In the previous two lessons students learned to represent proportional relationships with equations of the form \(y = k x\). In this lesson they continue to write equations, and they begin to see situations where using the equation is a more efficient way of solving problems than other methods they have been using, such as tables and equivalent ratios.
The activities introduce new contexts and, for the first time, do not provide tables; students who still need tables should be given a chance to realize that and create tables for themselves. The activities are intended to motivate the usefulness of representing proportional relationships with equations, while at the same time providing some scaffolding for finding the equations.
As students use the abstract equation \(y = kx\) to reason about quantitative situations, they engage in MP2.
- Generate an equation for a proportional relationship, given a description of the situation but no table.
- Interpret (orally) each part of an equation that represents a proportional relationship in an unfamiliar context.
- Use an equation to solve problems involving a proportional relationship, and explain (orally) the reasoning.
Let’s use equations to solve problems involving proportional relationships.
- I can find missing information in a proportional relationship using the constant of proportionality.
- I can relate all parts of an equation like $y = kx$ to the situation it represents.
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
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
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