A B C

Solutions for Session 10, Part B

See solutions for Problems: B1 | B2 | B3 | B4

 Problem B1 Measuring with nonstandard units provides students the opportunity to choose the unit they want and to gain a sense of the physical process of a measuring experience. When students use nonstandard units (e.g., paper clips, playing cards, or buttons) to measure area, they notice that they can't cover the surface completely. The other difficulty with using nonstandard measures is that students can't compare the measures; for example, 110 paper clips vs. 12 playing cards.

 Problem B2 Students made predictions about whether the rectangle or square was larger at the start of the lesson. Doing so helps them consider which attributes are important as well as visualize how much space the shapes take up. Students may even visualize cutting up one shape to cover the other. In the video, one group of students compares the dimensions of the two shapes. The students in this group notice that the length of the rectangle is longer than that of the square, and then they use finger span to estimate that the heights of both the rectangle and square are the same. They conclude that the rectangle must be larger.

 Problem B3 Ms. Guerino asked students to use both square inches and square centimeters to measure the shapes so that she could address the idea of conservation. Students may not realize that the areas will stay the same even if they measure with different-sized units. To help students understand conservation, ask them to predict which rectangle will have the largest area before they cover the shapes with the new unit. Also, asking students to explain why the larger shape is still larger, and if it will always be larger, can help them reason about conservation. Another measurement concept that Ms. Guerino explored when she asked students to cover the shapes with different-sized units is the concept that the smaller the unit, the greater the number needed to cover the shapes.

 Problem B4 It is not only important, but actually helpful when misconceptions surface in a lesson. Misconceptions often provide the contrast needed for students to understand an idea. One strategy to use when misconceptions arise is to present two opposing viewpoints and ask students which idea they agree with and why. You can also ask students to give a counterexample, or give the students a counterexample yourself and ask them what they think. Another reason why it is important to discuss misconceptions: By confronting a conception that turns out to be incorrect, students become engaged in understanding why it is a misconception.