Topics
2. Proportional Relationships
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United States of AmericaVirginia
Grade 7

2.05 Unit rate and slope

Lesson
Practice

Unit rate and slope

Recall that a rate is a special type of ratio that is used to compare different types of quantities.

A unit rate describes how many units of the first quantity corresponds to one unit of the second quantity. Some common unit rates are distance per hour, cost per item, earnings per week, etc.

Exploration

Move the slider to adjust m and observe what happens.

Loading interactive...

  1. What do you notice when m is negative? positive? zero?

  2. How does changing m affect the equation and the graph?

When we are looking at unit rates in tables and graphs, we want to know how much the dependent variable \left(y\right) will increase when the independent variable \left(x\right) is increased by one. The change in y for every change in x is called the slope of the line.

When x and y are related in a way where one is a constant multiple of the other, the two quantities are proportional. Proportional relationships are also an example of direct variation. We can represent this type of relationship as an equation:

\displaystyle y = m \cdot x
\bm{m}
slope

In a proportional relationship, the slope, also called the constant of proportionality, is the ratio of the y-values to the x-values \left(\dfrac{y}{x}\right). This means for proportional relationships the unit rate, slope, and constant of proportionality are all equivalent.

Consider the proportional equation:

y = 2\cdot x

The slope is 2, and we can create a table of values for this proportional relationship:

x-101234
y-202468

\,\\\,Notice this is a ratio table with a unit rate of 2 since every ratio y:x is equivalent to 2.

-2
-1
1
2
3
4
5
6
7
8
x
-2
-1
1
2
3
4
5
6
7
8
y

\,\\\,The table of values contains 6 ordered pairs:

\left(-1,-2\right) \text{, } \left(0,0\right) \text{, } \left(1,2\right) \text{, } \left(2,4\right) \text{, } \left(3,6\right) \text{, } \left(4,8\right)

We can plot these points on a coordinate plane.

\,\\\,Notice this relationship is proportional since it is linear and crosses the origin.

-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
x
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
y

Slope \left(m\right) represents the rate of change in a linear function or the "steepness" of a line. The slope, m, can be calculated by looking at the slope triangles:

m = \dfrac{\text{change in } y}{\text{change in } x}

The slope triangle can be drawn between any two points on the line. Using the larger triangle, we have:

m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{8}{4} = 2

\,\\\,Slope can be positive, negative or zero:

-4
-3
-2
-1
1
2
3
4
x
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
y
A proportional relationship with a negative slope y=-3x
-4
-3
-2
-1
1
2
3
4
x
-1
1
2
3
4
5
6
7
y
A proportional relationship with a slope of zero y=0 \cdot x = 0

As you move across a graph from left to right, a graph with a positive slope will increase, a graph with a negative slope will decrease, and a graph with a zero slope is a horizontal line.

Examples

Example 1

Determine whether the line on each graph represents a positive slope, a negative slope, or a zero slope.

a
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
Worked Solution
Create a strategy

As we move across the graph from left to right, determine whether the line is increasing, decreasing, or constant.

Apply the idea
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y

As we move across the graph, we can see the line is decreasing, similar to walking down stairs. Therefore, the slope of this graph is negative.

Reflect and check

We can also check the direction of the slope by graphing a point on the far left and far right of our graph:

-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y

Since the point on the right is lower, the slope is negative.

If the point on the right were to be higher, the slope would be positive.

b
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
Worked Solution
Create a strategy

As we move across the graph from left to right, determine whether the line is increasing, decreasing, or constant.

Apply the idea

As we move across the graph, we can see the y-values are constant. Therefore, the slope of this graph is zero.

c
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y
Worked Solution
Create a strategy

As we move across the graph from left to right, determine whether the line is increasing, decreasing, or constant.

Apply the idea
-8
-6
-4
-2
2
4
6
8
x
-8
-6
-4
-2
2
4
6
8
y

As we move across the graph, we can see the line is increasing, similar to walking up stairs. Therefore, the slope of this graph is positive.

Example 2

The graph shows the amount of time it takes Kate to make beaded bracelets.

1
2
3
4
5
6
\text{Time (hours)}
5
10
15
20
25
30
\text{No. of bracelets made}
a

Find the slope of the line.

Worked Solution
Create a strategy

For a 1 unit increase in x on the graph, find the increase of y.

Apply the idea
1
2
3
4
5
6
\text{Time (hours)}
5
10
15
20
25
30
\text{No. of bracelets made}

From the slope triangle, we see the change in y-values is 5 every time the change in x is 1.

To calculate slope, we know:

m = \dfrac{\text{chane in } y}{\text{change in } x} = \dfrac{5}{1}

The slope is 5.

b

Interpret the unit rate based on the slope of the line.

Worked Solution
Create a strategy

The slope of the graph is the unit rate.

Apply the idea

The slope 5 based on the graph means that 5 bracelets are made for every hour.

The unit rate is 5 bracelets per hour.

Example 3

Carl has kept a table of his reading habits which is shown below:

Number of weeks12243648
Number of books read20406080
a

Determine the unit rate of the number of books Carl reads for every week, rounding the answer in one decimal place.

Worked Solution
Create a strategy

Find the constant change in the y values for every change in the x values.

Apply the idea

Based on the table, for every 12 weeks (say between 24 and 12 weeks), Carl reads 20 books \left(40-20\right).

The unit rate is \dfrac{20}{12} books per week or 1.7 books per week.

b

Write an equation that represents this situation.

Worked Solution
Create a strategy

Recall that the equation of a proportional relationship is in the form:

y = mx

where m represents the slope.

Apply the idea

Notice we found the unit rate in part (a) to be \dfrac{20}{12} = \dfrac{5}{3}.

Using the equations y = mx, we have:

y = \dfrac{5}{3} x

Reflect and check

Since we found the decimal version of the unit rate to be 1.7, we could also write the equation as:

y = 1.7x

We often leave slope as a fraction. It helps us make the connection to rates and ratios and also allows for easier graphing.

Example 4

Jun needs to mix a batch of 'flamingo pink' paint to match his wall. 'Flamingo pink' is made by mixing 10 cans of white paint with 1 can of red paint.

a

Find the unit rate.

Worked Solution
Create a strategy

We know the unit rate is how much y changes for every 1 unit increase in x. We can also use the formula:

m = \dfrac{\text{change in } y}{\text{change in } x}

Apply the idea

m = \dfrac{\text{change in } y}{\text{change in } x} = \dfrac{10}{1}

The unit rate, or slope, is 10, since 10 cans of white paint are required to combine with 1 can of red paint to produce 'flamingo pink'.

b

Write an equation for the situation.

Worked Solution
Create a strategy

In the equation y=mx, m is the slope of the line. The slope is the unit rate. Let x be the number of cans of red paint and y is the number of cans of white paint needed.

Apply the idea

Since we found the unit rate in part (a) to be 10, we have the equation:

y = 10 \cdot x

Idea summary

A proportional relationship is represented by the equation:

y = mx \,\,\, \text{ where } m \text{ is the slope}

In a proportional relationship, the unit rate, slope, and constant of proportionality are all equivalent. They describe how many units of the first quantity corresponds to one unit of the second quantity.

On a graph, slope describes the steepness of a line, or how y changes and x changes:

m = \dfrac{\text{change in } y}{\text{change in } x}

Outcomes

7.PFA.1

The student will investigate and analyze proportional relationships between two quantities using verbal descriptions, tables, equations in y = mx form, and graphs, including problems in context.

7.PFA.1a

Determine the slope, m, as the rate of change in a proportional relationship between two quantities given a table of values, graph, or contextual situation and write an equation in the form y = mx to represent the direct variation relationship. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).

7.PFA.1b

Identify and describe a line with a slope that is positive, negative, or zero (0), given a graph.

7.PFA.1e

Make connections between and among representations of a proportional relationship between two quantities using problems in context, tables, equations, and graphs. Slope may include positive or negative values (slope will be limited to positive values in a contextual situation).

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