policies - numeracy

7/30/2019 Policies - Numeracy

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St Thomas Becket Catholic College

(Specialist in Humanities)

Numeracy Across the Curriculum Policy

November 2011(Draft)

A whole school approach to Mathematics


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Teachers of subjects other than mathematics should:

1. ensure that they are familiar with correct mathematical language, notation, conventions and techniques, relating

to their own subject, and encourage students to use these correctly.

2. be aware of appropriate expectations of students and difficulties that might be experienced with numeracy skills.

3. provide information for mathematics teachers on the stage at which specific numeracy skills will be required for

particular groups.

ROLE &USE OF CALCULATORS

In simple terms, each department needs to decide and then plan into each module of work whether calculators are

banned, ignored, allowed, encouraged or compulsory!

Whole school Policy on the use of calculators

The school expects all pupils to bring their own scientific calculator to lessons when required.

In deciding when pupils use a calculator in lessons we should ensure that:

pupils first resort should be mental methods;

pupils have sufficient understanding of the calculation to decide the most appropriate method: mental,

pencil and paper or calculator;

pupils have the technical skills required to use the basic facilities of a calculator constructively and efficiently,

the order in which to use keys, how to enter numbers as money, measures, fractions, etc.;

pupils understand the four arithmetical operations and recognise which to use to solve a particular problem; when using a calculator, pupils are aware of the processes required and are able to say whether their answer

is reasonable;

pupils can interpret the calculator display in context (e.g. 5.3 is 5.30 in money calculations);

we help pupils, where necessary, to use the correct order of operations especially in multi-step calculations,

such as (3.2 - 1.65) x (15.6 - 5.77).

Vocabulary

The following are all important aspects of helping pupils with the technical vocabulary of Mathematics:

Use of Word walls Using a variety of words that have the same meaning e.g. add, plus, sum

Encouraging pupils to be less dependent on simple words e.g. exposing them to the word multiply as a

replacement for times

Discussion about words that have different meanings in Mathematics from everyday life e.g. take away, volume,

product etc

Highlighting word sources e.g. quad means 4, lateral means side so that pupils can use them to help remember

meanings. This applies to both prefixes and suffixes to words.


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IV Transfer of Skills:

It is vital that as the skills are taught, the applications are mentioned and as the

applications are taught the skills are revisited.

The Mathematics team will deliver the National Curriculum knowledge, skills and understanding

through the Framework using direct interactive teaching, predominantly in 3 part lessons. Theywill make references to the applications of Mathematics in other subject areas and give contexts

to many topics. Other curriculum teams will build on this knowledge and help pupils to apply them

in a variety of situations. Liaison between curriculum areas is vital to pupils being confident with

this transfer of skills and the Maths team willingly offers support to achieve this.

The transfer of skills is something that many pupils find difficult. It is essential to start from the

basis that pupils realise it is the same skill that is being used; sometimes approaches in subjects

differ so much that those basic connections are not made.


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SECTION A: Number and calculation methods

Reading and writing numbers

Pupils should be encouraged to write figures simply and clearly. Zeros or sevens with a linethrough should be discouraged.

Most pupils should be able to write numbers up to a thousand in words, but there are oftenproblems with bigger numbers. It is now common practice to use spaces between each group ofthree figures in large numbers rather than commas

eg 56 000 rather than 56,000

In reading large numbers pupils should apply their knowledge of place value working from the rightin groups of three digits. So the first group contains hundreds, tens and units. This is repeated inthe next group as thousands and the next group as millions. The number is then read from left toright.

So

4 024 132 is read as Four Million Twenty Four Thousand One Hundred and Thirty Two

Language of operations

All pupils should be able to understand and use different terms for the four basic operations butsome pupils will have difficulty in associating terms with symbols.

+ - x add

increasemoreplussumtotal

decrease

differenceless

minusreduce

subtracttake

multiply

ofproduct

times

divide

sharequotient

Order of operations

When carrying out a series of operations pupils can often be confused about the order in which the

operations should be done

Eg Does 6 + 5 x 7 mean 11 x 7 or 6 + 35 ?

Some pupils will be familiar with the mnemonic BODMAS or BIDMAS

Brackets, powerOF orIndices, Division, Multiplication, Addition, Subtraction

Hence,

6 + 5 x 7 = 6 + 35 = 41 whereas

(6 +5) x 7 = 11 x 7 = 77

Calculators


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The school expects all pupils to bring their own scientific calculator to lessons when required.In deciding when pupils use a calculator in lessons we should ensure that:

pupils first resort should be mental methods;

pupils have sufficient understanding of the calculation to decide the most appropriate

method: mental, pencil and paper or calculator;

pupils have the technical skills required to use the basic facilities of a calculatorconstructively and efficiently, the order in which to use keys, how to enter numbers asmoney, measures, fractions, etc.;

pupils understand the four arithmetical operations and recognise which to use to solve aparticular problem;

when using a calculator, pupils are aware of the processes required and are able to saywhether their answer is reasonable;

pupils can interpret the calculator display in context (e.g. 5.3 is 5.30 in moneycalculations);

we help pupils, where necessary, to use the correct order of operations especially inmulti-step calculations, such as (3.2 - 1.65) x (15.6 - 5.77).

Some pupils are over-dependent on the use of calculators for simple calculations. Whereverpossible pupils should be encouraged to use mental or pencil and paper methods.

It is, however, necessary to give consideration to the ability of the pupil and the objectives of thetask in hand. In order to complete a task successfully it may be necessary for pupils to use a

calculator for what you perceive to be a relatively simple calculation. This should be allowed ifprogress within the subject area is to be made. Before completing the calculation pupils should be

encouraged to make an estimate of the answer.

Mental Calculations

Most pupils should be able to carry out the following processes mentally though the speed withwhich they do it will vary considerably.

recall addition and subtraction facts up to 20

recall multiplication and division facts for tables up to 10 x 10.

Pupils should be encouraged to carry out other calculations mentally using a variety of strategiesbut there will be significant differences in their ability to do so. It is helpful if teachers discuss withpupils how they have made a calculation. Any method which produces the correct answer isacceptable.


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Writing Calculations

Pupils have a tendency to use the = sign incorrectly and write mathematical expressions that donot make sense

Eg 3 x 2 = 6 + 4 = 10 7 = 3

It is important that pupils are encouraged to write such calculations correctly

Eg 3 x 2 = 66 + 4 = 1010 7 = 3

The = sign should only be used when both sides of the equation have the same value. There isno problem with calculations such as

34 + 28 = 30 + 20 + 4 + 8 = 50 + 12 = 62

Decimal notation

All pupils should be familiar with decimal notation for money although the may use it incorrectly.

Correct Incorrect2.35 2.35p4.60 4.60.25 0.25p32p 0.32p

Pupils should also be familiar with decimal notation in the context of measurement but willsometimes misinterpret the decimal part of the number they may need to be reminded for

example that 1.5 metres is 150 cm and not 105cm. Simple problems may occur when interpretinga calculator with respect to money.

Pupils often read decimal numbers incorrectly eg 8.72 is often read as eight point seventy twoinstead of eight point seven two.

They may also have problems with comparing the size of decimal numbers and may believethat 2.36 is bigger than 2.8 because 36 is bigger than 8. If they need to compare numbers it mayhelp to write them to the same number of decimal places eg 2.36 and 2.80


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Calculation methods

Below, you will find examples of problems involving the four operators, how they are taught, and

approached by pupils at different stages in their school lives. It is important to remember that,

pupils on entry to secondary school, may still be using these earlier methods of calculation.

Pupils are encouraged to use mental calculations to support their written recordings.

Year Addition Subtraction

3/4 Example 1: 29 + 39 = 30 + 40 - 2= 70 - 2= 68

or29 + 39 = 20 + 30 + 9 + 9

= 50 + 18= 50 + 10 + 8

= 60 + 8= 68

Example 1: 90 - 37 = 90 - 40 + 3= 53

or+3 +50

37 40 90 90 37 = 53

Example 2: 287 + 45200 + 80 + 7+ 40 + 5200 + 120 + 12 = 332

or287 287+45 +45200 12120 12012 200

332 332

Example 2: 567 243500 + 60 + 7200 + 40 + 3300 + 20 + 4 = 324

or767 619 By

50 exchanging

700 + 60 + 17 a ten for ten600 + 10 + 9 units100 + 40 + 8 = 48

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7/8

Example: 8642 + 7538000 + 600 + 40 + 2

700 + 50 + 38000 + 1300 + 90 + 5

= 9395or

8000 + (600 + 700) +(40 + 50) + (2 + 3)

= 8000 + 1300 + 90 + 5= 9395or

8642 By this time they+753 understand9395 carrying1

Example: 2410 - 482

1000 1300 100 10

2000 + 400 + 10 + 0400 + 80 + 2

1000 + 900 + 20 + 8= 1928

or2410

- 482 Using what they-2 know about-70 negative numbers

20001928

or

1 13 10 1 By now they2410 understand- 482 borrowing1928

By Year 5/6 pupils will be able to extend their experience of addition and subtraction into

a range of contexts, including calculations with money and measurement with decimals.


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Multiplication and Division

Multiplication is usually associated with the idea of repeated addition, e.g.:

7 x 6 = 6 + 6 + 6 + 6 + 6 + 6 + 6 = 42

Division is associated with repeated subtraction or sharing, e.g.:

42 7 = 42 7 7 7 7 7 7 7 = 0 (42 shared equally 6 times)

Year Multiplication Division

3/4 Pupils will be taught multiplication tablesup to 10 x 10 and associated facts, e.g.:

If 7 x 9 = 63then 9 x 7 = 63

63 9 = 763 7 = 9

4 + 4 + 4 + 4 + 4 + 4 = 6 x 4

8 + 8 + 8 = 8 x 3

6 x 4 = 248 x 3 = 24

7 x 29 = (7 x 30) (7 x 1)= (7 x 3 x 10 ) (7 x 1)= 210 7= 203

20 x 30 = (2 x 10) x (3 x 10)Using = 2 x 3 x 10 x 10multiples = 6 x 10 x 10of 10 = 600

Recognition that division is the inverseof multiplication, e.g.:

63 9 = 7became 7 x 9 = 63

5/6 Example: 275 x 8(200 x 8) + (70 x 8) + (5 x 8)

1600 + 560 + 40

= 2200or

275 x 2 = 550275 x 4 = 1100 (doubled)275 x 8 = 2200 (doubled)

orGridmethod

= 1600 + 560 + 40= 2200

Working on the idea that division isrepeated subtraction. e.g.:

458 3Since 3 x 100 = 300 and

3 x 200 = 600the answer must be between 100 and200 (estimation).

3 x 100 = 3002 x 50 = 150 (halving)3 x 150 = 4503 x 2 = 63 x 152 = 456

So: 458 3 = 152 remainder 2

x

8

200

1600

70

560

5

40


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Year Multiplication Division

5/6 Example: 24 x 1624 x 10 = 240

24 x 1624 x 6 = (20 x 6) + (4 x 6)

= 144240 + 144 = 384

or(24 x 20) (24 x 4)

=480 96= 384

or

20 4

10 200 40

6 120 24

26 x 16 = 200 + 120 + 40 + 24= 384

Note: This is a very popular method is

known as the box or grid method. This

is an algorithmic method that does require

understanding but is utilised very

effectively my many pupils.

Example: 432 15

1 5 4 3 2

- 1 5 0 1 0

2 8 2

- 1 5 0 1 0

1 3 2

- 6 0 4

7 2

- 6 0 4

1 2 2 8

432 15 = 28 remainder 12

The above method is referred to aschunking, as they are subtractingchunks of 15 at a time.

5/6 Example: 24x 16

24 (6 x 4)120 (6 x 20)40 (10 x 4)

200 (10 x 20)384

This comes direct from the grid method.

7/8 Standard method long multiplication24

x 16144 (6 x24)240 (10 x 24)384

Standard method long division28 r 1215 432

30132120

12

Many lower attaining pupils experience great difficulty in understanding some formal methods of

calculation. It is essential to build upon what the child knows, understands and can do. The

examples above are real examples of how pupils have approached the problems.


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Calculating with,

Fractions, Decimals and Percentages

Non-Calculator Methods

CALCULATING WITH FRACTIONS

Example: Calculate5

6of 48

Solution:

Now5

6

= 5 x1

6

, but before we can find5

6

, we need to find the value of1

6

, then multiply by 5.

To find1

6of 48 we need to divide 48 by 6.

So,1

6of 48 = 48 6 = 8

We can now replace1

6with 8.

So,5

6of 48 = 5 x

1

6= 5 x 8 = 40.

Final answer

5

6of 48 = 40

1

6is now replaced with the 8,

since1

6of 48 is 8


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CALCULATING WITH PERCENTAGES

Example: Calculate the VAT on 240 @ 17.5%

Solution:

Noting first that 17.5% = 10% + 5% + 2.5%

Now 10% of 240 = 240 10 = 24

Now, since 10% is equal to 24

and

of 10% is 5%

then

5% of 24

= of 24

=12

Similarly,

2.5% of 24 = 6.

We have then:

17.5% of 240

= [10% + 5% + 2.5%] of 240

= 24 + 12 + 6

= 42

Final answer:

17.5% of 240 = 42

Since 10% =1/10 , to find 10% of

240 we divide 240 by 10.


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We can of course use this method to calculate any percentage but this method is best used when

the numbers are nice. See the example below.

Example: Calculate 45% of 36

Solution: 45% of 36

= [4 x 10% + 5%] of 36

Now, 10% of 36

= 36 10 = 3.6

= 3.60,

and similarly

5% = 1.80

We then have

40% = 4 x 10%

= 4 x 3.6

= 14.4

= 14.40

So,

45% of 36

= [40% + 5%] of 36

= 14.40 + 1.80

= 16.20

Final answer: 45% of 36 = 16.20

Whilst it is possible, it would be inefficient to calculate 23.608% of 406.87kg using this method.

For this a calculator is best used!

Definitely worth reinforcing that 3.6means 3.60 and not 3.06


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Calculator Methods

Calculator methods generally involve changing the fraction or percentages into a decimal first.

Decimals are often referred to as multiplying factors since, having turned the fraction or

percentage into a decimal, we use it to multiply.

Changing a fraction into a decimal

To change a fraction into a decimal, you divide the top number (numerator) by the bottom number

(denominator). To help pupils to remember this it is often useful to remind them that a fraction

looks like a divide sign:

numerator

denominator

Changing a percentage into a decimal

To change a percentage into a decimal we aim to remember that percentage means per-

hundred. Therefore 23% means 23 per 100. This is written as 23/100, and to change this fraction

into a decimal we divide 23 by 100 , as above.

Note:

Most pupils simply need to remember that to change a percentage into a decimal; they should

divide the percentage by 100.

A starter to a lesson where this skill needs to be utilised might look like this, where the pupils

need to fill in the blank spaces with the appropriate value.

Fraction Decimal Percentage

4/5 ? ?

? 0.125 ?

? ? 65%

8/13 ? ?

This line means divide

Lower ability students willstill have a problem with this

skill area.


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Example: Calculate 5.35% of 23,456kg

Solution: 5.35% as a decimal?

5.35

100 = 0.0535

So,

0.0535 x 23,456 = 1,254.896

Final answer

5.35% of 23,456kg = 1,254.896kg

Depending upon the context, a suitable degree of accuracy would now be required.

This is themultiplying factor


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SECTION B: Handling Data

Throughout the mathematics curriculum the clear message is that data handling is best taught in

the context of real statistical enquiries, in a coherent way so that teaching objectives arise naturally

from the whole cycle, as represented in the following diagram:

Data Handling Cycle Process

Other subjects within the national curriculum have similar descriptions for the role of data handling

within their subject-specific contexts:

The new GCSE Mathematics curriculum, first examined in 2003, in addition to the overall data

handling picture presented above, requires candidates to produce one piece of extended work that

demonstrates an ability to handle and compare data. This is worth 10% of the total marks. This

data handling project also offers opportunities for candidates to produce a piece of work that will

satisfy the vast majority of requirements for Key Skills.

1. Specifying the problem and planning

In order to specify a problem, pupils need to suggest a hypothesis that could be investigated.

A hypothesis is a statement about something youre going to investigate which clearly states a

point of view, eg:

taller people jump higher

more expensive cars are faster

girls are more intelligent than boys

Specify theroblem and lan

Collect data from avariet of sources

Interpret anddiscuss data

Process andre resent data


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2. Collecting Data

It is important that data are collected for a purpose. Data are found as either:

a) Primary data data you collect yourself using a survey or experiment; or

b) Secondary data data that is already collected for you. You can find secondary data in

books or on the internet.

Example: Survey/Questionnaire

To decide whether traffic outside school can be reduced, the Maypole High Governors want to askdrivers:

How far is it to school? How many people do you bring to school?

Do you drive in every day? Do any other pupils live near you?

Why do you drive your children to school? What do you think of the traffic outside school?

How long does your car journey take? What buses go from near your house?

Some of the questions haveyes or no answers:

Others have numericalanswers:

These have many differentanswers:

Do you drive in every day?

Do any other pupils livenear you?

How far is it to school?

How long does your carjourney take?

How many people do youbring to school?

Why do you drive yourchildren to school?

What buses go from nearyour house?

What is your opinion on thetraffic outside school?

These areclosedquestions. They haveparticular answers. You could use tick boxes tocollect this data

These areopenquestions. They can includeanswers you havent thought of.

The governors develop a questionnaire for their questions:

Traffic Questionnaire

1. Do you drive to school every day of the week? Yes No

Yes/No answers give very limited information butthe data is easy to collect

2. How many people do you

bring to school? 1 2 3 4+ This question has an exact, or discrete, numberof answers

3. How far do you travel to school? _________ These questions have a range of numericanswers. The data is easier to use if you collectit in ranges in a frequency table4. How long does your car journey take? _____

5. Why do you drive your children to school? Whyquestions are very open so the responsesmay not be easy to analyse

6. What do you think about the traffic outsideschool? (1 = good, 5 = bad)

1 2 3 4 5

An open question can be closed down to specificresponses using a scale


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You can use a questionnaire to conduct a survey. Open questions invite any response. Closed

questions invite choice.

To understand how to collect data properly, it is necessary to consider different types of data, so

that collection and handling activities can take place. One key idea, important to the propercollection of data, is that of sampling.

The Vocabulary of Sampling

Population: The entire group of people, animals, or things about which we want information

Sample: A part of the populationfrom which we actually collect data/information, used todraw conclusions about the whole

IMPORTANT:

In order for a sample to be suitable,at least 30

pieces of information need to be collected.


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3. Representing data and interpretation

Representing data in an orderly and easy-to-read/understand form is paramount to Handling Data.

Charts and diagrams without headings, labels and an appropriate scale are useless.

The representations synthesise the raw data into summary information. We will be looking at how

to draw the most common charts: bar charts, pie charts and scatter diagrams. Also a brief look at

averages.

Bar Charts:

A bar chart uses bars to represent data. Each bar represents a category or class.

Title

Key if required

Both axes labelledclearly

These diagrams are most frequently used in areas of the curriculum other than mathematics. Theway in which the graph is drawn depends on the type of data to be processed.

Graphs should be drawn with gaps between the bars if the data categories are not numerical(colours, makes of car, names of pop star, etc). There should also be gaps if the data is numericbut can only take a particular value (shoe size, KS3 level, etc). In cases where there are gaps inthe graph the horizontal axis will be labelled beneath the columns.

The labels on the vertical axis should be on the lines.

Opinion about smoking in public places

66%

58%52%

27%

38%44%

0%

20%

40%

60%

80%

100%

Hotels & motels Workplaces Restaurants

Survey conducted during July 19-31, 2000

Percentage

ofresponses

Set aside areas

Totally ban


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eg.

Bar Chart to show representation of

non-numerical data

0

1

2

3

4

5

6

78

9

10

walk train car bus other

Mode of Transport

Numberofpupils

Where the data is continuous, eg. lengths, the horizontal scale should be like the scale used for agraph on which points are plotted.

eg

0 10 20 30 40 50 60

NOT

0 1 2 3 4 5 x

"Bar Chart" to show representation of continuous

data

0

2

4

6

8

10

0

Height (cm)

Frequency

0 10 20 30 40 50 60 70


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Line Graphs

Line graphs should only be used with data in which the order in which the categories are written is

significant.Points are joined if the graph shows a trend or when the data values between the plotted pointsmake sense to be included. For example the measure of a patients temperature at regularintervals shows a pattern but not a definitive value.

Incorrect Use of Line Graph

0

2

4

6

8

10

walk train car bus other

Mode of Transport

Numberofp

upils

USE OF COMPUTERS

Pupils throughout the school should be able to use Excel or other spreadsheets to draw graphs torepresent data. As it is easy to produce a wide variety of graphs there is a tendency to producediagrams which have little relevance. Pupils should be encouraged to write a comment explainingtheir observations from the graph

PIE CHARTS:

A pie chart uses a circle to show data. Each class or category has a slice of the circle.Example: Draw a pie chart to illustrate the following information.

Type of transport Train Coach Car Ship Plane

Frequency 48 28 125 22 27

We need to find the fraction of the total, which represents each type of transport, and express this

as a decimal. Many pupils prefer to remember that we simply divide each frequency by the total.

This decimal is called the multiplying factor. To find the angle we then multiply 360 degrees by the

multiplying factor.


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Type of transport Frequency Multiplying Factor Angle

Train 4848 250 = 0.192 360 0.192 69

Coach 2828 250 = 0.112

360 0.112 40

Car 125125 250 = 0.5

360 0..5 180

Ship 2222 250 = 0.088

360 0.088 32

Plane 27 27 250 = 0.108360 0.108 39

Totals 250 1 360o

Notice: We use the total of 250 to calculate each fraction

We round off each angle to the nearest degree

We check that the sum of all the angles is 360o

The pie chart can now be drawn. It is important to check that pupils know how to use the protractor

and are accurate with their measurement

INTERPRETATIONS

Most people travel to holiday by car

Less than a quarter go by train

If 1000 people went on holiday, about 160 would go bycoach

There is no information about who took part in the survey,so is the pie chart representative of the population?

SCATTER DIAGRAMS

A scatter diagram is a method of comparing two sets of data, and discovering if there is a link

(relationship) between them, e.g.

25 pupils' Mathematics & English scores

0

20

40

60

80

100

0 50 100

Mathematics marks

Englishmarks


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Looking at relationships, scatter diagrams tell us whether there is a correlation (link) between the

two data sets. It is quite common when using scatter diagrams to include a line of best fit (a

straight line), which goes through the middle of the data, passing as close to as many points as

possible. This would allow us to make estimates for certain cases.

Here are three statements that may or may not be true.

The taller people are, the wider their arm span is likely to be.

The older a car is, the lower its value will be.

The distance you live from your place of work will affect how much you can earn.

Collecting data and plotting the data on a scatter diagram could test these

relationships. For example, the first statement may give a scatter

diagram like that on the right. This has a positive correlation because

the data has a clear trend and we can draw a line of best fit that passes

quite close to most of the points. From such a scatter diagram we could

say that the taller someone is, the wider the arm span.

Testing the second statement may give a scatter diagram like that on the

right. This has a negative correlation because the data has a clear

trend, and we can draw a line of best fit that passes quite close to most

of the points. From such a scatter diagram we could say that as a car

gets older, its value decreases.

Testing the third statement may give a scatter diagram like that on the

right. This scatter diagram has no correlation. It is not possible to

draw a line of best fit. It could therefore say that there is no relationship

between the distance a person lives from his or her work and how much

the person earns.

x x

x x x

x x

x x x xx

Height (cm)

Armspan(cm

)

xx

x x x xx

Age (years)

Value

()

x xx x x

x x x xWa

ge()

Distance from work


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Averages

This is a number that is used to represent a set of data. There are three main averages used in

different circumstances. You have to choose the most appropriate average to use.

MEAN: The sum of all the values divided by the number of values, eg Find the mean of 6, 3, 1, 4

Mean = 6 + 3 + 1 + 44

= 14 4

= 3.5

MEDIAN: The value in the middle of the data after it has been arranged in size order. If we havean even number of data, then we find the mean of the middle two values.

Example 1. Find the median of 4, 6, 3, 2, 1

6, 4,, 2, 1 Median is 3

Example 2. Find the median of 4, 6, 3, 2, 1, 2

6, 4, 3, 2, 2, 1 Median = 3 + 22

= 2.5

Mode: The value in the data that occurs most frequently, e.g.

Find the mode of : 3, 15, 0, 3, 1, 0, 4, 3 Mode = 3

If there is no number that occurs most often, there is no mode.

The Rangeis the spread of data, i.e. the largest value subtract the smallest value, e.g.:

7, 6, 8, 12, 9 Range = 12 6 = 6

The meanis a good average when the range is small. The medianis a useful average when the

range is large.


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SECTION C: Algebra

Algebra

Algebra is often referred to as the language of mathematics. When working with algebra it is important

that before attempting to perform any calculations pupils translate the algebra into English.

Example: 5x+ 3 = 18 means five times a number plus three equals eighteen

Solving Equations

There are three main methods for solving equations.

METHOD ONEMACHINE/INVERSE METHOD

Example: Solve 5x + 3 = 18

Solution: First we look at what is happening to x, in the correct order, to get 18.

x 5 + 3 18

We then push the 18 back through the machine in the reverse order performing the inverse

operation e.g. + 3 becomes 3 and 5 becomes 5

We then get:

(15)

3 5 - 3 18so

x = 3

This method can also be used effectively when rearranging simple equations.

Example

Rearrange the following equation, writing v in terms ofu

142

u

v

First we look at what is happening to u, in the correct order, to get v. We then reverse the flowdiagram by putting the v back through the machine.

u 2 + 14 v


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Something plus 3 equals 18.

That something must be 15.

So what I have covered up

must be equal to 15.

Performing the inverse operations e.g. +14 becomes 14 and 2 becomes 2

(V14)

2(v 14) 2 14 v

so u = 2(v 14)

METHOD TWOCOVER UP METHOD

Example

Solve: 5x + 3 = 18

We first cover up the 5x with something:

+ 3 = 18

The thinking then goes something like this:

We can then write down: 5x = 15

Five times something equals 15

So

x = 3


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METHOD THREEBALANCING METHOD

Example

Solve: 5x + 3 = 18

The idea here is to consider the equals sign as a set of balancing scales, and therefore whatever

you do to one side of the equals sign you have to do to other if the scales are to remain balanced.

For example, if you add 3 to one side you must add 3 to the other; if you divide by 4 on one side,

you divide by 4 on the other. This is the mathematically conventional way of solving all equations

and thus the one least liked by middle/low attaining pupils.

So, to solve our equation: 5x + 3 = 18

subtract 3 from both sides

5x = 15

divide both sides by 5

x = 3

Pupils should check solutions by substituting answers back into the original equations

Substituting into formulae

Again, it is essential that pupils write out what the formula means in long hand, before replacing

the letters with numbers. Stressing the importance of method is essential to obtaining the correct

answer. It is expected that pupils show all of the following working out exactly as detailed below,

the equal signs all underneath each other.

Example: v= u + atmeans v = u + a x t (remembering to multiply first!)

So given u = 4, a = 5 , t= 10 , v = ?

We now literallyreplace the letters with the numbers and perform the calculation in

the normal way, not forgetting to multiply first!

v = 4 + (5) 10

v = 4 + (50)

v = 46

Note

Problems will occur if you provide the pupils with values for v, u and a and ask them to find the

value of t. This is quite a difficult question/concept for pupils and will need reminding of the

method.


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Transforming formulae

Pupils are taught the balance method of transforming formulae (see equations) only pupils inhigher ability groups are likely to be able to do this

Eg V = IR Change the subject of the formula to R

V = IR

V = R both sides by II

Eg2 v2 = u2 + 2as Make a the subject of the formula

v2 = u2 + 2as

v2 - u2 = 2as - u2 from both sides

v2 - u2 = a both sides by 2s2s

Triangle method

Some pupils will be familiar with the triangle method which works for specific types of questions such as speed or density calculations

eg Density = Mass

Volume

M M M

D V D V D V

Density = Mass , Mass = Density x Volume, Volume = MassVolume Density

Similarly with Distance, Speed and Time

D D D

S T T S T S T

Speed = Distance ,Distance = Speed x Time, Time = DistanceTime Speed


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SECTION D: Accuracy in Measurement & Drawing

Pupils should be expected to draw and measure accurately. It is an essential requirementin many subjects. For example:

Reading scales in science and technology

Measuring and cutting materials in textiles

Plotting points on graphs in geography

Measuring distances and times in physical education

Equipment:

Pupils should be actively encouraged to have with them at all timesa ruler, protractor, apair of compasses, a sharp pencil and an eraserso that they can work accurately.

Estimation:

Estimation is an important aspect of measurement and drawing. Pupils should beencouraged whenever possible to make sensible estimates before measurement.Estimation can help pupils avoid careless mistakes in measurement. Estimation can alsobe used to introduce discussion on appropriate and sensible degrees of accuracy.

Units:

The choice of units is also important particularly as many pupils confuse the units of

length, area and volume. (Note: in mathematics, cm3

is called cubic centimetres ratherthan centimetres cubed). Pupils also need to understand that in some contexts,millimetres are used as the principle unit of length rather than centimetres. Note: pupils aretaught about commonly used imperial units and their metric conversions.

Checking accuracy:

Pupils involved in measurement tasks need to be clear about the level of accuracyrequired so their work can be checked and marked fairly. Peer assessment is a very usefulstrategy for improving accuracy and promotes self-evaluation.


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Appendix One

Numeracy across the Curriculum

Below you will find the key mathematical skills that pupils develop throughout Key Stages 2 and 3.

This will also be a guide for teachers to see when certain key topics are taught.

Please note:

Pupils are expected to explain their methods and reasoning at every opportunity.

Years 5 and 6 Key Skills

Use all four operations to solve simple word problems involving number and quantities

based on real life money and measures

Multiply and divide positive whole numbers by 10, 100 and 1000

Order positive and negative numbers

Use decimal and negative numbers

Use decimal notation for tenths and hundredths

Find basic fractions of quantities, e.g. of, of, 1/10 of

Find simple percentages of quantities, e.g. 10%, 50%, 25%

Basic calculator skills but NO scientific calculator skills taught at primary school

Round a number with one or two decimal places to the nearest whole number

Use, read and write standard metric units, including their abbreviations, and conversions

between them

Suggest suitable units and measuring equipment to estimate or measure length, mass or

capacity

Measure and draw lines to the nearest millimetre

Calculate area and perimeter of simple shapes

Record estimates and readings from scales to a suitable degree of accuracy

Recognise perpendicular and parallel lines

Recognise where a shape will be after reflection

Read and plot co-ordinates in all four quadrants

Use a protractor to measure angles up to 180o

Find the mode and range of a set of data and begin to calculate a mean of a set of data

Able to extract and interpretdata in tables and bar charts


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Year 7 Key Skills

All of the above, plus

Know and use the order of operations, including brackets, ie brackets, and , then

and

Mental methods to calculate simple decimals, fractions and percentages

Check a result by considering whether it is of the right magnitude estimation

Use a letter to represent unknown numbers

Simplify algebraic expressions, e.g.x + 2y + 3xy =y + 4x

Use simple formulae

Calculate area, perimeter and volume for simple shapes

Design a data collection sheet or questionnaire to use in a simple survey; constructfrequency tables

Construct, on paper and using ICT, graphs and diagrams to represent data including bar-

line graphs, frequency diagrams

Use ICT to generate pie charts

Compare data using the range, mean, median and mode

Interpret diagrams and graphs (including pie charts) and draw simple conclusions based

on the shape of graphs and simple statistics

Year 8 Key Skills

All of the above plus

Add, subtract, multiply and divide positive and negative numbers

Calculate any fraction of any quantity

Use squares, square roots, cubes and cube roots

Find any percentage of any quantity

Reduce a ratio to its simplest form

Use a scientific calculator to carry out more difficult calculations

Solving simple algebraic equations, e.g. 2x + 3 = 15

Plot graphs of simple algebraic equations, including interpretation of the gradient of a

straight line

Use and read and write standard metric units to solve problems involving perimeter, area

and volume

Calculate area, perimeter and volume of simple shapes including triangles

Understand and use the language and notation associated with enlargement


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Collect data using a suitable method, including data logging using ICT

Able to construct, extract and interpret data in tables, bar charts, pie charts, simple line

graphs and scatter graphs (NB both on paper and using ICT)

Find the mean, median, mode and range of a set of data, both discrete and continuous,

and begin to calculate mean of a set of continuous data

Year 9 Key Skills

All of the above plus

Begin to use numbers in standard form

Add, subtract, multiply and divide fractions

Make and justify estimates and approximations of calculations

Use formulae from mathematics and other subjects including substitution of numbers into

those formulae and changing the subject of those formulae

Construct graphs arising from real life problems, for example: distance time graphs,

conversion graphs, e.g. temperature, currency

Circle facts - formulae for perimeter and area

Use and interpret maps and scale drawings

Suggest a problem to explore using statistical methods, frame questions and raise

conjectures/hypotheses

policies - numeracy

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