201. Finding Gradient of a Line Segment

Learning Intentions

• Calculate gradient Use rise over run.
• Interpret positive, negative, zero and undefined gradients.
• Relate gradient to the steepness and direction of a line segment.

Pre-requisite Summary

• Know that an ordered pair is written in the form .

• Know that horizontal movement is measured using changes in -coordinates and vertical movement is measured using changes in -coordinates.

• Know that rise means vertical change and run means horizontal change.

• Know that gradient can be calculated using:

• Know that a positive gradient rises from left to right and a negative gradient falls from left to right.

• Know that horizontal lines have gradient and vertical lines have undefined gradient.

Worked Examples

Worked Example 1

Calculate the gradient of the line segment joining the two points.

and

Worked Example 2

Calculate the gradient of the line segment joining the two points.

and

Worked Example 3

For each pair of points, calculate the rise, run and gradient.

a) and

b) and

c) and

Worked Example 4

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

a) A line segment rises from left to right.

b) A line segment falls from left to right.

c) A horizontal line segment.

d) A vertical line segment.

Worked Example 5

Calculate the gradient and describe the direction of the line segment.

a) and

b) and

c) and

Worked Example 6

Compare the steepness of the two line segments.

Line segment : gradient

Line segment : gradient

Which line segment is steeper? Explain your reasoning.

Worked Example 7

A ramp rises metres over a horizontal run of metres.

a) Calculate the gradient of the ramp.

b) Interpret the gradient in context.

Problems

Problem 1

Calculate the gradient of the line segment joining the two points.

and

Problem 2

Calculate the gradient of the line segment joining the two points.

and

Problem 3

For each pair of points, calculate the rise, run and gradient.

a) and

b) and

c) and

Problem 4

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

a) A line segment falls from left to right.

b) A horizontal line segment.

c) A line segment rises from left to right.

d) A vertical line segment.

Problem 5

Calculate the gradient and describe the direction of the line segment.

a) and

b) and

c) and

Problem 6

Compare the steepness of the two line segments.

Line segment : gradient

Line segment : gradient

Which line segment is steeper? Explain your reasoning.

Problem 7

A ramp rises metres over a horizontal run of metres.

a) Calculate the gradient of the ramp.

b) Interpret the gradient in context.

Exercises

Understanding and Fluency

Exercise 1

Calculate the rise, run and gradient for each pair of points.

a) and

b) and

c) and

d) and

Exercise 2

Calculate the rise, run and gradient for each pair of points.

a) and

b) and

c) and

d) and

Exercise 3

Calculate the gradient of each line segment.

a) and

b) and

c) and

d) and

Exercise 4

Calculate the gradient of each line segment.

a) and

b) and

c) and

d) and

Exercise 5

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

a)

b)

c)

d)

Exercise 6

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

a) and

b) and

c) and

d) and

Exercise 7

Calculate the gradient and describe the direction of each line segment.

a) and

b) and

c) and

d) and

Exercise 8

Copy and complete each gradient calculation.

a) From to :

b) From to :

c) From to :

Exercise 9

Compare the steepness of each pair of gradients.

a) and

b) and

c) and

d) and

Exercise 10

A ramp or path has the following rise and run. Calculate the gradient.

a) Rise m, run m

b) Rise m, run m

c) Rise m, run m

d) Rise m, run m

Reasoning

Exercise 11

Explain why the gradient of a horizontal line is .

Exercise 12

Explain why the gradient of a vertical line is undefined.

Exercise 13

A student says that the gradient from to is because the run is and the rise is .

Explain the mistake and give the correct gradient.

Exercise 14

A student calculates the gradient from to as:

Explain the sign error and give the correct gradient.

Exercise 15

Explain why a line segment with gradient is steeper than a line segment with gradient .

Exercise 16

Decide whether each statement is true or false. Justify your answer.

a) A positive gradient means the line rises from left to right.

b) A gradient of means the line is vertical.

c) A larger absolute value of gradient means a steeper line.

Problem-solving

Exercise 17

A cyclist travels along a straight path on a coordinate map from to .

a) Calculate the rise.

b) Calculate the run.

c) Calculate the gradient and interpret it in context.

Exercise 18

A walking trail goes from to .

a) Calculate the rise.

b) Calculate the run.

c) State whether the trail rises or falls from left to right.

Exercise 19

Two ramps have the following gradients.

Ramp A:

Ramp B:

a) Which ramp is steeper?

b) Explain your reasoning.

c) Write one sentence interpreting the steeper gradient.

Exercise 20

Create your own pair of coordinate points with a negative gradient.

Your response must include:

• the two coordinate points
• the rise
• the run
• the gradient
• one sentence describing the steepness and direction of the line segment

Potential Misunderstandings

• Students may reverse rise and run, calculating instead of .
• Students may subtract coordinates in an inconsistent order, such as using but .
• Students may calculate horizontal and vertical differences without considering direction.
• Students may think a negative gradient means the line is below the -axis, rather than falling from left to right.
• Students may think a zero gradient and an undefined gradient are the same.
• Students may write a vertical line’s gradient as instead of recognising that division by is undefined.
• Students may compare steepness using the sign only, instead of considering the absolute value of the gradient.
• Students may think is steeper than because is larger than .
• Students may not connect the gradient value to the visual direction and steepness of a line segment.

Next: 202. Finding Midpoints