Home | Chapter 19 |

There is a significant difference between being able to apply the cross product algorithm to solve a proportion and being a proportional thinker. As outlined in the book, those who reason proportionally have a number of distinct characteristics.

Proportional thinkers -

- Have a sense of covariation. They understand relationships where a change in one quantity causes a corresponding change in another.
- Recognize proportional relationships as distinct from non-proportional relationships.
- Have a wide variety of informal methods for solving proportions and comparing ratios.
- Understand a ratio as an entity distinct from the quantities it compares.

Identification of Multiplicative Relationships. Additive relationships are easy to understand and use. They simply reflect an absolute difference between two quantities or measures. They are also restricted to like quantities or measures. Research indicates that multiplicative relationships are not as easily understood. A good place to begin this topic is with settings that allow for both additive and multiplicative relationships so that the two can be contrasted. This short section of the text encourages students to make these contrasts. Selection of Equal Ratios. In these activities ratios are presented some of which are equal and some are not. Students are to group ratios that are equivalent and explain. Comparing Ratios. The "Lemonade" task (Activity 19.5) has proven to result in excellent discussion. The format in which it is posed (with two pitchers as shown here) tends to cause a wide variety of solution methods as well as arguments that the two pitchers will taste the same. This and the pizza task (Figure 19.7) are important ones to explore. From a mathematical viewpoint, these activities might be viewed as the same as the comparison of two fractions. Pedagogically, even the pizza problem will prove to be different from the lemonade task, indicating the influence of context on ratio comparison activities. Note also that ratios often compare unlike quantities (as in the juice/water example) whereas fractions are strictly part/whole ratios. Also, when comparing two fractions, the size of the whole for each is always the same. This is not at all true for ratios. In fact, that is one of the things that distinguish non-fraction ratios - they can compare a ratio within a set of 12 things, for example, to a corresponding ratio within a set of 10,000 things. This fact lends itself to methods that are quite different from fraction comparison strategies.

(WATER IMAGE HERE)

Scaling Activities. These activities (finding missing entries in a table of two related quantities) if designed correctly, can involve excellent thinking in covariation. As Lamon points out, some scaling activities simply require adding like amounts along each portion of the table. While this may solve a proportion, the process does not involve proportional thinking. The book explains this difference. An activity related to scaling is to make coordinate graphs from these charts. Proportional situations are always represented by linear graphs through the origin with the common ratio being the slope of this line. It is useful to point this out whenever possible.

Measurement Activities. These activities are the most engaging because it is here that students are constructing or building, or measuring things that may or may not be proportional and then observing or predicting the ratios.

The drawing of models to establish a proportion turns out to be especially useful for looking at percentage situations. In fact, these drawings are essentially the same as those suggested in the previous chapter. You may find it useful to look at equivalent fractions as an example of a proportion and to examine these using a small drawing (Figures19.20 and 19.21). If students can become comfortable with the fraction situations, the jump to percents is not at all difficult.