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In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three- dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

Rates and Proportions

Question 1

Y is directly proportional to x. When x = 5, y = 8. What does y equal when x = 9?

First, we set up our general equation. Because y is directly proportional to x, we have:

y = cx

where c is the constant of proportionality. In other words, when x goes up, y goes up, and when x goes down, y goes down.

The next thing we do is plug our values for x and y into the equation so we can solve for c:

8 =(c)(5)

Solving for c, we get c = 8/5 = 1.6 and we plug this into our equation:

y = 1.6x

Now, we can plug x = 9 into the equation to find out what y equals:

y = (1.6)(9)

y = 14.4

So, our answer is 14.4

Question 2

Y is directly proportional to the square of x. When x = 2, y = 32. What does y equal when x = 5?

This time, our general equation is slightly more complicated because x is squared:

Like before, we solve for our constant:

32 = (c)(22)

32 = (c)(4)

We get c = 8:

y= 8x2

Solving for y when x = 5, we get y = (8)(52) = (8)(25) = 200

Question 3

Y is inversely proportional to x. When x = 2, y = 8. What does y equal when y = 24?

This time, because y is inversely proportional to x, our general equation is different:

xy = c

so when x goes up, y goes down, and vise versa. But, other than that, we solve these kinds of problems the same way as direct proportion problems.

Solving for the constant, we get:

(2)(8)= c

So c = 16 and our equation is now:

xy = 16

Solving for y when x = 24 we get y = 16/24 = 2/3

Question 4

Y is inversely proportional to the square root of x. When x = 36, y = 2. What does y equal when x = 64?

As before, we set up our equation: