Mathematics and Numeracy

File

Written by the Mathematics Department and Numeracy group

Final version

1Prior permission required for use outwith John Ogilvie High School, SLC.©

Introduction 

Curriculum for Excellence has given the opportunity for all educators to work together. All teachers now have a responsibility for promoting the development of Numeracy. With an increased emphasis upon Numeracy for all young people, teachers will need to revisit and consolidate Numeracy skills throughout schooling. To this end I feel that it is important that “we” (all staff at John Ogilvie High School) deliver a consistent approach to “our” pupils. Pupils always have difficulties with transferable skills and if we can deliver consistent approaches of Numeracy across the school we will be helping our pupils become successful learners.

This information booklet has been produced to inform parents/carers and teachers how the Numeracy Outcomes from Curriculum for Excellence are taught within the Maths Department at John Ogilvie High School.

It is hoped that use of the information in the booklet will help our parents/carers. You will hopefully be given an insight into the way number topics are being taught to your children in the school, making it easier for you to help them with their homework, and as a result improve their progress.

 I would like to take this opportunity to thank the maths staff, the Numeracy Group, office staff, parents and pupils from John Ogilvie High School for their contribution in producing this booklet

 Carol McAninchPrincipal Teacher of MathematicsJohn Ogilvie High School

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Contents• 1. Estimation and Rounding • 2. Number and number processes• 3. Fractions• 4. Decimal Fractions• 5. Percentages• 6. Money• 7. Time• 8. Measure• 9. Data and analysis (graphs)• 10. Probability• 11. Algebra• 12. Which calculator to buy?

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1. Estimation and Rounding• Before rounding we must take the context of the question

into account. The information below is a guideline.• Round 8·573, a) To the nearest whole number Ans: 9, the number 5 is after the decimal point so round up

b) To one decimal place Ans: 8·6, the number 7 is after one decimal place so round

up So ‘5 or above round up’

c) To two decimal places, the number 3 is after the second decimal place so we do not round upAns: 8∙57

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2. Number and Number Processes

Some questions can be completed as read:• 4 – 2 + 6 Ans: 8• 12 ÷ 3 x ½ = 4 x ½ Ans: 2

However, To answer the questions below pupils would use an

order of operations:

Calculate 5 + 6 x 2Brackets /PowersDivide/Multiply

Addition/SubtractionAnswerMultiply comes first, 6 x 2 = 12 then 5 + 12= 17

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3. Fractions

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Multiplying and dividing fractions

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4. Decimal fractionsDefinition

• A fraction which has a denominator of 10 or 100 or 1000 or... • For example 3/10; 34/100; 5/100 are decimal fractions and these are written as

the decimal numbers 0.3, 0.34, 0.05 respectively, whereas 0.5 would mean 5/10

• Some examples:• Example 1 2∙5 + 0∙587 + 4∙63Working 2∙500

0∙587 + 4∙630

7∙717

• Example 2 0∙53 x 5000 = 0∙53 x 1000 x 5 = 530 x 5 = 2650• Example 3 48 ÷ 6000 = 48 ÷6 ÷ 1000 = 8 ÷ 1000 = 0∙008• Example 4 2∙4 ÷ 0∙006 = 2400 ÷ 6 (multiplying both numbers by 1000)

= 400• Example 5 0∙004 x 0∙7 We know that 4 x 7 = 28. There are four digits

after the decimal point in the question. There should be four digits after the decimal point in the answer, so the answer is 0∙0028.

Important to add in extra zeros!!!

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5. Percentages-Calculator

• Calculate 23% of £184.Note: Do not use percentage key on calculator.

• Ans:

• On calculator, 23 100 x 184 = £42·32

• In jotter, 23 x 184 = £42·32 100

÷

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5. Percentages-Non calculator

Pupils must remember:50% = ½ 25% = ¼ 75%

= ¾

10% = 20% = 33 % =

Pupils will use fractions when no calculators are allowed.

1

101

33

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5

10

Multiples of 10

Divide by 10

Divide by 5

2 lots of 10%

a) 10% of 80

= 8

b) 20% of 34

Ans: 34 ÷ 5 = 6·8 OR Ans: Find 10% of 34 = 3·4

20% of 34 23·4 = 6·8

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Another example: 30% of 32 Ans: Find 10% of 32

= 3·2

30% = 10% x 3

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So 30% of 32 3·2×3 = 9·6

Fractions and Percentages

Divide by 4

b) Find 75% of 120

Ans: ¾of 120 = 90

a) Find 25% of 84

Ans: ¼ of 84

= 21

Divide by 4 then multiply your answer by 3

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More examples

5% is half of 10%

1) Find 15% of 260

Ans: Find 10% of 260 = 26

5 % of 260 = 13

So 15% = 10% + 5% = 26 + 13 = 39

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3. 23% of 260

Ans: Find 10% of 120 = 120 ÷ 10 =12 30% of 120 3×12 = 36 5% of 120 = 6So 35% = 36 + 6 = 42

30% = 10% x 3

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2. 35% of 120

Ans: Find 10% of 260 = 26 20% of 260 2×26 = 52

1% = 260 ÷ 100 = 2·6 3% = 2·6 x 3 = 7·8

So 23% = 52 + 7·8 = 59·8

Note:20% can also be calculated

using 1

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7. MoneyBest Buys

Pupils are encouraged to use unit amounts (i.e. find the value of 1) to decide which is the better value for money.Example 1 The same brand of coffee is sold in two different sized jars as shown. Which jar represents the better value for money? Find the cost per gram for both jars.100g costs 186p so 186 ÷ 100 = 1.86p per gram.250g costs 247p so 247 ÷ 250 = 0.988p per gram.Since the large jar costs less per gram it is better value for money.

Wages and SalariesPupils will have to learn that people earn money in all sorts of ways, e.g. hourly, weekly, monthly or yearly (salary).Remember: 52 weeks per year, 12 months in a year and “annual” means yearly. Example 1 Isobel gets paid £19760 per annum. What is her weekly wage?

£19760 ÷ 52 = £380

Example 2 Duncan is a chef. His wage last week was £249 for working 30 hours.a) Calculate his hourly rate of pay?

Hourly rate = £249 ÷ 30 = £8.30 

b) This week he worked 38 hours. How much did he earn?

This week he earned 38 X £8.30 = £315.40

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Overtime In some jobs the rate of pay is higher for people working at night, weekends or holidays.Double time is the normal rate X 2Time and a half is the normal rate X 1.5 Example 1 Stuart is a long distance lorry driver with a basic wage of £14.50

per hour.His overtime pay is paid at double time.Calculate what he gets for 7 hours overtime.Overtime = 7 X (2 X £14.50)

= £203 Example 2 Janet works in a petrol station, getting £6 per hour. Her overtime rate is time and a half. Calculate her

total pay for a week in which she works 34 hours plus 5 hours overtime.

Basic wage = 34 X 6 = £204

Overtime = 5 X (1.5 X £6) = £45

Total pay = £204 + £45 = £249

CommissionSome people, particularly salespersons, receive a lower basic wage, but boost their earnings by adding on a percentage of

their total sales – this is called commission.  Example 1 Sally sells kitchens. She earns 13% commission on each kitchen she sells. How much is she paid for selling

a £3000 kitchen.Commission = 13% of £3000

= 13 ÷ 100 X 3000= £390

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Hire Purchase

Hire Purchase is a way of paying for a product over a period of time. This is useful as people can get a product and pay it off over time.

Hire purchase works as follows:-A deposit is paid and the product can be taken by the customer.The customer pays weekly or monthly instalments until the product is fully paid.When an item is bought through hire purchase, it usually ends up costing more than it would have if the product was paid for

outright.This extra money is sometimes called interest.

Example 1 The cash price for a sofa is £1100. To pay for the sofa through hire purchase a 15% deposit has to be paid then twelve monthly instalments of £90.

a) How much will the deposit be?Deposit = 15% of £1100

= 15÷ 100 X 1100= £165

b) How much would be paid for all 12 instalments?Instalments = 12 X £90

= £1080c) What is the total hire purchase price of the sofa?HP Price = £165 + £1080= £1245d) How much more is this than the cash price?Interest = £1245 - £1100= £145

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Foreign Exchange

The rate of exchange for each currency will normally be given by an amount per £ and it changes daily with the stock market. Great

Britain uses the pound (GBP) as its currency. Many European countries use the euro.Foreign Money = Number of Pounds X Exchange RateNumber of Pounds = Foreign Money ÷ Exchange RateIn May 2010 the exchange rate was: £1 1.15€

Example 1 Robert goes on holiday to Paris and takes £600 spending money with him. Using the exchange rate above how many euros would he get?

Euros = 600 X 1.15 = 690€

Example 2 Jim returns from a school trip to Germany with 85€. Use the exchange rate above to find out how many pounds he will

get back.Pounds = 85 ÷ 1.15 = £73.91

Gross Pay, Net Pay and DeductionsGross pay is the amount that an employer pays you.Deductions are taken from your gross pay and include things like:-Superannuation – a type of extra pension for when you retire.National Insurance (NI) – to pay for loss of earnings if you are sick / unemployed.Income Tax – paid to the government to pay for education, health, transport etc.Net pay is the amount that you take home after deductions are made.

Net Pay = Gross pay – Deductions

Example 1 Blair has a gross pay of £26000 per annum. He pays £4892 in deductions.

a) Calculate his annual net pay.Net pay = £26000 - £4892 = £21108

b) Calculate his monthly take home pay.Monthly pay = £21108 ÷ 12 = £1759

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Income Tax

Income tax calculation is a difficult and sometimes very confusing process. The Inland Revenue (H.M.R.C.) do not calculate your bill

purely on your gross income. Instead they give you allowances and relief on part of your income. The allowances change after we have a

budget. Taxable Income = Gross Pay – Allowances.

Value Added Tax (VAT) The government also raises money by charging V.A.T. Most items that you purchase include VAT (usually at 20%). Example 1 Find the total cost of a car costing £7800 + VAT.

Vat = 20% of £7800= 20 ÷ 100 X 7800= £1560

Total = £7800 + £1560= £9360

Three quick methods.

To find VAT only X 0.20 or 0.2

To find the price including the VAT X 1.20 To find the price without VAT ÷ 1.20 

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Insurance

Questions on insurance usually involve you reading values from tables and performing calculations. There are many different types of

insurance:

People take out Building Insurance policies to protect themselves against fire damage, storm damage, burst pipes etc. Household

Content cover protects us from e.g. theft or accidental damage to the living room carpet. The payment made each year is called a P

Premium and Insurance Companies usually quote the cost of insuring contents for each £1000 of value.  Some people take out Life Insurance policies (Whole Life) so that when they die their loved ones are left with some money.

There is also endowment policy scheme where people place savings to earn interest and take out the money later in life.  If you drive, you are required by law to have your car insured in case of accident or theft. The cost of your motor insurance

dependson:-The make of your car and engine size.Where you liveYour ageA No Claims Discount is money deducted from your annual premium the longer you are able to drive without making a claim. People are obliged to take out Travel Insurance mainly to cover against their holiday being cancelled or delayed, their

luggage being lost on the journey or to pay for medical attention if they take ill while on holiday. The cost of your insurance depends on

where you are going and for how long. Children under 12 years normally pay 50% less and under 3’s go free. 

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7. TimeConvert between 12 hour and 24 hour clock

12.00am 0000 hours 1.32pm 1332 hours11.42pm 2342hours

*If a question is given in 24 hour clock time the answer must also be given 24 hour

clock time and vice versa.

Time intervalsQuestion How long did the film last? Start 1.40pm Finish 3.15pm

Answer 1.40pm to 2pm = 20 minutes2pm to 3.15pm = 1 hour 15 minutesTotal 1 hour 35 minutes

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Converting hours and minutes to decimal hoursThe most common mistake made by pupils is converting 2 hours 30 minutes

to 2∙3hrs and thinking 0∙1 hour is 10 minutes. It is very important to highlight

common errors to pupils. Pupils should be aware of the basics:

15 mins changes to 0∙25 hours30 mins to 0∙5 hours45 mins to 0∙75 hours

To change 1 hour 23 minutes in a calculator:23÷60 = 0∙383 (to 3 decimal places) so 1∙383 hours.

If this conversion was part of a speed, distance and time question it is best to round to 3 decimal places to avoid losing any accuracy.

To change 1.2 in a calculator:0∙2 x 60 = 12 minutes. Answer 1 hour 12 minutes

• Examples of calculating speed, distance and time will follow:

Learning the triangle below is an easy way to remember all 3 formulas:

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Speed

Calculate the average speed of a plane which travels a distance of 343km in 2 hours 30 minutes.

td

s =

52

343

.s

hrkms /2.137

2hr 30min = 2·5 hrs

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Distance

Calculate to one decimal place, the distance travelled in 1 hour 37 minutes at an average speed of 58 miles per hour.

Convert to decimal hours37 ÷ 60 = 0·617

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d = s x td = 58 x 1·617d = 93·8 miles

Time

Calculate, in hours and minutes, the time taken to travel 56km at 24 km/hr.

s

dt

24

56t

min202

.....333.2

hrt

t

0·33˙ x 60 = 20 mins

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7. Measure

Commonly used units of measure:Length10 mm = 1 cm 100 cm = 1000 mm = 1 m1000 m = 1 km 1 mile = 1·609 kms (to 3

decimal places)

Mass (Weight)1000 g = 1 kg 1000 kg = 1 tonne

Volume1000 ml = 1 litre 10 ml = 1 cl (centilitre)1 cm³ = 1 ml 1000cm³= 1 litre1000 litres = 1 m³

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9. Data and analysis

Using the average to Compare DataThe average (or mean) of a set of scores is found by• Adding all the scores together• Then dividing by the number of scores

In any example you do, pupils should always show how they added the set of

numbers first, then show their division.

ExampleFind the mean of the four numbers, 5, 6, 2 and 7

Mean = 5 + 6 + 2 +7 = 20 = 5 4 4

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Types of dataA graph is a pictorial representation of a set of data.

When you plan to collect data, one of the first things to think about is what kind of data you will collect as this will influence the way you organise data.Will the data be words (qualitative data) or numbers (quantitative data) ?If they are numbers, will you be counting (discrete/discontinuous data) or measuring (continuous data)?ExampleYou are collecting data on TV programmes.The kind of programme (e.g. soap, drama, documentary) is qualitative.The length of a programme is quantitative and continuous.The number of TV programmes is quantitative and discrete/discontinuous.

Some examples of graphs will follow.

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• Checklist when drawing graphs

• Bar graph

• Histogram

• Line Graph

• Pie chart

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Drawing graphs

We expect pupils to:

• use a sharpened pencil and ruler at all times• give the graph a title• label the axes with quantity and unit• if it is a bar graph, to label the bars in the centre of the bar (each bar has

an equal width) and make sure to leave an even space between each bar• label the frequency (up the side ie vertical axes) on the lines not on the

spaces• if it is a line graph to plot the points neatly (using a cross or a dot)• if asked to draw a line of best fit then the line should have the same

number of points above the line as below it.• if necessary, make use of a jagged line to show that the lower part of the

graph has been missed out• label all the sections or include a key when drawing a pie chart

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Bar graphsBar charts can be used for continuous or discrete data. They are often used to present data in a pictorial form to illustrate the information collected and highlight important points. 

ExampleThe marks awarded for an assignment set for a class of 20 students were as follows:

     6     7     5     7     7     8     7     6     9     7     4     10   6     8     8     9     5     6     4     8

Step 1 - Present this information in a frequency table.

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Pupils MUST label the axes and have a title.

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HistogramsHistograms are used for continuous data only. Continuous Variables: can take

on unlimited number of different values e.g. time,temperature, heights and

weights.• Lets look at the important differences between a bar graph and histogram.• Bar Charts • All the bars are the same width.• The height or length of the bar indicates the frequency.• Only one axis has a scale.• Histograms• For histograms bars are together as data is continuous.• Bars can be different widths.• Note: Histograms are usually used with grouped data.

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HistogramExample: Yearly Precipitation in New York City. The following table shows the number of inches of (melted) precipitation, yearly, in New York City, (1869-1957)

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43.6

37.8

49.2

40.3

45.5

44.2

38.6

40.6

38.7

46.0

37.1

34.7

35.0

43.0

34.4

49.7

33.5

38.3

41.7

51.0

54.4

43.7

37.6

34.1

46.6

39.3

33.7

40.1

42.4

46.2

36.8

39.4

47.0

50.3

55.5

39.5

35.5

39.4

43.8

39.4

39.9

32.7

46.5

44.2

56.1

38.5

43.1

36.7

39.6

36.9

50.8

53.2

37.8

44.7

40.6

41.7

41.4

47.8

56.1

45.6

40.4

39.0

36.1

43.9

53.5

49.8

33.8

49.8

53.0

48.5

38.6

45.1

39.0

48.5

36.7

45.0

45.0

38.4

40.8

46.9

36.2

36.9

44.4

41.5

45.2

35.6

39.9

36.2

36.5

36

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Histogram from frequency table

Line Graph

• How to Manually Construct a Line Graph• Obtain squared or graph paper. • Identify variables • Determine the range of the two variables.  For each variable subtract

the lowest data value from the highest data value.  If you wish the origin of the graph to start at zero, use the highest data value for the range.

• Determine the scale of the graph.  Count the number of squares horizontally and vertically on the graph paper.  Divide the range of the x variable by the horizontal number of squares, then round off to the next larger convenient number.  (For example, if dividing gives 34.3, it might be convenient to allow each horizontal square to have a value of 50.)  Determine the scale for the y variable by dividing its range by the number of vertical squares, then rounding up.  This procedure should spread the graph to use MOST of the available space.

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• Number each axis.  X-axis-Starting with a number just smaller than the smallest data value (zero if that is desired for your origin or words can be used) label each LINE along the bottom of the graph from left to right by successive multiples of the chosen scale.  (For our example, 0, 50, 100, 150...)  It is critical that these numbers represent lines and not boxes.  That will allow the space between the lines to represent values in between.  Number upwards the lines along the left side of the vertical axis using the scale chosen for the y variable, starting just below the smallest data (or zero if so chosen for the origin).  If lines are so close together that the numbers will be crowded, omit writing some of the numbers but continue to space as if all lines were numbered.

• Label each axis. – Briefly describe the properties represented by the x and y variables.– In parentheses abbreviate the measuring units used to measure the

data along each axis.• Plot the data points.  For each pair of related data, use the value for the

x variable to plot horizontal location and the value for the y variable to plot vertical location using the number lines along the axes.  Make a tiny dot or mark x at the intersection of the horizontal and vertical locations to accurately show the two values.  If other sets of data are also plotted on the same graph for comparison, use a different marker (square, triangle etc.) for each set of data and use a legend box to show what each marker represents.

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• Draw the line.  Join dots to produce a line.• Add a title. The title should be selected to clearly but briefly tell what the

graph is about. • Examples of line graphs are on the next page.

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More information: In some curricular areas an independent variable (along the x-axis) and dependent variable (y-axis) would be used to construct a scatter graph then a line of best fit would be drawn onto graph. More detailed information can be found on the internet.

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Pie chartsPie Charts are used for discontinuous data.

ExampleMakes of car in a school car parkMake Vauxhall Ford Rover Volvo

Frequency 3 2 4 1

Construct a pie chart to show this information.

A full circle is 360o

There are a total of 3+2+4+1 = 10 cars

We use 36o to represent each car

360o

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Makes of car in a school car park

Make Vauxhall Ford Rover Volvo

Frequency 3 2 4 1

Angle in pie chart

We use 36o to represent each car so 3 parts Vauxhall are represented by 3x36o = 108o

Work out the angle needed for each of the other models of car

3x36o

=108o

2x36o

=72o

4x36o

=144o

1x36o

=36o

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Makes of car in a school car park

Make Vauxhall Ford Rover Volvo

Angle 108o 72o 144o 36o

Mark the centre of the pie and draw the circle.

Draw a line from the centre to the edge

Measure the first angle needed (108o)

Draw in the line and label the section

108oVauxhall

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Makes of car in a school car park

Make Vauxhall Ford Rover Volvo

Angle 108o 72o 144o 36o

From this line measure the next angle, draw the line and label the section.

Continue in this way

Check that the last section has the correct angle and label it.

72o

Vauxhall

Ford144o

Rover

36o

Volvo

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10. ProbabilityTo find the probability of an event, we use:

Probability (event) = number of favourable outcomes total number of possible outcomes

An example What is the probability of picking a black counter from a bag containing five

red, three blue and two black counters?

Number of favourable outcomes = 2 (number of black counters)Number of possible outcomes = 5 + 3 + 2 = 10 (total number of counters)

P(black) = = Always leave your fraction in its simplest form.

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10

2

5

1

11. Algebra

+ 4y 20

5y 15

What is the value of y?......

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Solving Equationsa) y + 3 = 5

- 3 - 3

3 from both sides

Balance!

How would you get y on

its own?

y = 2

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b) w 2 = 9 + 2 + 2

Add 2 to both sides

c) 3p = 9

÷3 ÷3

p = 3

w = 11

Divide both sides by 3!

Balance!

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Subtract 1 from both sides

Divide both sides by 2

Add 1 to both sides

Divide both sides by 4

d) 2y + 1 = 9 - 1 - 1

2y = 8 ÷2 ÷2 y = 4

e) 4t 1 = 15 + 1 + 1

4t = 16÷ 4 ÷ 4

t = 4

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12. Which calculator to buy?In the Mathematics department we use the

following calculators:

Casio fx-82 MS Casio fx 83-ES 51

The End

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Prior permission required for use outwith John Ogilvie High School,

South Lanarkshire Council. ©