Course Guide - USAID Teacher Education Project

Unit 1 Number and OperationsWeek 4, Session 2: Ratios6. Maths Concepts to be studieda. Proportional thinking, like multiplicative thinking, develops over time with its roots instudents’ understanding of fractions, decimals, and percents. Ratios and proportions areused to compare two quantities in order to answer questions such as, “What is the ratioof men to women in our class?” where we are comparing two parts of the whole to eachother (part-to-part).This is different from the type of comparison being made when asking the question,“What proportion of our class has a laptop computer?” where the comparison is part-towhole.Note that these types of part-to-whole questions could be rephrased as, “Whatpercent (or fraction) of our class has a laptop?”b. Models for Ratios and Proportions: Ratios and proportions can be numeric, or they canbe geometric. For a numeric example, consider the ratio of orange juice concentrate towater. Each can of concentrate has directions printed on the label that says it should bediluted with three cans of water. Thus, the resulting mix of 1 can of concentrate with 3cans of water gives a total of 4 “cans worth” of juice.This mixture can be thought of in several different ways. If we consider therelationship “part-to-part,” the mixture would have a ratio of 1:3 (1 part concentrate to 3parts water).We can also think of the mixture as “part-to-whole” where the concentrate is 1/4 of themixture (1:4) and the water is 3/4 of the mixture (3:4). Either relationship, part-to-partor part-to-whole, is valid, but we need to be clear about which type of relationship weare discussing.7. How children think about these conceptsa. Youngsters often are not aware that the order of terms in a ratio is important. Forexample, if I have four children and only one is a girl, the ratio of girls to boys is 1: 3,whereas the ratio of boys to girls is 3:1.b. Children also are confused by part-to-part vs. part-to-whole ratios that refer to the samesituation. For example in the above scenario, I have three times as many sons asdaughters (part-to-part) but 3/4 of my children are boys, where the 4 represents the totalnumber of children (“the whole”) in my family.c. As mentioned above, although youngsters can be taught cross-multiplying as a quickway to solve proportions, they usually have no idea why this works. Even though thewritten description of using equivalent fractions might seem involved, it actually makesthe mathematics of solving proportions more sensible to students. Loretta, where aboveis this cross multiplying?8. What is essential to know, and do in classa. Compare and contrast the “part-to-part” and “part-to-whole” models for thinkingabout ratios and proportions, referring back to the nature of “the whole” that wasdiscussed when studying fractions and percents