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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.

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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.

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Lesson details

Key learning points

  1. The gradient of a curve can be estimated.
  2. You can estimate by drawing a straight line between two points on the graph.
  3. 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.
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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

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Starter quiz

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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?
An image in a quiz
$$+2$$
$$+3$$
Correct answer: $$-4$$
$$-8$$
Q3.
What is the gradient of this line?
An image in a quiz
$$4$$
$$-6$$
$$-6/4$$
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?
An image in a quiz
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: $$-{{292}\over{155}}$$
$${{155}\over{292}}$$
$$-{{155}\over{292}}$$

Exit quiz

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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 __________.
An image in a quiz
Correct answer: an estimate
a calculation
a computation
Correct answer: an approximation
Q2.
Estimate the gradient of this curve between $$x=2$$ and $$x=6$$
An image in a quiz
Correct Answer: 2, +2
Q3.
Estimate the gradient of this curve between $$x=-6$$ and $$x=-4$$
An image in a quiz
$$2$$
$$5\over2$$
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?
An image in a quiz
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.
An image in a quiz
Correct Answer:$$x=0 \ \text{to} \ x=1$$,$$4$$
tick

$$4$$

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

$$-4$$

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

$$-6$$

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

$$-2$$

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

$$7$$

Q6.
The interval $$x=0$$ to $$x=$$__________ would be poor because it would give you an estimated gradient of $$0$$
An image in a quiz
Correct Answer: 2, 5