What is the relationship of the Volume of the pyramid to the Volume of a prism to the Volume?

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Students will explore the relationship between the volume of a prism and the volume of a pyramid by first estimating and then observing how many pyramids will fit in a prism with the same base and height.

Intentionality…

Through this problem based math lesson, students will determine how many pyramids will fit in a prism by either repeatedly filling and pouring water, sand, beads or another material from a physical pyramid and prism or by leveraging the included video clips as a means to first estimate and then observing the result of the investigation.

Some of the big ideas that may emerge from this problem based math lesson include:

• The volume of a 3-dimensional figure can be found by determining the number of cubic units that can be contained within the figure;
• The volume of a prism can be determined by finding the number of cubic units required to cover the base and multiply by the number of layers (i.e.: the height); and,
• The volume of a pyramid is one-third the volume of a prism with a congruent base and equivalent height. Therefore, the volume of a cone is also one-third the volume of a cylinder with a congruent base and equivalent height;

Spark

Show students the following video:

Then, ask students:

What do you notice?
What do you wonder?

Give students 60 seconds (or more) to do a rapid write on a piece of paper.

Then, ask students to share with their neighbours for another 60 seconds.

Finally, allow students to share with the entire group.

Some of the noticing and wondering that may arise include:

• I notice two 3D figures.
• I notice a prism and a pyramid.
• They both have a square for their base.
• I wonder why we’re looking at shapes?
• I wonder what they’re going to do with those figures?
• How much wrapping paper would it take to wrap those figures?
• How many of the pyramid will it take to fill the prism?
• And many more.

At this point, you can acknowledge some of the details that students noticed and answer any of the wonders that will not reveal too much information as to take away from the intentionality of the lesson.

Estimation: Video & Prompt

Share the following video:

Prompt students with:

How many square base pyramids will it take to fill the square base prism?

Make an estimate.

Be sure students are made aware that the prism and the pyramid both have a congruent square base (i.e.: the same base) and an equivalent height (i.e.: the same height).

We can now ask students to make an estimate using their spatial reasoning as to how many square base pyramids it will take to fill the square base prism. Encourage students to share their predictions and justification with neighbours before welcoming them to the group as a whole.

Update Estimate: Video & Prompt

Share the following video:

Prompt students with:
How many square base pyramids will it take to fill the square base prism?

Update your estimate.

Monitor student thinking by circulating around the room and listening to the mathematical discourse. Encourage students to use precise mathematical language where appropriate and listen for the use of additive thinking (it will take one more pyramid to fill the prism) or multiplicative thinking (the prism can hold 2 and ½ pyramids of water).
Allow students to share their estimates with neighbours first, then with the class. Write down their estimates on the chalkboard/whiteboard/chart paper so students feel their voices are being heard and so they feel they have a stake in solving this problem. At this point, encourage students to describe their thinking so that anyone could be convinced of their estimate.

Sense Making

Show students the productive struggle prompt video:

While the video is playing, you can narrate what students see to reveal the struggle prompt for this task:

Along with updating your estimate of how many square based pyramids will it take to fill the square based prism, also make an estimate of:

how many triangular based pyramids it will take to fill the triangular based prism; and,

how many cones will it take to fill the cylinder?

Note that each prism and pyramid we are comparing has a congruent base and equivalent height.

After you give students an opportunity to make an estimate and share with neighbours and to the group, prompt students to investigate the volume relationships between each pair of prisms and pyramids by filling and pouring water, sand, rice or similar into geometric solids from a math manipulative kit or by creating their own prisms and pyramids using the provided templates.

While it may be tempting for educators to skip over the opportunity for students to construct their own prisms and pyramids due to the time it will require, note that this is an opportunity for students to better spatially recognize each figure by exploring their nets and also to deeply investigate the volume relationships from each pyramid and prism with a congruent base and equivalent height.

During Moves

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Student Approach #1: Skip Counting

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Student Approach #2: Pouring And Updating

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Student Approach #3: Generalizing a Formula For Volume of a Pyramid

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Next Moves

Show students the following reveal video:

Consider leaving the final frame up for students to reflect on and discuss as you prepare to head into the consolidation of the lesson.

Consolidation: Making Connections

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Reflect

Provide students an opportunity to reflect on their learning by offering these consolidation prompts to be completed independently.

Consolidation Prompt #1:

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Consolidation Prompt #2:

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Consolidation Prompt #3:

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We suggest collecting this reflection as an additional opportunity to engage in the formative assessment process to inform next steps for individual students as well as how the whole class will proceed.

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Explore The Entire Unit of Study

This Make Math Moments Task was designed to spark curiosity for a multi-day unit of study with built in purposeful practice, and extensions to elicit and emerge mathematical models and strategies.

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Estimation: Video

Estimation: Image

Update Estimate: Image