This lesson takes students through a rather elongated chain of reasoning that eventually arrives at why the formula for the volume of a cylinder should be pi times radius squared times height. I explain to students that we will be establishing this chain of reasoning and that each link in the chain is equally important, as are the connections between them.
This section forms the first link in the chain of reasoning.
The section starts with each student receiving Postulating the Rectangular Prism Formula.
To begin, I have students try to recall what a postulate is. Then I have them do a quick pair-share before I discuss with them that a postulate is a statement accepted without proof that is often used as a basis for proving other statements. I explain, as the handout states, that we'll be treating V = lwh as a postulate, but in order to do so, we'll have to convince ourself of its truth so that we can really accept it without proof.
The first six items on the handout give students a concrete example illustrating V = lwh for a rectangular prism. I give my students 5 minutes to complete these six items before showing and discussing the answers with them. I emphasize thinking in terms arrays and multiplication rather than simply counting cubes. For example on each layer, we know there should be 4 rows of 2 cubes (or 2 rows of 4 cubes), so 8 cubes per layer. We also know that there should be 10 layers so 10x8 = 80 cubes in total. This type of thinking will prepare students for the abstract case on the second side whereas counting will not be useful.
Next, I give students 3 or 4 minutes to complete items 7 and 8 on the second side of the handout. I then have them share their answers with their seat partners before I call randomly on non-volunteers to share their answers with the class. As I'm hearing the answers, I provide feedback that will move students toward answers like, "There are lw unit cubes on the bottom layer because there will be l rows of w cubes for a total of lw cubes per layer. Similarly, there will be a total of h layers, and since there are lw cubes per layer, there will be a total of lwh cubes."
Finally, I give my students 3 or 4 minutes to complete items 9 and 10 in collaboration with their seat partners before I show and explain the answers and their implications. I basically explain that we have attempted to convince ourselves through a concrete example, and a consideration of the abstract case (with integer dimensions), that V = lwh. We then expanded our thinking to non-integer dimensions in hopes of convincing ourselves that V = lwh for any rectangular prism regardless of its dimensions. This is our postulate that we will use going forward to the next section.
After this explanation, I have the students do a quick pair-share to re-voice what we have accomplished in this section.
To prepare for this section I hand each student a copy of Reasoning from rectangular to triangular prisms.docx.
In this lesson, it's easy for students to lose sight of the forest for the trees. So I need to hold the big picture view in front of them along the way. I start this section by reminding my students that we are working our way toward the formula for the volume of a cylinder. However, I explain, we can't jump straight from rectangular prisms to cylinders. First we'll deal with triangular prisms which will be a key step in our working toward cylinders. Remember, each link in the chain of reasoning is equally important as are the connections between them.
We read the text on the first side of the handout together as a class and I will clarify concepts as the need arises. I will definitely explain, for example, why we choose to show the triangular prism as part of a rectangular prism. I explain that we have to consider where we just came from. Namely, establishing the volume formula for rectangular prisms. So if we can see a triangular prism as part of a rectangular prism, we can find a way to express the volume of the triangular prism.
Next, I have students complete item #1, which is a way for me to make sure that students are understanding the text and interacting appropriately with the diagrams. After students have completed item 1, I have them check with their seat partner before I reveal and briefly discuss the correct answers.
I follow this same pattern for items 2 through 5, giving students time to answer on their own, then time to consult with their seat partners before I reveal and discuss answers. Time permitting, I may also have students share their answers. For example, I like to have students explain what is meant in #2 when it says, "By properties of rectangles and SSS..."
When we've finished items 2 through 5 in this manner, I have students spend 3 minutes independently trying to make sense of the paragraph immediately following item 5. I direct them to read it and keep re-reading it until it makes perfect sense or time runs out.
Next, I'll clarify the meaning of the paragraph before moving on to the rest of the narrative at the bottom of the page. With regard to the last part of the narrative, I'll ask check for understanding questions. For example, "Where did the 1/2 (AC)(AD)(h) come from?"
Finally I'll play up the important accomplishment in this section, which is finding out that the volume of the triangular prism is the area of its base times the height of the prism. I explain that this is key because we will be showing that this fact is true for all prisms regardless of the shape of their bases.
In preparation for this section, I give each student a copy of Reasoning from Triangular Prisms to General Prisms.
I remind students that we are working our way closer to establishing the volume formula for cylinders. I explain that in this section connecting triangular prisms to general polygonal prisms will be an important advance toward that goal. I explain that they will be working, first on their own, and then with a partner to complete a partially completed argument that establishes this important connection between triangular prisms and general polygonal prisms. I explain that they will need to read and study diagrams in order make sense of things and they will need to persevere in order to get the job done.
Then I give the students 10 minutes to work independently (no collaboration) trying to complete both sides of the handout (in pencil...just in case). During these 10 minutes, I walk around checking in with any student who appears to be stuck or completely off course, giving them feedback to move them forward in the right direction.
When the 10 minutes have elapsed, I give the students another 5 minutes to consult with their seat partners and neighbors. During this time, I walk around monitoring the conversations. If I see that two seat partners have different answers, I'll point this out and start a discussion between them aimed at reconciling the disagreement.
Finally, I reveal and discuss the correct answers to make sure that all students understand the logic and math involved.
To begin this section, each student receives a copy of Reasoning from General Prisms to Cylinders. I give students 10 minutes to work in pairs to complete the handout using the following protocol:
When the 10 minutes have elapsed, I reveal and discuss the correct answers.
Next, I give students 10 minutes to work in pairs rehearsing the argument laid out in items 1 through 5 without looking at the text. In other words, the partner who is rehearsing should be referring only to the diagram as they explain to the other partner how they reason that the volume formula for a cylinder is pi times radius squared times height.
As a take-home assignment, I'll ask students to provide a written explanation of the role that each of the four parts of the argument played in establishing the volume formula for a cylinder.