In this lesson, students continue to work with proportional relationships represented by tables using contexts familiar from previous grades: unit conversion and constant speed. They recognize the constant of proportionality as the conversion factor or the speed, and use it to answer questions about the context. Although students might continue to reason with equivalent ratios to solve problems, the contexts are designed so that it is more efficient to use the constant of proportionality. For example, when converting length measurements from feet to inches, it is more convenient to know the rule “multiply by 12” than to use an equivalent ratio with a different scale factor every time: “1 foot is 12 inches, so multiplying both quantities by 3 I see that 3 feet is 36 inches, and multiplying both quantities by 5 I see that 5 feet is 60 inches.”
- Compare, contrast, and critique (orally and in writing) different ways to express the constant of proportionality for a relationship.
- Explain (orally) how to determine the constant of proportionality for a proportional relationship represented in a table.
- Interpret the constant of proportionality for a relationship in the context of constant speed.
Let’s solve more problems involving proportional relationships using tables.
- I can find missing information in a proportional relationship using a table.
- I can find the constant of proportionality from information given in a table.
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