Mathematics & Numeracy

Key Skills Booklet

Auchenharvie Academy

Mathematics Department

2019 to 2020

Respect Equality Determination	This booklet has been designed to support your learning in S1 and should be used, if necessary, to practice numerical skills; as a guide to some common methods used and for reference when attempting work at home.

You are now entering the world of Mathematics!

Universal Language

Mathematics often looks like a collection of symbols. It doesn't matter what country you are from, or what language you speak, if you are trained in Mathematics you will understand their meaning - it is the Universal Language. In fact it is quite possible that Mathematics will be the common link between ourselves and any aliens we may one day meet!

Where did Mathematics come from?

Nobody is certain, but Mathematics may simply be "part of us". Even people without mathematical training can use their fingers to count, can use basic logic to solve their problems, and can recognise different properties of shapes.

Logical

Above all Mathematics is logical. Most of advanced Mathematics is based on just a few established ideas, and the rest is created purely by logical thinking.

Fun and Challenging

Mathematics is challenging to your mind, and that is what makes it fun. There is nothing better than working on a difficult puzzle, and getting a breakthrough.

Applied Mathematics

Mathematics has wide applications in Engineering and the Sciences. The major discoveries, inventions, theories and applications would never be possible without Mathematics. Mathematics is widely used in both Information Technology and Communication. These would have been non-existent if there were no developments in Mathematics over the centuries. Accountants, Economists and Business people use it every day. The weather is predicted using powerful Mathematics Modelling and your favourite computer game has, at its heart, lots of mathematical equations that work out how everything moves and behaves.

An Adventure

Mathematics is an adventure ... there are always new things to discover and learn. It will never cease to amaze you how neatly and beautifully it all fits together. Mathematics is not always easy, but what good adventure has no challenges? Most of the fun is in learning new things, mastering new methods and solving new puzzles.

So enjoy it and have fun!

Welcome from the Auchenharvie Academy Maths Department!

In the Maths department at Auchenharvie Academy you will find committed, hardworking teachers who want the very best for the pupils they teach. That means of course that we set high standards for behaviour and effort. We realise that not everyone is brilliant at Maths, but we do expect everyone to try their best and to hopefully enjoy the subject. We all love Maths and we want you to love it too!!

---------------------------------------------------------------------

Auchenharvie Academy

Maths Department

Classroom Code

I will:

1.	Arrive on time

2.	Bring my jotter, pencil and calculator

3.	Work to the best of my ability

4.	Behave well and contribute to the lesson

5.	Attempt and hand homework in on time

---------------------------------------------------------------------

A typical Maths lesson

All lessons last for 50 minutes and most Maths lessons follow the following format:

5 mins	Starter questions

15 mins	Learning intentions & class lesson

25 mins	Time to practise

5mins	Plenary – review of learning

We try to incorporate a variety of teaching styles to make learning Maths more fun through using mini whiteboards, pair & share activities, maths games, projects, Tarsia, ICT, relays, treasure hunts and murder mysteries to name a few.

Homework & Revision

You don’t stop learning Maths when the bell rings. It is really important to get into a good routine for homework and private study right from day one. We use a variety of different homework styles including: written formal homework and creative tasks. We expect S1 pupils to plan for two 30 minute slots per week of personal study at home, in addition to homework. You will be issued with materials to be working on at home and copies will also be available on the Auchenharvie maths department Facebook page.

Please make sure you ask for help if you are finding Maths a challenge.

How well do you know your times tables?

If you really want to prepare yourself for secondary Maths then you have to know your times tables inside out. The best way to do that is - practise, practise, practise!!!

This website is fantastic at helping you learn your tables:

http://www.transum.org/Tables/Times_Tables.asp

Mental calculations

Being good at Mental Maths (Mental Agility) will certainly help you progress at Secondary. Various strategies can be used which make calculations easier to do in your head.

Addition	Example Calculate 54 + 27

Exercise 1

1.	Calculate mentally :-

a. 38 + 45	b. 43 + 49	c. 59 + 47	d. 7 + 108

e. 75 + 65	f. 66 + 85	g. 116 + 68	h. 70 + 89

i. 340 + 290	j. 310 + 490	k. 360 + 850	l. 730 + 990

m. 4600 + 7500	n. 2800 + 5200	o. 6800 + 2400	p. 1850 + 8150

2.	A train with 153 passengers stops at Stevenston station.

Another 28 passengers get on the train.

How many are now on the train?

3.	The local garage charged Mr Denny £199	for parts and £114 for labour to repair his car.

How much was Mr. Denny’s bill in total?

Subtraction	Example	Calculate 93 – 56

Method 3 Round up and subtract then add on the extra

e.g. subtract 60 then add on 4	93 – 60 = 33

33 + 4 = 37

Exercise 2

1. Write down the answers to :-

a. 73 – 41	b. 84 – 29	c. 76 – 28	d. 51 – 16

e. 100 – 25	f. 92 – 15	g. 110 – 33	h. 190 - 39

i. 370 – 280	j. 5400 - 1600	k. 7600 – 2700	l. 20000 - 3700

2. a. Of the 3616 miles from Glasgow to Dubai,

a plane had flown 2815 miles.

How much further had it to travel?

b.	Mrs Fletcher won £1400 on a scratch card.

She bought a new laptop for £790.

How much had Mrs Fletcher left?

Multiplication

Exercise 3

1.	Choose the easiest method to help you calculate the following:

a. 36 x 2	b. 56 x 3	c. 91 x 5	d. 103 x 7

e. 4 x £1.99	f. 8 x £3.99	g. 5 x £2.96	h. 9 x £5.20

2.	The owner of the Cross keys bought four

HD Smart TVs priced £999.

What did his bill come to?

Multiplying by 10, 100 or 1000

Exercise 4

1	Write down the answers to the following:-

a. 25 x 10	b. 21 x 10	c. 66 x 10	d. 10 x 88

e. 10 x 225	f. 402 x 10	g. 10 x 557	h. 10 x 2920

2.	Calculate mentally:-

a. 64 x 100	b. 87 x 100	c. 100 x 255	d. 100 x 800

e. 8020 x 100	f. 100 x 3004	g. 100 x 9600	h. 80500 x 100

3.	Write down the answers to the following :-

a. 26 x 1000	b. 214 x 1000	c. 850 x 1000	d. 1000 x 970

e. 1000x 800	f. 3460 x 1000	g. 1000 x 6300	h. 1000 x 58080

4.	Calculate the following using the 2-step approach:-

a. 22 x 30	b. 51 x 40	c. 30 x 80	d. 65 x 60

e. 30 x 114	f. 41 x 900	g. 700 x 52	h. 3000 x 36

Addition

Exercise 5

1.	a. 452	b. 569	c. 486	d. 878

+168	+ 357	+566	+213

e. 5003	f. 4456	g. 8374	h. 8777

+3087	+5678	+5895	+1999

i. 6389 + 364	j. 3345 + 6666	k. 6802 + 4299	l. 9429 + 368

2.	Andy earns £21 563 per year and Joan gets an

annual salary of £24 168.

What is their combined pay?

3.	Last year Mrs Eagleson sent 4835 text messages.	This year she sent 5596 text messages.

How many did she send altogether?

Subtraction – Written Method

We always use the decomposition method.

Example 2 2006 – 92	2 1 109 10 6	- 9 2	1 9 1 4

Exercise 6

1.	Copy the following and calculate the answers :-

a. 35	b. 469	c. 626	d. 905

-16	- 75	-386	-366

e. 8374	f. 4000	g. 7052	h. 20 000

-5895	- 893	-3463	- 8 409

i. 9527 – 5521	j. 9000 – 374	k. 8006 – 2967	l. 10 000 - 8381

2.	Of the 632 pupils in Auchenharvie Academy,

465 are girls.

How many boys are in the school?

3.	On Jan 1st 2013 Mrs Scott noticed that her car had

covered 29 312 miles.

On Jan 1st 2014, the milometer reading was 43 187.

How many miles had Mrs Scott driven over the year?

Multiplying by a single digit

Written Method

Example 1 245 x 8	Example 2 8251 x 4

2 4 5	8 2 5 1
x 3 4 8	x 1 2 4
1 9 6 0	3 3 0 0 4

Exercise 7

1. Write down the answer to the following :-

a. 5 x 7	b. 4 x 9	c. 3 x 6	d. 8 x 3

e. 6 x 8	f. 6 x 11	g. 5 x 12	h. 12 x 8

2. Copy and complete the following :-

a.	456	b.	127	c.	374	d.	8971
x 6	x 4	x 8	x 9

3. Set out the same as above and complete :-

a. 362 x 7	b. 498 x 5	c. 1479 x 3	d. 8691 x 6

4. Mrs Aitken decides to make and sell cupcakes for charity.

She makes 125 cupcakes, to sell, in one day.

How many cupcakes can she make in 5 days?

If she charges 40p per cake, how much will he raise?

5.	It was calculated that an average pupil	spends 959 minutes on the internet each week.

How many minutes would they have spent browsing	the internet in 8 weeks?
Dividing by a single digit

Exercise 8

1.	Copy the following and complete each calculation.

a. 7 56	b. 5 795	c. 6 756	d. 8 856

e. 6 6594	f. 4 9420	g. 7 81354	h. 9 88857

2.	Set the following down in the same way as above and complete each calculation :-

a. 90 ÷ 6	b. 592 ÷ 2	c. 474 ÷ 3	d. 774 ÷ 9

e. 749 ÷ 7	f. 5740 ÷ 5	g. 8948 ÷ 4	h. 6738 ÷ 3

i. 1143 ÷ 9	j. 7980	k. 2127	l. 4182 ÷ 6

5 3

3.	Show all your working in solving the following :-

a.	A bar of chocolate has 9 squares.

How many bars would you need to have 621 squares?

b.	A small jar contains 8 lollies.

How many jars are needed for

Mrs Hands secret stash of 136 lollies?!

Dividing by multiples of 10, 100 and 1000

Examples	a) 890 ÷ 10	b) divide 3800 by 100

H	T	U	●	t	Th	H	T	U	●	t	h
8	9	0	●	3	8	0	0	●
8	9	●	0	3	8	●	0	0
8	9	3	8

c) Divide 354 by 10	d) 50.6 ÷ 100

Exercise 9

1.	Write down the answers to the following :-

a. 350 ÷ 10	b. 470 ÷ 10	c. 690 ÷ 10	d. 4450 ÷ 10

e. 700 ÷ 100	f. 2200 ÷ 100	g. 4600 ÷ 100	h. 58000 ÷ 100

2.	Calculate the following :-

a. 43 ÷ 10	b. 96 ÷ 10	c. 32 ÷ 10	d. 8 ÷ 10

e. 790 ÷ 100	f. 230 ÷ 100	g. 47 ÷ 100	h. 5 ÷ 100

3.	Calculate the following using the 2 step approach :-

a. 60 ÷ 20	b. 750 ÷ 30	c. 3000 ÷ 500	d. 50400 ÷ 700

(Hint: ÷2 then ÷10)

Adding and Subtracting Decimals

Example 1 5234·5 + 952·4	Example 2	1679.4 – 83.1

5 2 3 4·5	15617 9·4
+ 1 9 5 2·4	- 8 3·1
6 1 8 6·9	1 5 9 6·3

Note: When adding or subtracting with decimals

the decimal point always stays in line.

Exercise 10

Copy the following and find :-

a.	6·6	b.	28·3	c.	23·7	d.	86·8
+ 2·5	+ 8·9	+ 66·8	34·9
e.	8·65	f.	5·28	g.	27·04	h.	59·37
- 2·41	- 3·07	- 4·31	- 48.45
i.	125·7	j.	356·23	k.	6793·6	l.	9264·25
+ 388·4	+ 451·64	+ 2324·3	+ 4162·97
m.	456·3	n.	784·2	o.	4529·3	p.	7482·5
- 145·1	- 355·5	- 2615·5	- 2563·6

2.	Miss Paton bought a pair of earrings for £33·75

and a bracelet at £54·60.

How much did she spend altogether?

3.	Mrs Urquhart saw a lawnmower priced at £79·58 in the Garden Centre.

Later she sees it on the Internet for £49·89.

How much would she save by buying online?

4.	Grace weighs 42·81 kilograms and Nicola weighs 37·89 kilograms.

a. What is their combined weight?

b. By how much is Nicola lighter than Grace?

Multiplying and Dividing Decimals

Example 1 5·62 x 4	Example 2 5·74 ÷ 2

5·62
x 4	2·87
22·48	2	5·74

Note: When dividing a decimal number by a whole number, The decimal points must stay in line.

Exercise 11

1.	Copy and complete the following calculations :-

a.	4·58	b.	8·98	c.	63·42	d.	323·76

x 3	x 2	x 5	x 7

e. 6 34·44	f. 4 96·48	g. 8 0·96	h. 9 229·23

2.	A bar of chocolate weighs 105·6 grams.

How much will 6 bars weigh?

3.	Michael works as an electrician.

If he earns £16·25 an hour how much

will he earn in 8 hours?

4.	Baby Molly gains 729·6 grams in 3 weeks.

How much does she gain, on average, each week?

Fractions of a Quantity

To find a unitary fraction of a quantity simply divide by the denominator.

Example 1 Find of £120	Example 2 Find of 138 km

of	120	of	138
=	120	÷	3	=	138	÷	6
=	£40	=	23 km

Exercise 12

1.	Find :-

a.	of	18	b.	of	28	c.	of	40
d.	of	490	e.	of	72	f.	of	680
g.	of	78	h.	of	160	i.	of	35

2.	There are 32 desks in Mr Hasting’s classroom.

of them need repaired.

How many desks need repaired?

3.	Each week the school canteen makes 3130 sandwiches.

of these are chicken.

How many chicken sandwiches do they produce?

4.	There are 42 children in Primary 7 at Glencairn Primary.

a third of them have brown hair.

half of them have blonde hair.

a sixth of them have red hair.

How many children have :-

a. brown hair	b. blonde hair	c. red hair?

Finding a Percentage of a Quantity

It is important that you remember the following

50%	=
25%	=
10%	=

Example 1	50%	of	£22	Example 2	10%	of	70 m
=	of	£22	=	of	70 m
=	£22	÷	2	=	70 m	÷	10
=	£11	=	7 m

Exercise 13

1.	Calculate each of the following :-

a.	50% of £14	b.	50% of 68 m	c.	50% of 162

d.	25% of 8p	e.	25% of 32 kg	f.	25% of 44 mm

g.	10% of 400 ml	h.	10% of £570	i.	10% of 6000 km

j.	50% of £450	k.	10% of 34 kg	l.	25% of 64p

m.	10% of £7.20	n.	50% of £20.40	o.	25% of 500 cm

2.	Derek had £550. He decided to spend 50% of it on a

holiday weekend away.

How much did he pay for his holiday?

3.	Emma weighed 70 kg. She decided to go on a diet and lost 10% of her weight.

How much weight did she lose?

Percentages, Decimals and Fractions

Percentages, decimals and fractions are all connected.

By remembering that percent just means out of one hundred

it makes it easy to spot the connection.

Example 1	Write 47% as a fraction and decimal.

47% = = 0·47

Example 2	Write 0·08 as a fraction and percentage.

0·08 = = 8%

Exercise 14

1.	Write each of the following as a fraction :-

34%	b. 58%	c. 72%	d. 7%

2.	Write each of the following as a decimal :-

42%	b. 98%	c. 14%	d. 2%

3.	Write each of these as a fraction and as a decimal :-

13%	b. 29%	c. 60%	d. 9%

4.	Write each fraction or decimal as a percentage :-

b. c. 0·67	d. 0.07

e.	f. 0·99	g. 0·6	h. 0·3

Some More Information

Length

Centimetres & Millimetres on a Ruler

Each centimetre (cm) is split into equal parts called millimetres (mm)	1 cm = 10 mm

The length of the line AB shown below is 7.8 cm or 78 mm or 7 cm 8 mm.

Main Units of Length

Kilometre	metre	centimetre	millimetre

Metre	About the distance from the floor to a door handle.

Centimetre	A metre divided into 100 parts – about the width of your pinky nail.

Millimetre	A centimetre divided into 10 parts – about the width of a thin needle.

Kilometre	1000 metres – about the distance 3 times around a football pitch.

Change Units of Length

÷10	÷100	÷1000

mm	cm	m	km

x10	x100	x1000

Perimeter

The perimeter of a shape is the total distance around its outside.

This simply means you add up ALL the sides.

Perimeter = 6 + 8 + 9 + 7 = 30cm

Area

Weight

The weight of heavier items is measured in kilograms (kg).

The weight of lighter items is measured in grams (g).	1 kg = 1000 g

A small cube 1cm by 1cm by 1cm weighs 1 g when filled with water.

Example	Example

A bag of sand weighs about 10kg.	a	2 kg = 2000 g

b	5 ½ kg = 5500 g

A packet of crisps weighs 30 grams.	c	1700 g + 300 g = 2000 g = 2 kg

Volume

Liquid volumes are usually measured in litres or millilitres.

Larger volumes – litres (l)	1 litre = 1000 millilitres

Smaller volumes – millilitres (ml)

Example	Example

A large tin of paint holds 5 litres.	a	3 litres = 3000 ml

b	7 ½ litres = 7500 ml

A can of cola holds 330 millilitres.	c	4600 ml + 400 ml = 5000 ml

= 5 litres

Solid volumes – cubes & cuboids

Volume = length x breadth x height

Example

V = l x b x h

= 6 x 4 x 5

= 120 cm³

2 and 3 Dimensions

We live in a 3-dimensional world where solid objects have depth as well as length and breadth.

Common 2D shapes

You should be able to recognise and name the following 3D shapes;

Angles

acute - less than 900	right – exactly 900

obtuse – more than 900 and less than 1800	straight - exactly 1800

reflex – more than 1800 and less than 3600	full turn – exactly 3600

Time

We usually think of the time of day in terms of midnight to noon, (ante-meridian – morning) and noon to midnight, (post-meridian – afternoon/night), but pilots and sailors need a system that causes no confusion.

The 24 hour clock

Examples

6.00 am = 0600	5.00 pm = 1700	midday = 1200

9.25 am = 0925	8.15 pm = 2015	midnight = 0000

11.40 am = 1140	11.50 pm = 2350

The Calendar

12 months in a year	30days has September, April

52 weeks in a year	June and November.

365 days in a year	All the rest have 31, except

366 days in a leap year (every 4 years)	February alone which has 28 days clear

And 29 in each leap year.

Time Intervals

Counting on is the easiest way to find how long something lasts.

Example

A TV programme starts at 7.45pm and ends at 10.25pm.

How long did it last?

or

Drawing Graphs

When drawing statistical graphs like Bar graphs, Line graphs or Pie Charts, it is important to label them correctly and show any scale used accurately.

TYPE OF HOUSING

Conducting a Survey

When carrying out a survey it is important to consider:

the type of questions you will ask

who you will ask

how you will organise your answers

how you will display your answers

When deciding how to present or display the results of your survey, you can choose to use a frequency table, bar graph, line graph or pie chart.

Algebra

Like Terms

We can simplify expressions where terms are the same.

Examples

Simplify:

3a+5a+2a	8b+4c+4b-2c	6m+5-3m+2r

=10a	=12b+2c	=3m+2r+5

(b’s and c’s are different so this

does not simplify further)

Equations

We always want to work out the value the symbol/letter should take.

Examples

Solve:

x+8=10	y-2=11	3xp=15	m = 8	2c+1=15	5q-3=12

x=2	y=13	p=5	4	2c=14	5q=15

m=32	c=7	q=3

(Where a number and letter are stuck together the instruction is to multiply

eg. 2a = 2 x a))

Evaluating

Replace the letters with values and work out to a number.

Examples

a=5, b=3 and c=10.

Evaluate:

a+4	8b	c²	bc	5b-c

=5+4	=8x3	=10x10	=3x10	=5x3-10

=9	=24	=100	=30	=15-10

=5

Answers

Exercise 1

1a) 83 b) 92 c) 106 d) 115 e) 140 f) 150 g) 184 h) 159 i) 630 j) 800 k)1210 l)1720

m) 12100 n) 8000 0) 9200 p) 10000

2) 181 3) £313

Exercise 2

1 a) 32 b) 55 c) 48 d) 35 e) 75 f) 77 g) 77 h) 151 i) 90 j) 3800 k) 4900 l) 16300

2a) 801 b) £610

Exercise 3

1a) 72 b) 168 c) 455 d) 721 e) £7.96 f) £31.92 g) £14.80 h) £46.80

2) £3996

Exercise 4

1a) 250 b) 210 c) 660 d) 880 e) 2250 f) 4020 g) 5570 h) 29200

2a) 6400 b) 8700 c) 25500 d) 80000 e) 802000 f) 300400 g) 960000 h) 850000

3a) 26000 b) 214000 c) 850000 d) 970000 e) 800000 f) 3460000 g) 6300000

h) 58080000

4a) 660 b) 2040 c) 2400 d) 3900 e) 3420 f) 36900 g) 36400 h) 108000

Exercise 5

1a) 620 b) 26 c) 1052 d) 1091 e) 8090 f) 10134 g) 14269 h) 10776 i) 6753

j) 10011 k) 11101 l) 9797

2) 45731 3) 10431

Exercise 6

1a) 19 b) 394 c) 240 d) 539 e) 2479 f) 3107 g) 3589 h) 11591 i) 4006 j) 8626

k) 5039 l) 1619

2) 167 3) 13875

Exercise 7

1a) 35 b) 36 c) 18 d) 24 e) 48 f) 66 g) 60 h) 96

2a) 2736 b) 508 c) 2992 d) 80739

3a) 2534 b) 2490 c) 4437 d) 52146

4) 625 cakes, £250 5) 7672

Exercise 8

1a) 8 b) 159 c) 126 d) 107 e) 1099 f) 2355 g) 11622 h) 9873

2a) 15 b) 296 c) 158 d)86 e) 107 f) 1148 g) 2237 h) 2246 i) 127 j) 1596

k) 709 l) 697

3a) 69 b) 17

Exercise 9

1a) 35 b) 47 c) 69 d) 445 e) 7 f) 22 g) 46 h) 580

2a) 4.3 b) 9.6 c) 3.2 d) 0.8 e) 7.9 f) 2.3 g) 0.47 h) 0.05

3a) 3 b) 25 c) 6 d) 72

Exercise 10

1a) 9.1 b) 37.2 c) 90.5 d) 121.7 e) 6.24 f) 2.21 g) 22.73 h) 10.92 i) 514.4

j) 807.87 k) 9117.9 l) 13427.22 m) 311.2 n) 428.7 o) 1913.8 p) 4918.9

2)£ 88.35 3) £29.69 4a) 80.7Kg b) 4.92Kg

Exercise 11

1a) 13.74 b) 17.96 c) 317.1 d) 2266.32 e) 5.74 f) 24.12 g) 0.12 h) 25.47

2) 633.6g 3) £130 4) 243.2g

Exercise 12

1a) 9 b) 7 c) 8 d) 70 e) 8 f) 68 g) 13 h) 20 i) 3.5

2) 4 3) 626 4a) 14 b) 21 c) 7

Exercise 13

1a) £7 b) 34m c) 81 d) 2p e) 8kg f) 11m g) 40ml h) £57 i) 600km

j) £225 k) 3.4kg l) 16p m) 72p n) £10.20 o) 125cm

2) £225 3) 7Kg 4a) £20.50 b) £61.50

Exercise 14

1a) 17/50 b) 29/50 c) 18/25 d) 7/100

2a) 0.42 b) 0.98 c) 0.14 d) 0.02

3a) 13/100 0.13 b) 29/100 0.29 c) 3/5 0.6 d) 9/100 0.09

4a) 35% b) 67% c) 14% d) 7% e) 5% f) 99% g) 60% h) 30%