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LESSON 1: Length, Width, and Depth • M1-
LEARNING GOALS
- Determine the volume of right rectangular prisms with fractional edge lengths using unit cubes with unit fractional dimensions.
- Connect the volume formulas V 5 lwh and V 5 Bh with a unit-cube model of volume for rectangular prisms.
- Apply the formulas V 5 lwh and V 5 Bh to determine volumes in real-world problems.
- Fluently add, subtract, and multiply multi-digit decimals using the standard algorithms.
KEY TERMS
You know about three-dimensional figures such as cubes and other rectangular prisms. You also know how to operate with positive rational numbers. How can you use what you know to calculate measurements of any rectangular prism, even one with fractional edge lengths?
WARM UP
Determine the least common multiple of the numbers in each pair.
- 2, 10
- 3, 8
- 6, 14
- 10, 15
Length,
Width, and
Depth
Deepening Understanding of Volume
- point
- line segment
- polygon
- geometric solid
- polyhedron
- face
- edge
- vertex
- right rectangular prism
- cube
- pyramid
- volume
M1-116 • TOPIC 3: Decimals and Volume
Common Figures
Cut out the cards found at the end of the lesson. Sort the figures into two or more groups. Name each category and be prepared to share your reasoning.
Getting Started
M1-118 • TOPIC 3: Decimals and Volume
Three polyhedra are shown.
face
face
face face
face
face
face
face
face
face
When you have
more than 1
vertex, you say
“vertices.”
Figure A Figure B Figure C
Figure A is a right rectangular prism. A right rectangular prism is a polyhedron with three pairs of congruent and parallel rectangular faces.
Figure B is an example of a cube , which is a special kind of right rectangular prism. A cube is a polyhedron that has congruent squares as faces.
Figure C is an example of a rectangular pyramid. A pyramid is a polyhedron with one base and the same number of triangular faces as there are sides of the base.
- Describe the different faces of each polyhedron.
- Study the right rectangular prism. Identify the three pairs of congruent parallel faces.
- Study the cube.
a. Describe the locations of the cube faces you can see and the locations of the faces you cannot see.
b. What do you know about the length, width, and height of the cube?
c. Describe how the cube is also an example of a right rectangular prism.
A unit cube is a cube whose sides are all 1 unit long.
NOTES
LESSON 1: Length, Width, and Depth • M1-
- Compare the numbers of faces, edges, and vertices of the cube and the other right rectangular prism. Write what you notice.
- Study the rectangular pyramid. How do the faces of the rectangular pyramid differ from the faces of the rectangular prisms?
- List examples in the real-world objects that are shaped like right rectangular prisms or pyramids.
Volume of Rectangular Prisms
AC T I V I T Y
Volume is the amount of space occupied by an object. The volume of an object is measured in cubic units.
The volume of a cube is calculated by multiplying the length times the width times the height.
Volume of a cube 5 l 3 w 3 h
- Calculate the volume of each cube with the given side length.
a. ___ 109 centimeter b. 1 __ 31 centimeters
- Suppose a cube has a volume of 27 cubic meters. What are the dimensions of the cube?
NOTES
LESSON 1: Length, Width, and Depth • M1-
- Interpret the worked example.
a. How was the number of cubes needed to pack the prism in each dimension determined?
b. Instead of cubes with a width of 1 __ 4 inch, suppose you used cubes each with a width of 1 __ 8 inch. How does this change the volume of the rectangular prism?
- Use the method from the worked example to determine the volume of each rectangular prism.
a. b. 1 __^34 in. by 2 __^13 in. by __^12 in.
2 12 in.
(^34) in. (^38) in.
M1-122 • TOPIC 3: Decimals and Volume
You have calculated the volume of a rectangular prism using the formula V 5 lwh , where V is the volume, l is the length, w is the width, and h is the height. You also know that the area of a rectangle can be calculated using the formula A 5 l? w.
Consider the two formulas:
V 5 l? w? h A 5 l? w
If B is used to represent the area of the base of a rectangular prism, then you can rewrite the formula for area: B 5 l? w.
Now consider the two formulas:
V 5 l? w? h B 5 l? w
Using both of these formulas, you can rewrite the formula for the volume of a rectangular prism as V 5 B? h , where V represents the volume, B represents the area of the base, and h represents the height.
In order to calculate the volume of various geometric solids you will need to perform multiplication. In this activity, you will calculate the volume of rectangular prisms with decimal side lengths.
Consider the right rectangular prism shown.
10.1 m
32.64 m
7.3 m
Volume Formulas
AC T I V I T Y
You can use the formula V 5 Bh to calculate the volume of any prism. However, the formula for calculating the value of B will change depending on the shape of the base.
It is good practice to estimate before you actually calculate. If you have an estimate, you can use it to decide whether your answer is correct.
- Each number sentence represents the base, B , times height, h , of different rectangular prisms. Complete each number sentence by inserting a decimal point to show the correct volume.
a. 53.6 sq. ft 3 0.83 ft 5 44488 cu. ft
b. 7.9 sq. cm 3 0.6 cm 5 474 cu. cm
c. 0.94 sq. m 3 24.9 m 5 23406 cu. m
- Casey thought that using a pattern would help her understand how to calculate the product in a decimal multiplication problem.
a. Complete the table.
Problem Product Problem Product Problem Product
b. Describe any patterns that you notice.
M1-124 • TOPIC 3: Decimals and Volume
LESSON 1: Length, Width, and DepthLESSON 1: Length, Width, and Depth • • M1-125M1-
- A rectangular prism with B 5 26 square centimeters and h 5 31 centimeters has a volume of 806 cubic centimeters. Use this information to determine the volume of the other rectangular prisms.
a. 2.6 sq. cm 3 31 cm b. 2.6 sq. cm 3 3.1 cm
c. 0.26 sq. cm 3 3.1 cm d. 2.6 sq. cm 3 0.31 cm
e. 0.26 sq. cm 3 31 cm f. 2.6 sq. cm 3 0.031 cm
g. 0.026 sq. cm 3 0.31 cm h. 0.26 sq. cm 3 0.31 cm
- Look at the patterns in Question 4.
a. How can some of the rectangular prisms have the same volume?
b. How can you tell without multiplying which rectangular prisms will have the same volume?
- LESSON 1: Length, Width, and Depth • M1-
- Figure 1 Figure
- Figure 3 Figure
- Figure 5 Figure
- Figure 7 Figure
Review
- Elena wants to put together some of her favorite songs on her computer. She wants to store 60 minutes worth of music. Elena wonders how many songs she will be able to include. She looks online and finds a source that says the average song length is 3 __^12 minutes. If this is true, about how many songs will Elena be able to store? Show your work.
- Ling is a camp counselor at a local summer camp. She is in charge of the weekly craft activity for 40 campers. Ling plans to make fabric-covered frames that each require 1 __ 6 yard of fabric. When Ling sets up for her craft activity, she measures the four separate fabric remnants her director gave her. The table shows how much of each fabric she has. How many campers can use plaid fabric? Show your work.
- Represent each product using an area model. Then calculate the product. a. 3 __ 4 3 1 __ 3 b. 1 __ 2 3 3 __ 5
Fabric Amount (yards)
Plaid ___ 1211
Tie-dyed 1 7 __ 9
Striped 2 2 __ 9
Polka-dotted 1 3 __ 4
1
0
0 1
1
0
0 1
M1-130 • TOPIC 3: Decimals and Volume
- Determine the GCF of each set of numbers. a. 72 and 30 b. 30 and 54 5. Determine the LCM of each set of numbers. a. 10 and 12 b. 8 and 9
