More: In-Depth Discussion of the Reasoning Activities in "Teaching Fractions and Ratios for Understanding"

By Susan J. Lamon | Go to book overview

Chapter 2
Relative and Absolute Thinking
1. Both the King family and the Jones family have two girls. However, the families have different numbers of children, so that if we think about the number of girls as compared to (or relative to) the number of boys or to the total number of children, then the situation looks different in each family. If we compare the number of girls to the total number of children, in the Jones family of the children are girls and in the King family, 2/4 or 1/2 of the children are girls. The ratio of girls to boys in the Jones family is 2:3, while in the King family, it is 2:2. Therefore, we would say that the King family has a greater proportion of girls or that more of the King family is girls.
2. Merely by counting the brown eggs in each container, we can tell that the 18-egg container has more brown eggs. However, we might consider the fact that the containers hold different numbers of eggs. 7 out of the 18 eggs or 7/18 are brown. 4 out of the 12 eggs or 1/3 of the dozen are brown. Because 7/18 is more than 1/3, the dozen-and-a half container has more brown eggs in both an absolute and in a comparative sense. This problem shows the importance of asking students to explain their reasoning. If a student said that the larger container has more brown eggs, we would not know if the student was or was not thinking relatively.
3. The answer to "how much" is not a number of slices. Pan B contains 3/8 of a pizza more than pan A contains. You could serve up the amount of pizza in A 2 1/2 times out of the pizza in B. Note that these comparisons are OK in this situation because the pizzas are the same size. If they were not, these questions would not be meaningful.
4. To decide which ramp is steeper, you cannot merely measure the part on which you ski. How steep something is depends on how the amount it goes up is spread over a horizontal distance. For example, it you walk up a vertical distance of 1200 feet but do it gradually, say, over the distance of 2 miles, you will not feel that you have been traveling a steep road. But if you rise 1200 feet in the course of half a mile, you would be walking a very steep path! So steepness is a comparison of how high you rise to the horizontal distance it takes to rise that distance. You

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