1. Volume of Prisms and Pyramids
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
Core Math
July 26, 2016
8:00 – 10:10

2. Learning Intention
We are learning to help children understand the volume measures of prisms and pyramids.
We will be successful when we can
Give precise definitions of prisms and pyramids;
Explain the formula for the volume of a prism;
Describe the relation between the volume of a prism, and the volume of the corresponding pyramid.

3. What does this first grader understand about composing 3-D figures?

4. What is the Volume of this Figure?
The picture shows a prism whose base is a right triangle. Find the volume of this prism.
Explain your reasoning.
We are saying this is a “prism”. Why is it a prism? (What is the definition of “prism”?)
On this first viewing, participants can say the volume is half the volume of the corresponding rectangular prism (as in the video), and can compute this using the volume formula for a rectangular prism, but need not get as far as “area of base) x height.
Picture credit:

5. Combining Prisms – Wooden Blocks
Choose a pair of identical (congruent) wooden blocks in the shape of right triangular prisms
Combine these two prisms in as many ways as you can to make prisms with triangular or quadrilateral bases with the same height as your original prisms. Record your combinations in your notebook.
How does the volume of each of your combined prisms compare with the volume of one of the original prisms?
How does the base area of each of your combined prisms compare with the base area of one of the original prisms?

6. Combining Prisms – Pattern Blocks
Combine triangular prisms to make a hexagonal prism (with the same height). How many triangular prisms does it take to make one hexagonal prism?
How does the volume of the hexagonal prism compare with the volume of one of the triangular prisms?
How does the base area of the hexagonal prism compare with the base area of one of the triangular prisms?
Find as many ways as you can to make a hexagonal prism by combining prisms from the pattern block set. Record your results in your notebook.
What can you conclude about the volumes and base areas of the various pattern block shapes?
Relate this to RP in Grades 6 and 7: if the height stays the same, the volume of a prism is proportional to its base area.

7. What is the Volume of this Figure?
The picture shows a prism whose base is a right triangle. Find the volume of this prism.
How is the volume of the prism related to the area of its base?
On this second viewing, participants should be pushed to (area of base) x height; this version of the formulas will then be successively generalized in the next few slides.
Picture credit:

8. Volume of any Triangular Prism
Explain why the volume of any triangular prism must be equal to the product of the area of its base, and its height
Hint: The triangular base can be decomposed into two right triangles.

9. What is the Volume of this Figure?
The picture shows a prism whose base is an isosceles trapezoid. Find the volume of this prism.
Explain your reasoning.
Picture credit:

10. Volume of an Arbitrary Prism
Explain why the volume of any prism must be equal to the product of the area of its base, and its height
Picture credit:

11. Learning Intention
We are learning to help children understand the volume measures of prisms and pyramids.
We will be successful when we can
Give precise definitions of prisms and pyramids;
Explain the formula for the volume of a prism;
Describe the relation between the volume of a prism, and the volume of the corresponding pyramid.

12. CCSSM Volume Measurement Standards
Geometry: Solve real-world and mathematical problems involving angle measure, area, surface area, and volume.
7.G.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Note the exact parallel with 3.MD.5 and 3.MD.6 for area!

13. What is the Volume of a Pyramid?
Sort the plastic shapes at your table into pairs, with one prism and the corresponding pyramid in each pair
What does “corresponding” mean in this context?
Do you have any shapes left over that you did not sort into pairs?
This would be a good time to point out that a cylinder is really just a ”circular prism” and a cone is just a “circular pyramid”. In particular, those two shapes should also be considered a pair. (And any relations we find between the volumes of a prism and the corresponding pyramid will also work for the cylinder/cone pair.)
Before moving to the next slide, ask “In each pair, which shape has the larger volume, the prism or the pyramid? How do you know?”

14. What is the Volume of a Pyramid?
For each pair of shapes,
Estimate how many times larger is the volume of the prism than the volume of the pyramid
Fill the pyramid with water, then pour the water into the prism, as many times as necessary to fil the prism completely.
What do you conclude about the volume of a pyramid?

15. Packing vs. Filling
Turn to page 26 in your Geometric Measurement Progressions. Start with “Understand concepts of volume and relate volume to multiplication” and read to the end of the first paragraph on page 27.
Turn and talk: Clarify the difference between packing and filling. How do today’s activities connect to that understanding?

16. Comparison with Area
In 2 dimensions, the area of a triangle is ½ times the area of the rectangle with the same base and height.
In 3 dimensions, the volume of a pyramid is 1/3 times the volume of the prism with the same base and height.
Coincidence?
Question: What would happen in 4 dimensions?

17. The Great Pyramid of Khufu
When it was built, the Great Pyramid of Cheops was meters tall. Each side of the square base is meters long. What was the volume of the pyramid?
If the stone used to build the pyramid weighs 2000 kilograms per cubic meter, what was the weight of the whole pyramid?
One estimate is that the pyramid was built in approximately 20 years. If that estimate is correct, how many kilograms of stone had to be transported to the site each year?

18. Learning Intention
We are learning to help children understand the volume measures of prisms and pyramids. We will be successful when we can Give precise definitions of prisms and pyramids; Explain the formula for the volume of a prism; Describe the relation between the volume of a prism, and the volume of the corresponding pyramid.

19. Core Mathematics Partnership Project
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.