Investigate slope direction, steepness, and parallel lines

Ontario Grade 10 Mathematics

Modelling Linear Relations

How a line’s movement across a graph tells you about its slope

Picture a path on a map. As you move to the right, the path might rise, fall, or stay level. A graph shows the same kinds of movement. Slope is a number that describes how a straight line moves up or down as it moves across. In this lesson, you will connect that number to a line’s direction and steepness, then use slope to identify parallel lines.

What you will learn

Describe a line’s direction using the sign of its slope.
Compare the steepness of lines using the size of their slopes.
Recognize and create parallel lines by comparing slopes.

1. Grade 9 bridge: read a line on a grid

A coordinate grid has a horizontal axis called the

x

-axis and a vertical axis called the

y

-axis. A point is named by its coordinates, such as

(2, 3)

. The first number tells how far to move horizontally from the origin. The second tells how far to move vertically. The origin is the point

(0, 0)

where the axes cross.

To compare two points, count the horizontal change and the vertical change. Moving from left to right is a positive horizontal change. Moving upward is a positive vertical change; moving downward is a negative vertical change. For example, moving right

4

units and up

2

units gives a rise of

2

and a run of

4

.

Slope describes the vertical change for each unit of horizontal change. The vertical change is often called the rise, and the horizontal change is called the run. These names describe movements on the graph, not the length of the line itself.

m=\frac{\text{rise}}{\text{run}}

Rise measures the up-or-down change.
Run measures the left-or-right change.
For slope, compare rise with run.

2. Direction and steepness

The letter

m

is commonly used to represent slope. A line that rises as you read it from left to right has a positive slope. A line that falls from left to right has a negative slope. A horizontal line has a slope of zero because it does not rise or fall. In this lesson, slope is considered for lines that are not vertical; a vertical line has no defined slope because its horizontal change is zero.

Steepness is about how quickly a line rises or falls as you move across. Compare the size of the slopes, without focusing on whether they are positive or negative. For example, a slope of

3

is steeper than a slope of

1

because it changes more vertically for each one unit across. A slope of

-3

is also steeper than a slope of

-1

. The negative sign shows direction, while the size of the number shows steepness.

When a slope is a fraction, compare its size carefully. A rise of

2

for a run of

5

gives a gentler line than a rise of

3

for a run of

4

. Both lines rise to the right, but the second gains more height over a shorter horizontal distance. For falling lines, the same steepness comparison applies to the sizes of the negative slopes.

Read the graph from left to right when deciding direction. Looking from right to left can make a rising line seem to fall. Also, do not judge steepness only by how a graph looks on a screen: the horizontal and vertical scales may differ. Use the grid or the slope values when they are available.

|m|=\text{steepness measure}

Positive slope: the line rises from left to right.
Negative slope: the line falls from left to right.
Zero slope: the line is horizontal.
Greater slope size means a steeper line.

3. Parallel lines and equal slopes

Parallel lines are lines in the same plane that stay the same distance apart and never meet. On a graph, non-vertical parallel lines have the same slope. They have the same direction and steepness, even though they may cross the

y

-axis at different places.

For example, two lines can both rise

2

units for every

3

units they move right. One might pass through one point, while the other passes through a different point. Their positions differ, but their equal slopes mean they are parallel. Horizontal lines are also parallel to one another because each has slope zero.

To investigate whether two lines are parallel, find or read each line’s slope and compare the values. If the slopes match, the lines are parallel, provided the lines are distinct. If the slopes have different signs or different sizes, they are not parallel. For instance, slopes

2

and

-2

have the same size but opposite directions, so those lines are not parallel.

A graph may provide the line directly, or a question may give two points on each line. For two points, calculate the vertical change and horizontal change using the same order for both. Dividing the vertical change by the horizontal change gives the slope. Using a consistent order matters because it keeps the changes matched.

m_1=m_2

Parallel non-vertical lines have equal slopes.
Equal slope means equal direction and steepness.
A different intercept or position does not change a line’s slope.

4. Guided example and independent practice

Use the rise-over-run idea before reaching for a formula. Mark two points on each line, then count the vertical and horizontal changes between them. You may also use a slope formula when the points are given as coordinates. In that formula, subtract the

y

-coordinates for the rise and the

x

-coordinates for the run, keeping the point order consistent.

For independent practice, answer these without looking at the quick check. First, a line moves right

5

units and down

2

units. State the sign of its slope. Next, compare slopes

\frac{1}{2}

and

\frac{3}{2}

for steepness. Finally, explain what you would check to decide whether two non-vertical lines are parallel. Use the key ideas in this lesson to check your reasoning.

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

Count changes or use coordinates to find slope.
Use the sign for direction and the slope’s size for steepness.
Compare slopes to decide whether non-vertical lines are parallel.

What slope tells you

Slope	Direction from left to right	Steepness clue
Positive	Rises	Larger slope size means steeper
Negative	Falls	Larger slope size means steeper
Zero	Horizontal	No rise or fall
Equal slopes	Same direction	Non-vertical lines are parallel

Worked example

Compare two lines and identify a parallel pair

Line A passes through

(1, 2)

and

(5, 4)

. Line B passes through

(0, -1)

and

(3, 0.5)

. Line C passes through

(2, 3)

and

(6, 1)

. Find each slope. Describe each line’s direction and steepness, then decide which lines are parallel.

Find Line A’s changes
From $(1, 2)$ to $(5, 4)$ , the horizontal change is $4$ and the vertical change is $2$ . The line rises as it moves right, so its slope is positive.
$m_A=\frac{4-2}{5-1}=\frac{1}{2}$
Find Line B’s changes
From $(0, -1)$ to $(3, 0.5)$ , the horizontal change is $3$ and the vertical change is $1.5$ . The line rises to the right. Simplifying the ratio gives a slope of one-half.
$m_B=\frac{0.5-(-1)}{3-0}=\frac{1.5}{3}=\frac{1}{2}$
Find Line C’s changes
From $(2, 3)$ to $(6, 1)$ , the horizontal change is $4$ and the vertical change is $-2$ . The line falls as it moves right, so its slope is negative.
$m_C=\frac{1-3}{6-2}=-\frac{1}{2}$
Compare direction, steepness, and parallel lines
Lines A and B have the same positive slope, so they rise in the same way and are parallel. Line C has the same slope size but the opposite sign. It is just as steep, but it falls rather than rises, so it is not parallel to A or B.
$m_A=m_B=\frac{1}{2},\quad m_C=-\frac{1}{2}$

Answer: Line A rises and has slope

\frac{1}{2}

. Line B rises and has slope

\frac{1}{2}

. Line C falls and has slope

-\frac{1}{2}

. All three have the same steepness, but only A and B are parallel.

Check: For A, a run of

4

matches a rise of

2

. For B, a run of

3

matches a rise of

1.5

. Both ratios are

\frac{1}{2}

. C has a rise of

-2

over a run of

4

, so its direction is opposite.

Common mistakes and how to avoid them

Calling every line that rises to the right a steep line.

Correction: Rising tells direction, not steepness. Compare the slope sizes to decide which line is steeper.

Treating a negative slope as less steep than a positive slope of the same size.

Correction: The sign gives direction. Compare the sizes without the signs to compare steepness.

Saying lines are parallel because their slopes have the same size.

Correction: Their signed slopes must be equal. A positive slope and a negative slope point in opposite directions.

Using the vertical change as the run or changing the point order halfway through.

Correction: Use vertical change over horizontal change. Keep the same point order in both coordinate differences.

Lesson summary

Slope compares vertical change with horizontal change.
A positive slope rises to the right; a negative slope falls to the right; a zero slope is horizontal.
Compare the size of slopes to compare steepness.
Distinct, non-vertical parallel lines have equal slopes.

Check your understanding

Question 1

A line falls as you read it from left to right. What sign does its slope have?

Positive
Negative
Zero
correctIndex operations

Show answer and explanation

Negative

A line that falls from left to right has a negative slope.

Question 2

Which line is steeper: one with slope

-\frac{3}{4}

or one with slope

\frac{1}{2}

?

The line with slope $-\frac{3}{4}$
The line with slope $\frac{1}{2}$
They have equal steepness
correctIndex operations

Show answer and explanation

The line with slope

-\frac{3}{4}

The slope sizes are

\frac{3}{4}

and

\frac{1}{2}

. Since

\frac{3}{4}

is greater, that line is steeper. Its negative sign only indicates that it falls.

Question 3

Two distinct non-vertical lines have slopes

2

and

2

. What can you conclude?

They are parallel.
They have opposite directions.
One is horizontal.
correctIndex operations

Show answer and explanation

They are parallel.

Distinct non-vertical lines with equal slopes have the same direction and steepness, so they are parallel.

Key terms

Coordinate: A number that tells the position of a point on an axis.
Rise: The vertical change between two points.
Run: The horizontal change between two points.
Slope: The ratio of vertical change to horizontal change for a line.
Parallel lines: Lines in the same plane that stay the same distance apart and do not meet.

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