Go through the enVision Math Common Core Grade 8 Answer Key Topic 8 Solve Problems Involving Surface Area And Volume regularly and improve your accuracy in solving questions.

enVision Math Common Core 8th Grade Answers Key Topic 8 Solve Problems Involving Surface Area And Volume

Topic 8 Essential Question
How can you find volumes and surface areas of three-dimensional figures?
Answer:
The “Surface area” is the sum of the areas of all faces (or surfaces) on a 3D shape.
Ex:
A cuboid has 6 rectangular faces. To find the surface area of a cuboid, add the areas of all 6 faces
We know that,
The volume of a three-dimensional figure = Cross-sectional area × length

3-ACT MATH

Measure Up
Have you ever heard of the terms griffin beaker, Erlenmeyer flask, or graduated cylinder? Maybe you’ve used them in your science class. Each piece of equipment in a chemistry lab has a specific purpose, so containers come in many shapes. It’s sometimes necessary to pour a solution from one container to another. Think about this during the 3-Act Mathematical Modeling lesson.

Topic 8 ënVision STEM Project

Did You Know?
The production of packaging is a huge industry employing over five million people with annual sales of more than 400 billion dollars.
Packaging materials protect and deliver food and products to consumers.

A plastic bottle takes 450–1,000 years to biodegrade.

Seabirds are dying of starvation with stomachs full of plastic and Styrofoam.

Polystyrene foam lasts forever!
Eco-friendly packaging materials are being made from mushrooms and bamboo.

There is even a drink bottle made from recyclable paper.

New technology results in packaging materials that are both affordable and biodegradable.
Environmentally friendly companies are producing sustainable packaging. In addition to using recyclable materials, they reduce the water, natural resources, and energy needed for production. They minimize waste when designing products.

Your Task: Wrap it Up!

Engineers consider several factors when designing product packaging. These factors include cost efficiency and eco-friendly design so that materials are disposable, recyclable, biodegradable, and not wasted. Suppose you are an engineer working for Liquid Assets, an environmentally friendly company that designs, builds, and packages water purifiers. You and your classmates will use your knowledge of volume and surface area to determine an environmentally sound way to package the purifiers.

Topic 8 GET READY!

Review What You Know!

Vocabulary
Choose the best term from the box to complete each definition.
base
diameter
radius
three-dimensional
two-dimensional

Question 1.
The __________ is the distance from the center to the edge of a circle.
Answer:
We know that,
The “Radius” is the distance from the center to the edge of a circle
Hence, from the above,
We can conclude that the best term to complete the given definition is: Radius

Question 2.
A shape that has length, width, and height is ___________.
Answer:
We know that,
A shape that has a length, width, and height is known as “Three-dimensional”
Hence, from the above,
We can conclude that the best term to complete the given definition is: Three dimensional

Question 3.
Any side of a cube can be considered a __________.
Answer:
We know that,
Any side of a cube can be considered a “Base”
Hence, from the above,
We can conclude that the best term to complete the given definition is: Base

Question 4.
A shape that has length and width, but not height, is ___________.
Answer:
We know that,
A shape that has length and width, but not height is known as “Two-dimensional”
Hence, from the above,
We can conclude that the best term to complete the given definition is: Two-dimensional

Question 5.
The _____________ of a circle is a line segment that passes through its center and has endpoints on the circle.
Answer:
We know that,
The “Diameter” of a circle is a line segment that passed through its center and has endpoints on the circle
Hence, from the above,
We can conclude that the best term to complete the given definition is: Diameter

Multiplying with Decimals

Find the product.
Question 6.
14 ∙ 3.5 = _______
Answer:
The given expression is:
14 × 3.5
So,
14 × 3.5 = 49.0
Hence, from the above,
We can conclude that the value for the given expression is: 49

Question 7.
9 ∙ 3.14 = _________
Answer:
The given expression is:
9 × 3.14
So,
9 × 3.14 = 28.26
Hence, from the above,
We can conclude that the value for the given expression is: 28.26

Question 8.
4.2 ∙ 10.5 = _________
Answer:
The given expression is:
4.2 × 10.5
So,
4.2 × 10.5 = 44.1
Hence, from the above,
We can conclude that the value for the given expression is: 44.1

Areas of Circles

Find the area of each circle. Use 3.14 for π.

Question 9.

A = ________
Answer:
The given circle is:

From the given circle,
The radius is: 8 cm
Now,
We know that,
The area of the circle = πr²
So,
The area of the given circle = 3.14 × 8²
= 200.96 cm²
Hence, from the above,
We can conclude that the area for the given circle is: 200.96 cm²

Question 10.

A = _________
Answer:
The given circle is:

From the given circle,
The diameter is: 12 cm
Now,
We know that,
Radius = \(\frac{Diameter}{2}\)
Radius = \(\frac{12}{2}\)
Radius = 6 cm
Now,
We know that,
The area of the circle = πr²
So,
The area of the given circle = 3.14 × 6²
= 113.04 cm²
Hence, from the above,
We can conclude that the area of the given circle is: 113.04 cm²

Use the Pythagorean Theorem

Find the missing side length of the triangle.
Question 11.

x = _________
Answer:
The given triangle is:

Now,
Fro the given figure,
We can observe that the triangle is a right triangle
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 side lengths
So,
13² = 12²+ x²
x²= 169 – 144
x² = 25
x = \(\sqrt{25}\)
x = 5 in.
Hence, from the above,
We can conclude that the missing side length of the given triangle is: 5 in.

Question 12.

x = __________
Answer:
The given triangle is:

Now,
Fro the given figure,
We can observe that the triangle is a right triangle
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 side lengths
So,
30² = 24²+ x²
x²= 900 – 576
x² = 324
x = \(\sqrt{324}\)
x = 18 in.
Hence, from the above,
We can conclude that the missing side length of the given triangle is: 18 in.

Language Development

Complete the word web. Write keywords, ideas, examples, or illustrations that connect to each new vocabulary term.

Topic 8 Pick A Project

PROJECT 8A
What makes a concert rock?
PROJECT: DESIGN PROPS OR STAGE STRUCTURES

PROJECT 8B
What is the most interesting museum you have visited?
PROJECT: MAKE A MODEL OF A MUSEUM

PROJECT 8C
Where around the United States can you find quarries?
PROJECT: POUR AND MEASURE SAND

PROJECT 8D
If you were cast in a play, would it be a comedy or a drama? Why?
PROJECT: WRITE A SKIT

Lesson 8.1 Find Surface Area of Three-Dimensional Figures

Explore It!
Andrea is designing the packaging for a tube-shaped container.

I can… find the surface areas of cylinders, cones, and spheres.

A. Model with Math What two-dimensional shape represents the top and bottom of the container? What two-dimensional shape represents the tube? Draw a net of the tube-shaped container.
Answer:
It is given that Andrea is designing the packaging for a tube-shaped container.
Now,
The given arrangement for tube shaped container is:

Now,
From the given arrangement,
We can observe that
The two-dimensional shape that represents the top and bottom of the container is: Circle
The two-dimensional shape that represents the tube is: Rectangle
Hence,
The representation of the tube-shaped container is:

B. Look for Relationships The circular top and bottom fit perfectly on the ends of the container. How are the measures of the circles and the rectangle related?
Answer:
The representation of the tube-shaped container is:

Now,
From the given figure,
We can observe that the tube-shaped container is made up of 2 circles and 1 rectangle
So,
The total surface area of the tube-shaped container is the sum of the areas of 2 circles and 1 rectangle
Now,
We know that,
The area of a circle = πr²
The area of a rectangle = Height× Base
Where,
The base of the rectangle is a circle
So,
The circumference of the circle = 2πr
Hence,
The surface area of the tube-shaped container = 2πr² + 2πrh

Focus on math practices
Model with Math
How can you check whether the net that you drew accurately represents the tube-shaped container?
Answer:
The representation of the tube-shaped container is:

Now,
When the top and bottom of the container correctly fit the tube,
That is the representation that the net you drew accurately represents the tube-shaped container

Essential Question
How are the areas of polygons used to find the surface area formulas for three-dimensional figures?
Answer:
We know that,
A three-dimensional figure is a combination of some two-dimensional figures
Ex:
We can make a “Cuboid” from the combination of “Rectangles”
We can make a “Sphere” from the combination of ‘Circles”
So,
The total surface area of a three-dimensional figure can be given as the sum of all the areas of the two-dimensional figures that are used to make the three-dimensional figure

Try it

What is the surface area of a cylinder with a height of 9.5 inches and a radius of 2.5 inches?
The surface area of the cylinder is __________ square inches.
Answer:
It is given that
The height of a cylinder is: 9.5 inches
The radius of a cylinder is: 2.5 inches
Now,
We know that,
The surface area of the cylinder (S.A) = 2πr² + 2πrh
= 2 × 3.14 × (2.5)² + 2 × 3.14 × 9.5 × 2.5
= 39.25 + 149.15
= 188.4 square inches
Hence, from the above,
We can conclude that the surface area of the given cylinder is: 188.4 square inches

Convince Me! How can you find the surface area of a cylinder if you only know its height and the circumference of its base?
S.A. = 2(πr²) + (2πr)h
= 2π(________²) + 2π(_______)(________)
= _______π + _______π
= _______π
Answer:
It is given that
We know only the height of the cylinder and the circumference of its base
Now,
We know that,
The surface area of a cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2π (r²) + 2π (r) (h)
S.A = 2π + π
S.A = 3π
Hence, from the above,
We can conclude that the surface area of a cylinder with only its height and the circumference of its base is: 3π

Try It!

a. What is the surface area of a cone with a radius of 7 feet and a slant height of 9 feet? Use \(\frac{22}{7}\) for π.
Answer:
It is given that
The radius of the cone is (r): 7 feet
The slant height of the cone is (l): 9 feet
Now,
We know that,
The surface area of the cone (S.A) = πr² + πrl
So,
S.A = \(\frac{22}{7}\) × 7² + \(\frac{22}{7}\) × 7 × 9
= 154 + 198
= 352 square feet
Hence, from the above,
We can conclude that the surface area of the given cone is: 352 square feet

b. What is the surface area of a sphere with a diameter of 2.7 inches? Use 3.14 for π.
Answer:
It is given that,
The diameter of a sphere is: 2.7 inches
Now,
We know that,
Radius = \(\frac{Diameter}{2}\)
So,
Radius of the sphere = \(\frac{2.7}{2}\)
So,
The radius of the sphere (r) = 1.35 inches
Now,
We know that,
The surface area of the sphere (S.A) = 4πr²
So,
S.A = 4 × 3.14 × (1.35)²
= 22.89 square inches
Hence, from the above,
We can conclude that the surface area of the given sphere is: 22.89 square inches

KEY CONCEPT

Formulas for finding the area of polygons can be used to find the surface areas of cylinders, cones, and spheres.

Do You Understand?
Question 1.
Essential Question How are the areas of polygons used to find the surface area formulas for three-dimensional figures?
Answer:
We know that,
A three-dimensional figure is a combination of some two-dimensional figures
Ex:
We can make a “Cuboid” from the combination of “Rectangles”
We can make a “Sphere” from the combination of ‘Circles”
So,
The total surface area of a three-dimensional figure can be given as the sum of all the areas of the two-dimensional figures that are used to make the three-dimensional figure

Question 2.
Reasoning Why is the length of the base of the rectangle the same as the circumference of the circles in the net of a cylinder?
Answer:
The representation of the cylinder is:

Now,
If you look at the net, the curved surface of the cylinder is rectangular in shape. The length of the rectangle is the same as the circumference of the circle. Since the length of the rectangle wraps around the circle rim, it is the same length as the circumference of the circle.

Question 3.
Construct Arguments Aaron says that all cones with a base circumference of 8 inches will have the same surface area. Is Aaron correct? Explain.
Answer:
It is given that
Aaron says that all cones with a base circumference of 8 inches will have the same surface area
Now,
We know that,
The surface area of a cone = πr² + πrl
Now,
From the above formula,
We can conclude that the surface area of the cone does not depend only on the circumference of the base but also we need the side length of the cone part as well
So,
All cones with a base circumference of 8 inches will not have the same surface area.
Hence, from the above,
We can conclude that Aaron is not correct

Do You Know How?
Question 4.
What is the surface area of the cylinder? Use 3.14 for π, and round to the nearest tenth.

Answer:
The given figure is:

From the given figure,
The diameter of the cylinder is: 2 mm
The height of the cylinder is: 10 mm
So,
Radius of the cylinder = \(\frac{Diameter of the cylinder}{2}\)
= \(\frac{2}{2}\)
= 1 mm
Now,
We know that,
The surface area of the cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2 × 3.14 × 1² + 2 × 3.14 × 1 × 10
= 6.28 + 62.8
= 69.08 mm²
Hence, from the above,
We can conclude that the surface area of the given cylinder is: 69.08 mm²

Question 5.
What is the surface area of the cone to the nearest tenth? Use 3.14 for π.

Answer:
The given figure is:

From the given figure,
The slant height of the cone is (l): 4 ft
The radius of the cone is: 3 ft
Now,
We know that,
The surface area of the cone (S.A) = πr² + πrl
So,
S.A = 3.14 × 3² + 3.14 × 3 × 4
= 28.26 + 37.68
= 65.94 ft²
Hence, from the above,
We can conclude that the surface area of the given cone is: 65.94 ft²

Question 6.
What is the surface area of the sphere in terms of π?

Answer:
The given figure is:

From the given figure,
The diameter of the sphere is: 2 cm
Now,
We know that,
The radius of the sphere (r) = \(\frac{Diameter of the sphere}{2}\)
= \(\frac{2}{2}\)
= 1 cm
Now,
We know that,
The surface area of the sphere (S.A) = 4πr²
So,
S.A = 4 × 3.14 × 1²
= 12.56 cm²
Hence, from the above,
We can conclude that the surface area of the given sphere is: 12.56 cm²

Practice & Problem Solving

Leveled Practice In 7-8, find the surface area.
Question 7.
What is the surface area of the cylinder? Use 3.14 for π, and round to the nearest tenth.

S.A. = 2(πr²) + (2πr)h
= 2π(________²) + 2 π(_______)(________)
= 2 π(________) + 2 π(_________)
= _______π + _______π
= _______π
≈ ________ cm²
Answer:
The given figure is:

From the given figure,
The radius of the cylinder is: 3 cm
The height of the cylinder is: 5 cm
Now,
We know that,
The surface area of the cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2 × 3.14 × 3² + 2 × 3.14 × 3 × 5
= 56.52 + 94.2
= 150.72 cm²
Hence, from the above,
We can conclude that the surface area of the given cylinder is: 150.72 cm²

Question 8.
What is the surface area of the cone? Use 3.14 for π.

S.A. = πr² + πlr
= π(________²) + π(_______)(________)
= ________ π + _________π
= _______π
≈ ________ cm²
Answer:
The given figure is:

From the given figure,
The radius of the cone (r) is: 7 cm
The slant height of the cone (l) is: 13 cm
Now,
We know that,
The surface area of the cone (S.A) = πr² + πrl
So,
S.A = 3.14 × 7² + 3.14 × 7 × 13
= 153.86 + 285.74
= 439.6 cm²
Hence, from the above,
We can conclude that the surface area of the given cone is: 439.6 cm²

Question 9.
Construct Arguments Sasha incorrectly claimed that the surface area of the cylinder is about 76.9 square inches. Explain her likely error and find the correct surface area of the cylinder.

Answer
The given figure is:
:
Now,
From the given figure,
The diameter of the cylinder is: 7 in.
The height of the cylinder is: 19 in.
Now,
We know that,
Radius = \(\frac{Diameter}{2}\)
So,
Radius of the circle = \(\frac{7}{2}\)
= 3.5 in.
Now,
We know that,
The surface area of the cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2 × 3.14 × (3.5)² + 2 × 3.14 × 3.5 × 19
= 76.93 + 417.62
= 494.55 in.²
Hence, from the above,
We can conclude that
The correct surface area of the given cylinder is: 494.55 in.²
The mistake made by Sasha is:
She adds only the area of the top and bottom but not the area of the rectangle and the area of the top and bottom of the cylinder

Question 10.
A theme park has a ride that is located in half a sphere. The ride goes around the widest part of the sphere, which has a circumference of 514.96 yards. What is the surface area of the sphere? Estimate to the nearest hundredth using 3.14 for π.

Answer:
It is given that
A theme park has a ride that is located in half a sphere. The ride goes around the widest part of the sphere, which has a circumference of 514.96 yards
Now,
The given sphere is:

Now,
We know that,
Circumference = 2πr
So,
Circumference of the theme park = 514.96 yd
2πr = 514.96
r = \(\frac{514.96}{2 × 3.14}\)
= 2.38 yd
Now,
We know that,
The surface area of the sphere (S.A) = 4πr²
So,
S.A = 4 × 3.14 × (2.38)²
= 71.14 yd²
Hence, from the above,
We can conclude that the surface area of the given sphere is: 71.14 yd²

Question 11.
Find the amount of wrapping paper you need to wrap a gift in the cylindrical box shown. You need to cover the top, the bottom, and all the way around the box. Use 3.14 for π, and round to the nearest tenth.

Answer:
The given figure is:

Now,
From the given figure,
The radius of the cylinder is 9 in.
The height of the cylinder is: 8 in
Now,
We know that,
The surface area of the cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2 × 3.14 × 9² + 2 × 3.14 × 9 × 8
= 508.68 + 452.16
= 960.84 in.²
Hence, from the above,
We can conclude that the amount of wrapping paper you need to wrap a gift in the cylindrical box is: 960.84 in.²

Question 12.
Donna paints ornaments for a school play. Each ornament is made up of two identical cones, as shown. How many bottles of paint does she need to paint 70 ornaments?

Answer:
It is given that
Donna paints ornaments for a school play. Each ornament is made up of two identical cones, as shown
Now,
From the given figure,
We can observe that
The radius of a cone is: 4.1 cm
The slant height of a cone is: 8.9 cm
Now,
We know that,
The surface area of a cone (S.A) = πr² + πrl
So,
S.A = 3.14 × (4.1)² + 3.14 × 4.1 × 8.9
= 52.78 + 114.57
= 167.35 cm²
So,
The surface area of the second cone (S.A) = 167.35 cm²
So,
The S.A of 70 ornaments = 70 × (167.35 × 2)
= 23,429 cm²
So,
The number of bottles of paint she needed to paint 70 ornaments = \(\frac{23,429}{2,000}\)
= 11.7
≅ 12 bottles
Hence, from the above,
We can conclude that she need 12 bottles of paint to paint 70 ornaments

Question 13.
Higher-Order Thinking
a. What is the surface area of the cone? Use 3.14 for π, and round to the nearest whole number.

Answer:
The given figure is:

Now,
From the given figure,
The diameter of the cone is: 6 cm
The slant height of the cone is: 12 cm
Now,
We know that,
Radius = \(\frac{Diameter}{2}\)
Radius = \(\frac{6}{2}\)
Radius = 3 cm
Now,
We know that,
The surface area of the cone (S.A) = πr² + πrl
So,
S.A = 3.14 × 3² + 3.14 × 3 × 12
= 28.26 + 113.04
= 141.3 cm²
Hence, from the above,
We can conclude that the surface area of the given cone is: 141.3 cm²

b. Reasoning Suppose the diameter and the slant height of the cone are cut in half. How does this affect the surface area of the cone? Explain.
Answer:
It is given that the diameter and the slant height of the cone are cut in half
So,
S.A = π(\(\frac{d}{2}\))² + π (\(\frac{d}{2}\)) (\(\frac{l}{2}\))
= 3.14 × \(\frac{6²}{4}\) + 3.14 × 3 × 6
= 28.26 + 56.52
= 84.78 cm²
Now,
From part (a),
The S.A of the cone is: 141.3 cm²
So,
The ratio of S.A of the cones obtained from part (a) and part (b) respectively = \(\frac{141.3}{84.78}\)
= 1.666
Hence, from the above,
We can conclude that the S.A of the cone we obtained in part (b) is 1.666 times of the S.A of the cone we obtained in part (b)

Assessment Practice
Question 14.
What is the surface area of the sphere? Use 3.14 for π, and round to the nearest tenth.

A 254.5 cm²
B. 56.55 cm²
C. 1,017.4 cm²
D. 4,071.5 cm²
Answer:
The given figure is:

From the given figure,
The radius of the sphere is: 9 cm
Now,
We know that,
The surface area of the sphere (S.A) = 4πr²
So,
S.A = 4 × 3.14 × 9²
= 1,017.36 cm²
≈ 1,017.4 cm²
Hence, from the above,
We can conclude that the S.A of the given sphere is: 1,017.4 cm²

Question 15.
What is the approximate surface area of the cone, in square inches? Use 3.14 for π, and round to the nearest whole number.

Answer:
The given figure is:

Now,
From the given figure,
The slant height of the cone (l) is 40 in.
The diameter of the cone (d) is 40 in.
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{40}{2}\)
r = 20 in.
Now,
We know that,
The surface area of the cone (S.A) = πr² + πrl
So,
S.A = 3.14 × 20² + 3.14 × 20 × 40
= 1,256 + 2,512
= 3,768 in.²
Hence, from the above,
We can conclude that the surface area of the given cone is: 3,768 in²

Lesson 8.2 Find Volume of Cylinders

Explain It!
Jenna and Ricardo are buying a new fish tank for the growing population of zebrafish in their science lab. Jenna says the tanks hold the same amount of water because they have the same dimensions. Ricardo says that he can fill the bottom of the rectangular tank with more cubes, so it can hold more water.

I can… use what I know about finding volumes of rectangular prisms to find the volume of a cylinder.

A. Look for Relationships How are the shapes of the two fish tanks alike? How are they different?
Answer:
When we observe the fish tanks of Jenna and Ricardo,
We can observe that
a. The heights of the two fish tanks are the same
b. The number of cubes that can be filled at the bottom is different
c. The number of cubes filled in Jenna’s fish tank is less than that of the number of cubes filled in Ricardo’s fish tank
d. The amount of water that can hold in Jenna’s fish tank is less than the amount of water that can hold in Ricardo’s fish tank

B. Critique Arguments Who do you think is correct, Ricardo or Jenna? Explain.
Answer:
It is given that
Jenna and Ricardo are buying a new fish tank for the growing population of zebrafish in their science lab. Jenna says the tanks hold the same amount of water because they have the same dimensions. Ricardo says that he can fill the bottom of the rectangular tank with more cubes, so it can hold more water.
Now,
To find which fish tank holds more water, find the volume of the two fish tanks
Now,
For Jenna’s fish tank:
The fish tank is in the form of a cylinder
Now,
We know that,
The volume of the cylinder (V) = πr²h
= \(\frac{πd²h}{4}\)
So,
V = \(\frac{3.14 × 24 × 24 × 48}{4}\)
= \(\frac{86,814.72}{4}\)
= 21,073.68 in³
For Ricardo’s fish tank:
The fish tank is in the form of a rectangular prism
Now,
We know that,
The volume of a rectangular prism (V) = Length × Width × Height
So,
V = 24 × 24 × 48
= 27,648 in³
Hence, from the above,
We can conclude that Ricardo is correct on the basis of volumes of their fish tanks

Focus on math practices
Use Structure How can you use what you know about areas of two-dimensional figures and volumes of prisms to compare the volumes of the fish tanks?
Answer:
We know that,
Volume = Area × Length (or) Height (or) Depth
Where,
“Area” is the area of two-dimensional figures like rectangles, circles, etc.

Essential Question
How is the volume of a cylinder related to the volume of a rectangular prism?
Answer:
Rectangular prisms and cylinders are somewhat similar because they both have two bases and a height.
The formula for the volume of a rectangular solid is
V=Bh
can also be used to find the volume of a cylinder
Where,
“B” in the rectangular prism is the area of the rectangle
“B” in the cylinder is the area of the circle

Try It!

The area of the base of the cylinder is 78.5 in.². What is the volume of the cylinder?

V = Bh
= _______ ∙ _______
= _______
The volume of the cylinder is ________ cubic inches.
Answer:
It is given that
The area of the base of the cylinder is 78.5 in.²
Now,
The given figure is:

Now,
We know that,
The volume of a cylinder (V) = Bh
Where,
“B” is defined as the area of the circle
So,
V = 78.5 × 11
= 863.5 in.³
Hence, from the above,
We can conclude that the volume of the given cylinder is: 863.5 cubic inches

Convince Me!
Why can you use the formula V = Bh to find the volume of a cylinder?
Answer:
The representation of the cylinder is:

Now,
From the given figure,
We know that,
The two circles that are in the top and bottom positions are congruent
So,
The area for both the circles is also the same
Now,
We know that,
Volume = Area × Height
Now,
We know that,
Area of the circle = πr²
Hence,
The volume of the cylinder (V) = πr²h

Try It!

Lin is building a cylindrical planter with a base diameter of 15 inches. She has 5,000 cubic inches of soil to fill her planter. What is the height of the largest planter Lin can build? Use 3.14 for π, and round to the nearest inch.
Answer:
It is given that
Lin is building a cylindrical planter with a base diameter of 15 inches. She has 5,000 cubic inches of soil to fill her planter.
So,
From the given information,
The volume of the cylindrical planter = 5,000 cubic inches
The diameter of the cylindrical planter = 15 inches
Now,
We know that,
The volume of a cylinder (V) = πr²h
= \(\frac{πd²h}{4}\)
So,
5,000 = \(\frac{3.14 × 15 × 15 × h}{4}\)
5,000 = 176.625h
h = \(\frac{5,000}{176.625}\)
h = 28.3 inches
Hence, from the above,
We can conclude that the height of the largest planter Lin can build is: 28.3 inches

KEY CONCEPT
The formula for the volume of a cylinder is the same as the formula for the volume of a prism. The formula for the volume of a cylinder is V= Bh, where B is the area of the circular base and h is the height of the cylinder.

Do You Understand?
Question 1.
Essential Question How is the volume of a cylinder related to the volume of a rectangular prism?
Answer:
Rectangular prisms and cylinders are somewhat similar because they both have two bases and a height.
The formula for the volume of a rectangular solid is
V=Bh
can also be used to find the volume of a cylinder
Where,
“B” in the rectangular prism is the area of the rectangle
“B” in the cylinder is the area of the circle

Question 2.
Use Structure What two measurements do you need to know to find the volume of a cylinder?
Answer:
We know that,
The volume of a cylinder (V) = πr²h
Now,
From the given formula,
We can observe that
π is a constant
Hence, from the above,
We can conclude that the two measurements that needed to be known are:
a. Radius of the cylinder
b. The height of the cylinder

Question 3.
Reasoning Cylinder A has a greater radius than Cylinder B. Does Cylinder A necessarily have a greater volume than Cylinder B? Explain.
Answer:
It is given that
Cylinder A has a greater radius than Cylinder B
Now,
We know that,
The volume of a cylinder (V) = πr²h
Now,
Fro the above,
We can observe that
Volume (V) ∝ Radius²
So,
When we increase the value of the radius, the value of the volume will automatically increase
Hence, from the above,
We can conclude that cylinder A has a greater volume than Cylinder B

Do You Know How?
Question 4.
What is the volume of the cylinder? Express your answer in terms of π.

Answer:
The given figure is:

Now,
We know that,
The volume of a cylinder (V) = Bh
Where,
B = πr²
So,
V = 4π × 10
V = 40π mm³
Hence, from the above,
We can conclude that the volume of the given cylinder in terms of π is: 40π mm³

Question 5.
What is the approximate height of the cylinder? Use 3.14 for π, and if necessary, round to the nearest tenth.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The volume of a cylinder (V) = 314 ft³
The radius of a cylinder = 10 ft
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
314 = 3.14 × 10 × 10 × h
h = \(\frac{314}{3.14 × 10 × 10}\)
h = 1 ft
Hence, from the above,
We can conclude that the height of the given cylinder is: 1 ft

Question 6.
What is the volume of the cylinder? Use 3.14 for π, and if necessary, round to the nearest tenth.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The height of a cylinder = 4 cm
The circumference of a circle = 22.4 cm
Now,
We know that,
The circumference of a circle = 2πr
So,
2πr = 22.4
r = \(\frac{22.4}{2 × 3.14}\)
r = 3.56 cm
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = 3.14 × 3.56 × 3.56 × 4
= 159.2 cm³
Hence, from the above,
We can conclude that the volume of the given cylinder is: 159.2 cm³

Practice & Problem Solving

Question 7.
Leveled Practice What is the volume of a cylinder with a radius of 5 centimeters and height of 2.5 centimeters? Use 3.14 for π.
V = π ________ ² ∙ ________
= π ______ ∙ ________
= _______ π
The volume of the cylinder is about _______ cubic centimeters.
Answer:
It is given that
The radius of a cylinder (r) = 5 cm
The height of a cylinder (h) = 2.5 cm
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = 3.14 × 5 × 5 × 2.5
= 196.25 cm
Hence, from the above,
We can conclude that the volume of the given cylinder is: 196.25 cm³ (or) 62.5π cm³

Question 8.
Find the volume of each cylinder in terms of. Which cylinder has the greater volume?
Cylinder A: Area of Base = 61 ft², height = 10 ft
Cylinder B: Circumference = 6π ft, height = 6 ft
Answer:
The given data is:
Cylinder A: Area of Base = 61 ft², height = 10 ft
Cylinder B: Circumference = 6π ft, height = 6 ft
Now,
We know that,
The volume of a cylinder (V) = Bh
Where,
B is the area of the circle
Now,
For Cylinder A:
V = 61 × 10
= 610 ft³
For Cylinder B:
We know that,
Circumference = 2πr
So,
2πr = 6π
r = \(\frac{6π}{2π}\)
= 3 ft
So,
V = 3.14 × 3 × 3 × 6
= 169.56 ft³
Hence, from the above,
We can conclude that Cylinder A has the greatest volume when we compare the volumes of Cylinder A and Cylinder B

Question 9.
The volume of a cylinder is 2251 cubic inches, and the height of the cylinder is 1 inch. What is the radius of the cylinder?
Answer:
It is given that
The volume of a cylinder is 2251 cubic inches, and the height of the cylinder is 1 inch
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
2251 = 3.14 × r² × 1
r² = \(\frac{2251}{3.14}\)
r² = 716.87 in²
r = 26.77 in
Hence, from the above,
We can conclude that the radius of the given cylinder is: 26.77 in.

Question 10.
A company is designing a new cylindrical water bottle. The volume of the bottle is 103 cubic centimeters. What is the radius of the water bottle? Estimate using 3.14 for π, and round to the nearest hundredth.

Answer:
It is given that
A company is designing a new cylindrical water bottle. The volume of the bottle is 103 cubic centimeters
Now,
The given figure is:

Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
103 = 3.14 × r² × 8.1
r² = \(\frac{103}{3.14 × 8.1}\)
r² = 4.05cm²
r = 2.01 cm
Hence, from the above,
We can conclude that the radius of the given cylinder is: 2.01 cm

Question 11.
Use the figure at the right.

a. Find the volume of the cylinder in terms of π.
Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The radius of the cylinder (r) = 4 in.
The height of the cylinder (h) = 3 in
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = π × 4 × 4 × 3
V = 48π in.³
Hence, from the above,
We can conclude that the volume of the given cylinder in terms of π is: 48π in.³

b. Is the volume of a cylinder, which has the same radius but twice the height, greater or less than the original cylinder? Explain.
Answer:
From part (a),
The radius of the cylinder (r) = 4 in.
The height of the cylinder (h) = 3 in
So,
Now,
For part (b),
The radius of the cylinder (r) = 4 in.
The height of the cylinder (h) = 6 in.
So,
V = π × 4 × 4 × 6
= 96π in.³
Hence, from the above,
We can conclude that the volume of the cylinder we obtained in part (b) is greater than the volume of the cylinder we obtained in part (a)

Question 12.
Reasoning A rectangular piece of cardboard with dimensions 6 inches by 8 inches is used to make the curved side of a cylinder-shaped container. Using this cardboard, what is the greatest volume the cylinder can hold? Explain.
Answer:
It is given that
A rectangular piece of cardboard with dimensions 6 inches by 8 inches is used to make the curved side of a cylinder-shaped container.
Now,
Let the height of the cylinder be: 6 inches (or) 8 inches
Now,
We know that,
The volume of a rectangular prism (V) = Length × Width × Height
So,
For h = 6 inches:
V = 6 × 8 × 6
= 288 cubic inches
For h = 8 inches:
V = 6 × 8 × 8
= 384 cubic inches
Hence, from the above,
We can conclude that the greatest volume the given cylinder can hold is: 384 cubic inches

Question 13.
The cylinder shown has a volume of 885 cubic inches.

a. What is the radius of the cylinder? Use 3.14 for π.
Answer:
It is given that
The cylinder shown has a volume of 885 cubic inches.
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The height of a cylinder (h) = 11.7 in.
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
885 = 3.14 × r² × 11.7
r² = \(\frac{885}{3.14 × 11.7}\)
r² = 24.08
r = 4.9 in.
Hence, from the above,
We can conclude that the radius of the given cylinder is: 4.9 in.

b. Reasoning If the height of the cylinder is changed, but the volume stays the same, then how will the radius change? Explain.
Answer:
We know that,
The volume of a cylinder (V) = πr²h
Now,
It is given that
Volume —–> Constant
Height —– > Changed
Now,
From the given formula,
If the volume is constant, then
h ∝ \(\frac{1}{r²}\)
Hence, from the above relation,
We can conclude that
If the value of height increases, then the value of radius decreases
If the value of height decreases, then the value of radius increases

Question 14.
Toy rubber balls are packaged in a cylinder that holds 3 balls. Find the volume of the cylinder. Use 3.14 for π, and round to the nearest tenth.

Answer:
It is given that
Toy rubber balls are packaged in a cylinder that holds 3 balls
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The height of the cylinder (h) = 20.7 cm
The diameter of the cylinder (d) = 6.9 cm
Now,
We know that,
The volume of a cylinder (V) = πr²h
= \(\frac{πd²h}{4}\)
So,
V = \(\frac{3.14 × 20.7 × 20.7 × 6.9}{4}\)
= 2,321 cm³
Hence, from the above,
We can conclude that the volume of the given cylinder is: 2,321 cm³

Question 15.
Higher-Order Thinking
An insulated collar is made to cover a pipe. Find the volume of the material used to make the collar. Let r = 3 inches, R= 5 inches, and h = 21 inches. Use 3.14 for π, and round to the nearest hundredth.

Answer:
It is given that
An insulated collar is made to cover a pipe.
Now,
The given data is:
r = 3 inches, R= 5 inches, and h = 21 inches
Now,
The radius of insulated collar = R – r
= 5 – 3
= 2 inches
Now,
The volume of a cylinder (V) = πr²h
So,
The volume of the material that is used to make the collar (V) = 3.14 × 2 × 2 × 21
= 263.76 cubic inches
Hence, from the above,
We can conclude that the volume of the material that is used to make the collar is: 263.76 cubic inches

Assessment Practice
Question 16.
The volume of a cylinder is 1,029π cubic centimeters. The height of the cylinder is 21 centimeters. What is the radius, to the nearest centimeter, of the cylinder?
Answer:
It is given that
The volume of a cylinder is 1,0291 cubic centimeters. The height of the cylinder is 21 centimeters.
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
1,029π = πr² × 21
r² = \(\frac{1,029}{21}\)
r² = 49
r = 7 cm
Hence, from the above,
We can conclude that the radius of the given cylinder is: 7 cm

Question 17.
The diameter of a cylinder is 7 yards. The height is 12 yards. What is the volume, in terms of π and to the nearest cubic yard, of the cylinder?
Answer:
It is given that
The diameter of a cylinder is 7 yards. The height is 12 yards
Now,
We know that,
The volume of a cylinder (V) = πr²h
= \(\frac{πd²h}{4}\)
So,
V = \(\frac{3.14 × 7 × 12 × 7}{4}\)
= 1,846.32 yards³ (or) 588π yards³
Hence, from the above,
We can conclude that the volume of the given cylinder is: 1,846.32 cubic yards (or) 588π cubic yards

Question 18.
A cylinder is shown. What statements about the cylinder are true?

☐ The radius of the cylinder is 2 ft.
☐ The diameter of the cylinder is 4 yd.
☐ The height of the cylinder is 8 in.
☐ The volume of the cylinder is 32 in.².
☐ The volume of the cylinder is 32π in.³.
Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The diameter of the cylinder = 4 in.
The height of the cylinder = 8 in.
So,
The radius of the cylinder = \(\frac{Diameter}{2}\)
= 2 in.
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = 3.14 × 2 × 2 × 8
= 32π in.³
= 100.48 in.³
Hence, from the above,
We can conclude that the statements that are true about the given cylinder are:

Topic 8 MID-TOPIC CHECKPOINT

Question 1.
Vocabulary Select all the statements that describe surface area and volume. Lessons 8-1 and 8-2
☐ Surface area is the sum of the areas of all the surfaces of a figure.
☐ Volume is the distance around a figure.
☐ The surface area is a three-dimensional measure.
☐ Volume is the amount of space a figure occupies.
☐ Volume is a three-dimensional measure.
Answer:
The above statements that describe the surface area and volume are:

In 2-4, use the figure at the right. Sallie packed a cone-shaped cup inside of a cylindrical package.

Question 2.
The cone-shaped cup is made out of paper. How much paper was used to make the cup, excluding the opening at the top of the cup? Use 3.14 for π, and round to the nearest tenth. Lesson 8-1
Answer:
It is given that
Sallie packed a cone-shaped cup inside of a cylindrical package and the cone-shaped cup is made out of paper
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The slant height of cone (l) is: 33 cm
The diameter of the cone (d) is: 20 cm
So,
The radius of cone (r) = \(\frac{Diameter}{2}\)
= 10 cm
Now,
We know that,
The surface area of a cone (S.A) = πr² + πrl
So,
S.A = 3.14 × 10 × 10 + 3.14 × 10 × 33
= 314 + 1,036.2
= 1,350.2 cm²
Hence, from the above,
We can conclude that
We have to use 1,350.2 cm² of paper was used to make the cup, excluding the opening at the top of the cup

Question 3.
The cylindrical package is made out of cardboard. In terms of π, how much cardboard was used to make the package? Lesson 8-1
Answer:
It is given that the cylindrical package is made out of cardboard
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The height of the cylinder (h) is: 33 cm
The diameter of the cylinder (d) is: 20 cm
So,
Radius = \(\frac{Diameter}{2}\)
= \(\frac{20}{2}\)
= 10 cm
Now,
We know that,
The surface area of a cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2π × 10² + 2π × 10 × 33
= 200π + 660π
= 860π cm²
Hence, from the above,
We can conclude that 860π cm² of cardboard was used to make the package

Question 4.
How much space does the package occupy in terms of π? Lesson 8-1
Answer:
The given figure is:

Now,
From the above,
We can observe that the given figure is a cylinder
Now,
From the given figure,
We can observe that
The height of the cylinder (h) is: 33 cm
The diameter of the cylinder (d) is: 20 cm
So,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{20}{2}\)
r = 10 cm
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = π × 10² × 33
= 3,300π cm³
Hence, from the above,
We can conclude that 3,300π cm³ of space does the package occupy in terms of π

Question 5.
What is the surface area of the sphere in terms of π? Lesson 8-1

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The radius of the sphere (r) is: 3 ft
Now,
We know that,
The surface area of a sphere (S.A) = 4πr²
So,
S.A = 4π × 3²
= 36π ft²
Hence, from the above,
We can conclude that the surface area of the given sphere is: 36π ft²

Question 6.
The volume of the cylinder is 400π cm³. What is the height of the cylinder? Lesson 8-2
A. 5 cm
B. 16 cm
C. 25 cm
D. 80 cm
Answer:
It is given that
The volume of the cylinder is 400π cm³
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The diameter of the cylinder (d) is: 10 cm
So,
Radius = \(\frac{Diameter}{2}\)
= \(\frac{10}{2}\)
= 5 cm
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
400π = π × 5² × h
h = \(\frac{400}{25}\)
h = 16 cm
Hence, from the above,
We can conclude that the volume of the given cylinder is: 16 cm

Topic 8 MID-TOPIC PERFORMANCE TASK

Melissa designed a sculpture in which a cylinder-shaped section was removed from a cube.

PART A
Before painting the surface of the sculpture, Melissa wants to sand the surface where the cylinder section was removed. What is the surface area of the section she will sand? Use 3.14 for π. Explain how you found the surface area.
Answer:
It is given that
Melissa designed a sculpture in which a cylinder-shaped section was removed from a cube.
Now,
The given figure is:
Now,
From the given figure,
The diameter of the cylinder (d) is: 5 cm
The height of the cylinder (d) is: 10 cm
So,
The radius of the cylinder (r) = \(\frac{Diameter of the cylinder}{2}\)
r = 2.5 cm
Now,
We know that,
We know that,
The surface area of a cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2 × 3.14 × 2.5 × 2.5 + 2 × 3.14 × 2.5 × 10
= 39.25 + 157
= 196.25 cm²
Hence, from the above,
We can conclude that 196.25 cm²  is the surface area of the section Melissa will sand

PART B
Melissa has a can of spray paint that covers about 6,500 square centimeters. Can Melissa apply two coats of paint to the entire sculpture? Explain. Use 3.14 for π.
Answer:
The given figure is:

Now,
From the above,
We can observe that
The scripture is the combination of the cube and the cylinder
Now,
We know that,
The surface area of a cube (S.A) = 6a²
Where,
a is the side of the cube
So,
S.A of the cube = 6 × 10²
= 600 cm²
Now,
From part (a),
The S.A of the cylinder = 196.25 cm²
So,
The S.A of the scripture = 600 + 196.25
= 796.25 cm²
So,
6,500 > 796.25 × 2
Hence, from the above,
We can conclude that Melissa can apply two coats of paint to the entire sculpture

PART C
What is the volume of the sculpture? Use 3.14 for π.
Answer:
The given figure is:

Now,
From the above,
We can observe that
The scripture is the combination of the cube and the cylinder
Now,
The diameter of the cylinder (d) is: 5 cm
The height of the cylinder (h) is: 10 cm
So,
The radius of the cylinder (r) is: 2.5 cm
Now,
We know that,
The volume of the cylinder (V) = πr²h
The volume of the cube (V) = a³
Where,
a is the side of the cube
So,
The volume of the scripture (V) = πr²h + a³
So,
V = 3.14 × 2.5 × 2.5 × 10 + 10³
= 196.25 + 1,000
= 1,196.25 cm³
Hence, from the above,
We can conclude that the volume of the scripture is: 1,196.25 cm³

Lesson 8.3 Find Volume of Cones

Solve & Discuss It!
A landscape architect uses molds for casting rectangular pyramids and rectangular prisms to make garden statues. He plans to place each finished pyramid on top of a prism. If one batch of concrete mix makes one prism or three pyramids, how does the volume of one pyramid compare to the volume of one prism? Explain.

I can… find the volume of cones.
Answer:
It is given that
A landscape architect uses molds for casting rectangular pyramids and rectangular prisms to make garden statues. He plans to place each finished pyramid on top of a prism and one batch of concrete mix makes one prism or three pyramids
Now,
We know that,
The prism is in the shape of the cylinder
The pyramid is in the shape of the cone
Now,
We know that,
The volume of a prism (V) = πr²h
The volume of a pyramid (V) = \(\frac{1}{3}\)πr²h
So,
The volume of a prism = 3 × The volume of a pyramid
Hence, from the above,
We can conclude that
The volume of a prism is 3 times the volume of a pyramid

Look for Relationships
What do you notice about the dimensions of the bases of the pyramid and prism? How are the heights of the two solids related?
Answer:
The given figure is:

Now,
From the given figure,
We can observe that
a. The dimensions of the bases of the prism and the pyramid are the same
b. The heights of the pyramid and the prism are the same

Focus on math practices
Make Sense and Persevere If the architect mixes 10 batches of concrete, how many sculptures combining 1 prism and 1 pyramid could he make? Explain.
Answer:
It is given that
one batch of concrete mix makes one prism or three pyramids
So,
The ratio of the mix of prism and pyramid in 1 batch is: 1 : 3
Hence,
For 10 batches of the mix of concrete,
We can make sculptures combining 1 prism and 1pyramid are: 10 prisms and 30 pyramids

Essential Question
How is the volume of a cone related to the volume of a cylinder?
Answer:
The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism is related. If the heights of a cone and a cylinder are equal, then the volume of the cylinder is 3 times as much as the volume of a cone

Try It!
Find the volume of the cone. Use 3.14 for π.

The volume of the cone is about ________ cubic inches.
V = ____ πr²h
≈ _______(3.14) (______)² (4)
= ________ (3.14) (______)(4)
= _________
Answer:
The given figure is:

From the above figure,
We can observe that
The radius of the cone (r) is 1.5 in.
The height of the cone (h) is 4 in.
Now,
We know that,
The volume of the cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × (1.5)² × 4
= 9.42 in.³
Hence, from the above,
We can conclude that the volume of the given cone is: 9.42 in.³

Convince Me!
If you know the volume of a cone, how can you find the volume of a cylinder that has the same height and radius as the cone?
Answer:
We know that,
If the cylinder and the cone has the same height and radius, then
The volume of cone = \(\frac{1}{3}\) × The volume of cylinder

Try It!

Find the volume of each cone.
a. Use \(\frac{22}{7}\) for π. Express the answer as a fraction.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The slant height (l) is: 5 mm
The height of the cone (h) is: 3 mm
Now,
We know that,
l² = r²+ h²
So,
5² = r² + 3²
r² = 25 – 9
r² = 16
r = 4 mm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 4² × 3
= \(\frac{1,056}{21}\) mm³
Hence, from the above,
We can conclude that the volume of the given cone in the fraction form is: \(\frac{1,056}{66}\) mm³

b. Express the volume in terms of π.

Answer:
The given figure is:

From the given figure,
We can observe that
The height of the cone (h) is: 21 ft
The circumference of the circle (C) is: 16π ft
Now,
We know that,
The circumference of the circle (C) = 2πr
So,
2πr = 16π
r = 8 ft
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × π × 8² × 21
= 448π ft³
Hence, from the above,
We can conclude that the volume of the given cone in terms of π is: 448π ft³

KEY CONCEPT

The volume of a cone is \(\frac{1}{3}\) the volume of a cylinder with the same base and height. The formula for the volume of a cone is V = \(\frac{1}{3}\)Bh, where B is the area of the base and his the height of the cone.

Do You Understand?
Question 1.
Essential Question How is the volume of a cone related to the volume of a cylinder?
Answer:
The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism is related. If the heights of a cone and a cylinder are equal, then the volume of the cylinder is 3 times as much as the volume of a cone

Question 2.
Use Structure What dimensions do you need to find the volume of a cone?
Answer:
To find the volume of a cone,
The dimensions we need are:
a. The radius of the cone
b. The height of the cone

Question 3.
Look for Relationships If you know a cone’s radius and slant height, what must you do before you can find its volume?
Answer:
If you know a cone’s radius (r) and slant height(l), then
The volume of a cone (V) is given as:
V = \(\frac{1}{3}\)πr² × \(\sqrt{l² – r²}\)

Do You Know How?
Question 4.
Wanda found a cone-shaped seashell on the beach. The shell has a height of 63 millimeters and a base radius of 8 millimeters. What is the volume of the seashell? Estimate 63 mm using \(\frac{22}{7}\) for π.

Answer:
It is given that
Wanda found a cone-shaped seashell on the beach. The shell has a height of 63 millimeters and a base radius of 8 millimeters
Now,
The given figure is:

Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 8² × 63
= 4,224 mm³
Hence, from the above,
We can conclude that the volume of the seashell is: 4,224 mm³

Question 5.
What is the volume of the cone? Estimate using 3.14 for π, and round to the nearest tenth.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The height of the cone (h) is: 40mm
The slant height of the cone (l) is: 41 mm
Now,
We know that,
l² = r² + h²
So,
41² = r²+ 40²
r² = 81
r = 9 mm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 9² × 40
= 3,391.2 mm³
Hence, from the above,
We can conclude that the volume of the given cone is: 3,391.2 mm³

Question 6.
What is the volume of the cone in terms of π if the circumference of the base is 1.4π feet?

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The height of the cone (h) is: 2.7 ft
Now,
We know that,
Circumference (C) = 2πr
So,
2πr = 1.4π
r = 0.7 ft
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × π × (0.7)² × 2.7
= 0.441π ft³
Hence, from the above,
We can conclude that the volume of the given cone is: 0.441π ft³

Practice & Problem Solving

Multimedia Leveled Practice In 7 and 8, find the volumes of the cones.
Question 7.
What is the volume of the cone? Write your answer in terms of π.

Answer:
From the given figure,
We can observe that
The radius of the cone (r) is: 3 cm
The height of the cone (h) is: 4 cm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × π × 3² × 4
= 12π cm³
Hence, from the above,
We can conclude that the volume of the given cone is: 12π cm³

Question 8.
What is the volume of the cone to the nearest hundredth? Use 3.14 for π.

Answer:
From the given figure,
We can observe that
The radius of the cone (r) is: 16 units
The height of the cone (h) is: 36 units
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 16² × 36
= 9,646.08 units³
Hence, from the above,
We can conclude that the volume of the given cone is: 9,646.08 units³

Question 9.
If a cone-shaped hole is 3 feet deep and the circumference of the base of the hole is 44 feet, what is the volume of the hole? Use \(\frac{22}{7}\) for π.
Answer:
It is given that
A cone-shaped hole is 3 feet deep and the circumference of the base of the hole is 44 feet
Now,
We know that,
Circumference (C) = 2πr
So,
2πr = 44
r = \(\frac{44}{2π}\)
r = 7 feet
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 7² × 3
= 154 feet³
Hence, from the above,
We can conclude that the volume of the given cone-shaped hole is: 154 feet³

Question 10.
The volume of the cone is 462 cubic yards. What is the radius of the cone? Use \(\frac{22}{7}\) for π.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The height of the cone (h) is: 9 yd
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
462 = \(\frac{1}{3}\) × \(\frac{22}{7}\) × r² × 9
r² = \(\frac{462}{9.428}\)
r = 7 yd
Hence, from the above,
We can conclude that the radius of the given cone is: 7 yd

Question 11.
A city engineer determines that 5,500 cubic meters of sand will be needed to combat erosion at the city’s beach. Does the city have enough sand to combat erosion? Use \(\frac{22}{7}\) for π. Explain.

Answer:
It is given that
A city engineer determines that 5,500 cubic meters of sand will be needed to combat erosion at the city’s beach.
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The slant height of the cone(l) is: 37 m
The height of the cone (h) is: 35 m
Now,
We know that,
l² = r²+ h²
37² = r² + 35²
r² = 37² – 35²
r = 12 m
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 12² × 35
= 5,280 m³
So,
5,500 > 5,280
Hence, from the above,
We can conclude that the city has enough sand to combat erosion

Question 12.
A water tank is shaped like the cone shown.

a. How much water can the tank hold? Use 3.14 for π, and round to the nearest tenth.
Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The slant height of the cone (l) is: 61 ft
The height of the cone (h) is: 60 ft
Now,
We know that,
l² = r²+ h²
61² = r² + 60²
r² = 61² – 60²
r = 11 ft
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 11² × 60
= 7,598.8 ft³
Hence, from the above,
We can conclude that the amount of water that the tank can hold is: 7,598.8 ft³

b. If water is drained from the tank to fill smaller tanks that each hold 500 cubic feet of water, how many smaller tanks can be filled?
Answer:
It is given that the water is drained from the tank to fill smaller tanks that each holds 500 cubic feet of water
Now,
From part (a),
We know that,
The amount of water that the tank can hold is: 7,598.8 ft³
Now,
The number of smaller tanks that can be filled = \(\frac{The amount of water that the tank can hold}{The amount of water that the smaller tank can hold}\)
= \(\frac{7,598.8}{500}\)
≅ 15
Hence, from the above,
We can conclude that the number of smaller tanks that can be filled is: 15

Question 13.
An ice cream cone is filled exactly level with the top of a cone. The cone has a 9-centimeter depth and a base with a circumference of 91 centimeters. How much ice cream is in the cone in terms of π?
Answer:
It is given that
An ice cream cone is filled exactly level with the top of a cone. The cone has a 9-centimeter depth and a base with a circumference of 91 centimeters
Now,
We know that,
Circumference (C) = 2πr
So,
2πr = 91
r = \(\frac{91}{2π}\)
r = 14.49 cm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × π × (14.49)² × 9
= 629.88π cm³
Hence, from the above,
We can conclude that the amount of ice cream that is in the cone in terms of π is: 629.88π cm³

Question 14.
In the scale model of a park, small green cones represent trees. What is the volume of one green cone? Use \(\frac{22}{7}\) for π.

Answer:
The given green cone is:

Now,
From the given green cone,
We can observe that
The slant height of the cone (l) is: 65 mm
The height of the cone (h) is: 63 mm
Now,
We know that,
l² = r²+ h²
65² = r² + 603²
r² = 65² – 63²
r = 16 mm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 16² × 63
= 16,896 mm³
Hence, from the above,
We can conclude that the volume of one green cone is: 16,896 mm³

Question 15.
Reasoning Compare the volumes of two cones. One has a radius of 5 feet and a slant height of 13 feet. The other one has a height of 5 feet and a slant height of 13 feet.
a. Which cone has the greater volume?
Answer:
The given data is:
Cone 1: Radius: 5 feet       Slant height: 13 feet
Cone 2: Height: 5 feet       Slant height: 13 feet
Now,
We know that,
l² = r²+ h²
So,
For cone 1:
13² = h² + 5²
h² = 13² – 5²
h = 12 feet
For cone 2:
13² = r² + 5²
r² = 13² – 5²
r = 12 feet
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
For Cone 1:
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 5² × 12
= 314.28 feet³
For Cone 2:
V = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 12² × 5
= 754.28 feet³
Hence, from the above,
We can conclude that Cone 2 has the greater volume

b. What is the volume of the larger cone in terms of π?
Answer:
From part (a),
We can observe that,
Cone 2 has a greater volume
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × π × 12² × 5
= 240π feet³

Question 16.
An artist makes a cone-shaped sculpture for an art exhibit. If the sculpture is 7 feet tall and has a base with a circumference of 24.492 feet, what is the volume of the sculpture? Use 3.14 for π, and round to the nearest hundredth.
Answer:
It is given that
An artist makes a cone-shaped sculpture for an art exhibit and the sculpture is 7 feet tall and has a base with a circumference of 24.492 feet
Now,
We know that,
Circumference (C) = 2πr
So,
2πr = 24.492
r = \(\frac{24.492}{2π}\)
r = 3.9 feet
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 3.9² × 7
= 111.43 feet³
Hence, from the above,
We can conclude that the volume of the given cone-shaped sculpture is: 111.43 feet³

Question 17.
Higher-Order Thinking A cone has a radius of 3 and a height of 11.
a. Suppose the radius is increased by 4 times its original measure. How many times greater is the volume of the larger cone than the smaller cone?
Answer:
It is given that
A cone has a radius of 3 and a height of 11
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 3² × 11
= 103.62 units³
Now,
If the radius is increased by 4 times of its original measure, then
V = \(\frac{1}{3}\) × 3.14 × (3 × 4)² × 11
= 1,657.92 units³
Now,
The number of times the volume of the larger cone than the smaller cone = \(\frac{1,657.92}{103.62}\)
= 16
Hence, from the above,
We can conclude that the volume of the larger cone is 16 times greater than the volume of the smaller cone

b. How would the volume of the cone change if the radius were divided by four?
Answer:
From part (a),
We know that,
The volume of the cone with a radius of 3 and a height of 11 units is: 103.62 units³
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × (3 ÷ 4)² × 11
= 6.476 units³
Now,
The ratio of the volume of the changed cone to the volume of the original cone = \(\frac{6.476}{103.62}\)
= 0.06
Hence, from the above,
We can conclude that the volume of the cone will be 0.06 times of the original cone

Assessment Practice
Question 18.
List the cones described below in order from least volume to greatest volume.
• Cone 1: radius 6 cm and height 12 cm
• Cone 2: radius 12 cm and height 6 cm
• Cone 3: radius 9 cm and height 8 cm
A. Cone 2, Cone 3, Cone 1
B. Cone 1, Cone 3, Cone 2
C. Cone 2, Cone 1, Cone 3
D. Cone 1, Cone 2, Cone 3
Answer:
The given data is:
a. Cone 1: radius 6 cm and height 12 cm
b. Cone 2: radius 12 cm and height 6 cm
c.  Cone 3: radius 9 cm and height 8 cm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
For Cone 1:
V = \(\frac{1}{3}\) × 3.14 × 6² × 12
= 452.16 cm³
For Cone 2:
V = \(\frac{1}{3}\) × 3.14 × 12² × 6
= 904.32 cm³
For Cone 3:
V = \(\frac{1}{3}\) × 3.14 × 9² × 8
= 678.24 cm³
Hence, from the above,
We can conclude that the order of the volumes from the least to the greatest is:
Cone 1 < Cone 3 < Cone 2

Question 19.
What is the volume, in cubic inches, of a cone that has a radius of 8 inches and a height of 12 inches? Use 3.14 for π, and round to the nearest hundredth.
Answer:
It is given that
A cone has a radius of 8 inches and a height of 12 inches
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 8² × 12
= 803.84 inches³
Hence, from the above,
We can conclude that the volume of the given cone is: 803.84 inches³

Lesson 8.4 Find Volume of Spheres

Explore It!
Marshall uses the beaker to fill the bowl with water.

I can… find the volume of a sphere and use it to solve problems.

A. Draw and label three-dimensional figures to represent the beaker and the bowl.
Answer:
The representation of the three-dimensional figures that represent the beaker and bowl are:

B. Marshall has to fill the beaker twice to completely fill the bowl with water. How can you use an equation to represent the volume of the bowl?
Answer:
The given figures are:

Now,
It is given that
Marshall has to fill the beaker twice to completely fill the bowl with water
Now,
From part (a),
We can observe that the beaker is in the form of a cone
Now,
We know that,
The volume of a cone = \(\frac{1}{3}\)πr²h
Now,
According to the given situation,
The volume of the bowl(V’) = 2 × The volume of the beaker(V)
V’ = 2 × \(\frac{1}{3}\)πr²h
Now,
From the given figures,
We can observe that
The height of a bowl is 2 times its radius
So,
V’ = \(\frac{2}{3}\)πr² (2r)
V’ = \(\frac{4}{3}\)πr³
Hence, from the above,
We can conclude that the volume of the bowl is: \(\frac{4}{3}\)πr³

Focus on math practices
Reasoning How is the volume of a sphere and the volume of a cone related? What must be true about the radius and height measurements for these relationships to be valid?
Answer:
We know that,
The relationship between the volume of the sphere and the volume of the cone is:
The volume of the sphere = 2 × The volume of the cone
Now,
For the above relationship to be valid,
a. The heights of the sphere and the cone must be the same
b. The radius of the sphere and the cone must be the same

Essential Question
How is the volume of a sphere related to the volume of a cone?
Answer:
The volume of a sphere is twice the volume of a cone that has the same circular base and height. i.e.,
The volume of the sphere = 2 × The volume of the cone

Try It!

What is the volume of a ball with a diameter of 6 centimeters? Use 3.14 for π.
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π _______³
≈ ________ ∙ ________
= __________
The volume of the ball is about _______ cm³
Answer:
It is given that
The diameter of a ball is: 6 cm
Now,
We know that,
The ball is in the form of a sphere
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{6}{2}\)
r = 3 cm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × 3.14 × 3³
= 113.04 cm³
Hence, from the above,
We can conclude that the volume of the given ball is about 113.04 cm³

Convince Me!
How is the volume of a sphere related to the volume of a cone that has the same circular base and height?
Answer:
The relationship between the volume of a cone and the volume of a sphere that has the same circular base and height is:
The volume of the sphere = 2 × The volume of the cone

Try It!

What is the volume of the composite figure shown? Use 3.14 for π.

Answer:
The given composite figure is:

Now,
We know that,
A composite figure is made up of 2 or more two-dimensional figures
Now,
The given composite figure is made up of Hemisphere and Cone
Now,
We know that,
The volume of the given composite figure = The volume of the Hemisphere + The volume of the cone
Now,
From the given composite figure,
We can observe that
Radius (r) = 2 in.
Height (h) = 6 in.
Now,
We know that,
The volume of the Hemisphere (V) = \(\frac{2}{3}\)πr³
So,
V = \(\frac{2}{3}\) × 3.14 × 2³
= 16.74 in.³
Now,
We know that,
The volume of the cone (V’) = \(\frac{1}{3}\)πr²h
So,
V’ = \(\frac{1}{3}\) × 3.14 × 2² × 6
= 25.12 in.³
So,
The volume of the given composite figure = 16.74 + 25.12
= 41.84 in.³
Hence, from the above,
We can conclude that the volume of the given composite figure is: 41.84 in.³

KEY CONCEPT

The volume of a sphere is twice the volume of a cone that has the same circular base and height. The formula for the volume of a sphere with radius r is V = \(\frac{4}{3}\) πr³.

Do You Understand?
Question 1.
Essential Question How is the volume of a sphere related to the volume of a cone?
Answer:
The volume of a sphere is twice the volume of a cone that has the same circular base and height. i.e.,
The volume of the sphere = 2 × The volume of the cone

Question 2.
Critique Reasoning Kristy incorrectly says that the volume of the sphere below is 144π cubic units. What mistake might Kristy have made?

Answer:
The given sphere is:

Now,
It is given that
Kristy incorrectly says that the volume of the sphere below is 144π cubic units
Now,
From the given sphere,
We can observe that
Radius (r) = 6 units
Now,
We know that,
The volume of a spher (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × π × 6³
= 288π units³
Hence, from the above,
We can conclude that the mistake done by Kristy is:
Considering the sphere as the hemisphere and calculated the volume of the hemisphere instead of the sphere

Question 3.
Generalize Mehnaj has a set of blocks that are all the same height. The cone-shaped block has a volume of 125 cubic inches. The sphere-shaped block has a volume of 250 cubic inches. What do you know about the radius of the base of the cone-shaped block? Explain.
Answer:
It is given that
Mehnaj has a set of blocks that are all the same height. The cone-shaped block has a volume of 125 cubic inches. The sphere-shaped block has a volume of 250 cubic inches
So,
From the given situation,
We can observe that
a. The heights of the two cone-shaped blocks are the same
b. The volumes of the two cone-shaped blocks are different
Now,
We know that,
The volume of a cone = \(\frac{1}{3}\)πr²h
So,
V₁ = \(\frac{1}{3}\)πr₁²h₁
V₂ = \(\frac{1}{3}\)πr₂²h₂
So,
V₁/V₂ = r₁²/ r₂²
r₁²/ r₂² = \(\frac{125}{250}\)
r₁² / r₂² = 0.5
r₁ / r₂ = 0.707
Hence, from the above,
We can conclude that the radius of the first cone-based block is 0.707 times the radius of the second cone-shaped block

Do You Know How?
Question 4.
Clarissa has a decorative bulb in the shape of a sphere. If it has a radius of 3 inches, what is its volume? Use 3.14 for π.
Answer:
It is given that
Clarissa has a decorative bulb in the shape of a sphere and it has a radius of 3 inches
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × 3.14 × 3³
= 113.04 inches³
Hence, from the above,
We can conclude that the volume of the given decorative bulb is: 113.04 inches³

Question 5.
A sphere has a surface area of about 803.84 square centimeters. What is the volume of the sphere? Use 3.14 for π and round to the nearest whole number.
Answer:
It is given that
A sphere has a surface area of about 803.84 square centimeters
Now,
We know that,
The surface area of a sphere (S.A) = 4πr²
So,
4πr² = 803.84
r² = \(\frac{803.84}{4π}\)
r² = 64
r = 8 cm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × 3.14 × 8³
= 2,143.57 cm³
≈2,144 cm³
Hence, from the above,
We can conclude that the volume of the given sphere is about 2,144 cm³

Question 6.
A water pipe is a cylinder 30 inches long, with a radius of 1 inch. At one end of the cylinder, there is a hemisphere. What is the volume of the water pipe? Explain.
Answer:
It is given that
A water pipe is a cylinder 30 inches long, with a radius of 1 inch. At one end of the cylinder, there is a hemisphere.
So,
The volume of the water pipe = The volume of the cylinder + The volume of the hemisphere
Now,
From the given water pipe,
We can observe that
Radius (r) = 1 in.
Height (h) = 30 in.
Now,
We know that,
The volume of the Hemisphere (V) = \(\frac{2}{3}\)πr³
So,
V = \(\frac{2}{3}\) × 3.14 × 1³
= 2.09 in.³
Now,
We know that,
The volume of the cylinder (V’) = πr²h
So,
V’ = 3.14 × 1² × 30
= 94.2 in.³
So,
The volume of the given water pipe = 2.09 + 94.2
= 96.29 in.³
Hence, from the above,
We can conclude that the volume of the given water pipe is: 96.29 in.³

Practice & Problem Solving

Question 7.
Leveled Practice What is the amount of air, in cubic centimeters, needed to fill the stability ball? Use 3.14 for π, and round to the nearest whole number. Use the formula

The volume of the stability ball is approximately __________ cubic centimeters.
Answer:
From the given figure,
We can observe that
The stability ball is in the form of a sphere
Now,
The diameter of the stability ball (d) is: 55 cm
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{55}{2}\)
r = 27.5 cm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × 3.14 × (27.5)³
= 87,069.53 cm³
≈87,070 cm³
Hence, from the above,
We can conclude that the volume of the stability ball is approximately 87,070 cm³

Question 8.
The spherical balloon has a 22-inch. diameter when it is fully inflated. Half of the air is let out of the balloon. Assume that the balloon remains a sphere. Keep all answers in terms of π.
a. Find the volume of the fully-inflated balloon.
Answer:
It is given that
The spherical balloon has a 22-inch. diameter when it is fully inflated. Half of the air is let out of the balloon.
Now,
From the given situation,
The diameter of the fully inflated balloon (d) is: 22 inches
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{22}{2}\)
r = 11 cm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × π × 11³
= 1,774.6π cm³
Hence, from the above,
We can conclude that the volume of the fully inflated balloon is: 1,774.6π cm³

b. Find the volume of the half-inflated balloon.
Answer:
It is given that
The spherical balloon has a 22-inch. diameter when it is fully inflated. Half of the air is let out of the balloon.
So,
The diameter of half-inflated balloon (d) = \(\frac{The diameter of fully inflated balloon}{2}\)
d = \(\frac{22}{2}\)
d = 11 cm
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{11}{2}\)
r = 5.5 cm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × π × (5.5)³
= 221.8π cm³
Hence, from the above,
We can conclude that the volume of the half-inflated balloon is: 221.8π cm³

c. What is the radius of the half-inflated balloon? Round to the nearest tenth.
Answer:
From part (b),
We can observe that
The diameter of half-inflated balloon (d) = \(\frac{The diameter of fully inflated balloon}{2}\)
d = \(\frac{22}{2}\)
d = 11 cm
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{11}{2}\)
r = 5.5 cm
Hence, from the above,
We can conclude that the radius of the half-inflated balloon is: 5.5 cm

Question 9.
Find the volume of the figure. Use 3.14 for π, and round to the nearest whole number.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The figure is a combination of a hemisphere and a cone
So,
From the given figure,
Diameter (d) =14 cm
Height (h) = 17 cm
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{14}{2}\)
r = 7 cm
Now,
We know that,
The volume of the figure (V) = The volume of a hemisphere + The volume of a cone
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 7² × 17
= 871.87 cm³
Now,
We know that,
The volume of a hemisphere (V’) = \(\frac{2}{3}\)πr³
So,
V’ = \(\frac{2}{3}\) × 3.14 × 7³
= 718.01 cm³
So,
The volume of the given figure = 871.87 + 718.01
= 1,589.88 cm³
≈1,590 cm³
Hence, from the above,
We can conclude that the volume of the given figure is: 1,590 cm³

Question 10.
The surface area of a sphere is about 2,826 square millimeters. What is the volume of the sphere? Use 3.14 for π, and round to the nearest whole number.
Answer:
It is given that
The surface area of a sphere is about 2,826 square millimeters
Now,
We know that,
The surface area of a sphere (S.A) = 4πr²
So,
4πr² = 2,826
r² = \(\frac{2,826}{4π}\)
r² = 225
r = 15 mm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × 3.14 × 15³
= 14,130 mm³
Hence, from the above,
We can conclude that the volume of the given sphere is: 14,130 mm³

Question 11.
A sphere has a volume of 1,837.35 cubic centimeters. What is the radius of the sphere? Use 3.14 for π, and round to the nearest tenth.
Answer:
It is given that
A sphere has a volume of 1,837.35 cubic centimeters
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
1,837.35 = \(\frac{4}{3}\)πr³
πr³ = \(\frac{1,837.35 × 3}{4}\)
πr³ = 1,378.01
r³ = \(\frac{1,378.01}{π}\)
r³ = 438.85
r = 0.333 cm
Hence, from the above,
We can conclude that the radius of the given sphere is: 0.333 cm

Question 12.
Find the volume of the solid. Use 3.14 for π, and round to the nearest whole number.

Answer:
The given figure is:

From the above,
We can observe that the given figure is a composite figure
Now,
From the given figure,
We can observe that
Radius (r) = 4 m
Height (h) = 17 m
Now,
We know that,
The volume of the given figure = The volume of a hemisphere + The volume of a cylinder
Now,
We know that,
The volume of the Hemisphere (V) = \(\frac{2}{3}\)πr³
So,
V = \(\frac{2}{3}\) × 3.14 × 4³
= 133.97 m³
Now,
We know that,
The volume of the cylinder (V’) = πr²h
So,
V’ = 3.14 × 4² × 17
= 854.08 m³
So,
The volume of the given figure = 133.9 + 854.08
= 987.98 m³
Hence, from the above,
We can conclude that the volume of the given figure is: 987.98 m³

Question 13.
Your friend says that the volume of a sphere with a diameter of 3.4 meters is 164.55 cubic meters. What mistake might your friend have made? Find the correct volume. Use 3.14 for π and round to the nearest hundredth.
Answer:
It is given that
Your friend says that the volume of a sphere with a diameter of 3.4 meters is 164.55 cubic meters
Now,
From the given information,
Diameter (d) = 3.4 m
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{3.4}{2}\)
r = 1.7 m
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × 3.14 × (1.7)³
= 20.56 m³
Hence, from the above,
We can conclude that
The mistake made by your friend is:
Consideration of diameter as radius and find the volume of the given sphere

Question 14.
A solid figure has a cone and hemisphere hollowed out of it. What is the volume of the remaining part of the solid? Use 3.14 for π, and round to the nearest whole number.

Answer:
It is given that
A solid figure has a cone and hemisphere hollowed out of it
So,
The volume of the remaining part of the solid = | The volume of a hemisphere – The volume of a cone |
Now,
From the given solid,
We can observe that
Radius (r) = 6 in.
Height = 23 in.
Now,
We know that,
The volume of a hemisphere (V) = \(\frac{2}{3}\)πr³
So,
V = \(\frac{2}{3}\)π × 6³
= 144π in.³
Now,
We know that,
The volume of a cone (V’) = \(\frac{1}{3}\)πr²h
So,
V’ = \(\frac{1}{3}\)π × 6² × 23
= 276π in.³
So,
The volume of the remaining part of the solid = |144π – 276π|
= 132π
= 132 × 3.14
= 414.48 in.³
≈415 in.³
Hence, from the above,
We can conclude that the volume of the remaining part of the given solid is about 415 in.³

Question 15.
Higher-Order Thinking A student was asked to find the volume of a solid where the inner cylinder is hollow. She incorrectly said the volume is 2,034.72 cubic inches.

Answer:
It is given that
A student was asked to find the volume of a solid where the inner cylinder is hollow. She incorrectly said the volume is 2,034.72 cubic inches.

a. Find the volume of the solid. Use 3.14 for π. Round to the nearest whole number.
Answer:
The given solid is:

Now,
From the given solid,
We can observe that it is a combination of a cone and a cylinder
So,
The volume of the given solid = The volume of a cylinder + The volume of a cone
Now,
From the given solid,
We can observe that
The radius of the cylinder is 3 in.
The height of the cylinder is 15 in.
The diameter of the cone is 12 in.
The slant height of the cone is 9 in.
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = 3.14 × 3² × 15
= 423.9 in.³
Now,
The volume of a cone:
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{12}{2}\)
r = 6 in.
Now,
We know that,
l² = r² + h²
So,
9² = 6² + h²
h² = 9² – 6²
h² = 45
h = 6.70 in.
Now,
We know that,
The volume of a cone (V’) = \(\frac{1}{3}\)πr²h
So,
V’ = \(\frac{1}{3}\) × 3.14 × 6² × 6.70
= 252.45 in.³
So,
The volume of the given solid = 423.9 + 252.45
= 676.3 in.³
≈676 in.³
Hence, from the above,
We can conclude that the volume of the given solid is about 676 in.³

b. What mistake might the student have made?
Answer:

Assessment Practice
Question 16.
The spherical boulder is 20 feet in diameter and weighs almost 8 tons. Find its volume. Use 3.14 for π. Round to the nearest cubic foot.
Answer:
It is given that
The spherical boulder is 20 feet in diameter and weighs almost 8 tons.
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{20}{2}\)
r = 10 feet
So,
V = \(\frac{4}{3}\) × 3.14 × 10³
= 4,186.6 feet³
≈ 4,187 feet³
Hence, from the above,
We can conclude that the volume of the given spherical boulder is: 4,187 feet³

Question 17.
A bowl is in the shape of a hemisphere (half a sphere) with a diameter of 13 inches. Find the volume of the bowl. Use 3.14 for π, and round to the nearest cubic inch.
Answer:
It is given that
A bowl is in the shape of a hemisphere (half a sphere) with a diameter of 13 inches. Find the volume of the bowl
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{13}{2}\)
r = 6.5 inches
Now,
We know that,
The volume of a hemisphere (V) = \(\frac{2}{3}\)πr³
So,
V = \(\frac{2}{3}\) × 3.14 × (6.5)³
= 574.88 inches³
≈ 575 inches³
Hence, from the above,
We can conclude that the volume of the given bowl is about 575 inches³

3-ACT MATH

3-Act Mathematical Modeling:
Measure Up

ACT 1
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.
Construct Arguments Predict an answer to this Main Question. Explain your prediction.
Answer:

Question 4.
On the number line below, write a number that is too small to be the answer. Write a number that is too large.

Answer:

Question 5.
Plot your prediction on the same number line.
Answer:

ACT 2
Question 6.
What information in this situation would be helpful to know? How would you use that information?

Answer:

Question 7.
Use Appropriate Tools What tools can you use to solve the problem? Explain how you would use them strategically.
Answer:

Question 8.
Model with Math
Represent the situation using mathematics. Use your representation to answer the Main Question.
Answer:

Question 9.
What is your answer to the Main Question? Is it higher or lower than your prediction? Explain why.

Answer:

ACT 3
Question 10.
Write the answer you saw in the video.

Answer:

Question 11.
Reasoning Does your answer match the answer in the video? If not, what are some reasons that would explain the difference?
Answer:

Question 12.
Make Sense and Persevere Would you change your model now that you know the answer? Explain.

Answer:

ACT 3 Extension
Reflect
Question 13.
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 14.
Make Sense and Persevere When did you struggle most while solving the problem? How did you overcome that obstacle?

Answer:

SEQUEL
Question 15.
Generalize Suppose you have a graduated cylinder half the height of the one in the video. How wide does the cylinder need to be to hold the liquid in the flask?
Answer:

Topic 8 REVIEW

Topic Essential Question
How can you find volumes and surface areas of three-dimensional figures?
Answer:
The “Surface area” is the sum of the areas of all faces (or surfaces) on a 3D shape.
Ex:
A cuboid has 6 rectangular faces. To find the surface area of a cuboid, add the areas of all 6 faces
We know that,
The volume of a three-dimensional figure = Cross-sectional area × length

Vocabulary Review
Complete each definition and then provide an example of each vocabulary word.
Vocabulary
composite figure
cone
cylinder
sphere

Answer:

Use Vocabulary in Writing
Draw a composite figure that includes any two of the following: a cylinder, a cone, a sphere, and a hemisphere. Label each part of your drawing. Then describe each part of your composite figure. Use vocabulary terms in your description.

Concepts and Skills Review

Lesson 8.1 Find Surface Area of Three-Dimensional Figures

Quick Review
Surface area is the total area of the surfaces of a three-dimensional figure. The chart gives formulas for finding the surface area of a cylinder, a cone, and a sphere.

Example
What is the surface area of the cylinder? Use 3.14 for π.
Answer:

Practice
Question 1.
What is the surface area of the cone? Use 3.14 for π.

Answer:
The given figure is:

From the given figure,
We can observe that
The slant height of the cone (l) is: 13 m
The radius of the cone (r) is: 5 m
Now,
We know that,
The surface area of a cone (S.A) = πr² + πrl
So,
S.A = 3.14 × 5² + 3.14 × 5 × 13
= 78.5 + 204.1
= 282.6 m²
Hence, from the above,
We can conclude that the surface area of the given cone is: 282.6 m²

Question 2.
What is the surface area of the sphere in terms of π?

Answer:
The given figure is:

From the given figure,
We can observe that
The diameter of the sphere (d) is: 10 cm
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{10}{2}\)
r = 5 cm
Now,
We know that,
The surface area of a sphere (S.A) = 4πr²
So,
S.A = 4 × 3.14 × 5²
= 314 cm²
Hence, from the above,
We can conclude that the surface area of the given sphere is: 314 cm²

Question 3.
What is the surface area of the cylinder in terms of π?

Answer:
The given figure is:

Now,
From the given figure,
We can observe that,
The diameter of the cylinder (d) is 12 in.
The height of the cylinder (h) is 15 in.
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{12}{2}\)
r = 6 in.
Now,
We know that,
The surface area of a cylinder (S.A) = 2πr² + 2πrh
So,
S.A = 2 × 3.14 × 6² + 2 × 3.14 × 6 × 15
= 226.08 + 565.2
= 791.28 in.³
Hence, from the above,
We can conclude that the surface area of the given cylinder is: 791.28 in.³

Lesson 8.2 Find Volume of Cylinders

Quick Review
The volume of a cylinder is equal to the area of its base times its height.
V = area of base · height, or V = πr²h

Example
What is the volume of the cylinder? Use 3.14 for π.
Answer:

Practice
Question 1.
What is the volume of the cylinder in terms of π?

Answer:
The given figure is:

From the given figure,
We can observe that
The diameter of the cylinder (d) is: 2 m
The height of the cylinder (h) is: 6 m
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{2}{2}\)
r = 1 m
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
V = 3.14 × 1² × 6
= 18.84 m³
Hence, from the above,
We can conclude that the volume of the given cylinder is: 18.84 m³

Question 2.
The volume of the cylinder is 141.3 cubic centimeters. What is the radius of the cylinder? Use 3.14 for π.

Answer:
It is given that
The volume of the cylinder is 141.3 cubic centimeters
Now,
The given figure is:

Now,
From the given figure,
We can observe that
The height of the cylinder (h) is: 5 cm
Now,
We know that,
The volume of a cylinder (V) = πr²h
So,
141.3 = 3.14 × r² × 5
r² = \(\frac{141.3}{3.14 × 5}\)
r²= 9
r = 3 cm
Hence, from the above,
We can conclude that the radius of the given cylinder is: 3 cm

Lesson 8.3 Find Volume of Cones

Quick Review
To find the volume of a cone, use the formula V = \(\frac{1}{3}\)πr²h.

Example
What is the volume of the cone? Use 3.14 for π.
Answer:

Practice
Question 1.
What is the volume of the cone in terms of π?

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The radius of the cone (r) is 3 in.
The height of the cone (h) is 8 in.
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 3² × 8
= 75.36 in.³
Hence, from the above,
We can conclude that the volume of the given cone is: 75.36 in.³

Question 2.
What is the volume of the cone? Use 3.14 for π.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The radius of the cone (r) is: 4 cm
The slant height of the cone (h) is: 5 cm
Now,
We know that,
l² = r² + h²
5² = 4² + h²
h² = 5² – 4²
h² = 9
h = 3 cm
Now,
We know that,
The volume of a cone (V) = \(\frac{1}{3}\)πr²h
So,
V = \(\frac{1}{3}\) × 3.14 × 4² × 3
= 50.24 cm³
Hence, from the above,
We can conclude that the volume of the given cone is: 50.24 cm³

Lesson 8.4 Find Volume of Spheres

Quick Review
To find the volume of a sphere, use the formula V = \(\frac{4}{3}\)πr³

Example
Find the volume of the composite figure. Use 3.14 for π.

Answer:
First, find the volume of the sphere. Use 3.14 for π.
V = \(\frac{4}{3}\)πr³
= \(\frac{4}{3}\)π(3.5)³ → Substitute 3.5 for r.
= 57.17π ≈ 179.5 cm³
Divide by 2 to find the volume of the hemisphere: 179.5 ÷ 2 ≈ 89.75 cubic centimeters.
Then, find the volume of the cone. Use 3.14 for π.
V = \(\frac{1}{3}\)πr² h
= \(\frac{1}{3}\)π(3.5)²(14) → Substitute 3.5 for r and 14 for h.
= 57.171 ≈ 179.5 cm³
The volume of the composite figure is approximately 89.75 + 179.5 ≈ 269.25 cubic centimeters.

Practice
Question 1.
What is the volume of the sphere? Use \(\frac{22}{7}\) for π.

Answer:
The given figure is:

Now,
From the given figure,
We can observe that
The radius of the sphere (r) is: 14 cm
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × \(\frac{22}{7}\) × 14³
= 11,498.6 cm³
Hence, from the above,
We can conclude that the volume of the given sphere is: 11,498.6 cm³

Question 2.
The surface area of a sphere is 1,017.36 square inches. What is the volume of the sphere? Use \(\frac{22}{7}\) for π.
Answer:
It is given that
The surface area of a sphere is 1,017.36 square inches.
Now,
We know that,
The surface area of a sphere (S.A) = 4πr²
So,
4πr² = 1,017.36
r² = \(\frac{1,017.36}{4π}\)
r² = 80.92
r = 8.99 inches
Now,
We know that,
The volume of a sphere (V) = \(\frac{4}{3}\)πr³
So,
V = \(\frac{4}{3}\) × \(\frac{22}{7}\) × (8.99)³
= 3,044.68 inches³
Hence, from the above,
We can conclude that the volume of the given sphere is: 3,044.68 inches³

Question 3.
What is the volume of the composite figure? Use \(\frac{22}{7}\) for π.

Answer:
The given figure is:

Now,
From the above,
We can observe that the given figure is a combination of a hemisphere and a cone
So,
The volume of the given figure = The volume of a hemisphere + The volume of a cone
Now,
From the given figure,
We can observe that
Diameter (d) = 4 cm
Height (h) = 10 cm
Now,
We know that,
Radius (r) = \(\frac{Diameter}{2}\)
r = \(\frac{4}{2}\)
r = 2 cm
Now,
We know that,
The volume of a hemisphere (V) = \(\frac{2}{3}\)πr³
So,
V = \(\frac{2}{3}\) × \(\frac{22}{7}\) × 2³
= 16.76 cm³
Now,
We know that,
The volume of a cone (V’) = \(\frac{1}{3}\)πr²h
So,
V’ = \(\frac{1}{3}\) × \(\frac{22}{7}\) × 2² × 10
= 41.90 cm³
So,
The volume of the given figure = 41.90 + 16.76
= 58.66 cm³
Hence, from the above,
We can conclude that the volume of the given figure is: 58.66 cm³

Topic 8 Fluency Practice

Hidden Clue
For each ordered pair, solve the equation to find the unknown coordinate. Then locate and label the corresponding point on the graph. Draw line segments to connect the points in alphabetical order. Use the completed picture to help answer the riddle below.

What do squares, triangles, pentagons, and octagons have in common?

Answer:
Step 1:
For each ordered pair, solve the equation to find the unknown coordinate.

Step 2:
Then locate and label the corresponding point on the graph.
Step 3:
Draw line segments to connect the points in alphabetical order.

Step 4:
Use the completed picture to help answer the riddle below.
All the squares, triangles, pentagons, and octagons have in common the sum of all the total angles i.e., 360°

Go through the enVision Math Common Core Grade 3 Answer Key Topic 11 Use Operations with Whole Numbers to Solve Problems regularly and improve your accuracy in solving questions.

enVision Math Common Core 3rd Grade Answers Key Topic 11 Use Operations with Whole Numbers to Solve Problems

Essential Question:
What are ways to solve 2-step problems?

enVision STEM Project: Engineering Design
Do Research Use the Internet or other sources to find information about kites. Find two designs for building a kite. What materials do you need for each design? How much do those materials cost?
Journal: Write a Report Include what you found. Also in your report:

• Find the total cost for each design.
• Decide which design is less expensive.
• Write an equation to show how much less expensive that design is.

Review What You Know

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

• equation
• product
• quotient
• unknown

Question 1.
The equal sign shows that the left side of a(n) __________ has the same value as the right side.
Answer:
The equal sign shows that the left side of a(n) equation has the same value as the right side.

Question 2.
A question mark can be a symbol for a(n) _________ value.
Answer:
A question mark can be a symbol for a(n) unknown value.

Question 3.
The answer to a division problem is the _________.
Answer:
The answer to a division problem is the Quotient .

Addition and Subtraction

Question 4.
739 – 104
Answer:

Question 5.
512 + 216
Answer:

Question 6.
710 – 569
Answer:

Question 7.
104 + 67
Answer:

Question 8.
664 + 78
Answer:

Question 9.
825 – 477
Answer:

Multiplication and Division

Question 10.
60 ÷ 6
Answer:
60 ÷ 6 = 10
Explanation :

Question 11.
40 × 4
Answer:
40 × 4 = 160
Explanation :

Question 12.
7 × 3
Answer:
7 × 3 = 21 .

Question 13.
(3 × 10) × 6 =
A. (3 × 10) + (6 × 10)
B. (3 × 6) + (10 × 6)
C. 3 × (10 × 6)
D. (10 + 10 + 10) + 6
Answer:
Option C.
(3 × 10) × 6 = 3 × 10 × 6 = 180 .
Explanation :
All signs are multiplication signs in the Equation so, just remove the brackets .

Question 14.
4 × (20 × 2) =
A. (4 × 20) × 2
B. (4 × 20) + (4 × 2)
C. 4 + (20 + 20)
D. (4 + 20) + (4 + 2)
Answer:
Option A .
4 × (20 × 2) = 4 × 20 × 2 = 160
Explanation :
All signs are multiplication signs in the equation so, just remove the brackets .

Model with Math

Question 15.
Caleb has 8 toy cars. Each car has 4 wheels. He wants to know how many wheels are on all of his cars. Represent this problem using a bar diagram and an equation. Then solve.
Answer:
Number of Car toys = 8
Number of wheels for each car toy = 4 wheels .
Total number of wheels for 8 car toys = 8 × 4 = 32 wheels .

Pick a Project

PROJECT 11A
Why do stores have sales and other promotions?
Project: Write a Skit About a Sale

PROJECT 11B
How did grapefruit first come to Florida?
Project: Create a Poster About Citrus Groves

PROJECT 11C
How would you make a budget for selling lemonade in a lemonade stand?
Project: Perform a Song About Lemonade

3-ACT MATH PREVIEW

Math Modeling
Cash Bucket

I can … model with math to solve a problem that involves computing with whole numbers..

Lesson 11.1 Solve 2-Step Word Problems: Addition and Subtraction

Solve & Share
Jen buys a backpack and a sleeping bag. She gets $10 off if the total is more than $200. What is the final cost of Jen’s order? Complete the bar diagram below. Then draw another diagram to solve the problem.
I can… draw diagrams and write equations to show how the quantities in a problem are related.

Look Back! How can you use an estimate to show that the final cost of the backpack and sleeping bag you found makes sense?
Answer :

Essential Question
How Can You Use Diagrams to Solve 2-Step Problems?

Visual Learning Bridge
The results of a car survey are shown in the table. How many fewer cars in the survey have poor fuel efficiency than have either good or average fuel efficiency? Use estimation to check the answer.

Find and answer the hidden question.
How many total cars in the survey have either good or average fuel efficiency?

Answer the original question.
How many fewer cars in the survey have poor fuel efficiency?

Convince Me! Critique Reasoning Jane used estimation to check the reasonableness of the work above. Explain whether Jane’s work makes sense.
165 and 98 is about 200.
224 minus 200 equals 24, which is close to 39.

Guided Practice

Do You Understand?
Question 1.
How do the bar diagrams help you write equations for the problem on the previous page?
Answer:
A bar is used to represent the whole. So we label it with the amount of the whole. .Finally, we indicate the required number of fractional parts with a question mark and determine the value represented.
In the above problem two quantities are given and total is calculated .

Do You Know How?
Question 2.
Josie has $145. She buys a bike for $127. The next week she saves $15. How much money does Josie have now? Complete the bar diagrams and write equations to solve.

Answer:

Explanation:
a = 145 – 127
so, a = 18
18 + 15 = $33
$33 money that Josie have now

Independent Practice

In 3, use the map. Draw diagrams and write equations to solve. Use letters to represent unknown quantities.

Question 3.
Manuel’s family drove from Louisville to Indianapolis to Detroit and then directly back to Louisville. How much farther did they drive going to Detroit than returning from Detroit?
Answer:
Distance from Louisville to Indianapolis  = 114 miles .
Distance from Indianapolis to Detroit = 283 miles .
Distance from Louisville to Indianapolis to Detroit =  t = 114 + 283 = 397 miles .

Question 4.
How can you estimate to check if your answer above is reasonable? Explain.
Answer:
Distance from Louisville to Indianapolis  = 114 miles, estimate is 100
Distance from Indianapolis to Detroit = 283 miles, estimate is 300
Distance from Louisville to Indianapolis to Detroit =  t = 114 + 283 = 397 miles, Estimated is  100+300 = 400
400 is close to 397 so, my answer is reasonable.

Problem Solving

Question 5.
Write equations to find how many more tickets were sold for the roller coaster on Saturday than for the swings on both days combined. Use letters to represent the unknown quantities. You can draw diagrams to help.

Answer:
swings on both days combined is 138 + 154 = 292
roller coaster on Saturday = 486
For more tickets let us assume as a
a = 486 – 292
a = 194

Question 6.
Higher Order Thinking Write a two-step problem about buying or selling books that can be solved using addition or subtraction. Solve your problem.
Answer: Kia bought 400 Science books Diana bought 200 Math’s books. How much more books Kia bought than Diana
Explanation:
400 – 200 = 200
200 more books Kia bought than Diana

Question 7.
enVision® STEM Lindsay has 6 boxes of toothpicks. There are 80 toothpicks in each box. Lindsay uses all of the toothpicks to build a model bridge. How many toothpicks does Lindsay use in all?
Answer:
80 x 6 = 480
Explanation:
Lindsay has 6 boxes of toothpicks.
There are 80 toothpicks in each box.
Lindsay uses all of the toothpicks to build a model bridge.
80 x 6 = 480 toothpicks that Lindsay use in all

Question 8.
Model with Math Matt had 327 plastic bottles for recycling. He recycled 118 bottles on Monday. He recycled 123 bottles on Tuesday. How many bottles does Matt have left to recycle? Use representations such as bar diagrams or equations to model with math. Use letters to represent the unknown quantities. Estimate to check your work.

Answer:
Let a be the Unknown quantity
118 + 123 = 241
a = 327 – 247
a = 80
Explanation:
80 bottles that Matt have left to recycle
Estimation:
120 + 120 = 240
320 – 240 = 80
The estimated value is close to the answer so, my answer is reasonable.

Assessment Practice

Question 9.
Ukie has 142 leaves in her collection. She gives 25 to her brother and then collects 19 more. Create and solve equations to find the number of leaves Ukie has now.
Answer:

Question 10.
Richard had $236 in his savings account. He got $45 for his birthday and saved all but $16 of it. Create and solve equations to find the amount in Richard’s savings account now.
Answer:
45 + 16 = $61
236 – 61 = $175
Explanation:
Richard had $236 in his savings account.
He got $45 for his birthday and saved all but $16 of it.
The amount in Richard’s savings account now is $175

Lesson 11.2 Solve 2-Step Word Problems: Multiplication and Division

Solve & Share
Two friends decide to share equally all of the apples they picked. They filled the bags shown with 4 apples in each bag. How many apples will each friend get?
I can … draw diagrams and write equations to show how the quantities in a problem are related.

Look Back! Tell why multiplication can be used to find the total number of apples.

Essential Question
How Can You Use Diagrams to Solve 2-Step Problems?

Visual Learning Bridge
The teams for the City Baseball Tournament are divided equally into 3 leagues. Each league is divided into 2 regions with the same number of teams in each region. How many teams are in each region?

Step 1
Find and answer the hidden question.
How many teams are in each league?

t = 24 ÷ 3
t = 8
There are 8 teams in each league.

Step 2
Use the answer to the hidden question to answer the original question.
How many teams are in each region?

r = 8 ÷ 2
r = 4
There are 4 teams in each region.

Convince Me! Make Sense and Persevere Another tournament has 2 leagues with 9 teams in each league. An equal number of teams will play on each of the 3 days of the tournament. Each team will play once. How many teams will play on each day?
Answer:
tournament has 2 leagues with 9 teams in each league.
2 x 9 = 18
An equal number of teams will play on each of the 3 days of the tournament.
18 ÷ 3 = 6
6 teams will play on each day.

Guided Practice

Do You Understand?
Question 1.
Why do you need to solve the hidden question first for the problem in Box A on the previous page?
Answer:
As it is divided in to 2 teams
first we have to know how many teams are there
so, First we found the hidden question first.

Question 2.
What multiplication equation could be written for the bar diagram in Box C on the previous page?

Answer:
3 x 8 = 24
They are inverse proportion equations.

Do You Know How?
In 3, complete the bar diagrams and write equations to solve.
Question 3.
There are 8 students in 2 school vans with the same number of students in each van. The admission fee for each student is $5. What is the total admission fee for all of the students in one van?

Answer:

Explanation:
There are 8 students in 2 school vans with the same number of students in each van.
8 ÷2 = 4
The admission fee for each student is $5.
8 x 5 = 40
What is the total admission fee for all of the students in one van

Independent Practice

In 4, draw diagrams and write equations to solve. Use letters to represent the unknown quantities.
Question 4.
Arif saves $4 each week. After 6 weeks, he spends all the money he saved on 3 items. Each item costs the same amount. How much does each item cost?
Answer: $8
Explanation:
Arif saves $4 each week. After 6 weeks,
6 x 4 = 24
he spends all the money he saved on 3 items.
24 ÷ 3 = 8
Each item costs the same amount.
Each item cost $8

Question 5.
Georgia says that because she knows 5 × 4 = 20, she knows Arif will have saved more than $20. Is Georgia correct? Explain.
Answer:  Yes, Georgia correct.
Explanation:
The information given is
Arif saves $4 each week. After 6 weeks,
6 x 4 = 24
According to the information 24 >20

Problem Solving

Question 6.
Be Precise One Grade 3 class collected 86 pounds of newspaper. The other Grade 3 class collected 65 pounds of newspaper. Which grade collected the greatest number of pounds of newspaper?

Answer:
One Grade 3 class collected 86 pounds of newspaper.
The other Grade 3 class collected 65 pounds of newspaper.
86 > 65
85 + 65 = 150
so,  Grade 3 class collected more.

Question 7.
Six pounds of pecans are shared equally between 2 friends. Each pound has 60 pecans. How many pecans does each friend get? Write equations to solve. Use letters to represent the unknown quantities.

Answer:
18 pecans that each friend get
Explanation:
Let the unknown quantity be a
Six pounds of pecans are shared equally between 2 friends.
6 ÷2 = 3
Each pound has 60 pecans.
6 x 60= 36 pecans.
Total friends are 2
36 ÷ 2 = 18
18 pecans that each friend get

Question 8.
Higher Order Thinking A store can buy boxes of 8 calculators each for $32 or boxes of 5 calculators each for $30. How much less I CICIC is each calculator if the store buys boxes of 8 calculators each instead of boxes of 5 calculators each? Complete the bar diagrams and solve.

Answer:

Explanation:
A store can buy boxes of 8 calculators each for $32 or boxes of 5 calculators each for $30.
32 ÷ 8 = 4
30 ÷ 5 = 6
2 less  is each calculator if the store buys boxes of 8 calculators each instead of boxes of 5 calculators each
6 – 4 = 2

Assessment Practice

Question 9.
In a basketball game, Morgan made 9 shots, each worth 2 points. Jim scored the same number of points in the game. Jim’s shots were worth 3 points each. How many shots did Jim make?
A. 4 shots
B. 6 shots
C. 8 shots
D. 9 shots
Answer: D
Explanation:
9 x 2 = 18
18 ÷ 3 = 9 Shots

Question 10.
Mia saved an equal amount of money each week for 6 weeks. Heather saved $3 a week for 8 weeks. Mia and Heather saved the same total amount. Which equation could you use to help find how much Mia saved each week?
A. s = 8 × 6
B. s = 6 × 3
C. s = 6 ÷ 3
D. s = 24 ÷ 6
Answer: D
Explanation:
Mia saved an equal amount of money each week for 6 weeks.
Heather saved $3 a week for 8 weeks.
8 x 3 = 24
Mia and Heather saved the same total amount.
s = 24 ÷ 6
s = 4

Lesson 11.3 Solve 2-Step Word Problems: All Operations

Solve & Share

An aquarium had 75 clownfish in a large water tank. The clownfish represented in the graph were added to this tank. How many clownfish are in the tank now? Write and explain how you found the answer.
I can … solve two-step word problems involving different operations.

Look Back! What operations did you use to solve this problem? Tell why you needed those operations.
Explanation:
9 x 5 = 45 Recent arrivals at the aquarium
An aquarium had 75 clownfish in a large water tank.
The clownfish represented in the graph were added to this tank.
75 + 45 = 120
120 clownfish are in the tank now.

Essential Question
How Can You Solve 2-Step Problems?

Visual Learning Bridge
Jill can rent a car and GPS device for $325 for 7 days. What is the cost to rent the car for a week without the GPS device? Use estimation to check the answer.
a = cost to rent the GPS for 7 days
b = cost to rent the car without the GPS for 7 days

To solve a two-step problem, you first have to find and answer the hidden question.
The hidden question is:
How much does it cost to rent the GPS for 7 days?
Write and solve an equation for the hidden question. Use a letter for the unknown quantity.
7 × $9 = a
$63 = a
It costs $63 to rent the GPS for 7 days.

Write and solve an equation for the problem. Use a letter for the unknown quantity
$325 – $63 = b
$262 = b
It costs $262 to rent the car without the GPS for 7 days.
You can use compatible numbers and mental math to estimate. 7 × 9 = 63. 63 is close to 75 and 325 – 75 is 250. 250 is close to 262, so the
answer is reasonable.

Convince Me! Use Structure Jill can rent a different car and a DVD player for $384 for 7 days. She wants to know the cost to rent the car for a week without the DVD player. Explain how this problem is different from the problem above. Then solve.

To solve a two-step problem, you first have to find and answer the hidden question.
The hidden question is:
How much does it cost to rent the DVD for 7 days?
Write and solve an equation for the hidden question. Use a letter for the unknown quantity.
7 × $6 = a
$42 = a
It costs $42 to rent the DVD for 7 days.

Write and solve an equation for the problem. Use a letter for the unknown quantity
$384 – $42 = b
$342 = b
It costs $342 to rent the car without the DVD for 7 days.
You can use compatible numbers and mental math to estimate. 7 x 6 = 42, 42 is close to 50 and 384 – 50 is 334. 334 is close to 342, so the
answer is reasonable.

Guided Practice

Do You Understand?
Question 1.
Dotty says you also can use $63 + b = $325 instead of $325 – $63 = b for the problem on the previous page. Can this equation be used to get the correct answer? Explain.
Answer:
Yes, this equation can be used to get the correct answer
When we bring Generally, the algebraic expression should be any one of the forms such as addition, subtraction, multiplication and division. To find the value of x, bring the variable to the left side and bring all the remaining values to the right side. Simplify the values to find the result.

Question 2.
When solving a two-step problem, why should you answer the hidden question first?
Answer:
For Finding the value we have to find the hidden value first.

Do You Know How?
In 3, write equations to solve. Use letters to represent unknown quantities.
Question 3.
Look at Box A on the previous page. How much would it cost Jill to rent the car for a week with the GPS and DVD player?
Answer:
Let the unknown quantity be a
Jill to rent the car for a week with the GPS and DVD player
a = 262 + 342
a = 604 it cost Jill to rent the car for a week with the GPS and DVD player

Independent Practice

In 4 and 5, write equations to solve. Use letters to represent the unknown quantities.
Question 4.
Trish bought 4 yards of rope to make a swing. Judy spent $18 on rope. How much did the two girls spend in all?

Answer: $30
Explanation:
Per yard is $3
4 x 3 = $12
The two girls spend in all
$12 + $18 = $30

Question 5.
Martha has 12 stamps. Toni has 21 stamps. Toni divides her stamps into 3 equal groups. She gives one group to Martha. How many stamps does Martha have now?

Answer:  19 stamps.
Explanation:
Martha has 12 stamps.
Toni has 21 stamps. Toni divides her stamps into 3 equal groups.
21 ÷ 3 = 7
She gives one group to Martha.
12 + 7 = 19
19 stamps that Martha have now

Problem Solving

In 6 and 7, use the fruit shown at the right.

Question 6.
Maurice needs 36 apples for his party. How much will the apples cost?
Answer:
9 apples = $3
36 ÷ 9 = 4
4 x 3 = $12
Explanation:
$12 will the apples cost

Question 7.
Higher Order Thinking Delia bought 24 lemons and 63 apples. How much did she spend on fruit?
Answer: 21 + 12 = 33 she spend on fruit
Explanation:
6 lemons = 3
9 apples = 3
Delia bought 24 lemons and 63 apples.
24 ÷  6 = 4
63 ÷  9 = 7
4 x 3 = 12
7 x 3 = 21
21 + 12 = 33 she spend on fruit

Question 8.
Vocabulary Fill in the blank. When you ________ 72 to the nearest ten, you get 70.
Answer: Regroup
Explanation:
When you Regroup 72 to the nearest ten, you get 70

Question 9.
Generalize Carla collected 328 shells. Dan collected 176 shells. How can you use compatible numbers to estimate how many shells they collected?
Answer:
The compatible numbers of 328 and 176 are 300 and 200
Explanation:
300 +200 = 500
328 + 176 = 504
504 is very close to 500
so, my estimation is reaasonable.

Question 10.
enVision® STEM Sasha is using the engineering design process to plan a scratching post for her cat. There will be 3 levels. Sasha will spend $10 on the pole. She will spend $7 on each level. Sasha’s plan to find the total cost is shown at the right. Is she correct? Explain.
3 × $10 = $30
$30 + $7 = $37
Answer: No,
Explanation:
Sasha is using the engineering design process to plan a scratching post for her cat.
There will be 3 levels. Sasha will spend $10 on the pole.
3 x 7 = $21
21 + 10 = 31$

Assessment Practice

Question 11.
Use the fruit from Exercises 6 and 7. Kaylie bought 4 bags of oranges and 1 bag of apples. How many pieces of fruit did she buy? Write equations to solve. Use letters to represent any unknown quantities.
Answer:
Explanation:
4 bags of oranges and 1 bag of apples
1 bag holds 10 oranges 4 x 10 = 40
1 bag holds 9 apples 1 x 9 = 9
40 + 9 = 49

Lesson 11.4 Problem Solving

Critique Reasoning
Solve & Share
Concert tickets for adults cost $12. Concert tickets for students cost $9. Marie has $190. She wants to buy 1 adult and 20 student tickets.

Skip says, “$190 is enough for all the tickets because $9 × 20 = $180 and $180 is less than $190.”

Does Skip’s reasoning make sense? Explain.

I can … critique the reasoning of others using what I know about estimating.
Explanation:
No, Skip’s reasoning doesn’t make sense
Skip says, “$190 is enough for all the tickets because $9 × 20 = $180 and $180 is less than $190.”
He want to buy 1 adult ticket more
that cost $12
He need $2 more.

Thinking Habits
Be a good thinker! These questions can help you.

• What questions can I ask to understand other people’s thinking?
• Are there mistakes in other people’s thinking?
• Can I improve other people’s thinking?
  

Look Back! Critique Reasoning What did Skip get correct in his reasoning and what did he get wrong?

Essential Question
How Can You Critique the Reasoning of Others?

Visual Learning Bridge
Gina has $68. She earns $9 an hour babysitting. She wants to buy a computer program for $130.
Will Gina have enough money to buy the program if she babysits for 6 hours?
Danielle solved this problem.
Her work is shown at the right.

6 × $9 = $54, which is about $60.
Gina has $68, which is about $70.
$60 + $70 = $130
Gina can buy the program.

What is Danielle’s reasoning to support her conclusion?
Danielle used an estimate to add the amount Gina made babysitting to the amount she already had.

How can I critique the reasoning of others?
I can

• ask questions for clarification.
• decide if the strategy used makes sense.
• look for flaws in estimates or calculations.

Here’s my thinking…
Danielle’s reasoning has flaws. Because Danielle rounded up each time, her total estimate adds up to more than the actual amount Gina will have.
6 × 9 = $54
$54 + $68 = $122
Danielle’s conclusion is not correct because the actual amount Gina will have is less than $130.

Convince Me! Critique Reasoning Tony says that if Gina babysits for 8 hours, she will have enough money. He reasons that 8 × $9 = $72, which rounds to $70, and $70 + $60 = $130. Does his reasoning make sense? Explain.
Explanation:
Yes his decision make sense.
130 = 130
he reaches the goal.

Guided Practice

Critique Reasoning Miguel’s and Nita’s goal is to collect 600 box tops. When you critique Miguel collects 253 box tops in January and 158 box reasoning, you can look for tops in February. Nita collects 209 box tops in all.

Teri estimates they have already collected more than clarify or improve their goal. She estimates that 250 + 150 = 400 and 400 + 200 = 600.
the reasoning.

Question 1.
What is Teri’s argument? How does she support it?
Answer:
Yes, Teri’s argument is true. She rounded to the lowest values than the expected
so, by this she can fill the goal
She estimates that 250 + 150 = 400 and 400 + 200 = 600.
253 + 158 = 411 and 411 + 209 = 620
620 > 600

Question 2.
Does Teri’s conclusion make sense? Explain.
Answer: Yes
As 620 > 600
Miguel’s and Nita’s goal is to collect 600 box tops will satisfy

Independent Practice

Critique Reasoning
Gill gets 24 stickers on Monday. She gets the same number of stickers on Tuesday. Gill then shares all of her stickers equally among 8 friends.

Liam concluded that each friend gets fewer than 5 stickers. His work is shown at the right.
Liam’s work
8 × 3 = 24
So, 24 ÷ 8 = 3.
Each friend gets 3 stickers. 3 is less than 5.

Question 3.
What is Liam’s argument? How does he support it?
Answer:
Liam’s work
8 × 3 = 24
So, 24 ÷ 8 = 3.
Each friend gets 3 stickers. 3 is less than 5.

Question 4.
Does Liam’s reasoning make sense? Explain.
Answer:
No, Gill gets 24 stickers on Monday. She gets the same number of stickers on Tuesday.
According to the statement
24 + 24 = 48
48 ÷ 8 = 6
Liam concluded that each friend gets fewer than 5 stickers.
But each friend get 6 stickers each

Question 5.
Explain the strategy you would use to improve Liam’s work.
Answer:
He did not added the Tuesday work
Thus his estimation does not make sense.

Problem Solving

Performance Task
Selling Buttons A Grade 3 class is going to buy buttons like the ones shown. Each package costs $8. Each package is 40 cm long. They need to know if $50 is enough money to buy 200 buttons.

Jim’s work
6 × 30 = 180 buttons, which is not enough.
7 × 30 = 210 buttons, so the class needs to buy 7 packages.
7 × $8 = $48
48 < 50
$50 is enough.

Question 6.
Make Sense and Persevere Which given information do you NOT need to solve this problem?
Answer:
length of the packet
Explanation:
Each package is 40 cm long. Is the irrelevant information
Which is not used to solve the equation.

Question 7.
Critique Reasoning Jim solved the problem as shown above. Does Jim’s strategy make sense? Explain.

Answer: Yes. His Critique Reasoning make sense
Jim’s work
6 × 30 = 180 buttons, which is not enough.
7 × 30 = 210 buttons, so the class needs to buy 7 packages.
7 × $8 = $48
48 < 50
$50 is enough.
By this information we can conclude that $50 is enough for 200 buttons.

Question 8.
Be Precise Are Jim’s calculations correct? Explain how you decided.
Answer: Yes, Jim’s calculations are correct
6 × 30 = 180 buttons, which is not enough.
7 × 30 = 210 buttons, so the class needs to buy 7 packages.
7 × $8 = $48
48 < 50
$50 is enough.
According to this information.

Question 9.
Use Appropriate Tools Can place-value blocks be used to check whether Jim’s math is correct? Explain.
Answer:
No, We cannot use the place value blocks

Topic 11 Fluency Practice Activity

Find a Match
Work with a partner. Point to a clue. Read the clue.

Look below the clues to find a match. Write the clue letter in the box next to the match.

Find a match for every clue.
I can… multiply and divide within 100.

Clues
A. The product is between 55 and 60.
B. The product is equal to 10 × 2.
C. The quotient has two digits.
D. The product is between 50 and 55.
E. The quotient is less than 5.
F. The product is between 30 and 40.
G. The quotient is a multiple of 3.
H. The quotient is equal to the divisor.

Explanation:

Explanation:
The table is completed according to the given clues.

Topic 11 Vocabulary Review

Understand Vocabulary
Choose the right term from the Word List. Write it in the blank.
Word List

• difference
• dividend
• divisor
• equation
• factor
• product
• quotient
• sum
• unknown

Question 1.
The missing number in an equation is a(n) ___________.
Answer: Unknown
Explanation:
The unknown numbers are called missing values.

Question 2.
A multiplication problem has more than one __________.
Answer: Factors
Explanation:
Multiplying two whole numbers gives a product. The numbers that we multiply are the factors of the product.

Question 3.
A bar diagram can help you write a(n) _________.
Answer: Sum
Explanation:
A bar diagram can be used to represent math problems. Bar diagrams show how the numbers in a word problem are related. This bar diagram models addition. The parts represent the addends.

Question 4.
In a division problem, you divide the _______ by the ________.
Answer: Dividend by the divisor
Explanation:
In a division problem, you divide the Dividend by the Divisor.

Draw a line to match the term to the result of a relationship between the numbers 80 and 4.

Answer:

Explanation:
80 – 4 = 76
80 x 4 = 320
80 ÷ 4 = 20
80 + 4 = 84

Use Vocabulary in Writing
Question 9.
There are 52 cards in a deck. Al turns 4 cards face up. Then he deals the rest of the cards to 6 people. How many cards does each person get? Explain how to solve this problem. Use at least 2 terms from the word list.
Answer: 8 cards
Explanation:
There are 52 cards in a deck. Al turns 4 cards face up.
Then he deals the rest of the cards to 6 people.
52 – 4 = 48
48 ÷ 6 = 8
Each person get 8 Cards

Topic 11 Reteaching

Set A pages 409-412

You can use more than one step to solve problems.
Use bar diagrams to help write equations.
A ship has 439 passengers. One hundred seventy-nine new passengers get on at a port. Then 250 passengers get off. How many passengers are on the ship now?

Remember to check for reasonableness after each step.

In 1, draw bar diagrams and write equations to solve the problem.
Question 1.
Mr. Sato has $800. He spends $600 on rent. Then he spends $85 on groceries. How much money does he have left?
Answer:

Explanation:
800 – 600 = 200
200 -85 = 115 money that Mr. Sato has left

Set B pages 413-416

You can use bar diagrams and equations to show how numbers are related.
Roger reads a poem with 16 lines. Each line has 7 words. The poem is on 2 pages with the same number of lines on each page. How many words are on each page?

16 ÷ 2 = a lines on each page
a = 8 lines

8 × 7 = w words on each page
w = 56 words

Remember to use unknowns to stand for the numbers you need to find.

In 1, draw bar diagrams and write equations to solve the problem.
Question 1.
A rancher has 24 cows. He puts an equal number of cows in 4 fields. Each cow produces 5 gallons of milk. How much milk do the cows in one field produce?
Answer:

24 ÷8 = 3
3 x 5 = 15 gallons of milk the cows in one field produce

Set C pages 417-420

Ryan reads a book with 420 pages. He also reads 7 magazine articles. Each article has 6 pages. How many pages does Ryan read?

First, write and solve an equation for the hidden question. Use a letter for the unknown quantity.
n = number of article pages
7 × 6 = n
42 = n
Then, write and solve an equation for the problem. Use a letter for the unknown quantity.
t = total pages read
420 + 42 = t
462 = t
Ryan reads 462 pages.

Remember to perform the operations in the correct order for the problem.

In 1 and 2, write equations to solve. Use letters to stand for unknown quantities.
Question 1.
Destiny gets $168 for selling mint cookies and chocolate cookies. She sells 8 boxes of chocolate cookies at $9 per box. How much does Destiny get for selling mint cookies?
Answer: $96 that Destiny get for selling mint cookies
Explanation:
Let the unknown value be n
8 x 9 = $72
n = 168 – 72
n = $96

Question 2.
Mary buys three books that cost $7 and one book that costs $12. How much did Mary spend on books?
Answer: $33 Mary spend on books
Explanation:
Let the total be n
3 x 7 = 21
n = 21 + 12
n = 33

Set D pages 421-424

Think about these questions to help you critique the reasoning of others.
Thinking Habits

• What questions can I ask to understand other people’s thinking?
• Are there mistakes in other people’s thinking?
• Can I improve other people’s thinking?
  

Remember to consider all parts of an explanation.

Pat needs to practice guitar for at least 40 hours this month. He has practiced 9 hours a week for the last 3 weeks. He practiced for 15 hours this week. Pat says, “3 × 9 is less than 40. So I have not practiced enough.”
Question 1.
Does Pat’s reasoning make sense? Explain.
Answer: No
Explanation:
9 x 3 = 27
27 + 15 = 42
42 > 40
He did not estimated this week practicing hours
That is the pat’s mistake.

Question 2.
How can you clarify or improve Pat’s reasoning?
Answer:
Pat says, “3 × 9 is less than 40. So I have not practiced enough.”
Explanation:
But last week he practiced 15 hours
3 × 9 is less than 40. that is 27
27 + 15 = 42

Topic 11 Assessment Practice

Question 1.
Emma works at a diner. On Monday, she served 7 tables with 6 people at each table. On Tuesday, she served 72 people. She wants to know how many more people she served on Tuesday than on Monday.

Choose the correct operations to represent this problem using equations. Write each operation on the blanks.
7 ____ 6 = m
72 _____ 42 = d.
+ – × ÷
Answer:
7 x 6 = 42
72 – 42 = 30
Explanation:
multiplication and subtraction operations are used.

Question 2.
Madison has a jar of 160 jelly beans. She saves 88 for herself. She divides the rest equally among 8 friends. She wants to find how many jelly beans each friend gets.
Which equations should she use? Select all that apply.
☐ 80 ÷ 8 = a
☐ 88 – 8 = b
☐ 160 – 88 = c
☐ 160 ÷ 80 = d
☐ 72 ÷ 8 = e
Answer:
160 – 88 = 72
72 ÷ 8 = 9
Explanation:
72 ÷ 8 = e is the equation used
80 ÷ 8 = 10
88 – 8 = 80
160 – 88 = 72
160 ÷ 80 = 2
72 ÷ 8 = 9

Question 3.
Alberto sells magazine subscriptions to his neighbors. Three neighbors each buy 3 subscriptions. There are 8 issues in each subscription. Write and solve equations to find the total number of magazine issues all of his neighbors get.
Answer: 72 issues
Explanation:
Alberto sells magazine subscriptions to his neighbors.
Three neighbors each buy 3 subscriptions
3 x 3 = 9
. There are 8 issues in each subscription.
9 x 8 = 72
the total number of magazine issues all of his neighbors get is 72

Question 4.
Ro’s diner has a jar of 585 toothpicks. At lunch, 315 toothpicks are used. At dinner, 107 toothpicks are used. Ro says that 315 – 107 = t toothpicks are used in all, and that 585 – 208 = r toothpicks are left. Is her answer reasonable? Explain why or why not.
Answer: Yes, her answer is reasonable.
Explanation:
315 – 107 = 208
585 – 208 = 377 toothpicks are left.

Question 5.
Tyler has 127 trading cards in his collection. Chloe has 63 cards in her collection. Then Chloe gives her collection away. She divides it equally among Tyler and 8 other friends. How many cards does Tyler have now?
Answer: 134 cards

Explanation:
63 ÷ 9 = 7
127 + 7 = 134 cards that Tyler have now

Question 6.
Ten people each bring 4 platters of food to a family reunion. The 120 guests all share the platters equally.
A. How many guests share 1 platter?
Answer:
10 x 4 = 40
120 ÷ 40 = 30
30 guests shares 1 platter

B. Landon thinks that the answer is 12 guests. He says, “10 + 4 = 14, and 14 rounds to 10. Then 10 × 12 = 120.” Do you agree with his reasoning? Explain.
Answer: No
Explanation:
His reasoning does not make sense
10 x 4 = 40
120 ÷ 40 = 30
30 guests shares 1 platter
He made the error.

Question 7.
Jeri collects flags of the United Nations. She wants to know how many cases she needs to display her collection. She can fit 9 flags in one case. There are 193 countries in the United Nations. Jeri does not have flags for 130 countries. Which equation could she use first to solve this problem?
A. c = 193 + 130
B. c = 193 – 130
C. f = 72 ÷ 9
D. f = 9 × 7
Answer: D,B
Explanation:
B and D both the equations are used.
193 – 130 = 63
63 ÷ 9 = 7

Question 8.
Mrs. Lazio lives 8 miles from her office. She drives to her office and back 5 days each week. On Saturday she also drives 173 miles to visit her sister. How many miles does Mrs. Lazio drive each week?
Answer:
5 x 8 = 40
40 + 173 = 213
Explanation:
Mrs. Lazio drive each week 213 miles.

Question 9.
José counted beetles for a project. He saw 5 beetles at the park each month and 3 beetles in his backyard each month. How many beetles did he count in 4 months?
A. 7
B. 32
C. 30
D. 8
Answer: B
Explanation:
5 x 4 = 20
3 x 4 = 12
20 + 12 = 32 beetles that he count in 4 months

Question 10.
Gail wants to save $68 for a jacket. She saves money for 3 weeks. Each week she saves $5. What is a good estimate for how much more money Gail needs to save? Select all that apply.
☐ $55
☐ $50
☐ $60
☐ $40
☐ $35
Answer:

Explanation:
3 x 5 = 15
68 – 15 = 53
50 and 55 are the good estimates

Topic 11 Performance Task

Filmmaking Camp
Mrs. Radner and Mr. Yu teach filmmaking at a summer camp. The students work in crews to make movies. The summer ends with the crew and actors watching all the movies.

Class Details

• Mrs. Radner helps the students who make the action and drama films.
• Mr. Yu helps the students who make the comedy films.
• There are 246 actors in all.

Use the Class Details list and Film Types table to answer Questions 1-3.

Question 1.
There are 20 actors working on each drama film. How many actors are not working on drama films?
Answer:
246 – 20 = 226
Explanation:
Total actors are 246
20 are working for Drama
226 actors are not working on drama films

Question 2.
Mrs. Radner has read 139 pages of the action-film scripts. How many more pages does she need to read to finish reading all of the action-script pages?
Answer:
In 1 film there are 126 pages
2 x 126 = 252
252 – 139 = 113
Explanation:
113 more pages that she need to read to finish reading all of the action-script pages

Question 3.
Mr. Yu says, “I have read 169 pages of the scripts for each comedy film. I need to read 27 more pages to finish reading all of the pages.” Do you agree with his reasoning? Explain why or why not.
Answer: Yes, we agree with his reasoning
Explanation:
Each film has 138
3 x 178 = 534 is the total pages
3 x 169 = 507
507 + 27 = 534
534 = 534

Question 4.
Mrs. Radner wants to find t, the total time to watch all the student films. Use the Film Lengths table to answer the following questions.

Part A
Mrs. Radner estimates t = 810 minutes. She reasons, “There are 9 films. Most of the films are 90 minutes long. Nine times 90 equals 810.” Do you agree with her reasoning? Explain why or why not.
Answer: No, We cannot agree with her statement
Explanation:
9 out of 5 are 90 minutes
Remaining 4 can be estimated to 45 minutes
half of ninety
5 x 90 = 450
1 x 30 = 30
3 x 60 = 180
450 + 180 + 30 = 660
There is a lot of difference in his estimation to the answer
so, we cannot agree

Part B
Use bar diagrams or equations to represent t. Then find t.
Answer:
5 x 90 = 450
1 x 30 = 30
3 x 60 = 180
450 + 180 + 30 = 660
t = 660

Question 5.
Mrs. Radner sets up chairs for the crew and actors to watch the films. Use the Film Lengths table and Class Details list to answer the following questions.
Part A
How many students are in the crew in all?
Answer:
12 + 10 + 20 = 42
Explanation:
42 students are in the crew in all.

Part B
Find the number of chairs Mrs. Radner needs if 147 students cannot watch the films. Use estimation to check your work.
Answer:
The total number of actors are 246 and 42 students
246 + 42 = 288
288 – 147 = 141
Explanation:
Mrs. Radner needs 141 chairs if 147 students cannot watch the films.

Go through the enVision Math Common Core Grade 2 Answer Key Topic 8 Work with Time and Money regularly and improve your accuracy in solving questions.

enVision Math Common Core 2nd Grade Answers Key Topic 8 Work with Time and Money

Essential Question:
How can you solve problems about counting money or telling time to the nearest 5 minutes?

enVision STEM Project: Money Matters
Find Out Collect examples of different types of coins and dollar bills. Describe how different coins and bills look and feel. Sort the money by size, color, and whether or not you can bend it.
Journal: Make a Book Show what you find out in a book. In your book, also:

• Tell how different types of coins are alike. Tell how they are different.
• Show as many different ways as you can to make 25¢.

Review What You Know

Vocabulary
Question 1.
Draw the hands to show 8 o’clock.

Answer:

The hands in the above clock shows 8 o’clock.
Explanation:
Clocks have 2 hands one is a little hand and other is a big hand. The little hand marks the hour. The bigger hand marks the minutes. Whichever number it’s pointing to is the hour of the day. Here the little hand is pointing to “8” it’s in the 8 o’clock hour.

Question 2.
Circle the number of minutes in one hour.
30 minutes
50 minutes
60 minutes
Answer:

There are 60 minutes in one hour.
Explanation:
A minute is a unit of time. The symbol for minute is min. There are 60 minutes in an hour.

Question 3.
Write the time below to the half hour.

Answer:
In the above clock we can see the time as 3 : 30 means 3hours 30 minutes.
Explanation:
A minute and hours are unit of time. The symbol for minute is min and for hour is hr. There are 60 minutes in an hour. There are 30 minutes in an half hour. The above clock shows the time as 3 : 30. It can be written as 3 hours 30 minutes.

Doubles Facts
Question 4.
Write each sum.
7 + 7 = ________
9 + 9 = ________
10 + 10 = ________
Doubles facts are fun.
Answer:
7 + 7 = 14
9 + 9 = 18
10 + 10 = 20
Explanation:
“Double facts” are additions in which a number is added to itself. Here seven is added to seven then we got fourteen. Nine is added to nine then we got eighteen. Next ten is added to ten then we got twenty.

Array
Question 5.
Use mental math.
How many squares are in the array?

_________ squares
Answer:
5 x 5 = 25
There are 25 squares in the array.
Explanation:
In the above image we can observe some squares. There are 5 rows and 5 columns. Multiply 5 rows of squares with 5 columns of squares. Then we got 25 squares. There are 25 squares in the array.

Math Story
Question 6.
Some pennies are in a cup. Jan takes out 22 of the pennies.
Now, 14 pennies are left in the cup. How many pennies were in the cup at the start?
_________ pennies
Answer:
Some pennies are in a cup. Jan takes out 22 of the pennies. Now, 14 pennies are left in the cup.
? – 22 = 14
? = 14 + 22
14 + 22 = 36
There are 36 pennies were in the cup at the start.
Explanation:
Some pennies are in a cup. Jan takes out 22 of the pennies. Now, 14 pennies are left in the cup. Add 14 pennies with 22 pennies then we got 36 pennies. At the start there are 36 pennies were in the cup.

Pick a Project

PROJECT 8A
What does a train schedule show?
Project: Make a Train Schedule Poster

PROJECT 8B
How much do things cost?
Project: Create an Advertisement

PROJECT 8C
Do you have the same schedule every day?
Project: Write a Daily Journal

PROJECT 8D
What would you sell if you ran your own store?
Project: Set Up a Store

Lesson 8.1 Solve Problems with Coins

Solve & Share
Kelsey had 10 cents in her piggybank. She finds 5 cents more and puts it in her bank. Then Kelsey’s mother gives her 20 cents to put in her bank. How many cents does Kelsey have in her bank now?
I can … solve problems with coins.

Answer:

Now Kelsey have 35 cents in her bank.
Explanation:
Kelsey had 10 cents in her piggybank. She finds 5 cents more and puts it in her bank. Then Kelsey’s mother gives her 20 cents to put in her bank. Add 10 cents with 5 cents and 20 cents then we get 35 cents. Now Kelsey have 35 cents in her bank.

Visual Learning Bridge

Convince Me! How many quarters have the same value as a half-dollar?
How many dimes have the same value as a half-dollar?
How many cents would Micah have if he didn’t have the half-dollar?
Answer:
1 quarter = 25￠
2 quarters = 50￠
Half-dollar = 50￠
2 quarters have the same value as a half-dollar.

1 dime = 10￠
5 dimes = 50￠
Half-dollar = 50￠
5 dimes have the same value as a half-dollar.

Micah would have 50 cents if he didn’t have the half-dollar.

Guided Practice

Count on to find each total value.
Question 1.
Li has these coins. How many cents does Li have?

Answer:

Li has 31￠.
Explanation:
Li has some coins. In the above image we can observe some coins. In that first coin is 10 cents, second coin is 10 cents, third coin is 5 cents, fourth coin is also 5 cents, fifth coin is 1 cent. By adding these coins value then we got 31 cents.

Question 2.
Manny has these coins. How many cents does Manny have?

Answer:

Manny have 86￠.
Explanation:
Manny have some coins. In the above image we can observe some coins. In that first coin is 50 cents, second coin is 25 cents, third coin is 10 cents, fourth coin is 1 cent. By adding these coins value then we got 86 cents.

Independent Practice

Count on to find each total value.

Question 3.
Jan has these coins. How many cents does Jan have?

Answer:

Jan have 20￠.
Explanation:
Jan have some coins. In the above image we can observe some coins. In that first coin is 5 cents, second coin is 5 cents, third coin is 5 cents, fourth coin is 5 cent. By adding these coins value then we got 20 cents.

Question 4.
Tim has these coins. How many cents does Tim have?

Answer:

Tim have 42￠.
Explanation:
Tim have some coins. In the above image we can observe some coins. In that first coin is 25 cents, second coin is 10 cents, third coin is 5 cents, fourth coin is 1 cent and fifth coin is 1 cent. By adding these coins value then we got 42 cents.

Question 5.
Manny has these coins. How many cents does Manny have?

Answer:

Manny have 60￠.
Explanation:
Manny  have some coins. In the above image we can observe some coins. In that first coin is 10 cents, second coin is 10 cents, third coin is 10 cents, fourth coin is 10 cent and fifth coin is 10 cents, sixth coin is 10 cents. By adding these coins value then we got 60 cents.

Question 6.
Algebra Stacey had 92¢ this morning. Write the name of the coin she lost.

Answer:

She lost the coin Nickel.
Explanation:
Algebra Stacey had 92¢ this morning. She has some coins as we can observe in the above image. She has one half-dollar is equal to 50¢, one quarter is equal to 25¢, one dime is equal to 10¢ and 2 pennies is equal to 2¢. Add these coins value. Add 50¢ with 25¢,10¢ and 2¢ then the sum is 87¢. Subtract 87¢ from 92¢ then the difference is 5¢. The missing coin in the above image is nickel. One nickel is equal to 5¢.

Problem Solving

Solve each problem below.
Question 7.
Explain Tori has 2 quarters, 1 dime, and 1 nickel. How many cents does Tori have? Show how you found your answer.
Answer:
Here Tori has 2 quarters, 1 dime and 1 nickel.

1 quarter = 25￠
2 quarters = 50￠
1 dime = 10￠
1 nickel = 5￠
Tori have 65￠.
Explanation:
Tori has 2 quarters, 1 dime, and 1 nickel. Here we have to add 50￠with 10￠and 5￠then we got 65￠. Tori have 65￠.

Question 8.
Higher Order Thinking Write a story about what coins you could use to buy an orange for 60¢.
Answer:
Ram have some coins. He has 2 quarters, 1 dime, 1 nickel and 1 penny. what coins he could use to buy an orange for 60¢.

He can use 2 quarters and 1 dime to buy an orange for 60￠.
Explanation:
Ram have some coins. He has 2 quarters, 1 dime, 1 nickel and 1 penny. He used 2 quarters and 1 dime to buy an orange for 60￠.Here 2 quarters value is 50￠ and 1 dime value is 10￠.

Question 9.
Assessment Practice Lucas has these coins.

He is buying a gift for his brother. If he had one more nickel, which item would he have exactly enough money to buy?

Answer:

Explanation:
Lucas has some coins. The coins value is 36 cents. He is buying a gift for his brother. If he had one more nickel then his coins value is will increase. 36cents plus 5 cents is equal to 41 cents. He has enough money to buy a item which was in option A.

Question 10.
Assessment Practice Jamie has 31¢. Which shows 31¢? Choose all that apply.

Answer:

Jamie has 31￠.
Explanation:
In the above image we can observe some coins. Here Jamie has 31￠. Here we have to find which option have 31￠. First option coins value is 30￠and second option coins value is 31￠,third option coins value is 31￠,fourth option coins value is 51￠. So option 2 and 3 are correct because they have 31￠.

Lesson 8.2 Continue to Solve Problems with Coins

Solve & Share
Grace has 40 cents. She gives Barry 24 cents. What coins could Grace have to start? How many cents does Grace have left? Explain.
I can … solve problems with coins.
Answer:

Grace has 40 cents. She gives Barry 24 cents.
40 – 24 = ?
40 – 24 = 16
Grace have left with 16 cents.
Explanation:
Grace has 40 cents. She gives Barry 24 cents. So Subtract 24 cents from 40 cents then we got 16 cents. Grace have to start with these coins one is quarter and one is dime, one is nickel.

Visual Learning Bridge

Convince Me! Why is subtracting 75¢ – 68¢ like subtracting 75 – 68? Explain.

Guided Practice

Solve each problem. Show your work.
Question 1.
Dora had some money in her pocket. She went to the mall and spent 3 dimes. Now Dora has 34¢. How much money was in her pocket before she went to the mall?

Answer:
Before she went to the mall she has 64¢ in her pocket.
Explanation:
Dora had some money in her pocket. She went to the mall and spent 3 dimes. Now Dora has 34¢. So add 30¢ with 34¢ then we got 64¢. Before she went to the mall she has 64¢ in her pocket.

Question 2.
Peter has 58¢. His brother gave him 2 nickels and 3 pennies. How much money does Peter have now?
_______ ¢
Answer:
1 nickel = 5¢
2 nickels = 10¢
1 penny = 1¢
3 pennies = 3¢
Peter has 58¢. His brother gave him 2 nickels and 3 pennies.
58¢ + 10¢ + 3¢ = ?
58¢ + 10¢ + 3¢ = 71¢
Now Peter have 71¢.
Explanation:
Peter has 58¢. His brother gave him 2 nickels and 3 pennies. Here 2 nickels value is equal to 10¢ and 3 pennies value is equal to 3¢. Add 58¢ with 10¢ and 3¢ then we get 71¢. Now Peter have 71¢.

Independent Practice

Solve each problem. Show your work.
Question 3.
Marco buys an apple for 42¢. He pays with 4 dimes and I nickel. How much change should Marco get?
Answer:
1 dime = 10¢
4 dimes = 40 ¢
1 nickel = 5¢
Marco buys an apple for 42¢. He pays with 4 dimes and I nickel.
40¢ + 5¢ = 45¢
45¢ – 42¢ = 3¢
Macro get 3¢ back.
Explanation:
Marco buys an apple for 42¢. He pays with 4 dimes and I nickel. Here 4 dimes value is equal to 40¢ and 1 nickel value is equal to 5¢. He payed 4 dimes and 1 nickel. So add 40¢ with 5 ¢ it results 45¢. Macro buys apples for 42¢. So subtract 42¢ from 45¢ then he got 3¢. Macro get 3¢ back.

Question 4.
Trina buys a ring. She pays for it with 9 dimes. She receives 8 pennies in change. How much does the ring cost?
Answer:
1 dime = 10￠
9 dimes = 90￠
1 penny = 1￠
8 pennies = 8￠
90￠ – 8￠ = ?
90￠ – 8￠ = 82￠
The ring cost is 82￠.
Explanation:
Trina buys a ring. She pays for it with 9 dimes. Here 9 dimes value is equal to 90￠. She receives 8 pennies in change. Here 8 pennies value is equal to 8￠. So subtract 8￠ from 90￠ which results 82￠. The ring cost is 82￠.

Question 5.
How many dimes have the same value as 2 quarters?
________ dimes
a half-dollar? _______ dimes
a dollar? _________ dimes
Answer:
1 quarter = 25￠
2 quarters = 50￠
1 dime = 10￠
2 quarters = 5 dimes 
a half-dollar = 5 dimes
a dollar = 10 dimes

Question 6.
How many pennies have the same value as
a nickel? _________ pennies
a dime? _________ pennies
a quarter? _________ pennies
a dollar? ________ pennies
Answer:
1 penny = 1￠
1 nickel = 5￠
1 dime = 10￠
1 quarter = 25￠
half-dollar = 50￠
dollar = 100￠
a nickel = 5 pennies
a dime = 10 pennies
a quarter = 25 pennies
a dollar = 100 pennies

Question 7.
Number Sense What 5 coins have a total value of a quarter? Draw the coins. Label the value of each coin.
Answer:

The five coins of nickel have a total value of a quarter.
1nickel = 5￠
5￠ + 5￠ +5￠ + 5￠ + 5￠ = 25￠
Explanation:
The five coins of nickel have a total value of a quarter. Five nickels is equal to 25￠. One quarter value is equal to 25￠.

Problem Solving

Solve each problem.
Question 8.
enVision® STEM Greg’s science class wants to sort these coins by their color. What is the total value of the silver coins?

______ ¢
Answer:

The total value of silver coins is 50￠+ 25￠ = 75￠.
Explanation:
Greg’s science class wants to sort these coins by their color. They have to find the value of silver coins. Here first two coins are silver coins. Add the two silver coins value. Add 50￠with 25￠ then we got 75￠.The total value of silver coins is 75￠.

Question 9.
Model Rodney buys a banana for 43¢ and an orange for 37¢. Draw the fewest number of coins to show how Rodney could pay for the fruit.
Answer:

Question 10.
Higher Order Thinking Ella has 4 dimes and 15 pennies. Marie has 3 quarters. Ella says she has more money because she has more coins. Is Ella correct? Explain.
Answer:
1 dime = 10¢
4 dimes = 40¢
1 penny = 1 ¢
15 pennies = 15¢
Ella has 4 dimes and 15 pennies.
40¢ +15¢ = ?
40¢ + 15¢ = 55¢
Ella has 55¢.
Marie has 3 quarters.
1 quarter = 25¢
3 quarters = 75¢
Marie has 75¢.
Ella has 55¢ and Marie has 75¢. So Ella is not correct because Marie has more money.
Explanation:
Ella has 4 dimes and 15 pennies. Add 40¢ with 15¢ which results 55¢. Marie has 3 quarters. Here 3 quarters value is equal to 75¢. Ella has 55¢ and Marie has 75¢. So Ella is not correct because Marie has more money.

Question 11.
Assessment Practice Lydia buys a toy car for 59¢. She pays with 6 dimes. Which shows how much change Lydia should get?

Answer:

option A is correct.
Lydia should get 1¢ back.
Explanation:
Lydia buys a toy car for 59¢. She pays with 6 dimes. Here 6 dimes means 60¢. Lydia should get 1¢ back.

Lesson 8.3 Solve Problems with Dollar Bills

Solve & Share
What is one way you can show 100¢ with coins? Use coins to model. Draw and label the coins you use.
I can … solve problems with money.

Answer:

Explanation:
In the above image 100￠can be shown in four different ways with coins.
In First step two half-dollars is equal to 100￠.
In second step four quarters is equal to 100￠.
In third step two quarters and five dimes is equal to 100￠.
In fourth step one quarter and five dimes and five nickel is equal to 100￠.

Visual Learning Bridge

Convince Me! How is counting dollar bills like counting coins? How is it different?

Guided Practice

Solve each problem.
Question 1.
Mr. Park has these dollar bills. Count on to find the total value.

Answer:

$20 $30 $40 $45 $50
20 + 10 + 10 + 5 + 5
The total value is $50. Mr. Park has $50
Explanation:
Mr. Park has the above dollar bills. Count from the greatest bill to the least bill. Add $20 with $10, $10, $5, $5. The total value is $50. Mr. Park has $50.

Question 2.
Ms. Lenz has these dollar bills. Count on to find the total value.

Answer:

Ms. Lenz has $36.
Explanation:
Ms. Lenz has the above dollar bills. Count from the greatest bill to the least bill. Add $10 with $10, $10, $5, $1. The total value is $36. Ms. Lenz has $36.

Independent Practice

Solve each problem.
Question 3.
Mr. Higgins has these dollar bills. Count on to find the total value.

Answer:

The total value is $27.
Mr. Higgins has $27.
Explanation:
Mr. Higgins has above dollar bills. Count from the greatest bill to the least bill. Add $10 with $10, $5, $1, $1. The total value is $27. Mr. Higgins has $27.

Question 4.
Ms. Nguen has these dollar bills. Count on to find the total value.

Answer:

The total value is $45.
Ms. Nguen has $45.
Explanation:
Ms. Nguen has above dollar bills. Count from the greatest bill to the least bill. Add $20 with $10, $5, $5, $5. The total value is $45. Ms. Nguen has $45.

Question 5.
Mr. Abreu has these dollar bills. Count on to find the total value.

Answer:

The total value is $23.
Mr. Abreu has $23
Explanation:
Mr. Abreu has above dollar bills. Count from the greatest bill to the least bill. Add $10 with $10, $1, $1, $1. The total value is $23. Mr. Abreu has $23.

Question 6.
Ms. Wills has these dollar bills. Count on to find the total value.

Answer:

The total value is $300.
Ms. Wills has $300.
Explanation:
Ms. Will has above dollar bills. Count from the greatest bill to the least bill. Add $100 with $100 and $100. The total value is $300. Ms. Will has $300.

Question 7.
Number Sense Mr. Anson has $26 in his wallet. What is the least number of bills he can have? Draw the bills.
________ bills
Answer:

Mr. Anson has the least number of bills are $1, $5, $10, $10.
Explanation:
Mr. Anson has $26 in his wallet. He can have the least number of bills which are $1, $5, $10, $10.

Problem Solving

Solve each problem.
Question 8.
Model Diana buys shoes on sale for $28. Draw dollar bills that she could use to pay for the shoes.
Answer:

Diana uses these dollar bills $20, $5, $1, $1, $1 to pay for the shoes.
Explanation:
Diana buys shoes on sale for $28. She uses the above dollar bills which are $20, $5, $1, $1, $1 to pay for the shoes.

Question 9.
Mrs. Baker has two $10 bills and three $5 bills in her purse. Does she have enough money to buy a dress that costs $33? Explain.
Answer:
Mrs. Baker has two $10 bills and three $5 bills in her purse.
$10 + $10 + $5 + $5 + $5 = ?
$10 + $10 + $5 + $5 + $5 = $35
Mrs. Baker has $35 in her purse. Yes she has enough money to buy a dress that costs $33.
Explanation:
Mrs. Baker has two $10 bills and three $5 bills in her purse. Add two $10 bills and three $5 bills which results $35. Mrs. Baker has $35 in her purse. Yes she has enough money to buy a dress that costs $33.

Question 10.
Higher Order Thinking Roger buys a baseball bat that costs $27. He pays the clerk with two $20 bills. What bills can the clerk give him back as change?
Answer:
Two $20 bills is equal to $40.
$40 – $27 = ?
$40 – $27 = $13
The bills that clerk gives him back as change are $10, $1, $1, $1.
Explanation:
Roger buys a baseball bat that costs $27. He pays the clerk with two $20 bills. That means he pays $40. The base ball bat is $27. Subtract $27 from $40 the difference is $13. The bills that clerk gives him back as change are $10, $1, $1, $1.

Question 11.
Assessment Practice The dollar bills below show the total cost of tickets for a soccer game.

How much do the tickets cost?
A. $5
B. $31
C. $40
D. $41
Answer:

The tickets cost for a soccer game is $41.
Explanation:
Count from the greatest bill to the least bill. Add $20 with $10, $5, $5, $1. The total value is $41. The tickets cost for a soccer game is $41.

Lesson 8.4 Continue to Solve Problems with Dollar Bills

Solve & Share
Billy, Sarah, and Maria have the same amount of money. Billy has a $1 bill. Sarah has only dimes. Maria has only nickels.
How many dimes does Sarah have?
How many nickels does Maria have?
Explain how you found your answers.
I can … solve problems about money.

Sarah has ________ dimes.
Maria has _________ nickels.
Answer:
Billy, Sarah, and Maria have the same amount of money.
Billy has a $1 bill.
$1 bill = 100￠
Sarah has only dimes.
1 dime = 10￠
10 dimes = 100￠
Sarah has 10 dimes.
Maria has only nickels.
1 nickel = 5￠
20 nickel = 100￠
Maria has 20 nickels.
Explanation:
Billy, Sarah, and Maria have the same amount of money. Billy has a $1 bill. Here $1 bill is equal to the 100￠ as we can observe in the above image. Sarah has only dimes. Here 1 dime is equal to the 10￠ then 10 dimes is equal to the 100￠. Sarah has 10 dimes. Maria has only nickels. Here 1 nickel is equal to the 5￠ then 20 nickels is equal to 100￠. Maria has 20 nickels.

Visual Learning Bridge

Convince Me! Tina solves the problem above by subtracting $100 – $25 and then subtracting $21 from that difference. Do you think she got the same answer as Tammy? Explain.
Answer :
In the above image we can observe that Tammy has $100 bill. She buys the the game with $25 and the toy dog with $21.
Tina Subtracted the cost of the game $25 from total number of bill $100. Then we got the difference as $75.
$100 – $25 = ?
$100 – $25 = $75
Now Tammy has $75 bill.
After that he subtracted the cost of the toy dog is $21 from the difference which is $75.
$75 – $21 = ?
$75 – $21 = $54
Tammy has $54 left.
Tina got the same answer as Tammy.

Guided Practice

Solve each problem. Show your work.
Question 1.
Sam had some money in his wallet. He went to the carnival and spent $12. Now Sam has $5. How much was in his wallet before the carnival?

Answer:
$12 + $5 = ?
$12 + $5 = $17
Before carnival Sam has $17 in his wallet.
Explanation:
Sam had some money in his wallet. He went to the carnival and spent $12. Now Sam has $5. So add $12 with $5 then the sum is $17. Sam had $17 in his wallet before the carnival.

Question 2.
Morgan has $7. Her grandmother gives her a $10 bill and a $5 bill. How much money does Morgan have now?
$ ________
Answer:
$7 + $10 + $5 = ?
$7 + $10 + $5 = $22
Morgan have $22.
Explanation:
Morgan has $7. Her grandmother gives her a $10 bill and a $5 bill. Perform addition operation on $7 with $10 and $5 then the sum is $22. Morgan have $22.

Independent Practice

Solve each problem. Show your work.
Question 3.
Cindy buys a carton of milk for 38¢. She pays with 2 quarters. How much change should Cindy get?
Answer:
1 quarter = $25
2 quarters = $50
$50 – $38 = ?
$50 – $38 = $12
Cindy get $12.
Explanation:
Cindy buys a carton of milk for 38¢. She pays with 2 quarters. Here 2 quarters is equal to $50. Subtract 38$ from $50 then the difference is $12. Cindy get $12.

Question 4.
Roberto buys a baseball mitt. He pays for it with a $100 bill and receives $29 in change. How much does the baseball mitt cost?
Answer:
$100 – $29 = ?
$100 – $29 = $71
The baseball mitt cost is $71.
Explanation:
Roberto buys a baseball mitt. He pays for it with a $100 bill and receives $29 in change. Subtract $29 from $100 then the difference is $71. The baseball mitt cost is $71.

Question 5.
Latrice has bills that have a total value of $50. Draw the bills that Latrice could have.
Answer:

Latrice could have $20, $20, $10.
Explanation:
Latrice has bills that have a total value of $50. The above image shows the Latrice bills. Latrice could have $20, $20, $10.

Question 6.
How many $1 bills have the same value as
a $5 bill? _______ $1 bills
a $10 bill? _______ $1 bills
a $20 bill? _______ $1 bills
a $100 bill? _______ $1 bills
Answer:
A $5 bill have the same value as five $1 bills. 5x$1=$5
A $10 bill have the same value as ten $1 bills. 10x$1=$10
A $20 bill have the same value as twenty $1 bills. 20x$1=$20
A $100 bill have the same value as hundred $1 bills. 100x$1=$100

Question 7.
Higher Order Thinking Jeff buys a book that costs $34. He pays for it with a $100 bill. Draw the bills that Jeff could receive as change.
Answer:

$100 – $34 = ?
$100 – $34 = $66
Jeff receive $66.
Explanation:
Jeff buys a book that costs $34. He pays for it with a $100 bill. Subtract $34 from $100 then the difference is $66. Jeff receive $66. Jeff receives the change as $20, $20, $20, $5 and $1.

Problem Solving

Solve each problem. Show your work.
Question 8.
Make Sense Lily has two $10 bills, three $5 bills, and one $1 bill. She gives Grace $11. How much money does Lily have left?

$ _______
Answer:
Two $10 bills is equal to $20.
Three $5 bills is equal to $15.
one $1 bill is equal to $1.
$20 + $15 + $1 = $36
Lilly gives Grace $11.
$36 – $11 = $25
Lily have $25
Explanation:
Lily has two $10 bills, three $5 bills, and one $1 bill. By adding $20 with $15 and $1 then the sum is $36. She gives Grace $11. Subtract $11 from $36 then the difference is $25. Lily left with $25.

Question 9.
Make Sense Isaac wants to buy a backpack for $20. He has two $5 bills and nine $1 bills. How much more money does he need to buy the backpack?
$ ________
Answer:
Two $5 bills is equal to $10.
Nine $1 bills is equal to $9.
$10 + $9 = ?
$10 + $9 = $19
Isaac wants to buy a backpack for $20.
$20 – $19 = ?
$20 – $19 = $1
He need more $1 to buy the backpack.
Explanation:
Isaac wants to buy a backpack for $20. He has two $5 bills and nine $1 bills. Add $10 with $9 then the sum is $19. Subtract $19 from $20 then the difference is $1. He need more $1 to buy the backpack.

Question 10.
Higher Order Thinking Henry has two $10 bills, two $5 bills, and three $1 bills. Mr. Harper has one $100 bill. Henry says he has more money because he has seven bills and Mr. Harper only has one bill. Is Henry correct? Explain.
Answer:
Henry
Two $10 bills is equal to $20.
Two $5 bills is equal to $10
Three $1 bills is equal to $3.
$20 + $10 + $3 = $33
Mr. Harper
one $100 bill is equal to $100.
Henry is not correct because Henry has $33 and Mr. Harper has $100.
Explanation:
Henry has two $10 bills, two $5 bills, and three $1 bills. Add $20 with $10 and $3 then the sum is $33. Mr. Harper has one $100 bill. Mr. harper has $100. Henry is not correct because Henry has $33 and Mr. Harper has $100.

Question 11.
Assessment Practice Olivia has one $20 bill, three $5 bills, and nine $1 bills. How much more money does Olivia need to buy a coat that costs $49? Explain.
What bills can you use to show how much money Olivia needs?
Answer:
one $20 bill is equal to $20.
Three $5 bills is equal to $15.
Nine $1 bills is equal to $9.
$20 + $15 + $9 = $44
Olivia need to buy a coat that costs $49.
$49 – $44 = ?
$49 – $44 = $5
Oliva needs more $5 to buy a coat.
Explanation:
Olivia has one $20 bill, three $5 bills, and nine $1 bills. Add $20 with $15 and $9 then the sum is $44. Olivia need to buy a coat that costs $49. Subtract $44 from $49 then the difference is $5. Oliva needs more $5 to buy a coat.

Lesson 8.5 Problem Solving

Reasoning
Solve & Share
Suppose you want to buy a pencil that costs 35¢. How many different ways can you use nickels, dimes, or quarters to make 35¢? Show each way. Tell how you know.
I can … reason about values of coins and find different ways to make the same total value.
Answer:

25¢ + 10¢ = 35¢
10¢ + 10¢ + 5¢ + 5¢ + 5¢ = 35¢
25¢ + 5¢ + 5¢ = 35¢
10¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ = 35¢
Explanation:
I buy a pencil that costs 35¢ in four different ways.
First way add one quarter with one dime then the sum is 35¢.
Second way add two dimes with three nickels then the sum is 35¢.
Third way add one quarter with two nickel then the sum is 35¢.
Fourth way add one dime with five quarters then the sum is 35¢.

Thinking Habits
What do the numbers and symbols in the problem mean?
How do the values of the coins relate to the total?

Visual Learning Bridge

Convince Me! Use the chart above. Write equations to show the ways to make 25¢ using dimes and nickels.
Answer:
The equations to show the ways to make 25¢ using dimes and nickels are as below.
10¢ +10¢ + 5¢ = 25¢
10¢ + 5¢ + 5¢ + 5¢ = 25¢
5¢ + 5¢ + 5¢ + 5¢ + 5¢ = 25¢
Explanation:
First way add two dimes with one nickel then the sum is 25¢.
Second way add one dime with three nickels then the sum is 25¢.
Third way five nickels is equal to 25¢.

Guided Practice

Use reasoning. Complete the table.
Question 1.
Tony wants to buy a pencil.

Answer:

50¢ + 5¢ = 55¢
25¢ +25¢ + 5 = 55¢
25¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ = 55¢
5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ + 5¢ = 55¢
Explanation:
First way add one half dollar with one nickel then the sum is 55¢.
Second way add two quarters with one nickel then the sum is 55¢.
Third way add one quarter with six nickels then the sum is 55¢.
Fourth way eleven quarters is equal to 55¢.

Independent Practice

Use reasoning. Complete each table.
Question 2.
Sue needs $1 to buy a book. She has nickels, dimes, quarters, and half-dollars. Find 3 more ways Sue can make $1.

Answer:

25¢ + 25¢ + 10¢ + 10¢ + 10¢ + 10¢ + 5¢ + 5¢ = $1
50¢ + 25¢ + 10¢ + 10¢ +5¢ = $1
50¢ + 25¢ + 25¢ = $1
50¢ + 10¢ + 10¢ + 10¢ + 10¢ + 5¢ +5¢ = 1
Explanation:
Sue needs $1 to buy a book. She has nickels, dimes, quarters, and half-dollars. The other 3 more ways Sue can make $1.
First way add one half-dollar with one quarter, two dimes and 1 nickel then the sum is $1.
Second way add one half-dollar with two quarters then the sum is $1.
Third way add one half-dollar with four dimes and two nickels then the sum is $1.

Question 3.
Raul wants to buy a bookmark for 14¢. He has dimes, nickels, and pennies. Find all of the ways he can make 14¢.

Answer:

10¢ + 1¢ + 1¢ + 1¢ + 1¢ =14¢
5¢ + 5¢ + 1¢ + 1¢ + 1¢ + 1¢ = 14¢
5¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ = 14¢
1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢ + 1¢= 14¢
Explanation:
Raul wants to buy a bookmark for 14¢. He has dimes, nickels, and pennies. All of the ways he can make 14¢.
First way add one dime with four pennies then the sum is 14¢.
Second way add two nickel with four pennies then the sum is 14¢.
Third way add one nickel with nine pennies then the sum is 14¢.
Fourth way fourteen pennies is equal to 14¢.

Number Sense What is the least number of coins that you could use to make each amount? You can use the tables above to help.
Question 4.
$1
Number of coins: __________
Coins I would use: __________
Answer:
Number of coins: 3.
Coins I would use: one half-dollar and two quarters.
Explanation:
In the above table of $1. There are different ways to get total amount $1. The least number of coins that I used are 3 coins. The three coins are one half dollar and two quarters. These three coins value is equal to $1.

Question 5.
14¢
Number of coins: __________
Coins I would use: __________
Answer:
Number of coins: 5
Coins I would use: one dime and 4 pennies.
Explanation:
In the above table of 14¢. There are different ways to get total amount 14¢. The least number of coins that I used are 5 coins. The five coins are one dime and four pennies. These five coins value is equal to 14¢.

Question 6.
Write an equation to show the total value of 2 nickels and 4 pennies.
Answer:
2 nickels = 10¢
4 pennies = 4¢
10¢ +4¢ = 14¢
The total value is 14¢.
Explanation:
Perform addition operation on 2 nickels and 4 pennies then the sum is 14¢.

Problem Solving

Performance Task

Carnival Game Money Don wants to use these coins to play as many carnival games as he can. Each game costs 40¢.

How can Don spend the coins that are shown at the right?

Question 7.
Use Models Choose from the coins shown. Show one way Don can spend 40¢ on one game. Write an equation.
Answer:

The one way Don can spend 40¢ on one game.
25¢ + 10¢ + 5¢ = ?
25¢ + 10¢ + 5¢ = 40¢
Explanation:
In the above image we can observe some coins. The one way Don can spend 40¢ on one game with some coins. In the above image I take three coins from first row at the right side part. The coins are one quarter and one dime and one nickel add these coins. Add 25¢ with 10¢ and 5¢ then the sum is 40¢.

Question 8.
Make Sense You used some of the coins for one of the ways. How will you know which coins are left to spend?
Answer:
I used some coins to spend 40¢ on one game. The coins I used from first row at the right side part. So I can know the coins left to spend on game.

Question 9.
Reasoning Show how Dan can spend the coins to play games. Use tally marks in the table.
Which coin is left over?

Answer:

One penny coin is left over.
Explanation:
Don can spend the coins in three different ways to play games.
First way he spend the coins to play game are one quarter, one dime and one nickel. Add 25￠ with 10￠ and 5￠then the sum is 40￠.
Second way he spend the coins to play game are three dimes and two nickels. Add 30￠ with 10￠ then the sum is 40￠.
Third way he spend the coins to play game are three dimes and one nickel and 5 pennies. Add 30￠ with 5￠and 5￠ then the sum is 40￠.
One penny coins is left over.

Lesson 8.6 Tell and Write Time to Five Minutes

Solve & Share
What is something you do that takes about 15 minutes? What is something you can do in less than 15 minutes?
I can … tell time to the nearest 5 minutes.

Answer:

Visual Learning Bridge

Convince Me! The time is 9:35. Where is the hour hand?
Where is the minute hand?
Answer:
The hour hand should be between 9 and 10. The minute hand will be pointing to number 7 in the clock. So the time is 9:35.

Guided Practice

Complete the clocks so both clocks show the same time.
Question 1.

Answer:
The time is 6 : 45 which means 6 hours and 45 minutes.
Explanation:
The time is showing 6 hours and 45 minutes. So, hour hand should be after 6 and near to 7. Minute hand should be pointing to number 9 in the clock.

Question 2.

Answer:

The time is 3 : 25 which means 3hours and 25 minutes.
Explanation:
The time is showing 3 hours and 25 minutes. So, hour hand should be after 3 and near to 4. Minute hand should be pointing to number 5 in the clock.

Question 3.

Answer:

The time is 8 : 00 which means the 8 hours and 0 minutes.
Explanation:
The time is showing 8 hours and 0 minutes. So, hour hand should be at 8. Minute hand should be pointing to number 12 in the clock.

Question 4.

Answer:

The time is 5 : 40 which means 5 hours and 40 minutes.
Explanation:
The time is showing 5 hours and 40 minutes. So, hour hand should be after 5 and near to 6. Minute hand should be pointing to number 8 in the clock.

Independent Practice

Complete the clocks so both clocks show the same time.
Question 5.

Answer:

The time is 9 : 00 which means 9 hours and 0 minutes.
Explanation:
The time is showing 9 hours and 0 minutes. So, hour hand should be at 9. Minute hand should be pointing to number 12 in the clock.

Question 6.

Answer:

The time is 2 : 25 which means 2 hours and 25 minutes.
Explanation:
The time is showing 2 hours and 25 minutes. So, hour hand should be after 2 and near to 3. Minute hand should be pointing to number 5 in the clock.

Question 7.

Answer:

The time is 12 : 40 which means 12 hours and 40 minutes.
Explanation:
The time is showing 12 hours and 40 minutes. So, hour hand should be after 12 and near to 1. Minute hand should be pointing to number 8 in the clock.

Question 8.

Answer:

The time is 4 : 40 which means 4 hours and 40 minutes.
Explanation:
The time is showing 4 hours and 40 minutes. So, hour hand should be after 4 and near to 5. Minute hand should be pointing to number 8 in the clock.

Question 9.

Answer:

The time is 1 : 05 which means 1 hour and 5 minutes.
Explanation:
The time is showing 1 hour and 5 minutes. So, hour hand should be at 1. Minute hand should be pointing to number 1 in the clock.

Question 10.

Answer:

The time is 11 : 10 which means 11 hours and 10 minutes.
Explanation:
The time is showing 11 hours and 10 minutes. So, hour hand should be after 11 and near to 12. Minute hand should be pointing to number 2 in the clock.

Question 11.
Number Sense Complete the pattern.

Answer:

The pattern is 7 : 00 ,7 : 05, 7 : 10, 7 : 15.
Explanation:
In the above image we can observe four timing. The pattern is 7 : 00 ,7 : 05, 7 : 10, 7 : 15.

Problem Solving

Solve each problem.
Question 12.
Reasoning One of the clocks is running a little slow. The other clock is running a little fast. Estimate the correct time.

Answer:

One of the clocks is running a little slow. The time is 2 : 40 which means 2 hours and 40 minutes.
The other clock is running a little fast. The time is 2 : 50 which means 2 hours and 50 minutes.
The correct time is 2 : 45 which means 2 hours and 45 minutes.
Explanation:
One of the clocks is running a little slow. The time is 2 hours and 40 minutes. The other clock is running a little fast. The time is 2 hours and 50 minutes. The estimated correct time is 2 : 45 which means 2 hours and 45 minutes.

Question 13.
Number Sense Look at the time on the first clock. What time will it be in 5 minutes? Write that time on the second clock.

Answer:

The time on the second clock is 8:40 which means 8 hours and 40 minutes.
Explanation:
The time on the first clock is 8:35 which means 8 hours and 35 minutes. After 5 minutes the time on the second clock is 8:40 which means 8 hours and 40 minutes.

Question 14.
Higher Order Thinking Draw a clock that shows your favorite time of the day. Explain why it is your favorite time.
Answer:

Question 15.
Assessment Practice What time does the clock show?

A. 1:05
B. 2:10
C. 2:01
D. 2:05
Answer:
The clock is showing 2:05 which means 2 hours and 5 minutes. So option D is correct.
Explanation:
In the above image we can observe a time. The time is  2:05 which means 2 hours and 5 minutes. So option D is correct.

Lesson 8.7 Tell Time Before and After the Hour

Solve & Share
Both of these clocks show the same time. How many different ways can you say this time? Write each way.
I can … say the time in different ways.

Answer:

Explanation:
In the above image both clocks shows the same time as 6:45 which means 6 hours and 45 minutes.
There are two different ways to say 6:45. One is 45 minutes after 6. Second one is 15 minutes before 7 and quarter to 7.

Visual Learning Bridge

Convince Me! Write two ways to say 5:30.
Answer :
There are two ways to say 5:30.
one is 30 minutes after 5 or half past 5.
Second one is 30 minutes before 6.

Guided Practice

Complete so both clocks show the same time. Then circle another way to say the time.
Question 1.

Answer :
The both clocks show the same time as 2:30. The another way to say the time is half past 2.
Explanation:
In the above image we can observe two clocks. The two clocks are showing the same time as 2:30. The another way to say the time is half past 2. So, draw the circle for half past 2.

Question 2.

Answer:

The both clocks show the same time as 6:45. The another way to say the time is quarter to 7.
Explanation:
In the above image we can observe two clocks. The two clocks are showing the same time as 6:45. The another way to say the time is quarter to 7. So, draw the circle quarter to 7.

Independent Practice

Complete so both clocks show the same time. Then write the time before or after the hour.
Question 4.

________ minutes before 5
Answer:

15 minutes before 5 or 45 minutes after 4.
Explanation:
In the above image we can observe two clocks. The two clocks are showing the same time as 4:45. The another way to say the time is 15 minutes before 5 or 45 minutes after 4.

Question 5.

quarter past _______
Answer:

quarter past 12 or 15 minutes after 12.
Explanation:
In the above image we can observe two clocks. The two clocks are showing the same time as 12:15. The another way to say the time is quarter past 12 or 15 minutes after 12.

Question 6.

25 minutes after _______
Answer:

25 minutes after 7.
Explanation:
In the above image we can observe two clocks. The two clocks are showing the same time as 7:25. The another way to say the time as 25 minutes after 7.

Higher Order Thinking Look at the clock to solve each problem.

Question 6.
What time will it be in 30 minutes? Write this time in two different ways.

Answer:

After 30 minutes the time will be 5:45. There are two ways to say 5:45. One is 15 minutes before 6. second way is quarter to 6.
Explanation:
In the above image we can observe a clock. The clock is showing the time as 5:15. After 30 minutes the time will be 5:45. There are two ways to say 5:45. One is 15 minutes before 6. second way is quarter to 6.

Question 7.
What time will it be in 50 minutes? Write this time in two different ways.

Answer:

After 50 minutes the time will be 10:30. There are two ways to say 10:30. One is 30 minutes after 10 or half past 10. Second way is 30 minutes before 11.
Explanation:
In the above image we can observe a clock. The clock is showing the time as 9:40. After 50 minutes the time will be 10:30. There are two ways to say 10:30. One is 30 minutes after 10 or half past 10. Second way is 30 minutes before 11.

Solve each problem.
Question 8.
Vocabulary Miguel is meeting a friend at half past 4.
Complete both clocks to show this time.

Answer:

Miguel is meeting a friend at 4:30
Explanation:
In the above image we can observe two clocks. Miguel is meeting a friend at half past 4.The two clocks are showing the same time as 4:30. The another way to say the time as half past 4 or 30 minutes after 4. Miguel is meeting a friend at 4:30

Question 9.
Generalize A train left the station at 6:55. What are two other ways to say this time?

Answer:
A train left the station at 6:55. The two ways to say this time are 55 minutes after 6 and 5 minutes before 7.
Explanation:
A train left the station at 6:55. There are two ways to say the time 6:55. one is 55 minutes after 6 and second is 5 minutes before 7.

Question 10.
Higher Order Thinking Draw a clock with hands that show 11:45. Then write two ways to say the time.
Answer:

The time is 11:45 which means 11 hours and 45 minutes. The are two ways to say the time 11:45. One is quarter to 12 and another one is 45 minutes after 11.
Explanation:
The time is showing 11 hours and 45 minutes. So, hour hand should be after 11 and near to 12. Minute hand should be pointing to number 9 in the clock. The are two ways to say the time 11:45. One is quarter to 12 and another one is 45 minutes after 11.

Question 11.
Assessment Practice James gets home at 6:00. He starts his homework at quarter past 6. At what time does James start his homework?

Answer:

James start his homework at 6:15.
Explanation:
James gets home at 6:00. He starts his homework at quarter past 6. James start his homework at 6:15.

Lesson 8.8 A.M. and P.M.

Solve & Share
What is something you do in the morning? In the evening? What are some things you do in the morning and evening? Write or draw your answers.
I can … tell time and use reasoning to state if the event is happening in the a.m. or p.m.

Answer:

The events happening in morning time are denoted with a.m.
The events happening in evening time are denoted with p.m.

Visual Learning Bridge

Convince Me! What might you be doing at 6:15 a.m.? At 6:15 p.m.?
Answer:
I do brushing my teeth at 6:15 a.m.
I eat snacks at 6:15 p.m.

Guided Practice

Complete the clocks so both clocks show the same time. Then circle a.m. or p.m. to tell when you would do each activity.
Question 1.
Ride the bus to school

Answer :
I ride the bus to school at 8 : 45 am
Explanation:
In the above image both clocks shows the same time as 8:45. I ride the bus to school at 8:45 a.m. So, draw a circle for a.m.

Question 2.
Do your homework

Answer:

I do my homework at 4:30 p.m.
Explanation:
In the above image both clocks shows the same time as 4:30. The time is showing 4 hours and 30 minutes. So, hour hand should be after 4 and near to 5. Minute hand should be pointing to number 6 in the clock. I do my homework at 4:30 p.m. So, draw a circle for p.m.

Independent Practice

Complete the clocks so both clocks show the same time. Circle a.m. or p.m. to tell when you would do each activity.
Question 3.
Take the bus home from school

Answer:

I take the bus from school to home at 3:45 p.m.
Explanation:
In the above image both clocks shows the same time as 3:45. The time is showing 3 hours and 45 minutes. So, hour hand should be after 3 and near to 4. Minute hand should be pointing to number 9 in the clock. I take the bus from school to home at 3:45 p.m. So, draw a circle for p.m.

Question 4.
Walk the dog before breakfast

Answer:

I take the dog for walk before breakfast at 8:05 a.m.
Explanation:
In the above image both clocks shows the same time as 8:05. The time is showing 8 hours and 5 minutes. So, hour hand should be at 8. Minute hand should be pointing to number 1 in the clock. I take the dog for walk before breakfast at 8:05 a.m. So draw a circle for a.m.

Question 5.
Read a book before bedtime

Answer:

I read a book before bedtime at 7:50 p.m.
Explanation:
In the above image we can observe two clocks. Both clocks show the same time as 7:50. The time is showing 7 hours and 50 minutes. So, hour hand should be after 7 and near to 8. Minute hand should be pointing to number 10 in the clock. I read a book before bed time at 7:50p.m. So, draw a circle for p.m.

Question 6.
Take swimming lessons on Saturday morning

Answer:

Take swimming lessons on Saturday morning at 10 :15 a.m.
Explanation:
In the above image we can observe two clocks. Both clocks show the same time as 10:15. The time is showing 10 hours and 15 minutes. So, hour hand should be after 10 and near to 11. Minute hand should be pointing to number 3 in the clock. Take swimming lessons on Saturday morning at 10 :15 a.m. So draw a circle for a.m.

Question 7.
Watch a movie on Friday night

Answer:

Watch a movie on Friday night at 7:20 p.m.
Explanation:
In the above image we can observe two clocks. Both clocks show the same time as 7:20. The time is showing 7 hours and 20 minutes. So, hour hand should be after 7 and near to 8. Minute hand should be pointing to number 4 in the clock. Watch a movie on Friday night at 7:20 p.m. So draw a circle for p.m.

Question 8.
Go to a party on Saturday afternoon

Answer:

Go to a party on Saturday afternoon at 2:00 p.m.
Explanation:
In the above image we can observe two clocks. Both clocks show the same time as 2:00. The time is showing 2 hours and 0 minutes. So, hour hand should be at 2. Minute hand should be pointing to number 12 in the clock. Go to a party on Saturday afternoon at 2:00 p.m. Draw a circle for p.m.

Question 9.
Higher Order Thinking Jen and Maria have dance lessons at the time shown on the clock.

Write the time two different ways. Is it a.m. or p.m.? Explain.
Answer:
Jen and Maria have dance lessons at 3:45 p.m.
Explanation:
Jen and Maria have dance lessons at the time shown on the above clock. The time is 3:45 p.m. Because morning time is not possible to have a dance lessons. The time 3:45 can be written as two different ways. One way is 45 minutes after 3. Second way is quarter to 4.

Problem Solving

Solve each problem.
Question 10.
Be Precise Draw hands on the clock to show what time your school begins each day. Then write the time. Include a.m. or p.m.

Time: ________
Answer:

My school begins each day at 9:00 a.m.
Explanation:
My school begins each day at 9:00 a.m. The time is showing 9 hours and 0 minutes. So, hour hand should be at 9. Minute hand should be pointing to number 12 in the clock.

Question 11.
enVision® STEM Stargazing is looking at the stars. The best time to stargaze is on a clear moonless night.
Gina went outside to stargaze at 9:00. Is this 9:00 a.m. or 9:00 p.m.? Explain.
Answer:
It is 9:00 p.m.. One can look at stars at night time and and if it is a clear moonless night, then the stars will be visible with a better view.

Question 12.
Higher Order Thinking Grace starts her homework at 4:15. She finishes her homework 45 minutes later. Draw the hands on the clocks to show both times. Write both times on the digital clocks.

Circle a.m. or p.m. below to tell when Grace does her homework.
a.m.
p.m.
Answer:

Grace done her homework at 5:00p.m.
Explanation:
Grace starts her homework at 4:15p.m. So first clock shows the time 4:15p.m. In first digital clock the time is 4:15p.m. She finishes her homework 45 minutes later. Second clock shows the time as 5:00p.m. Second digital clock shows the time as 5:00p.m. Grace done her homework at 5:00p.m. So draw a circle for p.m.

Question 13.
Assessment Practice Circle a.m. or p.m. to tell when you would do each activity.
Brush your teeth before bedtime
a.m.
p.m.
Answer:

Explanation:
I brush my teeth before bedtime. So draw a circle for p.m.

Walk the dog before dinner
a.m.
p.m.
Answer:

Explanation:
I go to walk with the dog before dinner. So, draw a circle for p.m.

Go to soccer practice after school
a.m.
p.m.
Answer:

Explanation:
I go to Soccer practice after school. After school means evening. So, draw a circle for p.m.

Watch the sunrise
a.m.
p.m.
Answer:

Explanation:
I watch the sunrise at morning time. So, draw a circle for a.m.

Topic 8 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. is equal to 12 – 5
B. Is equal to 9 + 2
C. is equal to 12 – 10
D. is equal to 8 + 9
E. Is equal to 3 + 3
F. is equal to 15 – 7
G. Is equal to 8 + 6
H. is equal to 12 – 8
I. Is equal to 17 – 8
J. Is equal to 6 + 9
K. is equal to 8 – 5
L. is equal to 9 + 9

Answer:

Topic 8 Vocabulary Review

Understand Vocabulary

Question 1.
Circle the name of the coin with the greatest value.
quarter
nickel
dime
Answer:

The coin with the greatest value is quarter.
Explanation:
One quarter is equal to 25￠and one nickel is equal to 5￠, one dime is equal to 10￠. The coin with the greatest value is quarter. So, draw a circle for quarter.

Question 2.
Circle the name of the coin with the least value.
half-dollar
penny
quarter
Answer:

The coin with the least value is penny.
Explanation:
Half-dollar is equal to 50￠, one penny is equal to 1￠ and one quarter is equal to 25￠. The coin with the least value is penny. So draw a circle for penny.

Question 3.
Cross out the time that is NOT quarter past 5 or quarter to 5.
4:45
5:15
5:25
Answer:

The time 5:25 is Not quarter past 5 or quarter to 5.
Explanation:
The time quarter past 5 means 5 :15 and quarter to 5 means 4 :45. The time 5:25 is Not quarter past 5 or quarter to 5. So put a cross mark for 5:25.

Question 4.
Cross out the time that is NOT half past 8 or quarter past 8.
8:30
8:45
8:15
Answer:

The time 8:45 is NOT half past 8 or quarter past 8.
Explanation:
The time half past 8 means 8:30 and quarter past 8 means 8:15. The time 8:45 is NOT half past 8 or quarter past 8. So put a cross mark for 8:45.

When does each event happen? Write a possible time.
Use a.m. or p.m.
Question 5.
school ends
Answer:
My school ends at 4 p.m.

Question 6.
eat breakfast
Answer:
I eat breakfast at 7:45 a.m.

Question 7.
sunrise
Answer:
Sun rises at 6:00 a.m..

Use Vocabulary in Writing
Question 8.
Explain how you can show ways to make 1 dollar (100¢) using coins. Use terms from the Word List. Give examples.
Answer:

Topic 8 Reteaching

Set A

When you count coins, start with the coin of greatest value.

Randi has the coins shown below. Count on to find the total value.

Another Example
Duane has the coins below. How much money does Duane have?

Solve each problem. Count on to find the total value.

Question 1.
These coins are in a jar. How many cents are in the jar?

Draw the coins in order.
Count on.
There is _________ in the jar.
Answer:

There is 45￠ in the jar.
Explanation: The above coins are in a jar. We have to keep the coins in order.
In the above image first coin is quarter. One quarter is equal to 25￠.
The second coin is dime. One dime is equal to 10￠.
The third coin is also dime. one dime is equal to 10￠.
Add these coins. Add 25￠ with 10￠ and 10￠ then the sum is 45￠. There is 45￠ in the jar.

Question 2.
The coins shown below are in a box. How much money is in the box?

There is ________ in the box.
Answer:

There is 54￠ in the box.
Explanation:
The above coins are in a box. First we have to count from the greatest value to the smallest value of the coin. Here we have one quarter, one dime, three nickels and 4 pennies.
One quarter is equal to 25￠. One dime is equal to 10￠. Three nickel is equal to 15￠. Four pennies is equal to 4￠.
By adding these coins the sum is 54￠. There is 54￠ in the box.

Set B

Dollar bills are paper money and can have different dollar values.

Matt has $56. Two of his bills are $20 bills. What other bills could Matt have? You can count on to get to $56.

Solve each problem.
Question 3.
Mr. Park has these dollar bills. Count on to find the total value.

Answer:

$20,  $40,  $50,  $55,  $56
+$20  +$10 +$5 +$1
Mr. Park has $56.
Explanation:
Mr. Park has above dollar bills. Mr. park has two $20 and one $10 and one $5 and one $1. Add these dollars $20 with $20, $10, $5, $1 then the sum is $56. Mr. Park has $56.

Question 4.
A cookbook costs $36. Mrs. Beeson has a $10 bill and a $5 bill. How much more money does Mrs. Beeson need to buy the cookbook?
$ _________
Answer:
Mrs. Beeson has a $10 bill and a $5 bill.
$10 +$5 = ?
$10 +$5 = $15
A cookbook costs $36.
$15 + ? = $36
$36 – $15 = ?
$36 – $15 = $21
Mrs. Beeson needs $21 to buy the cookbook.
Explanation:
A cookbook costs $36. Mrs. Beeson has a $10 bill and a $5 bill. Add $10 with $5 then the sum is $15. Subtract $15 from $36 then the difference is $21. Mrs. Beeson needs $21 to buy the cookbook.

Set C

Thinking Habits
Reasoning
What do the numbers and symbols in the problem mean?
How do the values of the coins relate to the total?

Use reasoning. Finish the table.

Question 5.
Mitch has dimes, nickels, and pennies. Find ways he can make 11¢. Show a tally mark for each coin you use.

Answer:

Explanation:
1 dime = 10￠
1 nickel = 5￠
1 penny = 1￠
First way add one dime with one penny then the sum is 11￠.
Second way add two nickel with one penny then the sum is 11￠.
Third way add one nickel with 6 pennies then the sum is 11￠.
Fourth way 11 pennies is equal to 14￠.

Set D

It takes 5 minutes for the minute hand to move from one number to the next. Count on by 5s.

Read the time.
Write the same time on the digital clock.

Question 6.

Answer:

Explanation:
The above two clocks show the time as 4:45 which means 4 hours and 45 minutes or quarter to 5.

Set E

You can say the number of minutes before the hour or after the hour.

Circle the time each clock shows.

Question 7.

5 minutes before 3
5 minutes after 3
Answer:

Explanation:
The above clock shows the time 2:55. The time 2:55 which means 5 minutes before 3. So, draw a circle for 5 minutes before 3.

Question 8.

15 minutes before 10
15 minutes after 10
Answer:

Explanation:
The above clock shows the time 10:15. The time 10:15 which means 15 minutes after 10. So, draw a circle for 15 minutes after 10.

Set F

Use a.m. from midnight to noon.
Use p.m. from noon to midnight.

Circle a.m. or p.m. to tell when you would do each activity.

Question 9.
Afternoon recess

Answer:

Afternoon recess break at 2:05 p.m. So, draw a circle for p.m.
Explanation:
The above clock shows the time 2:05. Afternoon recess break at 2:05 p.m. So, draw a circle for p.m.

Question 10.
Feeding fish after breakfasts

Answer:

Feeding fish after breakfasts at 9:00 a.m. So, draw a circle for a.m.
Explanation:
The above clock shows the time 9:00. Feeding fish after breakfasts at 9:00 a.m. So, draw a circle for a.m.

Topic 8 Assessment Practice

Question 1.
Chen has these coins. How much money does Chen have? If he spends a quarter and a penny, how much money will he have? Count on to find the total.

Chen has _____ cents. He will have _____ cents
Answer:
First coin is quarter. One quarter is equal to 25¢.
Second coin is dime. One dime is equal to 10¢.
Third coin is nickel. One nickel is equal to 5¢.
Fourth coin is penny. one penny is equal to 1¢.
Fifth coin is penny. one penny is equal to 1¢.
25¢ + 10¢ + 5¢ + 1¢ + 1¢ = 42¢
42¢ – 25¢ – 1¢ = ?
42¢ – 25¢ – 1¢ = 16¢
Chen has 42 cents. He will have 16 cents.
Explanation:
Chen has these coins quarter, dime, nickel, penny. Add these coins value. Add 25¢ with 10¢, 5¢, 1¢, and 1¢ then the sum is 42¢. Chen spends quarter and penny. Subtract 25¢ and 1¢ from 42¢ then the difference is 16¢. Chen has 42 cents. He will have 16 cents.

Question 2.
Ellen has 4 coins that equal 46¢. She has 1 quarter, 1 dime, and 1 penny. Which is her fourth coin?

Answer:

Fourth coin is dime. So option C is correct.
Explanation:
Ellen has 4 coins that equal 46¢. She has 1 quarter, 1 dime, and 1 penny.
One quarter is equal to 25¢. One dime is equal to 10¢. one penny is equal to 1¢. Add these coins value. Add 25¢ with 10 ¢ and 1¢ then the sum is 36¢. The fourth coin is dime. By adding 10¢ to 36¢ then the sum is 46¢. So option C is correct.
Question 3.
Nan has 36¢. Which coins show this?

Answer:

Option A is correct because the three coins value is equal t0 36¢.
Explanation:
Nan has 36¢.
Option A has three coins. Add 10¢ with 25¢ and 1¢ then the sum is 36¢. The three coins value is equal to 36¢.
Option B has four coins. Add 1¢ with 5¢, 10¢ and 10¢ then the sum is 26¢. The four coins value is equal to 26¢.
Option C has four coins. Add 5¢ with 5¢, 10¢ and 1¢ then the sum is 21¢. The four coins value is equal to 21¢.
Option D has four coins. Add 10¢ with 10¢, 1¢ and 10¢ then the sum is 31¢. The four coins value is equal to 31¢.
Option A is correct because the three coins value is equal t0 36¢.

Question 4.
George has quarters, dimes, and nickels. Show all the ways he can make 25¢. Use tally marks.
Circle the way that uses the least number of coins.

Answer:

Explanation:
George has quarters, dimes, and nickels. There are four different ways he can make 25¢.
First way one quarter is equal to 25¢.
Second way two dimes is equal to 20¢ and one nickel is equal to 5¢. Add 20¢ with 5¢ then the sum is 25¢.
Third way one dime is equal to 10¢ and 3 nickels is equal to 15¢. Add 10¢ with 15¢ then the sum is 25¢.
Fourth way five nickels is equal to 25¢.
So draw a circle for quarter because least number of coins.

Question 5.
Mr. Zink has the dollar bills shown. How many dollars does Mr. Zink have? If he spends $16 on breakfast and does not receive any change, which bills could he have used to pay for it?

Answer:
$10 + $10 + $5 + $5 + $1 = ?
$10 + $10 + $5 + $5 + $1 = $31
Mr. Zink have $31.
He spends $16 on breakfast
$31 – $16 = ?
$31 – $16 = $15
Mr. Zink used $10 and $5 ,$1.
Explanation:
Mr. Zink has the dollar bills as we can observe in the above image. Add $10 with $10, $5, $5, $1 then the sum is $31. Mr. Zink have $31. He spends $16 on breakfast. Subtract $16 from $31 then the difference is $15. Mr. Zink used $10 and $5, $1 he does not receive any change back.

Question 6.
A. Claire has two $20 bills, two $5 bills, and three $1 bills. Terrance has one $100 bill. How much more money does Terrance have? Explain.
Answer:
Claire has two $20 bills, two $5 bills, and three $1 bills.
Two $20 bills is equal to $40.
Two $5 bills is equal to $10.
Three $1 bills is equal to $3.
$40 + $10 + $3 = $53
Claire has $53.
Terrance has one $100 bill.
One $100 bill is equal to $100.
$100 – $53 = $47
Terrance have more $47.
Explanation:
A. Claire has two $20 bills, two $5 bills, and three $1 bills. Add the bills. Add $40 with $10 and $3 then the sum is $53. Terrance has one $100 bill. Subtract $53 from $100 then the difference is 47. Terrance have more $47.