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Teaching Math: A Video Library, 5-8

Building Viewpoints

Video Overview
Pam Hardaway begins by showing the class a blueprint of a building. She asks what it is and what it's used for. Then she explains that they will be emulating Emily Hawkins, a fictional archaeologist, in learning what ancient buildings looked like based on their plans.
She leads the students through creating a "building mat," on which they'll make models out of cubes.
Then the students work with printed worksheets to build several structures and answer questions about them. Their tasks include drawing side views and describing relationships between them.

An Exploration for Teacher Workshops
Materials: cubes, markers, grid paper
Working with a partner, build a structure with six cubes. The cubes must share whole faces (as with Multilinks, no partial overlaps).
Then, using grid paper, make four silhouettes of the structure: one from the front, one from the back, one from the left side, and one from the right. Be sure to label the silhouettes.
Trade your set of silhouettes with another group. See if you can reconstruct their structure from the silhouettes. (It is not always possible from the silhouettes alone.)
Note: This exploration uses silhouettes partly to sidestep the issue of whether to draw visible edges or hidden lines.

Some Questions

Is there only one structure that fits a set of silhouettes, or are there many structures?
What do you think determines whether a set of silhouettes will be unique?
How can you devise a structure with six cubes so that all four silhouettes are identical? (There are at least two.)
Suppose the silhouettes were not labeled back, front, left, and right. Could you still reconstruct the structure? How?

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

This is the only lesson in this video collection in which students work through a printed worksheet. Does it affect the "feel" of the lesson? What do you like about it? What would you change?
This lesson was going to continue on the following day. At the end of class, Ms. Hardaway simply stopped the students and had them clean up. She chose not to bring closure—for example, with brief student presentations. Discuss the pros and cons of bringing closure in these situations. What would you do?
This lesson works with individuals as well as groups; in fact, one of Ms. Hardaway's students was working alone. What aspects of the lesson make that possible? How do you decide whether students work alone or in groups?
How would you assess students' understanding as they are working on this lesson? Can you think of specific examples from the video?

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.
Discuss Ms. Hardaway's questioning strategy.
Pam Hardaway usually answers a question with a question. Many teachers would agree with this philosophy—but what purpose does it serve?
When is giving the answer effective? When—and in what way—is giving the answer ineffective?
Why is the content of this activity important in the middle grades?
In this activity, students construct three–dimensional objects from a two–dimensional plan, then make a different set of two–dimensional representations of the object. Finally, they discuss and write about relationships among the two–dimensional drawings.
Where will students use those skills in real life? Where will students use them later in the mathematics curriculum?
How could you extend or enrich this activity?
What are the most important features of this activity? What other kinds of structures might you have students make plans for? How else could they represent three–dimensional structures? How might they build on what they have done? How else could you make connections outside math?