The purpose of this lesson is to introduce the concept of a proportional relationship by looking at tables of equivalent ratios. Students learn that all entries in one column of the table can be obtained by multiplying entries in the other column by the same number. This number is called the constant of proportionality. The activities use contexts that make using the constant of proportionality the more convenient approach, rather than reasoning about equivalent ratios.
In any proportional relationship between two quantities \(x\) and \(y\), there are two ways of viewing the relationship; \(y\) is proportional to \(x\), or \(x\) is proportional to \(y\). For example, the two tables below represent the same relationship between time elapsed and distance traveled for someone running at a constant rate. The first table shows that distance is proportional to time, with constant of proportionality 6, and the second table, representing the same information, shows that time is proportional to distance, with constant of proportionality \(\frac16\).
|time (h)||distance (mi)||constant of proportionality|
|distance (mi)||time (h)||constant of proportionality|
These tables illustrate the convention that when we say “\(y\) is proportional to \(x\)” we usually put \(x\) in the left hand column and \(y\) in the right hand column, so that multiplication by the constant of proportionality always goes from left to right. This is not a hard and fast rule, but it prepares students for later work on functions, where they will think of \(x\) as the independent variable and \(y\) as the dependent variable.
- Comprehend that the phrase “proportional relationship” (in spoken and written language) refers to when two quantities are related by multiplying by a “constant of proportionality.”
- Describe (orally and in writing) relationships between rows or between columns in a table that represents a proportional relationship.
- Explain (orally) how to calculate missing values in a table that represents a proportional relationship.
Let’s solve problems involving proportional relationships using tables.
A measuring cup and a tablespoon is optional—they may be handy for showing students who are unfamiliar with these kitchen tools.
- I can use a table to reason about two quantities that are in a proportional relationship.
- I understand the terms proportional relationship and constant of proportionality.
constant of proportionality
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity. This number is called the constant of proportionality.
In this example, the constant of proportionality is 3, because \(2 \boldcdot 3 = 6\), \(3 \boldcdot 3 = 9\), and \(5 \boldcdot 3 = 15\). This means that there are 3 apples for every 1 orange in the fruit salad.
number of oranges number of apples 2 6 3 9 5 15
Two ratios are equivalent if you can multiply each of the numbers in the first ratio by the same factor to get the numbers in the second ratio. For example, \(8:6\) is equivalent to \(4:3\), because \(8\boldcdot\frac12 = 4\) and \(6\boldcdot\frac12 = 3\).
A recipe for lemonade says to use 8 cups of water and 6 lemons. If we use 4 cups of water and 3 lemons, it will make half as much lemonade. Both recipes taste the same, because and are equivalent ratios.
cups of water number of lemons 8 6 4 3
In a proportional relationship, the values for one quantity are each multiplied by the same number to get the values for the other quantity.
For example, in this table every value of \(p\) is equal to 4 times the value of \(s\) on the same row.
We can write this relationship as \(p = 4s\). This equation shows that \(s\) is proportional to \(p\).
\(s\) \(p\) 2 8 3 12 5 20 10 40