A B C 

a. 

The teachers discussed the following ideas that would be important to explore with their students:

b. 

Answers will vary. Students may have difficulty understanding measurement relationships, measurement formulas, indirect measurement, and the ideas of accuracy and precision.

c. 

To help students make sense of these concepts and skills, you can help them explore the relationship between area and perimeter of various shapes. They can also estimate the area of irregular shapes. They can see the effect that using different-sized square units has on a measurement and how using smaller units can help make a better approximation.

d. 

To deepen and extend their understanding of area, students can explore the effects of a change in dimension, surface area, and volume on the other attributes of a three-dimensional object.

Problem A2

Answers will vary. To deepen students' understanding of area, you can have them examine the effect that changing dimensions will have on the surface area of a rectangular prism. Give students 24 unit cubes and challenge them to make a rectangular solid that has the least possible surface area and one that has greatest possible surface area.

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Problem A3

Even older students can benefit from having the experience of physically measuring. Just as number play helps students develop number sense, measuring helps them develop measurement or unit sense. Measurement activities also help students see measurement shortcuts and develop measurement formulas. For example, filling a rectangular box with layers of unit cubes can help students see that a layer is equal to the area of the base, and also that volume can be determined by multiplying the area of the base by the height. The reason for conducting an activity like going outside to measure shadows is that it allows students to see how to apply indirect measurement concepts to the solution of problems. The skills involved relate measurement ideas to proportionality, which is another important topic for middle school students. Hands-on experiences like these help students form a strong conceptual foundation upon which to develop more abstract and complex forms of thinking and analysis.

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