175e. Gradient of a Straight Line

Learning Intentions

• Recognise that the gradient is a number measuring the slope of a line
• Recognise that the gradient can be positive, negative, zero or undefined
• Solve the gradient of a straight line

Pre-requisite Summary

• A straight line can rise, fall or stay level as you move from left to right
• Coordinates are written as ordered pairs
• The horizontal change between two points is the change in
• The vertical change between two points is the change in
• Division can be used to compare vertical change with horizontal change
• A horizontal line has equation
• A vertical line has equation

Worked Examples

Worked Example 1

State whether the gradient of each line is positive, negative, zero or undefined.

a) a line rising from left to right

b) a line falling from left to right

c) a horizontal line

d) a vertical line

Worked Example 2

Find the gradient of the line through the points and .

Worked Example 3

Find the gradient of the line through the points and .

Worked Example 4

Find the gradient of the line through the points and .

Worked Example 5

Explain why the line through the points and has undefined gradient.

Worked Example 6

A student says the gradient of the line through and is .

Test the claim and Determine the correct gradient.

Problems

Problem 1

State whether the gradient of each line is positive, negative, zero or undefined.

a) a line rising from left to right

b) a horizontal line

c) a vertical line

Problem 2

Find the gradient of the line through the points and .

Problem 3

Find the gradient of the line through the points and .

Problem 4

Find the gradient of the line through the points and .

Problem 5

Explain why the line through the points and has undefined gradient.

Problem 6

A student says the gradient of the line through and is .

Test the claim and determine the correct gradient.

Exercises

Understanding and Fluency

Exercise 1

State whether each line has positive, negative, zero or undefined gradient.

a) a line that rises from left to right

b) a line that falls from left to right

c) a horizontal line

d) a vertical line

Exercise 2

State whether each equation describes a line with zero or undefined gradient.

a)

b)

c)

d)

Exercise 3

Find the gradient of each line through the given points.

a) and

b) and

c) and

Exercise 4

Find the gradient of each line through the given points.

a) and

b) and

c) and

Exercise 5

Find the gradient of each line through the given points.

a) and

b) and

c) and

Exercise 6

Find the gradient of each line through the given points.

a) and

b) and

c) and

Exercise 7

Match each description to a possible gradient.

a) rising steeply from left to right

b) falling gently from left to right

c) horizontal line

d) vertical line

Possible gradients: , undefined, ,

Exercise 8

For each pair of points, decide whether the line has positive, negative, zero or undefined gradient, then find the gradient if possible.

a) and

b) and

c) and

d) and

Reasoning

Exercise 9

Explain why a line with positive gradient rises from left to right.

Exercise 10

Explain why a horizontal line has gradient .

Exercise 11

Explain why a vertical line has undefined gradient.

Exercise 12

A student says that every sloping line has a positive gradient. Explain why this is incorrect.

Exercise 13

A student calculates the gradient of the line through and as . Explain the error.

Exercise 14

Explain why the gradient compares vertical change with horizontal change.

Problem-solving

Exercise 15

A road rises from height m to height m over a horizontal distance of m.

Find the gradient.

Exercise 16

A line on a Draw passes through the points and .

Find the gradient and state whether it is positive or negative.

Exercise 17

A line passes through the points and .

Find the gradient.

Exercise 18

A builder draws a level beam represented by the points and .

Find the gradient and explain what it means about the beam.

Exercise 19

A student says that the line through and has gradient because the -value does not change.

Exercise 20

Two points on a straight line are and .

Find the gradient and Describe the slope of the line.

Potential Misunderstandings

• Thinking gradient is the same as the -intercept
• Confusing positive gradient with lines that fall from left to right
• Thinking every straight line has a defined gradient
• Forgetting that horizontal lines have gradient
• Forgetting that vertical lines have undefined gradient
• Reversing the order of the changes when calculating gradient
• Use change in over change in instead of change in over change in
• Believing a larger gradient always means a higher line rather than a steeper slope
• Thinking zero gradient and undefined gradient mean the same thing

Next: 176e. Gradient and y-Intercept Form