Shetland Islands Council Schools’ Service documents/AHS... · Order of Operations or BODMAS...

Shetland Islands Council Schools’ Service

Numeracy and Mathematics – Across the Curriculum

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

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 1000................................18 Estimating .............................................................................................................................................19 Types of Data ..................................................................................................................................... 20 Data Analysis........................................................................................................................................21 Pictogram.............................................................................................................................................. 22

• Bar Graph............................................................................................................................ 23 • Pie Charts............................................................................................................................ 24 • Line Graphs......................................................................................................................... 25 • Scattergraphs (scatter diagrams, scatterplot) ..................................................... 27 • Five-figure summary........................................................................................................ 28 • Boxplot (box and whisker diagram)............................................................................. 29 • Dot Plots.............................................................................................................................. 30 • Stem-and-leaf Diagrams .................................................................................................31 • Frequency Polygon ............................................................................................................ 32

Using Formulae ................................................................................................................................... 33 Equations .............................................................................................................................................. 34 Changing the Subject of a Formula.............................................................................................. 36

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

Introduction to Topics

Where the document refers to the stage at which pupils will be introduced or re- introduced 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

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

4 0 0 7 4 -

3 2 6

3 9 1

3 8

1 2 7 1

- 2 3 3

6

“ borrow and pay back”


Numeracy

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 2 + 3 x 5 is? Is it (2 + 3) x 5 = 5 x 5 = 25? or 2 + (3 x 5) = 2 + 15 = 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 2 + 3 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 1. 10 + 2 x 7 not 12 x 7 2. 12 – 10 ÷ 2 not 2 ÷ 2 3. (5 + 4) x 3 = 10 + 14 = 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

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

1. 46 + 2.8 + 0.23 2. 7.1 – 1.84 3. 8 x 3.24

46 7 . 1 0 ←Add in a zero 3.24 2.8 - 1 . 8 4 x 8

+ 0.23 5 . 2 6 25.92 49.03

4. 2.4 x 23.7 5. 0.6944 ÷ 0.08 (0.08 x 100 = 8)

23.7 = 69.44 ÷ 8 x 2.4 8148 = 4740

128.88 2 figures after d.p. in answer

2 figures after point 0.08 0.6944

8.68 8 69.44


Numeracy

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.) 5.31 →5.3 (to 1 d.p.) 11.97→12.0 to 1 d.p.)

§ Rounding to more than 1 decimal place 6.2459→6.246 (to 3 d.p.)

§ Money is always rounded to 2 decimal places, but be careful of context £245.672→£245.67

§ 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.

2.6→£2.60 4.07→£4.07 (this one is OK)

Always round up for 5 or more


Numeracy

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

§ “20000 people attended a football match” – there will most likely not have been exactly 20000 people, so this number has only 1 significant figure

§ “19695 people exactly, attended a football match” – this is exact and therefore has 5 significant figures

§ 19695→19700 (to 3 s.f.) § However, 20000 people exactly, will be 5 s.f.


Numeracy

Fractions

§ at Second Level we expect pupils to: o do simple fractions of 1 or 2 digit numbers e.g.

(Divide by bottom)

o do simple fractions of up to 4 digit numbers e.g.

(Divide by bottom, multiply by top)

§ at Third and Fourth Levels we……

o use equivalences of widely used fractions and decimals e.g. 3 0.3 10

=

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

denominators equal Multiply top and multiply bottom Invert the second

fraction and multiply

=

=

=

=

Cancelling first

= =

=

1 9 3 3 of = (9 3) ÷

1 70 14 5 of = (70 5) ÷

3 176 132 4 of = 176 4 44 ÷ =

44 3 132 × =

1 1 2 3

+

3 2 6 6

+

5 6

2 3 3 4

×

6 12 1 2

2 3 3 4

×

1 2

1

1

1

2

3 2 4 5

÷

3 5 4 2

×

15 7 1 8 8

=


Numeracy

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: 1 25% 4

= 1 1 33 % 3 3

= 1 1% 100

=

1 50% 2

= 2 2 66 % 3 3

= 1 10% 10

=

3 75% 4

= 5% 10% 2 = ÷

1 2 % 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 = 300÷100x23 =

£69); o recognise that “of” means multiply.

This is the same as finding a fraction of a value, e.g. 23 £300 100

of OR

300 100 3 ÷ = 23 3 69 × =

23 100 0.23 ÷ = 0.23 300 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 £12.50 (10% ÷ 2) 1% is £2.50 (10% ÷ 10) 36% is £90 (30% + 5% + 1%)

§ Express two fifths as a percentage

Either 2 4 40 40% 5 10 100

= = = OR 2 100% 5 of OR 2 2 5 0.4 40%

5 = ÷ = =

=100 5 20 ÷ = 2 20 40 × =


Numeracy

Percentages (continued)

§ You buy a car for £5000 and sell it for £3500, what is the percentage loss?

Loss = £5000 - £3500 = £1500 1500 15 30 30% 5000 50 100

= = =

§ Increase £350 by 15%

15% of 350 = 350 ÷ 100 x 15 = £52.50 ( ….. to find the increase) (then add on for the new total….) £350 + £52.50 = £402.50

use the % button on the calculator. Inconsistencies between models can result in

incorrect answers.


Numeracy

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 7

by to get 35? Multiply 4 by the same number.

20 35

There are 20 cats.

2. £35 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 = 2 + 3 = 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: £14 + £21 = £35)

Pupils should always be encouraged to check their answers by adding the value of the shares.


Numeracy

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) 5 ⇒ 80 1 ⇒ 80 ÷ 5 = 16 3 ⇒ 16 × 3 = 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) 60 ⇒ 30 1 ⇒ 30 × 60 = 1800 minutes

50 ⇒ 1800 ÷ 50 = 36 minutes

A common sense approach should be stressed – as the number of items increase the cost

increases and vice versa.


Numeracy

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

WORKED EXAMPLES

Eg.1 The classroom floor is 10m by 8m. Eg.2 A map of Britain. Scale 1cm to 1m. So 1cm on the

Drawing represents 1m on the floor.

Scale Drawing 10cm by 8cm.

Scale 1:100 000 Map 1cm on the map represents

100 000cm, 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.

Drawing length Actual length

N

N

N

P

S

Start

1cm represents 10km

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

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)

X

X

X

X M(5,2)

X A(7,0)

T(0,4)

H(-4,2)

S(-3,-2)

1

2

3

4

-1 -2 -3

-3

-2

-1

4 8 3 2 1 5 6 7 -4 x

y


Numeracy

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:

24 500 000 = 2.45 x 10 7

0.000988 = 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

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 = 11.27 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?

0755 0800 0900 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 = 27 ÷ 60 = 0.45 hours

teach time as a subtraction intervals


Numeracy

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

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

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

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 = (3 + 3 + 4 + 4 + 4 + 5 + 6 + 6 + 7 + 8) ÷ 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

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

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

1 2 3 4

Num

ber

of p

upils

brown

blue

green

Colour of eyes

5

0 5

10 15 20 25

freq

uenc

y

Types of litter

paper

plastic

Shoe Size

Freq

uenc

y

3.5 4 4.5 5 5.5 0 2 4 6 8

10 12


Numeracy

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 = 108° 7 x 18º = 126º 2 x 18º = 36 º

Transport to School Favourite Subject

bus 30%

car 10%

walk 55%

cycle 5%

108°

18°

36°

198º

Art

Science

Maths

English

108° 126º

36º


Numeracy

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) 0 5 10 15 20 25 30 Distance (cm) 0 15 30 45 60 75 90

Distance travelled by an object over time

0 5 10 15 20 25 30 Time (sec)

10 20 30

40

50 60

70 80 90

100

Dis

tanc

e (c

m)


Numeracy

Eg.2

§ 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

Exercise and Pulse Rates

Puls

e Ra

te in

bea

ts p

er m

inut

e

Time in minutes

0 12 24 36 48 60

16

32

48

64

80

96

112

128

144

160

Mabel Albert


Numeracy

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

Negative correlation No correlation

50 55 60 70 75

1.5

1.6

1.7

1.8

Hei

ght

(met

res)

Weight (kg)


Numeracy

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 (Q2) o The upper quartile, the median of the upper half (Q3) o The lower quartile, the median of the lower half (Q1)

WORKED EXAMPLES

Eg.1 2 4 5 5 6 7 7 7 8 9 10

Eg.2 2 4 5 5 6 7 7 7 8 9 10 11

§ NB. These values are then used graphically in a box-plot (see next page)

L H Q1 Q2 Q3

L Q1 Q2 Q3 H 5 5 2

5

+ =

=

7 7 2

7

+ =

=

8 9 2

8.5

+ =

=


Numeracy

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 Highest Value

Lower quartile

Median

Upper quartile

2 3 4 5 6 7 8 9 10 11


Numeracy

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:

4 6 8 10 12 12 10 8 6 4 A Symmetric Distribution A distribution skewed to the right

12 10 8 6 4 A distribution skewed to the left

12 10 8 6 4 A widely spread distribution

12 10 8 6 4A tightly clustered distribution

12 10 8 6 4A uniform distribution


Numeracy

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 stem-

and-leaf diagram. § A stem and leaf diagram should :

o have a title o have a key o ‘leaves‛ should be ordered

Example:

Stem Leaves (ordered)

Key

Minutes Spent Eating Lunch

2

3

4

2 3 7 7

2 2 5 7 9

0 3 6 8 8

n = 14 2 2 means 22 minutes

Minutes Spent Eating Lunch

2

3

4

2 3 7 7

2 2 5 7 9

0 3 6 8 8

n = 14 2 2 means 22 minutes

Class 1A Class 1B

9 7 4

8 8 4 3 2 1 1

7 6 6 3 3 3 0 0

n = 18


Numeracy

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

0 20 40 60 80 100 Examination Marks

Num

ber

of s

tude

nts

2 4 6 8

10 12 14 16


Numeracy

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 = 16 + 3W

(a) Find S when W = 3 g S = 16 + 3W (write formula) S = 16 + 3 x 3 (replace letters by numbers) S = 16 + 9 S = 25

Length of string is 25 mm. (interpret result in context)

(b) Find W when S = 20.5 mm S = 16 + 3W (write formula) 20.5 = 16 + 3W (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

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

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 = -18 – 6 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 = 30 + 18 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

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.

2 5 f e = + 5 2 f e − = subtract 5 from both sides 5

2 f e −

= divide both sides by 2

5 2 f e −

=

(b) Change the subject of the formula to m. k mn = ² k mn = square both sides ² k m n

= divide both sides by n

² k m n

=

(c) Change the subject of the formula to r. 2 (6 4) 3

p r = −

3 2(6 4) p r = − multiply both sides by 3 3 12 8 p r = − remove brackets 3 8 12 p r + = add 8 to both sides 3 8 12 p r +

= divide both sides by 12

3 8 12 p r +

=

Techniques for inverse operations should be emphasised.

Shetland Islands Council Schools’ Service documents/AHS... · Order of Operations or BODMAS...

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