Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

## The Largest Container

Video Overview

Ruth Ann Duncan begins the lesson with a review of the volume formulas for rectangular prisms and cylinders.

Then she poses the task: to make the largest container you can from a single letter-sized sheet of paper. Students work in parallel in their groups for a while, then Ms. Duncan asks each group to choose one container. Using a transparency and an overhead pen, the students then explain their container and report on its volume and surface area. At the end of the video, students make presentations to the class.

An Exploration for Teacher Workshops

Materials: paper, tape, rulers, calculators, scissors

Recall and review how to find the areas of plane figures such as rectangles and triangles, and how to find the volumes of solids such as cubes and pyramids.

Now, working in pairs, make the largest pyramid you can using only one sheet of paper. You may use any shape for its base. Then measure the pyramid to find its surface area and volume. Write the surface area and volume on the pyramid itself.

Discussion Topics
1. How did you and your partner decide what pyramid to build? How did you decide it was the largest?
2. What was the hardest part of doing the construction?
3. How did you find the volume and surface area? What was the hardest part of figuring them out?
4. Explain why the formula for the area of a triangle works. Can you find the area without using the formula?
5. Explain why the formula for the volume of a pyramid works. Can you find the volume without using the formula?

Extension

Record surface area and volume data for all of the pyramids in the room, then display these data on a scatter plot. Discuss the graph.

Discussion Questions

These questions appear at the end of the video. Here are some follow-up ideas and prompts to help get a discussion going.

#### When should you intervene to help students correct computational errors?

Some students make computational errors in this lesson, yet, in this video, Ms. Duncan doesn't correct them. Why do you suppose she doesn't? Would you? When and how? What are the pitfalls of correcting a student? What are the pitfalls of not correcting?

What's the role of the calculator here? Are there benefits? Pitfalls?

#### How could you restructure this lesson for your students?

Consider the components of Ms. Duncan's lesson. There's an introduction, a period of work in groups as individuals, a refocusing by the teacher, another period of work, and then presentations. How would you organize the lesson? Where would you spend more time? Less?

Also consider the goal of the lesson. "Largest" is undefined in Ms. Duncan's instructions. She is happy to let the students go for surface area or volume--- or, at the end, for "unique." Should they get a more specific goal, or is it all right to leave it open?

#### Discuss Ms. Duncan's contention that mathematics is for all students.

In every math class, some students "get it" more easily than others. In the video, some students are reasoning about unusual shapes while others are confused about volume, surface area, and measurement.

Do some students need more skills practice and less open-ended work? Do some need more challenging problems? Discuss the extent to which this activity is accessible to all students while challenging the most experienced?

Here are some additional ideas for discussion that arise in the video:

• Ms. Duncan, explaining the activity, concludes, "this finally was an activity they worked on somewhat independently." This implies that they've done previous, more structured work with containers and formulas for surface area and volume. What do you think your students would need in order to understand this activity and make it useful for them?
• What's difficult about doing activities in three dimensions? Why is it important?

Measurement and Formulas

• What do you think of the way Ms. Duncan explains the formula for the volume of a cylinder? (She moves the tennis-ball-can lid through space.) What other kinds of explanations might make sense to your students?
• What do you think of the way she shows a student how to find the surface area of a cylinder? (She wraps grid paper around it.)
• What strategies did you see students using in this task?
• What evidence do you see that students are developing a better understanding of the concepts of surface area and volume?
• Ms. Duncan says that the formulas "come alive" for the students. What evidence do you see that shows students are using and understanding formulas?