The Pyramid Stack (Part 1)
Lesson 2 of 6
Objective: SWBAT give an informal argument for the formula for the volume of a pyramid. SWU how the volume of a pyramid is related to the volume of a prism with the same base and altitude.
Opener - Team Warm-Up
Class begins in the regular classroom. As soon as I complete administrative tasks, the entire class will move to a computer lab.
I display the warm-up problem with an LCD projector using the slide show for this lesson. This classroom routine follows our Team Warm-up format. Students work the problem in their learning journals, and one member of each team writes the team’s answer on the white board at the front of the classroom. While the students work, I complete administrative tasks.
The warm-up problem for today’s lesson asks students to make a conjecture: What is the smallest number of pyramids that can be stacked so that they completely fill a triangular prism
(with no overlap and no space left over)?
Purpose: I want students to discuss the question with their team mates and make a conjecture. They will test their hypothesis in the computer lab.
I warn the class that they must quickly agree on a conjecture as I will enforce the 5 minute time limit strictly. Any thoughtful hypothesis is acceptable. As soon as I finish administrative tasks, I give the class instructions to pick up the handouts (Parts 1 and 2) for the investigation and meet me in the computer lab. It takes about 5 minutes to move to the computer lab.
Provisioning: Request that the WinGeom software be loaded onto a shared drive where it can be
accessed by all students. (See my notes on Wingeom in my Strategies folder for more information on this freeware application. Reserve a classroom with enough computers for the class. If necessary, 2 students could work together on one computer, if there is room to sit. Reproduce the handouts for the WinGeom activity (Part 1 and Part 2), 1 per student. Print out the screen shots and make about a dozen copies (for students who are absent or who do not finish during class). Print out the nets and assemble the pyramids and prism with clear tape. (This will probably take 20-25 minutes the first time.)
I give the students 5 minutes to complete step 1 of Part 1 of the investigation. Questions a-c ask students to infer the meanings of the vocabulary words (written in bold, italicized font) and name the parts of the pyramids. Question d requires students to calculate simple measurements using the 3-dimensional coordinates of the vertices of a pyramid. As they are working, I circulate. At the end of the time limit, I review the answers with the class, answering any questions.
Next, students work individually or with a partner to complete Part 1 of the Wingeom investigation. I tell them that they should try to finish in 10 minutes. As they are working, I circulate. Students who finish Part 1 early should move on to Part 2, but I tell them that I am
going to need their attention when we go over Part 1.
At the end of the time limit, (20 minutes into class), I call the class together to go over Part 1 of the investigation.
BOLOs (Things to be on the lookout for): Are all students taking time to answer the questions as they work through the lab. Do students understand that the volume of a pyramid does not change when the apex is translated parallel to the base of the pyramid (so that the altitude does not change)? Do students understand that the vertices of the base of pyramid ABC-D all lie in the same plane (z-coordinates of 0), while the apex lies in a parallel plane 2 units above the plane of the base (z-coordinate of 2)? Do students understand the definition of the altitude of a pyramid?
- What happened to the volume of the pyramid when we shifted the apex?
- Could we move the apex any old way and still have the same result? What was special about
the way we moved the apex? How do you know? (MP6)
- What is the advantage of using a coordinate system here? (MP5)
- What is the altitude of a pyramid? Can you define it in your own words? (MP6)
- Do you think we would have the same result with any pyramid? Or, is there something special
about the pyramids we are working with? (MP6)
- In Part 2 of the investigation, we will use the computer software to ‘stack’ pyramids to make a prism. We want to use only pyramids with the same volume. Why is that important? (Keep in mind that we are trying to find the volume of a single pyramid.) (MP5)
Purpose: I want students to see for themselves that the volume of a pyramid does not change
when the apex is shifted (altitude remaining constant). We will use this property in Part 2, when we cleverly choose 3 pyramids all of the same volume that fit perfectly together to make a triangular prism. This part of the lesson also gives students a chance to apply important vocabulary.
Students work individually or with a partner to complete Part 2 of the Wingeom investigation. I tell them that they should try to finish in 15 minutes. As they are working, I circulate. Students who finish Part 2 early can handle the paper model of the pyramid stack, which I made using the printable nets.
At the end of the time limit, (45 minutes into class), I call the class together to go over Part 2 of the investigation. (See the questioning strategies.) Using the pyramid stack constructed out of the paper nets, I show that three triangular pyramids fit perfectly within the triangular prism. I then show (by placing two pyramids next to
one another on a table or other surface) that two pyramids could be formed by
shifting the apex of the original pyramid without changing its volume.
BOLOs (Things to be on the lookout for): Do students understand that three triangular pyramids can be stacked together to make a triangular prism? Do students understand that the pyramids are not congruent, yet all have the same volume? Do students understand that one pyramid (AA’B-D aka A’B’D-A) has one face congruent to the base of each of the other two pyramids? Do students understand that, to find the volume of a triangular pyramid, the steps are: translate the base of the pyramid along the altitude, find the volume of the triangular prism that has
for its bases the base of the pyramid and its translated image, divide by 3. Do students
understand that the procedure they performed in this activity can be performed using any triangular pyramid?
- How many vertices does this solid have? If we regard the triangle formed by these 3 vertices as a base, how many vertices are left over? What is the name for this sort of solid? (MP1)
- What if, instead, we regard the rectangle formed by these 4 vertices as a base. How many vertices are then left over? (MP1)
- If we cut a rectangle along its diagonal, what figures result? What properties do these triangles have? Do they have any sides congruent? What are the properties of the original rectangle? (MP1)
- How could you prove that bases AA’B’ and ABB’ are congruent? (The triangles are formed by dissecting a rectangle along one diagonal; SSS congruence.) (MP3)
- How do we know that pyramids AA’B’-D and ABB’-D have the same altitude? (Bases are coplanar, and they share an apex. There is one and only one shortest distance between point D and plane ABB’A’.) (MP3)
- How do we know that bases ABC and A’B’D are congruent? (One triangle is the translated image of the other.) (MP3)
- How do we know that pyramids ABC-D and A’B’D’-A have the same altitude? (Because of the way we translated the original base.) (MP3)
- How can we use the steps of this investigation to find the volume of a pyramid? (MP5)
- Do you think we would have the same result with any pyramid? Or, is there something special about the pyramids we are working with? Did we all work with the same pyramid? (MP6)
Purpose: I want students to see that when we create a triangular prism by translating the base of a triangular pyramid along its altitude, the prism is made up of 3 triangular pyramids whose volumes can be shown to be equal.
The homework provided with this lesson consists of three problems. The first two problems ask students to reproduce the procedure they learned during this lesson and use it to find the volumes of pyramids. The third problem asks students to explain how Cavalieri’s Principle can be used to solve a practical problem.
I display the Lesson Close prompt using the slide show for this lesson. Students write their answers in their learning journals. I remind students to turn in their Wingeom Investigations as an exit
slip. After class, I look over the students’ work to see who is getting this and what problems others are having.