## Functional Relationships

Functional relationships exist all around us in our everyday experiences. As adults, we experience these relationships when we are spending our money. For example, a large retail store is having a sale on bananas. The sale price is 4 pounds for a dollar. The regular price is \$0.35 per pound. The sale price of 4 pounds per dollar is actually \$0.25 per pound. Therefore, you are getting more banana for your dollar buying while the bananas are on sale.

The difficulty is realizing that the word "sale" does not always mean that the consumer saves money. Teaching our students to understand the use of functional relationships is the key to developing educated consumers. With the advent of large wholesale stores, our society is in the habit of thinking that the larger the container you buy, the better the deal you are getting. This is not always the case. The use of functional relationships allows an educated consumer to make the smart decision by getting "the most for their buck." As adults, we need to buy staple products such as laundry detergent. Some stores sell the "economy" box of detergent (276 ounces). Since this is the largest box, many consumers think it is the better deal. A closer look sometimes reveals the opposite. For example, the prices of three different sized boxes are as follows:

 276 ounce box \$12.95 184 ounce box \$6.25 92 ounce box \$3.00

If we look at the relationship between the number of ounces and the price, we can determine how much we are actually paying for each ounce. By dividing the cost of each box by the number of ounces we find:

 Box price approx. cost per ounce 276 ounce box \$13.95 \$0.05 184 ounce box \$7.50 \$0.04 92 ounce box \$3.00 \$0.03

An educated consumer finds that the smaller box is actually a better buy. Three smaller boxes would cost you \$9.00 for a total of 276 ounces. Bigger is not always better.

As we teach children about the important and usefulness of functional relationships, it is important to use examples that they can relate to. Although buying detergent is a common chore for most adults, searching for a better price on CDs might make more sense to your students.

There are many types of relationships that you will encounter in everyday life. A linear relationship is an example that we encounter; for example, taking a test in school. The table below shows the results of a ten-question test.

 Number correct Points 1 5 2 10 3 15 4 20 5 25 6 30 7 35 8 40 9 45 10 50

Based on the information in the table, each problem was worth 5 points. A graph of the data is shown below.

Notice that when all the data points are connected, the result is a straight line. This indicates a linear relationship. The word linear means line. A linear relationship indicates that as the number of questions answered correctly increases, the number of points will also increase by a value of 5 points per problem. The rate the number of points increases is constant.

The goal is for students to recognize the relationship between the number of questions answered correctly and the number of points awarded, and that for each correct answer, the student is awarded five points. Although students at this level will not see linear relationships on graphs, they will deal with linear relationships in word problems and in function tables.

Due to the level of mathematics of your students, you will not be able to describe some of these relationships in a mathematically sophisticated fashion. Not all relationships are linear. For example, look at the table below. The table represents the number of minutes students studied for a spelling test and their actual grade on the test.

 Time (minutes) Grade (percent correct) 10 15 20 20 30 30 40 50 50 75 60 100

A graph of the data is shown below.

Notice that if you connect the dots, you would not form a straight line, but a curve. Therefore, this is not a linear relationship. In other words, as the number of minutes increases, the percent correct does not increase at the same rate throughout the graph. The number of minutes increases by 10 each time, but the percent correct increases by values such as 5, 10, 20 and even 25. The goal is for students to recognize that the rate of increase is not constant (linear). An important characteristic of the relationship of study time versus percent correct is that the more the person studied, the greater the percent correct on the test. In other words, there was a positive result for the increase in study time.