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      Improving the estimate of the gradient of a curve

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      Lesson details

      Learning outcome

      I can improve the estimate of the gradient by considering the gradient of the tangent at a fixed point.

      Key learning points

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

      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.

      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.

      Teacher tip

      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.

      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

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      Prior knowledge starter quiz

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      6 Questions

      Q1.
      $$5$$ is __________ for this curve between $$x=-10$$ and $$x=-8$$

      An image in a quiz
      the gradient
      Correct answer: the estimated gradient
      the rate of difference
      the intercept

      Q2.
      Estimate the gradient of this curve between $$x=-4$$ and $$x=0$$

      An image in a quiz
      Correct Answer: -2

      Q3.
      Estimate the gradient of this curve between $$x=-2$$ and $$x=2$$

      An image in a quiz
      Correct Answer: -4

      Q4.
      Match these intervals to the estimated gradient of the curve.

      An image in a quiz
      Correct Answer:$$x=-2 \ \text{to} \ x=-1$$,$$4$$
      Correct Answer:$$x=-1 \ \text{to} \ x=1$$,$$1$$
      Correct Answer:$$x=0 \ \text{to} \ x=2$$,$$5\over2$$
      Correct Answer:$$x=2 \ \text{to} \ x=3$$,$$10$$
      Correct Answer:$$x=-1 \ \text{to} \ x=2$$,$$2$$

      Q5.
      Order the estimated gradients of these intervals from highest to lowest.

      An image in a quiz
      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 $$0$$?

      An image in a quiz
      $$x=1$$ 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 $$x=4$$

      Assessment exit quiz

      Download exit quiz questions (PDF)

      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?

      An image in a quiz
      Correct Answer: An image in a quiz
      An image in a quiz
      An image in a quiz

      Q4.
      Use this tangent to calculate the gradient of this curve at $$x=0$$

      An image in a quiz
      Correct Answer: 3

      Q5.
      Use this tangent to calculate the gradient of this curve at $$x=1$$

      An image in a quiz
      Correct Answer: 5

      Q6.
      This tangent was drawn by hand. The triangle enables us to estimate the gradient at $$x=5$$ to be __________.

      An image in a quiz
      $$5.3$$
      $$9$$
      Correct answer: $$9\over5$$
      $$5\over9$$
      $$-{5\over9}$$

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