1. Numeracy in Science a common approach
Have matching grid settler activity on tables to engage participants on entry
Have PowerPoint handouts distributed on tables
Have register for signing
Deal with housekeeping, if required by ASE
Introduce self
Became interested in Numeracy in Science several years ago when undertaking consultancy in a Worcestershire high school. A question arose when I was undertaking a joint monitoring exercise with the HoD that led to the simple question- why can’t pupils transfer their numeracy skills into science lessons?
The answers to this question are what we are going to explore in this session and the next slide gives an overview of the areas we will look at
Click to display next slide
A resource to support the development of numeracy skills required for KS3 & 4 Sciences

2. Using the resource
The areas of mathematics that arise naturally from the science content in science GCSEs are listed in slide 3.
This is not a checklist for each question paper or Controlled Assessment, but assessments reflect these mathematical requirements, covering the full range of mathematical skills over a reasonable period of time.
Each area is hyperlinked to a series of slides that models a strategy

3. Numerical skills required for GCSE Sciences
Understand number size and scale and the quantitative relationship between units.
Understand when and how to use estimation.
Carry out calculations involving +, – , x, ÷, either singly or in combination, decimals, fractions, percentages and positive whole number powers.
Provide answers to calculations to an appropriate number of significant figures.
Understand and use the symbols =, <, >, ~.
Understand and use direct proportion and simple ratios.
Calculate arithmetic means.
Understand and use common measures and simple compound measures such as speed.
Plot and draw graphs (line graphs, bar charts, pie charts, scatter graphs, histograms) selecting appropriate scales for the axes.
Substitute numerical values into simple formulae and equations using appropriate units.
Translate information between graphical and numeric form.
Extract and interpret information from charts, graphs and tables.
Understand the idea of probability.
Calculate area, perimeters and volumes of simple shapes.
In addition, higher level candidates must also
Interpret, order and calculate with numbers written in standard form.
Carry out calculations involving negative powers (only -1 for rate).
Change the subject of an equation.
Understand and use inverse proportion.
Understand and use percentiles and deciles.

4. Maths grades of the numerical skills required
Understand number size and scale and the quantitative relationship between units./grade G+
Understand when and how to use estimation./grade F-E
Carry out calculations involving +, – , x, ÷, either singly or in combination, decimals, fractions, percentages and positive whole number powers./grade G-C
Provide answers to calculations to an appropriate number of significant figures./ grade D-C
Understand and use the symbols =, <, >, ~./grade G+
Understand and use direct proportion and simple ratios./grade D+
Calculate arithmetic means./grade E+
Understand and use common measures/ grade G-F and simple compound measures/grade D+ such as speed.
Plot and draw graphs (line graphs/grade E+, bar charts/ grade G+, pie charts/ grade E+, scatter graphs/grade D+, histograms/ grade A) selecting appropriate scales for the axes.
Substitute numerical values into simple formulae and equations using appropriate units./grade F+
Translate information between graphical and numeric form./grade D+
Extract and interpret information from charts, graphs and tables./ dependant on diagram
Understand the idea of probability./ grade F+
Calculate area, perimeters and volumes of simple shapes. / grade dependant on shape calculated
In addition, higher level candidates must also
Interpret, order and calculate with numbers written in standard form./ grade B+
Carry out calculations involving negative powers (only -1 for rate)./grade B+?
Change the subject of an equation. /grade C+
Understand and use inverse proportion. /grade B+
Understand and use percentiles and deciles./ not formally covered at KS4 maths

5. Writing one number as a percentage of another
Calculating percentages 3
To calculate the percentage of an amount
E.g % of 72
= 0·35 ×
= 25·2
Writing one number as a percentage of another
E.g. What percentage is 48 boys out of a year group of 180?
48 out of 180 =
48 ÷180 = 0·27
= 27%
of means times
Convert to decimal 48
180

6. Rounding 4
When the answer to your calculation has many decimal places it might be appropriate to round the number
E.g. If your answer was 6· …
To 1 d.p. = 6·
So we write 6·2
To 2 d.p. = 6·
So we write 6·25
The number after the line is less than 5 so the number at 1 decimal place does not need changing
The number after the line is more than 5 so the number at 2 decimal place needs to be rounded up
next

7. Significant figures 4
Digits of a number are kept in place by zeros where necessary.
Zeros at the beginning or end don’t usually count, but zeros ‘inside’ the number do.
The rounded answer should be a suitable reflection of the original number
E.g. 24,579 to 1 s.f could not possibly be 2
24,579 to 1 s.f is 20,000
Further examples on next page…

8. Deciding on Significant figures4
Number
1 sf
2sf
3456
3000
3500
345·5
300
350
34·56 30 35
3·456 3
3·5
0·3456
0·3
0·35
0·03456
0·03
0·035
0·003456
0·003
0·0035

9. Ratio: what does this mean?6
Ratio is used to compare the sizes of two (or more) quantities
Example 1:
A drink is made by mixing two parts orange juice with five parts water.
This relationship of 2 to 5 can be written as the ratio 2:5.
Example 2
A steel alloy rod is made up of 60% iron and 40% copper.
So the ratio of iron to copper is 60:40.
We could simplify this to give 3:2
next

10. Simplifying a ratio Oil : Petrol 4 : 100 2 : 50 1 : 256
Simplifying a ratio
Use a similar approach to simplifying fractions
What would divide into both sides?
Oil : Petrol
4 : 100
2 : 50
1 : 25
Keep going until no further simplification is possible
next

11. Dividing a quantity in a ratio6
Dividing a quantity in a ratio
Total number of parts = 7
(3 + 4)
Divide 560kg in the ratio 3:4
To find 1 part, divide the amount by the total number of parts: 560 ÷ 7 = 80kg
Multiply to calculate each share:
3 × 80 = 240kg
4 × 80 = 320kg
So dividing 560kg in the ratio 3:4 is 240kg and 320kg

12. Calculating averages There are 3 different averages commonly used.7
There are 3 different averages commonly used.
MEAN, MODE and MEDIAN
You also have RANGE
The range of a set of data is the difference between the highest and the lowest data values.
In an examination the highest mark is 80% and the lowest mark is 45%
Range is 35% because 80% - 45% = 35%
 It is not acceptable to leave the answers as an interval.
eg 45% → 80%
next

13. Mean = adding up all the values and dividing by the number of values.
Calculating mean 7
Mean = adding up all the values and dividing by the number of values.
For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Mean = = 44 = 4·4
next

14. Mode is the most common value. It is sometimes called the modal group.
Calculating mode 7
Mode is the most common value. It is sometimes called the modal group.
For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Mode is 3
because 3 is the value which occurs most often 
next

15. For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Calculating median 7
Median is the middle value when a set of values has been arranged in order.
For the following values: 3, 2, 5, 8, 4, 3, 6, 3, 2, 8
Median – is 3·5
because 3·5 is in between 3 and 4 when the values are put in order (2, 2, 3, 3, 3, 4, 5, 6, 8, 8)
If there are an even number of values it is the mean of the centre two).

16. d s t Rearranging formula8
The ‘triangle trick’ is used to help rearrange formula for speed/distance/time and density/mass/volume
Speed = distance ÷ time
Time = distance ÷ speed
Distance = speed × time d s t
next

17. Calculating gradient of a line8
Draw a triangle on your graph
Gradient = Y X Y
6 squares ÷ 3 units = 2 X

18. Bar Charts 9
If your data is continuous the bars should be next to each other so that all possible values are included
If your data is discrete
(in categories) the bars must not touch
Eye Colour
Frequency
Brown 10
Blue 5
Green 2
Height
Frequency
0 ≤ h < 5 12
5 ≤ h < 10 4
10 ≤ h < 15 3
15 ≤ h < 20 2
next

19. 1 person’s share of the pie chart
Pie charts 9
As a pie chart is based on a circle if the numbers involved are simple it will be possible to calculate simple fractions of 360˚.
Mode of Transport
Frequency
Angle calculation
Angle
Walk 10
10 × 12
120˚
Train 3
3 × 12
36˚
Car 5
5 × 12
60˚
Bus 6
6 × 12
72˚
Other
Total 30
360˚
1 person’s share of the pie chart
360 ÷ total frequency
360 ÷ 30 = 12°
next

20. 1 person’s share of the pie chart
Pie charts 9
With more difficult numbers we would expect to use a calculator. However, the approach remains the same.
Any calculations of angles should be rounded to the nearest degree only at the final stage of the calculation
Subject
Number of pupils
Pie Chart Angle
English 53
53 × 2·4 = 127·2˚
Mathematics 65
65 × 2·4 = 156˚
Science 32
32 × 2·4 = 76·8˚
Total
150
360˚
1 person’s share of the pie chart
360 ÷ total frequency
360 ÷ 150 = 2·4°
next

21. The graph shows a positive correlation between the two variables.
Scatter graphs 9
These are used to compare two sets of numerical data. The two values are plotted on two axes labelled as for continuous data.
The graph shows a positive correlation between the two variables.
However you need to ensure that there is a reasonable connection between the two, e.g. ice cream sales and temperature. Plotting use of mobile phones against cost of houses will give two increasing sets of data but are they connected?
next

22. No correlation comes from a random distribution of points
Scatter graphs 9
If possible a ‘line of best fit’ should be drawn. If students are expected to draw the line of best fit, this is done 'by eye'
Negative correlation depicts one variable increasing as the other decreases.
No correlation comes from a random distribution of points
next

23. Histograms 9
The area of each bar represents the class or group frequency of the bar. This means each bar can have a different width.
Price (P) in pounds, £
Frequency
Frequency density
0 < P ≤ 5 40
40 ÷ 5 = 8
5 < P ≤ 10 60
60 ÷ 5 = 12
10 < P ≤ 20
60 ÷ 10 = 6
20 < P ≤ 40
40 ÷ 20 = 2
Frequency density = Frequency of class interval
Width of class interval
next

24. The vertical axis is always labelled ‘Frequency density’
Histograms 9
A histogram never has gaps between the bars
The vertical axis is always labelled ‘Frequency density’
The horizontal axis has a continuous scale since it represents continuous data, such as time, weight or length.

25. Substitution 10
Substitution – replacing the letter representing a variable by a number
Need to know the order of operations:
Brackets
Indices
Division
Multiplication
Addition
Subtraction
This equally applies to all number as well as algebra
These have the same precedence
These have the same precedence
next

26. Work out the brackets first: (68 – 32) = 36
Substitution 10
This formula allows you to substitute any °F temperature to find its equivalent temperature in °C
c = 5(f – 32) where f represents the temperature in °F.
c represents the temperature in °C
Example
When f = 68
5(f - 32) ÷ 9 = 5( ) ÷ 9
5(36) ÷ 9 = (5 × 36) ÷ 9 = 180 ÷ 9 = 20
So 68°F = 20°C 9
Work out the brackets first: (68 – 32) = 36
A number next to anything in brackets means the contents of the brackets should be multiplied, so 5(36) means
5 × 36:

27. Line graphs 12
Line graphs are used in Science and maths to show how data changes over a period of time. In maths, these are also called ‘trend graphs’
next

28. 12
The lines connecting the points give estimates of the values between the points
The title of the line graph tells us what the graph is about
The points or dots on the graph show us the facts
The horizontal scale across the bottom and the vertical scale along the side tell us how much or how many. These must go up in equal jumps
The horizontal label across the bottom and the vertical label along the side tells us what kinds of facts are listed

29. Calculating area under a graph12
Separate the area under the graph into rectangles and triangles A B C
Total area = A + B + C
Area of a rectangle = length × width
Area of a triangle = (base × height) ÷ 2

30. Theoretical probability13
Based on the concept that the probability scale runs from 0 (impossible) to 1 (certain)
Can be expressed as a fraction, a decimal or a percentage
Theoretical probability
P(event) = Number of success
Total number of outcomes
E.g.: when tossing a coin P(head) = ½
Relative frequency
Relative Freq = Number of successful trials
Total number of trials
E.g.: an experiment found that a tossed coin landed on heads 12 out of 20 throws. P(head) = 12⁄20
next

31. Simple Shapes – area and perimeter14
next

32. Simple Shapes – area and perimeter14
next

33. Simple Shapes – area and perimeter14
next

34. Simple Shapes – surface area and volume14
next

35. Simple Shapes – surface area and volume14
next

36. Simple Shapes – surface area and volume14

37. Negative Powers A negative power is best shown by: a2 ÷ a5 = a × a = 116
A negative power is best shown by:
a2 ÷ a5 = a × a =
a × a × a × a × a a3
We also know that:
a2 ÷ a5 = a 2-5 = a-3. So = a-3 a3
Example
52 ÷ 54 = 5(2-4) = 5-2 = 1 = 1

38. Standard Index Form 15
Standard (Index) Form is a simple way to write very large and very small numbers.
The ‘Standard’ refers to the fact that all numbers have a similar appearance when written in ‘standard form’. The ‘Index’ refers to the index or power.
All numbers in Standard Form take the form:
a × 10x
Where 1 ≤ a < 10 and x is an integer (positive or negative whole number)
next

39. Standard (Index) Form Adding and subtracting numbers Example 115
Adding and subtracting numbers
Convert them into ordinary numbers, do the calculation, then change them back if you want the answer in standard form.
Example 1
4·5 × ·45 × 105
= 45, ,000
= 690,000
= 6·9 × 105
next

40. Standard (Index) Form Multiplying and dividing numbers15
Multiplying and dividing numbers
Here you can use the rules for multiplying and dividing powers.
Remember these rules:
To multiply powers you add,
E.g. 105 × 103 = 108
To divide powers you subtract
E.g. 105 ÷ 103 = 102
Example 2
Simplify (2 × 103) × (3 × 106)
Solution
(2 × 103) × (3 × 106) = 6 × 109
Use rule to multiply powers
Multiply together to give 6

41. Changing the subject of the formula17
a + b = c
c is the subject
To rearrange the formula to make b the subject:
a + b = c
The
method
is the
same as
solving
equations -a -a
-a from both sides
b = c - a
next

42. Changing the subject of the formula17
bx + c = a
To rearrange so that b is on the right
bx + c = a -c -c
-c from both sides
bx = a - c ÷x ÷x
÷x from both sides
b = a – c x
next

43. Changing the subject of the formula17
n = m - 3s
To rearrange to make s the subject
n = m - 3s
+3s to both sides
+3s
+3s
n + 3s = m
-n from both sides
3s = m - n
Divide both sides by 3
s = m - n 3

44. Inverse proportion 18
Inverse proportion is when one value increases as the other value decreases
Example
y is inversely proportional to x. When y = 3, x = 12 .
Find the constant of proportionality, and the value of x when y = 8.
If y is inversely proportional to x we can write it as y ∝ 1/x
Or, y = k/x where k is a constant
So xy = k Substituting the values of x and y gives:
3 × 12 = 36
So k = 36
To find the value of x when y = 8, substitute k = 36 and y = 8 into xy = k
8x = 36
So x = 4·5