Using Tables & Graphs to Explore Ratios & Rates

Instructor: Elizabeth Popelka-Brown
In this lesson, students will learn how to determine the ratio or rate of change of a relation from a table or graph. Students will also compare and contrast ratios, rates, and rates of change.

Using Ratios and Rates

Jayshawn has recently started running. Although his main purpose is to be healthy, he also wants to be a good runner. He can use ratios and rates to assess his current ability and make goals for improvement. Let's explore how!

Ratios

Before diving into Jayshawn's situation, it is helpful to review some basic concepts. A ratio is a comparison of two different numbers. Examples of the three most common ways to represent a ratio are seen here.


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Pay close attention to the fact that, in ratios, order matters. For example, if a classroom has 17 boys and 13 girls in it, and a teacher wants to write the ratio of girls to boys as a fraction, she should write


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The number of girls should be in the numerator, or the top, of the fraction.

Ratios in a Table

Ratios can be pulled from a table of values as seen here. In addition, we frequently simplify ratios the same way that we would simplify any other fraction.


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Rates

A rate is a special kind of ratio. The two numbers being compared must represent two different things. The ratio


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is a rate because it compares the number of miles traveled to minutes. Since miles are a unit of distance and minutes are a unit of time, they are two different measures.

Rates and Speed

Jayshawn has an app that tracked his most recent running times and distances in a table. He pulled the rates out of his table, as seen here, but, unfortunately, they are not helpful for comparing, because none of the denominators, bottom numbers, are the same.


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In order to more easily compare, we should find the unit rate, the rate when the denominator is a 1. To obtain the unit rate, divide the numerator and denominator by the value of the denominator, as seen here.


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Each of these rates provides Jayshawn with information about his speed. For example, the first row of the table provided the unit rate 13.3 minutes per mile. This means, at that rate, he would take 13.3 minutes to run a mile. The second row of the table shows that he must have picked up his pace a bit, because it produced a rate of 11.5 minutes per mile.

Rate of Change

Jayshawn might also choose to look at a special kind of rate. Rate of change compares the difference, or change, between two sets of data points. Here, Jayshawn took the first two data points from the table and subtracted to find the difference between the first two distances and the first two times.


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When calculating rate of change, mathematicians have come to an agreement that the dependent variable, the one that depends on the other, should be in the numerator of the rate. Since the number of miles Jayshawn runs depends on how long he runs, the number of miles are in the top of the rate.


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Frequently, the rate of change is also converted into a unit rate.


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Interpreting the Rate of Change

The meaning of the rate of change between the first two data points on the table can be seen here.


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For Jayshawn's run, the rate of change was not the same between every set of data points.


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Constant Rate of Change

If we were to graph Jayshawn's run on a graph, as we have here, it would be non-linear, or not a straight line. That is because the rate of change is not constant, the same, between all sets of data points.


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A Linear Graph

Sam is a good friend of Jayshawn's who also runs. One thing that Sam does well is controlling his running pace, so that it is very consistent. The graph below is a screenshot Sam sent from his running app. The graph is perfectly linear, which means that his rate of change is constant throughout the whole run.


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