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Year 11
Higher

Efficiently estimating journeys from non-linear graphs

I can efficiently estimate the distance travelled for non-linear speed-time graphs.

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New
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Year 11
Higher

Efficiently estimating journeys from non-linear graphs

I can efficiently estimate the distance travelled for non-linear speed-time graphs.

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Share activities with pupils
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Lesson details

Key learning points

  1. The more trapezia used, the better the estimate.
  2. Multiple trapezia can be calculated efficiently.
  3. A formula for this can be deduced from the sum of the areas of multiple trapezia.

Keywords

  • Estimate - A quick estimate for a calculation is obtained from using approximate values, often rounded to 1 significant figure.

  • Generalisation - A generalisation is a statement or rule that applies correctly to all relevant cases.

Common misconception

When pupils see four, five or six trapezia they want to work each one out individually, then, do another calculation adding those four, five, six areas together.

This is hugely inefficient. Write out the sum of the area of multiple trapeziums then, from there, look for efficiencies. This is where applying knowledge of factorisation can be beneficial as it reduces the work.

A starter reminding pupils about factorising will set them up for the key moments where they are asked to factorise in this lesson, such as making calculations more efficient, and coming up with the simplified generalisation.
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).

Lesson video

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Starter quiz

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6 Questions

Q1.
We could estimate the distance travelled by the vehicle modelled in this speed-time graph by mapping __________ onto the graph.
An image in a quiz
a linear graph
Correct answer: polygons
triangles
a second vehicle
Q2.
We could improve the accuracy of this estimation by __________.
An image in a quiz
finding the area of the trapezium
breaking the shape down into a rectangle and triangle
Correct answer: mapping more polygons onto the graph
Q3.
Map one polygon onto the graph to estimate the distance travelled between $$0$$ and $$4$$ seconds. metres.
An image in a quiz
Correct Answer: 12
Q4.
Map two polygons onto the graph to estimate the distance travelled between $$0$$ and $$8$$ seconds. Give your answer to $$1$$ d.p. metres.
An image in a quiz
Correct Answer: 40.4
Q5.
Fully factorise $$15ab-9abc+3a^2$$
$$a(15b-9bc+3a)$$
$$3(5ab-3abc+a^2)$$
Correct answer: $$3a(5b-3bc+a)$$
$$3ab(5-3c+a)$$
Q6.
What are the common factors in this calculation? $${1\over4}(3+7)\times{h}+{1\over4}(7+11)\times{h}+{1\over4}(11+20)\times{h}$$
Correct answer: $${1\over4}$$
$$4$$
$$7$$
Correct answer: $$h$$
$$(7+11)$$

Exit quiz

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6 Questions

Q1.
To be more efficient, before calculating $${1\over2}(3+7)\times{5}+{1\over2}(7+11)\times{5}+{1\over2}(11+20)\times{5}$$ we could __________ and __________.
Correct answer: factorise
Correct answer: simplify
expand
commute
substitute
Q2.
Use four strips of equal width to estimate the distance travelled between $$0$$ and $$8$$ seconds. Give your answer to $$1$$ d.p. metres.
An image in a quiz
Correct Answer: 38.2
Q3.
What are the hidden labels on this diagram?
An image in a quiz
$$y$$
Correct answer: $$y_0$$
Correct answer: $$h$$
$$h_2$$
Correct answer: $$y_4$$
Q4.
Which of the below is the Trapezium Rule?
Correct answer: $$A\approx{1\over2}h(y_0+y_n+2(y_1+y_2+...+y_{n-1})$$
$$A\approx{1\over2}h(y_0+y_{n-1}+2(y_1+y_2+...+y_n)$$
$$A\approx{1\over2}h(y_0+y_n+2(y_1+y_2+...+y_{5})$$
$$A\approx{1\over2}h(y_0+y_n+2(y_1+y_{n-1})$$
Q5.
When using the Trapezium Rule which of the below calculates $$h$$, the width of the strips? $$n$$ is the number of trapezia.
An image in a quiz
Correct answer: $$h={{b-a}\over{n}}$$
$$n={{b-a}\over{h}}$$
$$h={{n}\over{b-a}}$$
$$h={{b-n}\over{a}}$$
Q6.
Use the Trapezium Rule to estimate the area under this curve between $$t = 0$$ and $$t = 20$$ using $$5$$ trapeziums. metres.
An image in a quiz
Correct Answer: 66