Comparing Quantities
Ratios and proportions are important mathematical tools. With an understanding of these concepts, students can solve a wide variety of problems and understand situations in which relationships between quantities play an important part.
Ratios are comparisons of two quantities. For example, in a certain group there are 5 girls for every 7 boys. This means that in each group of 12 students, there are 5 girls and 7 boys; the girls are in a 5 to 7 (5:7) ratio with the boys. The boys are in a 7 to 5 (7:5) ratio with the girls. Mathematicaly, the quantities of the boys and girls are being compared by using a ratio:
girls : boys :: 5 : 7
In order to portray this ratio with a model, we can use something as simple as circles and squares, or any other shapes. One shape represents the boys, and the other the girls:
This model is known as a 'set' of the minimal ratio - 'minimal' because no common factors exist for five and seven - they are 'relatively prime'.
We can use the model to represent various situations involving a group in which the ratio of boys to girls is 7 to 5. Here's an example.
Suppose the group contains 21 boys. We know that for every seven boys there are five girls, and that if there are 21 boys, there must be three groups of seven, represented by the blue squares. For every set of seven blue squares, there are five red circles representing the girls. The model would develop like this:
By observing the model, we can answer questions like: 'How many girls are in the group?" (15), and 'How many people are on the group all together?" (36). We simply have to create the appropriate number of sets to describe the situation.
Here's another example to think about.
Suppose there are sixty people in the group. How many boys and girls are there? We know that each 'set' of boys and girls contains twelve people (7 + 5), so there must be 5 (60 ÷ 12) sets in all. The model looks like this:
We can see the answers to our question in the model. There are 35 boys (5 x 7) and 25 girls (5 x 5). After working with a number of situations like this, students begin to see some generalized procedures that can be represented by arithmetic operations. In this problem, divide the total number of people by the number of people in one set. Multiply that answer by the number of boys and girls in each set to get the totals for each. (60 ÷ 12) x 5 = girls, and (60 ÷ 12) x 7 = boys.
We can also represent ratios with a linear model. A ratio of 3 to 4 could be represented this way:
This model is particularly good for exploring proportion, the comparison of two ratios. Proportions are written as a:b::c:d - read as "a is to b as c is to d". Typical mathematics problems often involve finding a missing part of the proportion in which all but one of the letters, representing pieces of information, are known. We can explore these situations with examples like this one:
We can approach this two ways - (1) find the number of same-sized groups of small segments in a and b, or (2) divide a and b into an equal number of groups. Then we can use the same ratio to construct segment d.
Now that we know the ratio of a:b, we can use it with segment c to determine the length of segment d.
Again, by exploring a variety of similar situations, students can generalize procedures for finding the missing information.
Many mathematical situations can be modeled by the two types of models described here. Here are a few that we explore in class. How would you approach them?
