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      Advanced problem solving with real-life graphs

      Lesson details

      Learning outcome

      I can use my knowledge of interpreting real-life graphs to solve problems.

      Key learning points

      1. Different speed/time graphs can be interpreted and compared.
      2. The area under the graph can also be interpreted in context.
      3. Circle theorems can be demonstrated using coordinate geometry.

      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.

      • Tangent - A tangent of a circle is a line that intersects the circle exactly once.

      Common misconception

      The intersection of two different graphs on a speed-time graph is the point where the faster particle overtakes the slower moving particle.

      Remind pupils that this is a speed-time graph and ask, "How do we find the distance travelled on a speed-time graph?". Then, ask pupils to calculate how far each particle has travelled at the point of intersection.

      Teacher tip

      A powerful starter is to give students a distance-time graph and ask, "How much information can we obtain from this graph? What does it tell us?". Then, do the same activity for a speed-time graph.

      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

      6 Questions

      Q1.
      This speed-time graph shows __________ rate of change between time and speed.

      An image in a quiz
      Correct answer: a constant
      an increasing
      a decreasing
      a fast

      Q2.
      What speed is this vehicle travelling at after $$10$$ seconds? m/s

      An image in a quiz
      Correct Answer: 50

      Q3.
      What distance has this vehicle travelled after $$10$$ seconds? m/s

      An image in a quiz
      Correct Answer: 250

      Q4.
      Which equation generalises how far this vehicle has travelled after $$x$$ seconds?

      An image in a quiz
      $$A={{x^2}\over{2}}$$
      $$A=5x$$
      $$A=5x^2$$
      $$A={{5x}\over{2}}$$
      Correct answer: $$A={{5x^2}\over{2}}$$

      Q5.
      What is the $$y$$-coordinate at the point of intersection of the two linear equations $$y=2x-10$$ and $$y=-{1\over2}x+5$$ ?

      Correct Answer: 2, y=2

      Q6.
      What is the equation of the tangent to the circle at point A?

      An image in a quiz
      $$y=-{2\over5}x+{34\over5}$$
      $$y=-{5\over2}x+1$$
      Correct answer: $$y={5\over2}x+1$$
      $$y={5\over2}x+11$$
      $$y={5\over2}x-{27\over2}$$

      6 Questions

      Q1.
      The __________ to a circle from an external point are equal in length.

      radii
      Correct answer: tangents
      perpendiculars

      Q2.
      What is the tangent to the circle at A?

      An image in a quiz
      $$y=4x-14$$
      $$y={1\over4}x+{29\over4}$$
      Correct answer: $$y=-{1\over4}x+{29\over4}$$
      $$y=-{1\over4}x-{29\over4}$$
      $$y=-{1\over4}x+{13\over2}$$

      Q3.
      Find the point of intersection of the tangents to the circle from points A and B.

      An image in a quiz
      Correct answer: $$(21,2)$$
      $$(2,21)$$
      $$(19,2)$$
      $$(23,2)$$

      Q4.
      Tangents from A and B meet at point C. To show that AC $$=$$ BC we would use __________.

      An image in a quiz
      a ruler
      Correct answer: Pythagoras theorem
      circle theorem
      $$A=\pi \times r^2$$

      Q5.
      How far have these cars travelled between $$0$$ and $$6$$ seconds?

      An image in a quiz
      Both have travelled $$30$$ metres each
      Both have travelled $$90$$ metres each
      'a' has travelled $$90$$ metres and 'b' has travelled $$120$$ metres
      Correct answer: 'a' has travelled $$90$$ metres and 'b' has travelled $$135$$ metres
      'a' has travelled $$250$$ metres and 'b' has travelled $$275$$ metres

      Q6.
      After how many seconds will car 'a' overtake car 'b'? seconds.

      An image in a quiz
      Correct Answer: 12

      To help you plan your 11 maths lesson on: Advanced problem solving with real-life graphs, download all teaching resources for free and adapt to suit your pupils' needs...