This week your student will learn about proportional relationships. This builds on the work they did with equivalent ratios in grade 6. For example, a recipe says, “for every 5 cups of grape juice, mix in 2 cups of peach juice.” We can make different-sized batches of this recipe that will taste the same.

The relationship between the quantities of grape juice and peach juice is a proportional relationship. In a table of a proportional relationship, there is always some number that we can multiply by the number in the first column to get the number in the second column for any row. This number is called the constant of proportionality.

Here is a task you can try with your student:

Using the recipe “for every 5 cups of grape juice, mix in 2 cups of peach juice,”

1. How much peach juice would be mixed with 20 cups of grape juice?
2. How much grape juice would be mixed with 20 cups of peach juice?

Solution:

1. 8 cups of peach juice. We can multiply any amount of grape juice by 0.4 to find the corresponding amount of peach juice: \(20 \boldcdot (0.4) = 8\).
2. 50 cups of grape juice. We can divide any amount of peach juice by 0.4 to find the corresponding amount of grape juice: \(20 \div (0.4) = 50\).

This week your student will learn to write equations that represent proportional relationships. For example, if each square foot of carpet costs \$1.50, then the cost of the carpet is proportional to the number of square feet.

We can represent this relationship with the equation \(c = 1.5f\), where \(f\) represents the number of square feet and \(c\) represents the cost in dollars. Remember that the cost of carpeting is always the number of square feet of carpeting times 1.5 dollars per square foot. This equation is just stating that relationship with symbols.

The equation for any proportional relationship looks like \(y = kx\), where \(x\) and \(y\) represent the related quantities and \(k\) is the constant of proportionality. Some other examples are \(y = 4x\) and \(d = \frac13 t\). Examples of equations that do not represent proportional relationships are \(y = 4 + x\), \(A = 6s^2\), and \(w=\frac{36}{L}\).

Here is a task to try with your student:

1. Write an equation that represents the relationship between the amounts of grape juice and peach juice in the recipe “for every 5 cups of grape juice, mix in 2 cups of peach juice.”
2. Select all the equations that could represent a proportional relationship:
   1. \(K = C + 273\)
   2. \(s = \frac14 p\)
   3. \(V = s^3\)
   4. \(h = 14 - x\)
   5. \(c = 6.28r\)

Solution:

1. Answers vary. Sample response: If \(p\) represents the number of cups of peach juice and \(g\) represents the number of cups of grape juice, the relationship could be written as \(p = 0.4g\). Some other equivalent equations are \(p = \frac25 g\), \(g = \frac52 p\), and \(g = 2.5p\).
2. B and E. For the equation \(s = \frac14 p\), the constant of proportionality is \(\frac14\). For the equation \(c = 6.28r\), the constant of proportionality is 6.28.

This week your student will work with graphs that represent proportional relationships. For example, here is a graph that represents a relationship between the amount of square feet of carpet purchased and the cost in dollars.

Notice that the points on the graph are arranged in a straight line. If we buy 0 square feet of carpet, it would cost \$0. Graphs of proportional relationships are always parts of straight lines including the point \((0, 0)\).

Here is a task to try with your student:

Create a graph that represents the relationship between the amounts of grape juice and peach juice in different-sized batches of fruit juice using the recipe “for every 5 cups of grape juice, mix in 2 cups of peach juice.”

Solution: