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

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

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.

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

Writing one number as a percentage of another Calculating percentages 3 To calculate the percentage of an amount E.g. 35% of 72 = 0·35 × 72 = 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

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 6·24823567… To 1 d.p. = 6·24823567 So we write 6·2 To 2 d.p. = 6·24823567 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

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…

Deciding on Significant figures 4 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

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

Simplifying a ratio Oil : Petrol 4 : 100 2 : 50 1 : 25 6 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

Dividing a quantity in a ratio 6 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

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

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 = 3 + 2 + 5 + 8 + 4 + 3 +6 + 3 + 2 + 8 = 44 = 4·4 10 10 next

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

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).

d s t Rearranging formula 8 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

Calculating gradient of a line 8 Draw a triangle on your graph Gradient = Y X Y 6 squares ÷ 3 units = 2 X

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

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

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

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

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

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

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.

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

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(68 - 32) ÷ 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:

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

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

Calculating area under a graph 12 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

Theoretical probability 13 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

Simple Shapes – area and perimeter 14 next

Simple Shapes – area and perimeter 14 next

Simple Shapes – area and perimeter 14 next

Simple Shapes – surface area and volume 14 next

Simple Shapes – surface area and volume 14 next

Simple Shapes – surface area and volume 14

Negative Powers A negative power is best shown by: a2 ÷ a5 = a × a = 1 16 A negative power is best shown by: a2 ÷ a5 = a × a = 1 a × a × a × a × a a3 We also know that: a2 ÷ a5 = a 2-5 = a-3. So 1 = a-3 a3 Example 52 ÷ 54 = 5(2-4) = 5-2 = 1 = 1 52 125

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

Standard (Index) Form Adding and subtracting numbers Example 1 15 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 × 104 + 6·45 × 105 = 45,000 + 645,000 = 690,000 = 6·9 × 105 next

Standard (Index) Form Multiplying and dividing numbers 15 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

Changing the subject of the formula 17 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

Changing the subject of the formula 17 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

Changing the subject of the formula 17 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

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