Lesson 5

Side Lengths of Rectangular Prisms

Warm-up: Notice and Wonder: Prism Print (10 minutes)

Narrative

The purpose of this warm-up is for students to notice that each face of a prism can be the base, which will be useful when students use a base of a prism to find the prism’s volume in a later activity. While students may notice and wonder many things about these images, the relationship between the images of the prism and the images of the rectangles are the important discussion points.

Launch

  • Group size: 2
  • Display the images for all to see.

Activity

  • Ask students to think of at least one thing they notice and at least one thing they wonder.
  • 1 minute: quiet think time
  • 2 minute: partner discussion
  • Share and record responses.

Student Facing

What do you notice? What do you wonder?

Prism. 2 by 1 by 6 cubes.
Prism. 6 by 1 by 2 cubes.
Prism. 2 by 6 by 1 cubes.

Rectangle. 2 squares by 1 squares.
Rectangle. 6 squares by 1 square. 
Rectangle. 2 squares by 6 squares.

Student Response

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Activity Synthesis

  • “These rectangles show different faces of the prisms. Any face of a prism can be a base. We are going to learn more about this in the activities.”
  • “Where do we see each base in the prism?”

Activity 1: All About That Base (10 minutes)

Narrative

The purpose of this activity is for students to recognize that a base of a prism is a two-dimensional rectangle and any face of a prism can be a base. Students may start with a possible rectangular base and try to visualize which face of a given prism matches the base or they may start with the prism, study the faces, and try to find an appropriate base to match. In either case, they need to persevere and think through all of the possible bases for each prism systematically in order to solve this problem (MP1).

Engagement: Provide Access by Recruiting Interest. Provide choice and autonomy. Provide access to various tools that can be used to solve the problem. For example, connecting cubes can be used to build the prisms and colored pencils can be used to shade the faces of each prism.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing

Required Materials

Materials to Gather

Required Preparation

  • Have connecting cubes available for students who need them.

Launch

  • Groups of 2
  • Give students access to connecting cubes.

Activity

  • 4 minutes: independent work time
  • 2 minutes: partner discussion
  • Monitor for students who:
    • Match each prism to a rectangle that represents the base the prism is resting on.
    • Recognize that the 4 by 3 rectangle represents a base on each of the prisms.

Student Facing

Here are 3 rectangular prisms.

1Prism. 3 by 4 by 4 cubes. 
2Prism. 3 by 6 by 4 cubes. 
3Prism. 3 by 5 by 4 cubes.

These rectangles represent bases of the prisms.

ARectangle. 3 by 6 squares.
BRectangle. 3 by 4 squares.
CRectangle. 5 squares by 3 squares.
  1. Match each prism with a rectangle that represents its base. Note: Some prisms may match more than 1 rectangular base.
  2. Find the volume of each prism. Explain or show your reasoning.

Student Response

For access, consult one of our IM Certified Partners.

Advancing Student Thinking

If students do not correctly match prisms to bases, ask them to build each prism and describe how a given face does or does not match each base that is represented with rectangles A, B, and C.

Activity Synthesis

  • Invite previously identified students to share.
  • Display student work that shows it is possible for rectangle B to be a base for any of the prisms.
  • “Where do we see this rectangle as a base in each prism?” (If we rotated each prism to be resting on the face that is facing us, each prism would be resting on base B.)
  • “If prism 2 was resting on the 4 by 3 base, how many layers tall would the prism be?” (6 layers)
  • “We can use the word height to describe how tall a prism is.”
  • “If prism 3 was resting on the 4 by 3 base, what would the height of the prism be?” (5 cubes)

Activity 2: Growing Prism (15 minutes)

Narrative

The purpose of this activity is for students to describe the layered structure of rectangular prisms using the side lengths of the prism. Instead of diagrams of rectangular prisms built from cubes, students are shown a diagram of one of the bases of a prism and are asked to find the volume of the prism with different heights. Students may still use informal language, such as layers, to describe the prisms and find their volume. During the lesson synthesis, connect their informal language o the more formal math language of length, width, height, and area of the base.

MLR2 Collect and Display. Amplify words and phrases such as: length, width, taller, pattern, base, number of layers.
Advances: Conversing, Reading

Required Materials

Materials to Gather

Required Preparation

  • Have connecting cubes available for students who need them.

Launch

  • Groups of 2
  • Give students access to connecting cubes.

Activity

  • 2 minutes: independent work time
  • 10 minutes: partner work time
  • As students work, monitor for students who:
    • describe and use layers of the prism.
    • multiply 40 by the height of the prism to determine the volume of the prisms.

Student Facing

Here is a base of a rectangular prism.

Rectangle. 10 squares by 4 squares.
  1. Fill out the table for the volumes of rectangular prisms with this base and different heights.
    height multiplication expression to represent the volume volume
    1
    2
    3
    10
    25

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Ask previously selected students to share their solutions for the first problem.
  • Display a completed table or use the one given in student solutions:
    number of layers multiplication expression for volume volume
    1 \(4\times10\times1\) or \(40\times1\) 40 cubes
    2 \(4\times10\times2\) or \(40\times2\) 80 cubes
    3 \(4\times10\times3\) or \(40\times3\) 120 cubes
    10 \(4\times10\times10\) or \(40\times10\) 400 cubes
    25 \(4\times10\times25\) or \(40\times25\) 1,000 cubes
  • “How does the volume of the prism change in the table?” (The volume is increasing by 40 cubes for each layer added to the prism.)
  • “How is the change in volume represented by the multiplication expressions in each row?” (The expressions in each row shows more groups of 40.)
  • Display the expression \(3 \times 40\).
  • “How does the expression represent the volume of the prism?” (There are 40 cubes in the base layer of the prism or 4 rows of 10 cubes. Then there are 3 of these layers.)

Activity 3: What is the Question? (10 minutes)

Narrative

This activity provides students an opportunity to interpret a calculation in the context of the situation (MP2) when a scenario is given with an equation that shows solutions to unknown questions. Students have to interpret the equations and ask the question whose answer is given. Numbers are chosen specifically to prompt students to consider the structure of rectangular prisms.

Launch

  • groups of 2

Activity

  • 2 minutes: quiet think time
  • 4 minutes: partner work time
  • Monitor for students who:
    • use informal language, such as layers.
    • use the terms length, width, height, and base in their questions.

Student Facing

This is the base of a rectangular prism that has a height of 5 cubes.

Diagram. Rectangle partitioned into 3 rows of 4 of the same size squares.

These are answers to questions about the prism. Read each answer and determine what question it is answering about the prism.  

  1. 3 is the answer. What is the question?
  2. 5 is the answer. What is the question?
  3. \(3\times4=12\). The answer is 12. What is the question?
  4. \(12\times5=60\). The answer is 60 cubes. What is the question?
  5. 3 by 4 by 5 is the answer. What is the question?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Ask previously selected students to share their solutions.
  • Connect the informal language to the math terms length, width, height, and area of a base.
  • “How does the expression \(3\times4\times5\) represent the prism described in the second question?” (The area of the base is \(3\times4=12\), and the height is 5, so \(3\times4\times5\) represents the product of length, width, and height.)

Lesson Synthesis

Lesson Synthesis

Display the poster of language from a previous lesson.

“What information do you need to measure the volume of any rectangular prism?” (We need to know the area of a base and how tall it is with that base or we need to know the length, width, and height.)

As student share responses, update the display, by adding (or replacing) language, diagrams, or annotations.

“What language can we add to our poster to explain how to find the volume of a prism when we can’t see the cubes?”(We can multiply the area of the base and the height. We can multiply the length, width, and height.)

“What is the connection between the number of layers and the height of the prism?”(The number of layers is the number of cubes high, or the height.)

Update the display.

Math Community

After the Cool-down, give students 2–3 minutes to discuss in small groups any revisions to the “Norms”​ ​section.
Collect and record any revisions.

Cool-down: Determine the Volume (5 minutes)

Cool-Down

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