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Lesson 15 of 16

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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Lesson 15 of 16

New

Year 11
Higher

Efficiently estimating journeys from non-linear graphs

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

Download all resources

Share activities with pupils

These resources will be removed by end of Summer Term 2025.

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Lesson details

Key learning points

The more trapezia used, the better the estimate.
Multiple trapezia can be calculated efficiently.
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.

To help you plan your year 11 maths lesson on: Efficiently estimating journeys from non-linear graphs, download all teaching resources for free and adapt to suit your pupils' needs...

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.

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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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Worksheet

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Prior knowledge 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.

a linear graph

Correct answer: polygons

triangles

a second vehicle

Q2.
We could improve the accuracy of this estimation by __________.

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.

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.

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: $$h$$

$$(7+11)$$

Assessment 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.

Correct Answer: 38.2

Q3.
What are the hidden labels on this diagram?

$$y$$

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.

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.

Correct Answer: 66

Efficiently estimating journeys from non-linear graphs

Efficiently estimating journeys from non-linear graphs

These resources will be removed by end of Summer Term 2025.

Lesson slides

Lesson details

Key learning points

Keywords

Common misconception

How to plan a lesson using our resources

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.We could estimate the distance travelled by the vehicle modelled in this speed-time graph by mapping __________ onto the graph.

Q2.We could improve the accuracy of this estimation by __________.

Q3.Map one polygon onto the graph to estimate the distance travelled between $$0$$ and $$4$$ seconds. metres.

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.

Q5.Fully factorise $$15ab-9abc+3a^2$$

Q6.What are the common factors in this calculation? $${1\over4}(3+7)\times{h}+{1\over4}(7+11)\times{h}+{1\over4}(11+20)\times{h}$$

Assessment exit quiz

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 __________.

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.

Q3.What are the hidden labels on this diagram?

Q4.Which of the below is the Trapezium Rule?

Q5.When using the Trapezium Rule which of the below calculates $$h$$, the width of the strips? $$n$$ is the number of trapezia.

Q6.Use the Trapezium Rule to estimate the area under this curve between $$t = 0$$ and $$t = 20$$ using $$5$$ trapeziums. metres.

Q1.
We could estimate the distance travelled by the vehicle modelled in this speed-time graph by mapping __________ onto the graph.

Q2.
We could improve the accuracy of this estimation by __________.

Q3.
Map one polygon onto the graph to estimate the distance travelled between $$0$$ and $$4$$ seconds. metres.

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.

Q5.
Fully factorise $$15ab-9abc+3a^2$$

Q6.
What are the common factors in this calculation? $${1\over4}(3+7)\times{h}+{1\over4}(7+11)\times{h}+{1\over4}(11+20)\times{h}$$

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 .

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.

Q3.
What are the hidden labels on this diagram?

Q4.
Which of the below is the Trapezium Rule?

Q5.
When using the Trapezium Rule which of the below calculates $$h$$, the width of the strips? $$n$$ is the number of trapezia.

Q6.
Use the Trapezium Rule to estimate the area under this curve between $$t = 0$$ and $$t = 20$$ using $$5$$ trapeziums. metres.