Category: Envision Math

Go through the enVision Math Common Core Grade 2 Answer Key Topic 1 Fluently Add and Subtract Within 20 regularly and improve your accuracy in solving questions.

enVision Math Common Core 2nd Grade Answers Key Topic 1 Fluently Add and Subtract Within 20

Essential Question:
What are strategies for finding addition and subtraction facts?
Answer:
The Split and Jump strategies are used for finding Addition and Subtraction.
Explanation:
You can add or subtract the tens separately to the ones (or units).
For example, using the split strategy to add 46 + 23, you would split each number (decompose) into tens and ones: 46 + 23 = 40 + 6 + 20 + 3. rearrange the tens and ones: 40 + 20 + 6 + 3.

 Using a Number Line or Number Chart. This is a very popular and visual way to help students work out subtraction problems.

enVision STEM Project: Material Math
Find Out Collect different types of paper. Talk about the uses of paper. Tell how strong each type of paper is. Tell how the paper feels. Tell if the paper can soak up water.
Journal: Make a Book Show what you find out in a book. In your book, also:

• Glue samples of paper and tell what you found.
• Choose a type of paper to make flash cards of addition and subtraction facts.
  Answer:
  Types of Paper:
  • Bond paper. This type of paper is stronger and more durable than the average sheet of paper.
  • Gloss coated paper, it is typically used for flyers and brochures as it has a high shine.
  • Matt coated paper, has a semi-gloss finish, produces high quality prints.
  • Recycled paper, saves landfill space and reduces the amount of pollution in the air. 
  • Silk coated paper, used commonly for leaflets and flyers.
  • Uncoated paper, used for books, letterheads, envelopes, note-pads and a number of other paper products. 

  USES OF PAPER :
  It is a versatile material with many uses, including printing, packaging, decorating, writing, cleaning, filter paper, wallpaper, book endpaper, conservation paper, laminated worktops, toilet tissue, currency and security paper and a number of industrial and construction processes.

  Tell how the paper feels:
  Answer may vary, but paper think poetically
  u can write ur feelings,
  scribble ur anger,
  use me 2 absorb tears.
  Don’t throw me after use but
  when u feel cold burn me
  to feel warm cause I Love you.
  Tell if the paper can soak up water:
  When paper soaked in water it readily absorbs water because it is made of cellulose, which water molecules like to cling to.

  1. Make a List of the Journals Available. …
  2. Determine the Impact of the Journal. …
  3. Make Sure the Journal Scope and Policies match your Needs. …
  4. Check the Journal Requirements and Distribution. …
  5. Collect Information about the Journal’s Peer Review Process. …
  6. Glue samples of paper and tell what you found.
  7. Check the “Instructions for Authors” thoroughly.
     Journal: Make a Book Show what you find out in a book. In your book.

• 1. Make Your Own Flash Cards.
  2. Mix Pictures and Words.
  3. Use Mnemonic Devices to Create Mental Connections.
  4. Write Only One Question Per Card.
  5. Break Complex Concepts Into Multiple Questions.Observations of Glue samples of paper.
     PVA Glue. The most versatile adhesive! PVA can be used on a wide range of surfaces: paper, card, fabric, wood, plaster and much more.
     Preparation of flash cards:
     Collect your materials together. This means that you should have your flash cards and textbook ready. Have a good pen, markers, highlighters, and whatever other writing implements you’d like to use.

Question 1.
Circle the symbol for equal
+
=
–
Answer:

Question 2.
Circle the symbol for minus.
–
+
=
Answer:

Question 3.
Circle the number that is the whole.

4 + 2 = 6
Answer:

Subtraction Stories
Question 4.
There are 7 birds on a fence. 2 fly away. How many birds are left?

_______ birds
Answer:
5 birds
Explanation:
Number of birds on fence = 5
Number of birds fly away = 2
Number of birds left = (7-2) = 5

Addition Stories
Question 5.
Write an equation to solve the problem.
Kate draws 4 big stars. Then she draws 2 small stars. How many stars does Kate draw in all?
______ + _____ = ______
Answer:
6 stars
Explanation:
__4___ + __2___ = ___6___
Number of big stars =4
Number of small stars =2
Total number of stars =6

Make 10
Question 6.
Write an equation that shows this way to make a 10.

Answer:
8+2=10
___8___ + ___2__ = __10____
Explanation:

By adding 2 to 8 we get 10.

Pick a Project

PROJECT 1A
What has 8 legs and is big and hairy?
Project: Make an Insect and Spider Poster

Answer:
Spider
YARN SPIDER CRAFT – MATERIAL
Materials Required:
1. YARN
2. STRYO FOAM BALL
3.PIPE CLEANERS
4. CHOP STICKS
5.HOT GLUE
6.GOOGLY EYES
Step 1. Wrap yarn around your styro foam ball. Once you’re done, secure the yarn with a bit of hot glue.

Step 2. Take your skinny long object like a chopstick and pierce two holes, one on each side of your yarn ball.

Step 3. Insert your pipe cleaners (cut into “legs”) you should have 4 legs per side.

Step 4. Glue on your googly eyes.

 

Project: Collect Water Data

Water source	Water volume, in cubic miles	Water volume, in cubic kilometers	Percent of freshwater	Percent of total water
Oceans, Seas, & Bays	321,000,000	1,338,000,000	—	96.54
Ice caps, Glaciers, & Permanent Snow	5,773,000	24,064,000	68.7	1.74
Groundwater	5,614,000	23,400,000	—	1.69
Fresh	2,526,000	10,530,000	30.1	0.76
Saline	3,088,000	12,870,000	—	0.93
Soil Moisture	3,959	16,500	0.05	0.001
Ground Ice & Permafrost	71,970	300,000	0.86	0.022
Lakes	42,320	176,400	—	0.013
Fresh	21,830	91,000	0.26	0.007
Saline	20,490	85,400	—	0.006
Atmosphere	3,095	12,900	0.04	0.001
Swamp Water	2,752	11,470	0.03	0.0008
Rivers	509	2,120	0.006	0.0002
Biological Water	269	1,120	0.003	0.0001

PROJECT 1C
Have you seen this different kind of art?
Project: Create a Shape Collage

Answer:
BRINGING THE CONCEPT OF DIFFERENT COLORS AND SHAPES
PROJECT;
BRINGING THE CONCEPT OF DIFFERENT COLORS AND SHAPES
Materials required:

• Tissue paper (circles and squares)
• Old art or paper scraps
• Scissors
• Cardstock
• Clear glue or white glue
• Bows, Brushes
  Process:
  Depending on the time you have or the interest of your child, you can either set out some pre-cut shapes, or you can let you child cut their own shapes. Each child’s artwork is unique and reflects their personality. I love how one child overlapped her tissue and created new colors. Another child was keen to keep everything very orderly. And another barely used any pre-cut shapes and instead, used the scissors like Matisse and cut his own shapes.
  

• By making their own choices about color and composition, they are practicing their critical thinking skills.
  They are exploring basic shapes and abstract shapes while creatively putting together an image.
  This gives them an opportunity to associate shapes with real life objects.
• When using the tissue paper, they are learning about transparency, and how colors can mix and overlap to make new colors.

3-ACT MATH PREVIEW

Math Modeling
Losing Marbles

Lesson 1.1 Addition Fact Strategies

Solve & Share
Use cubes. Show 2 + 5 and 5 + 2. Solve both problems. Explain how the problems are alike and how they are different.
I can … count on to add and add in any order.

Visual Learning Bridge

Convince Me! Does 5 + 2 = 2 + 5? How do you know?
Answer:
5+2=2+5=7
In additions there is no difference in addends, either from left to right the answer will be the same.

Guided Practice

Count on to find the sum. Then change the order of the addends.
Question 1.

Question 2.

Answer:
4+3=7
Explanation:
you can add numbers in any order and the sum is same, as shown below.

Independent Practice

Count on to find the sum. Then change the order of the addends. Use cubes if needed.
Question 3.
8 + 2 = ________
_______ + _______ = ________
Answer:
8+2=10
2+8=10
Explanation:

An equation uses an equal sign (=) to show that the value on the left is the same as the value on the right.

Question 4.
8 + 5 = ________
________ + ________ = ________
Answer:
8+5=13
5+8=13
Explanation:

An equation uses an equal sign (=) to show that the value on the left is the same as the value on the right.

Question 5.
9 + 3 = ________
________ + ________ = ________
Answer:
9+3=11
3+9=11
Explanation:

An equation uses an equal sign (=) to show that the value on the left is the same as the value on the right.

Question 6.
8 + 7 = ________
________ + ________ = ________
Answer:
8+7=15
7+8=15
Explanation:

An equation uses an equal sign (=) to show that the value on the left is the same as the value on the right.

Question 7.
7 + 10 = ________
________ + ________ = ________
Answer:
7+10=17
10+7=17
Explanation:

An equation uses an equal sign (=) to show that the value on the left is the same as the value on the right.

Question 8.
7 + 9 = ________
________ + ________ = ________
Answer:
7+9=16
9+7=16
Explanation:

An equation uses an equal sign (=) to show that the value on the left is the same as the value on the right.

Question 9.

Answer:
7+2=9
Explanation:
you can add numbers in any order and the sum is same, as shown below.

Question 10.

Answer:
6+2=8
Explanation:
you can add numbers in any order and the sum is same, as shown below.

Question 11.

Answer:
5+6=11
Explanation:
you can add numbers in any order and the sum is same, as shown below.

Question 12.
Algebra Write the missing numbers.
6 + ____ = 4 + 6
8 + 2 = ____ + 8

6 + ____ = 5 + 6
____ + 7 = 7 + 4

9 + 3 = 3 + ____
____ + 8 = 8 + 4
Answer:

Explanation:
you can add numbers in any order and the sum is same, in the above pattern.

Problem Solving

Solve the problems below.

Question 13.
Model Joy has 8 bean plants and 6 corn plants in her garden. How many plants does she have in all? Draw a picture to explain your thinking. Then write facts for this story with the addends in a different order.
____ + ____ = ____
____ + ____ = ____
Answer:
8+6=14
Explanation:

Question 14.
Higher Order Thinking Find the objects in Box I and Box 2 that are the same. Write an equation to show how many of each object there are. Then change the order of the addends.

Answer:

Question 15.
Assessment Practice which shows how to count on to find 7 + 5?
A. 7… 8, 9, 10, 11, 12
B. 1, 2, 3, 4, 5
C. 7 – 5
D. 7…8, 9, 10, 11
Answer:
Option A
Explanation:
Keep 7 in your mind and five fingers open, then count 8,9,10,11,12

Lesson 1.2 Doubles and Near Doubles

Solve & Share
Use counters. Show 6 + 6 = 12. Then show and explain how knowing that fact can help you find 6 + 7.
I can … use doubles and near doubles to add quickly and accurately.

Visual Learning Bridge

Convince Me! How could you use the doubles fact 7 + 7 to find 7 + 9?
Answer:
You can use a double fact in additions to add the addends.
Explanation:
7+8 is 1 more than 7+7
7+8=15
7+9 is 2 more than7+7
7+9=15
So these are near double facts.

Guided Practice

Complete the doubles facts. Use the doubles facts to solve the near doubles. Use cubes if needed.
Question 1.
2 + 2 = _______
2 + 3 = _______
Answer:
2+2=4
2+3=5
Explanation:
So these are near double facts.

Question 2.
_______ = 4 + 4
_______ = 4 + 5
Answer:
4+4=8
4+5=9
Explanation:
So these are near double facts.

Question 3.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 3+4 is considered a near double because it is close to the doubles fact 3+3.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 6 and 7 are the nearest double facts.

Question 4.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 5+7 is considered a near double because it is close to the doubles fact 5+5.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 10 and 12 are the nearest double facts.

Independent Practice

Complete the doubles facts. Use the doubles facts to solve the near doubles. Use cubes if needed.
Question 5.
6 + 6 = _______
6 + 7 = _______
Answer:
6+6=12
6+7=13
Explanation:

Question 6.
5 + 5 = _______
5 + 6 = _______
Answer:
5+5=10
5+6=11
Explanation:

Question 7.
8 + 8 = _______
8 + 10 = _______
Answer:
8 + 8 = 16
8 + 10 = 18
Explanation:

Question 8.
_______ = 1 + 1
_______ = 1 + 3
Answer:
2 = 1 + 1
4 = 1 + 3
Explanation:

Question 9.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 2+2 is considered a near double because it is close to the doubles fact 2+3.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 4 and 5 are the nearest double facts.

Question 10.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 4+4 is considered a near double because it is close to the doubles fact 4+5.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 8 and 9 are the nearest double facts.

Question 11.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 8+8 is considered a near double because it is close to the doubles fact 8+9.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 16 and 17 are the nearest double facts.

Question 12.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 3+73is considered a near double because it is close to the doubles fact 3+5.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 6 and 8 are the nearest double facts.

Question 13.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 5+5 is considered a near double because it is close to the doubles fact 5+7.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 10 and 12 are the nearest double facts.

Question 14.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
For example, 7+7 is considered a near double because it is close to the doubles fact 7+9.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
So 14 and 16 are the nearest double facts.

Question 15.
Algebra Complete. Then explain how you solved the problem.

Answer:

Explanation:
A near double is a math fact that is close to a doubles fact.
9+9is considered a near double because it is close to the doubles fact 9+10.
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one fact

Problem Solving
Solve the problems below.
Question 16.
Model John drew 4 houses. Then he drew 5 more houses. How many houses did John draw in all?
Draw a picture and write an equation to explain your thinking.
_______ houses
Answer:
9 houses
Explanation:

Question 17.
Higher Order Thinking Choose a doubles fact. Use that doubles fact to draw a picture that shows a near doubles story. Write the near doubles fact.
Answer:
Near doubles could be doubles plus one facts, doubles plus two facts, or doubles minus one facts.
Let’s take a look at the doubles fact 5+5. 5+6 would be the double plus one (5+5 plus one more).
5+7 would be the double plus two (5+5 plus two more).23-Jul-2020

Question 18.
Assessment Practice Kate wants to find 6 + 7.

Which doubles fact will help her the most?
A. 4 + 4 = 8
B. 6 + 6 = 12
C. 8 + 8 = 16
D. 9 + 9 = 18
Answer:
Option B
6+6=12
6+7=13
6+7 is one more then 6+6

Lesson 1.3 Make a 10 to Add

Solve & Share
How can thinking about 10 help you find 9 + 3? Use the ten-frames and counters to show how.
I can … make a 10 to help me add quickly and accurately.

Visual Learning Bridge

Convince Me! Why do you move 2 counters to add 8 + 5?
Answer:
Addition can be thought of as combining two quantities and finding the total to add two plus eight combined five yellow counters and eight red counters. There are 13 counters in total.
Explanation:
8+5=13
10+3=13

Guided Practice

Make a 10 to add. Use counters and ten-frames.
Question 1.

Answer:
7+4=11
Explanation:

Independent Practice

Make a 10 to add. Use counters and ten-frames.
Question 2.

Answer:
8+4=12
Explanation:

Question 3.

Answer:
3+9=12
Explanation:

Question 4.

Answer:
6+7=13
Explanation:

Question 5.

Answer:
5+8=13

Question 6.

Answer:
7+5=12
Explanation:

Question 7.
5 + 9 = _______

Answer:
5+9=14
Explanation:

Question 8.
3 + 8 = ________

Answer:
3+8=11
Explanation:

Question 9.
4 + 9 = ______

Answer:
4+9=13
Explanation:

Question 10.
7 + 9 = _______

Answer:
7+9=16
Explanation:

Algebra Which number is missing?
Question 11.
8 + 5 = + 3
Answer:
10 is missing
Explanation:
8+5=13
10+3=13

Question 12.
6 + 9 = 10 +
Answer:
5 is missing
Explanation:
6+9=15=10+5

Question 13.
8 + 9 = 10 +
Answer:
7 is missing
Explanation:
8+9=17=10+7

Question 14.
Higher Order Thinking Can you make a 10 to help you add 7 + 4 + 5? Explain.
Answer: 16

Explanation:
The make–ten strategy is great for addition! It helps students understand place value and the relationships between numbers.
Ten-frames help students develop a good “mind picture” for the make–ten strategy because our place-value system is based on making groups of ten.

Problem Solving

Solve the problems below. Show your work.
Question 15.
Tan’s team scored 16 points in a game. During the first half, they scored 9 points. How many points did the team score in the second half of the game?
_________ points
Answer:
Tan’s team score 7 points in second half.
Explanation:
Total points scored by Tan’s team = 16
In first half they scored = 9 points
In second half they scored (16-9)=7 points.

Question 16.
Make Sense The school has a clothing drive for charity. Ana’s class donates 8 coats. Nico’s class donates 5 hats. Adam’s class donates 8 coats. How many coats were donated in all?
_________ coats
Answer:
16 coats were donated in all.
Explanation:
No of coats donated by Ana’s class = 8
No of hats donated by Adam’s class = 5
No of coats donated by Nico’s class = 8
Total coats donated in all (8+8) = 16

Question 17.
Higher Order Thinking Draw a picture to show how you can make a 10 to help you add 3 + 5 + 9. Explain your picture to a friend.
Answer:

Question 18.
Assessment Practice Use the ten-frames. Show how to find 5 + 7 by making a 10. Then fill in the gray boxes.

Answer: 12
Explanation:

Lesson 1.4 Addition Fact Patterns

Solve & Share
Look at the sums on an addition facts table for addends 0 to 5. Describe one of the patterns that you see. Use words, colors, or addition facts to describe the patterns.
I can … use the patterns on an addition facts table to help me remember the addition facts.

Visual Learning Bridge

Convince Me! How can patterns on an addition facts table help you remember the addition facts? Explain.
Answer:
The addition facts are all of the sums from 0+1 or 1+0 up to 10+10.
Explanation:
It’s essential that children master in sums, because they’re the foundation for the rest of elementary arithmetic.
Here’s the full addition facts chart is above shown picture.

Guided Practice

Use fact patterns to complete each equation.
Question 1.

Question 2.
10+ ______ = 14
______ + 5 = 14
8 + 6 = ______
7 + ______ = 14
______ + 8 = 14
5 + ______ = 14
______ + 10 = 14
Answer:

Explanation:

By adding 4 to 10,9 to5,8 to 6, 7 to 7, 6 to 8, 5 to 9, 4 to 10 the answer will be 10 only.
So it tells the facts about number patterns.

Question 3.
9 + ______ = 9
______ + 1 = 9
7 + 2 = ______
6 + ______ = 9
______ + 4 = 9
5 + ______ = 9
______ + 3 = 9
______ + 2 = 9
8 + 1 = ______
______ + 0 = 9
Answer:

9+0=9
8+1=9
7+2=9
6+3=9
5+4=9
4+5=9
3+6=9
2+7=9
1+8=9
9+0=9
Explanation:

Independent Practice

Use fact patterns to complete each equation.
Question 4.
10 + ______ = 12
______ + 3 = 12
8 + 4 = ______
7 + ______ = 12
6 + 6 = ______
______ + 7 = 12
4+ ______ = 12
______ + 9 = 12
Answer:

10+2=12
9+3=12
8+4=12
7+5=12
6+6=12
5+7=12
4+8=12
3+9=12
Explanation:

Question 5.
7 + ______ = 7
______ + 1 = 7
5 + 2 = ______
______ + 3 = 7
______ + 4 = 7
______ + 5 = 7
______ + 6 = 7
0 + ______ = 7
Answer:
7+0=7
6+1=7
5+2=7
4+3=7
3+4=7
2+5=7
1+6=7
0+7=7
Explanation:

Question 6.
10+ ___ = 15
______ + 5 = 15
9 + 6 = ______
9 + ______ = 15
7 + ______ = 15
______ + 8 = 15
Answer:
10+ _5__ = 15
_10__ + 5 = 15
9 + 6 = _15__
9 + __6__ = 15
7 + __8__ = 15
_7_ + 8 = 15
Explanation:

Question 7.
Number Sense Find the 8 in the top row of the addition facts table. Complete all of the equations using the 8-column on the table. What pattern do you see?
8 + ______ = 8
8 + ______ = 9
8 + ______ = 10
8 + 3 = ______
8 + ______ = 12
8 +            =
8 + 6 = ______
8 + ______ = 15
8 + 8 = ______
8 + 10 = ______
Answer:

Explanation:

Problem Solving
Solve the problems below.
Question 8.
Reasoning Lucy is sorting items by their texture. She finds 6 items that are bumpy. Lucy finds 5 items that are smooth. How many items does Lucy find in all?
_________ items
Answer:
Lucy find 12 items in all.
Explanation:
Number of items that are bumpy = 6
Number of items that are smooth =5
Total items that Lucy had = 12

Question 9.
Vocabulary Look at the equation below. Circle the addends. Draw a square around the sum.
9 + 5 = 14
Answer:

Explanation:
In the above equation 9 and 5 are addends.
14 is the sum.
Question 10.
Higher Order Thinking Write 8 addition facts that have a sum of 12. How can the addition patterns help you? Explain.
Answer:
8+4=12
4+8=12

Explanation:

Question 11.
Assessment Practice which have a sum of 17? Choose all that apply.
Think about fact patterns.
☐ 10 + 7 = ?
☐ 7 + 6 = ?
☐ 9 + 8= ?
☐ 8 + 8 = ?
Answer:
10+7=17
9+8=17
Explanation:

Lesson 1.5 Count On and Count Back to Subtract

Solve & Share
How can counting help you find 12 – 4? Use the number line to show your work.
I can … count on and count back on a number line to subtract.

Answer:
12-4=8
Explanation:

Start with greater number and count back 8 moves to find the difference.

Visual Learning Bridge

Convince Me! How can you count back on a number line to find 9 – 5?
Answer:
9-5=4
Start with greater number and count back 5 moves to find the difference.
Explanation:

Subtraction of numbers using number line will help us to learn how a number line can be used for subtracting one number from the another number. Subtraction of numbers can be well understood with the help of the number line.

Guided Practice

Count on or count back to subtract. i.e Show your work on the number line.
Question 1.

Question 2.
14 – 7 =

Answer:
14-7=7
Explanation:

Independent Practice

Count on or count back to subtract. Show your work on the number line.
Question 3.
14 – 8 =

Answer:
14-8=6
Explanation:

Question 4.
12 – 7 =

Answer:
12-7=5
Explanation:

Question 5.
9 – 7 =

Answer:
9-7=2
Explanation:

Question 6.
15 – 6 =

Answer:
15-6=9
Explanation:

Question 7.
Higher Order Thinking How can you count on to find 13 – 4? Explain.
Answer:
13-4=9
Explanation:

Start with lesser number and count on to 13 to find the difference.

Problem Solving

Solve the problems below.
Question 8.
Callie had 18 seeds. She planted 10 of the seeds. How many seeds does Callie have now?
________ seeds
Answer:
After planting10 seeds Callie had 8 seeds.

Explanation:
Number of seeds = 18
Number seeds planted = 10
Total number of seeds left = 18-10=8

Question 9.
Make Sense Peter has 16 grapes. He eats some of the grapes. Peter has 10 grapes left. How many grapes did Peter eat?
________ grapes
Answer:
Peter ate 6 grapes.
Explanation:
Total number of grapes = 16
Number of grapes left = 10
Some of the grapes eaten by Peter = 16-10=6

Question 10.
Higher Order Thinking Choose 2 numbers. Use the numbers to write or draw a subtraction story. Write the equation you used to solve your story.
______ – _______ = ________
Answer:
9 – 7 = 2
Explanation:
Start with greater number and count back 2 moves to find the difference.

Question 11.
Assessment Practice Jake wants to find 6 – 3.
Use the numbers on the cards. Write them in the boxes to show how to use a number line to count on to solve.

6 – 3 = _______
Answer:
6-3=3
Explanation:
Begin with greater number 6 and count on to lesser number 3, then difference is 3.

Lesson 1.6 Think Addition to Subtract

Solve & Share
How can you use an addition fact to find 14 – 6? Use counters to help show how.
I can … use addition to help me subtract quickly and accurately.
_____ + _______ = ________
So, 14 – 6 = _______

Answer:
6 + 8 = 14
14 – 6 =8
Explanation:
The basic facts of addition are those equations in which two single-digit numbers are combined by addition to give a sum. The missing number is the same in both equation.
Addition Fact  6+8=14
Subtraction Fact 14- 6 =8

Question 3.

Visual Learning Bridge

Convince Me! How do you know which addition fact to use to complete the subtraction fact?
Answer:
In the case of addition and subtraction, you use three numbers and get four facts.

Guided Practice

Think addition to help you subtract.
Question 1.

Question 2.
9 – 3 = ?
3 + _______= 9
So, 9 – 3 = _______
Answer: 6
9 – 3 = 6
3 + 6= 9
So, 9 – 3 = 6
Explanation:
The basic facts of addition are those equations in which two single-digit numbers are combined by addition to give a sum. The missing number is the same in both equation.
Addition Fact 3+6=9
Subtraction Fact 9 – 3 = 6
So,9-3=6

Question 3.
14 – 5 =
5 + _____=4
So, 14 – 5 = _____
Answer: 9
14 – 5 = 9
5 +  9  = 14
So, 14 – 5 = 9
Explanation:
The basic facts of addition are those equations in which two single-digit numbers are combined by addition to give a sum. The missing number is the same in both equation.
Addition Fact 14-6=9
Subtraction Fact 5+9=14
So 14-5=9

Question 4.
12 – 4 = ?
4 + _______ = 12
So, 12 – 4 = _______
Answer: 8
12 – 4 = 8
4 + 8 = 12
So, 12 – 4 = 8
Explanation:
The basic facts of addition are those equations in which two single-digit numbers are combined by addition to give a sum. The missing number is the same in both equation.
Addition Fact 12-4=8
Subtraction Fact 4+8=12
So,12-4=8

Independent Practice

Subtract. Complete the addition fact that can help you.
Question 5.
8 – 1 = _______
1 + _______ = 8
Answer: 7
8 – 1 = 7
1 + 7 = 8
Explanation:
The basic facts of addition are those equations in which two single-digit numbers are combined by addition to give a sum.
Hence they range from 0+0=0 to 9+9=18.
For each basic addition fact there is a related basic subtraction fact.
Question 6.
10 – 2 = _______
2 + _______ = 10
Answer: 8
10 – 2 = 8
2 + 8= 10
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 2 and 8 to get 10 is the opposite of 10 minus 2, leaving 8.

Question 7.
15 – 6 = _______
6 + _______ = 15
Answer: 9
15 – 6 = 9
6 + 9= 15
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 6 and 9 to get 15 is the opposite of 15 minus 6, leaving 9.

Question 8.
17 – 7 = _______
7 + _______ = 17
Answer: 10
17 – 7 = 10
7 + 10 = 17
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 7and 10 to get 17 is the opposite of 17 minus 7, leaving 17.

Question 9.
14 – 8 = _______
8 + _______ = 14
Answer: 6
14 – 8 = 6
8 + 6 = 14
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 8 and 6 to get 14 is the opposite of 14 minus 8, leaving 6.

Question 10.
9 – 5 = _______
5 + _______ = 9
Answer: 4
9 – 5 = 4
5 + 4  = 9
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 5 and 4 to get 9 is the opposite of 9 minus 5, leaving 4.

Question 11.

Answer: 10

Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 8 and 10 to get 18 is the opposite of 18 minus 8, leaving 10
Question 12.

Answer: 7

Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 9 and 7 to get 16 is the opposite of 16 minus 9, leaving 7

Question 13.

Answer: 10

Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 10 and 9 to get 19 is the opposite of 19 minus 9, leaving 10
Higher Order Thinking Write a related addition fact to complete the subtraction fact.
Question 14.
11 – _______ = 5
______ + ______ = _______
Answer:
11 – 6 =5
5 + 6 = 11
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 5 and 6 to get 11 is the opposite of 11 minus 6, leaving 5

Question 15.
7 – _______ = 2
_______ + _______ = _______
Answer:
7 – 5 =2
2 + 5 = 7
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 2 and 5 to get 7 is the opposite of 7 minus 5, leaving 2

Question 16.
12 – _______ = 8
_______ + _______ = _______
Answer:
12 – 4 =8
8 + 4 = 12
Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.
Adding 8 and 4 to get 12 is the opposite of 12 minus 4, leaving 8

Problem Solving

Solve the problems below. Write an equation.
Question 17.
Reasoning Kate had 6 pens. She got 5 more pens from John. How many pens does Kate have in all?

______ pens
Answer:11 pens

Explanation:
Number of pens Kate had 6
John gave 5 more pens
Total pens Kate has 11

Question 18.
Reasoning John had 11 pens. He gave 5 pens to Kate. How many pens does John have now?

_______ pens
Answer: 6 pens

Explanation:
John had 11 pens.
John gave 5 pens to Kate.
Total pens does John have now is 6

Question 19.
Higher Order Thinking Write a subtraction story using the numbers 18 and 10. Then write an addition fact that can help you solve the problem in your story.
Answer:
Peter has 18 chocolates.
He eats 10 chocolates.
Now Peter has 18 chocolates in total.
18 – 10 = 8
10 + 8 = 18

Question 20.
Assessment Practice Pam has 16 cherries. She eats 7 cherries. Which addition fact can help you find how many cherries Pam has left?

A. 7 + 4 = 11
B. 7 + 9 = 16
C. 7 + 6 = 13
D. 9 + 9 = 18
Answer:
Option B
Explanation:
16-7=11
11+7=16

Lesson 1.7 Make a 10 to Subtract

Solve & Share
14 ladybugs are on a leaf. 6 ladybugs fly away. How can thinking about 10 help you find how many ladybugs are left? Explain.
I can … make a 10 to help me subtract quickly and accurately.

Visual Learning Bridge

Convince Me! Do you prefer to add first to get to 10 or subtract first to get to 10? Explain.
Answer:
Yes, i prefer addition first.
Explanation:
Add first to get 10 as shown in the above,
Then add 3 more to make 13.

Guided Practice

Make a 10 to subtract. Use counters and your workmat.
Question 1.
First add to get to 10.

Answer:7

Question 2.
First subtract to get to 10.

Answer: 9
Explanation:

Independent Practice

Make a 10 to subtract. Use counters and your workmat.

Think of the ways you know to make 10.
Question 3.

Answer: 7
Explanation:

Question 4.

Answer: 6
Explanation:

Question 5.

Answer: 5
Explanation:

Question 6.

Answer: 8
Explanation:

Question 7.

Answer: 9
Explanation:

Question 8.

Answer: 9
Explanation:

Question 9.

Answer: 8
Explanation:

Question 10.

Answer: 9
Explanation:

Question 11.

Answer: 6
Explanation:

Question 12.

Answer: 7
Explanation:

Question 13.

Answer: 4
Explanation:

Question 14.

Answer: 7
Explanation:

Question 15.
Higher Order Thinking Carol subtracts 6 from 15. First, she adds to get to 10. Then she adds again to find her answer. Her answer is 10. Is Carol correct? Explain.
Answer: 9
No, her answer is wrong
Explanation :
15-6=9
she added 1 + 9 to get 10
She added again 5 +10 = 15

Problem Solving

Solve each problem. Which tool would you use?
Question 16.
Use Tools Chen had 12 animal stickers. He gave 5 of the stickers away. How many animal stickers does Chen have now?
________ animal stickers
Answer: 7 stickers
Explanation:

Question 17.
Use Tools Angie bought 13 strawberries. She ate 8 of the strawberries. How many strawberries does Angie have now?
________ strawberries
Answer: 5
Explanation:

Question 18.
Higher Order Thinking Show how you can make a 10 to find 17 – 9. Then explain your work.
Answer: 8
Explanation:

Question 19.
Assessment Practice Use the ten-frames. Show how to make a 10 to find 15 – 9. Start by subtracting to get to 10. Then complete the equations.

15 – _____ = 10
10 – _____ = _____
So, 15 – 9 =_____;
Answer:

15 – 5 = 10
10 – 4 = 6
15 – 9 =6
Explanation:

Lesson 1.8 Practice Addition and Subtraction Facts

Solve & Share
Write four related facts that use both the numbers 7 and 9 as quickly as you can. Hold up your hand when you are done. Then, tell how you found each fact.
I can … add and subtract quickly and accurately using mental math strategies.

9 + 7 = 16                       16 – 7 = 9
7 + 9 = 16                       16 – 9 = 7

Visual Learning Bridge

Convince Me! How can thinking about 10 help you find 14 – 8?
Answer : 6
Explanation:

Guided Practice

Add or subtract. Use any strategy.
Question 1.

Answer : 5
Explanation:

Question 2.

Answer: 8
Explanation:

Question 3.

Answer: 12
Explanation:

Question 4.

Answer: 5
Explanation:

Question 5.

Answer: 6
Explanation:

Question 6.

Answer: 18
Explanation:

Question 7.

Answer: 8
Explanation:

Question 8.

Answer: 20
Explanation:

Question 9.

Answer: 7
Explanation:

Question 10.

Answer: 10
Explanation:

Question 11.

Answer: 8
Explanation:

Question 12.

Answer: 8
Explanation:

Independent Practice

Add or subtract. Use any strategy.
Question 13.
14 – 7 = ________
Answer:
14-7=7
Explanation:
By subtracting smaller number 7 from greater number 14 we get the difference 7.

Question 14.
3 + 0 = ________
Answer:
3+0=3
Explanation:
By adding 3 to 0 we get the sum 3.

Question 15.
8 + 7 = ________
Answer:
8+7=15
Explanation:
By adding 8 to 7 we get the sum 15.

Question 16.
13 – 6 = ________
Answer:
13-6=7
Explanation:
By subtracting smaller number 6 from greater number 13 we get the difference 7.

Question 17.
10 + 9 = ________
Answer:
10+9=19
Explanation:
By adding 10 to 9 we get the sum 19.

Question 18.
17 – 8 = ________
Answer:
17-8=9
Explanation:
By subtracting smaller number 8 from greater number 17 we get the difference 9.

Question 19.
18 – 9 = ________
Answer:
18-9=9
Explanation:
By subtracting smaller number 9 from greater number 18 we get the difference 9.

Question 20.
9 – 1 = ________
Answer:
9-1=8
Explanation:
By subtracting smaller number 1 from greater number 9 we get the difference 8.

Question 21.
7 + 4 = ________
Answer:
7+4=11
Explanation:
By adding 7 to 4 we get the sum 11.

Question 22.
6 + 6 = ________
Answer:
6+6=12
Explanation:
By adding 6 to 6 we get the sum 12.

Question 23.
16 – 9 = ________
Answer:
16-9=7
Explanation:
By subtracting smaller number 9 from greater number 16 we get the difference 7.

Question 24.
20 – 10 = ________
Answer:
20-10=10
Explanation:
By subtracting smaller number 10 from greater number 20 we get the difference 10.

Question 25.
16 – 7 = ________
Answer:
16-7=9
Explanation:
By subtracting smaller number 7 from greater number 16 we get the difference 9.

Question 26.
15 – 8 = ________
Answer:
15-8=7
Explanation:
By subtracting smaller number 8 from greater number 15 we get the difference 7.

Question 27.
7 + 3 = ________
Answer:
7+3=10
Explanation:
By adding 3 to 7 we get the sum 10.

Question 28.
2 + 7 = ________
Answer:
2+7=9
Explanation:
By adding 2 to 7 we get the sum 9.

Question 29.
9 + 6 = ________
Answer:
9+6=15
Explanation:
By adding 9 to 6 we get the sum 15.

Question 30.
10 – 2 = ________
Answer:
10-2=8
Explanation:
By subtracting smaller number 2 from greater number 10 we get the difference 8.

Higher Order Thinking Write the missing number.
Question 31.
6 + ☐ = 14 – 5
Answer: 3
6 + 3 = 14 – 5

Question 32.
12 – 4 = ☐ + 2
Answer: 6
12 – 4 = 6 + 2

Question 33.
14 – ☐ = 5 + 4
Answer: 5
14 – 5 = 5 + 4

Problem Solving

Solve the problems below.
Question 34.
en Vision® STEM Danielle had 17 pieces of paper. She changed 8 of the pieces by cutting them. How many pieces were not changed? Write an equation to solve.

__________ pieces of paper
Answer: 9

Explanation:
Danielle had 17 piece.
She used 8 pieces for cutting.
After cutting no. of pieces left with her 9.
Question 35.
Model Diego saw 5 frogs on a rock. He also saw 7 frogs in the grass. How many frogs did Diego see in all? Write an equation to solve.

_______ frogs
Answer: 12

Explanation:
Number of frogs Diego saw on rock = 5
Number of frogs Diego saw in the grass = 7
Total number of frogs he saw together = 12
Question 36.
Higher Order Thinking Glen counts on to solve 9 + ☐ = 14. Explain how he can do this. What is the missing addend?
Answer: 5

Explanation:
Addition and subtraction are inverse operations because one operation can “undo” the other operation
Question 37.
Assessment Practice Choose a strategy to find 11 – 8. Explain how you used the strategy to find the difference.
Which strategy will you use to solve this problem?
Answer: 3
Explanation:

By using number line strategy, counting from bigger number to smaller number with back moves

Lesson 1.9 Solve Addition and Subtraction Word Problems

Solve & Share
Diego has 6 apples. Leslie has 9 apples. How many more apples does Leslie have than Diego? Will you add or subtract to solve this problem? Explain.
Answer:3
Leslie has 3 more apples.
Explanation:
By subtracting smaller number 6 from greater number 9 we get the difference 3.
I can … use addition and subtraction to solve word problems.

Visual Learning Bridge

Convince Me! Why can you use addition OR subtraction to solve the problem above?
Answer:
Addition and subtraction are inverse operations because one operation can “undo” the other operation.

Guided Practice

Write an equation to solve each problem. Use any strategy.
Question 1.
Sam has 5 red tomatoes and 3 green tomatoes. How many tomatoes does he have in all?

Answer: 8

Explanation:
By adding 3 to 5 we get the sum 8.

Question 2.
There are 16 party hats in a box. There are 10 party hats in a bag. How many fewer hats are in the bag than in the box?

______ fewer hats
Answer: 6

Explanation:
By subtracting smaller number 10 from bigger number 16, we get the difference 6.
Independent Practice

Write an equation to solve each problem. Use any strategy.
Question 3.
Cho has 3 more toy horses than Hakeem. Cho has 9 toy horses. How many toy horses does Hakeem have?

________ toy horses
Answer: 6

Explanation:
By subtracting smaller number 9 from bigger number 15, we get the difference 6.
Question 4.
There are 12 peaches in a bowl. The children eat some of them. Now there are 8 peaches. How many peaches did the children eat?

________ peaches were eaten.
Answer: 4

Explanation:
By subtracting smaller number 8 from bigger number 12, we get the difference 4.
Question 5.
Juan reads 5 books. Susan reads some books. They read 11 books in all. How many books did Susan read?

__________ books
Answer: 6

Explanation:
By subtracting smaller number 5 from bigger number 11, we get the difference 6.
Question 6.
Jack has 13 brushes. Igor has 6 brushes. How many fewer brushes does Igor have than Jack?

_______ fewer brushes
Answer: 7

Explanation:
By subtracting smaller number 6 from bigger number 13, we get the difference 7.
Question 7.
Number Sense Jen had 3 animal stickers in her collection. Her friend gave her 5 more stickers. Jen bought 7 more stickers. How many stickers does Jen have now? Show your work.
Answer:
15 stickers.
Explanation:
Jen had 3 animal stickers first and her friend gave 5 more, now she has total 8 stickers. Next
3 + 5 = 8
Jen bought 7 more and added to 8 animal stickers, now total 15 animal stickers.
8 +7 = 15

Problem Solving

Solve the problems below.
Question 8.
Higher Order Thinking Sandy has 8 markers. Alex has 6 fewer markers than Sandy. Jill has 2 markers. How many markers do they have in all? Show your work. Then explain how you found the answer.
Answer:
12 markers.
Explanation:
Sandy has 8 markers
Alex has 8 – 6 = 2 markers (subtract 6 from Sandy 8 markers as fewer or less)
Jill has 2 markers
Total 8 + 2 + 2 = 12

Use counters to solve.
Question 9.
Make Sense Annika saves 13 dimes. She put some of the dimes in a box and the rest in a jar. Write an equation to show one way she could have sorted the dimes.

________ dimes in a jar
________ dimes in a box
Answer:
Addition can be thought of as combining two quantities and finding the total to add 6 dimes plus 7 dimes combined counters. There are 13 counters in total.
Explanation:
6 + 7 = 13
7 + 6 =13
6 dimes in a jar
7 dimes in a box

Question 10.
Assessment Practice Maria had 5 rings. She bought some more rings. Now she has 12 rings. Choose Yes or No to show if the equation can be used to find how many more rings Maria bought.

Answer:
Maria brought 7 rings.
Explanation:

Lesson 1.10 Problem Solving

Construct Arguments
Solve & Share
How can you use the make a 10 strategy to find 7 + 9?
Explain your thinking and work. Use pictures, numbers, or words.
I can … use pictures, numbers, and words to explain why my thinking and work are correct.

Thinking Habits
How can I use math to explain why my work is correct?
Is my explanation clear?

Visual Learning Bridge

Convince Me! Are both math arguments above clear and complete? Explain.
Answer: YES
Explanation:
Double fact method is used
6+7 is one more then 6+6
6+6+1=13
6+7=13

Guided Practice

Use the picture to help you solve the problem. Then use words and numbers to make a math argument.
Question 1.
Is the sum of 9 + 5 the same as the sum of 10 + 4?

Answer:
9+5=14
10+4=14
Explanation:

Independent Practice

Solve each problem. Use words, pictures, and numbers to make a math argument.
Question 2.
Lynn had 14 grapes. She ate 8 of them. She wants to eat 6 more grapes. Will Lynn have enough grapes? Explain.
Answer: yes

Question 3.
The Lions scored 11 runs in a baseball game. The Tigers scored 7 runs. Did the Tigers score 3 fewer runs than the Lions? Explain.
Answer: yes
Explanation:
Number of runs scored by Lions = 11
Number of runs scored by Tigers = 7
By using equation 7 + 3 = 11
And 11 – 7 = 3

Question 4.
Complete the explanation below for how to find 8 + 9. Use pictures, words, or numbers to complete the explanation.
I know that 8 + 8 = 16.
Answer: 17

Problem Solving

Performance Task Puppies Sold The Sunset Pet Store sells puppies. The table shows how many puppies were sold Monday through Thursday.

Is the total number of puppies sold on Tuesday and Wednesday less than the number of puppies sold on Monday?
Answer : NO
Explanation:
Number of puppies sold Tuesday and Wednesday = 10 + 7 = 17
Number of puppies sold on Monday = 16
So the difference = 17 – 16 = 1
Question 5.
Make Sense Will you use all the numbers in the table to solve the problem? Explain.
Answer: NO

Explanation:
Puppies sold on Monday is 16
puppies sold on Tuesday and Wednesday 10+7=17

17 is 1 more then 16, So Puppies the total number of puppies sold on Tuesday and Wednesday not less than the number of puppies sold on Monday.

Question 6.
Model Write an equation to find the total number of puppies sold on Tuesday and Wednesday. Solve.
Answer:
10 + 7 = 17

Question 7.
Explain Solve the problem. Use words, pictures, and . numbers to explain your work and thinking.
Answer:
First count the puppies sold on Monday it is 16
then count the puppies sold on Tuesday and then add it to the puppies sold to Wednesday( 10 + 7 = 17)
comparing the puppies sold on Monday to the Tuesday and Wednesday
the answer is
the total number of puppies sold on Tuesday and Wednesday is one mote than the number of puppies sold on Monday.

Topic 1 Fluency Practice Activity

Find a Match
Find 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 within 20.

Clues
A. Near doubles with sums near 8
B. Every difference is 6.
C. Ways to make 12
D. Exactly two differences equal 9.
E. Every sum is greater than 14.
F. Exactly three differences are equal.
G. Near doubles with sums near 6
H. Every difference equals 14 – 7.

Read the clues carefully, you will get all answers
Answer:

Explanation:

A. Near doubles with sums near 8
Explanation:
3+4 = 7 or 4+3 = 7
3+4 is one more then 3+3 and the sum is near to 8
4+5 = 9 or 5+4 = 9
4+5 is one more then 4+4 and the sum is near to 8
So the clue is matched

B. Every difference is 6.
Explanation:
11-5 =6, 10-4 =6, 12-6 =6 and 9-3 =6
the difference is 6
So, the clue is Matched.

C. Ways to make 12
Explanation:
5+7=12, 6+6=12, 8+4=12 and 9+3=12
So, the clue is matched.

D. Exactly two differences equal 9.
Explanation:
9-0=9 and 16-7=9
So, the clue is matched.

E. Every sum is greater than 14.
Explanation:
8+9=17, 7+8=15, 8+7=15 and 6+9=15
So, the clue is matched

F. Exactly three differences are equal.
Explanation:
10-5=5, 12-7=5 and 13-8=5
So, the clue is matched.

G. Near doubles with sums near 6
Explanation:
4+3=7 or 3+4=7, 3+2=5 or 2+3=5
So, the clue is matched.

H. Every difference equals 14 – 7.
Explanation:
9-2=7, 13-6=7, 8-1=7 and 15-8=7
So, the clue is matched.

 

Topic 1 Vocabulary Review

Understand Vocabulary
Word List

• addend
• bar diagram
• difference
• doubles
• equation
• near doubles
• sum

Question 1.
Circle a doubles fact.
7 + 7 = 14
6 + 7 = 13
7 + 0 = 7
Answer:

Question 2.
Circle a near doubles fact.
4 + 4 = 8
4 + 1 = 5
4 + 5 = 9
Answer:

Question 3.
Write a subtraction equation using numbers and symbols.
Answer:
9 – 4 = 5
7 – 4 = 3
Explanation:
By subtracting smaller number from bigger number we get the difference.

Question 4.
Find the sum of 8 + 6.
Answer: 14
8 + 6 = 14
Explanation:
By adding 6 to 8bwe get 14.

Question 5.
Find the difference of 12 – 5.
Answer: 7
12 – 5 = 7
Explanation:
By substracting smaller number 5 from bigger number 12 we get the difference 7.

Use Vocabulary in Writing

Question 6.
Describe how you can make a 10 to add 7 + 4. Use a term from the Word List.
Answer:
7 + 4 =11
Explanation:
7 and 4 are addends
11 is the sum
7 + 3 = 10
10 + 1 =11

Topic 1 Reteaching

Set A

You can count on to find a sum.

Count on to find the sum. Then change the order of the addends.
Question 1.
9 + 3 = ______
______ + ______ = ______
9 + 3 = ______ + ______
Answer:12
9 + 3 = 12
3 + 9 = 12
9 + 3 = 3 + 9
Explanation:
In additions if we change the addends from left to right or right to left the sum will be the same.

Question 2.
6 + 4 = ______
______ + ______ = ______
6 + 4 = ______ + ______
Answer: 10
6 + 4 = 10
4 + 6 = 10
6 + 4 = 4 + 6
Explanation:
In additions if we change the addends from left to right or right to left the sum will be the same.

Set B

You can use doubles to help you add a near double.

Complete the doubles facts. Use the doubles facts to solve the near doubles.
Question 3.
8 + 8 = ______
So, 7 + 8 = ______
Answer:
8 + 8 = 16
So, 7 + 8 =15
7 + 8 is one less then 8 + 8
Explanation:
According to double fact

Question 4.
5 + 5 = ______
So, 6 + 5 = ______
Answer:
5 + 5 = 10
6 + 5 = 11
6+5 is one more then 5+5
Explanation:

Set C

You can make a 10 to help you add 8 + 6.

Make a 10 to add.
Question 5.
8 + 4 = ?

Answer: 12
Explanation:

Set D

You can count on or count back to find 11 – 4.

Count on or count back to subtract. Show your work on the number line.
Question 6.
8 – 5 = ☐

Answer: 3

Explanation:
Start at 8 and count back 5 moves to get 3.
Question 7.
15 – 6 = ☐

Answer:9

Explanation:
Start at 15 and count back 6 moves to get 9.

Set E

You can think addition to help you subtract.
Find: 16 – 9 = ?
Think: 9 + 7
So, 16 – 9 = 7

Subtract. Write the addition fact that helped you.
Question 8.
13 – 7 = ______
7 + ______ = 13
Answer: 6
Explanation:
Double fact
13 – 7 = 6
7 + 6 = 13
6 + 7 = 13

Question 9.
17 – 9 = ______
9 + ______ = 17
Answer: 8
Explanation:
Double fact
17 – 9 = 8
9 + 8 = 17
8 + 9 = 17

Set F

You can make a 10 to subtract. Find 17 – 8.

Make a 10 to find 13 – 8. Draw counters to show your work.
Question 10.

13 – ______ = 10
10 – ______ = ______
13 – 8 = ______
Answer:
13 – 3 = 10
10 – 5 = 5
13 – 8 = 5
Explanation:

Set G

You can use addition or subtraction to solve word problems.
11 grapes are in a bowl. 9 grapes are in a cup. How many fewer grapes are in the cup?
11 – 9 = 2,
9 + 2 = 11
So, 2 fewer grapes are in the cup.

Write an equation to solve each problem.
Question 11.
13 shirts are in a closet. 8 shirts are in a box. How many more shirts are in the closet?

___5_____ more shirts
Answer:
5 shirts are there in the closet.
Explanation:

Question 12.
Drake has 10 more books than Yuri. Yuri has 10 books. How many books does Drake have?

________ books
Answer:
Drake have 20 books.
Explanation:

Set H

Thinking Habits
Construct Arguments
How can I use math to explain why my work is correct?
Did I use the correct numbers and symbols?

Solve. Use words, pictures, or numbers to construct arguments.

Question 13.
Tyler read 15 pages of a book. Ann read 9 pages of the same book. Did Tyler read 4 more pages than Ann? Explain.
Answer: No
Tyler read 6 more pages then Ann
Explanation:
15 – 9 = 6
Tyler read 15 pages
Ann read 9 pages
The difference is 6 pages

Topic 1 Assessment Practice

Question 1.
Tom draws 7 bugs. Gina draws 4 bugs. How many bugs did they draw in all?
Which shows how to count on to solve the problem?
A. 7…8, 9, 10, 11
B. 4…5, 6, 7
C. 7 + 4
D. 4 + 7
Answer: A
Explanation:
Keep 7 in mind 4 on fingers
after 7…count 8,9,10,11
Total 7+4=11 bugs

Question 2.
Lilly has 7 fish. Jack has 1 more fish than Lilly.
Which equations show how many fish in all? Choose all that apply.
☐ 7 + 7 + 7 = 21
☐ 7 + 1 = 8
☐ 7 + 7 + 1 = 15
☐ 7 + 7 = 14
☐ 7 + 8 = 15

Answer: 15
Explanation:
Double fact method
7+8 is one more then 7+7
7 + 8 = 15

Question 3.
Use the ten-frames. Show how to find the sum of 8 + 7 by making a 10. Then fill in the gray boxes.

Answer: 15
Explanation:

Question 4.
Which have a sum of 18? Choose all that apply.
☐ 10 + 8
☐ 8 + 10
☐ 9 + 8
☐ 8 + 8
☐ 9 + 9
Answer:
10 + 8 = 18
8 + 10 = 18
9 + 9 = 18

Question 5.
7 friends go to the movies. They have 4 tickets. How many more tickets do they need?
Draw lines to match each solution to how it was solved.

Answer:

Question 6.
Nita has 14 grapes. She eats 6 grapes.
Which addition fact can help you find how many grapes Nita has left?
A. 6 + 6 = 12
B. 6 + 7 = 13
C. 6 + 8 = 14
D. 8 + 8 = 16
Answer:
Option C
Explanation:
6 + 8 = 14
By adding 8 to 6 we get the sum 14.

Question 7.
Use the ten-frames. Show how to make a 10 to find 13 – 7. Then complete the equation.

13 – 7 = ______
Answer: 6

Question 8.
Bruce has some coins. He gives 4 coins to his brother. Now Bruce has 9 coins. How many coins did Bruce have at first? Write an equation to solve.

_________ coins
Answer:
Bruce have at first 13 coins.
Equation,

Question 9.
Maria has 4 pears. She buys some more pears. Now she has 12 pears. How many pears does Maria buy?
Part A
Draw a picture to model the problem.
Answer:
Maria brought 8 pears

Part B
Write an equation to solve the problem.

_________ pears
Answer:
8 pears

Question 10.
The team has 9 players. Then 2 players quit. After that, 5 players join the team. How many players does the team have now?
Use the numbers on the cards. Complete both equations to solve the problem.

Answer:
12 players

Question 11.
Choose Yes or No to show if 7 will make each equation true.

Answer:

Question 12.
Matt finds 9 sticks at the park. Mabel finds 7 sticks. How many sticks do they find in all? Write an equation to explain.

_________ sticks in all
Answer:
16 sticks

Question 13.
Choose Yes or No to show if 8 will make each equation true.

Answer: 8

Question 14.
Josh ate 6 fewer cherries than Gail. Gail ate 15 cherries. How many cherries did Josh eat?
Answer:
Josh ate 9 cherries
Explanation:
Number of cherries Gail ate = 15
Josh ate 6 fewer cherries than Gail.
Total number of cherries Josh ate = 15 – 6 = 9

Part A
Draw a picture to model the problem.
Answer:

Part B
Write an equation to solve the problem.

________ cherries
Answer:
9 cherries
Equation:

cherries

Question 15.
The table shows how many pictures 3 friends made.

Choose one of the friends. Write the name of the friend you choose.
How many pictures did that friend make?
Write an equation to solve the problem.

_________ pictures
Answer:
Brian, 13 pictures
Equation:

Topic 1 Performance Task

Farm Kittens
Many kittens are born each summer at the Sunshine Farm. The table shows the number of kittens born at the farm from June to August.

Question 1.
During which two months were a total of 20 kittens born?
Answer:
June + August
Explanation:

During the month of June and July
Answer: 21
Explanation:

Question 2.
Write an equation to find the total number of kittens born in July and August. Solve.
Answer: 15
Equation:

Question 3.
Joy said 5 more kittens were born in June than in August. Do you agree? Circle Yes or No. Show your work to explain.
Yes
No
Answer:
6 more kittens were born

Question 4.
Use the clues to complete the table below.

• No kittens were born in December, January, and February.
• In March, 6 kittens were born.
• Three kittens were born in April and in September.
• In May, 4 kittens were born.
• Two kittens were born in October and in November
  
  Answer:
  

Question 5.
Joy says that more kittens were born in the summer than in all other seasons combined. Is she correct? Explain.
Answer:
Yes, its correct
Explanation:
First add 13 + 4 = 17 (Kittens were born in the spring and Fall seasons)
Now subtract the above sum from the kittens born in summer
28 – 17 = 11

Question 6.
How many more kittens were born in the spring than in the fall? Show how to solve the problem with a subtraction equation.
Answer:
9 kittens
Equation:
13 – 4 = 9

Go through the enVision Math Common Core Grade 3 Answer Key Topic 7 Represent and Interpret Data regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 7 Represent and Interpret Data

Essential Question:
How can data be represented, analyzed, and interpreted?

enVision STEM Project: Seasons
Do Research Use the Internet or other sources to find information about patterns of temperature in the different seasons where you live. Include information about the average monthly temperatures and the record low and high temperatures.

Journal: Write a Report Include what you found. Also in your report:

• For one week, record the daily high and low temperatures in the area where you live. Make a graph displaying this information.
• Find the difference between the highest and lowest daily temperatures from your graph.

Review What You Know

Vocabulary
Choose the best term from the box. Write it on the blank.
• equal groups
• multiples
• multiplication
• number line

Question 1.
_______ have the same number of items.
Answer:
_Equal Groups__ have the same number of items.

Question 2.
A ________ can be used to help compare numbers.
Answer:
A _multiples_ can be used to help compare numbers.

Question 3.
________ is used to find a total when joining equal groups.
Answer:
_Multiplication__ is used to find a total when joining equal groups.

Multiplication

In 4 and 5, complete the equation.
Question 4.
5 × 3 = _______
Answer:
5 × 3 = 15.
Explanation:
When we multiply 5 with 3 times we get 15.
When you add 5 three times we get 15.
5 + 5 + 5 = 15.

Question 5.
3 × _______ = 21
Answer:
3 × 7 = 21
Explanation:
Given
one multiple and the product.
one multiple = 3
Product = 21
To find the other multiple we divide product by multiple
21 ÷ 3 = 7
3 × 7 = 21.

Question 6.
Make a bar diagram to represent 4 × 6.
Answer:

Multiplication on the Number Line

Question 7.
Ed bought 2 bags of grapefruit. There are 6 grapefruit in each bag. How many grapefruit did he buy? Draw jumps on the number line to find the answer.

Answer:

Question 8.
Show the multiplication fact 3 x 4 on the number line. Write the product.

Answer:

Finding Area
Question 9.
Find the area of the rectangle.

☐ = 1 square inch
Answer:
Number of squares = 4 × 5 = 20 squares.
Area = 20 square inches.

Pick a Project

PROJECT 7A
How is a book printed?
Project: Collect Data and Create Picture Graphs

PROJECT 7B
Would you like to live in a city?
Project: Make a Bar Graph About Cities and Towns

PROJECT 7C
What is your favorite animal?
Project: Develop a Picture Graph About Animals

3-ACT MATH PREVIEW

Math Modeling
Swings and Slides

+ `1

Solve & Share
Students in Jorge’s class took a survey of their favorite cereals and made this graph to show the results. Name at least three facts about the information in the graph.
I can … use picture graphs and bar graphs to answer questions about data sets.

Look Back! What do the two different symbols on the graph stand for?

Essential Question
How Can You Read Picture Graphs?

Visual Learning Bridge

Use the key
Look at the data for East Falls League.
There are 3 and 1 .
The 3 represent 3 × 2 = 6 teams.
The 1 represents 1 × 1 = 1 team.
6 + 1 = 7
There are 7 teams in the East Falls League.

How many more teams does the East Falls League have than the South Falls League?
Use the picture graph to write equations and compare the two rows.

The East Falls League has 3 more teams than the South Falls League.

Convince Me! Be Precise Tell something about each league you can find out from the picture graph.

Another Example!
A scaled bar graph uses bars to represent and compare information. This bar graph shows the number of goals scored by different players on a hockey team. The scale shows the units used.

On this bar graph, each horizontal grid line represents two units. Every other grid line is labeled: 0, 4, 8, and so on. For example, the line halfway between 4 and 8 represents 6 goals.

Guided Practice

Do You Understand?
In 1 and 2, use the bar graph above.
Question 1.
How many goals in all did Alex and Reggie score?
Answer:

Number of goals Alex scored = 18 goals.
Number of goals Reggie scored = 16 goals.
Total number of goals both scored in all = 18 + 16 = 34 goals.

Do You Know How?
Question 2.
Explain how to find how many fewer goals Cindi scored than Alex.
Answer:
Number of goals Alex scored = 18 goals
Number of goals Cindi scored = 14 goals
Number of fewer goals Cindi scored than Alex = 18 – 14 = 4 goals.
Cindi scored 4 fewer goals than Alex.

Independent Practice

In 3-5, use the picture graph.

Question 3.
Which area has lights on for exactly 50 hours each week?
Answer:
full bulb = 10 hours    Half bulb = 5 hours
Number of hours of light on each week in Exercise room = 7 full bulb and 1 half bulb
(7 × 10 hours) + (1 × 5 hours) = 70 + 5 = 75 hours.
Number of hours of light on each week in Locker room = 9 full bulbs
9 × 10 hours = 90 hours
Number of hours of light on each week in Swimming Pool = 5 full and 1 half
(5 × 10 hours) + (1 × 5 hours) = 50 + 5 = 55 hours.
Number of hours of light on each week in Tennis Court = 5 full
5 × 10 hours = 50 hours.
Tennis Court area has lights on for exactly 50 hours each week

Question 4.
What does the half bulb in the data for the exercise room represent?
Answer:
Half bulb in the data for exercise room represents 5 hours.

Question 5.
In one week, how many more hours are lights on in the exercise room than at the swimming pool?
Answer:
Number of hours of light on each week in Exercise room = 7 full bulb and 1 half bulb
(7 × 10 hours) + (1 × 5 hours) = 70 + 5 = 75 hours.
Number of hours of light on each week in Swimming Pool = 5 full and 1 half
(5 × 10 hours) + (1 × 5 hours) = 50 + 5 = 55 hours.
Number of more hours of light on in the exercise room than the swimming Pool = 75 hours – 55 hours = 20 hours.
20 hours of more light on in the exercise room than the swimming Pool

Problem Solving

In 6-8, use the picture graph.

Question 6.
Reasoning For which days can you use the expression 9 x 7 to find how many points were scored?
Answer:

From the data above
Number of points scored by a football team on October 3 = 9 × 7 = 63 points
Number of points scored by a football team on October 10 = 9 × 7 = 63 points
Number of points scored by a football team on October 17 = 6 × 7 = 42 points
Number of points scored by a football team on October 24  = 7 × 7 = 49 points
October 3  and October 10 days can  use the expression 9 x 7 to find number of  points  scored.
On both the days the team scored 9 × 7 points = 63 points.

Question 7.
On which days did the football team score fewer than 50 points?
Answer:
Number of points scored by a football team on October 17 = 6 × 7 = 42 points
Number of points scored by a football team on October 24  = 7 × 7 = 49 points
On October 17 and October 24 the football team scored points fewer than 50.

Question 8.
Higher Order Thinking How many more points were scored on October 10 and 24 combined than on October 3 and 17?
Answer:
Number of points scored by a football team on October 3 = 9 × 7 = 63 points
Number of points scored by a football team on October 17 = 6 × 7 = 42 points
Number of points scored on October 3 and October 17 combined = 63 + 42 = 105 points.
Number of points scored by a football team on October 10 = 9 × 7 = 63 points
Number of points scored by a football team on October 24  = 7 × 7 = 49 points
Number of points scored on October 10 and October 24 combined =63 + 49 = 112 points.
Number of more points scored on October 10 and 24 combined than on October 3 and 17 combined = 112 – 105 = 7 points.
7 more points scored on October 10 and 24 combined than on October 3 and 17 combined.

Assessment Practice

In 9 and 10, use the bar graph at the right.

Question 9.
How many more miles per hour is the top running speed of the cheetah than of the Cape hunting dog?
A. 25 miles per hour
B. 30 miles per hour
C. 35 miles per hour
D. 40 miles per hour
Answer:

Given the Top running speeds of animals
mile per hour – mph
The running speed of Cape Hunting Dog = 35 mph
The running speed of Cheetah = 70 mph
The running speed of Grizzly Bear = 30 mph
The running speed of Coyote = 40 mph
The top running speed of the Cheetah  more than of the Cape hunting dog = 70 mph – 35 mph
= 70 mph – 35 mph  = 35 mph.
Answer B. 35 mph more miles per hour is the top running speed of the Cheetah than of the Cape hunting dog.

Question 10.
How many fewer miles per hour is the top running speed of the grizzly bear than of the coyote and the Cape hunting dog combined?

A. 40 miles per hour
B. 45 miles per hour
C. 50 miles per hour
D. 55 miles per hour
Answer:
The top running speed of Grizzly Bear = 30 mph
The top running speed of Coyote = 40 mph
The top running speed of Cape Hunting Dog = 35 mph
The top running speed of Coyote and Cape Hunting Dog combined = 40 mph + 35 mph = 75 mph
The top running speed of the grizzly bear fewer than of the coyote and the Cape hunting dog combined = 75 mph – 30 mph
= 75 mph – 30 mph
= 45 mph
Answer B. 45 mph is fewer miles per hour is the top running speed of the grizzly bear than of the coyote and the Cape hunting dog combined.

Lesson 7.2 Make Picture Graphs

Solve & Share
Mary is helping her teacher count school playground equipment. She records the data in a frequency table. Use the data in the table to complete the picture graph. Write two statements about your completed graph.
I can … make a picture graph to record information and answer questions about a data set.

Look Back! How did you know the number of symbols to draw for jump ropes?

Essential Question
How Do You Make a Picture Graph?

Visual Learning Bridge
Sam recorded the number of each kind of bicycle a store sold during one month. He made a frequency table. Use the table to make a picture graph.
You can also collect data with a survey by asking people questions.

Write a title for the picture graph.
The title is “Kinds of Bicycles Sold.”
Choose a symbol for the key. Decide what each whole symbol and half-symbol will represent.

Set up the graph and list the kinds of bicycles. Decide how many symbols you need for each number of bicycles sold. Draw the symbols.

Convince Me! Model with Math Suppose 25 mountain bicycles were also sold. Draw symbols to show a row in the picture graph for mountain bicycles. Explain how you decided.

Guided Practice

Do You Understand?
In 1 and 2, use the picture graph on the previous page.
Question 1.
Explain the symbols that were used for the number of training bicycles sold.
Answer:

Question 2.
If the scale used in the key were = 2 bicycles, how many symbols would be used for the number of road bicycles sold? For the number of track bicycles sold?
Answer:

Do You Know How?
Question 3.
Use the table to complete the picture graph.

Answer:

Independent Practice

In 4-6, use the data in the chart.

Question 4.
Complete the picture graph.

Answer:

Question 5.
Which two teams scored more goals, the Cubs and the Lions or the Hawks and the Roadrunners?
Answer:

Question 6.
Explain how you decided the number of each symbol to draw to show the goals for the Roadrunners.
Answer:

Problem Solving

In 7-9, use the frequency table at the right.

Question 7.
Use Appropriate Tools Select and use appropriate tools to help make a picture graph to show the data in the table.
Answer:

Question 8.
What is the difference between the odd number and the sum of the even numbers in the table?
Answer:

Question 9.
Ask six students in your class which of the three vegetables is their favorite. Record the answers in your picture graph.
Answer:

Question 10.
Higher Order Thinking Suppose you are going to make a picture graph to show the data in the Simon’s Book Shop table. Choose a symbol to stand for 5 books sold. Draw the row for fiction books sold. Justify your drawing.

Answer:

Assessment Practice

Question 11.
The Garden Shop sold 25 plants in April, 30 plants in May, and 35 plants in June. Complete the picture graph for this data. Choose the symbols you will use.

Answer:

Lesson 7.3 Make Bar Graphs

Solve & Share
Use the data in the table below to complete the bar graph. What conclusions can you make by analyzing the bar graph?
I can … make a bar graph to record information and answer questions about a data set.

Look Back! How can tools such as a ruler help you create a bar graph?

Essential Question
How Do You Make a Bar Graph?

Visual Learning Bridge
Greg made a table to show the amount of money he saved each month from tutoring. Use the data in the table to make a bar graph.

Write a title. Use the same title as in the table.
The title of this bar graph is
Amount Greg Saved Each Month.
Choose the scale. Decide how many units each grid line will represent.

Set up the graph with the scale, each month listed in the table, and labels. Draw a bar for each month.

Convince Me! Be Precise Write new amounts for how much Greg saved in 4 later months. Consider the scale.

In May, Greg saved __________.
In June, Greg saved __________.
In July, Greg saved __________.
In August, Greg saved __________.
Draw bars on the graph to show your new data.

Guided Practice

Do You Understand?
In 1-3, use the bar graph on the previous page.
Question 1.
Explain why the bar for January ends between 20 and 30.
Answer:
The bar for January ends between 20 and 30 because in January Greg  saved $25.
Since the grid line represents $10.
Half of $10 is = $10 ÷ 2 = $5
So, $25 is represented between 20 and 30.

Question 2.
Suppose Greg saved $35 in May. Between which grid lines would the bar for May end?
Answer:
$35 would be between the grid lines of 30 and 40.

Question 3.
How can you tell how much more Greg saved in February than in April?
Answer:
Total Amount Greg saved in February = $50
Total Amount Greg saved in April = $40
Difference of amount Greg saved in February and April = $50 – $40 = $10.
Greg saved $10 more in February than in April.

Do You Know How?
Question 4.
Use the table to complete the bar graph.

Answer:

Explanation:
The bar graph has number of people who signed up for classes on one side and the classes on the other side.
Here Chess class is represented with Red color
Guitar class is represented with Orange color
Painting  class is represented with Yellow color
Writing class is represented with Green color.

Independent Practice

In 5, use the table at the right.
Question 5.
Complete the bar graph to show the data.

Answer:

Deal Mart represented with Blue color
Jane’s represented with Purple color
Parker’s represented with Indigo blue color
Trends represented with Brown color.
Number of votes are represented on one side of the bar with  5 multiples.
Other side represents the stores.

Problem Solving
In 6-8, use the table at the right.

Question 6.
Make a bar graph to show the data.
Answer:

Question 7.
Construct Arguments Which two kinds of movies received about the same number of votes? Explain how to use your bar graph to find the answer.
Answer:
Cartoon and Science Friction movies received about same number of votes.
The bar graph represents the votes the kind of movies received.
The vertical side of the bar graph represents the number of votes received.
Votes are represented with 2 multiples.
The horizontal line represents the kind of movies.
Looking at the graph
Number of votes for Cartoon kind of movies = 7
Number of votes for Science Friction kind of movies = 6
They both received about the same number of votes.

Question 8.
Each movie ticket costs $8. Jo buys tickets for the number of people who voted for science fiction. How much change does she get from $50?
Answer:
Cost of each movie ticket = $8
Number of votes for science fiction movie = 6
Total amount Jo have = $50
Cost for 6 science fiction movie tickets = 6 × $8 = $48
The change Jo get for her $50 = $40 – $48 = $2
Jo get $2 change from $50.

Question 9.
Higher Order Thinking Suppose you are going to make a bar graph to show the data in the table at the right. What scale would you choose? Explain.

Answer:

Assessment Practice

Question 10.
Tanji collected data on the colors of his friends’ shoes. Eight friends have black shoes. Five friends have blue shoes. Seven friends have white shoes. Use Tanji’s data to complete the bar graph.

Answer:

Lesson 7.4 Solve Word Problems Using Information in Graphs

Solve & Share
The students in Ms. Seymour’s class voted for their favorite kind of sandwich. How many more students voted for peanut butter than cheese? How many fewer students voted for tuna than peanut butter?
I can … use graphs and other tools to solve word problems.

Look Back! What is the scale for this graph? How do you know the number of votes a bar represents when it is between two lines on this graph?

Essential Question
How Can You Solve Problems Question Using Graphs?

Visual Learning Bridge
Angela wants Karli and Monique to have a total of 60 paper cranes. The bar graph shows how many paper cranes her friends already have. How many more paper cranes does Angela need to make for Karli and Monique to have 60 paper cranes in all?

This problem has a hidden question!

Solve the hidden question.
How many paper cranes do Karli and Monique already have?
Use the scale to find how many paper cranes Karli and Monique each have. Then add.

Karli has 30 paper cranes. Monique has 10 paper cranes.
30 + 10 = 40
Together they have 40 paper cranes.

Solve the main question.
How many paper cranes does Angela need to make?
Subtract the number of cranes the friends already have from the total.
60 – 40 = 20
Angela needs to make 20 paper cranes.

Convince Me! Critique Reasoning Angela says, “I want llana and Elspeth to also have 60 cranes in all. I can subtract two times to find how many more cranes I need to make for them.” Is Angela correct? Explain.

Guided Practice

Do You Understand?
Question 1.
Look at the graph on the previous page. Explain whether you would add, subtract, multiply, or divide to find how many more paper cranes Karli already has than Monique.
Answer:

Number of Paper Cranes Karli has = 30
Number of Paper Cranes Monique has = 10
To find how many more paper cranes Karli has than Monique
we subtract Monique paper cranes from Karli’s paper cranes
30 – 10 = 20

Karli has 20 more paper cranes than Monique.

Question 2.
How does a bar graph help you compare data?
Answer:
A bar diagram makes it easy to compare sets of data between different groups at a glance. The graph represents categories on one axis and a discrete value in the other. The goal is to show the relationship between the two axes.

Do You Know How?
In 3, use the bar graph.

Question 3.
How many more miles did Hal and Victor ride together than Rosita?
Answer:
Number of miles Hal rode = 20 miles
Number of miles Victor rode = 20 miles
Total number of miles Hal and Victor rode together = 20 + 20 = 40 miles.
Number of miles Rosita rode = 35 miles
35 miles because it is represented between 30 and 40 miles bar line in the bar graph.
Number of more miles Hal and victor ride together than Rosita = 40 miles – 35 miles = 5 miles.

Independent Practice

In 4-6, use the picture graph at the right.

Question 4.
How many more red T-shirts were sold at Ultimate T than at Jazzy’s?
Answer:
You see
Full T-shirt = 10 T-shirts
Half T-shirt = 5 T-shirts
Number of red T-shirts sold at Ultimate T = 2 full T-shirts and 1 half T-shirt = 10+10+5 = 25 T-shirts
Number of red T-shirts sold at Jazzy’s =2 full T-shirts = 10 +10 =20 T-shirts
Number of more red T-shirts sold at Ultimate T than at Jazzy’s = 25 – 20 = 5 T-shirts.

Question 5.
How many fewer green T-shirts were sold at Jazzy’s than at Ultimate T?
Answer:
Full T-shirt = 10 T-shirts
Half T-shirt = 5 T-shirts
Number of green T-shirts sold at Ultimate T = 3 full T-shirts = 10 + 10 + 10 = 30 T-shirts
Number of green T-shirts sold at Jazzy’s = 1 half T-shirt = 5 T-shirt
Number of fewer green T-shirts were sold at Jazzy’s than at Ultimate T = 30 – 5 = 25 T-shirts.
25 fewer green T-shirts were sold at Jazzy’s than at Ultimate T.

Question 6.
How many more blue and red T-shirts combined were sold at Jazzy’s than green T-shirts were sold at Ultimate T?
Answer:
Number of Blue T-shirts sold at Jazzy’s = 2 full T-shirts and 1 half T-shirt = 10+10+5 = 25 T-shirts
Number of red T-shirts sold at Jazzy’s =2 full T-shirts = 10 +10 = 20 T-shirts
Total number of Blue and red T-shirts combined were sold at Jazzy’s = 25 + 20 = 45 T-shirts
Number of green T-shirts were sold at Ultimate T = 3 full T-shirts = 10 + 10 + 10 = 30 T-shirts
Number of more blue and red T-shirts combined were sold at Jazzy’s than green T-shirts were sold at Ultimate T =
= 45 – 30 = 15 T-shirts.

Problem Solving

In 7-9, use the bar graph at the right.

Question 7.
Number Sense How many people voted for their favorite type of exercise? How can you find the answer?
Answer:
Here three types of exercises are given
Jogging
Swimming
Gymnastics
Number of people voted for Jogging = 6
Number of people voted for Swimming = 14
Number of people voted for Gymnastics = 12
Total number of people who voted for their favorite exercises = 6 + 14 + 12 = 32.
32 people voted for their favorite exercises.
By adding the votes of all the exercises gives the answer to find people voted for their favorite exercise.

Question 8.
Construct Arguments How many more people voted for gymnastics than for jogging? How do you know?
Answer:
Number of people voted for Jogging = 6
Number of people voted for Gymnastics = 12
By subtracting votes of jogging from gymnastics
we find
Number of more people voted for gymnastics than for jogging = 12 – 6 = 6 votes.
6 more people voted for gymnastics than for jogging.

Question 9.
How many fewer people voted for swimming than for gymnastics and jogging combined?
Answer:
Number of people voted for Jogging = 6
Number of people voted for Gymnastics = 12
Total number of votes for gymnastics and jogging combined = 12 + 6 = 18 votes.
Number of people voted for Swimming = 14
Subtracting votes for Swimming from gymnastics and jogging combined
Number of  fewer people voted for swimming than for gymnastics and jogging combined = 18 – 14 = 4
4 fewer people voted for swimming than for gymnastics and jogging combined.

Question 10.
Leslie delivers papers on weekdays and Saturdays. She delivers 6 papers each weekday and 16 papers on Saturday. How many papers does Leslie deliver during the entire week?
Answer:
A week has 7 days
5 weekdays
2 = weekends -Saturday and Sunday
Number of papers Leslie delivers on weekday = 6 each
Total number of papers she delivers on all weekdays = 6 × 5 = 30 papers
Number of papers Leslie delivers on Saturday = 16 papers
Total number of papers Leslie deliver during the entire week = 30 + 16 = 46 papers.
Leslie delivers 46 papers during the entire week.

Question 11.
Higher Order Thinking What kinds of comparisons can you make when you look at a bar graph or a picture graph?
Answer:
Fewer , more , difference, adding

Assessment Practice

Question 12.
Daryl made a bar graph to record the number of books read by each member of a reading club. How many fewer books did Alice read than Sandra and Daryl combined?

A 6 books
B. 8 books
C. 10 books
D. 12 books
Answer:
Number of books Alice read = 16 books
Number of books Sandra read = 8 books
Number of books Daryl read = 18 books
Total number of books Sandra and Daryl read combined = 18 + 8 = 26 books
Number of fewer books  Alice read than Sandra and Daryl combined = 26 – 16 = 10 books.
Alice read 10 fewer books than Sandra and Daryl combined.

Lesson 7.5 Problem Solving

Precision
Solve & Share
Action books and mystery books cost $5 each. Biography books cost $10 each. A librarian has $100 to spend on new books. She collected some information about the kinds of books that students checked out to read last month.
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 did you use words and symbols to explain your answer?

Essential Question
How Can You Be Precise When out Solving Math Problems?

Visual Learning Bridge
Bella has a bakery. She will use the bakery items at the right to make a gift basket worth $40. Bella wants the basket to have more than one of each bakery item. Show one way to make a gift basket.

What do I need to do to make a gift basket?
I need to be precise. I will decide how many of each item to put in a basket so the total is exactly $40.

Here’s my thinking…

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.

I will start with $40.
I know how many of each item is available.
3 wheat loaves × $4 = $12
$40 – $12 = $28

9 cinnamon buns × $2 = $18
$28 – $18 = $10

10 muffins × $1 = $10
$10 – $10 = $0
All calculations are correct. My gift basket has 3 wheat loaves, 9 cinnamon buns, and 10 muffins. The total is exactly $40.

Convince Me! Be Precise Is there another way to make a gift basket that totals exactly $40? Explain.

Guided Practice

Be Precise Use the graph on the previous page. Suppose Bella wanted to make a gift basket worth $25 instead. The gift basket must also have more wheat loaves than muffins. Show one way Bella can make the gift basket.

Question 1.
What given information will you use to solve?
Answer:

Given
Bella wants a gift basket worth of $25
She wants more wheat loaves than muffins.
Using the list of items in the Bakery and the cost of the items. I will calculate the cost of the items and compare with the basket worth.

Question 2.
Show and explain one way Bella can make the gift basket.
Answer:
Total worth of the gift basket = $25
Cost of Muffins = $1 each
Cost of Cinnamon Buns = $2 each
Cost of  Wheat loaves = $4 each
Cost of  4 Wheat loaves = 4 × $ 4 = $16
Cost of the basket – cost of 4 Wheat loaves
= $25 – $16
= $9
Remaining Amount = $9
Cost of 3 Cinnamon Buns = 3 × $2 = $6
Remaining Amount – Cost of 3 Cinnamon Buns
= $9 – $6
= $3
Remaining Amount
Cost of 3 Muffins =3 × $1 = $3
Remaining Amount – Cost of 3 Muffins
= $3 – $3
= $0
The gift basket worth of $25 have 4 Wheat loaves , 3 Cinnamon Buns  , 3 Muffins.

Independent Practice

Be Precise Derek is making a tile pattern that will be 30 inches long. The graph shows how many of each length of tile Derek has. He wants to use more than one of each length of tile in his pattern. Show one way to make the pattern.

Question 3.
What given information will you use to solve?
Answer:
The length of the tile pattern = 30 inches
Type of tiles Derek have
3 in tile
2 in tile
4 in tile.
Number of tiles he have.
Number of 2 in tile he have = 5 tiles
Number of 3 in tile he have = 7 tiles
Number of 4 in tile he have = 6 tiles
Use these information to solve the problem.

Question 4.
Show and explain one way Derek can make the pattern.
Answer:
The length of the tile pattern = 30 inches
Type of tiles Derek have
3 in tile
2 in tile
4 in tile.
Number of tiles he have.
Number of 2 in tile he have = 5 tiles
Number of 3 in tile he have = 7 tiles
Number of 4 in tile he have = 6 tiles
To make the tile pattern he wants to use more than one of each length of tile in his pattern.
if we use 1 of each tile
length of all types of tiles  = 2 inch + 3 inch + 4 inch = 9 inch
Total length of tile pattern = 30 inches
= 30 inches – 9 inches = 21 inches.
Now,
To make more 21 inches
2 time length of all tiles = 2 × 9 = 18 inches
= 18 inches +3 inch
= 21 inches
Total types of tiles needed for the 30 inch tile pattern = Three 4 in tiles + Four 3 inch tile + Three 2 inch tiles
=Three 4 in tiles + Four 3 inch tile + Three 2 inch tiles
=( 3 × 4) + (4 ×3) + (2 × 3)
= 12 + 12 + 6
= 30 inches.

Problem Solving

Performance Task
Picture Planning
Marta has $50 to spend on sketches. She wants to display them in an array of 3 rows, with 4 sketches in each row. Marta wants to include each type of sketch at least two times in her array.

Question 5.
Reasoning How many sketches does Marta want?
Answer:
Marta wants a array of 3 rows.
Each row has 4 sketches in each row.
Number of sketches in the array = 3 × 4 = 12 sketches.
Marta wants 12 sketches.

Question 6.
Make Sense and Persevere What is a good plan for solving the problem?
Answer:
As Marta need 12 sketches. and she has  $50 to spend on sketches.
There are three types of sketches and their cost
Portrait = $10 each
Animals = $4 each
Landscape =$2 each
She want each type at least two times in the array.
Calculating the cost of each sketches and plan the sketches in the array.

Question 7.
Be Precise Show one way Marta can buy sketches to make the array. Use math words and symbols to explain.
Be precise when analyzing the symbols, words, and numbers displayed in a picture graph.
Answer:
As Marta need 12 sketches.
At least 2 sketches of each type in the array
2 Portrait sketches = 2 × $10 = $20
2 Animal sketches = 2 × $4 = $8
2 Landscape sketches = 2 × $2 = $4
Total number of sketches = 2 + 2 + 2 = 6
6 more sketches remaining.
Adding all the costs
= $20  + $8 + $4 = $32
Total amount Marta has = $50
$50 – $32 = $18 remaining.
$18 remaining for 6 more sketches
we can add
=3 Animal sketch + 3 Landscape sketches
$18 = (3 × $4) + (3 × $2)
$18 = $12 + $6
$18 = $18.
Total number of sketches in the Marta’s array = 2 Portrait + 5 Animal + 5 Landscape
=(2 × $10) + (5 × $4) + (5 × $2 )
=$20 + $20 + $10
=$50.

Question 8.
Generalize Suppose Marta wants to make an array of 4 rows, with 3 sketches in each row. Would your answer still work? Explain.
Answer:
If Marta wants to make a array of 4 rows and 3 sketches in each row.
Total sketches = 4 × 3 = 12 sketches.
In both the cases the total sketches are same. So, the answer will work in this case as well.
Just the order of sketches changes.

Topic 7 Fluency Practice Activity

Point & 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. Both partners multiply those numbers.

If the answer is on your color, you get a tally mark. Partners then write the remaining facts in the fact family. Work until one partner has seven tally marks.
I can… multiply and divide within 100.

Topic 7 Vocabulary Review

Understand Vocabulary

Graph A

Word List
• data
• frequency table
• graph
• key
• scale
• scaled bar graph
• scaled picture graph
• survey

Complete each sentence with scaled picture graph, scaled bar graph, key, or scale.
Question 1.
The _________ in Graph A shows that each umbrella represents 2 days.
Answer:
The _Key__ in Graph A shows that each umbrella represents 2 days.

Question 2.
Graph A is a _________
Answer:
Graph A is a __Scaled picture graph___.

Question 3.
The ________ in Graph B increases by 5.
Answer:
The _Scale__ in Graph B increases by 5

Question 4.
Graph B is a _________.
Answer:
Graph B is a __scaled bar graph___.

Write T for True or F for False.

Question 5.
________ A survey is the only way to collect data.
Answer:
True A survey is the only way to collect data.

Question 6.
________ A scaled bar graph has a key.
Answer:
False A scaled bar graph has a key.

Question 7.
________ Data from a frequency table can be used to make a scaled bar graph.
Answer:
True Data from a frequency table can be used to make a scaled bar graph.

Use Vocabulary in Writing
Question 8.
Suppose you found out the number and type of pets your classmates have. Explain how you can display that information. Use at least 3 terms from the Word List in your answer.
Answer:
Given data
Number of pets
Type of pets.
I will be using a scaled bar graph with number of pets on one axis and type of pets on the other axis. The scale range and key depends on the data collected by the survey.

Topic 7 Reteaching

Set A pages 253-256

Picture graphs use pictures or parts of pictures to represent data.

Bar graphs use bars to represent data. You can use a scale to find how much a bar represents.

Each line in this bar graph represents 2 votes.

Remember to use a key or scale. The number of pictures in picture graphs and the lengths of bars in bar graphs help to compare data.

In 1-3, use the picture graph on the left.
Question 1.
How many more baseball caps are on sale than racing caps?
Answer:
1 full cap represents 10 caps
1 half cap represents 5 caps
Number of baseball caps on sale = 3 full + 1 half = 10+10+10+5= 35 caps
Number of Racing caps on sale = 2 full caps = 10 + 10 = 20 caps.
Subtracting racing caps from baseball caps
Number of more baseball caps are on sale than racing caps =  35 caps – 20 caps = 15 caps
There are 15 more baseball caps are on sale than racing caps.

Question 2.
How many more baseball caps are on sale than basketball and racing caps combined?
Answer:
Number of basketball caps on sale = 1 full cap = 10 caps
Number of Racing caps on sale = 2 full caps = 10 + 10 = 20 caps.
Total number of basketball and racing caps combined = 10 + 20 = 30 caps.
Number of baseball caps on sale = 3 full + 1 half = 10+10+10+5= 35 caps
Number of more baseball caps are on sale than basketball and racing caps combined = 35 caps – 30 caps = 5 caps.
There are 5 more baseball caps are on sale than basketball and racing caps combined.

Question 3.
How many fewer basketball caps are on sale than baseball caps?
Answer:
Number of basketball caps on sale = 1 full cap = 10 caps
Number of baseball caps on sale = 3 full + 1 half = 10+10+10+5= 35 caps.
Number of fewer basketball caps are on sale than baseball caps = 35 – 10 = 25 caps
There are 25 fewer basketball caps are on sale than baseball caps.

In 4-7, use the bar graph on the left.
Question 4.
Which color got the most votes? How many votes did that color get?
Answer:

Red color got the most votes.
11 people voted for red color.

Question 5.
How many fewer votes were for yellow than for green?
Answer:
Number of votes for yellow = 3 votes
Number of votes for green = 10 votes
Number of fewer votes for yellow than for green = 10 – 3 = 7 votes.
There are 7 fewer votes for yellow than for green.

Question 6.
How many more votes were for red than for blue?
Answer:
Number of votes for red  = 11 votes
Number of votes for  blue = 8 votes
Number of more votes to red than for blue = 11 – 8 = 3 votes.
There are 3 more votes for red than for blue.

Question 7.
What is the difference between the votes for red and the votes for blue and yellow combined?
Answer:
Number of votes for  blue = 8 votes
Number of votes for yellow = 3  votes
Total number of votes for blue and yellow combined = 8  + 3 = 11 votes
Number of votes for red  = 11 votes
The difference between the votes for red and the votes for blue and yellow combined = 11 – 11 = 0 votes.
There is 0 difference between the votes for red and the votes for blue and yellow combined.

Set B pages 257-264

This frequency table shows data about the number of coins Mark has.

You can use the data to make a picture graph. Picture graphs include a title, symbol, and a key to show the scale.

In this picture graph, each symbol equals 2 coins. You also can use the data to make a bar graph.
1. Label the bottom and side of the graph.
2. Choose a scale.
3. Draw a bar for each type of coin.
4. Include a title.

Remember that a frequency table includes tally marks or numbers. A picture graph uses pictures to show data.

In 1 and 2, use the frequency table below.
Dan’s class voted for their favorite pet. The results are shown in this frequency table.

Question 1.
Complete the frequency table.
Answer:

Question 2.
Use the data in the frequency table to make a picture graph.
Answer:

Question 3.
Use the data in the table to make a bar graph.
Answer:

Set C pages 265-268

You can use data from bar graphs or picture graphs to draw conclusions.
In a picture graph, Erica recorded the number of magazines she read. How many more magazines did she read in April and May combined than in June?

You can solve 2-step data problems.
There are 6 symbols for April and May.
There are 3 symbols for June.
6 – 3 = 3. There are 3 more symbols for April and May.
Each symbol represents 2 magazines. 3 × 2= 6. Erica read 6 more magazines in April and May than in June.

Remember you can use tables and graphs to make comparisons. Sometimes you need to find and answer hidden questions.

In 1-6, use the picture graph below.

Question 1.
How many more maple trees than elm trees are there?
Answer:
1 Tree symbol = 6 trees
Number of Maple trees = 6 Tree = 6 × 6 = 36 trees.
Number of elm trees = 2 Tree = 2 × 6 = 12 trees.
Number of more maple trees than elm trees = 36 – 12 = 24 trees.
There are 24 more maple trees than elm trees.

Question 2.
How many fewer beech trees are there than maple trees?
Answer:
Number of beech trees = 5 Tree = 5 × 6 trees = 30 trees
Number of Maple trees = 6 Tree = 6 × 6 trees = 36 trees.
Number of fewer beech trees are there than maple trees = 36 – 30 = 6 trees.
There are 6 fewer beech trees are there than maple trees.

Question 3.
How many trees are NOT maple trees?
Answer:
Number of elm trees = 2 Tree = 2 × 6 = 12 trees.
Number of beech trees = 5 Tree = 5 × 6 trees = 30 trees.
Number of Oak trees = 3 Tree = 3 × 6 trees = 18 trees.
Total number of trees which are NOT Maple trees = elm trees + beech trees + Oak trees
= 12 + 30 + 18
= 60 trees.
There are 60 trees which are NOT Maple   trees.

Question 4.
How many more maple and beech trees combined are there than oak trees?
Answer:
Number of beech trees = 5 Tree = 5 × 6 trees = 30 trees.
Number of Maple trees = 6 Tree = 6 × 6 trees = 36 trees.
Total number of maple and beech trees combined = 36 + 30 = 66 trees.
Number of Oak trees = 3 Tree = 3 × 6 trees = 18 trees.
Number of more maple and beech trees combined are there than oak trees = 66 – 18 = 48 trees.

Question 5.
How many fewer oak trees are there than beech and elm trees combined?
Answer:
Number of beech trees = 5 Tree = 5 × 6 trees = 30 trees.
Number of elm trees = 2 Tree = 2 × 6 = 12 trees.
Total number of beech and elm trees combined = 30 + 12 = 42 trees.
Number of Oak trees = 3 Tree = 3 × 6 trees = 18 trees.
Number of fewer oak trees are there than beech and elm trees combined = 42 – 18 = 24 trees.
There are 24 fewer oak trees are there than beech and elm trees combined.

Question 6.
If the city wants to have 24 elm trees, how many more elm trees does it need to plant? Explain how to solve.
Answer:
Number of elm trees = 2 Tree = 2 × 6 = 12 trees.
Total number of elm trees city wants = 24 trees.
Number of more elm trees  need to plant = 24 – 12 = 12 trees.
The city needs 12 more elm trees to plant. and
2 TREE to be added to the picture graph to make it 24 trees.

Set D pages 269-272

Think about these questions to help you attend to precision.
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 use words, numbers, and symbols to show your reasoning.

In 1 and 2, use the bar graph at the left to solve.
Jackie has $50 to spend on books. She has made a bar graph to show the number of each type of book the store has in stock. Jackie wants to buy at least 2 of each type of book. Show one way Jackie can spend $50 on books.
Action books cost $5.
Biographies cost $10.
Mysteries cost $5.
Question 1.
What given information will you use to solve the problem?
Answer:
From the bar graph we know
Types of books and the number of books.
Cost of each type of books in the store.
Total amount Jackie can spend on books = $50.

Question 2.
Show one way Jackie can spend $50 on books. Use math words and symbols to explain your thinking.
Answer:

• 
  Action books cost $5.
  Biographies cost $10.
  Mysteries cost $5.

Number of Action books = 4
Number of Mystery books = 5
Number of Biography books = 7
Total money Jackie have = $50
Given that Jackie wants to buy at least 2 of each type of book.
cost of 2 action books = 2 × $5 = $10
$50 – $10 = $40
Remaining money =$40
Cost of 2 Mystery books = 2 × $5 = $10
$40 – $10 = $30
Cost of 2 Biography books = 2 × $10 = $20
$30 – $20 = $10
Remaining money after buys 2 books of each type = $10.
Here are two probabilities
Jackie may buy 2 Action books or 2 Mystery books or 1 Action and 1 mystery book or 1 Biography book.
Here are the ways Jackie spend her $50 buying books.
4 Action books + 2 Mystery books + 2 Biography books = 20 + 10 + 20 = 50
2 Action books + 4 Mystery books + 2 Biography books = 10 + 20 + 20 = 50
3 Action books + 3 Mystery books + 2 Biography books = 15 + 15 + 20 = 50
2 Action books + 2 Mystery books + 3 Biography books = 10 + 10 + 30 = 50

Topic 7 Assessment Practice

Question 1.
Use the data from the frequency table to make a picture graph.

Answer:

B. Draw a picture graph.
Answer:

Question 2.
Use the data from the picture graph you made in Question 1. How many students did NOT choose turkey as their favorite sandwich?
A. 10
B. 13
C. 12
D. 14
Answer:
There are 4 types of sandwiches.
Turkey, Ham, Tuna, Egg
Number of Ham sandwiches = 6
Number of Tuna sandwiches = 2
Number of Egg sandwiches = 4
Total number of sandwiches which are not Turkey = 6 + 2 + 4 = 12 sandwiches.

Question 3.
Jamie’s class made a picture graph to show how many hours they volunteered each week. In which week or weeks did the class volunteer 9 hours?

A. Week 1
B. Week 2
C. Week 3
D. Weeks 1 and 3
Answer:
1 full clock = 2 hours
1 half clock = 1 hour
Number of hours volunteered in week 1 = 5 full clock = 5 × 2 hours = 10 hours.
Number of hours volunteered in week 2 = 4 full and 1 half clock = (4 × 2 hours) + (1 × 1 hour) = 8 + 1 = 9 hours.
Number of hours volunteered in week 3 = 3 full and 1 half clock = (3 × 2 hours ) + (1 × 1 hour ) = 6 + 1 = 7 hours.
In Week 2  Jamie’s class volunteered 9 hours.

Question 4.
Look at the picture graph above. How many total hours did the class volunteer?
Answer:
Number of hours volunteered in week 1 = 5 full clock = 5 × 2 hours = 10 hours.
Number of hours volunteered in week 2 = 4 full and 1 half clock = (4 × 2 hours) + (1 × 1 hour) = 8 + 1 = 9 hours.
Number of hours volunteered in week 3 = 3 full and 1 half clock = (3 × 2 hours ) + (1 × 1 hour ) = 6 + 1 = 7 hours.
Total number of hours class volunteer = 10 hours + 9 hours + 7 hours = 26 hours.

Question 5.
How many more hours did the class volunteer in Weeks 2 and 3 combined than in Week 1?
Answer:
Number of hours volunteered in week 1 = 5 full clock = 5 × 2 hours = 10 hours.
Number of hours volunteered in week 2 = 4 full and 1 half clock = (4 × 2 hours) + (1 × 1 hour) = 8 + 1 = 9 hours.
Number of hours volunteered in week 3 = 3 full and 1 half clock = (3 × 2 hours ) + (1 × 1 hour ) = 6 + 1 = 7 hours.
Number of hours class volunteer in Weeks 2 and 3 combined = 9 hours + 7 hours = 16 hours.
Number of more hours the class volunteer in Weeks 2 and 3 combined than in Week 1 = 16 hours – 10 hours = 6 hours 
The class volunteer in Weeks 2 and 3 combined than in Week 1

Question 6.
Mr. Thomas’s class made a bar graph of the number of brothers and sisters each student has. How many students in the class have 1 brother or sister?

A. 1
B. 2
C. 5
D. 7
Answer:
5 students have 1 brother or sister. The bar is represented with purple color.

Question 7.
Compare students with 3 brothers and sisters and students with O brothers and sisters. How many more students have 3 brothers and sisters?
________ more student(s)
Answer:
Number of students having 3 brothers and sisters = 7 students.
Number of students having 0 brothers and sisters = 2 students.
Number of more students have 3 brothers and sisters = 7 – 2 = 5 students.
5 more students have 3 brothers and sisters.

Question 8.
How many students in the class have 2 or more brothers and sisters?
Answer:
Number of students having 2 brother and sisters = 7 students
Number of students having 3 brothers and sisters = 7 students
Number of students having 4 brothers and sisters = 2 students
Number of students having 5 brothers and sisters = 1 students
Total number of students having 2 or more brothers and sisters = 7 + 7 + 2 + 1 = 17 students.

Question 9.
Beth is making a bar graph to compare how many marbles of each color she has. She has 25 blue marbles, 35 red marbles, 5 green marbles, and 15 yellow marbles. Which scale makes the most sense for Beth to use with her graph?
A. Each grid line equals 1 marble.
B. Each grid line equals 2 marbles.
C. Each grid line equals 5 marbles.
D. Each grid line equals 20 marbles.
Answer:
25 , 35, 5, 15 all these are in the multiples of 5. So, the grid line equals 5 marbles.
25 = 5 × 5 grid line
35 = 5 × 7 grid line
5 =  5 × 1 grid line
15 = 5 × 3 grid line

Question 10.
Use the information in Question 9 to make a bar graph of Beth’s marbles.

Answer:

Question 11.
Select all the statements that are true. Use the information from Question 9.
☐ Beth has more green and yellow marbles combined than blue ones.
☐ Beth has as many blue and yellow marbles combined as red ones.
☐ Beth has more red marbles than blue and green ones combined.
☐ Beth has fewer yellow marbles than blue and green ones combined.
☐ Beth has fewer red marbles than green and yellow ones combined.
Answer:
25 blue marbles, 35 red marbles, 5 green marbles, and 15 yellow marbles
1. Beth has more green and yellow marbles combined than blue ones.
Green and yellow = 5 + 15 = 20 , blue = 25
25 – 20 = 5 marbles.
5 fewer marbles.
False Beth has  more green and yellow marbles combined than blue ones.
2. Beth has as many blue and yellow marbles combined as red ones
blue and yellow = 25 + 15 = 40 marbles , red = 35 marbles.
False Beth has as many blue and yellow marbles combined as red ones.
3. Beth has more red marbles than blue and green ones combined.
Red marbles = 35 marbles , blue and green combined = 25 + 5 = 30 marbles.
35 – 30 = 5 marbles
True Beth has 5 more red marbles than blue and green ones combined.
4 . Beth has fewer yellow marbles than blue and green ones combined.
blue and green combined = 25 + 5 = 30 marbles , yellow marbles = 15 marbles
30 – 15 = 15 marbles.
True Beth has 15 fewer yellow marbles than blue and green ones combined.
5. Beth has fewer red marbles than green and yellow ones combined.
green and yellow ones combined = 5 + 15 = 20 marbles , red marbles = 35 marbles.
35 – 20 = 15 marbles.
False .

Question 12.
The school had a fundraiser in the first part of the school year. In which month did the school make the most money?

A. September
B. October
C. November
D. December
Answer:
Amount of fund raised in September = $35
Amount of fund raised in October = $25
Amount of fund raised in November = $50
Amount of fund raised in December = $40
The school make the most money in November of $50.

Question 13.
Look at the bar graph above. Suppose $45 was raised in January. Where would the bar end?
Answer:
If the money raised $45 in January it would end put between 40 and 50 grid line.

Question 14.
Did the school earn more money in September and October combined than in November? Explain.
Answer:
Amount of fund raised in September = $35
Amount of fund raised in October = $25
Total money earned in September and October combined  = $35 +$25 = $60.
Amount of fund raised in November = $50
Amount of more money earned in September and October combined than in November = $60 – $50 = $10.
School earned $10 of more money earned in September and October combined than in November.

Question 15.
The frequency table below shows the time Kelly jumped rope during the week. Use the data to make a picture graph.

Answer:

B. Draw a picture graph.

Answer:

Question 16.
Look at the picture graphs below. Which type of book was chosen by the same number of students in each class?

A. Adventure
B. Mystery
C. Nonfiction
D. Not here
Answer:
Full book = 2 students and Half book = 1 student
In Class A picture graph
Number of students choose Nonfiction types of books  = 2 full and 1 half = (2 × 2) + 1 = 5 students.
In Class B picture graph
Number of Nonfiction types of books choose = 2 full and 1 half = (2 × 2) + 1 = 5 students.

Question 17.
How many students in Classes A and B chose mystery as their favorite type of book?
Answer:
Number of students in class A chose Mystery type of book = 5 Full = 5 × 2 = 10 students
Number of students in class B chose Mystery type of book = 4 full and 1 half = (4 × 2) + 1 = 8 + 1 = 9 students.
Total number of students in Classes A and B chose mystery as their favorite type of book = 10 + 9 = 19 students.

Question 18.
A. Ellie has $22 to spend on art supplies. She wants to buy at least one canvas, one tube of paint, and one brush. Which art supplies can she buy if she spends all of her money?

Answer:

She can buy 3 canvas , 2 tube of paint and 2 brushes.

B. Describe the given information and solve the problem. Explain your thinking and show the supplies she can buy on the bar graph.
Answer:

Topic 7 Performance Task

Twisting Balloons Miles twisted balloons into different animals at his daughter’s birthday party. The Balloons Used picture graph shows the different color balloons he used.

Use the Balloons Used picture graph to answer Questions 1 and 2.

Question 1.
How many more green balloons than yellow balloons did Miles use? Explain.
Answer:
one Balloon = 2 balloons
Number of green balloons Miles used = 3 Balloons = 3 × 2 balloons = 6 balloons
Number of yellow balloons Miles used = 2 Balloons = 2 × 2 balloons = 4 balloons
Number of more green balloons  Miles used  than yellow balloons  = 6 – 4 = 2 balloons.
Miles used 2 more green balloons than yellow balloons.

Question 2.
How many fewer blue balloons were used than all of the other colors combined? Explain.
Answer:
Number of blue balloons Miles used =5 Balloons = 5 × 2 balloons = 10 balloons
Number of brown balloons Miles used = 2 Balloons = 2 × 2 balloons = 4 balloons
Number of green balloons Miles used = 3 Balloons = 3 × 2 balloons = 6 balloons
Number of yellow balloons Miles used = 2 Balloons = 2 × 2 balloons = 4 balloons
Total number of green, yellow and brown balloons Miles used combined = 4 + 6 + 4 = 14 balloons.
Number of fewer blue balloons were used than all of the other colors combined = 14 – 10 = 4 balloons.
4 fewer blue balloons were used than all of the other colors combined.

Use the Balloons Used picture graph and the Balloons Bought bar graph to answer Question 3.

Question 3.
How many balloons does Miles have left? Complete the table below.

Answer:

Use your response to Question 3 to answer Question 4.
Question 4.
Complete the bar graph to show how many balloons are left.

The Balloon Shapes and Colors table shows the number and color of balloons Miles needs to use to make each balloon animal. Use your response to Question 4 and the Balloon Shapes and Colors table to answer Question 5.

Answer:

Question 5.
Miles plans to use all of the balloons that are left and wants to make at least one of each balloon animal. Make a picture graph to show one way Miles can finish using the balloons.

Part A
Circle the key you will use.

Answer:

Part B
Complete the picture graph and explain how you solved the problem.

Answer:

Go through the enVision Math Common Core Grade 3 Answer Key Topic 8 Use Strategies and Properties to Add and Subtract regularly and improve your accuracy in solving questions.

enVision Math Common Core 2nd Grade Answers Key Topic 8 Use Strategies and Properties to Add and Subtract

Essential Question: How can sums and differences be estimated and found mentally?

EnVision STEM Project: Traits and the Environment
Do Research Use the Internet or other sources to find out how the environment can influence plants or animals. Describe a trait in an animal or plant that can change due to the environment.
Journal: Write a Report Include what you found. Also in your report:

• Make a table that includes the plant or animal, the trait, and changes in the environment. Record any related data about the environment, such as temperature or rainfall.
• Include information about why the trait is useful.
• Write and solve addition problems using your data. Use estimation to check for reasonableness.

Review What You Know

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

• difference
• equation
• number line
• sum

Question 1.
The amount that is left after you subtract is the ___.
Answer: The amount that is left after you subtract is the difference.

Question 2.
A line that shows numbers in order from left to right is a(n) ___.
Answer:  A line that shows numbers in order from left to right is a number line

Question 3.
The total when you add is the ___.
Answer: The total when you add is the sum.

Question 4.
Both sides of a(n) ___ are equal.
Answer: Both sides of an equation are equal.

Addition and Subtraction Strategies
Find the sum or difference. Show your work.

Question 5.
32 + 58
Answer:
32 + 58
(30 + 2) + (50 +8) = 90

Question 6.
27 + 46
Answer:
27 + 46 = 73
(20 + 7) + (40 + 6) = 73

Question 7.
73 – 52
Answer:
73 – 52 = 21
(70 + 3) – ( 50 + 2) = 21

Question 8.
63 +16
Answer:
63 + 16 = 79
(60 + 3) + (10 + 6) = 79

Question 9.
88 – 28
Answer:
88 – 28 = 60
(80 + 8) – (20 + 8) = 60

Question 10.
76 – 49
Answer:
76 – 49 = 27
(70 + 6) – (40 + 9) = 27

Numerical Expressions

Question 11.
Atif puts 45 rocks in a display box. He has 54 rocks in all. Which expression can be used to find how many rocks are not in the display box?
A. 45 + 54
B. 45 + 45
C. 54 – 45
D. 54 – 54
Answer:
Option C is the correct answer because
Total rocks = 54
Atif puts in a display box = 45
Then total rocks – Atif puts in a display box = rocks are not in a display box.

Counting Money no

Question 12.
Tony has the coins shown at the right. Does he have enough money to buy a toy car that costs 86¢? Explain.

Answer:
No, because the sum of all cents is not enough to buy a toy car.

Pick a Project

PROJECT 8A
How much citrus is grown in Florida?
Project: Plan a Citrus Grove

PROJECT 8B
Would you like to travel across the country?
Project: Create and Perform Skits

PROJECT 8C
How can you add and subtract large numbers without a calculator?
Project: Make a Mental Math Game

PROJECT 8D
How many people live in our country?
Project: Design a Class Census and Give an Estimation Test

Lesson 8.1 Addition Properties

Activity

Solve & Share

Olivia arranges cups of buttons on three trays. She records the number of buttons on each cup. Which tray has the most buttons? Use place-value blocks or drawings to help solve the problem.

Are you making the same calculations more than once? How can you use structure to help solve the problem?

Look Back! Olivia pours all the buttons on Tray A into a bowl. She then divides the buttons equally into 8 cups. How many buttons are in each cup? Explain.

Visual Learning Bridge

Essential Question
What Are Some Properties of Addition?

A.
You can use properties of addition to join groups.

Parentheses show what to do first.

Associative (Grouping) Property of Addition: You can group addends in any way and the sum will be the same.

В.
Commutative (Order) Property of Addition: You can add numbers in any order and the sum will be the same.

57 + 35 = 35 + 57

C.
Identity (Zero) Property of Addition: The sum of zero and any number is that same number.

39 + 0 = 39

Convince Me!
Use Structure Pick one of the properties above. Explain how you can use a number line to show an example of that property.

Guided Practice

Do You Understand?

Question 1.
Ralph says he can rewrite (4 + 5) + 21 as 9 + 21. Do you agree? Why or why not?
Answer:
Yes I  agree
because 4 + 5 = 9
So, In place of 4 + 5, we can write has 9

Question 2.
What property is shown with this equation? How do you know?
65 + (18 + 38) = (18 + 38) + 65
Answer:
This equation is in the form of Associative property. This property allows us to change the groupings of addition or multiplication and keep the same value.

Do You Know How?

In 3 and 4, identify each property.

Question 3.
4 + (15 + 26) = (4 + 15) + 26
Answer:
The equation is in the form of Associative property.

Associative property allows us to change groupings of addition or multiplication and keeps the same value.

Question 4.
17 + 0 = 17
Answer:
This equation is in the form of the Identity property of addition.

The identity property of 1 says that any number multiplied by 1 keeps its identity.

In 5-7, write each missing number.

Question 5.
__ + 90 = 90
Answer:
0 + 90 = 90

Question 6.
42 + 23 = 23 + __
Answer:
42 + 23 = 23 + 42

Question 7.
(2 + __) + 36 = 2 + (23 + 36).
Answer:
(2 + 23) + 36 = 2 + (23 + 36)

Independent Practice

In 8-11, identify each property.

Question 8.
19 + 13 = 13 + 19
Answer:
The equation is in the form of
a + b = b + a
It is a commutative property.

Commutative property states that the change in the order of numbers in the addition or multiplication operation does not change the sum or the product.

Question 9.
18 + 0 = 18
Answer:
It is a identity property.

The identity property of 1 says that any number multiplied by 1 keeps its identity.

Question 10.
16 + (14 + 13) = (16 + 14) + 13
Answer:
The equation is in the form of Associative property.This allows us to change groupings of addition or multiplication and keep the same value.

Question 11.
(39 + 12) + 8 = (12 + 39) + 8
Answer:
The equation is in the form of the commutative property.

Commutative property states that the change in the order of numbers in an addition or multiplication and keeps the same value.

In 12-19, write each missing number.

Question 12.
25 + 62 = __ + 25
Answer:
25 + 62 = 62 + 25
Thus the missing number is 62.

Question 13.
(22 + 32) + 25 = ___ + (22 + 32)
Answer:
(22 + 32) + 25 = 25 + (22 + 32)
Thus the missing number is 25.

Question 14.
23 + __ + 11 = 23 + 11
Answer:
23 + 0 + 11 = 23 + 11
Thus the missing number 0.

Question 15.
10 + (45 + 13) = (__ + 45) + 13
Answer:
10 + ( 45 + 13 ) = (10 + 45) + 13
Thus the missing number is 10.

Question 16.
(__ + 0) + 14 = 7 + 14
Answer:
(7 + 0) + 14 =  7 + 14
Thus the missing number is 7.

Question 17.
(12 + 2) + 20 = ___+ 20
Answer:
(12 + 2) + 20 = 14 + 20
Thus the missing number is 14.

Question 18.
34 + (2 + 28) = (___ + 28) + 34
Answer:
34 + (2 + 28) = ( 2 + 28 ) + 34
Thus the missing number is 2.

Question 19.
(50 + 30) + __ = 50 + (30 + 20)
Answer:
(50 + 30) + 20 = 50 + (30 + 20)
Thus the missing number is 20.

Problem Solving

Question 20.
Make Sense and Persevere Gino packs his blue and green pencils into boxes. He puts 8 pencils in each box. How many boxes does Gino use?

Answer:
Blue pencils = 23
Green pencils =17
Gino puts pencils in each box = 8
Blue pencils + Green pencils = 40
Therefore 40 ÷ 5 = 5
Gino uses 5 boxes.

Question 21.
Group the addends below in a different way to get the same sum. Write the new equation.
(42 + 14) + 6 = 62
Answer:
New equation is  (6+ 14) + 42 = 62

Question 22.
Vocabulary How is the Commutative Property of Addition like the Commutative Property of Multiplication?
Answer:
The commutative property states that the numbers on which we operate can be moved from their position without making any difference to the answer.
The property holds for addition and multiplication.

Question 23.
enVision® STEM A lionfish has 13 spines on its back, 2 more spines near its stomach, and 3 more near its tail. Using a property of add isition, write two different equations to find how many spines a lionfish has. What property did you use?
Answer:
Lionfish has spines on its back = 13
More spines near it’s stomach = 2
More near it’s tail = 3
To find the total spines on lionfish we are using two different properties of addition.
1. Associative property of addition
(13 + 2) + 3 = 13 + (2 + 3)
2. Commutative property of addition
13 + 2 + 3 = 3 + 2 + 13
Total spines on lionfish = 18

Question 24.
Higher Order Thinking Barry says he can subtract numbers in any order and the difference will stay the same. Is Barry correct? Give an example to support your answer.
Answer: No
3 – 2 = 1
2 – 3 = -1
By this example, we can say that the subtraction of numbers in any order will not be the same.

Assessment Practice

Question 25.
Use place value to find the sum of 33 +42 + 17.
A. 89
B. 90
C. 91
D. 92
Answer:
D is the answer because
33 + 42 + 17 = 92 

Question 26.
Use properties of operations to find the sum of 22 + 30 + 28.
A. 80
B. 70
C. 60
D. 50
Answer:
To find the sum of 22 + 30 + 28 we are using commutative property.
We write the as
22 + 28 + 30; 30 + 28 +  30,
From this equation, we can calculate 22,28 in an easy way.

Lesson 8.2 Algebra: Addition Patterns

Activity

Solve & Share

Shade three sums that are next to each other on the addition table. Add the first and third sums you shaded. Find a pattern using that total and the second sum you shaded. How are the total and the second sum related? Is this true for other sets of three sums next to each other?

You can look for relationships in the addition table. The numbers in the shaded column and shaded row are addends. The other numbers are the sums.

Look Back! Explain how you can test to see if the relationship among the three sums that are next to each other is a pattern.

Essential Question How Can You Find Addition Patterns?

Visual Learning Bridge

A.
Helen found the sum of the purple numbers in the red square. Then she found the sum of the green numbers. The sums form a pattern. Find the sums and describe the pattern.

You can use a variety of strategies to find the sums!

B.
Use the Associative Property.
44 + 48 = 44 + (2 + 46)
= (44 + 2) + 46
= 46 + 46

C.
Use mental math.
44 +48 = (44 + 2) + (48 – 2)
= 46 +46
44 +48 = 46 + 46

The sum of the purple numbers is equal to the sum of the green numbers. That’s a pattern!

D.
Use the Commutative and Associative Properties.
44 + 48 = (10 + 34) + (12 + 36)
46 + 46 = (12 + 34) + (10 + 36)
Use the properties to rearrange the addends.
(10 + 34) + (12 + 36) = (10 + 34) + (12 + 36)
The sum of the purple numbers is 92.
The sum of the green numbers is 92.
The sums are double the middle number in the red square. That is another pattern.

Convince Me!
Generalize The red square above is 3 squares tall by 3 squares wide. Sebastian says there are other size squares in the addition table that have patterns. Describe a different-size square and its patterns.

Guided Practice

Do You Understand?

Question 1.
Are the sums of any two sets of diagonal corner numbers in a 3-by-3 square in a standard addition table always equal? Explain.
Answer: Yes

Do You Know How?

Question 2.
Look at the addition table in Box A on the previous page. Why do the numbers going down to the right on the diagonal increase by 2? Explain.
Answer: In the box A the numbers from left to right is sequence numbers so, in the diagonal box the numbers are going from down to right.

Independent Practice

In 3 and 4, use the table at the right.

Question 3.
Look at the sums that are shaded the same color. Describe a pattern shown by these pairs of sums. Explain why this pattern is true.

Answer:

Question 4.
Find other pairs of sums with a similar pattern. Shade them on the table. Explain why you chose those sums.

In 5 and 6, use the table at the right.

Question 5.
Shade the table to show a pattern you see. Describe your pattern.

Answer:

Question 6.
Explain why your pattern is true.
Answer:

Problem Solving

Question 7.
Look for Relationships Greg drew a rectangle on the addition table at the right. He colored the corners. Find the sum of the green corners. Find the sum of the orange corners. What pattern do you notice?

Answer:

Question 8.
Draw another rectangle on the addition table. See if Greg’s pattern is true for this rectangle.
Answer:

Question 9.
Explain why Greg’s pattern works.
Answer:

Question 10.
Which multiplication fact does the number line show? Write a related division fact.

Answer:
multiple of 5.

Question 11.
Higher Order Thinking Pierre made an addition table. He skip counted by 3 for the addends. Find and describe a pattern in Pierre’s table.

Answer:

Assessment Practice

Question 12.
Look at the shaded cells  the addition table below.

Which pattern and property of operations are shown in the shaded cells?
A. Each orange sum is equal to zero plus the other addend; The Identity Property of Addition
B. Each green sum is 10 greater than one of its addends; The Identity Property of Addition
C. Each green sum is ten greater than the sum before; The Associative Property of Addition
D. There are no patterns or properties.
Answer:

Lesson 8.3 Mental Math: Addition

Activity

Solve & Share

A school store sold 436 pencils last week and 7 packages that each had 4 pencils today. Use mental math to find how many pencils were sold in all. Explain how you found your answer.

You can use structure by examining the quantities in the problem.

Look Back! What is another way you can find the sum of 436 pencils plus 7 packages of 4 pencils each using mental math?

Visual Learning Bridge

Essential Question
How Can You Add with Question Mental Math?

A.
Dr. Gomez recorded the number of Northern Right Whales, Atlantic Spotted Dolphins, and Western Atlantic Harbor Seals she saw off the coast of Florida in two different years. How many whales did Dr. Gomez see during the two years?

You can use an open number line, mental math strategies, and properties of operations to solve this problem.

B.
One Way Find 325 + 114. Use the adding on strategy.

Start at 325. Break apart 114.
Add 100 to 325.
Add 10 to 425.
Add 4 to 435.
325 + 114 = 439
Dr. Gomez saw 439 whales.

C.
Another Way
Find 325 + 114. Use the make ten strategy. Break apart
114. 114 = 5 + 100 + 9
Add 5 to 325 to make ten.
325 + 5 = 330
Then, add 100.
330 + 100 = 430
Break 114 apart to find a number that makes a ten when added to 325.

Finally, add 9.
430 + 9 = 439
325 + 114 = 439
Dr. Gomez saw 439 whales.

Convince Me!
Model with Math Show two ways to find the total number of dolphins seen.

Guided Practice

Do You Understand?

Question 1.
Compare the One Way and on the previous page. How are they the same? How are they different?

Question 2.
Use mental math to find how many animals Dr. Gomez saw during Year 2. Show your work.
Answer:
Dr.Gomez saw whales in second year = 114
Dr.Gomez saw Dolphins in second year = 171
Dr. Gomez saw seals in second year = 212
Total number of animals that Dr.Gomez saw during second year is
114 + 171 +212
100 + 10 + 4 + 100 + 70 + 1 + 200 + 10 + 2
400 + 90 + 5 + 2
400 + 90 + 7
400 + 97 = 497

Do You Know How?

Question 3.
Use the make ten strategy to add 738 + 126.
126 = 2 + 24 + 100
738 + ___ = 740
740 + __ = 764
764 + __ = 864
So, 738 + 126 = ___.
Answer:
126 = 2 + 24 + 100
738 + 2 = 740
740 + 24 = 764
764 + 100 = 864
So, 738 + 126 = 864

Question 4.
Use the adding  strategy to add 325 + 212.
212 = 200 + 10 + 2
325 + 200 = ___
525 + 10 = ___
___ + 2 = 537
So, 325 + 212 = __.

Answer:
212 = 200 + 10 + 2
325 + 200 = 525
525 + 10 = 535
535 + 2 = 537
So, 325 + 212 = 537

Independent Practice

In 5-12, find each sum using mental math or an open number line.

Question 5.
252 +44
Answer:
To find the sum here we are using mental math.
252 + 44
200 + 50 + 2 + 40 + 4
200 + 90 + 6
200 + 96 = 296

Question 6.
236 + 243
Answer:
To find the sum here we are using mental math.
236 + 243
200 + 30 + 6 + 200 +40 + 3
400 + 70 + 9
400 + 79 = 479

Question 7.
651 + 150
Answer:
To find the sum here we are using mental math.
651 + 150
600+ 50 + 1 +100 + 50
700 + 100 + 1
600 + 1 = 601

Question 8.
378 + 542
Answer:
To find the sum here we are using mental math.
378 + 542
300 + 70 + 8 + 500 + 40 + 2
800 + 110 + 10
800 + 120 = 920

Question 9.
473 + 198
Answer:
To find the sum here we are using mental math.
473 + 198
400 + 70 + 3 + 100 + 90 + 8
500 + 160 + 11
500 + 171 = 671

Question 10.
319 + 339
Answer:
To find the sum here we are using mental math.
319 + 339
300 + 10 + 9 + 300 + 30 + 9
600 + 40 + 18
600 + 58 = 658

Question 11.
208 + 511
Answer:
To find the sum here we are using mental math.
208 + 511
200 + 8 + 500 + 10 + 10 + 1
700 + 18 + 10 +1
710 + 19 = 729

Question 12.
523 + 169
Answer:
To find the sum here we are using mental math.
523 + 169
500 + 20 + 3 + 100 + 60 + 9
600 + 80 + 12
600 + 92 = 692

Problem Solving

Question 13.
Higher Order Thinking Maxine earns $8 each hour that she works as a cashier. She starts with $233. Today she cashiers for 6 hours. How much does she have at the end of the day? Explain how you found your answer.
Answer:
Given,
Maxine earns $8 each hour that she works as a cashier.
She starts with $233. Today she cashiers for 6 hours.
6 × $8 = $48
$233 + $48 = $281
Therefore Maxine have $281 at the end of the day.

Question 14.
Lauren sorted the 4 solids into 2 groups. Use mathematical terms to explain how she sorted the solids.

Answer:

Question 15.
The Rodriguez family drives 229 miles on Friday and 172 miles on Saturday. Explain how you can use the adding on strategy to find the total number of miles the Rodriguez family drives.
Answer:
Rodriguez family drives on Friday = 229 miles
Rodriguez family drives on Saturday = 172
Total number of miles that Rodriguez family drives = 229 + 172
200 + 20 + 9 + 100 + 70 + 2
300 + 90 + 11
300 + 101 = 401

Question 16.

Critique Reasoning Bill added 438 + 107. He recorded his reasoning below. Critique Bill’s reasoning. Are there any errors? If so, explain the errors.
Find 438 + 107.
I’ll think of 7 as 2 + 5.
438 + 2 = 440
440 + 7 = 447
447 + 100 = 547
So, 438 + 107 is 547.
Answer:
Above equation has an error, In place of adding 7 with 440  we want to add 5 then it is correct.

Assessment Practice

Question 17.
Find 153 + 121. Break apart 121 by place value, and then use the adding on strategy. Select numbers to complete the equations.

Answer:

Question 18.
Find 123 + 176. Break apart 176 by place value, and then use the adding on strategy. Select numbers to complete the equations.

Answer:

Lesson 8.4 Mental Math: Subtraction

Activity

Solve & Share

Peyton wants to buy one item that costs $425 and another item that costs $210. If she gets the discount shown on the sign below, what is the sale price? Explain how you can use mental math to find your answer.

Even when you use mental math, you can still show your work! You can construct arguments using mental math.

Look Back! What is another way you can use mental math to solve the problem?

Visual Learning Bridge

Essential Question
How Can You Subtract with Mental Math?

A.
A store is having a sale on jackets. A jacket is on sale for $197 less than the original price. What is the sale price?

You can use mental math and the relationship between addition and subtraction to solve this problem.

The difference is the answer when subtracting two numbers.

B.
One Way
Count Back on the Number Line Find 352 – 197.
To subtract 197 on an open number line, you can subtract 200, and then add 3.
352 – 200 = 152
152 + 3 = 155

So, 352 – 197 = 155. The sale price is $155.

C.
Another Way
Count Up on the Number Line
Find 352 – 197.
To find 352 – 197, you can think addition: 197 + ? = 352
197 + 3 = 200
200 + 100 = 300
300 + 52 = 352
Addition and subtraction are inverse operations.

3+ 100 + 52 = 155
197 + 155 = 352, so 352 – 197 = 155.
The sale price is $155.

Convince Me! Construct Arguments Which of the two ways above would you use to solve 762 – 252? Explain.

Another Example!
You can make a simpler problem to find 352 – 197.
Add 3 to both numbers.
352 + 3 = 355 and 197 + 3 = 200.
Then you have 355 – 200.
So, 355 – 200 = 155 and 352 – 197 = 155.

If you get stuck using one strategy, another strategy may be easier!

Guided Practice

Do You Understand?

Question 1.
In the One Way example on the previous page, why do you add 3 to 152 instead of subtracting 3 from 152?
Answer: We have to add or subtract the same numbers for making the correct answer.

Question 2.
Suppose a computer costs $573. If you buy it today, it costs $498. What is the discount? Show your work.
Answer:
Given,
Suppose a computer costs $573. If you buy it today, it costs $498.
$573 – $498 = $92
Thus the discount is $92.

Do You Know How?

In 3-6, solve using mental math.

Question 3.
846 – 18
848 – 20 =___
Answer:
846 – 18
848 – 20
800 + 40 + 8 – 20
800 + 48 – 20
848 – 20 = 828

Question 4.
534 – 99
535 – 100 = __
Answer:
534 – 99
535 – 100
500 + 30 + 5 – 100
500 + 35 – 100
535 – 100 = 435

Question 5.

873 – 216
877 – 220 = ___
Answer:
873 – 216
877 – 220
800 + 70 + 7 – 200 + 20
800 + 77 – 220
877 – 220 = 657

Question 6.
782 – 347
785 – 350 = ___
Answer:
782 – 347
785 – 350
700 + 80 + 5 – 300 + 50
700 + 85 – 350
785 – 350 = 435

Question 7.
Find 400 – 138 using the make ten strategy
138 + __ = 140
140 + ___ = 200
200 + __ = 400
___ + __ + ___ = ___
Answer:
138 + 2 = 140
140 + 60 = 200
200 + 200 = 400
2 + 60 + 200 = 262

Independent Practice

In 8-15, use an open number line or the think addition strategy to find each difference.

Question 8.
128 – 19
Answer: 109

Question 9.
887 – 18
Answer: 869

Question 10.
339 – 117
Answer: 222

Question 11.
468 – 224
Answer: 244

Question 12.
784 – 515
Answer: 269

Question 13.
354 – 297
Answer: 57

Question 14.
853 – 339
Answer: 514

Question 15.
638 – 372
Answer: 266

Problem Solving

Question 16.
Sarah has $350. How much money will she have after buying the computer at the sale price?

Answer:
Given,
Sarah has $350.
$350 – $58 = $292

Question 17.
Model with Math Jessica has an array with 9 columns. There are 36 counters in the array. How many rows does her array have? Show how to represent the problem and find the answer.
Answer:
Given,
Jessica has an array with 9 columns.
There are 36 counters in the array.
9 × 4 = 36
Thus there are 4 rows and 9 columns.

Question 18.
Of the students at Paul’s school, 270 are girls and 298 are boys. There are 354 students at Alice’s school. How many more students are there at Paul’s school than at Alice’s school?
Answer:
Girls at Paul’s school = 270
Boys at Paul’s school = 298
Students at Alice’s school = 354
Students at Paul’s school = 568
568 – 354 = 214
Paul’s school has 214 students more than Alice’s school students.

Question 19.
Higher Order Thinking To find 357 – 216, Tom added 4 to each number and then subtracted. Saul added 3 to each number and then subtracted. Will both ways work to find the correct answer? Explain.
Answer:
To find 357 – 216, Tom added 4 to each number and then subtracted.
357+4 = 361
216 + 4 = 220
361 – 220 = 141
Saul added 3 to each number and then subtracted.
357+3 = 360
216 + 3 = 219
360 – 219 = 141

Assessment Practice

Question 20.
Use the relationship between addition and subtraction to find 233 – 112. Select numbers from the box to complete the work on the open number line and the equations.

Answer:

Lesson 8.5 Round Whole Numbers

Activity

Solve & Share
Think about ways to find numbers that tell about how much or about how many. Derek has 277 stickers. What number can you use to describe about how many stickers Derek has? Explain how you decided.

Think about whether you need to be precise.

Look Back! Derek gets 3 more packages of 10 stickers. About how many stickers does Derek have now?

Essential Question
How Can You Round Numbers?

Visual Learning Bridge

A.
About how many rocks does Tito have? Round 394 to the nearest ten.

Place value is the value of the place a digit has in a number. Think about the place value of the digits in 394.

When you round to the nearest ten, you are finding the closest multiple of ten for a given number.

B.
You can use place-value understanding and a number line to round to the nearest ten.

394 is closer to 390 than 400, so 394 rounds to 390.
Tito has about 390 rocks.

C.
About how many rocks does Donna have? Round 350 to the nearest hundred.

If a number is halfway between, round to the greater number
350 is halfway between 300 and 400, so 350 rounds to 400.
Donna has about 400 rocks.

Convince Me! Make Sense and Persevere Susan says, “I am thinking of a number that has a four in the hundreds place and a two in the ones place. When you round it to the nearest hundred, it is 500.” What number could Susan be thinking of? What other numbers could be Susan’s number?

Another Example
About how many rocks does Carl have? Round 345 to the nearest ten and hundred.
Round to the nearest ten.
345 is halfway between 340 and 350, so 345 rounds to 350.

Round to the nearest hundred.
345 is closer to 300 than 400, so 345 rounds to 300.

Guided Practice

Do You Understand?

Question 1.
What number is halfway between 200 and 300?
Answer:
The halfway between 200 and 300 is 250.

Question 2.
Sheri rounds 678 to 680. What place does she round to?
Answer:

Here the digits in the one’s place is  8 in 678, now round to the nearest multiple of 10 which is greater than the number.
Given,
Sheri rounds 678 to 680.
The nearest ten to 678 is 680.

Question 3.
Tito adds one more rock to his collection on the previous page. About how many rocks does he have now, rounded to the nearest ten? Rounded to the nearest hundred? Explain.
Answer:

Do You Know How?

In 4-6, round to the nearest ten.

Question 4.

Answer: 517 to the nearest ten is 520.

Question 5.
149
Answer:

Here the digits in the one’s place of 149 is 9 , now round to the nearest multiple of 10 which is greater than the number.
149 to the nearest ten is 150.

Question 6.
732
Answer:

Here the digits in the one’s place of 732 is 2, now round to the nearest multiple of 10

732 to the nearest ten is 730.

In 7-9, round to the nearest hundred.

Question 7.

Answer:

When rounding to the nearest hundred, look at the ten’s digit. If the numbers is 0,1,2,3,4 you will round to previous hundred.

640 to the nearest hundred is 600.

Question 8.
305
Answer:

When rounding to the nearest hundred, look at the ten’s digit. If the numbers is 0,1,2,3,4 you will round to the previous hundred.

305 to the nearest hundred is 300.

Question 9.
166
Answer:

When rounding to the nearest hundred , look at the ten’s digit.If the numbers is 5,6,7,8,9 you will round to the next hundred.

166 to the nearest hundred is 200.

Independent Practice

In 10-12, round to the nearest ten.

Question 10.
88
Answer:

Here the digits in the one’s place of 88 is 8 , now round to the nearest multiple of 10 which is greater than the number.
88 to the nearest ten is 90.

Question 11.
531
Answer:

Here the digits in the one’s place of 531 is 1, now round to the nearest multiple of 10 which is less than the number.
531 to the nearest ten is 530.

Question 12.
855
Answer:

Here the digits in the one’s place of 855 is 5, now round to the nearest multiple of 10 which is greater than the number.
855 to the nearest ten is 860.

In 13-15, round to the nearest hundred.

Question 13.
428
Answer:

When rounding to the nearest hundred, look at the ten’s digit of the number.If the numbers are 0,1,2,3,4, you will round down to the previous hundred.

428 to the nearest hundred is 400.

Question 14.
699
Answer:

When rounding to the nearest hundred look at the ten’s digit of the given number.If the digits are 5,7,8,9 you will round to the previous hundred.

699 to the nearest hundred is 700.

Question 15.
750
Answer: when rounding to the nearest hundred, look at the ten’s digit of the number. If the digits are 5,6,7,8,9 you will round to the next hundred.

750 to the nearest hundred is 700. or 800.

Problem Solving

Question 16.
The Leaning Tower of Pisa in Italy has 294 steps. To the nearest ten, about how many steps are there? To the nearest hundred, about how many steps are there?

Answer:
Given,
The Leaning Tower of Pisa in Italy has 294 steps.
294 to the nearest ten is 290.
294 to the nearest hundred is 300.
Thus there are about 300 steps.

Question 17.
Critique Reasoning Zoe says 247 rounded to the nearest hundred is 300 because 247 rounds to 250 and 250 rounds to 300. Is Zoe correct? Explain.
Answer:
Zoe says 247 rounded to the nearest hundred is 300 because 247 rounds to 250 and 250 rounds to 300.
247 rounded to the nearest hundred is 200.
So, Zoe is not correct.

Question 18.
Use the number line to show a number that rounds to 200 when it is rounded to the nearest ten.

Answer:

Question 19.
Name the least number of coins you can use to show $0.47. What are the coins?
Answer:

Question 20.
Suppose you are rounding to the nearest hundred. What is the greatest number that rounds to 600? What is the least number that rounds to 600?
Answer:
The greatest number that round 600 to the nearest 100 would be 600 + 50 – 1 = 649. It won’t be 650 because of 650 rounds to 700.

Question 21.
Higher Order Thinking A 3-digit number has the digits 2,5, and 7. To the nearest hundred, it rounds to 800. What is the number? Show how you found the answer.
Answer:
For the number to be rounded off to 800, it needs to be either 750 or above or 850 and below.
With the given numbers, we can only have the following combinations:
257~300
275~300
527~500
572~600
725~700
752~800
And hence, 752 is the only combination of numbers that fits.

Question 22.
Emil says, “I am thinking of a number that is greater than 142, rounds to 100 when rounded to the nearest hundred, and has a 5 in the ones place.” What is Emil’s number?

What else can you try if you get stuck?

Answer: 542

Assessment Practice

Question 23.
Select all the numbers that will equal $100 when rounded to the nearest hundred.
$10
$110
$ 89
$150
$91
Answer:
$110, $89, $91 are rounded to the nearest hundred.

Question 24.
Select all the numbers that will equal 70 when rounded to the nearest ten.
62
75
72
83
73
Answer:
The number 70 rounded to the nearest ten is 75, 72, 73.

Lesson 8.6 Estimate Sums

Activity

Solve & Share

Look at the table below. Is the mass of a female and male sun bear together more or less than the mass of one female black bear? Without finding an exact answer, explain how you can decide.

You can use symbols, numbers, and words to be precise in your explanation.

Look Back! Why is an exact answer not needed to solve the problem?

Essential Question
How Can You Estimate Sums?

Visual Learning Bridge

A.
Do the two pandas together weigh more than 500 pounds?
Estimate 255 + 329.

You can estimate to find about how much the two pandas weigh.

B.
One Way
Round to the nearest hundred.

255 + 329 is about 600.
600 > 500
The pandas together weigh more than 500 pounds.

C.
Another Way
Use compatible numbers.

Compatible numbers are numbers that are close to the addends, but easy to add mentally

255 + 329 is about 575 and 575 > 500.
The total weight is more than 500 pounds.

Convince Me! Be Precise Sandy said, “Just look at the numbers. 200 and 300 is 500. The pandas weigh over 500 pounds because one panda weighs 255 pounds and the other weighs 329 pounds.” What do you think she means? Use numbers, words, or symbols to explain.

Another Example!
Suppose one panda ate 166 pounds of bamboo in a week and another ate 158 pounds. About how many pounds of bamboo did the two pandas eat?

You can estimate 166 + 158 by rounding to the nearest ten.

The pandas ate about 330 pounds of bamboo in a week.

Guided Practice

Do You Understand?

Question 1.
Two addends are rounded to greater numbers. Is the estimate greater than or less than the actual sum?
Answer:

Question 2.
Mary and Todd estimate 143 + 286. They have different answers. Can they both be correct? Explain why or why not.
Answer:
Mary and Todd has only one answer that is 143 + 286 = 429

Do You Know How?

Round to the nearest ten to estimate.

Question 3.
218 + 466 ___+___= ___
Answer:

Here the digit in the one’s place of 218 is 8, now round to the nearest multiple of 10 which is greater than the ten.
218 round to the nearest ten to estimate is 220.

The digit in the one’s place of 466 is 6, now round to the nearest multiple of ten which is greater than the number.
466 round to the nearest ten to estimate is 470
220 + 470 = 690

Question 4.
108 + 223 ___ + ___ = ____
Answer:

Here the digit in the one’s place of 108 is 8, now round to the nearest multiple of 10 which is greater than the number.
108 round to the nearest ten to estimate is 110.

The digit in the one’s place of 223 is 3, now round to the nearest multiple of 10 which is less than the number.
223 round to the nearest ten to estimate is 220.
110 + 220 = 330

Round to the nearest hundred to estimate.

Question 5.
514 + 258 ___ + ___ = ____
Answer:

Here the digit in the one’s place of 514 is 4, now round to the nearest multiple of 10 which is less than the number.
514 round to the nearest hundred to estimate is 500.

The digit in the one’s place of 258 is round to the nearest ten which is greater than the number.
258 round to the nearest hundred to estimate is 300.
500 + 300 = 800

Question 6.
198 + 426 ___ + ___ = ____
Answer:

Here the digit in the one’s place of 198 is 8,now round to the nearest multiple of ten which is greater than the number.
198 round to the nearest hundred to estimate is 200.

The digit in the one’s place of 426 is 6, now round to the nearest multiple of 10 which is greater than the number.
426 round to the nearest hundred to estimate is 400.
200 + 400 = 600

Independent Practice

In 7-10, round to the nearest ten to estimate.

Question 7.
138 + 435
Answer:

Here the digit in the one’s place of 138 is 8, now round to the nearest multiple of ten which is greater than the number.
138 round to the nearest ten to estimate is 140.

The digit in the one’s place of 435 is 5 now round to the nearest multiple of ten which is greater than the number.
435 round to the nearest ten to estimate is 440.
140 + 440 = 580

Question 8.
563 + 289
Answer:

Here the digit in the one’s place of 563 is 5 now round to the nearest multiple of ten which is less than the number.
563 round to the nearest ten to estimate is 560.

The digit in the one’s place of 289 is 9 now round to the nearest multiple of ten which is greater than the number.
289 round to the nearest ten to estimate is 290
560 + 290 = 850

Question 9
644 + 172
Answer:

Here the digit in the one’s place of 644 is 4 now round to the nearest multiple of ten which is less than the number.
644 round to the nearest ten to estimate is 640.

The digit in the one’s place of 172 is 2, now round to the nearest multiple of ten which is less than the number.
172 round to the nearest ten to estimate is 170
640 + 170 = 810

Question 10.
376 + 295
Answer:

Here the digit in the one’s place of the number 376 is 6 now round to the nearest ten which is greater than the number.
376 round to the nearest ten to estimate is 380.

The digit in the one’s place of 295 is 5 now round to the nearest multiple of ten which is greater than the number.
295 round to the nearest ten to estimate is 300
380 + 300 = 680

In 11-14, round to the nearest hundred to estimate.

Question 11.
403 + 179
Answer:

When rounding to the nearest hundred look at the ten’s digit of the number which the number is 0,1,2,3,4 you will round to the previous hundred.
403 round to the nearest hundred to estimate is 400.

Here the ten’s place of the 179 is 7 you will round to the next hundred.
179 round to the nearest hundred to estimate is 200.
400 + 200 = 600

Question 12.
462 + 251
Answer:

Here the ten’s place of the number is 462 is 6 you will round to the previous hundred.

462 round to the nearest hundred to estimate is 500.

Here the ten’s place of the number 251 is 5 you will round to the next hundred.
251 round to the nearest hundred to estimate is 300.
500 + 300 = 800

Question 13.
274 + 443
Answer:

Here the digit in the ten’s place of 274 is 7 you will round to the next hundred.
274 round to the nearest hundred to estimate is 300.

The digit in the ten’s of 443 is  4 you will round to the previous hundred.
443 round to the nearest hundred to estimate is 400.
300 + 400 = 700

Question 14.
539 + 399
Answer:

The digit in the ten’s place of 539 is 3 you will round to the previous hundred.
539 round to the nearest hundred to estimate is 500.

The digit in the ten’s place of 399 is 9 you will round to the next hundred.
399 round to the nearest hundred to estimate is 400.
500 + 400 = 900

In 15-18, use compatible numbers to estimate.

Question 15.
175 + 126
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply or division mentally.

Here 170 and 130 are the compatible numbers to the 175 and 126.

Question 16.
167 + 27
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply or division mentally.

Here 170 and 30 are the compatible number for 167 and 27.

Question 17.
108 + 379
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply or division mentally.

Here 110 and 380 are the compatible numbers for 108 and 379.

Question 18.
145 + 394
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply or division mentally.

Here 150 and 390 are the compatible numbers for 145 and 394.

Problem Solving

Use the table to answer 19 and 20.

Question 19.
Ms. Tyler drove from Albany to Boston, and then from Boston to Baltimore. To the nearest ten miles, about how many miles did she drive in all?
Answer:

Question 20.

Ms. Tyler drove from Boston to New York City and back again. To the nearest hundred miles, about how many miles did she drive?

Answer:

Given,
Ms. Tyler drives from Boston to New York City  = 211
Ms.Tyler drives back to nearest hundred miles to Boston  = 211 – 100
= 111
How many miles did she drive  equals to 211 + 111 = 322

Question 21.
Reasoning Jen has $236. Dan has $289. Do Jen and Dan have more than $600 in all? Estimate to solve. Explain.
Answer:
Jen has $236
Don has $286
Jen and Don has 236 + 289 = 525
No, They did have more than $600

Question 22.
Ralph has 75¢. How much more money does he need to buy a pencil for 90¢? Complete the diagram.

Answer:
Ralph has 75 cents.
He need to buy a pencil for 90 cents.
Ralph need to buy a pencil = 90 cents – 75 cents =15 cents

Question 23.
Higher Order Thinking Susan drove 247 miles on Wednesday morning. Then she drove 119 miles on Wednesday afternoon. On Thursday, Susan drove 326 miles. About how far did Susan drive in all? Explain the method you used to estimate.

Remember that you learned different estimation strategies.

Assessment Practice

Question 24.
Round to the nearest 10 to estimate the sums.

Answer:
273 round to the nearest ten to estimate is 270
616 round to the nearest ten to estimate is 620
270 + 620 = 890
542 round to the nearest ten to estimate is 540
338 round to the nearest ten to estimate is 340
540 + 340 = 880
435 round to the nearest ten to estimate is 440
441 round to the nearest ten to estimate is 440
440 – 440 = 0

Question 25.
Round to the nearest 100 to estimate the sums.

Answer:
173 round to the nearest hundred to estimate is 200
139 round to the nearest hundred to estimate is 100
200 – 100 = 100
155 round to the nearest hundred to estimate is 200
177 round to the nearest hundred to estimate is 200
200 – 200 = 0
289 round to the nearest hundred to estimate is 300
18 round to the nearest hundred to estimate is 0
300 – 0 = 300

Lesson 8.7 Estimate Differences

Activity

Solve & Share

Sara collected 220 cans on Monday, 80 cans on Tuesday, and 7 cartons with 8 cans each on Wednesday to recycle. Pierre collected 112 cans. About how many more cans did Sara collect than Pierre?

You can make sense and persevere. What is a good strategy for solving this problem?

Look Back! Which strategy gives an estimate that is closest to the exact answer? How did you decide?

Essential Question
How Can You Estimate Differences?

A.
All of the tickets for a concert were sold. So far, 126 people have arrived at the concert. About how many people who have tickets have not arrived?
Estimate 493 – 126 by rounding.

You can estimate to find about how many.

B.
One Way
Round each number to the nearest hundred and subtract.

About 400 people have not yet arrived.

C.
Another Way
Round each number to the nearest ten and subtract.

About 360 people have not yet arrived.

Convince Me! Model with Math Suppose 179 people have arrived at the concert. Estimate how many people have not arrived.

Another Example!
You can use compatible numbers to estimate differences.

375 and 150 are compatible numbers for 372 and 149.

Guided Practice

Do You Understand?

Question 1.
Does rounding to the nearest ten or nearest hundred give an estimate closer to the exact answer for 295 – 153?

Answer:

Here the digit of the one’s place of 295 is 5 now round to the nearest multiple of ten which is greater than the number.
295 round to the nearest ten to estimate is 290

The digit in the one’s place of 153 is3 now round to the nearest multiple of ten which is less than the number.
153 round to the nearest ten to estimate is 140
290 – 140 =150

Question 2.
A theater sells 408 tickets. Two hundred seventy-three people arrive. About how many more people are expected to arrive? Use compatible numbers. Show your work.

Answer:

Given,

A theater sells tickets = 408

People arrive to theater = 253

Expected people to arrive = 408 – 253 = 155

The compatible numbers for 408 and 253 is 410 and 250

410  – 250 = 150

Do You Know How?

In 3-6, estimate. Use rounding or compatible numbers. Tell how you made each estimate.

Question 3.
321 – 182
Answer:

Compatible numbers are pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 321 and 182 is 320 and 180.

Question 4.
655 – 189
Answer:

Compatible numbers are pair of numbers that are easy to add, subtract, multiply or division mentally.

The compatible numbers for 655 and 189 is 660 and 190

Question 5.
763 – 471
Answer:

Compatible numbers are pair of numbers that are easy to add, subtract multiply, and division mentally.

The compatible numbers for 763 and 471 are 760 and 470.

Question 6.
816 – 297
Answer:

Compatible numbers are pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 816 and 297 are 820 and 300.

Independent Practice

In 7-10, round to the nearest hundred to estimate.

Question 7.
286 – 189
Answer:

When rounding to the nearest hundred look at the ten’s place.The ten’s place of 286 is 8 now round to the next hundred.
286 round to the nearest hundred to estimate is 300.

The ten’s place of the number 189 is 8 now round to the next hundred
189 round to the nearest hundred to estimate is 200.
300 – 200 = 100

Question 8.
461 – 216

Answer:

When rounding to the nearest hundred look at the ten’s place.The ten’s place of the number 461 is 6 now round to the next hundred.

461 round to the nearest hundred to estimate is 500.

The ten’s place of the number 216 is 1 now round to the  previous hundred.
216 round to the nearest hundred to estimate is 200.
500 – 200 = 300

Question 9.
891 – 686
Answer:

When rounding to the nearest hundred look at the ten’s place of the number 891 is 9 now round to the next hundred.
891 round to the nearest hundred to estimate is 900.

The ten’s place of the number 686 is 8 now round to the next hundred.
686 round to the nearest hundred to estimate is 700.
900 – 700 = 200

Question 10.
724 – 175
Answer:

When rounding to the nearest hundred look at the ten’s place of the number 724 is 2 now round to the previous hundred.
724 round to the nearest hundred to estimate is 600.

The ten’s place of the number 175 is 7 now round to the next hundred.
175 round to the nearest hundred to estimate is 200
600 – 200 = 400

In 11-14, round to the nearest ten to estimate.

Question 11.
766 – 492
Answer:

Here the digit of the one’s place of the number 766 is 6 now round to the nearest ten which is greater than the number.
766 round to the nearest ten to estimate is 770.

The one’s place of the number 492 is 2 now round to the nearest ten which is less than the number.
492 round to the nearest ten to estimate is 500.
770 – 500 = 270

Question 12.
649 – 487
Answer:

Here the digit of the one’s place of the number 649 is 9 now round to the nearest hundred which is greater than the number.
649 round to the nearest ten to estimate is 650.

The one’s place of the number 487 is 7 now round to the nearest hundred which is greater than the number.
487 round to the nearest ten to estimate is 490.
650 – 490 = 169

Question 13.
241 – 117
Answer:

Here the digit in the one’s place of the number 241 is 1 now round to the nearest hundred which is less than the number.
241 round to the nearest ten to estimate is 240.

The one’s place of the number 117 is 7 now round to the nearest hundred which is greater than the number.
117 round to the nearest ten to estimate is 120.
240 – 120 = 120

Question 14.
994 – 679
Answer:

Here the digit of the one’s place of the number 994 is 4 now round to the nearest hundred which is less than the number.
994 round to the nearest ten to estimate is 1000.

The digit in the one’s place of the number 679 is 9 which is greater than the number.
679 round to the nearest ten to estimate is 680.
1000 – 680 = 320

In 15-18, use compatible numbers to estimate.

Question 15.
760 – 265
Answer:

Compatible numbers are pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 760 and 265 are 760 and 270.

Question 16.
355 – 177
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally

Compatible numbers for 355 and 177 is 360 and 180.

Question 17.
481 – 105
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 481 and 105 is 480 and 110.

Question 18.
794 – 556
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 794 and 556 are 800 and 560.

Problem Solving

Question 19.
The Grand Concert Hall sold 100 more tickets on Sunday than on Friday. On what day did it sell about 150 tickets more than it sold on Sunday?

Use estimation strategies to help find your answers.

Question 20.
Model with Math Find the total number of tickets sold on Thursday and Friday. Explain what math you used.
Answer:
Number of tickets sold on Thursday = 323
Number of tickets sold on Friday = 251
Total number of tickets sold on Thursday and Friday equal to
321 + 251
300 + 20 + 1 + 200 + 50 +1
500 + 70 + 2
500 + 72 = 572

Question 21.
Algebra Anna and Joe write reports for their science class. Anna’s report is 827 words long. Joe’s report is 679 words long. Round each report length to the nearest ten and estimate how many more words Anna’s report is. Then write an equation that shows exactly how many more words Anna’s report is.
Answer:
Anna report is 827 long words.
Joe’s report is 679 long words.
827 round to the nearest ten to estimate is 830
679 round to the nearest ten to estimate is 680
830 – 680 = 150
Anna report is 150  long words more than Joe’s long report

Question 22.
Higher Order Thinking One week Mrs. Runyan earned $486, and the next week she earned $254. If Mrs. Runyan’s goal was to earn $545, by about how much did she exceed her goal? Show how you used estimation to find your answer.
Answers:
Mrs Runyan earned in one week = $486
Mrs Runyan earned in next week = $254
Total she earned in two weeks = $ 486 + $255
=$740
But she goal to earn is $545
She exceed her goal = $740 – $545
She exceed her goal is $ 195

Question 23.
About how many inches longer was a Brachiosaurus than a Tyrannosaurus rex?

Answer:
Brachiosaurus length = 972 inches.
Tyrannosaurus length = 468 inches.
972 – 468 = 504
Brachiosaurus length is 504 longer than Tyrannosaurus length.

Assessment Practice

Question 24.
Estimate 753 – 221 by rounding each number to the nearest ten.
A. 540
B. 530
C. 520
D. 510
Answer:
753 round to the nearest ten to estimate is 750.
221 round to the nearest ten to estimate is 220
750 – 220 = 530
Thus the correct answer is option B.

Question 25.
Estimate 812 – 369 by rounding each number to the nearest hundred.
A. 500
B. 400
C. 300
D. 200
Answer:
812 round to the nearest hundred to estimate is 810.
369 round to the nearest hundred to estimate is 370.
810 – 370 = 440
440 to the nearest hundred is 400.
Thus the correct answer is option B.

Lesson 8.8 Model with Math

Activity

Solve & Share

A pond has 458 rosy red minnows, 212 white cloud minnows, and 277 goldfish. How many more minnows than goldfish live in the pond?

Thinking Habits
Be a good thinker! These questions can help you.
• How can I use 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?

Look Back! Model with Math Explain what math you used to solve this problem.

Essential Question How Can You Model With math?

A.
David has $583 to spend on soccer uniforms. He buys this soccer jersey and 2 soccer shorts. How much money does David spend?

What math do I need to use to solve the problem?

I need to show what I know and then choose the needed operations.

B. How can I model with math? I can

• apply math I know to solve the problem.
• use a bar diagram and equations to represent the problem.
• use an unknown to represent the number I am trying to find.

Here’s my thinking…

C.
I will use a bar diagram and an equation.
The hidden question is: How much does David spend on shorts?

$35 + $35 = ?
$35 + $35 = $70. The shorts cost $70.
So I need to find the total including the jersey.

$70+ $109 = ?
$70 + $109 = $179. David spends $179.

Convince Me! Model with Math How does the bar diagram help you model with math?

Guided Practice

Model with Math
Harris’s office building has 126 windows. Morgan’s bank has 146 windows. Devon’s bank has 110 windows. How many more windows do the banks have altogether than the office building?

One way you can model  math is by using bar diagrams to represent each step of a two-step problem.

Question 1.
What is the hidden question you need to answer before you can solve the problem?
Answer:
How many windows do the banks have in all?

Question 2.
Solve the problem. Complete the bar diagrams. Show the equations you used.
Answer:
Morgan’s bank has 146 windows.
Devon’s bank has 110 windows.
Morgan’s and Devon’s office has 146 + 110 = 256 windows
Another office windows = 256 -126=130

Independent Practice

Model with Math
Regina’s bakery made 304 pies in January. Her bakery made 34 fewer pies in February. How many pies did her bakery make in both months?

Question 3.
What is the hidden question you need to answer before you can solve the problem?

Answer:
How many pies did the bakery make in February?

Question 4.
Solve the problem. Complete the bar diagrams. Show the equations you used.
Answer:

304 – 34 = ?
304 – 34 = 270
304 + 270 = ?
304 + 270 = 574

Question 5.
How would your equations change if the bakery made 34 more pies in February than in January?
Answer:
I would add 304 + 34 = ? instead of subtracting 304 – 34 = ? to find the number of pies baked in February.

Problem Solving

Performance Task

Skyscraper Heights
The Empire State Building in New York is 159 meters taller than the Republic Plaza in Denver. The John Hancock Building in Chicago is 122 meters taller than the Republic Plaza. The Empire State Building is 712 miles away from the Hancock Building. The Hancock Building is 920 miles away from the Republic Plaza. Manuel wants to know how tall the Hancock Building is. Answer Exercises 6-9 to solve the problem.

Answer:

Question 6.
Critique Reasoning Tom said to solve the problem, you should add 159 to the height of the Empire State Building. Do you agree? Explain why or why not.
Answer:

Question 7.
Model with Math What is the hidden question you need to solve in this problem? How can you represent the hidden question?
Answer:

Question 8.
Model with Math Solve the problem. Show the equations you used.
Answer:

Question 9.
Use Appropriate Tools Which tool would you use to represent and explain how to solve the problem: counters, cubes, or place-value blocks? Explain.

Model with math means you apply math you have learned to solve problems.

Answer:

Topic 8 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. Once you find the matches, write the fact families for each of the facts.

Clues
A The missing number is 9.

B The missing number is 10.

C The missing number is 3.

D The missing number is 6.

E The missing number is 7.

F The missing number is 4.

G The missing number is 8.

H The missing number is 5.

Answer:

Topic 8 Vocabulary Review

Glossary

Understand Vocabulary

Circle the property of addition shown in the following examples.

Word List
• Associative Property of Addition
• Commutative Property of Addition
• compatible numbers
• estimate
• Identity Property of Addition
• inverse operations
• mental math
• place value
• round

Question 1.
17 + 14 = 14 + 17
Associative Property Commutative Property Identity Property
Answer: Associative property.

Question 2.
93 + 0 = 93
Associative Property Commutative Property Identity Property
Answer: Identity property.

Question 3.
8 + (5 + 9) = (8 + 5) + 9
Associative Property Commutative Property Identity Property
Answer: Commutative property.

Question 4.
65 + 0 = 0 + 65
Associative Property Commutative Property Identity Property
Answer: Identity property.

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

Question 5.
You ___ when you use the nearest multiple of ten or a hundred.
Answer: You Estimate sum and differences in the number when you use the nearest multiple of ten or a hundred.

Question 6.
Addition and subtraction are ___
Answer:
Addition and subtraction are the arithmetic operations.

Question 7.
___ is the value given to the place a digit has in a number.
Answer: Place value is the value given to the place a digit has in a number.

Question 8.
Numbers that are easy to compute mentally are ___
Answer: Numbers that are easy to compute mentally are mental math.

Question 9.
You do not need pencil and paper when using ___
Answer: You do not need pencil and paper when using mental math.

Use Vocabulary in Writing

Question 10.
Jim found that 123 + 284 is about 400. Explain what Jim did. Use at least 3 terms from the Word List in your answer.
Answer: Jim used rounding numbers to add the given numbers.

Topic 8 Vocabulary Reteaching

Set A
pages 289-292
You can use properties of addition to help solve addition problems.
The Commutative Property of Addition
12 + = 15 + 12
You can order addends in any way, and the sum will be the same.
12 + 15 = 15 + 12

The Associative Property of Addition
3 + (7 + 8) = (3 + ) + 8
You can group addends in any way, and the sum will be the same.
3 + (7 + 8) = (3 + 7) + 8
The Identity Property of Addition
30 + = 30
The sum of any number and zero is that same number.
30 + 0 = 30

Remember that both sides of the equal sign must have the same value.

In 1-6, write each missing number.

Question 1.
18 + __ = 18
Answer:
18 + 0 = 18

Question 2.
14 + (16 + 15) = (___ +16) + 15
Answer:
14 + ( 16 + 15) = ( 14 + 16) + 15

Question 3.
___ + 13 = 13 + 17
Answer:
17 + 13 = 13 + 17

Question 4.
28 + (__ + 22) = 28 + (22 + 25)
Answer:
28 + (25 + 22) = 28 + ( 22 + 25 )

Question 5.
62 + 21 + 0 = 62 + ___
Answer:
62 + 21 + 0 = 62 + 21

Question 6.
___ + (26 + 78) = (31 + 26) + 78
Answer:
31 + (26 + 78) = (31 + 26) + 78

Question 7.
Use 78 and 34 to write an equation that shows the Commutative Property of Addition.

Answer:
78 + 34 = 34 + 78

Set B
pages 293-296
You can find patterns using an addition table.

The green sums increase by 2 down the column and are even numbers.
The yellow sums increase by 2 down the column and are odd numbers.
Use examples to make generalizations!

Remember that properties can help you understand patterns.

Question 1.
Find the doubles-plus-2 facts. What pattern do you notice about the sums?
Answer:

Question 2.
Explain why your pattern works.
Answer:

Set C
pages 297-300
Use mental math to find 374 + 238.
Break apart 238: 200 + 30+ 8.
Add hundreds, tens, and ones.
374 + 200 = 574
574 + 30 = 604
604 + 8 = 612
So, 374 + 238 = 612.

Remember that you can break apart addends when finding sums mentally.

Question 1.
302 + 56
Answer:
Using mental math
300 + 2 + 50 + 6
300 + 50 + 8
300 + 58 = 358

Question 2.
463 + 418
Answer:
Using mental math
463 + 418
400 + 60 + 3 + 400 + 10 + 8
800 + 70 + 11
800 + 81 = 881

Question 3.
222 + 725
Answer:
Using mental math
222 + 725
200 + 20 + 2 + 700 + 20 + 5
900 + 40 + 7
900 + 47 = 947

Question 4.
689 + 115
Answer:
Using mental math
689 + 115
600  + 80 + 9 + 100 + 10 + 5
700 + 90 + 14
700 + 104 = 804

Set D
pages 301-304
Think addition to find 400 – 168.
Count on.
168 + 2 = 170
170 + 30 = 200
200 + 200 = 400
2 +30 + 200 = 232
So, 400 – 168 = 232.

Remember that you can count on when subtracting mentally.

Question 1.
523 – 163
Answer:
Using mental math
523 – 163
500 + 20 + 3 – 100 + 60 + 3
400 – 40 = 360

Question 2.
847 – 372
Answer:
Using mental math
847 – 372
800 + 40 + 7 – 300 + 70 + 2
500 – 30 – 5
500 – 25 = 575

Question 3.
768 – 259
Answer:
Using mental math
768 – 259
700 + 60 + 8 – 200 + 50 + 9
500 + 10 – 1
500 + 9 = 509

Question 4.
282 – 125
Answer:
Using mental math
282 – 125
200 + 80 + 2 – 100 + 20 + 5
100 + 60 – 3
100 + 57 = 157

Set E
pages 305-308
You can use a number line to round.

Nearest ten: 437 rounds to 440.

Nearest hundred: 437 rounds to 400.
Think about place value when you round.

Remember that if a number is halfway between, round to the greater number.

Question 1.
Round 374 to the nearest ten and the nearest hundred.
Answer:
374 round to the nearest ten to estimate is 370.
374 round to the nearest hundred to estimate is 400.

Question 2.
Round 848 to the nearest ten and the nearest hundred.
Answer:
842 round to the nearest ten to estimate is 850.
842 round to the nearest hundred to estimate is 800.

Question 3.
Mark’s family traveled 565 miles. Rounded to the nearest ten, about how many miles did they travel?
Answer:
Mark’s family traveled 565 miles
565 round to the nearest ten to estimate is 570.
Mark’s family traveled 570 miles.
Question 4.
Sara collected 345 shells. Rounded to the nearest hundred, about how many shells did she collect?
Answer:

Set F
pages 309-312

Estimate 478 + 112.
Round each addend to the nearest ten.

Round each addend to the nearest hundred.

Use compatible numbers.

Remember that compatible numbers are numbers close to the actual numbers and are easier to add mentally.

Round to the nearest hundred.

Question 1.
367 + 319
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

Compatible numbers for 367 and 319 is 400 and 300.

Question 2.
737 + 127
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally,

The compatible numbers for 737 and 127 is 700 and 100

Round to the nearest ten.

Question 3.
298 + 542
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

Compatible numbers for 298 and 542 is 300 and 540.

Question 4.
459 + 85
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 459 and 85 is 460 and 90

Use compatible numbers.

Question 5.
372 + 173
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 372 and 173 is 370 and 170

Question 6.
208 + 164
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 208 and 164 is 210 and 160

Question 7.
Will rounding to the nearest ten or the nearest hundred give a closer estimate of 314 + 247? Explain your answer.
Answer:
314 + 247 = 561
314 round to the nearest ten to estimate is 310
247 round to the nearest ten to estimate is 250
310  +250 = 560

Set G
pages 313-316
Estimate 486 – 177.
Round each number to the nearest hundred.

Round to the nearest hundred.

Use compatible numbers.

Remember that an estimate is close to the actual answer.

Question 1.
527 – 341
Answer:

Question 2.
872 – 184
Answer:

Round to the nearest ten.

Question 3.
387 – 298
Answer:

Here the digit in the one’s place of the number 387 is 7 now round to the nearest ten which is greater than the number.

387 round to the nearest ten is 390

The digit in the one’s place of the number 298 is 8 now round to the nearest ten which is greater than the number.

287 round to the nearest ten is 290

390 – 290 = 100

Question 4.
659 – 271
Answer:

Here the digit in the one’s place of the number 659 is 9 now round to the nearest ten which is greater than the number.

659 round to the nearest ten is 660

The digit in the one’s place of the number 271 is 1 which is less than the number.

271 is round to the nearest ten to 270

660 – 270 = 390

Use compatible numbers.

Question 5.
472 – 228
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

The compatible numbers for 472 and 228 is 470 and 230

Question 6.
911 – 347
Answer:

Compatible numbers are the pair of numbers that are easy to add, subtract, multiply and division mentally.

Compatible numbers for 911 and 347 is 910 and 350

Question 7.
Will rounding to the nearest ten or the nearest hundred give a closer estimate of 848 – 231? Explain your answer.
Answer:
848 -231 = 617
848 round to the nearest hundred to estimate is 800
231 round to the nearest hundred to estimate is 200
800 – 200 = 600

Set H
pages 317-320
Think about these questions to help you model with math.

Thinking Habits
• How can I use 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?

Remember to apply the math you know to solve problems.
Elena has $265. She buys a jacket that costs $107 and a sweater that costs $69. How much money does Elena have left?

Question 1.
What is the hidden question you need to answer before you can solve the problem?
Answer:

Question 2.
Solve the problem. Draw bar diagrams to represent the problem. Show the equations you used.
Answer:

Toni read 131 pages on Monday, 56 pages on Tuesday, and some pages on Wednesday. She read 289 pages in all. How many pages did Toni read on Wednesday?

Ans:

Toni read pages on Monday = 131

Toni read pages on Tuesday = 56

Toni read pages on Wednesday = 289 – 131 + 56 = 102

Question 1.
What is the hidden question you need to answer before you can solve the problem?
Answer:

Question 2.
Solve the problem. Use equations to represent your work.
Answer:

Topic 8 Assessment Practice

Question 1.
What is the sum of 243, 132, and 157?
Answer:
243 + 132 + 157 = 532

Question 2.
Find a reasonable estimate for the sum of 171 and 69. Select all that apply.
175 + 70 = 245
100 + 60 = 160
170 + 70 = 240
175 + 75 = 250
130 + 70 = 200
Answer:

For the sum of 171 and 69 is we will appy 175 + 70 = 245 in the equation we are using the  compatible numbers

170 + 70 = 240 in this equation we are using the nearest ten.

Question 3.
Subtract 382 – 148 mentally. Which of the following should you do first to find the difference?
A. Add 2 to 148 and add 2 to 382.
B. Add 2 to 148 and subtract 2 from to 382.
C. Subtract 8 from 148 and subtract 2 from 382.
D. Subtract 12 from 382 and add 12 to 148.
Answer:
To find the difference first to subtract 12 from 382 and add 12 to 148.

Question 4.
Estimate the difference of 765 and 333. Explain your estimate.
Answer:
The difference of 765 and 333 is
765 – 333 = 432
Our estimate is to Subtract the lowest value from the highest value.

Question 5.
Look at the sums in the shaded column. Look at the addends. What pattern do you see? Explain why this pattern is always true.

Answer:

Question 6.
Use mental math to add 332 and 154. Which of these shows how to break apart the numbers into hundreds, tens, and ones?
A. Break 332 into 320 + 12. Break 154 into 125 + 29.
B. Break 332 into 100+ 230 + 2. Break 154 into 100 + 52 + 2.
C. Break 332 into 300 + 16 + 16. Break 154 into 100 + 27 + 27.
D. Break 332 into 300 + 30+ 2. Break 154 into 100 + 50 + 4.
Answer:
D is the correct answer .

Question 7.
Use mental math to find 634 – 528.
Answer:
634 – 528
600 + 30 + 4 – 500 + 20 + 8
100 +10 – 4
110 – 4 = 106

Question 8.
Select all the equations that are true.
32 + 56 + 10 = 10 + 56 + 32
(49 + 28) + 5 = 49 + (28 + 5)
56 + 890 = 890 + 56
82 + 0 = 82
45+ 27 = 27 + 35
Answer:

Yes all the equations are true they are in the associative property, commutative property and identity property.

Question 9.
Look at the table.

Write the letters A, B, C, and D above the number line to show each number rounded to the nearest ten.

Answer:

Question 10.
Use the subtraction work shown.

Which strategy shows a way to check the work using inverse operations?
A. Subtract 57 from 169.
B. Add 57 and 112.
C. Add 112 and 169.
D. Add 169 and 57.
Answer:
Inverse means reverse operation
Inverse operation is the Add 169 and 57

Question 11.
Estimate the sum of 405 and 385 by rounding to the nearest hundred. Explain your estimate.
Answer:
The sum of 405 + 385 = 790

When rounding to the nearest hundred look at the ten’s place of the number. The ten’s place of the 790 is 9 now round to the next hundred.
790 round to the nearest hundred to estimate is 800.

Question 12.
What addition equation can be used to check the answer for 456 – 342 = 114? Draw a bar diagram to show how the numbers in this problem are related.
Answer:

Question 13.
Find the sum of 350, 62, and 199.
A. Draw a bar diagram that represents the problem.
B. What is the first step you would do to solve this problem using mental math?
Answer:

The sum of 350 + 62 + 199 = 611

Question 14.
Subtract 341 – 97 mentally. First add 3 to 97 to get 100. What should the next step be? What is the difference?
A. Add 6 to 341. The difference is 247.
B. Add 200 to 100. The difference is 100.
C. Subtract 3 from 341. The difference is 238.
D. Add 3 to 341. The difference is 244.
Answer: The next step is Subtract 3 from 341. The difference is 238.

Question 15.
Round each number on the left to the nearest hundred. Select the appropriate answer.

Answer
668 round to the nearest hundred to estimate is 700.
404 round to the nearest hundred to estimate is 400.
649 round to the nearest hundred to estimate is 600.
489 round to the nearest hundred to estimate is 500.

Question 16.
Explain how to use mental math to find 620 – 278.
Answer:
Using the mental math
620 – 278
600 + 20 – 200 + 70 + 8
600 + 20 – 200 + 78
600 + 20 – 278
620 – 278 = 332

Question 17.
How can you check the answer for 693 – 231 = 462?
A. Subtract 639 – 300 = 339.
B. Subtract 462 – 231 = 231.
C. Add 693 + 231 = 924.
D. Add 462 + 231 = 693.
Answer:
The correct answer is D.
693 – 231 = 462
462 + 231 = 693
Question 18.
Consider the sum of 123, 201, and 387.
A. Estimate the sum by rounding to the nearest ten. Explain your estimate.
B. What is the difference between the exact sum of 123, 201, and 387 and your estimate? Explain your answer.
Answer:
The sum of 123 + 201 + 387 = 711
A. 711 round to the nearest ten to estimate is 710
B. The difference between the exact sum and the estimate sum is 711 – 710 is 1.

Question 19.
Consider the equation 360 = __ + 84.
Use a bar diagram to represent the equation. Then solve for the unknown value.
Answer:
360 = 276 + 84

Topic 8 Performance Task

Vacation Trip
Mia is planning a vacation in Orlando, FL.
The Mia’s Route table shows her route and the miles she will drive.

Use the Mia’s Route table to answer Questions 1-3.

Question 1.
Round each distance to the nearest ten to show about how many miles Mia will drive on each part of her trip.

Memphis, TN, to Birmingham, AL:

Birmingham, AL, to Gainesville, FL:

Gainesville, FL, to Orlando, FL:
Answer:
Mia’s drive from Memphis,TN, to Birmingham, AL is 237 miles.
237 is round to the nearest ten is to estimate is 240.
Mia’s drives from Memphis, TN, to Birmingham, AL is 240 miles.
Mia’s drive from Birmingham, AL to Gainesville, FL is 422
422 round to the nearest ten to estimate is 420
Mia’s drive from Birmingham, AL to Gainesville, FL is 420 miles.
Mia’s drive from Gainesville, FL to Orlando, FL is 183 miles.
183 round to the nearest ten to estimate is 180.
Mia’s drive from Gainesville, FL to Orlando, FL is 180 miles.

Question 2.
Use mental math to find the actual number of miles Mia will drive to reach Gainesville. Show your work.
Answer:
Mia’s started a drive from Memphis, TN to Birmingham, AL is 237 miles.
From Birmingham, AL to Gainesville, FL is 422 miles.
Mia’s drive to reach Gainesville is 237 + 422
By using mental math
237 +422
200 + 30 + 7 + 400 + 20 + 2
600 + 50 + 9
600 + 59 = 659

Question 3.
Mia says, “Birmingham is 185 miles closer to Memphis than to Gainesville.” Her brother says, “No, it is 175 miles closer.” Use mental math to decide who is correct. Show your work.
Answer:
Birmingham  to Memphis is 237 miles.
Birmingham to Gainesville is 422 miles.
Mia’s says the correct
422 – 237
400 + 20 + 2 – 200 + 30 + 7
200 – 10 – 5
200 – 15 = 185

Mia has to book a hotel and buy theme park tickets. The Hotel Prices and Theme Park Prices tables show the total prices for Mia’s stay. The Mia’s Options list shows two plans that Mia can choose from.

Question 4.
Mia has $600 to spend on a hotel and tickets.
Part A

Which option does Mia have enough money for? Explain using estimation.
Part B
Create a new option for Mia. Fill out the table with a hotel and a theme park. Explain why Mia has enough money for this plan.

Question 5.
One theme park has a special offer. For each ticket Mia buys, she gets another ticket free. Shade the squares in the table at the right to show this pattern. Explain why the pattern is true.

Answer:

Go through the enVision Math Common Core Grade 8 Answer Key Topic 7 Understand And Apply The Pythagorean Theorem regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 7 Understand And Apply The Pythagorean Theorem

Topic Essential Question
How can you use the Pythagorean Theorem to solve problems?
Answer:
The Pythagorean Theorem is used to calculate the steepness of slopes of hills or mountains. A surveyor looks through a telescope toward a measuring stick a fixed distance away, so that the telescope’s line of sight and the measuring stick form a right angle.

3-ACT MATH OOO

Go with the Flow
You may have noticed that when you double the base and the height of a triangle, the area is more than doubled. The same is true for doubling the sides of a square or the radius of a circle. So what is the relationship? Think about this during the 3-Act Mathematical Modeling lesson.

Topic 7 enVision STEM Project

Did You Know?
Over two billion people will face water shortages by 2050 according to a 2015 United Nations Environment Program report.

Rainwater can be collected and stored for use in irrigation, industrial uses, flushing toilets, washing clothes and cars, or it can be purified for use as everyday drinking water.
This alternative water source reduces the use of fresh water from reservoirs and wells.

Using water wisely saves money on water and energy bills and extends the life of supply and wastewater facilities.

Roofs of buildings or large tarps are used to collect rainwater.
A rainwater collection system for a building roof that measures 28 feet by 40 feet can provide 700 gallons of water-enough water to support two people for a year—from a rainfall of 1.0 inch.

Even a 5 foot by 7-foot tarp can collect 2 gallons of water from a rainfall total of only 0.1 in.

The rainwater harvesting market is expected to grow 5% from 2016 to 2020.

Your Task: Rainy Days

Rainwater collection is an inexpensive way to save water in areas where it is scarce. One inch of rain falling on a square roof with an area of 100 ft² collects 62 gallons of water that weighs over 500 pounds. You and your classmates will research the necessary components of a rainwater collection system. Then you will use what you know about right triangles to design a slanted roof system that will be used to collect rainwater.
Answer:
It is given that
Rainwater collection is an inexpensive way to save water in areas where it is scarce. One inch of rain falling on a square roof with an area of 100 ft² collects 62 gallons of water that weighs over 500 pounds
Now,
The necessary components of a rainwater collection system are:
A) Catchments B) Coarse mesh C) Gutters D) Conduits E) First-flushing F) Filter G) Storage facility H) Recharge Structures

Topic 7 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
cube root
diagonal
isosceles triangle
perimeter
right triangle
square root

Question 1.
The __________ of a number is a factor that when multiplied by itself gives the number.
Answer:
We know that,
The “Square root” of a number is a factor that when multiplied by itself gives the number
Hence, from the above,
We can conclude that the best term to complete the given definition is a “Square root”

Question 2.
A _________ is a line segment that connects two vertices of a polygon and is not the side.
Answer:
We know that,
A “Diagonal” is a line segment that connects two vertices of a polygon and is not the side
Hence, from the above,
We can conclude that the best term to complete the given definition is a “Diagonal”

Question 3.
The _________ of a figure is the distance around it.
Answer:
We know that,
The “Perimeter” of a figure is the distance around it
Hence, from the above,
We can conclude that the best term to complete the given definition is the “Perimeter”

Question 4.
A ___________ is a triangle with one right angle.
Answer:
We know that,
A “Right triangle” is a triangle with one right angle
Hence, from the above,
We can conclude that the best term to complete the given definition is a “Right angle”

Simplify Expressions with Exponents

Simplify the expression.
Question 5.
3² + 4²
Answer:
The given expression is: 3² + 4²
So,
3² + 4²
= (3 × 3) + (4 × 4)
= 9 + 16
= 25

Question 6.
2² + 5²
Answer:
The given expression is: 2² + 5²
So,
2² + 5²
= (2 × 2) + (5 × 5)
= 4 + 25
= 29

Question 7.
10² – 8²
Answer:
The given expression is: 10² – 8²
So,
10² – 8²
= (10 × 10) – (8 × 8)
= 100 – 64
= 36

Square Roots

Determine the square root.
Question 8.
\(\sqrt {81}\)
Answer:
The given expression is: \(\sqrt{81}\)
Hence,
\(\sqrt{81}\) = 9

Question 9.
\(\sqrt {144}\)
Answer:
The given expression is: \(\sqrt{144}\)
Hence,
\(\sqrt{144}\) = 12

Question 10.
\(\sqrt {225}\)
Answer:
The given expression is: \(\sqrt{225}\)
Hence,
\(\sqrt{225}\) = 15

Distance on a Coordinate Plane

Determine the distance between the two points.
Question 11.

Answer:
The given graph is:

From the given graph,
The given points are: (2, 5), (7, 5)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
We know that,
Distance between 2 points = √(x₂ – x₁)² + (y₂ – y₁)²
= √(7 – 2)² + (5 – 5)²
= \(\sqrt{5²}\)
= 5 units
Hence, from the above,
We can conclude that the distance between the given points is: 5 units

Question 12.

Answer:
The given graph is:

From the given graph,
The given points are: (3, 2), (3, 9)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
We know that,
Distance between 2 points =√(x₂ – x₁)² + (y₂ – y₁)²
= √(3 – 3)² + (9 – 2)²
= \(\sqrt{7²}\)
= 7 units
Hence, from the above,
We can conclude that the distance between the given points is: 7 units

Language Development

Complete the word map using key terms, examples, or illustrations related to the Pythagorean Theorem and its Converse.


Answer:

Topic 7 PICK A PROJECT

PROJECT 7A
Where would you like to bike ride in your neighborhood?
PROJECT: PLAN A METRIC CENTURY RIDE

PROJECT 7B
What designs have you seen on kites?
PROJECT: BUILD A KITE

PROJECT 7C
What buildings in your community have unusual shapes as part of their structure or design?
PROJECT: MAKE A SCRAPBOOK

PROJECT 7D
What geometric designs have you noticed on your clothes?
PROJECT: DESIGN A FABRIC TEMPLATE

3-ACT MATH

3-Act Mathematical Modeling: Go with the Flow

АСТ 1
Question 1.
After watching the video, what is the first question that comes to mind?
Answer:

Question 2.
Write the Main Question you will answer.
Answer:

Question 3.
Make a prediction to answer this Main Question.
_________ % will fit in the third square.
Answer:

Question 4.
Construct Arguments Explain how you arrived at your prediction.

Answer:

ACT 2
Question 5.
What information in this situation would be helpful to know? How would you use that information?
Answer:

Question 6.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 7.
Model With Math
Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 8.
What is your answer to the Main Question? Does it differ from your prediction? Explain.

Answer:

АСТ 3
Question 9.
Write the answer you saw in the video.
Answer:

Question 10.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 11.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

Act 3
Extension
Reflect
Question 12.
Model with Math
Explain how you used a mathematical model to represent the situation. How did the model help you answer the Main Question?

Answer:

Question 13.
Reason Abstractly How did you represent the situation using symbols? How did you use those symbols to solve the problem?

Answer:

SEQUEL
Question 14.
Construct Arguments Explain why you can use an area formula when the problem involves comparing volumes.

Answer:

Lesson 7.1 Understand the Pythagorean Theorem

Explain It!
Kelly drew a right triangle on graph paper. Kelly says that the sum of the areas of squares with side lengths a and b is the same as the area of a square with side length c.

I can… use the Pythagorean Theorem to find unknown sides of triangles.

A. Do you agree with Kelly? Explain.
Answer:
It is given that
Kelly drew a right triangle on graph paper. Kelly says that the sum of the areas of squares with side lengths a and b is the same as the area of a square with side length c.
We know that,
According to the Pythagorean theorem,
Hypotenuse² = Side length 1² + Side length 2²
So,
From the given graph,
We can observe that
Side length 1 is: a
Side length 2 is: b
The hypotenuse is: c
So,
c² = a² + b²
Hence, from the above,
We can conclude that we can agree with Kelly

B. Sam drew a different right triangle with side lengths a = 5, b = 12, and c = 13. Is the relationship Kelly described true for Sam’s right triangle? Explain.
Answer:
It is given that
Sam drew a different right triangle with side lengths a = 5, b = 12, and c = 13
Now,
From part (a),
The relation according to Kelly is:
c² = a² + b²
So,
13² = 12² + 5²
169 = 144 + 25
169 = 169
Hence, from the above,
We can conclude that the relationship Kelly described is true for Sam’s right-angled triangle

Focus on math practices
Generalize Kelly draws another right triangle. What would you expect to be the relationship between the areas of the squares drawn on each side of the triangle? Explain.
Answer:
It is given that Kelly draws another right triangle
Hence,
If in a triangle, the square on one of the sides equals the sum of the squares on the remaining two sides of the triangle, then the angle contained by the remaining two sides of the triangle is right.”

Essential Question
How does the Pythagorean Theorem relate to the side lengths of a right triangle?
Answer:
The Pythagorean equation relates the sides of a right triangle in a simple way so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides but less than their sum.

Try It!

A right triangle has side lengths 15 centimeters, 25 centimeters, and 20 centimeters. How can you use the Pythagorean Theorem to write an equation that describes how the side lengths are related?
a² + b² = c²
_______² + ________² = _________²
_________ + _________ = ________
Answer:
It is given that
A right triangle has side lengths 15 centimeters, 25 centimeters, and 20 centimeters.
We know that,
According to the Pythagorean Theorem,
The hypotenuse has the greatest length in the right triangle
Now,
Let the hypotenuse be c
Let the other two sides be a and b
So,
From the given information,
c = 25 centimeters, a = 15 centimeters,and b = 20 centimeters
So,
According to the Pythagorean Theorem,
25² = 15² + 20²
625 = 225 + 400
625 = 625
Hence, from the above,
We can conclude that we proved how the Pythagorean Theorem relates to the lengths of the right triangle

Convince Me!
How do you know that the geometric proof of the Pythagorean Theorem shown above can be applied to all right triangles?
Answer:
It can be proven using the law of cosines or as follows: Let ABC be a triangle with side lengths a, b, and c, with a² + b² = c².  Therefore, the angle between the side of lengths a and b in the original triangle is a right angle. This proof of the converse makes use of the Pythagorean theorem itself.

Try It!

A right triangle has a hypotenuse length of 32 meters. It has one leg with a length of 18 meters. What is the length of the other leg? Express your answer as a square root.
Answer:
It is given that
A right triangle has a hypotenuse length of 32 meters. It has one leg with a length of 18 meters.
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the legs
Now,
Let the other leg be x
So,
32² = 18² + x²
x² = 32² – 18²
x² = 1024 – 324
x² = 700
x = \(\sqrt{700}\)
Hence, from the above,
We can conclude that the length of the other leg is: \(\sqrt{700}\)

KEY CONCEPT

The Pythagorean Theorem is an equation that relates the side lengths of a right triangle, a² + b² = c², where a and b are the legs of a right triangle and c is the hypotenuse.

Do You Understand?
Question 1.
Essential Question How does the Pythagorean Theorem relate to the side lengths of a right triangle?
Answer:
The Pythagorean equation relates the sides of a right triangle in a simple way so that if the lengths of any two sides are known the length of the third side can be found. Another corollary of the theorem is that in any right triangle, the hypotenuse is greater than any one of the other sides but less than their sum.

Question 2.
Use Structure A side of each of the three squares forms a side of a right triangle.
Would any three squares form the sides of a right triangle? Explain.

Answer:
It is given that
A side of each of the three squares forms a side of a right triangle.
Now,
We know that,
The length of all the sides in a square is equal
Now,
From the given figure,
We can observe that
Each side of a square from the three squares form a right triangle
Hence, from the above,
We can conclude that the three squares form the sides of a right triangle

Question 3.
Construct Arguments Xavier said the missing length is about 18.5 units. Without calculating, how can you tell that Xavier solved incorrectly?

Answer:
It is given that
Xavier said the missing length is about 18.5 units
Now,
We know that,
According to the Pythagorean Theorem,
The length of the hypotenuse is the greatest
Now,
The given right triangle is:

So,
According to the Pythagorean Theorem,
The length of the missing side should be greater than 21 and 26
Hence, from the above,
We can conclude that Xavier calculated incorrectly

Do You Know How?
Question 4.
A right triangle has leg lengths of 4 inches and 5 inches. What is the length of the hypotenuse? Write the answer as a square root and round to the nearest tenth of an inch.
Answer:
It is given that
A right triangle has leg lengths of 4 inches and 5 inches
Now,
We know that,
According to the Pythagorean theorem,
c² = a²+ b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
c² = 4² + 5²
c² = 16 + 25
c² = 41
c = \(\sqrt{41}\)
c = 6.4 inches
Hence, from the above,
We can conclude that the length of the hypotenuse is: 6.4 inches

Question 5.
Find the missing side length to the nearest tenth of afoot.

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean theorem,
c² = a²+ b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
14² = 8² + b²
b²= 14² – 8²
b² = 196 – 64
b² = 132
b = \(\sqrt{132}\)
b = 11.5 feet
Hence, from the above,
We can conclude that the length of the missing side is: 11.5 feet

Question 6.
Find the missing side length to the nearest tenth of a millimeter.

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean theorem,
c² = a²+ b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
c² = (3.7)² + (7.5)²
c² = 13.69 + 56.25
c² = 69.94
c = \(\sqrt{69.94}\)
c = 8.4 mm
Hence, from the above,
We can conclude that the length of the missing side is: 8.4 mm

Practice & Problem Solving

Leveled Practice In 7 and 8, find the missing side length of each triangle.
Question 7.

The length of the hypotenuse is ________ units.
Answer:

Question 8.

The length of leg b is about ________ inches.
Answer:

Question 9.
What is the length of the hypotenuse of the triangle when x = 15? Round your answer to the nearest tenth of a unit.

Answer:
The given right angle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
c² = (3x)² + (4x + 4)²
Now,
When x = 15,
c² = (3 × 15)² + (4 × 15 + 4)²
c² = 45² + 64²
c² = 2,025 + 4,096
c² = 6,121
c = \(\sqrt{6,121}\)
c = 78.2 units
Hence, from the above,
We can conclude that the length of the hypotenuse when x= 15 is: 78.2 units

Question 10.
What is the length of the missing side rounded to the nearest tenth of a centimeter?

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a²+ b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
a² = (12.9)² + (15.3)²
a² = 166.41 + 234.09
a² = 400.5
a = \(\sqrt{400.5}\)
a = 20 cm
Hence, from the above,
We can conclude that the length of the side a is: 20 cm

Question 11.
Use the Pythagorean Theorem to find the unknown side length of the right triangle.

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a²+ b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
c² = (10)² + (24)²
c² = 100 + 576
c² = 676
c = \(\sqrt{676}\)
c = 26 m
Hence, from the above,
We can conclude that the length of the side a is: 26 m

Question 12.
What is the length of the unknown leg of the right triangle rounded to the nearest tenth of afoot?

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
9² = 2² + b²
b² = 9² – 2²
b² = 81 – 4
b² = 77
b = \(\sqrt{77}\)
b = 8.8 foot
Hence, from the above,
We can conclude that the length of the unknown leg is: 8.8 foot

Question 13.
A student is asked to find the length of the hypotenuse of a right triangle. The length of one leg is 32 centimeters, and the length of the other leg is 26 centimeters. The student incorrectly says that the length of the hypotenuse is 7.6 centimeters.
a. Find the length of the hypotenuse of the right triangle to the nearest tenth of a centimeter.
Answer:
It is given that
A student is asked to find the length of the hypotenuse of a right triangle. The length of one leg is 32 centimeters, and the length of the other leg is 26 centimeters. The student incorrectly says that the length of the hypotenuse is 7.6 centimeters.
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
c² = 32² + 26²
c² = 1,024 + 676
c² = 1,700
c = \(\sqrt{1,700}\)
c = 41.2 centimeters
Hence, from the above,
We can conclude that the length of the hypotenuse of a right triangle is: 41.2 centimeters

b. What mistake might the student have made?
Answer:
It is given that
The student incorrectly says that the length of the hypotenuse is 7.6 centimeters.
But,
From part (a),
The length of the hypotenuse is: 41.2 centimeters
Hence, from the above,
We can conclude that the mistake the student might make is the misinterpretation of the length of the hypotenuse

Question 14.
Find the length of the unknown leg of the right triangle.

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
(37.25)² = (12.25)² + b²
b² = (37.25)² – (12.25)²
b² = 1,387.56 – 150.06
b² = 1,237.5
b = \(\sqrt{1,237.5}\)
b = 35.17 units
Hence, from the above,
We can conclude that the length of the unknown leg is: 35.17 units

Question 15.
Higher-Order Thinking A right triangle has side lengths 12 centimeters and 14 centimeters. Name two possible side lengths for the third side, and explain how you solved for each.
Answer:
It is given that
A right triangle has side lengths of
12 centimeters and 14 centimeters.
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
Now,
Let the length of the third side be x
So,
The possible lengths of the third side may be:
x < 12 centimeters and x > 14 centimeters
Hence, from the above,
We can conclude that the two possible side lengths for the third side are:
x < 12 centimeters and x > 14 centimeters

Assessment Practice
Question 16.
Which right triangle has a hypotenuse that is about 39 feet long?

Answer:
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
For Options A and D:
A) c² = 30² + 15²                     D) c² = 30² + 14²
= 33.54 ft                                 = 33.10 ft
For Options B and C:
B) c² = 36² + 12²                      C) c² = 36² + 15²
= 37.94 ft                                    = 39 ft
Hence, from the above,
We can conclude that Option C matches the given situation

Question 17.
Which right triangle does NOT have an unknown leg length of about 33 cm?

Answer:
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the hypotenuse
a and b are the lengths of the legs
So,
For Options A and D:
A) b² = 35² – 11²                     D) b² = 34² – 9²
= 33.22 cm                              = 32.78 cm
For Options B and C:
B) b² = 30² – 10²                      C) b² = 35² – 12²
= 28.28 cm                              = 32.87 cm
Hence, from the above,
We can conclude that Option B matches the given situation

Lesson 7.2 Understand the Converse of the Pythagorean Theorem

Solve & Discuss It!
Kayla has some straws that she will use for an art project. She wants to glue three of the straws onto a sheet of paper, without overlapping, to make the outline of a right triangle. Which three straws could Kayla use to make a right triangle? Explain.

I can… use the Converse of the Pythagorean Theorem to identify right triangles.
Answer:
It is given that
Kayla has some straws that she will use for an art project. She wants to glue three of the straws onto a sheet of paper, without overlapping, to make the outline of a right triangle.
Now,
We know that,
The converse of the Pythagorean Theorem states that if the square of the third side of a triangle is equivalent to the sum of its two shorter sides, then it must be a right triangle i.e.,
If c² = a²+ b², then the given triangle is a right triangle
So,
From the given straws,
we can observe that
The straws numbered 3, 4, and 5 can be glued to make the outline of a right triangle
The straws numbered 5, 12, and 13 can be glued to make the outline of a right triangle
Hence, from the above,
We can conclude that there are 2 pairs of straws i.e., (3, 4, 5) and (12, 5, 13) to make the outline of a right triangle

Look for Relationships
How could you use the Pythagorean Theorem to determine whether the lengths form a right triangle?
Answer:
According to the converse of the Pythagorean Theorem,
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Focus on math practices
Use Structure Could Kayla use the straws that form a right triangle to make a triangle that is not a right triangle? Explain.
Answer:
From the given straws,
We can observe that the pair (6, 7, 4) can’t form a right triangle but (3, 4, 5) can form a right triangle
Hence, from the above,
We can conclude that Kayla can use the straws that form a right triangle to make a triangle that is not a right triangle

Essential Question
How can you determine whether a triangle is a right triangle?
Answer:
We can determine the triangle is a right triangle by using the converse of the Pythagorean Theorem
Hence,
According to the converse of the Pythagorean Theorem,
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Try It!

A triangle has side lengths 4 inches, 5 inches, and 7 inches. Is the triangle a right triangle?

Is a² + b² equal to c²? _______
Is the triangle a right triangle? _________
Answer:
It is given that
A triangle has side lengths 4 inches, 5 inches, and 7 inches.
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
So,
c² = a² + b²
Where,
c is the hypotenuse that has the longest side length
a and b are the lengths of the legs
So,
7² = 4² + 5²
49 = 16 + 25
49 = 41
So,
49 ≠ 41
Hence, from the above,
We can conclude that
c² ≠ a²+ b²
The given triangle is not a right triangle

Convince Me!
Explain the proof of the Converse of the Pythagorean Theorem in your own words.
Answer:
The converse of the Pythagorean Theorem is:
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Try It!

A triangle has side lengths 10 feet, \(\sqrt {205}\) feet, and \(\sqrt {105}\) feet. Is this a right triangle? Explain.
Answer:
It is given that
A triangle has side lengths 10 feet, \(\sqrt {205}\) feet, and \(\sqrt {105}\) feet.
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b², then
The given triangle is a right triangle
We know that,
c is the length of the hypotenuse that has the longest side in a right triangle
So,
(\(\sqrt{205}\))² =(\(\sqrt{105}\))² + 10²
205 = 105 + 100
205 = 205
So,
c² = a² + b²
Hence, from the above,
We can conclude that the triangle with the given side lengths is a right triangle

Try It!

A triangle is inside a trapezoid. Is the triangle a right triangle? Explain.

Answer:
It is given that a triangle is inside a trapezoid
Now,
The given trapezoid is:

From the given trapezoid,
The sides of the triangle are: 17 in., 15 in., \(\sqrt{514}\) in.
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then, the given triangle is a right triangle
So,
(\(\sqrt{514}\))² = 17² + 15²
514 = 289 + 225
514 = 514
So,
c² = a² + b²
Hence, from the above,
We can conclude that the triangle that is in the trapezoid is a right triangle

KEY CONCEPT

The Converse of the Pythagorean Theorem states that if the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, the triangle is a right triangle.

Do You Understand?
Question 1.
Essential Question How can you determine whether a triangle is a right triangle?
Answer:
We can determine the triangle is a right triangle by using the converse of the Pythagorean Theorem
Hence,
According to the converse of the Pythagorean Theorem,
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

Question 2.
Construct Arguments A triangle has side lengths of 3 centimeters, 5 centimeters, and 4 centimeters. Abe used the Converse of the Pythagorean Theorem to determine whether it is a right triangle.
3² + 5² \(\underline{\underline{?}}\) 4²
9 + 25 \(\underline{\underline{?}}\) 16
34 ≠ 16
Abe concluded that it is not a right triangle. Is Abe correct? Explain.
Answer:
It is given that
A triangle has side lengths of 3 centimeters, 5 centimeters, and 4 centimeters
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then, the triangle is  a right triangle
We know that,
The hypotenuse has the length of the longest side
So,
5²= 3² + 4²
25 = 9 + 16
25 = 25
So,
c² = a² + b²
Hence, from the above,
We can conclude that the given triangle is a right triangle and Abe is not correct

Question 3.
Use Structure When you are given three side lengths for a triangle, how do you know which length to substitute for a, b, or c in the Pythagorean Theorem?
Answer:
First, we will find the squares of all lengths, then we will check which two squares of sides are equal to the square of the third side as per Pythagoras theorem. Hence the two sides would according be a and b and the third side will become c

Do You Know How?
Question 4.
Is the triangle a right triangle? Explain.

Answer:
The given triangle is:

We know that,
According to the converse of the Pythagorean Theorem,
If c² = a² + b²,
then, the triangle is a right triangle
So,
10² = 6² + 8²
100 = 36 + 64
100 = 100
So,
c² = a² + b²
Hence, from the above,
We can conclude that the given triangle is a right triangle

Question 5.
Is the triangle a right triangle? Explain.

Answer:
The given triangle is:

We know that,
According to the converse of the Pythagorean Theorem,
If c² = a² + b²,
then, the triangle is a right triangle
So,
8² = (\(\sqrt{26}\))² + (\(\sqrt{28}\))²
64 = 26 + 28
64 = 54
So,
c² ≠ a² + b²
Hence, from the above,
We can conclude that the given triangle is not a right triangle

Question 6.
Is the purple triangle a right triangle? Explain.

Answer:
The given triangle is:

We know that,
According to the converse of the Pythagorean Theorem,
If c² = a² + b²,
then, the triangle is a right triangle
So,
(20.8)² = 14² + (15.5)²
432.64 = 196 + 240.25
432.64 = 436.25
So,
c² ≠ a² + b²
Hence, from the above,
We can conclude that the purple triangle is not a right triangle

Practice & Problem Solving

Leveled Practice In 7 and 8, determine whether each triangle is a right triangle.
Question 7.

Is the triangle a right triangle? ________
Answer:

Hence, from the above,
We can conclude that the given triangle is not a right triangle

Question 8.

Is the triangle a right triangle? _________
Answer:

Hence, from the above,
We can conclude that the given triangle is a right triangle

Question 9.
Can the sides of a right triangle have lengths 5, 15, and \(\sqrt {250}\)? Explain.
Answer:
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then, the triangle is a right triangle
The hypotenuse has the longest side length
So,
(\(\sqrt{250}\))² = 5² + 15²
250 = 25 + 225
250 = 250
Hence, from the above,
We can conclude that the sides of a right triangle have lengths 5, 15, and \(\sqrt {250}\)

Question 10.
Is ∆PQR a right triangle? Explain.

Answer:
The given triangle is:

We know that,
According to the converse of the Pythagorean Theorem,
If c² = a² + b²,
then, the triangle is a right triangle
So,
(6.25)² = (3.75)² + 5²
39.0625 = 14.0625 + 25
39.0625 = 39.0625
So,
c² = a² + b²
Hence, from the above,
We can conclude that ΔPQR is a right triangle

Question 11.
The green triangle is set inside a rectangle. Is the green triangle a right triangle? Explain.

Answer:
It is given that
The green triangle is set inside a rectangle
Now,
The given figure is:

We know that,
According to the converse of the Pythagorean Theorem,
If c² = a² + b²,
then, the triangle is a right triangle
So,
21² = (\(\sqrt{282}\))² + (\(\sqrt{159}\))²
441 = 282 + 159
441 = 441
So,
c² = a² + b²
Hence, from the above,
We can conclude that the given green triangle is a right triangle

Question 12.
The side lengths of three triangles are shown. Which of the triangles are right triangles?

Answer:
It is given that
The side lengths of the three triangles are shown in the table
Now,
The given table is:

Now,
We know that,
According to the converse of the Pythagorean Theorem,
If c² = a² + b²,
then, the triangle is a right triangle
So
For Triangle 1,
(\(\frac{5}{7}\))² = (\(\frac{3}{7}\))² + (\(\frac{4}{7}\))²
25 = 16 + 9
25 = 25
For Triangle 2,
15² = 8² + 8²
225 = 64 + 64
225 = 128
For Triangle 3,
(\(\frac{13}{17}\))² = (\(\frac{12}{17}\))² + (\(\frac{5}{17}\))²
169 = 144 + 25
169 = 169
So,
The condition
c² = a² + b²
is true for the Triangles 1 and 3
Hence, from the above,
We can conclude that Triangle 1 and Triangle 3 is a right triangle

Question 13.
Construct Arguments Three students draw triangles with the side lengths shown. All three say that their triangle is a right triangle. Which students are incorrect? What mistake might they have made?
Student 1: 22, 33, 55
Student 2: 44, 33, 77
Student 3: 33, 44, 55
Answer:
It is given that
Three students draw triangles with the side lengths shown. All three say that their triangle is a right triangle.
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then, the triangle is a right triangle
Now,
For student 1,
55² = 33² + 22²
3,025 = 1,089 + 484
3,025 = 1,573
For student 2,
77²= 44² + 33²
5,929 = 1,936 + 1,089
5,929 = 3,025
For student 3,
55² = 44² + 33²
3,025 = 1,936 + 1,089
3,025 = 3,025
So,
The condition
c² = a² + b²
is false for the side lengths of the triangles that are drawn by students 1 and 2
Hence, from the above,
We can conclude that student 1 and student 2 are incorrect

Question 14.
Model with Math
∆JKL is an isosceles triangle. Is KM the height of ∆JKL? Explain.

Answer:
It is given that
ΔJKL is an isosceles triangle
Now,
To find whether KM is the height of ΔJKL,
Find out whether ΔKLM is a right triangle or not
Now,
The given triangle is:

Now,
We know that,
According to the converse of the Pythagorean Theorem,
c² = a²+ b²
So,
From ΔKLM,
(\(\sqrt{340}\))² = 13² + 14²
340 = 169 + 196
340 = 365
So,
c² ≠ a² + b²
So,
ΔKLM is not a right triangle
Hence, from the above,
We can conclude that KM is not the height of ΔJKL

Question 15.
Higher-Order Thinking The side lengths of three triangles are given.
Triangle 1: \(\sqrt{229}\) units, \(\sqrt{225}\) units, 22 units
Triangle 2: \(\sqrt{11 \frac{1}{3}}\) units, \(\sqrt{13 \frac{2}{3}}\) units, 5 units
Triangle 3: 16 units, 17 units, \(\sqrt{545}\) units
a. Which lengths represent the side lengths of a right triangle?
Answer:
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²
then, the triangle is a right triangle
Now,
For Triangle 1,
(\(\sqrt{229}\))² = (\(\sqrt{225}\))² + 22²
229 = 225 + 484
229 = 709
For Triangle 2,
(\(\sqrt{\frac{41}{3}}\))² + (\(\sqrt{\frac{34}{3}}\))² = 5²
\(\frac{41}{3}\) + \(\frac{34}{3}\) = 25
25 = 25
For Triangle 3,
(\(\sqrt{545}\))² = 16² + 17²
545 = 256 + 289
545 = 545
So,
The condition
c² = a²+ b²
is true for triangle 2 and triangle 3
Hence, from the above,
We can conclude that Triangle 2 and Triangle 3 represent the side lengths of a triangle

b. For any triangles that are not right triangles, use two of the sides to make a right triangle.
Answer:

Assessment Practice
Question 16.
Which shaded triangle is a right triangle? Explain.

Answer:
The given figure is:

Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a²+ b²,
then the triangle is a right triangle
Now,
For ΔABC,
144² = 63² + (\(\sqrt{9}\))²
20,736 = 3,969 + 9
20,736 = 3,978
For ΔXYZ,
(\(\sqrt{144}\))² = (\(\sqrt{63}\))² + 9²
144 = 63 + 81
144 = 144
So,
The condition c² = a²+ b² is true for the shaded triangle XYZ
Hence, from the above,
We can conclude that the shaded triangle ΔXYZ is a right triangle

Question 17.
Which triangle is a right triangle?

A. Triangle I only
B. Triangle II only
C. Triangle I and Triangle II
D. Neither Triangle I nor Triangle II
Answer:
The given triangles are:

Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then the triangle is a right triangle
Now,
For Triangle I,
52² = 40² + 48²
2,704 = 1,600 + 2,304
2,704 = 3,904
For Triangle II,
65² = 60² + 25²
4,225 = 3,600 + 625
4,225 = 4,225
Hence, from the above,
We can conclude that Triangle II only is a right triangle

Topic 7 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary How are the hypotenuse and the legs of a right triangle related? Lesson 7-1
Answer:
The relation between the sides and angles of a right triangle is the basis for trigonometry. The side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called legs

Question 2.
Given that ∆QPR has side lengths of 12.5 centimeters, 30 centimeters, and 32.5 centimeters, proves ∆QPR is a right triangle. Lesson 7-2
Answer:
It is given that
∆QPR has side lengths of 12.5 centimeters, 30 centimeters, and 32.5 centimeters
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then, the triangle is a right triangle
So,
(32.5)² = (12.5)² + 30²
1,056.25 = 156.25 + 900
1,056.25 = 1,056.25
So,
The condition
c² = a² + b²
is true for the given side lengths of a triangle
Hence, from the above,
We can conclude that ΔQPR is a right triangle

Question 3.
Ella said that if she knows the lengths of just two sides of any triangle, then she can find the length of the third side by using the Pythagorean Theorem. Is Ella correct? Explain. Lesson 7-1
Answer:
Ella said that if she knows the lengths of just two sides of any triangle, then she can find the length of the third side by using the Pythagorean Theorem.
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the legs
Now,
If we know a and b, then we can find c
If we know b and c, then we can find a
If we know a and c, then we can find b
Hence, from the above,
We can conclude that Ella is correct

Question 4.
Find the unknown side length. Round to the nearest tenth. Lesson 7-1

Answer:
The given triangle is:

From the given triangle,
We can observe that it is a right triangle
So,
Now,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the legs
Now,
8² = 4² + b²
64 = 16 + b²
b² = 64 – 16
b² = 48
b = \(\sqrt{48}\)
b = 6.9 cm
Hence, from the above,
We can conclude that the length of the unknown side is: 6.9 cm

Question 5.
The height of a shed is 6 m. A ladder leans against the shed with its base 4.5 m away, and its top just reaching the roof. What is the length of the ladder? Lesson 7-1

Answer:
It is given that
The height of a shed is 6 m. A ladder leans against the shed with its base 4.5 m away, and its top just reaching the roof.
Now,
From the given figure,
We can observe that the ladder looks like the hypotenuse of a right triangle for the shed
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the legs
In this situation,
c is the length of the ladder
a is the height of the shed
b is the length of the base
So,
c² = 6² + (4.5)²
c² = 36 + 20.25
c² = 56.25
c = \(\sqrt{56.25}\)
c = 7.5 m
Hence, from the above,
We can conclude that the length of the ladder is: 7.5 m

Question 6.
Select all the sets of lengths that could represent the sides of a right triangle. Lesson 7-2
☐ 5 cm, 10 cm, 15 cm
☐ 7 in., 14 in., 25 in.
☐ 13 m, 84 m, 85 m
☐ 5 ft, 11 ft, 12 ft
☐ 6ft, 9 ft, \(\sqrt {117}\) ft
Answer:
Let the given options be named as A, B, C, D, and E
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse and has the longest length
a and b are the legs
So,
A)
15² = 5² + 10²
225 = 25 + 100
225 ≠ 125
B)
25² = 7² + 14²
625 = 49 + 196
625 ≠ 245
C)
85² = 84² + 13²
7,225 = 7,056 + 169
7,225 = 7,225
D)
12² = 11² + 5²
144 = 121 + 25
144 ≠ 146
E)
(\(\sqrt{117}\))² = 6² + 9²
117 = 36 + 81
17 = 117
Hence, from the above,
We can conclude that the side lengths present in options C and E represent the side lengths of a right triangle

Topic 7 MID-TOPIC PERFORMANCE TASK

Javier is standing near a palm tree. He holds an electronic tape measure near his eyes and finds the three distances shown.

PART A
Javier says that he can now use the Pythagorean Theorem to find the height of the tree. Explain. Use vocabulary terms in your explanation.
Answer:
It is given that
Javier says that he can now use the Pythagorean Theorem to find the height of the tree
Now,
According to the Pythagorean Theorem,
In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named Perpendicular (The height of the tree), Base, and Hypotenuse.

PART B
Find the height of the tree. Round to the nearest tenth. Show your work.

Answer:
From the given figure,
We can observe that
To find the height of the tree, we have to find the perpendicular distances of the two right triangles
Now,
Let x be the perpendicular height of the first right triangle
Let y be the perpendicular height of the second right triangle
So,
The height of the tree = x + y
Now,
The side lengths of the first right triangle are: 25 ft, 7 ft, x ft
Now,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the side lengths
So,
25² = 7² + x²
x² = 625 – 49
x² = 576
x = \(\sqrt{576}\)
x = 24 ft
Now,
The side lengths of the second right triangle are: 9 ft, 7ft, y ft
So,
9² = 7² + y²
y² = 81 – 49
y² = 32
y = \(\sqrt{32}\)
y = 5.6 ft
So,
The height of the tree = 24 + 5.6
= 29.6 ft
Hence, from the above,
We can conclude that the height of the tree to the nearest tenth is: 29.6 ft

PART C
Javier moves backward so that his horizontal distance from the palm tree is 3 feet greater. Will the distance from his eyes to the top of the tree also be 3 feet greater? Explain.
Answer:
Yes it will be greater, he is moving back 3 feet so what you are doing is taking the leg (a²) and multiplying it by 3. Once you do, you see triangle 1 has double and so did triangle 2. Triangle 2 was originally 5.6 (rounded to 6) then went up to 26.1. Triangle 1 was originally 24 and went up to 74.7 (or 75)
Step-by-step explanation:
For Triangle 1 (when multiplied by 3):
We know that,
a² + b² = c²
7² + b² = 27²
49 + b² = 729
b² = 680
b  = \(\sqrt{680}\)
b = 26.1 ft
For Triangle 2:
We know that,
a² + b² = c²
7² + b² =75²
b² = 5576
b = \(\sqrt{5,576}\)
b = 74.7 ft

PART D
Could Javier change his horizontal distance from the tree so that the distance from his eyes to the top of the tree is only 20 feet? Explain.
Answer:
Yes, Javier can change his horizontal distance from the tree so that the distance from his eyes to the top of the tree is only 20 feet by moving forward 5 ft

Lesson 7.3 Apply the Pythagorean Theorem to Solve Problems

Solve & Discuss It!
Carlos is giving his friend in another state a new umbrella as a gift. He wants to ship the umbrella in a box he already has. Which box can Carlos use to ship the umbrella? Explain.

I can… use the Pythagorean Theorem to solve problems.
Answer:
It is given that
Carlos is giving his friend in another state a new umbrella as a gift. He wants to ship the umbrella in a box he already has.
Now,
From the given figure,
We can observe that the umbrella is 37.5 inches long.
Now,
If you observe, each box doesn’t have 37 inches wide, however, they are tridimensional figures, which means they have a certain height.
So,
The smaller box is too small to fit the umbrella.
But,
The medium box is perfect because it has 27 inches wide and 27 inches in height, which is enough to fit the umbrella. These are the best dimensions to send the umbrella.
Hence, from the above,
We can conclude that the box that can be used by Carlos to ship the umbrella has the dimensions 27 in × 14 in × 27 in

Make Sense and Persevere
How will the umbrella fit inside any of the boxes?
Answer:
We know that,
The box is a 3-d figure
So,
To fit the whole umbrella in the box, we have to put it in a diagonal manner i.e., like the hypotenuse of a right triangle

Focus on math practices
Construct Arguments Tim says that the diagonal of any of the boxes will always be longer than the sides. Is Tim correct? Explain.
Answer:
We know that,
If we consider a square or any 2-d figure or any 3-d figure that consists of 1 right angle,
then, the diagonal will divide a figure into 2 right triangles
We know that,
We can apply the Pythagorean Theorem for any right triangle
We know that,
In a right triangle,
The hypotenuse is the longest side
We know that,
The hypotenuse in a right triangle is considered as a diagonal in a figure that consists of 1 right angle
Hence, from the above,
We can conclude that Tim is correct

Essential Question
What types of problems can be solved using the Pythagorean Theorem?
Answer:
The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle. Even though it is written in these terms, it can be used to find any of the sides as long as you know the lengths of the other two sides

Try It!

What is the length of the diagonal, d, of a rectangle with length 19 feet and width 17 feet?
leg² + leg² = hypotenuse²
______ ² + _______² = d²
______ + _______ = d²
_______ = d²
________ ≈ d
Answer:
It is given that a rectangle has a length of 19 feet and a width of 17 feet
Now,
We know that,
In a rectangle,
If a diagonal is drawn, then it divides the rectangle into 2 right angles
So,
According to the Pythagorean Theorem,
d² = a² + b²
Where,
d is the diagonal or hypotenuse
a and b are the lengths of the legs
So,
d²= 19² + 17²
d² = 361 + 289
d² = 650
d = \(\sqrt{650}\)
d = 25.4 ft
Hence, from the above,
We can conclude that the length of the diagonal is: 25.4 ft

Convince Me!
If the rectangle were a square, would the process of finding the length of the diagonal change? Explain.
Answer:
We know that,
For any figure i.e., either 2-d figure or 3-d figure with 1 right angle, the diagonal will divide that figure into 2 right angles
We know that,
We will use the Pythagorean Theorem to find the length of any unknown side in the right triangle
Hence, from the above,
We can conclude that even if the rectangle were a square, the process of finding the length of the diagonal will not change

Try It!

A company wants to rent a tent that has a height of at least 10 feet for an outdoor show. Should they rent the tent shown at the right? Explain.

Answer:
It is given that
A company wants to rent a tent that has a height of at least 10 feet for an outdoor show.
Now,
The given figure is:

Now,
To find whether the tent should be rented or not,
We have to find the value of h
We can also observe that the triangle that contains the value of h is a right triangle
Now,
The base of the right triangle = \(\frac{24}{2}\)
= 12 ft
Now,
The side lengths of the right triangle are: h, 15 ft, 12 ft
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
So,
15² = 12² + h²
h² = 225 – 144
h² = 81
h = \(\sqrt{81}\)
h = 9 ft
But,
The given height of the tent is: 10 ft
So,
9 ft < 10 ft
Hence, from the above,
We can conclude that the company should not rent the tent

KEY CONCEPT

You can use the Pythagorean Theorem and its converse to solve problems involving right triangles.

Do You Understand?
Question 1.
Essential Question What types of problems can be solved using the Pythagorean Theorem?
Answer:
The Pythagorean theorem is a way of relating the leg lengths of a right triangle to the length of the hypotenuse, which is the side opposite the right angle. Even though it is written in these terms, it can be used to find any of the sides as long as you know the lengths of the other two sides

Question 2.
Look for Structure How is using the Pythagorean Theorem in a rectangular prism similar to using it in a rectangle?
Answer:
We know that,
The rectangular prism and the rectangle have at least 1 right angle
We know that,
If a 3-d or 2-d figure has 1 right angle, then the diagonal of that figure divides the figure into the right triangles
So,
If we have the right triangle, then we can use the Pythagorean Theorem irrespective of the overall shape of the figure

Question 3.
Construct Arguments Glen found the length of the hypotenuse of a right triangle using \(\sqrt{a^{2}+b^{2}}\). Gigi used \(\sqrt{(a+b)^{2}}\). Who is correct? Explain.
Answer:
It is given that
Glen found the length of the hypotenuse of a right triangle using \(\sqrt{a^{2}+b^{2}}\). Gigi used \(\sqrt{(a+b)^{2}}\).
Now,
We know that,
We can use the Pythagorean Theorem only for the right triangles
The condition for a triangle to become the right triangle is:
Hypotenuse² = Side² + Side²
c² = a² + b²
Hence, from the above,
We can conclude that Glen is correct

Do You Know How?
Question 4.
You are painting the roof of a boathouse. You are going to place the base of a ladder 12 feet from the boathouse. How long does the ladder need to be to reach the roof of the boathouse?

Answer:
It is given that
You are painting the roof of a boathouse. You are going to place the base of a ladder 12 feet from the boathouse.
Now,
From the figure,
We can observe that the roof of a boathouse, ladder, and the base form a right triangle
So,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the height of the boathouse
a is the length of the base
b is the length of the ladder
So,
35² = 12² + b²
b² = 1,225 – 144
b² = 1,081
b = \(\sqrt{1,081}\)
b = 32.9 ft
Hence, from the above,
We can conclude that the length of the ladder that is needed to reach the roof of the boathouse is: 32.9 ft

Question 5.
A box-shaped like a right rectangular prism measures 5 centimeters by 3 centimeters by 2 centimeters. What is the length of the interior diagonal of the prism to the nearest hundredth?
Answer:
It is given that
A box-shaped like a right rectangular prism measures 5 centimeters by 3 centimeters by 2 centimeters.
So,
The dimensions of a rectangular prism is: 5 cm × 3 cm × 2 cm
So,
The length of the rectangular prism is: 5 cm
The width of the rectangular prism is: 3 cm
The height of the rectangular prism is: 2 cm
Now,
We know that,
The length of the diagonal of the prism = \(\sqrt{Length^{2} + Width^{2} + Height^{2}}\)
So,
The length of the diagonal of the rectangular prism = \(\sqrt{5^{2} + 3^{2} + 2^{2}}\)
= \(\sqrt{25 + 9 + 4}\)
= \(\sqrt{38}\)
= 6.16 cm
Hence, from the above,
We can conclude that the length of the interior diagonal of the rectangular prism is: 6.16 cm

Question 6.
A wall 12 feet long makes a corner with a wall that is 14 feet long. The other ends of the walls are about 18.44 feet apart. Do the walls form a right angle? Explain.

Answer:
It is given that
A wall 12 feet long makes a corner with a wall that is 14 feet long. The other ends of the walls are about 18.44 feet apart
Now,
The given figure is:

From the given figure,
We can observe that the given situation forms the right triangle
Now,
We know that,
According to the Pythagorean Therem,
c² = a² + b²
So,
c² = 12² + 14²
c² = 144 + 196
c² = 340
c = \(\sqrt{340}\)
c = 18.44 feet
Hence, from the above,
We can conclude that the walls form a right angle

Practice & Problem Solving

Leveled Practice In 7 and 8, use the Pythagorean Theorem to solve.
Question 7.
You are going to use an inclined plane to lift a heavy object to the top of a shelving unit with a height of 6 feet. The base of the inclined plane is 16 feet from the shelving unit. What is the length of the inclined plane? Round to the nearest tenth of a foot.

The length of the inclined plane is about ________ feet.
Answer:

Question 8.
Find the missing lengths in the rectangular prism.

Answer:

Question 9.
A stainless steel patio heater is shaped like a square pyramid. The length of one side of the base is 19.8 inches. The slant height is 92.8 inches. What is the height of the heater? Round to the nearest tenth of an inch.
Answer:
It is given that
A stainless steel patio heater is shaped like a square pyramid. The length of one side of the base is 19.8 inches. The slant height is 92.8 inches
Now,
We know that,
The slant height is nothing but a length of the diagonal
Now,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the slant height
a is the length of the base of the steel patio heater
b is the height of the heater
So,
(92.8)² = (19.8)² + b²
b² = 8,611.84 – 392.04
c² = 8,219.8
c = \(\sqrt{8,219.8}\)
c = 90.6 inches
Hence, from the above,
We can conclude that the height of the heater is: 90.6 inches

Question 10.
Reasoning What is the measurement of the longest line segment in a right rectangular prism that is 16 centimeters long, 9 centimeters wide, and 7 centimeters tall? Round to the nearest tenth of a centimeter.
Answer:
It is given that
A right rectangular prism is 16 centimeters long, 9 centimeters wide, and 7 centimeters tall
So,
The dimensions of a right rectangular prism are: 16 cm × 9 cm × 7 cm
So,
The length of the right rectangular prism is: 16 cm
The width of the right rectangular prism is: 9 cm
The height of the right rectangular prism is: 7 cm
We know that,
The longest line segment in any 2-d or 3-d figure is a “Diagonal”
Now,
We know that,
The length of the diagonal of the right rectangular prism = \(\sqrt{Length^{2} + Width^{2} + Height^{2}}\)
So,
The length of the diagonal of the right rectangular prism = \(\sqrt{16^{2} + 9^{2} + 7^{2}}\)
= \(\sqrt{256 + 49 + 81}\)
= \(\sqrt{386}\)
= 19.64 cm
Hence, from the above,
We can conclude that the length of the longest line segment in the right rectangular prism is: 19.64 cm

Question 11.
Felipe is making triangles for a stained glass window. He made the design shown but wants to change it. Felipe wants to move the purple triangle to the corner. The purple piece has side lengths of 4.5 inches, 6 inches, and 7 inches. Can the purple piece be moved to the corner? Explain.

Answer:
It is given that
Felipe is making triangles for a stained glass window. He made the design shown as above but wants to change it. Felipe wants to move the purple triangle to the corner. The purple piece has side lengths of 4.5 inches, 6 inches, and 7 inches.
Now,
From the given figure,
We can observe that
If the purple figure is moving to the corner, then one side of the purple figure will become a right angle
So,
Now,
We have to find whether the purple figure will be a right triangle or not if it moves to a corner
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²,
then, the triangle is a right triangle
We know that,
The hypotenuse has the longest side length
So,
7² = 4.5² + 6²
49 = 20.25 + 36
49 = 56.25
So,
The condition
c² = a² + b²
is false
Hence, from the above,
We can conclude that the purple figure should not

Question 12.
a. What is the longest poster you could fit in the box? Express your answer to the nearest tenth of an inch.

Answer:
The given figure is:

From the given figure,
We can observe that there will be two longest sides for two pairs of different side lengths
we know that,
The longest side is nothing but a diagonal
Now,
The two pairs of side lengths are: (20, 12, x), and (8, 12, y)
Where,
x and y are the lengths of the longest sides
Now,
We know that
Since the figure consists of the right triangles,
According to the Pythagorean Theorem,
c² = a² + b²
So,
x² = 20² + 12²
x² = 400 + 144
x² = 544
x = \(\sqrt{544}\)
x = 23.32 in.
So,
y² = 12² + 8²
y² = 144 + 64
y² = 208
y = \(\sqrt{208}\)
y = 14.42 in.
Hence, from the above,
We can conclude that the longest poster that you can fit in the box is: 23.32 in.

b. Explain why you can fit only one maximum-length poster in the box, but you can fit multiple 21.5-inch posters in the same box.

Answer:
It is given that
you can fit multiple 21.5-inch posters in the same box.
Now,
From part (a),
We can observe that the length of the longest poster that can fit into the box is: 23.32 in.
So,
21.5 in. < 23.32 in.
Hence, from the above,
We can conclude that since the given size of the posters is less than the maximum length of the poster,
We can fit multiple 21.5-inch posters in the same box instead of 1 poster that is of the maximum length

Question 13.

The corner of a room where two walls meet the floor should be at a right angle. Jeff makes a mark along each wall. One mark is 3 inches from the corner. The other is 4 inches from the corner. How can Jeff use the Pythagorean Theorem to see if the walls form a right angle?
Answer:
It is given that
The corner of a room where two walls meet the floor should be at a right angle. Jeff makes a mark along each wall. One mark is 3 inches from the corner. The other is 4 inches from the corner.
Now,
To see whether the walls form a right angle or not,
We have to see whether the length along the walls is greater than the lengths of the marks from the corners
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length along the walls
a is the length of one mark from the corner
b is the length of another mark from the corner
So,
c² = 3² + 4²
c² = 9 + 16
c² = 25
c = \(\sqrt{25}\)
c = 5
So,
From the above value,
We can observe that
c > a and c > b
Hence, from the above,
We can conclude that the walls form a right angle

Question 14.
Higher-Order Thinking It is recommended that a ramp have at least 6 feet of horizontal distance for every 1 foot of vertical rise along an incline. The ramp shown has a vertical rise of 2 feet. Does the ramp show match the recommended specifications? Explain.

Answer:
It is given that
It is recommended that a ramp have at least 6 feet of horizontal distance for every 1 foot of vertical rise along an incline. The ramp shown has a vertical rise of 2 feet
Now,
6 feet of horizontal distance for every 1 foot,
The ramp shown has a vertical rise of 5 feet.
So,
The rate of change (m)  ≤ \(\frac{1 foot}{6 feet}\)
Convert both to a single unit of inches
We know that,
1 foot = 12 inches
So,
m ≤ \(\frac{12 inches}{72 inches}\)
Divide by 4 into both sides
m ≤ \(\frac{4 inches}{18 inches}\)
Now,
For a ramp that has a vertical distance of 2 feet and the same horizontal distance
m = \(\frac{12 inches}{24 inches}\)
m = \(\frac{4 inches}{6 inches}\)
So,
The rate of change when the vertical distance is 6 feet > The rate of change when the vertical distance is 2 feet
Hence, from the above,
We can conclude that the ramp shown above matched the given specifications

Assessment Practice

Question 15.
A machine in a factory cuts out triangular sheets of metal. Which of the triangles are right triangles? Select all that apply.

☐ Triangle 1
☐ Triangle 2
☐ Triangle 3
☐ Triangle 4
Answer:
It is given that
A machine in a factory cuts out triangular sheets of metal.
Now,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²
then, the triangle is a right triangle
Now,
For Triangle 1,
(\(\sqrt{505}\))² = 12² + 19²
505 = 505
For Triangle 2,
(\(\sqrt{467}\))² = 16² + 19²
467 ≠ 617
For Triangle 3,
(\(\sqrt{596}\))² = 14² + 20²
596 = 596
For Triangle 4,
(\(\sqrt{421}\))² = 11² + 23²
421 ≠ 650
Hence, from the above,
We can conclude that Triangle 1 and Triangle 3 are the right triangles

Question 16.
What is the length b, in feet, of the rectangular plot of land shown?

Answer:
It is given that the given figure is a rectangular plot of land
We know that,
In a rectangle,
The opposite sides are equal and the angles are all 90°
So,
A diagonal forms 2 right triangles in a rectangle
Now,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the length of the diagonal
a is the width
b is the length
Now,
325² = 300² + b²
b²= 1,05,625 – 90,000
b² = 15,625
b = \(\sqrt{15,625}\)
b = 125 ft
Hence, from the above,
We can conclude that the length of b is: 125 ft

Lesson 7.4 Find Distance in the Coordinate

Explore It!
Thomas and Jim are outside the haunted castle ride and want to get to the clown tent in time for the next show.

I can… use the Pythagorean Theorem to find the distance between two points in the coordinate plane.

A. How can you represent the starred locations on a coordinate plane?

Answer:
It is given that
Thomas and Jim are outside the haunted castle ride and want to get to the clown tent in time for the next show.
Now,
The given route of the castle is:

Now,
From the given figure,
To represent the route in the coordinate plane,
The given scale is:
X-axis: 1 cm = 500 feet
Y-axis: 1 cm = 500 feet
Now,
From the given figure,
The coordinates of the starred locations can be:
The coordinates of the haunted house are: (500, 1,500),
The coordinates of the clown tent are: (2,000, 500)
Hence,
The representation of the starred locations in the coordinate plane is:

B. Jim says that the marked yellow paths show the shortest path to the tent. Write an expression to represent this and find the distance Jim walks from the haunted mansion to the clown tent.
Answer:
From part (a),
We know that,
The coordinates of the haunted house are: (500, 1,500),
The coordinates of the clown tent are: (2,000, 500)
Now,
We know that
The linear equation in the slope-intercept form is:
y = mx + b
Where,
m is the slope
b is the initial value (or) y-intercept
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
Slope (m) = y₂ – y₁ / x₂ – x₁
So,
Slope (m) = \(\frac{500 – 1,500}{2,000 – 500}\)
Slope (m) = –\(\frac{1,000}{1,500}\)
Slope (m) = –\(\frac{2}{3}\)
So,
The equation in the slope-intercept form is:
y = –\(\frac{2}{3}\)x + b
So,
3y = -2x + 3b
Now,
Substitute (500, 1,500) or (2,000, 500) in the above equation
So,
1,500 = -4000 + 3b
3b = 5,500
So,
The equation that represents the shortest path to the tent is:
3y = -2x + 5,500
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the haunted mansion and the clown tent = \(\sqrt{(2,000 – 500)^{2} + (500 – 1,500)^{2}}\)
= \(\sqrt{1,500^{2} + 1,000^{2}}\)
= \(\sqrt{2,250,000 + 1,000,000}\)
= 1,802.77 feet
Hence, from the above,
We can conclude that
The equation that represents the shortest path to the clown tent is:
3y = -2x + 5,500
The distance between the haunted mansion and the clown tent is: 1,802.77 feet

Focus on math practices
Construct Arguments Why is the distance between two nonhorizontal and nonvertical points always greater than the horizontal or vertical distance?
Answer:
Let us consider a coordinate plane
Now,
When we draw either a horizontal line or the vertical line,
We can observe that the length will be constant
But,
When we draw non-vertical and non-horizontal lines,
We can observe that the lengths are unknown and not constant
Hence, from the above,
We can conclude that the distance between two nonhorizontal and nonvertical points always greater than the horizontal or vertical distance

Essential Question
How can you use the Pythagorean Theorem to find the distance between two points?
Answer:
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the side lengths or lengths of the legs
Now,
Graphically,
The terms of the Pythagorean Theorem can be expressed as:
c is the distance between two points
a and b are the points
So,
c = \(\sqrt{a^{2} + b^{2}}\)

Try It!

What is the distance between points A and B?
The distance between point A and point B is about ________ units.

Answer:
The given graph is:

The representation of the graph in the coordinate plane is:

From the given graph,
The coordinates of A are: (2, 3)
The coordinates of B are: (4, 1)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points A and B = \(\sqrt{(4 – 2)^{2} + (1 – 3)^{2}}\)
= \(\sqrt{2^{2} + 2^{2}}\)
= \(\sqrt{4 + 4}\)
= 2.83 units
Hence, from the above,
We can conclude that the distance between points A and B is about 2.83 units

Convince Me!
Why do you need to use the Pythagorean Theorem to find the distance between points A and B?
Answer:
The representation of points A and B in the coordinate plane are:

Now,
When we observe the graph,
We can see that A and B can form a right triangle
Now,
We know that,
The Pythagorean Theorem is only applicable to the right triangles
So,
According to the Pythagorean Theorem,
c² = a² + b²
c = \(\sqrt{a^{2} + b^{2}}\)
where,
c is the distance between points A and B
A and B are the given points

Try It!
Find the perimeter of ∆ABC with vertices (2, 5), (5, -1), and (2, -1).
Answer:
It is given that
∆ABC with vertices (2, 5), (5, -1), and (2, -1)
Now,
The names of the vertices are:
A (2, 5), B (5, -1), and C (2, -1)
We know that,
The perimeter of a triangle is the sum of all the side lengths of a triangle
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
In ∆ABC,
AB and BC are the side lengths
Ac is the hypotenuse
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points A and B = \(\sqrt{(5 – 2)^{2} + (-1 – 5)^{2}}\)
= \(\sqrt{3^{2} + 6^{2}}\)
= \(\sqrt{9 + 36}\)
= 6.70 units
The distance between the points B and C (BC) = \(\sqrt{(2 – 5)^{2} + (-1 + 1)^{2}}\)
= \(\sqrt{3^{2} + 0^{2}}\)
= \(\sqrt{9 + 0}\)
= 3 units
The distance between the points A and C (AC) = \(\sqrt{(2 – 2)^{2} + (-1 – 5)^{2}}\)
= \(\sqrt{0^{2} + 6^{2}}\)
= \(\sqrt{0 + 36}\)
= 6 units
So,
The perimeter of ∆ABC = AB + BC + AC
= 6 + 3 + 6.70
= 15.7 units
Hence, from the above,
We can conclude that the perimeter of ∆ABC is about 15.7 units

Try It!

What are the coordinates, to the nearest tenth, of the third vertex in an isosceles triangle that has one side length of 2 and two side lengths of 5, with vertices at (1, 0) and (1, 2)? The third vertex is in the first quadrant.
Answer:
It is given that
An isosceles triangle that has one side length of 2 and two side lengths of 5, with vertices at (1, 0) and (1, 2)
Now,
Let the third vertex be (x, y)
Now,
The given vertices are:
A (1, 0), B (1, 2), and C (x, y)
It is also given that
BC = 2 units, and AC = 5 units
We know that,
An isosceles triangle has any 2 equal side lengths
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points A and B = \(\sqrt{(2 – 0)^{2} + (1 – 1)^{2}}\)
= \(\sqrt{2^{2} + 0^{2}}\)
= \(\sqrt{4 + 0}\)
= 4 units
The distance between the points B and C = \(\sqrt{(x – 1)^{2} + (y – 2)^{2}}\)
Squaring on both sides
So,
BC² = (x – 1)² + (y – 2)²
The distance between the points A and C = \(\sqrt{(x – 1)^{2} + (y – 0)^{2}}\)
Squaring on both sides
So,
BC² = (x – 1)² + y²
So,
(x – 1)² + (y – 2)² = 4 —– (1)
(x – 1)² + y² = 25 —— (2)
So,
From eq (1) and eq (2),
25 – y² + (y – 2)² = 4
-y² + y² – 4y + 4 = -21
-4y = -25
y = \(\frac{25}{4}\)
So,
(x – 1)² = |25 – (\(\frac{25}{4}\))²|
x = \(\frac{19}{4}\)
Hence, from the above,
We can conclude that the third vertex is: (\(\frac{19}{4}\), \(\frac{25}{4}\))

KEY CONCEPT

You can use the Pythagorean Theorem to find the distance between any two points, P and Q, on the coordinate plane.

Do You Understand?
Question 1.
Essential Question How can you use the Pythagorean Theorem to find the distance between two points?
Answer:
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the side lengths or lengths of the legs
Now,
Graphically,
The terms of the Pythagorean Theorem can be expressed as:
c is the distance between two points
a and b are the points
So,
c = \(\sqrt{a^{2} + b^{2}}\)

Question 2.
Model with Math
Can you use a right triangle to represent the distance between any two points on the coordinate plane? Explain.
Answer:
Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. We know that,
According to the Pythagorean Theorem,
a²+b²=c²
where,
a and b are the lengths of the legs adjacent to the right angle
c is the length of the hypotenuse.

Question 3.
Generalize How does the fact that the points are on opposite sides of the y-axis affect the process of finding the distance between the two points?
Answer:
The fact that the points are on opposite sides of the y-axis affects the process of finding the distance between the two points because  We need to find the distance between the two points by adding the distances from each of them to the y-axis.

Do You Know How?
In 4-6, use the coordinate plane below.

Answer:
The given coordinate plane is:

From the given coordinate plane,
The given points are:
C (1, 2), D (2, -1), and E (-2, 1)

Question 4.
Find the distance between points C and D. Round to the nearest hundredth.
Answer:
Compare the points C and D with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points C and D = \(\sqrt{(2 – 1)^{2} + (-1 – 2)^{2}}\)
= \(\sqrt{1^{2} + 3^{2}}\)
= \(\sqrt{1 + 9}\)
= 3.16 units
Hence, from the above,
We can conclude that the distance between points C and D is: 3.16 units

Question 5.
Find the perimeter of ∆CDE.
Answer:
We know that,
The “Perimeter” is defined as the sum of all the side lengths
So,
The perimeter of ∆CDE = CD + DE + CE
So,
The distance between the points C and D = \(\sqrt{(2 – 1)^{2} + (-1 – 2)^{2}}\)
= \(\sqrt{1^{2} + 3^{2}}\)
= \(\sqrt{1 + 9}\)
= 3.16 units
The distance between the points D and E = \(\sqrt{(-2 – 2)^{2} + (1 + 1)^{2}}\)
= \(\sqrt{4^{2} + 2^{2}}\)
= \(\sqrt{16 + 4}\)
= 4.47 units
The distance between the points C and E = \(\sqrt{(-2 – 1)^{2} + (1 – 2)^{2}}\)
= \(\sqrt{3^{2} + 1^{2}}\)
= \(\sqrt{1 + 9}\)
= 3.16 units
So,
The perimeter of ∆CDE = 3.16 + 4.47 + 3.16
= 10.79 units
Hence, from the above,
We can conclude that the perimeter of ∆CDE is: 10.79 units

Question 6.
Point B is plotted on the coordinate plane above the x-axis. ∆BDE is equilateral. What are the coordinates of point B to the nearest hundredth?
Answer:
It is given that
Point B is plotted on the coordinate plane above the x-axis. ∆BDE is equilateral.
Now,
Let the unknown vertex be B (x, y)
So,
The given points are:
B (x, y), D (2, -1), and E (-2, 1)
It is given that ΔBDE is equilateral
So,
BD = DE = EB
BD² = DE² = EB²
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points B and D = \(\sqrt{(x – 2)^{2} + (y – 1)^{2}}\)
Squaring on both sides
So,
BD² = (x – 2)² + (y + 1)²
The distance between the points D and E = \(\sqrt{(1 + 1)^{2} + (-2 – 2)^{2}}\)
= \(\sqrt{2^{2} + 4^{2}}\)
= \(\sqrt{4 + 16}\)
= 4.47 units
The distance between the points E and B = \(\sqrt{(x + 2)^{2} + (y – 1)^{2}}\)
Squaring on both sides
So,
EB² = (x + 2)² + (y – 1)²
Now,
(x – 2)² + (y + 1)² = 4.47 —- (1)
(x + 2)² + (y – 1)² = (x – 2)² + (y + 1)²
x² + 2x + 4 + y² + 1 – 2y = x² + 4 – 4x + y² + 1 + 2y
2x + 4 + 1 – 2y = 4 – 4x + 1 + 2y
6x + 5 = 4y + 5
6x = 4y
3x = 2y
x = \(\frac{2}{3}\)y
Now,
From eq (1),
x² + 4 – 4x + y² + 1 + 2y = 4.47
x² + y² -4x + 2y = -0.47
(\(\frac{2}{3}\)y)² + y² – 4 (\(\frac{2}{3}\))y + 2y = -0.47
4y² + 15y + 4.23 = 0
So,
y = -0.30 (or) y = -3.44
So,
x = \(\frac{2}{3}\) (-0.30) (or) x = \(\frac{2}{3}\) (-3.44)
x = -0.2 (or) x = -2.29
Hence, from the above,
We can conclude that the coordinates of point B are: (-0.2, -0.30) or (-2.29, -3.44)

Practice & Problem Solving

Question 7.
Leveled Practice Use the Pythagorean Theorem to find the distance between points P and Q.
Label the length, in units, of each leg of the right triangle.

The distance between point P and point Q is __________ units.
Answer:
From the given coordinate plane,
There are only 2 vertices
Let the third vertex be R(x, y) and the coordinates of R can be found from the coordinate plane
Now,
From the given coordinate plane,
The vertices are:
P (3, 2), Q (9, 10), and R (9, 2)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points Q and R = \(\sqrt{(9 – 9)^{2} + (10 – 2)^{2}}\)
= \(\sqrt{0^{2} + 8^{2}}\)
= \(\sqrt{0 + 64}\)
= 8 units
The distance between the points P and R = \(\sqrt{(9 – 3)^{2} + (2 – 2)^{2}}\)
= \(\sqrt{0^{2} + 6^{2}}\)
= \(\sqrt{0 + 36}\)
= 6 units
Now,
From the given coordinate plane,
We can observe that P, Q, R form the right triangle
So,
According to the Pythagorean Theorem,
PQ² = QR² + PR²
PQ² = 8² + 6²
PQ² = 64 + 36
PQ² = 100
PQ = \(\sqrt{100}\)
PQ = 10 units
Hence, from the above,
We can conclude that the length of PQ is: 10 units

Question 8.
Find the perimeter of triangle QPR. Round to the nearest hundredth.
Answer:
From the coordinate plane,
The vertices of ΔPQR are:
P (-5, -2), Q (2, -2), and R (-1, 3)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points P and Q = \(\sqrt{(2 + 5)^{2} + (2 – 2)^{2}}\)
= \(\sqrt{0^{2} + 7^{2}}\)
= \(\sqrt{0 + 49}\)
= 7 units
The distance between the points Q and R = \(\sqrt{(-1 – 2)^{2} + (3 + 2)^{2}}\)
= \(\sqrt{3^{2} + 5^{2}}\)
= \(\sqrt{9 + 25}\)
= 5.83 units
The distance between the points P and R = \(\sqrt{(1 – 5)^{2} + (-3 – 2)^{2}}\)
= \(\sqrt{4^{2} + 5^{2}}\)
= \(\sqrt{16 + 25}\)
= 6.40 units
So,
The perimeter of ΔPQR = PQ + QR + PR
= 7 + 5.83 + 6.40
= 19.23 units
Hence, from the above,
We can conclude that the perimeter of ΔPQR is: 19.23 units

Question 9.
Determine whether the triangle is equilateral, isosceles, or scalene.
Answer:
We know that,
On the basis of the side lengths,
Scalene Triangle – All the side lengths are different
Equilateral Triangle – All the side lengths are the same
Isosceles Triangle – Any two of the side lengths are the same
So,
From Exercise 8,
We can observe that all the side lengths are different
Hence, from the above,
We can conclude that ΔPQR is a scalene Triangle

Question 10.
You walk along the outside of a park starting at point P. Then you take a shortcut represented by \(\overline{P Q}\) on the graph.

a. What is the length of the shortcut in meters? Round to the nearest tenth of a meter.
Answer:
It is given that
You walk along the outside of a park starting at point P. Then you take a shortcut represented by \(\overline{P Q}\) on the graph.
Now,
From the given figure,
We can observe that
The vertices are:
P (0, 0), Q (40, 85), and R (40, 0)
The shortest path is represented as PQ
Now,
Compare the points P and Q with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points P and Q = \(\sqrt{(40 – 0)^{2} + (85 – 0)^{2}}\)
= \(\sqrt{40^{2} + 85^{2}}\)
= \(\sqrt{1,600 + 7,225}\)
= 93.9 m
Hence, from the above,
We can conclude that the length of the shortest path is: 93.9 meters

b. What is the total length of your walk in the park? Round to the nearest tenth of a meter.
Answer:
We know that,
The total length is nothing but the “Perimeter”
So,
The perimeter of the given triangle = PQ + QR + PR
Now,
The distance between the points P and Q = \(\sqrt{(40 – 0)^{2} + (85 – 0)^{2}}\)
= \(\sqrt{40^{2} + 85^{2}}\)
= \(\sqrt{1,600 + 7,225}\)
= 93.9 m
The distance between the points Q and R = \(\sqrt{(40 – 40)^{2} + (85 – 0)^{2}}\)
= \(\sqrt{0^{2} + 85^{2}}\)
= \(\sqrt{0 + 7,225}\)
= 85 m
The distance between the points P and R = \(\sqrt{(40 – 0)^{2} + (0 – 0)^{2}}\)
= \(\sqrt{0^{2} + 40^{2}}\)
= \(\sqrt{0 + 1,600}\)
= 40 m
So,
The total length of your walk in the park = PQ + QR + PR
= 93.9 + 85 + 40
= 218.9 meters
Hence, from the above,
We can conclude that the total length of your walk in the park is: 218.9 meters

Question 11.
Suppose a park is located 3.6 miles east of your home. The library is 4.8 miles north of the park. What is the shortest distance between your home and the library?
Answer:
It is given that
A park is located 3.6 miles east of your home. The library is 4.8 miles north of the park
So,
The representation of the above situation is:

Now,
According to the Pythagorean Theorem,
(The shortest distance between home and library)² = (The distance from home to park)² + (he distance from park to library)²
(The shortest distance between home and library)² = 3.6² + 4.8²
(The shortest distance between home and library)² = 12.96 + 23.04
The shortest distance between home and library = 6
Hence, from the above,
We can conclude that the shortest distance between home and library is: 6 miles

Question 12.
Use Structure Point B has coordinates (2, 1). The x-coordinate of point A is -10. The distance between point A and point B is 15 units. What are the possible coordinates of point A?
Answer:
It is given that
Point B has coordinates (2, 1). The x-coordinate of point A is -10. The distance between point A and point B is 15 units
Now,
Let the coordinates of A be:
(x, y) = (-10, y)
Now,
Compare the points A and B with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points A and B = \(\sqrt{(-10 – 2)^{2} + (y – 1)^{2}}\)
15 = \(\sqrt{12^{2} + (y – 1)^{2}}\)
15 = \(\sqrt{144 + (y – 1)^{2}}\)
Now,
Squaring on both sides
So,
144 + (y – 1)² = 225
(y  1)² = 225 – 144
(y – 1)² = 81
y – 1 = \(\sqrt{81}\)
y – 1 = 9 (or) y – 1 = -9
y = 9 + 1 (or) y = -9 + 1
y = 10 (or) y = -8
Hence, from the above,
We can conclude that the possible coordinates of A are: (-10, 10), and (-10, -8)

Question 13.
Higher-Order Thinking ∆EFG and ∆HIJ have the same perimeter and side lengths. The coordinates are E(6, 2), F(9, 2), G(8, 7), H(0, 0), and I(0, 3). What are the possible coordinates of point J?
Answer:
It is given that
∆EFG and ∆HIJ have the same perimeter and side lengths. The coordinates are E(6, 2), F(9, 2), G(8, 7), H(0, 0), and I(0, 3)
So,
According to the side lengths,
EF = HI, FG = IJ, and GE = JH
So,
EF² = HI², FG²= JI², and GE² = JH²
Now,
Let the unknown vertex be J (x, y)
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points E and F = \(\sqrt{(9 – 6)^{2} + (2 – 2)^{2}}\)
= \(\sqrt{3^{2} + 0^{2}}\)
= \(\sqrt{9 + 0}\)
= 3 units
The distance between the points F and G = \(\sqrt{(8 – 9)^{2} + (7 – 2)^{2}}\)
= \(\sqrt{1^{2} + 5^{2}}\)
= \(\sqrt{1 + 25}\)
= 5.09 units
The distance between the points G and E = \(\sqrt{(8 – 6)^{2} + (7 – 2)^{2}}\)
= \(\sqrt{2^{2} + 5^{2}}\)
= \(\sqrt{4 + 25}\)
= 5.38 units
Now,
The distance between the points H and I = \(\sqrt{(0 – 0)^{2} + (3 – 0)^{2}}\)
= \(\sqrt{3^{2} + 0^{2}}\)
= \(\sqrt{9 + 0}\)
= 3 units
The distance between the points I and J = \(\sqrt{(x – 0)^{2} + (y – 3)^{2}}\)
5.09 = \(\sqrt{x^{2} + (y – 3)^{2}}\)
Squaring on both sides
So,
x² + (y – 3)² = 25.90 units
The distance between the points J and H = \(\sqrt{(x – 0)^{2} + (y – 0)^{2}}\)
5.38 = \(\sqrt{x^{2} + y^{2}}\)
Squaring on both sides
So,
x² + y² = 28.94 units
Now,
28.94 – y² + (y – 3)² = 25.90
y² + 9 – 6y – y² = 25.90 – 28.94
9 – 6y = -3.04
-6y = -3.04 – 9
6y = 12.04
y = 2
Now,
Substitute the value of y in eq 2
x² + 4 = 28.94
x² = 24.94
x = 4.99 (or) x = -4.99
Hence, from the baove,
We can conclude that the missing vertex is: J (4.99, 2) or J (-4.99, 2)

b. Explain why there can be different possibilities for the coordinates for point J.
Answer:

Assessment Practice
Question 14.
Find the distance, in units, between P and R. Round to the nearest tenth.

Answer:
The given coordinate plane is:

From the given coordinate plane,
The points are:
P (5, 10), and R (12, 14)
Now,
Compare the points P and R with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points P and R = \(\sqrt{(14 – 10)^{2} + (12 – 5)^{2}}\)
= \(\sqrt{4^{2} + 7^{2}}\)
= \(\sqrt{16 + 49}\)
= 8.1 units
Hence, from the above,
We can conclude that the distance between P and R is: 8.1 units

Question 15.
Find the distance, in units, between A(1, 5) and B(5.5, 9.25). Round to the nearest tenth.
Answer:
The given points are:
A (1, 5), and B (5.5, 9.25)
Now,
Compare the points A and B with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points A and B = \(\sqrt{(9.25 – 5)^{2} + (5.5 – 1)^{2}}\)
= \(\sqrt{4.25^{2} + 4.5^{2}}\)
= \(\sqrt{18.06 + 20.25}\)
= 6.2 units
Hence, from the above,
We can conclude that the distance between points A and B is: 6.2 units

Topic 7 REVIEW

Topic Essential Question
How can you use the Pythagorean Theorem to solve problems?
Answer:
Step 1:
Draw a right triangle and then read through the problems again to determine the length of the legs and the hypotenuse. Step 2:
Use the Pythagorean Theorem (a² + b² = c²) to write an equation to be solved.
Step 3:
Simplify the equation by distributing and combining like terms as needed.

Vocabulary Review

Complete each definition and then provide an example of each vocabulary word.
Vocabulary
The converse of the Pythagorean Theorem
hypotenuse
leg
proof
Pythagorean Theorem

Answer:

Use Vocabulary in Writing
All faces of the figure are rectangles. Explain how to find the length of d. Use vocabulary terms in your description.

Answer:
The given figure is:

From the given figure,
We can observe that there are 2 right triangles
Now,
For the first right triangle,
The given side lengths are:
a = 3, b = 4, and c = x
Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
So,
x² = 3² + 4²
x² = 9 + 16
x² = 25
x = \(\sqrt{25}\)
x = 5
Now,
For the second right triangle,
a = 5, b = 12, and c = d
So,
d² = 5² + 12²
d² = 25 + 144
d² = 169
d = \(\sqrt{169}\)
d = 13
Hence, from the above,
We can conclude that the length of d is: 13 units

Concepts and Skills Review

Lesson 7.1 Understand the Pythagorean Theorem

Quick Review
The Pythagorean Theorem states that, in a right triangle, the sum of the squares of the lengths of the legs, a and b, is equal to the square of the length of the hypotenuse, c. So, a² + b² = c².

Example
Find the length of the hypotenuse of a triangle with legs of 7 meters and 24 meters.
Answer:
Substitute 7 for a and 24 for b. Then solve for c.
a² + b² = c²
49 + 576 = c²
\(\sqrt {625}\) = C
The length of the hypotenuse is 25 meters.

Practice
Question 1.
Find the length of the hypotenuse.

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the side lengths
So,
c² = 12² + 5²
c² = 144 + 25
c² = 169
c = \(\sqrt{169}\)
c = 13
Hence, from the above,
We can conclude that the length of the hypotenuse is: 13 cm

Question 2.
Find the unknown side length. Round to the nearest tenth.

Answer:
The given right triangle is:

Now,
We know that,
According to the Pythagorean Theorem,
c² = a² + b²
Where,
c is the hypotenuse
a and b are the side lengths
So,
9² = 7² + b²
b² = 81 – 49
b² = 32
b = \(\sqrt{32}\)
b = 5.65
Hence, from the above,
We can conclude that the length of the hypotenuse is: 5.65 in.

Lesson 7.2 Understand the Converse of the Pythagorean Theorem

Quick Review
For a triangle with side lengths a, b, and c, if a² + b² = c², then the triangle is a right triangle by the Converse of the Pythagorean Theorem.

Example
Is a triangle with side lengths of 8 m, 15 m, and 17 m a right triangle? Explain.
Answer:
Substitute 8 for a, 15 for b, and 17 for c.
a² + b² \(\underline{\underline{?}}\) c²
8² + 15² \(\underline{\underline{?}}\) 17²
289 = 289 ✓
Because a² + b² = c², the triangle is a right triangle.

Practice
Question 1.
Is the triangle a right triangle? Explain.

Answer:
The given triangle is:

Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²
then, the given triangle is a right triangle
So,
Now,
37² = 35² + 12²
1,369 = 1,225 + 144
1,369 = 1,369
So,
The condition c² = a² + b² is true
Hence, from the above,
We can conclude that the given triangle is a right triangle

Question 2.
A triangle has side lengths of 1.5 inches, 2 inches, and 3 inches. Is the triangle a right triangle? Explain.
Answer:
It is given that
A triangle has side lengths of 1.5 inches, 2 inches, and 3 inches
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²
then, the given triangle is a right triangle
We know that,
The longest side is the hypotenuse
So,
Now,
3² = (1.5)² + 2²
9 = 2.25 + 4
9 = 6.25
So,
The condition c² = a² + b² is false
Hence, from the above,
We can conclude that the given triangle is not a right triangle

Question 3.
A triangle has side lengths of 9 feet, 40 feet, and 41 feet. Is the triangle a right triangle? Explain
Answer:
It is given that
A triangle has side lengths of 9 feet, 40 feet, and 41 feet
Now,
We know that,
According to the converse of the Pythagorean Theorem,
If
c² = a² + b²
then, the given triangle is a right triangle
We know that,
The longest side is the hypotenuse
So,
Now,
41² = 40² + 9²
1,681 = 1,600 + 81
1,681 = 1,681
So,
The condition c² = a² + b² is true
Hence, from the above,
We can conclude that the given triangle is a right triangle

Lesson 7.3 Apply the Pythagorean Theorem to Solve Problems

Quick Review
The Pythagorean Theorem can be used to find unknown side lengths of an object that is shaped like a right triangle. It also can be used to find diagonal measures in certain two-dimensional and three-dimensional objects.

Example
A shipping box is 20 inches long along the diagonal of its base. Each diagonal of the box is 29 inches long. How tall is the box?

Answer:
Substitute 20 for a and 29 for c. Then solve for b.
a² + b² = c²
20² + b² = 29²
400 + b² = 841
b = \(\sqrt {441}\)
The height of the shipping box is 21 inches.

Practice
Question 1.
A basketball court is in the shape of a rectangle that is 94 feet long and 50 feet wide. What is the length of a diagonal of the court? Round to the nearest tenth.
Answer:
It is given that
A basketball court is in the shape of a rectangle that is 94 feet long and 50 feet wide
We know that,
By drawing a diagonal in the rectangle, it will become 2 right triangles
The diagonal will be the hypotenuse of the right triangle
Now,
We know that,
According to the Pythagorean Theorem,
c² = a²+ b²
Where,
c is the length of the diagonal
a and b are the side lengths
So,
c² = 94²+ 50²
c²= 8,836 + 2,500
c² = 11,336
c = \(\sqrt{11,336}\)
c = 106.4 feet
hence, from the above,
We can conclude that the length of the diagonal is: 106.4 feet

Question 2.
A packaging box for a metal rod is 7.5 inches along a diagonal of the base. The height of the box is 18 inches. What is the length of a diagonal of the box?

Answer:
It is given that
A packaging box for a metal rod is 7.5 inches along a diagonal of the base. The height of the box is 18 inches.
Now,
From the given figure,
We can observe that it looks like a right triangle
Now,
We know that,
According to the Pythagorean Theorem,
c² = a²+ b²
Where,
c is the length of the diagonal
a and b are the side lengths
So,
c² = 18² + (7.5)²
c²= 324 + 56.25
c² = 380.25
c = \(\sqrt{380.25}\)
c = 19.5 inches
Hence, from the above,
We can conclude that the length of the diagonal of the box is: 19.5 inches

Lesson 7.4 Find Distance in the Coordinate Plane

Quick Review
The Pythagorean Theorem can be used to find the distance between any two points on the coordinate plane.

Example
Find the distance between the two points on the coordinate plane. Round to the nearest tenth.

Answer:
Draw a right triangle. Determine the lengths of its legs.

The length of the horizontal leg is 5 units.
The length of the vertical leg is 5 units.
Use the relationship a² + b² = c². Substitute 5 for a and 5 for b. Then solve for C.
a² + b² = -2
5² + 5² = c²
25 + 25 = c²
50 = c²
\(\sqrt {50}\) = C
7.1 ≈ c
The distance between the two points is about 7.1 units.

Practice
Question 1.
Points C and D represent the location of two parks on a map. Find the distance between the parks if the length of each unit on the grid is equal to 25 miles. Round to the nearest mile.

Answer:
It is given that
Points C and D represent the location of two parks on a map
Now,
The given coordinate plane is:

From the given coordinate plane,
The points are:
C (-2, 2), and D (4, -1)
Now,
Compare the points C and D with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points C and D = \(\sqrt{(-1 – 2)^{2} + (4 + 2)^{2}}\)
= \(\sqrt{3^{2} + 6^{2}}\)
= \(\sqrt{9 + 36}\)
= 6.70 units
It is given that the length of each unit in the grid is: 25 miles
So,
The distance between the points C and D = 6.25 × 25
= 156.25 miles
Hence, from the above,
We can conclude that the distance between points C and D is: 156.25 miles

Question 2.
Find the perimeter of ∆ABC. Round to the nearest tenth.

Answer:
The given coordinate plane is:

Now,
From the given coordinate plane,
The vertices of ΔABC are:
A (1, 2), B (7, 9), and C (7, 2)
We know that,
The perimeter of a triangle is the sum of all the side lengths of a triangle
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
From the given coordinate plane,
We know that,
In ∆ABC,
AC and BC are the side lengths
AB is the hypotenuse
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points A and C = \(\sqrt{(2 – 2)^{2} + (7 – 1)^{2}}\)
= \(\sqrt{0^{2} + 6^{2}}\)
= \(\sqrt{36 + 0}\)
= 6 units
The distance between the points B and C (BC) = \(\sqrt{(7 – 7)^{2} + (-2 + 9)^{2}}\)
= \(\sqrt{0^{2} + 7^{2}}\)
= \(\sqrt{0 + 49}\)
= 7 units
The distance between the points A and B (AB) = \(\sqrt{(9 – 2)^{2} + (7 – 1)^{2}}\)
= \(\sqrt{7^{2} + 6^{2}}\)
= \(\sqrt{49 + 36}\)
= 9.21 units
So,
The perimeter of ∆ABC = AB + BC + AC
= 6 + 7 + 9.21
= 22.2 units
Hence, from the above,
We can conclude that the perimeter of ∆ABC is about 22.2 units

Question 3.
Triangle JKL is an equilateral triangle with two of its vertices at points J and K. What are the coordinates of point L? Round to the nearest tenth as needed.

Answer:
It is given that
Triangle JKL is an equilateral triangle with two of its vertices at points J and K
Now,
The given coordinate plane is:

Now,
From the given coordinate plane,
The two vertices of ΔJKL are:
J (3, 2), K (9, 2)
Now,
Let the third vertex be L (x, y)
We know that,
In an equilateral triangle, all the side lengths are the same
So,
JK = KL = JL
JK² = KL² = JL²
So,
Now,
Compare the given points with (x₁, y₁), (x₂, y₂)
Now,
We know that,
The distance between two points = √(x₂–  x₁) + (y₂ – y₁)²
So,
The distance between the points J and K = \(\sqrt{(9 – 3)^{2} + (2 – 2)^{2}}\)
= \(\sqrt{0^{2} + 6^{2}}\)
= \(\sqrt{36 + 0}\)
= 6 units
The distance between the points K and L = \(\sqrt{(x – 9)^{2} + (y – 2)^{2}}\)
6= \(\sqrt{(x – 9)^{2} + (y – 2)^{2}}\)
Squaring on both sides
So,
(x – 9)² + (y – 2)² = 36
The distance between the points J and L = \(\sqrt{(x – 3)^{2} + (y – 2)^{2}}\)
6= \(\sqrt{(x – 3)^{2} + (y – 2)^{2}}\)
Squaring on both sides
So,
(x – 3)² + (y – 2)² = 36
Now,
(x – 9)² + 36 – (x – 3)² = 36
(x – 9)² = (x – 3)²
x² + 81 – 18x = x² + 9 – 6x
18x – 6x = 81 – 9
12x = 72
x = \(\frac{72}{12}\)
x = 6
So,
Substitute the value of x in eq (2)
(6 – 3)² + (y – 2)² = 36
(y – 2)² = 36 – 9
y – 2 = \(\sqrt{27}\)
y = 7.2
Hence, from the above,
We can conclude that the third vertex is: L (6, 7.2)

Topic 7 Fluency Practice

Riddle Rearranging
Solve each equation. Then arrange the answers in order from least to greatest. The letters will spell out the answer to the riddle below.

I can… solve multistep equations.

Why did the coffee shop server love the job? Because there were so

Answer:
The Ascending order of the solutions of the equations is:
M < N < Y < P < A < E < R < K < S