Shetland Islands Council Schools Service

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1 Updated 07/10/11 A guide for teachers of all subjects on how various Numeracy and Mathematic topics are approached within Mathematics Departments in Shetland Numeracy and Mathematics Across the Curriculum Page 1

2 Contents Page Introduction to Topics...3 Basic calculations vocabulary...3 Subtraction...4 Order of Operations or BODMAS...5 Decimals...6 Rounding to Decimal Places (including money)...7 Rounding to Significant Figures...8 Fractions...9 Percentages (with and without the calculator)...10 Ratio and Proportion...12 Scale and Scale Drawing...14 Co-ordinates...15 Scientific Notation or Standard Form...16 Time Calculations...17 Units of Measurement, multiplying and dividing by 10, 100 and Estimating...19 Types of Data Data Analysis...21 Pictogram Bar Graph Pie Charts Line Graphs Scattergraphs (scatter diagrams, scatterplot) Five-figure summary Boxplot (box and whisker diagram) Dot Plots Stem-and-leaf Diagrams...31 Frequency Polygon Using Formulae Equations Changing the Subject of a Formula Early Level pre-school + P1 or later for some First Level To the end of P4, but earlier or later for some Second Level To the end of P7, but earlier or later for some Third and Fourth Levels S1 to S3, but earlier for some. Fourth level equates to SCQF Level 4 Numeracy and Mathematics Across the Curriculum Page 2

3 Introduction to Topics Where the document refers to the stage at which pupils will be introduced or reintroduced to a topic, the following definitions apply: many about 50% of a particular year group most about 80% of a particular year group Basic calculations vocabulary Addition (+) sum of more than eg. what is 6 more than 10? add total and plus increase Multiplication (x) multiply times product lots of sets of of Equals (=) is equal to same as makes will be means answer is Subtraction (-) less than eg. how many less than 12 is 7? take away minus subtract difference between decrease Divison ( ) divide share split groups of how many Numeracy Page 3

4 Subtraction From Second Level onwards we do.. subtraction using decomposition (as a written method) check by addition promote alternative mental methods where appropriate WORKED EXAMPLES Decomposition: Counting on: To solve mentally 41 27, count on from 27 until you reach 41 Break up the number being subtracted: e.g. To solve mentally 41 27, subtract 20 then subtract 7 borrow and pay back Numeracy Page 4

5 Order of Operations or BODMAS BODMAS is the mnemonic which we teach in maths to enable pupils to know exactly the right sequence for carrying out mathematical operations. Scientific calculators use this rule to know which answer to calculate when given a string of numbers to add, subtract, multiply, divide, etc. e.g. What do you think the answer to x 5 is? Is it (2 + 3) x 5 = 5 x 5 = 25? or 2 + (3 x 5) = = 17? We use BODMAS to give the correct answer: (B)rackets (O)f (D)ivision (M)ultiplication (A)ddition (S)ubtraction According to BODMAS, multiplication should always be done before addition, therefore 17 is the correct answer and should also be the answer which your calculator will give if you type in x 5 = O can be used to stand for Of, Off or Order. A sign goes with the number immediately after it. Sometimes the mnemonic BIDMAS is used where I stands for Index meaning work out powers and roots before dividing/multiplying. WORKED EXAMPLES x 7 not 12 x not (5 + 4) x 3 = = 12 5 = 9 x 3 = 14 = 7 = 27 Any calculation within brackets must be done first. Multiplication and division have equal priority. Addition and subtraction have equal priority. Numeracy Page 5

6 Decimals When adding, subtracting and dividing, the decimal points are always in line. When multiplying, decimal points need not be in line. Multiply as if whole numbers, count total number of figures after the point in numbers being multiplied, and put same number in answer. When dividing a decimal by a decimal, multiply the number you are dividing by, by 10, 100 or 1000, so that you always divide by a whole number. WORKED EXAMPLES x Add in a zero x x (0.08 x 100 = 8) 23.7 = } 2 figures after point x = figures after d.p. in answer Numeracy Page 6

7 Rounding to Decimal Places (including money) Pupils will be re-introduced to this topic during S1 We expect pupils to at First Level round 2 digit whole numbers to the nearest 10 at Second Level round 3 digit whole numbers to the nearest 10 at Second Level round any number to the nearest whole number, 10 or 100 at Third and Fourth Levels round any number to 1 decimal place at Third and Fourth Levels round to any number of decimal places or significant figures WORKED EXAMPLES Rounding to 1 decimal place (to 1 d.p.) (to 1 d.p.) to 1 d.p.) Rounding to more than 1 decimal place (to 3 d.p.) Money is always rounded to 2 decimal places, but be careful of context Money calculations do not necessarily have 2 d.p on the screen of a calculator. This has to be remembered when transferring answers from a calculator to paper (this one is OK) Always round up for 5 or more Numeracy Page 7

8 Rounding to Significant Figures We expect pupils to at Third and Fourth Levels round to any number of decimal places or significant figures The significance of a number is the accuracy of the number. The rules of rounding are the same as for decimals always round up for 5 or more. The abbreviation for Significant figures is sig. fig. or s.f. Worked Examples people attended a football match there will most likely not have been exactly people, so this number has only 1 significant figure people exactly, attended a football match this is exact and therefore has 5 significant figures (to 3 s.f.) However, people exactly, will be 5 s.f. Numeracy Page 8

9 Fractions at Second Level we expect pupils to: o do simple fractions of 1 or 2 digit numbers e.g of = (9 3) o do simple fractions of up to 4 digit numbers e.g of = = = 132 at Third and Fourth Levels we of = (70 5) (Divide by bottom) (Divide by bottom, multiply by top) o use equivalences of widely used fractions and decimals e.g = o find widely used fractions mentally o find fractions of a quantity with a calculator o use equivalences of all fractions, decimals and percentages o add, subtract, multiply and divide fractions with and without a calculator Vocabulary: Numerator number on the top Denominator number on the bottom Simplify - cancel WORKED EXAMPLES = = Add and Subtract Multiply Divide Make the Multiply top and multiply bottom Invert the second denominators equal fraction and multiply = = Cancelling first = = = = Numeracy Page 9

10 Percentages (with and without the calculator) Most pupils will be re-introduced to this topic during S1 Pupils will be shown to calculate amounts using the following: % = 33 % = 1% = % = 66 % = 10% = % = 5% = 10% % = 5% 2 2 At Third and Fourth Levels we expect pupils to: o find 50%, 25%, 10% and 1% without a calculator and use division to find other amounts; o find percentages with a calculator (e.g. 23% of 300 = x23 = 69); o recognise that of means multiply. This is the same as finding a fraction of a value, e.g = 0.23 of 300 OR = = = 69 o Express a fraction as a percentage via the decimal equivalent. WORKED EXAMPLES Find 36% of 250 without a calculator 10% is 25 30% is 75 ( 25 x 3) 5% is (10% 2) 1% is 2.50 (10% 10) 36% is 90 (30% + 5% + 1%) Express two fifths as a percentage Either = = = 40% OR 100% of OR % 5 = = = =100 5 = = 40 Numeracy Page 10

11 Percentages (continued) You buy a car for 5000 and sell it for 3500, what is the percentage loss? Loss = = = = = 30% Increase 350 by 15% 15% of 350 = x 15 = (.. to find the increase) (then add on for the new total.) = use the % button on the calculator. Inconsistencies between models can result in incorrect answers. Numeracy Page 11

12 Ratio and Proportion Many pupils will be re-introduced to this topic during S1 Ratios are used to compare different quantities. Pupils will be shown to simplify ratios like fractions. A ratio in which one of its values is 1 is called a unitary ratio eg. 1:2 If sharing eg. money in given ratios, pupils must Calculate the number of shares by adding the parts of the ratio together Divide the given quantity by the number of shares to find the value of one share Multiply each ratio by the value of one share to find how the money has been split WORKED EXAMPLES 1. The ratio of cats to dogs in an animal shelter is 4:7. If there are 35 dogs in the shelter, how many cats are there? Cats Dogs 4 7 What do you multiply by to get 35? Multiply 4 by the same There are 20 cats. number is split between Jack and Jill in the ratio 3:2. How much does Jack receive and how much does Jill receive? Number of shares = = 5 value of 1 share = 35 5 = 7 Jack s share = 2 7 = 14 Jill s share = 3 7 = 21 (check by adding the values of the shares: = 35) Pupils should always be encouraged to check their answers by adding the value of the shares. Numeracy Page 12

13 Ratio and Proportion (cont.) At Third and Fourth Levels we expect pupils to Identify direct and inverse proportion Record appropriate headings with the unknown on the right Use the unitary method (i.e. find the value of one first then multiply by the required value) where appropriate If rounding is required, we do not round until the last stage WORKED EXAMPLES Direct Unitary Method If 5 bananas cost 80p, then what do 3 bananas cost? bananas Cost (pence) = = 48 Inverse Unitary Method The journey time at 60km/h = 30 minutes, so what is the journey time at 50km/h? Speed (km/h) Time (mins) = 1800 minutes = 36 minutes A common sense approach should be stressed as the number of items increase the cost increases and vice versa. Numeracy Page 13

14 Scale and Scale Drawing A scale is used for making drawings or models of distances or objects that are too large or too small to copy exactly. The scale can be given by the ratio Drawing length Actual length WORKED EXAMPLES Eg.1 The classroom floor is 10m by 8m. Scale 1cm to 1m. So 1cm on the Drawing represents 1m on the floor. Scale Drawing 10cm by 8cm. Eg.2 A map of Britain. Scale 1: Map 1cm on the map represents cm, or 1000m, or 1km on the ground. Eg.3 In a desert race a Landrover set out and travelled 73km on a bearing of 038º, then changed direction to 144º and travelled a further 39km. (a) What direction would it take to return to the starting point? (b) How far would it have to travel? Step 1 Make a rough sketch Step 2 Pick a scale Step 3 Make and accurate drawing Step 4 Make accurate measurements and scale up. N 1cm represents 10km N N P Start S PS measures 7cm, so the distance to the start is 74km. Reflex angle NPS measures as 249º, so the bearing to return to the start is 249º. Numeracy Page 14

15 Co-ordinates At Second Level we expect pupils to Use a co-ordinate system to locate a point on a grid number the grid lines rather than the spaces use the terms across/back and up/down for the different directions use a comma and rounded brackets e.g. 3 across 4 up = (3, 4) At Third and Fourth Levels we expect pupils to use co-ordinates in all four quadrants to plot positions WORKED EXAMPLE: Plot the following points: M(5, 2), A(7, 0), T(0, 4), H(-4, 2), S(-3, -2) y 4 X T(0,4) H(-4,2) X 3 2 M(5,2) X A(7,0) X x -1 X S(-3,-2) -2-3 Numeracy Page 15

16 Scientific Notation or Standard Form In maths we introduce Scientific Notation in the Third and Fourth Levels. It is part of the General and Credit Standard Grade course and taught at the beginning of S3 or earlier for some. We teach that a number in scientific notation consists of a number between one and ten multiplied by 10 to some power. For example: = 2.45 x = 9.88 x 10-4 Other subjects may approach this differently. At third and fourth levels we introduce the terms: kilo meaning one thousand milli meaning one thousandth. Third and Fourth level pupils should be able to use powers and square roots. Numeracy Page 16

17 Time Calculations All pupils will be re-introduced to this topic during S1 At Second Level we expect pupils to: Convert between the 12 and the 24 hour clock (2327 = pm) Calculate duration in hours and minutes by counting on the required time At Third and Fourth Levels Convert between hours and minutes (multiply by 60 for hours into minutes) Recognise simple fractions of hours eg. 30min = 0.5hrs Convert between minutes and hours (divide by 60 for minutes into a decimal of an hour) WORKED EXAMPLES Second Level How long is it from 0755 to 0948? (5 min) + (1 hr) + (48 min) Total time is 1 hr 53 minutes Third and Fourth Levels Change 27 minutes into hours 27 min = = 0.45 hours teach time as a subtraction intervals Numeracy Page 17

18 Units of Measurement, multiplying and dividing by 10, 100 and 1000 Pupils will be re-introduced to this topic during S1 At Third and Fourth Levels we expect pupils to: Convert between the following: o mm cm (multiply by 10 to convert from cm to mm and divide by 10 to convert from mm to cm) o cm m (multiply by 100 to convert from m to cm and divide by 100 to convert from cm to m) o m km (mulitiply by 1000 to convert from km to m and divide by 1000 to convert from m to km) o mg g (multiply by 1000 to convert from g to mg and divide by 1000 to convert from mg to g) o g kg (multiply by 1000 to convert from kg to g and divide by 1000 to convert from g to kg) o ml l (multiply by 1000 to convert from l to ml and divide by 1000 to convert from ml to l) Multiply and divide by 10, 100 and 1000 o Whole numbers only- To multiply by 10 add 1 zero, by 100 add 2 zeros, by 1000 add 3 zeros. If sufficient zeros are in place at the end of a number, to divide by 10 remove 1 zero, by 100 remove 2 zeros, by 1000 remove 3 zeros Otherwise: o Although pupils should be shown that multiplying or dividing by 10, 100 and 1000 causes the number to move position (units to tens, etc) with the decimal point remaining fixed, in practice, many pupils find it easier to move the decimal point the appropriate number of places. The addition of zeros when multiplying whole numbers can cause confusion when multiplying decimals by 10, 100 and 1000 e.g. 3.7 x 10 does not equal 3.70 Numeracy Page 18

19 Estimating Most pupils will be re-introduced to this topic during S1 At Second Level we expect pupils to Estimate height and length in cm, m, 1 2 m, 1 10 m eg. length of a pencil = 10cm width of a desk = 1 2 m At Second Level we expect pupils to estimate small weights, small areas, small volumes eg. bag of sugar = 1kg At Third and Fourth Levels estimate areas in square metres, lengths in mm and m eg. area of a blackboard = 4 m² (said as four square metres) diameter of 1p = 15mm In real life, measurements of length tend to be stated in mm e.g. worktop heights for kitchen units. Numeracy Page 19

20 Types of Data Definitions Continuous Data can have an infinite number of possible values within a selected range eg. temperature, height, length Discrete Data can only have a finite or limited number of possible values. Shoe sizes are an example of discrete data since 37 and 38 mean something, however size 37.3 does not. Non-numerical data (Nominal data) data which is non-numeric e.g. favourite animal, colours of cars. Numeracy Page 20

21 Data Analysis At Third and Fourth Levels we expect pupils to analyse ungrouped data using a tally table and frequency column or an ordered list calculate range of a data set. In maths this is taught as the difference between the highest and lowest values of the data set. (Range is expressed differently in Biology) calculate the mean (average) of a set of data At Third and Fourth Levels we expect pupils to use a stem and leaf diagram calculate the mean (average) find the median (middle value of an ordered list) of a data set find the mode (most common value) of a data set be able to obtain all of the above values from an ungrouped frequency table Probability is always expressed as a fraction P (event) = number of favourable outcomes total number of possible outcomes WORKED EXAMPLES The result of a survey of the number of pets pupils owned were: 3, 3, 4, 4, 4, 5, 6, 6, 7, 8 Mean = ( ) 10 = 5 Median = the middle = (4 + 5) 2 = 4.5 Mode = most common = 4 Range = highest lowest = 8 3 = 5 P (pupil with 4 pets) = 3 10 Numeracy Page 21

22 Pictogram A pictogram/pictograph is usually used to display non-numerical data. Each pictogram should have the following: A title Appropriate labels on each axis A key (if each picture represents a value more than one) eg. Favourite Ice Cream Flavours Vanilla Chocolate Strawberry Chocolate Peanut Butter Key: each =50 Numeracy Page 22

23 Bar Graph We expect pupils to use a pencil give the graph a title label the axes label the bars in the centre of the bar (each bar has an equal width) label the frequency (up the side) on the lines, not on the spaces At First Level construct bar graphs with frequency graduated in single units At Second Level construct bar graphs with frequency graduated in multiple units At Second Level construct bar graphs involving simple fractions or decimals WORKED EXAMPLES First Level Second Level Second Level Colour of eyes Quantities of litter Class shoe sizes Number of pupils green blue brown Colour of eyes frequency paper plastic Types of litter Frequency Shoe Size Numeracy Page 23

24 Pie Charts We expect pupils to use a pencil label all the sectors (slices) or insert a key as required give the pie chart a title At Second Level construct pie charts involving simple fractions or decimals At Third and Fourth Levels construct pie charts of data expressed in percentages construct pie charts of raw data WORKED EXAMPLES Third and Fourth Levels 30% of pupils travel to school by bus, 10% by car, 55% walk and 5% cycle. Draw a pie chart of the data. Third and Fourth Levels 20 pupils were asked What is your favourite subject? Replies were Maths 5, English 6, Science 7, Art 2. Draw a pies chart of the data. 10% of 360º = 36º 360º 20 (the total) = 18º Bus Car Walk Cycle 30% = 3 x 36º = 108º 10% = 1 x 36º = 36º 55% = 5.5 x 36º = 198 5% = 0.5 x 36º = 18 Maths 5 English 6 Science 7 Art 2 5 x 18º = 90 6 x 18º = x 18º = 126º 2 x 18º = 36 º Transport to School Favourite Subject cycle 5% 18 bus 30% Art 36º Maths 198º walk 55% Science 126º 108 English car 10% Numeracy Page 24

25 Line Graphs Line graphs are used to show a trend. If there is more than one graph on the same axis you also need a key. From Second Level we expect pupils to use a sharpened pencil and a ruler choose an appropriate scale for the axes to fit the paper label the axes give the graph a title number the lines not the spaces plot the points neatly (using a cross or dot) fit a suitable line At Third and Fourth Levels If necessary, make use of a jagged line to show that the lower part of the graph has been missed out WORKED EXAMPLES Eg. 1 The distance an object travels over time has been recorded in the table below: Time (s) Distance (cm) Distance travelled by an object over time Distance (cm) Time (sec) Numeracy Page 25

26 Eg Exercise and Pulse Rates 144 Pulse Rate in beats per minute Mabel Albert Time in minutes Good Graph Guide: are both axes labelled the same as the table headings from which the data was taken? axes with numbers should have units next to the label does the scale go up in even jumps? check values have been plotted/drawn correctly Numeracy Page 26

27 Scattergraphs (scatter diagrams, scatterplot) A scattergraph allows you to compare two quantities. It allows you to see if there is a connection (correlation) between the two quantities. Correlation can be positive, negative or there may be no correlation. For each piece of data a point is plotted on the diagram. The points are not joined up. Each diagram needs a title an appropriate label on each axis. Example: Positive Correlation 1.8 Height (metres) Weight (kg) Negative correlation No correlation Numeracy Page 27

28 Five-figure summary When a list of n numbers are put in order, it can be summarised by quoting five figures: o The highest number (H) o The lowest number (L) o The median, the number which halves the list (Q 2 ) o The upper quartile, the median of the upper half (Q 3 ) o The lower quartile, the median of the lower half (Q 1 ) WORKED EXAMPLES Eg L Q 1 Q 2 Q 3 H Eg L Q 1 Q 2 Q 3 H = = = = 5 = 7 = 8.5 NB. These values are then used graphically in a box-plot (see next page) Numeracy Page 28

29 Boxplot (box and whisker diagram) A boxplot is a way of summarising a set of data measured on an interval scale. It is often used in exploratory analysis. It is a type of graph which is used to show the shape of the distribution, its central value and variability. The picture consists of the most extreme values in the data set, the lower and upper quartiles and the median. Boxplots are very useful when large number of observations are involved and when two or more data sets are being compared. Boxplots can be drawn horizontally or vertically. The box can be any height as this is not significant. Example Lowest Value Lower quartile Median Upper quartile Highest Value Numeracy Page 29

30 Dot Plots For non-numerical data, a dot plot is similar to a bar chart, with the bars replaced by a series of dots. A dot plot is a way of summarising data, often used in exploratory data analysis to illustrate the major features of distribution of the data in a convenient form. Example: A Symmetric Distribution A distribution skewed to the right A distribution skewed to the left A uniform distribution A widely spread distribution A tightly clustered distribution Numeracy Page 30

31 Stem-and-leaf Diagrams A stem-and-leaf diagram is another way of displaying discrete or continuous data. It is useful for finding the median and mode of a large set of data. If you have two sets of data to compare, you can draw a back-to-back stemand-leaf diagram. A stem and leaf diagram should : o have a title o have a key o leaves should be ordered Example: Minutes Spent Eating Lunch Stem Leaves (ordered) n = means 22 minutes Key Minutes Spent Eating Lunch Class 1A Class 1B n = 18 n = means 22 minutes Numeracy Page 31

32 Frequency Polygon A frequency polygon is drawn by joining the mid-points of the top of each bar of a histogram. It is optional whether you show or remove the bars. Frequency polygons are useful when comparing two sets of data. Example Number of students Examination Marks Numeracy Page 32

33 Using Formulae We expect pupils to construct and use simple formulae at Third and Fourth Levels by writing down the formula first rewriting the formula replacing the letters by the appropriate numbers (substitution) solving the equation interpreting the answer and putting the appropriate units back into context WORKED EXAMPLES: The length of a string S mm for the weight W g is given by the formula: S = W (a) Find S when W = 3 g S = W S = x 3 S = S = 25 Length of string is 25 mm. (write formula) (replace letters by numbers) (interpret result in context) (b) Find W when S = 20.5 mm S = W (write formula) 20.5 = W (replace letters by numbers) 4.5 = 3W (solve the equation) 1.5 = W The weight is 1.5 g (interpret result in context) Rearrange the formula before substitution (too difficult) State the answer only. Working must be shown Numeracy Page 33

34 Equations Pupils will be re-introduced to this topic during S1 Pupils begin by solving simple equations using the cover-up method. eg. x + 5 = 8 (cover up x and ask which number would you add to 5 to give 8? ) x = 3 eg. 2y = 12 ( cover up y and ask which number when multiplied by 2 gives 12? ) y = 6 eg. 3b 4 = 23 (cover up 3b and ask which number would you subtract 4 from to leave 23? ) 3b = 27 (cover up b and ask which number when multiplied by 3 gives 27? ) b = 9 This is the only method for solving equations that some pupils will experience. Numeracy Page 34

35 Solving Equations (continued) At the third and fourth levels we expect pupils to solve simple equations by... balancing Performing the same operations on each side of the equation doing Undo operations eg. undo + with -, undo with + undo x with, undo with x encouraging statements like: add something to both sides multiply both sides by something We prefer: the letter x to be written differently from a multiplication sign; one equals sign per line; equals signs beneath each other; we discourage bad form such as 3 x 4 = 12 2 = 6 x 3 = 18. WORKED EXAMPLES: Third and fourth levels 2 x + 3 = 9 take away 3 from both sides 2 x = 6 divide both sides by 2 x = 3 3 x + 6 = 2 (x 9) 3 x + 6 = 2 x 18 change side, change sign 3 x 2x = x = -24-6(y + 3) = 12 both sides by -6 OR -6 y - 18 = 12 x out brackets y + 3 = -2-3 from both sides -6 y = to both sides y = -5 y = -5 by -6 Pupils should be advised to be particularly careful when expanding brackets with a negative multiplier. Numeracy Page 35

36 Changing the Subject of a Formula Many pupils will be introduced to this topic during S4 Pupils should be given the following guidance: 1. If desired variable lies under a root, reverse the operation 2. Remove fractions by multiplying through on both sides 3. Remove brackets 4. Use normal rules for solving equations 5. Apply one step at a time to show each step clearly Examples (a) Change the subject of the formula to e. f = 2e + 5 (b) Change the subject of the formula to m. f 5 = 2 e subtract 5 from both sides f 5 = e 2 divide both sides by 2 f 5 e = 2 k = mn k² k ² m n = m = = mn square both sides k ² n divide both sides by n (c) Change the subject of the formula to r. 2 p = (6 r 4) 3 3p = 2(6r 4) multiply both sides by 3 3p = 12r 8 remove brackets 3p + 8 = 12 r add 8 to both sides 3p + 8 = r 12 divide both sides by 12 3p + 8 r = 12 Techniques for inverse operations should be emphasised. Numeracy Page 36