Connect rate of change to slope as rise over run

Ontario Grade 10 Mathematics

Modelling Linear Relations

Connecting how much a quantity changes to rise over run

A rate of change tells how one quantity changes as another quantity changes. For example, it can describe how far you travel as time passes. On a graph, this comparison is shown by slope. In this lesson, you will connect these ideas and use rise over run to find and explain slope.

What you will learn

Explain rate of change as the change in one quantity compared with the change in another.
Find slope from two points using rise over run.
Interpret positive, negative, and zero slope.
Use units to explain what a slope means.

1. Grade 9 bridge: read changes on a graph

A coordinate grid has a horizontal axis and a vertical axis. A point is written as an ordered pair: the first number gives its horizontal position, and the second gives its vertical position. For example,

(2, 5)

means 2 units across and 5 units up.

To compare two points, find how much the vertical value changes and how much the horizontal value changes. The vertical change is called the rise. The horizontal change is called the run. Rise and run are changes between points, not simply the coordinates of one point.

A positive change means the value increased. A negative change means it decreased. When moving from left to right along a graph, the run is positive. The rise may be positive, negative, or zero.

\text{change}=\text{final value}-\text{starting value}

Rise is the change in the vertical coordinate.
Run is the change in the horizontal coordinate.
Use the same direction of travel for both changes.

2. Rate of change is slope as rise over run

A rate compares changes in two quantities. If a person travels 12 metres in 3 seconds, distance changes by 12 metres while time changes by 3 seconds. The rate is 4 metres per second. The phrase “per second” shows that the distance change is compared with the time change.

For points on a graph, slope is the rate of change of the vertical quantity compared with the horizontal quantity. It is found by dividing rise by run. If a graph shows distance against time, slope tells the change in distance for each unit of time. Include the context and units when explaining what the rate means.

A non-vertical straight line has a constant slope. This means equal horizontal changes produce equal vertical changes. A positive slope rises from left to right. A negative slope falls from left to right. A zero slope is horizontal, so the vertical quantity does not change as the horizontal quantity increases. A vertical line has no defined slope because its run is zero, and division by zero is not defined.

The slope formula uses two points. The letters

x

and

y

name the horizontal and vertical coordinates. The subscripts 1 and 2 distinguish the first point from the second. Subtract in matching order: second

y

minus first

y

for rise, and second

x

minus first

x

for run.

m=\frac{y_2-y_1}{x_2-x_1}

Slope is rise divided by run.
Rate-of-change units are vertical units per horizontal unit.
A non-vertical line’s slope describes how the vertical value changes as the horizontal value changes.
A vertical line has no defined slope because its run is zero.

3. Read slope and its meaning

A table and a graph can show the same relationship. In a table, choose two rows and compare the change in the output with the change in the input. On a graph, choose two points on the line and compare their vertical and horizontal changes.

Slope can be a whole number, fraction, or decimal. A slope of

\frac{3}{2}

means a rise of 3 units for a run of 2 units. It does not mean that the line rises 3 units for every 1 unit across.

For a real situation, identify what each axis represents before interpreting slope. If the vertical axis is distance in kilometres and the horizontal axis is time in hours, the slope’s units are kilometres per hour. If the quantities are not named, describe slope as vertical change per horizontal change.

Pick two distinct points on the line.
Keep the point order consistent in the numerator and denominator.
Attach units to the rate when the graph or situation provides them.

4. Independent practice

Try these questions without looking back at the worked example. For each one, identify the rise and run before calculating. Then explain the result in context if units are given.

A line passes through

(1, 4)

and

(5, 12)

. Find its slope. A second line passes through

(2, 9)

and

(6, 3)

. Find its slope and state whether it rises or falls from left to right. Finally, a graph shows a horizontal line. What is its slope, and what does that mean about its vertical value?

Check that you divided the vertical change by the horizontal change.
Use the sign of the slope to describe the line’s direction.

Same rate shown in a table and as changes

Time (h)	Distance (km)	Change in time (h)	Change in distance (km)
1	5	—	—
4	17	3	12

Worked example

Find and interpret a rate from two points

A cyclist’s distance from the starting point is 5 km after 1 hour and 17 km after 4 hours. Treat time as the horizontal quantity and distance as the vertical quantity. Find the slope and explain its meaning.

Name the points
Write time first and distance second because time is on the horizontal axis and distance is on the vertical axis. The first point is $(1, 5)$ and the second is $(4, 17)$ .
$(1,5),\ (4,17)$
Find the rise and run
The rise is the change in distance: final distance minus starting distance. The run is the change in time: final time minus starting time. Both changes follow the same order, from the first point to the second.
$\text{rise}=17-5=12,\qquad \text{run}=4-1=3$
Divide rise by run
Slope is rise divided by run. Dividing the distance change by the time change gives the change in distance for each hour.
$m=\frac{12}{3}=4$
Interpret the units
Distance is measured in kilometres and time in hours, so the slope is 4 kilometres per hour. This means the cyclist’s distance from the start increases by 4 km for each hour over this interval.
$4\text{ km/h}$

Answer: The slope is 4 km/h. The cyclist’s distance from the starting point increases by 4 km per hour over the stated interval.

Check: The distance change is 12 km over a time change of 3 hours, and

12\div 3=4

. The units are kilometres per hour.

Common mistakes and how to avoid them

Dividing the horizontal change by the vertical change.

Correction: Slope is rise over run, so divide the vertical change by the horizontal change.

Subtracting in different orders, such as second minus first for rise but first minus second for run.

Correction: Use a consistent order for both changes. If you reverse the point order, reverse both subtractions.

Using a coordinate value as the change.

Correction: Rise and run are differences between two coordinates, not the coordinates themselves.

Giving a rate without units when the context provides units.

Correction: State vertical units per horizontal unit, such as kilometres per hour.

Lesson summary

Rate of change compares how much one quantity changes with how much another quantity changes.
On a graph, slope is rise divided by run.
Find each change by subtracting the starting coordinate from the ending coordinate.
Interpret slope using the graph’s axes and units.
A non-vertical straight line has a constant slope. A vertical line has no defined slope.
Positive slope rises left to right, negative slope falls, and zero slope is horizontal.

Check your understanding

Question 1

A line passes through

(2, 3)

and

(6, 11)

. What is its slope?

$\frac{1}{2}$
$2$
$4$
$-2$

Show answer and explanation

2

The rise is

11-3=8

and the run is

6-2=4

. The slope is

8\div4=2

.

Question 2

A graph has time in minutes on the horizontal axis and water volume in litres on the vertical axis. What do the slope’s units mean?

Minutes per litre
Litres per minute
Litres plus minutes
The total litres shown

Show answer and explanation

Litres per minute

Slope compares the vertical change with the horizontal change, so its units are litres per minute.

Question 3

A line slopes downward from left to right. What can you say about its slope?

It is positive.
It is negative.
It is zero.
It cannot be described as a rate.

Show answer and explanation

It is negative.

As the horizontal value increases, the vertical value decreases. That gives a negative rise for a positive run, so the slope is negative.

Key terms

Rate of change: A comparison of the change in one quantity with the change in another quantity.
Rise: The change in the vertical coordinate between two points.
Run: The change in the horizontal coordinate between two points.
Slope: The rate of change of the vertical quantity compared with the horizontal quantity; it is rise divided by run.
Coordinate: A number that gives a point’s position on one axis of a graph.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic G5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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