Arguments for Volume Formulas for Pyramids and Cones
Lesson 4 of 6
Objective: SWBAT give an informal argument for the volume formulas for pyramids and cones.
Central to this lesson and central to the study of statistics is the concept of central tendency. In this section of the lesson we'll be exploring the an important idea related to the mean. More specifically, we'll be exploring the big idea that the mean is the single number that could replace all entries in a data set and leave the sum of the entries unchanged.
In order to get students thinking in this way, I have them work on Concept Development-Central Tendency. This resource has three problems, with the last being slightly more challenging than the first two.
I start by giving students 5 minutes to do as much as they can do. I allow the students to collaborate during this time.Most students will finish at least #1 and begin #2. Some will finish #1 and #2. At the end of this time, I will explain #1 thoroughly. I make sure to emphasize the big idea that the arithmetic mean (aka average) is the number that could replace all entries in the data set and leave the sum of the entries unchanged.
Next I give students a couple of minutes to continue working on #2. When this time has elapsed, I call on students at random to share their answers and rationales.
When we've adequately addressed #2, I'll give students 5 minutes to work on #3. As students are working, I'll walk around observing how students are approaching the problem. When the five minutes have elapsed, I'll demonstrate how I would solve the problem. Usually there will be students who have solved it another way. These students usually say something like "I got the same answer, but I solved it another way. Is that ok?", at which time I ask them to come to the document camera and share their approach with the class.
See the reflection in this section on Multiple Pathways for my answer to the question "Should we treat all correct pathways as equal?"
I begin this section with a demonstration and a question. I have large fillable demonstration prisms and pyramids like the ones shown here.
I show the students, with emphasis, that the prism and pyramid have the same base area and the same height. I then pose the question: "If I were to fill the pyramid completely with water and then pour all of the water from the pyramid into the prism, what fraction of the prism would be filled?"
I give students a minute to think about this as I channel my inner flight attendant, miming the act of filling the pyramid, then pouring its contents into the prism and observing the fraction of it that has been filled.
When students have had time to think, I have them call out their answers one at a time so that I can get a quick survey of the class' thinking. Alternatively, I might also have students write their estimates on mini whiteboards. In any case, I want to know what all of them are thinking.
At this point, I'll write the formula for the volume of a pyramid (V = 1/3 Bh) on the board next to the formula for a prism (V=Bh). Students will then know that the fraction is 1/3. However, they may be simply saying 1/3 because it's the only fraction that appears on the board. For this reason, I want to take care to focus students' attention on the structure of the formulas, and have them understand that when the two formulas have equal base areas and heights as their inputs, the pyramid formula will yield a value exactly 1/3 of the value yielded by the prism formula.
Next I explain that our goal in this lesson is to construct an argument that will convince us, and others, that 1/3 is the correct fraction.
In order to structure the process for reaching that goal, I give each student a copy of Argument for Pyramid Volume Formula. We read through the handout together as a class and I stop at various points for pair-shares to make sure that students are understanding what we are reading. I also elaborate and clarify what is being said on the handout as I sense the need.
When we reach the second page of the handout, students will be working to complete the table. I give them 10 minutes working in pairs to do this and I allow them to use a calculator. At the end of the 10 minutes, I show the correct values. Then I give students 5 minutes individually to answer questions 1 and 2 on the handout.
Next I have students exchange papers in pairs and give each other constructive criticism aimed at improving their answers. Finally, I'll call 5-7 non-volunteers randomly to share what they have written and what they would add/modify based on the feedback from their partner.
Once students have answered the reflection questions in the previous section, I call their attention to the front of the room as I demonstrate how we can use spreadsheets to perform the types of calculations they've just finished performing with a calculator. I explain that spreadsheets are a very useful tool for these types of calculations, especially when we need to perform a large number of similar ones.
In the following video, I explain how this demonstration goes.
I use an exit ticket here to make sure that students understand, and can articulate, the logic that is inherent in the informal limit argument that has been laid out in the lesson.
Before having them write, I have the students discuss with their seat partners everything we did during the lesson focusing on the following questions:
What was the purpose? How did it support our goal of validating the pyramid volume formula?
I do this to give students a chance to organize their thoughts and have things clarified before having to write.
When this is done, I have students write independently in response to the following prompt.
Today we made an informal limit argument to establish that the volume of a pyramid is given by V=1/3Bh. Summarize that argument. Be sure to explain how our findings and calculations supported the argument.
At the end of class, I will collect these to take home and read. I will look for misconception and incomplete understandings that I need to address. I will also look for exemplary responses that I can share with the class at the next class meeting.
As a take home assignment, I will have students create an argument for the formula for the volume of a cone. The argument will be very similar to the one we made for the pyramid volume formula so this will be an opportunity for students to transfer what they learned in the lesson to a new, although not very novel, context.
The main elements of the argument are:
- The volume of a cylinder is (pi)*r^2*h
- A cone is composed of infinitely many circular cross-sections and a point (the vertex).
- A cylinder with the same height as the cone and a base with the average cross-sectional area of the cone would have the same volume of the cone.
- Informally using limits, we can look at greater and greater numbers of equally spaced cross-sections to approximate the average cross-sectional area. The limiting value seems to be 1/3 *pi*r^2
- So the volume of the pyramid is 1/3 *pi*r^2*h