Efficiently estimating journeys from non-linear graphs | Maths

Starter quiz

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

a linear graph

polygons ✓

triangles

a second vehicle
We could improve the accuracy of this estimation by __________.

finding the area of the trapezium

breaking the shape down into a rectangle and triangle

mapping more polygons onto the graph ✓
Map one polygon onto the graph to estimate the distance travelled between $0$ and $4$ seconds. ______ metres.

'12' ✓
Map two polygons onto the graph to estimate the distance travelled between $0$ and $8$ seconds. Give your answer to $1$ d.p. ______ metres.

'40.4' ✓
Fully factorise $15ab-9abc+3a^2$

$a(15b-9bc+3a)$

$3(5ab-3abc+a^2)$

$3a(5b-3bc+a)$ ✓

$3ab(5-3c+a)$
What are the common factors in this calculation? ${1\over4}(3+7)\times{h}+{1\over4}(7+11)\times{h}+{1\over4}(11+20)\times{h}$

${1\over4}$ ✓

$4$

$7$

$h$ ✓

$(7+11)$

Exit quiz

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 __________.

factorise ✓

simplify ✓

expand

commute

substitute
Use four strips of equal width to estimate the distance travelled between $0$ and $8$ seconds. Give your answer to $1$ d.p. ______ metres.

'38.2' ✓
What are the hidden labels on this diagram?

$y$

$y_0$ ✓

$h$ ✓

$h_2$

$y_4$ ✓
Which of the below is the Trapezium Rule?

$A\approx{1\over2}h(y_0+y_n+2(y_1+y_2+...+y_{n-1})$ ✓

$A\approx{1\over2}h(y_0+y_{n-1}+2(y_1+y_2+...+y_n)$

$A\approx{1\over2}h(y_0+y_n+2(y_1+y_2+...+y_{5})$

$A\approx{1\over2}h(y_0+y_n+2(y_1+y_{n-1})$
When using the Trapezium Rule which of the below calculates $h$ , the width of the strips? $n$ is the number of trapezia.

$h={{b-a}\over{n}}$ ✓

$n={{b-a}\over{h}}$

$h={{n}\over{b-a}}$

$h={{b-n}\over{a}}$
Use the Trapezium Rule to estimate the area under this curve between $t = 0$ and $t = 20$ using $5$ trapeziums. ______ metres.

'66 ' ✓

Worksheet

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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.

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.

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.