New

Lesson 9 of 16

Year 11
Higher

Estimating the gradient of a curve

I can estimate the gradient of a curved part of the graph by considering the gradient of a straight line connecting two points.

Download all

View unit

Lesson 9 of 16

New

Year 11
Higher

Estimating the gradient of a curve

I can estimate the gradient of a curved part of the graph by considering the gradient of a straight line connecting two points.

Download all

These resources were made for remote use during the pandemic, not classroom teaching.

Switch to our new teaching resources now - designed by teachers and leading subject experts, and tested in classrooms.

View new resources

Lesson slides

Download lesson slides

Skip lesson slides

Lesson details

Key learning points

The gradient of a curve can be estimated.
You can estimate by drawing a straight line between two points on the graph.
The closer the points, the more accurate the estimate is.

Keywords

Gradient - The gradient is a measure of how steep a line is. It is calculated by finding the rate of change in the $$y$$-direction with respect to the positive $$x$$-direction.

Common misconception

Pupils may estimate negative gradients incorrectly by drawing gradient triangles the wrong way round or ignoring the fact that the $$y$$ values are decreasing as $$x$$ increases.

Pupils should look at the shape of the graph between two points and decide if it is positive or negative before calculating. If $$y$$ is decreasing as $$x$$ increases then the gradient is negative.

To help you plan your year 11 maths lesson on: Estimating the gradient of a curve, download all teaching resources for free and adapt to suit your pupils' needs...

Pupils can use this skill to estimate average speed from non-linear distance-time graphs. There is an example in learning cycle 2 but you may wish to use graphs from lessons 5 and 7 of this unit as further practice for pupils.

Teacher tip

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

Lesson video

Skip lesson video

Worksheet

Download worksheet

Skip worksheet

Prior knowledge starter quiz

Download quiz pdf

6 Questions

Q1.
Gradient is the __________.

Correct answer: measure of how steep a line is

measure of the length of a line

Correct answer: rate of change in $$y$$-direction with respect to positive $$x$$-direction

rate of change in $$y$$-direction with respect to negative $$x$$-direction

Q2.
What is the gradient of this line?

$$+2$$

$$+3$$

Correct answer: $$-4$$

$$-8$$

Q3.
What is the gradient of this line?

$$4$$

$$-6$$

$$-6/4$$

Correct answer: $$2\over3$$

$$3\over2$$

Q4.
What is the gradient of the line going through coordinate pairs $$(18,73)$$ and $$(50,233)$$?

Correct answer: $$5$$

$$1\over5$$

$$-{1\over5}$$

$$-5$$

Q5.
Which statements are true of this curve?

The gradient is positive.

Correct answer: The gradient is positive sometimes.

The gradient is negative.

Correct answer: The gradient is negative sometimes.

Correct answer: The gradient is zero sometimes.

Q6.
What is the gradient of the line going through coordinate pairs $$\left(-{{1}\over{4}},{{5}\over{3}}\right)$$ and $$\left({{7}\over{3}},-{{16}\over{5}}\right)$$?

$${{292}\over{155}}$$

Correct answer: $$-{{292}\over{155}}$$

$${{155}\over{292}}$$

$$-{{155}\over{292}}$$

Assessment exit quiz

Download quiz pdf

6 Questions

Q1.
The gradient of the hypotenuse of the triangle is not the exact gradient of the curve between $$x=2$$ and $$x=6$$, it is only __________.

Correct answer: an estimate

a calculation

a computation

Correct answer: an approximation

Q2.
Estimate the gradient of this curve between $$x=2$$ and $$x=6$$

Correct Answer: 2, +2

Q3.
Estimate the gradient of this curve between $$x=-6$$ and $$x=-4$$

$$2$$

$$5\over2$$

Correct answer: $$-{{5\over2}}$$

$$-{{2\over5}}$$

$$-5$$

Q4.
What is wrong with using the coordinate pairs (-7, 12) and (-2, 1) to estimate the gradient of this curve?

Correct answer: The points are too far apart.

Nothing. It is quick and accurate to estimate this gradient this way.

Correct answer: The distance between the hypotenuse and the curve means it lacks accuracy.

The changes in $$x$$ and $$y$$ are not both $$+11$$

Q5.
Match the intervals to the estimated gradient of the curve in those intervals.

Correct Answer:$$x=0 \ \text{to} \ x=1$$,$$4$$

$$4$$

Correct Answer:$$x=1 \ \text{to} \ x=2$$,$$-4$$

$$-4$$

Correct Answer:$$x=2 \ \text{to} \ x=3$$,$$-6$$

$$-6$$

Correct Answer:$$x=3 \ \text{to} \ x=4$$,$$-2$$

$$-2$$

Correct Answer:$$x=4 \ \text{to} \ x=5$$,$$7$$

$$7$$

Q6.
The interval $$x=0$$ to $$x=$$__________ would be poor because it would give you an estimated gradient of $$0$$

Correct Answer: 2, 5

Estimating the gradient of a curve

Estimating the gradient of a curve

These resources were made for remote use during the pandemic, not classroom teaching.

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

How to plan a lesson using our resources

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Gradient is the __________.

Q2.What is the gradient of this line?

Q3.What is the gradient of this line?

Q4.What is the gradient of the line going through coordinate pairs $$(18,73)$$ and $$(50,233)$$?

Q5.Which statements are true of this curve?

Q6.What is the gradient of the line going through coordinate pairs $$\left(-{{1}\over{4}},{{5}\over{3}}\right)$$ and $$\left({{7}\over{3}},-{{16}\over{5}}\right)$$?

Assessment exit quiz

6 Questions

Q1.The gradient of the hypotenuse of the triangle is not the exact gradient of the curve between $$x=2$$ and $$x=6$$, it is only __________.

Q2.Estimate the gradient of this curve between $$x=2$$ and $$x=6$$

Q3.Estimate the gradient of this curve between $$x=-6$$ and $$x=-4$$

Q4.What is wrong with using the coordinate pairs (-7, 12) and (-2, 1) to estimate the gradient of this curve?

Q5.Match the intervals to the estimated gradient of the curve in those intervals.

Q6.The interval $$x=0$$ to $$x=$$__________ would be poor because it would give you an estimated gradient of $$0$$

Q1.
Gradient is the __________.

Q2.
What is the gradient of this line?

Q3.
What is the gradient of this line?

Q4.
What is the gradient of the line going through coordinate pairs $$(18,73)$$ and $$(50,233)$$?

Q5.
Which statements are true of this curve?

Q6.
What is the gradient of the line going through coordinate pairs $$\left(-{{1}\over{4}},{{5}\over{3}}\right)$$ and $$\left({{7}\over{3}},-{{16}\over{5}}\right)$$?

Q1.
The gradient of the hypotenuse of the triangle is not the exact gradient of the curve between $$x=2$$ and $$x=6$$, it is only __________.

Q2.
Estimate the gradient of this curve between $$x=2$$ and $$x=6$$

Q3.
Estimate the gradient of this curve between $$x=-6$$ and $$x=-4$$

Q4.
What is wrong with using the coordinate pairs (-7, 12) and (-2, 1) to estimate the gradient of this curve?

Q5.
Match the intervals to the estimated gradient of the curve in those intervals.

Q6.
The interval $$x=0$$ to $$x=$$__________ would be poor because it would give you an estimated gradient of $$0$$