d. Describe proportional relationships mathematically using y = kx, where k is the constant of proportionality.

There are many quantities around us that vary in relationship to each other. For example, here are some examples of pairs of quantities that vary simultaneously:

b) the number of pages of a book that have been read and the number of pages to be read

c) the distance traveled and the time of travel (at a constant speed)

d) the speed and the time it takes to travel a fixed distance

e) the length of a candle that has been burned and the remaining length

f) the amount of meat and the price of meat

g) the length and the width of a rectangle with a fixed area

h) time of the day in Atlanta and Los Angels

Let's look at these situations a little more carefully. How are the ways the quantities change similar or different? One thing you notice is that in situations a, c, f, and h, as one quantity increases the other also increases. We can cal these increase-increase situations. In contrast, in situations b, d, e, and g, as one quantity increases, the other decreases. So at one level, we can sort these situations into increase-increase and increase-decrease situations.

But, let's dig a little deeper. Let's look at each group more carefully. How are the ways quantities changing different from each other? Let's look at the increase-decrease situations (b, d, e, and g) first. As the two quantities in each of these situations change, is there anything that is not changing - mathematically, the idea of "invariance" is a very important one. You notice that in situations b and e, the sum of the two quantities remain the same. For example, the total number of pages in a book is the sum of the number of pages already read and the number of pages to be read. In contrast, in situations d and g, what stays constant is the product of the two quantities, the distance traveled in d and the area in g.

Now, let's look at a, c, f, and h. As the two quantities in each situation change, is there anything that is staying the same. In situations a and h, what stays the same is the difference between the two quantities. For example, the difference between the ages of two siblings on January 1 will always be the same no matter how old they become. In contrast, in situations c and f, what stays the same is the quotient of the quantities.

So, these situations can be sorted into four categories based on what stays constant in each situation: constant sum, constant difference, constant product, and constant quotient. Based on this way of sorting, we can also express the relationship between the two quantities using mathematical equations in the following ways (k is a constant):

constant difference: x - y = k

constant product: x*y = k

constant quotient: y÷x=k

Of these four ways two quantities change simultaneously, we call the last situation, i.e., constant quotient, a proportional relationship. This relationship can be written in mathematical equation as y÷x = k, or y = kx (M6A2 d & e). Moreover, the constant product relationship, xy = k, or y = k÷x, is called an inverse proportional relationship.

When we want students to understand a new concept, it is very important and useful if we provide situations to compare and contrast several cases - examples and non-examples. Clearly there are many other quantities that change in relationship to each other that do not necessarily fit into these four categories - for example, the amount of time you study for a test and your score on a test. Thus, restricting the situations to examine to these four types may be a bit arbitrary. However, sometimes we may want to investigate only those situations that will allow us to analyze them in a particular way. It does not mean that we should investigate other, more messy situations. However, we may not need non-examples that are too complicated.

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