Since the gradient is improved by moving the points closer together, you could consider a point.
By drawing the tangent to the graph at a given point, you can estimate the gradient at that point.
The gradient at that point is estimated by calculating the gradient of the tangent.

Common misconception

Pupils may think a tangent to a curve at a given point cannot intersect the curve at another point.

Define the tangent at a point as the line which has the same gradient as the curve at that point. If the graph is cubic then it is possible for the tangent at a point to intersect the graph again. There are examples in the lesson that show this.

Keywords

Tangent - A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same gradient at the given point.

This lesson builds directly on the previous lesson. Pupils could revisit their previous examples and improve their estimates using this skill. On a distance-time graph this skill can be used to estimate speed at a given point. On a speed-time graph this can be used to estimate acceleration rates.

Teacher tip

Licence

This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Worksheet

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Starter quiz

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6 Questions

Q1.

$$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$

the gradient

Correct answer: the estimated gradient

the rate of difference

the intercept

Q2.

Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$

Correct Answer: -2

Q3.

Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$

Correct Answer: -4

Q4.

Match these intervals to the estimated gradient of the curve.

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

$$4$$

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

$$1$$

Correct Answer:$$x=0 \ \text{to} \ x=2$$,$$5\over2$$

$$5\over2$$

Correct Answer:$$x=2 \ \text{to} \ x=3$$,$$10$$

$$10$$

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

$$2$$

Q5.

Order the estimated gradients of these intervals from highest to lowest.

1 - $$x=2$$ to $$x=3$$

2 - $$x=1$$ to $$x=3$$

3 - $$x=1$$ to $$x=2$$

4 - $$x=0$$ to $$x=2$$

5 - $$x=-2$$ to $$x=1$$

Q6.

Which of these intervals give this curve an estimated gradient of $$0$$?

$$x=1$$ and $$x=3$$

Correct answer: $$x=0$$ and $$x=2$$

$$x=2$$ and $$x=3$$

Correct answer: $$x=0$$ and $$x=4$$

Correct answer: $$x=2$$ and $$x=4$$

Exit quiz

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6 Questions

Q1.

A tangent to a curve at a given point is a line that intersects the curve at that point. Both the tangent and the curve have the same __________ at the given point.

Correct answer: gradient

length

radius

shape

Q2.

When using two points on a curve to estimate the gradient we can improve the accuracy of our estimate by __________ the distance between the two points.

Correct answer: reducing

increasing

calculating

Q3.

If you wanted to calculate the gradient of the curve at a given point, which diagram is likely to be the most helpful?