6600.

Question 8.
0.79 × 51

The product is 51.

Explanation:
In the above-given question,
given that,
the numbers are 0.70 and 51.
0.7 x 51.
0.7 is equal to 1.
51 x 1 = 51.
so the product is 51.

Independent Practice

In 9-16, estimate each product.

Question 9.
0.12 × 105

The product is 10.5.

Explanation:
In the above-given question,
given that,
the numbers are 0.12 and 105.
0.12 x 105.
0.12 is equal to 0.1.
105 x 0.1 = 10.5.
so the product is 10.5.

Question 10.
45.3 × 4

The product is 180.

Explanation:
In the above-given question,
given that,
the numbers are 45.3 and 4.
45.3 x 4.
45.3 is equal to 45.
45 x 4 = 180.
so the product is 180.

Question 11.
99.2 × 82

The product is 8118.

Explanation:
In the above-given question,
given that,
the numbers are 99.2 and 82.
99.2 x 82.
99.2 is equal to 99.
99 x 82 = 8118.
so the product is 8118.

Question 12.
37 × 0.93

The product is 37.

Explanation:
In the above-given question,
given that,
the numbers are 37 and 0.93.
37 x 0.93.
0.93 is equal to 1.
37 x 1 = 37.
so the product is 37.

Question 13.
1.67 × 4

The product is 8.

Explanation:
In the above-given question,
given that,
the numbers are 1.67 and 4.
1.67 x 4.
1.67 is equal to 2.
4 x 2 = 8.
so the product is 8.

Question 14.
3.2 × 184

The product is 552.

Explanation:
In the above-given question,
given that,
the numbers are 3.2 and 184.
3.2 x 184.
3.2 is equal to 3.
184 x 3 = 552.
so the product is 552.

Question 15.
12 × 0.37

The product is 3.6.

Explanation:
In the above-given question,
given that,
the numbers are 12 and 0.3.
0.3 x 12.
0.3 is equal to 0.3.
12 x 0.3 = 3.6.
so the product is 3.6.

Question 16.
0.904 × 75

The product is 75.

Explanation:
In the above-given question,
given that,
the numbers are 0.904 and 75.
0.904 x 75.
0.9 is equal to 1.
75 x 1 = 75.
so the product is 75.

Problem Solving

Question 17.
About how much money does Stan need to buy 5 T-shirts and 10 buttons?

The cost to buy 5 T-shirts and 10 buttons = $82. Explanation: In the above-given question, given that, the cost of a button is$1.95.
the cost of the T-shirt is $12.50. 1.95 x 10 = 19.5. 12.50 x 5 = 62.5. 19.5 + 62.5 = 82. so the cost to buy 5 T-shirts and 10 buttons =$82.

Question 18.
Joseph buys a pair of shorts for $17.95 and 4 T-shirts. About how much money does he spend? Answer: The amount of money he spends =$67.95.

Explanation:
In the above-given question,
given that,
Joseph buys a pair of shorts for $17.95 and 4 T-shirts.$12.50 x 4 = 50.
$17.95 + 50 = 67.95. so the money he spend =$67.95.

Question 19.
Marcy picked 18.8 pounds of peaches at the pick-your-own orchard. Each pound costs $1.28. About how much did Marcy pay for the peaches? Write an equation to model your work. Answer: The amount did Marcy pay for the peaches =$24.064.

Explanation:
In the above-given question,
given that,
Marcy picked 18.8 pounds of peaches at the pick-your-own orchard.
Each pound costs $1.28. 18.8 x 1.28 = 24.064. so the amount did Marcy pay for the peaches =$24.064.

Question 20.
Be Precise Joshua had $20. He spent$4.58 on Friday, $7.43 on Saturday, and$3.50 on Sunday. How much money does he have left? Show how you found the answer.

The money does he have left = $15.51. Explanation: In the above-given question, given that, Be Precise Joshua had$20.
He spent $4.58 on Friday.$7.43 on Saturday, and $3.50 on Sunday. 4.58 + 7.43 + 3.50 = 15.51. so the money does he have left =$15.51.

Question 21.
Higher Order Thinking Ms. Webster works 4 days a week at her office and 1 day a week at home. The route to Ms. Webster’s office is 23.7 miles. The route home is 21.8 miles. About how many miles does she drive for work each week? Explain how you found your answer.

The number of miles does she drive for work each week = $116.6. Explanation: In the above-given question, given that, Ms. Webster works 4 days a week at her office and 1 day a week at home. The route to Ms. Webster’s office is 23.7 miles. The route home is 21.8 miles. 23.7 x 4 = 94.8. 21.8 x 1 = 21.8. 94.8 + 21.8 = 116.6 so the number of miles does she drive for work each week =$116.6.

Assessment Practice

Question 22.
Rounding to the nearest tenth, which of the following give an underestimate?
39.45 × 1.7
27.54 × 0.74
9.91 × 8.74
78.95 × 1.26
18.19 × 2.28

The equations are 39.45 x 1.7 and 27.54 x 0.74.

Explanation:
In the above-given question,
given that,
the equations are:
39.45 x 1.7 = 67.065.
27.54 x 0.74 = 20.3796.
9.91 x 8.74 = 86.6134.
78.95 x 1.26 = 99.477.
18.19 x 2.28 = 41.4732.

Question 23.
Rounding to the nearest whole number, which of the following give an overestimate?
11.6 × 9.5
4.49 × 8.3
12.9 × 0.9