Today's warm-up gives students a preview of a sub-problem they will be encounter as they find the dimensions (length, width, and area) of rectangular sections of pyramids. It gives me a chance to see whether the class is ready and to make a few fixes, if necessary. Solution: DE = 9 ft.
This Lesson Opener follows our Team Warmup routine.
BOLOs: Do students see that corresponding angles formed by parallel lines are congruent? Do they see that congruent corresponding angles together with the shared angle shows that the triangles are similar?
- How might the two triangles be related? What property do they seem to have?
- How can we be sure that the triangles are in fact similar? What theorems or postulates could we use? (MP3)
- It is given that the bases of the triangles are parallel. What can this fact tell us? (MP1)
This activity helps students to see that examining cross-sections of a solid can help them to visualize its shape and to compare its volume with that of another solid.
I distribute the Treasure of Runegaard Activity handout with instructions that students should read it quietly to themselves and answer the questions on the front. I ask them not to look at the back side until instructed to do so. As students are reading, I use the Treasure of Runegaard Activity slide show to display larger views of the two heaps of treasure.
After about 3 minutes, I use the slide show for the lesson to display the rules for math ball. I tell the class that I want to hear how they are thinking about this problem, so they will have the chance to earn points by playing math ball. Referring to the rules, I tell them that they have 1 minute now to share their ideas as a team.
After 1 minute, I hold up a large googly ball, signaling that the game of math ball has
begun. I say, “Which heap of treasure has the greater volume, and why do you think it is so?”
Typical statements from students:
- Treasure A has the greater volume, because Treasure B is ‘flatter’.
- Both have the same volume, just as two triangles with the same base and height have the same area.
After about 3 minutes (3-5 comments from students), I ask students to turn over their handouts and read the situation described on the back side. I use the Treasure of Runegaard Activity slide show to display larger views of the two heaps of treasure with sections exposed by the work of the dwarves. I ask students to write their answers to questions 2 and 3 before sharing their ideas in their teams. I give students about 3 minutes before signaling that the game of math ball is about to resume.
I say, “Now which heap of treasure appears to have the greater volume, and why do you
think it is so? Also, I want to hear your answers to questions 2 and 3.”
I try to speak as little as possible during a game of math ball, and I never give an indication whether students’ statements are correct or not. However, I can ask for the ball back, and I do to keep the discussion on track.
BOLOs: Do students make conjectures and support them with informal arguments (MP3)? Do students build on what other students have said and explain why they either agree or disagree (MP3)? (To encourage this, I often offer double points for respectfully refuting another students’ claim.) Does the class recognize that the two pyramids have the same base and altitude? Do students use vocabulary words that were introduced in the handout: pyramid, apex? Do students recognize that sections of the two pyramids at the same height have the same shape and area?
- What seems to be true about the areas uncovered on top of each treasure heap at each level?
How can we check whether this is true?
- How does looking at cross-sections of the treasure help you?
Before class: Reproduce the Treasure of Runegaard Activity handout, 1 copy for every student.
I distribute the Slicing Pyramids Activity handout and tell the class that I want them to work in their teams to compare the measurements of the sections of the pyramids at different levels. Using slide show for the lesson, I display the rules for this cooperative learning activity, which follows our Team Jigsaw routine.
As students are working, I circulate around the classroom to help students keep moving forward and to identify student work that can be used as examples when we wrap up the activity. After all teams have completed the problem (or after 10 minutes), I call the class’s attention to the front board, where I use student work (or, if necessary, my own work) to summarize the activity (see the questioning strategies below). I end with the questions:
- How are corresponding sections of the two pyramids (sections taken at the same height) related? What does this seem to say about the volumes of the two pyramids?
BOLOs: Are students using proportional reasoning to find the lengths of the sides of each section of a pyramid? Do students see that parallel sections of a pyramid are similar? Do they recognize that Angle Angle similarity guarantees that this will be the case for any pyramid? Do students notice that the length of a side of a section of a pyramid is proportional to the distance from the apex, but the area of a section varies with the square of the distance? Do students notice that corresponding sections of the two different pyramids have the same area (and in fact are congruent)?
Questioning Strategies (MP1)
- Look at the section cut just at the apex of the pyramid. What is the shape of that section? What are its measurements? Area?
- Look at the section cut just at the base of the pyramid. What is the shape of that section? What are its measurements? Area?
- Look at the section that is half-way between the base and apex of the pyramid. What would
seem to be true about its measurements? Is the area also midway between the areas of the sections taken at the base and apex of the pyramid? (MP2)
- What property do all the sections of a pyramid have? Do you think that this is true of
all pyramids? Would it still be true if the sections of the pyramid were not all parallel to the base? Or parallel to one another?
- How could we use theorems and postulates we have learned before to prove/ verify that parallel sections of a pyramid are similar? (MP3) (Look at the front and side views of the pyramids. Do these drawings look like another problem we have seen before (Lesson opener)?
- How are corresponding sections of the two pyramids (sections taken at the same height) related? What does this seem to say about the volumes of the two pyramids? (Hopefully, a student will say that the volumes are the same, which is opens the door to the next part of the lesson.)
- (Extensions) Can you write an equation that relates the length of a section of the
pyramid to its height above the base, h? Can you write an equation that relates the area of a section of the pyramid to its height above the base? (MP2)
Before class: Reproduce the Slicing Pyramids Activity handout, 1 copy per team of 3-4 students. Cut into half-sheets.
As we wrap up the last activity, I am looking for a student to suggest that the pyramids have the same volume. This excellent idea is my cue to tell the (very short) story of Bonaventura Cavalieri and to display Cavalieri’s Principle using the slide show for the lesson.
I ask the following questions, giving students 1-2 minutes to discuss each one within their learning teams before calling on students to share their team’s answers.
- What effect does shifting the apex of a pyramid (parallel to the base) have on its volume?
- How could you use Cavalieri’s Principle and the work you did in teams today to explain that fact to an 8th grader? (MP3)
After hearing student answers, I use the computer animations to demonstrate how Cavalieri’s Principle shows that the volume of a pyramid does not change when the apex is translated parallel to the base. The animations are available at the NRICH math website
The homework provided with this lesson consists of three problems. The first provides additional practice in visualizing and finding the measurements of sections of pyramids. The second and third ask students to explain how Cavalieri’s Principle can be used to solve problems.
I display the prompt using the slide show for this lesson. Students write their answers in their learning journals.
I remind students to turn in their Slicing Pyramids Activity work as an exit slip. After class, I look over the students’ work to see who is getting this and what problems others are having.