The Pyramid Stack (Part 2)
Lesson 3 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 and Goal Setting
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 consider the volume of a hexagonal pyramid. In the last lesson, we saw that the volume of a triangular pyramid is 1/3 the volume of the prism with the same base and altitude. How can we find the volume of the hexagonal pyramid?
I am looking for students to collaborate to come up with a reasonable strategy (MP1) and explain their thinking (MP3). As I review the team solutions, I reward all answers that include a reasonable attempt at justifying a method even if incorrect. For example, a team may reason that the volume of a hexagonal pyramid is 2/3 the volume of the corresponding prism, since a hexagon has twice as many sides as a triangle. This is a fairly weak argument, but reasoning by analogy is a problem-solving heuristic.
Since students often think of dissecting a hexagon into triangles, at least one group of students should think of dissecting a hexagonal pyramid into triangular pyramids (MP5). I draw attention to this “good idea”. If no team provides the example I can supply it myself using the spare slide in the slide show. I tell the class that this method will work, and we are going to prove it using algebra. What is more, we will see that it doesn’t even require any extra work!
I use the slide show to display the lesson agenda and learning targets. During the second half of the lesson, we will be taking another look at how three triangular pyramids can fit together to make a prism—this time using hands-on models rather than computer software.
I begin the activity by summarizing the purpose. We know that the volume of any triangular pyramid is 1/3 the volume of a triangular prism with the same base and altitude. This activity will show us how to handle pyramids with any sort of base. We will actually work with an oblique hexagonal pyramid (not the right hexagonal pyramid of the warm-up, so students will have to do some re-thinking). That shape is non-special enough to require us to find methods that we can apply to any pyramid.
In fact, the final result will not be a variation of the method we use to find the volume of a triangular pyramid. We will actually find that the same method applies equally well to all pyramids. This is a nice surprise, and I try to set things up so that students will discover it themselves. I believe that students are more likely to be motivated to discover a method of solution than to confirm a method with a proof once they know it. If a student guesses at the end result, I tell them that they could very well be right, but they will have to work through the algebra to be sure. (MP1)
Without the details, the proof requires just four lines (algebraic equations). However, writing an algebraic proof is not an easy task for my geometry students, so I have carefully scaffolded the
activity. The accompanying resource is a puzzle proof, which my students complete using the Team Jigsaw format (MP1). More information on how I use these methods can be found in my Strategies folder.
One member of each team collects materials for the team: one part of the problem for each
student (designated by role name) and one copy of the plan for proof. I display the goals and rules for the activity using the slide show. I start a digital timer: 10 minutes to work, plus 5 for set-up, leaves 5 minutes to recap (or to finish the proof as a class). As teams get to work, I circulate through the classroom.
BOLOs (Things to be on the lookout for): Are students trying to draw 3-dimensional cutting planes to show how the pyramid and prism can be dissected? (They can simply draw lines
on the bases of the figures, as in the teacher solution.) Are students confused because the equations they wrote do not seem to fit together as lines of a proof? (They must use substitution to write equivalent equations.) Are students trying to end the proof with an equation that relates the volume of a hexagonal pyramid to the volumes of the triangular pyramids into which it was
divided? (Students have visualized the end product incorrectly, which is reasonable since I did not tell them just how far the algebra will take us. I just tell them that they should conclude the proof with an equation that gives the volume of a hexagonal pyramid in terms of the volume of a hexagonal prism.)
- Remember that an equation tells how two quantities are related—in fact, that they are equal. As you read this sentence, what two quantities are described as being equal?
- How can you represent the quantity described here as an algebraic expression?
- You have tried different ways to put the equations you wrote in a logical order, but none seem to make sense. That is because these equations do not all say the same thing: they are not equivalent. In an algebra problem, if you are given two equations that say different
things about a single problem, that is called a system of equations. How do you solve a system of equations? That is, what are some ways to combine two equations to make something new?
- What clues have you found in the plan for proof? How have you tried to use them?
- The plan for proof refers to the distributive property. What does that mean to you? How could the distributive property apply to one of the equations you have written?
- How can you use the result? Can you write a formula for the volume of a pyramid?
Purpose: This activity shows that the relationship we found in the last lesson between the volumes of triangular pyramids and prisms can be extended to any pyramid. It also gives students practice in algebraic reasoning (which they will use again soon in a later lesson: the Sphere and the Cylinder 1.)
Provisioning: Before class, I reproduce the resource for this activity (1 copy for every team) and cut into half sheets.
I distribute copies of the nets (one member of each team gets the copies from the Resource Center) with a challenge: each team has 10 minutes to assemble the three triangular pyramids and the prism, find the volume of the prism, and identify which pairs of pyramids have the same bases and heights. To find the volume of the prism, they must measure its dimensions with a ruler. Measurements should be made in inches to the nearest 1/8 inch. To match up bases,
they must clearly mark the pairs of bases that are congruent (for example, by marking each base with an ‘A’ or ‘B’).
Students will inevitably try to fit the pyramids into the prism, so I don’t need to tell them. The open-faced prism makes this easier: place the first pyramid snuggly into the bottom of the prism (no forcing should be necessary), then the second (matching up the vertices), then the third.
The base of the prism measures 3 inches by 4 inches by 5 inches. The altitude is 2 inches. Volume of the prism is 12 cubic inches.
This kinetic activity is designed to help students remember the relationship between the volumes of a pyramid and its circumscribing prism. It also gives students who had trouble visualizing the pyramids a chance to handle them and see how they can be fit together to make a prism.
Provisioning: Before class, I reproduce the resource for this activity (1 copy for every team).
This activity is conducted using the Rally Coach format, a structure for cooperative learning. This is a classroom routine, so students know what to do. More information on how I use cooperative learning and Rally Coach can be found in my Strategies folder.
I display the rules for the activity using the slide show. I give the class 10 minutes to work and start a timer. As students get to work, I circulate through the classroom.
BOLOs (Things to be on the lookout for): Do students understand that to find the volume of a pyramid they must find the volume of a prism with the same base and height, then divide by three? (Students may be looking for a formula, which I avoid teaching them at
this point. I want them to see that understanding the relationships between the volumes of different solids is more powerful than memorizing a different formula for each type of solid. If a student needs a formula, I supply it and point out that it describes the relationship we discovered in class.) Do students understand that to find the volume of a cylinder, they must find the area of the base and multiply by the altitude? (The second problem is chosen to test how well students understand this idea. If they are simply trying to operate on the given dimensions, as with a formula, they will probably not find the correct area for the base, a trapezoid. I emphasize that V = Bh is the only formula to remember, but they must understand the relationship it describes correctly.) Are students using the wrong altitude? (We have not explicitly covered the definition of the altitude of a pyramid, but the general concept was covered in a previous unit. If we stack layers one unit thick, the altitude represents the number of layers—or fractions of layers—in the stack. Height is always measured perpendicular to the base, as slant height over-states the number of layers.)
- What do we know? The relationship between the volumes of pyramids and prisms. (My students are used to hearing me say that math is about relationships.) How can we use a prism to find the volume of a pyramid?
- Think about why we multiply base by height to find the volume of a prism. What does the height, or altitude, of the prism represent? Which of these two measurements would it make
more sense to use?
Provisioning: Before class, I reproduce the resource for this activity (1 copy for every 2 students) and cut into half sheets.
The homework provided with this lesson consists of three problems. The first two problems are about finding the volume of pyramids. The third problem asks students to verify that the volume of a cylinder quadruples when the radius of its bases is doubled. This reviews the R: R^2: R^3
rule, which students will apply in the next lesson.
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 work from the Finding Volume of a
Pyramid activity as an exit slip. After class, I look over the students’ work to see who is getting this and what problems others are having.