Skills practice A - Hodder Education

1© Hodder & Stoughton Ltd 2014

Skills practice A1 a 5 cm b 3 cm c 7 cm d 2 cm e 4 cm f 8 cm2 a 80 mm b 40 mm c 100 mm d 160 mm e 70 mm3 a 2 cm b 5 cm c 9 cm d 14 cm e 3 cm4 a 10 cm b 8 cm c 12 cm d 16 cm e 5 cm f 15 cm Line d is longest.5 6 cm6 a 5 cm b 6 cm c 5.5 cm7 5 km8 The hamster, because the mouse is 7 cm which is 1 cm

shorter.

Strand 1 Unit 1 Answers


Strand 1 Units and scales Unit 1 Length Band c

Skills practice B1 a A 27 mm, B 36 mm, C 46 mm, D 41 mm, E 50 mm, F 43 mm, G 36 mm, H 70 mm b Eel H c Eel A d 43 mm2 a A 3 cm, B 4 cm, C 5 cm b Right-angled triangle3 a Buckingham Palace b Either

Bow Street – Piccadilly Circus – Trafalgar Square – Houses of Parliament or Houses of Parliament – Trafalgar Square – Piccadilly Circus – Bow Street

c Waterloo 4 a 2 cm b 15 cm c 4 cm d 10 cm e 9 cm f 5 cm5 20 times6 a 125 mm b 96 mm c 270 mm d 21 mm e 78 mm7 a 7.5 cm b 3.2 cm c 8.3 cm d 11.1 cm e 2900 cm8 a AB = 5 cm, BC = 5 cm, CD = 5 cm, DA = 5 cm

They are all equal. b AM = 16 mm, BM = 48 mm, CM = 16 mm, DM = 48 mm

AM = CM and BM = DM M bisects both diagonals.

9 a PQ = 20 mm, QR = 20 mm, SR = 30 mm, SP = 30 mm There are two pairs of adjacent equal sides.

b OP = 14 mm, OQ = 14 mm, OR = 14 mm, OS = 27 mm The longer diagonal bisects the shorter one.

10 4.5 cm is equal to 45 mm, which is greater than 35 mm.11 a 30 cm ruler b 1 metre ruler c micrometer12 A fi nger nail ≈ 1 cm, a railway carriage ≈ 20 m, a tall man ≈ 2 m, the width of this book ≈ 20 cm13 Own answers14 Own answers15 Own answers



Wider skills practice1 a i 160 mm

ii 112 mmiii 80 mmiv 56 mm

b 40 mm 2 a OA = 4 cm, OB = 5 cm, AB = 3 cm b OP = 8 cm, OQ = 10 cm, PQ = 6 cm c The lengths in part b are twice those in part a.3 a For 1 cm you use AB or CD or DE.

For 2 cm you use BC or CE or FG. For 3 cm you use AC or BD or EF. For 4 cm you use AD or BE or DF. For 5 cm you use AE or CF or EG. For 6 cm you use DG. For 7 cm you use BF or CG. For 8 cm you use AF. For 9 cm you use BG. For 10 cm you use AG.

b Here is one possible way to make all the measurements using only six marks (at 0, 1, 3, 4, 8 and 10):

A B DC E F

0 1 2 3 4 5 6 7 8 9 10



Applying skills1 The distance from London to Edinburgh – 700 km

The length of a swimming pool – 50 mThe width of a man’s hand – 10 cmThe height of a fully grown woman – 1.6 mThe width of a sewing needle – 1 mmThe width of your maths book – 20 cm

2 a i 21 cmii 210 mm

b 210 mm 3 Own answers



Reviewing skills1 a AB = 4 cm, BC = 6 cm, CA = 5 cm b XY = 20 mm, YZ = 30 mm, ZX = 25 mm c Triangle XYZ is half the size of triangle ABC.2 a Lime – length = 4.3 cm, width = 3.0 cm

Birch – length = 2.2 cm, width = 1.2 cm Oak – length = 3.6 cm, width = 2.1 cm

b Lime c Lime3 a Part J b Part E c Longest to shortest: parts J, C, G, D, B, A, F, H, E4 a 25 mm b 3 cm c 5.5 cm5 a i 4 cm

ii 7.2 cm b i 46 mm

ii 103 mm6 Own answers


Skills practice A1 a 600 g b 400 g2 a 5 g b 2 g c 130 g d 9 kg e 11 kg3 Growltiger, by 2 kg4 a 500 g + 200 g + 100 g b 2 kg + 1 kg + 500 g c 2 kg + 200 g d 100 g with the object and 1 kg on the other pan5 a 2 kg b 2 000 000 mg6 They both weigh the same.



Strand 1 Units and scales Unit 2 Mass Band c

Skills practice B1 a 1000 g b 1000 mg2 a i 5000 g

ii 20 000 giii 25 000 giv 500 g

b i 3000 mgii 8000 mgiii 500 mgiv 5 000 000 mg

3 20 days4 a 100 g b 30 apples5 a 35 g, 45 g, 150 g, 750 g, 1.4 kg, 2.3 kg b 1595 g6 a Industrial scales b Bathroom scales c Scientifi c balance d Kitchen scales7 Own answers8 Less than 1000 g (1 kg) More than 1000 g (1 kg)

Tin of beans Vacuum cleaner

Mobile phone Computer

Doll Dog

Apple Desk

Pencil Textbook*

* The textbook could be in either column.

9 a A 20p, B 50p, C 2p, D £2 b 1p, 20p, £1, 10p, 2p, 50p, £2

20 mm 30 mm

1p 20p £1 £22p10p 50p

c 1p, 20p, 10p, 2p, 50p, £1, £2 d 20p, and 10p or 2p e i Bag A 360 g, bag B 355 g, bag C 65 g, bag D 25 g ii Lightest to heaviest: bags D, C, B, A



Wider skills practice1 a $6.00 b i $2.00

ii $4.00iii $1.00iv $4.00v $3.00

c i $8.00ii $6

d Own answers e Package A: 230 g, 600 g, 1 kg 50 g, 900 g making

2 kg 780 g, which costs $6.00Package B: 1.5 kg, which costs $4.00Total: $6.00 + $4.00 = $10.00



Applying skills1 a i 2700 g

ii 2.7 kg b Own answers



Reviewing skills1 a 3 mg b 900 mg2 a 1 kg b 6 kg c 25 kg d 50 kg e 100 g f 450 g3 a 385 g b 25 g4 a i 9000 g

ii 4000 giii 2500 g

b i 6000 mgii 2000 mgiii 5500 mg


Skills practice A1 Hour hand pointing to Minute hand pointing to

a Just after 7 3

b Just after 11 1

c Just after halfway between 1 and 2

7

d Just before 9 11

e Nearer to 5 than 4 8

f Just before 2 10

2 a Twenty past nine, 9:20 b Five past six, 6:05 c Quarter to eleven, 10:45 d Twenty-five to nine, 8:35 e Five to two, 1:553 a 0925 b 1215 c 0015 d 15504 a 7.28 a.m. b 1.05 p.m. c 11.11 p.m.5 a 1930 b 1120 c 2055 d 1405 e 05356 a 1 hour 15 minutes b 2125 and 2325 c Only if the film lasts less than 135 mins7 a 220 minutes (3 hours 40 minutes) b 9.20 a.m.8 a 8 p.m. b 9.15 p.m. c 1 hour 15 minutes9 a Monday and Thursday b No c 40 hours d Friday, 6 12 hours

10 Norman, because films are more likely to be shown in the afternoon than the middle of the night.



Strand 1 Units and scales Unit 3 Time Band c

Skills practice B1 a Friday b 5 Wednesdays c 4 weekends d Wednesday e Tuesday f May 6th2 27th January3 a 3 April b 8 October c 7 January d 7 March e 8 March4 Flight from Time due Minutes late/early Time now due

Barcelona 2055 100 minutes late 2235

New York 2125 135 minutes late 2340

Sydney 2205 150 minutes late 0035

Tokyo 2240 50 minutes early 2150

5 a 2 years b 200 minutes c 3 years d The month of June6 a 2 hour 30 minutes b 3 hours 15 minutes c 1620 d 25 minutes7 a Hourly (except 1535) b 26 minutes (except the 1535 bus which takes 34 mins) c The bus turns round and comes back.8 a Calendar b Stop watch c Wall clock d Mobile phone9 a Years b Minutes c Hours d Seconds e Days10 a 105 seconds is 1 minute 45 seconds, as there are 60 seconds in one minute and not 100 as Sophie believes. b 18 minutes + 1 minute 45 seconds = 19 minutes 45 seconds

Tim works out the time as follows: Firstly he adds up all the whole minutes. He then adds up all the seconds and converts them to minutes and

seconds before adding all the data together.11 1835, 6.35 pm



Wider skills practice1 a 8 years b 20 years c 7 orbits d 18 is not a multiple of 4. e 9 years f 2024 g Yes, because this is 60 years later and 60 is a

multiple of 12. h No, because this is 64 years later and 64 is not a

multiple of 12.



Applying skills1 a 1461 days b 365.25 days2 The fl ight has crossed fi ve time zones, or London is

5 hours ‘ahead’ of NewYork.



Reviewing skills1 a Minutes b Years c Months, weeks or days d Hours e Seconds2 a Five past eight, 8:05 b Twenty to ten, 9:40 c Ten to fi ve, 4:50 d Twenty-fi ve to seven, 6:353 a 1930 b 1120 c 2055 d 1405 e 05354 a 35 minutes b Honiton and Feniton c 55 minutes


Skills practice A1 a 300 ml b 200 ml c 2000 ml d 800 ml2 a 2 ml b 3 ml c 4 ml3 a i 500 ml

ii 750 mliii 1000 ml

b 2 glasses c i 1000 ml

ii 4 glasses4 a i 200 ml

ii 400 mliii 550 ml

b 1150 ml c i 2 litres

ii 850 ml

5 Millilitre, litre a i 2 litres

ii 3 litresiii 7 litresiv 2.5 litres

b i 4000 mlii 6000 mliii 9000 mliv 3500 ml

7 a 14 litres b 730 litres (732 in a leap year)8 Own answers



Strand 1 Units and scales Unit 4 Volume Band c

Skills practice B1 a 105 ml b 16 days2 a Containers A, B and C b Containers B, C and D c Containers A, C and D d Container D three times e Container A and container B twice3 40 times4 One 70 cl bottle and six 30 cl bottles5 a 2000 ml b 4 bottles c 800 ml6 a measuring jug b 5 litre can c medicine spoon d gauge on a pump7 a 1 litre, measuring jug b Typically, about 250 ml [accept 200 ml to 300 ml]8 a

200100

300400500600700800900

1000 ml

b

400200

600800

100012001400160018002000 ml

9 Bathroom sink 15 litres Teaspoon 5 ml Bath 80 litres Mug 400 ml Kettle 112 litres Bottle of ketchup 250 ml



Wider skills practice1 a Flour, butter, yeast, raisins, sugar b Milk, water2 a 600 ml b Yes, because 2000 ÷ 5 is 400 ml and 600 ml is

greater than 400 ml.3 a 10 eggs, 1.25 kg fl our, 2.5 litres milk, 10 pinches

of salt b 2 boxes c 500 g d One full jug and two marks up the second time4 Michelle’s grandfather is a large man. He is

1.85 metres tall and weighs 90 kilograms. He is building a shed in his garden.

The shed is 3 metres long and 2 metres wide.To build it he is using 7 centimetre nails and planks of wood that are 3 metres long and 10 centimetres wide.

He has a pond in his garden. It holds 300 litres of water.There are a few large fi sh in the pond that weigh about 4 kilograms each.

For his lunch, Michelle’s grandfather usually walks to the pub.It is 1.7 kilometres away so sometimes he takes the bus.



Applying skills1 a 1000 ml or 1 litre b 640 miles c No, because he can only drive 48 miles.2 a i Fill the 9, pour from the 9 into the 4, empty

the 4, pour from the 9 into the 4; the 9 now contains 1 litre.

ii Fill the 5, pour from the 5 into the 3, empty the 3, pour from the 5 into the 3, fi ll the 5, pour from the 5 into the 3; the 5 now contains 4 litres.

iii 1 litre: fi ll the 5, pour from the 5 into the 2, empty the 2, pour from the 5 into the 2; the 5 now contains 1 litre.2 litres: fi ll the 2.3 litres: fi ll the 5, pour from the 5 into the 2; the 5 now contains 3 litres.4 litres: fi ll the 2, pour from the 2 into the 5, fi ll the 2, pour from the 2 into the 5; the 5 now contains 4 litres.5 litres: fi ll the 5.6 litres: repeat the instructions for 1 litre, pour from the 5 into the 2, fi ll the 5.7 litres: fi ll both.

b Own answer, e.g. 1 litre and 9 litres



Reviewing skills1 a i 2 litres

ii 120 ml b Beaker ii would be empty.

2 40 litres


Skills practice A1 a A 35 mph b B 38 mph, C 44 mph c P 78 mph, Q 86 mph d R 43 mph, S 56 mph2 a Temperature b Degrees Celsius c 1 °C d 0.1 °C e 37.8 °C3 a 70 g, 100 g b Each division is 10 g not 1 g.4 a K = 2.8, L = 4.6, M = 5.2 b P = 7.6, Q = 8.2, R = 9.6 c X = 2.63, Y = 2.75 d V = 5.03, W = 4.965 a 2.7 kg b 1.2 kg c 0.6 kg d 4.8 kg6 a A = 70, B = 72, C = 75, D = 72, E = 73, F = 72.5,

G = 75, H = 73.5, J = 70, K = 73 So the four pairs are A and J, B and D, C and G, and E and K.

b Own answer7 Own answers, e.g. a 5 minutes b 12 divisions c Because 5 × 12 = 60 and there are 60 minutes in

one hour.8 a 97.8°F b 98.4°F c 100.2°F d 100.8°F e b



Strand 1 Units and scales Unit 5 Interpreting scales Band d

Skills practice B1 a

0.4

0.43

0.5 0.6

0.47 0.52 0.58

b

0.02

0.024

0.03 0.04 0.05

0.036 0.042 0.048

2 a 10 mph b 70 mph3 a 0.02 d A = 0.04, B = 0.16, C = 0.22, D = 0.284 74 kg 5 W = 50, X = 58, Y = 64, Z = 76

Own advice, e.g.For Jacqui – Each division is worth double what you worked out.For Ben – The division is a whole number.

6 a i 177 °C (±2 °C) ii 108 °C (±2 °C) iii 133 °C (±2 °C) b i 66–73 °C (±2 °C) ii 84–91 °C (±2 °C)7 a They weigh masses in the ranges i 0 to 1kg, ii 0 to 5 kg and iii 0 to 10 kg (respectively) b i ii iii

SilverBreamDace

Roach

Eel

Tench

Perch

8 a i 86 °F ii 30 °C b i 40 °F ii 4 °C c i 77 °F ii 25 °C d i 108 °F ii 43 °C



Wider skills practice1 a i He has read the third dial in the wrong

direction. ii 0793.2 b 7314.82 a, b

3

0

kg 2

1

4

5

c The piece that weighs 3.3 kg3 a i 50 mph ii 80 km/h b The ‘rev’ counter shows 2500 revolutions per

minute.The fuel tank is three-quarters full.



Applying skills1 a i 20 ml ii 250 ml or 25 cl iii 100 ml or 10 cl b 100 ml2 a A 3000 m, B 9500 m, C 6800 m b D 300 km/h, E 380 km/h, F 250 km/h c i 1500 kg ii 500 kg iii 2500 kg iv 2100 kg d Own answers3 a i £60 ii £12.50 iii £17.50 iv £97.50 v £42.50 b £66, £13.75, £19.25, £107.25, £46.75 c Own answers d Own answers e.g. about £25 per kilo when paid

online. Bag is carried for much longer.



Reviewing skills1 a 1000 ml or 1 litre b 20 ml c 360 ml2 a 42 ml b 430 ml c 184 ml3 a 2800 rpm, 1643, 102 km/h, 68341 km, 213.7 km, 70 °C, 3/4 tank, +2 amps b engine speed, time of day, road speed, total distance travelled, trip distance, water temperature, fuel in tank, use

of electrical current (charge/discharge)


Skills practice A1

�10

mm cm m km

�100�10

�100 �1000

�1000

2 a 200 cm b 300 cm c 150 cm3 a 2.12 m b 3 m c 4.5 m4 a 100 cm b 10 mm5 a 3 km b 1.5 km c 0.8 km6 a 1.2 kg b 2.4 kg c 0.5 kg7 a 2 litres b 1.5 litres c 0.5 litres8 a 120 mm b 45 cm c 7 m d 2 kg e 3000 ml f 6 km9 a i 3500 mm ii 3.5 m b 35 dm10 a 14 cm b 3 cm c 4 km d 1 m e 8100 g f 3 litres g 1 m h 4100 g11 a 8500 ml ÷ 10 (iv) b 85 g × 100 (vi) c 140 cm × 10 (i) d 3600 m (v) e 85 g × 10 (iii) f 36 cm × 10 (ii)12 a °C b Millilitres c Grams d Metres or centimetres e Minutes f Millimetres



Strand 1 Units and scales Unit 6 The metric system Band d

Skills practice B1 C = 4 mm, F = 2000 g ÷ 10, G = 850 litres, I = 5000 ml, K = 2500 mm, M = 370 kg ÷ 10, P = 50 m2 5000 paces3 a 2.56 km b 567 km c 75 litres d 2 metres4 a 3000 ml b 20 litres of lemonade, 4500 ml orange juice, 3000 ml grapefruit juice, 2500 ml pineapple juice c i 12 glasses ii 120 glasses5 a 180 cm, 1.8 m b 15 600 g, 15.6 kg6 a 960 mm b 120 sequins7 a 3000 kg b 7000 kg c 4500 kg d 500 kg8 a 1000 micrometres b 10 000 micrometres c 1 000 000 micrograms9 a 5 megagrams b 24 megagrams c 1 megagram10 10 tonnes



Wider skills practice1 a 312 hours

b 214 hours c i 2 kg ii 2000 g2 a 1000 N b 1000 mN



Applying skills1 a 3750 g of sugar

3750 g of butter 2500 g of golden syrup 7500 g of rolled oats 1750 g of dried fruit

b i 11.25 kg of sugar 11.25 kg of butter 7.5 kg of golden syrup 22.5 kg of rolled oats 5.25 kg of dried fruit

ii He will need 1.25 kg sugar, 0.95 kg butter, 3.5 kg golden syrup, 5.6 kg rolled oats and 4.8 kg dried fruit.

2 a 541.8 kg b 361.2 m c 1.5 km



Reviewing skills1 a 3000 ml b 7400 ml c 230 ml d 821 ml e 9500 ml2 a 360 cm b 4.5 cm c 50 cm3 a 17 mm b 24 mm c 200 cm d 48 cm e 4000 m f 1200 m4 a 6.5 kg b 6500 g5 a 2 litres b 64 glasses


Skills practice A1 1 micrometer, length, micrometres 2 medicine spoon, capacity, millilitres 3 trundle wheel, length, metres 4 bathroom scales, mass, kilograms 5 stopwatch, time, seconds/minutes 6 calendar, time, days 7 wall clock, time, hours 8 measuring jug, capacity, millilitres 9 ruler, length, centimetres or inches 10 petrol can, capacity, litres 11 kitchen scales, mass, grams 12 scientific balance, mass, grams 13 tape measure, length, inches 14 mobile phone, time, hours/minutes 15 (petrol) gauge, capacity, litres 16 metre rule, length, metre/centimetres2 a 10 litres b 20 litres c 30 litres d 50 litres3 a 2 inches b 8 inches c 20 inches d 3 inches4 a 5 cm b 25 cm c 50 cm d 100 cm5 a 56 g b 140 g c 252 g d 168 g6 a 50 miles b 500 miles c 250 miles7 (142 g) (140 g) mixed whole spices

1 large red cabbage (1.94) litres (2 litres) vinegar (57 g) (56 g) sugar, optional

8 a 3200 pints b 1800 – 2000 litres9 a 100 miles b 150 miles c 612 feet d 132 lb e 10 feet, 13 feet f 5 oz, 12 pt g 12 inches, 4 oz10 a i Read up from 4 pints to the red line and across

to 2.3 litres. ii Multiply previous answer by 3, or read off at

6 pints and double the answer. b i Read up from 7 pints to the red line and across

to 4 litres. ii Read across from 2 litres to the red line and

down to 3.5 pints. iii Read up from 2.8 pints to the red line and

across to 1.6 litres. iv Read across from 1.5 litres to the red line and

down to 2.6 pints.



Strand 1 Units and scales Unit 7 Metric–imperial conversions Band e

Skills practice B1 ... morning with their cases packed to bursting – there wasn’t 2.5 cm to spare! After travelling 96 kilometres they stopped at a service station and fi lled the car with 35 litres of petrol. Sue went

into the shop and bought 1.7 litres of water to drink and 0.45 kilograms of chocolate. She also bought a book that was 3.75 cm thick!

They continued on their way at an average speed of 120 km/h, arriving at the airport in plenty of time for the fl ight. The car park was busy and the car had to be squashed into a small space with only 60 cm on each side to spare!

In the airport building they checked in their luggage. It weighed 15 kilograms! The fi nal job to do before they could relax was to change their money into euros. At last they could enjoy their holiday!

2 a i 1 metre ii 50 km b i 2 pints

ii 28 litres c 3.5 pints d 6 gallons e 25 litres f 6.4 miles g 10 metres h 17.6 pounds3 Yes. T he trousers would fi t a 26 inch waist.4 3 to 3.1 m5 a A metre is longer than a yard. There are 36 inches in a yard and 39 in a metre. b i 63 360 inches ii 60 000 inches (58 500) iii The mile is longer.6 a 17.5 miles b 45 litres7 Joanne’s size is 75 cm, so the trousers will be 5 cm too long.8 160 cm

No, the coat does not fi t.9 4 pears

280 g cream112 g (110 g) brown sugar4 vanilla pods285 ml water

10 a 6.6 lb b 11 lb c 3.6 kg d 4.5 kg11 30 cm ≈ 1 foot



12 a i 256 km ii 128 km

b i 10 miles ii 120 miles

13 a 2 kg b 4.4 lb14 a

00

10

20

Poun

ds (£

)

30

40

50

60

Euros (€)10 20 30 40 50 60 70 80 90

b £46.67 c €37.50 d £35.33



Wider skills practice1 27 to 30 metres2 a 17.6 lb b 91.4 cm c 6.4 km d 12.2 m



Applying skills1 Distance to drive = 160 km + 80 miles

= 1608 5×

miles + 80 miles

= (100 + 80) miles= 180 miles

Petrol needed to cover (180 − 50) = 130 miles No of gallons needed = 13050 = 2.6 gallons No of litres needed = 2.6 × 5 litres

= 13 litres Cost = 13 × € 1.70

= € 22.10 Cost in pounds = 22.10 × 0.88

= £ 19.45



Reviewing skills1 a 18 kg, 39.6 lb b No, it is less than a foot.2 a 25 cm b 60 mm c 3 kg3 a 7.7 gallons b 34 litres c i 2.2 gallons

ii 17.6 pintsiii 1.76 pints or 1.8 pints (1 d.p.)


Skills practice A1 a 050° b 120° c 235° d 330°2 a i Own accurate diagram (±1°) ii x = 105°, y = 255° iii 075° iv 255° b i Own accurate diagram (±1°) ii x = 60°, y = 300° iii 120° iv 300° c i Own accurate diagram (±1°) ii x = 80°, y = 280° iii 100° iv 280°3 a–d Own accurate diagram e 5 cm f 270° g Equilateral

4 Own accurate diagrams

5 a 000° b 270° c 135° d 315°6 a South b North-East c South-West d East7 a 310° b 130°8 a Own accurate diagram b 7.7 km c 180° d 000°9 a 326° b 165°



Strand 1 Units and scales Unit 8 Bearings Band e

Skills practice B1 a 058° b 250° c 270° d 160° e 130°2 a They lie in a straight line. b 157°3 a 166° b 346°4 a 86° b 274° c 180° d 061°5 a 293° and 067° b 113° and 247° c 023° and 157° d 158° and 292°6 Helen did not return, because her return journey was not on a back bearing.



Wider skills practice1 a i 031° ii 153° iii 326° iv 259° v 098° vi 270° (due West) b i 211° ii 333° iii 146° iv 079° v 278° vi 090° (due East) c Carham to Egwell and Filwood to Durton

Egwell to Durton and Carham to Filwood(and the reverses of these pairs of journeys)

d They have the same slope/gradient.2 a Own accurate diagram b 292° c 9.90 km (3 s.f.)3 a No, because the three lines do not meet at the

same point. b Blackbeard must be one of those telling the truth,

so there are two possible places where the treasure could be.

c The truth teller could be any of the three, so we do not know where the treasure is.



Applying skills1 a Accept distances ± 5 km, Course ± 1°

Leg Start End Distance (km) Course

1 Penzance Swansea 200 034°

2 Swansea Liverpool 230 019°

3 Liverpool Carlisle 160 000°

4 Carlisle Glasgow 140 330°

5 Glasgow Inverness 170 359°

6 Inverness Aberdeen 135 110°

7 Aberdeen Leeds 370 174°

8 Leeds Norwich 240 125°

9 Norwich Bristol 285 242°

10 Bristol Penzance 265 237°

b Penzance, Swansea, Liverpool, Carlisle, Glasgow, Inverness, Aberdeen, Leeds, Norwich, Bristol

c 2195 km (Accept 2195 ± 50)



Reviewing skills1 a i 225° ii 090° b i North ii North-West2 a i 052° ii 052°

They lie in a straight line. b i North ii 000° c i 60° ii 120°


Skills practice A1 a i Circle ii 60 cm b i 4 cm ii 4 cm iii Own accurate scale drawing c i 80 cm ii 80 cm iii That depends on a number of factors,

including the materials used in its construction.2 a 2 : 3 b 3 : 1 c 1 : 2 d Zat 1 m, Human 0.8 m, Blobbit 1.29 m3 Racket 5 cm : 80 cm = 1 : 16 Car 57 mm : 3400 mm = 1 : 604 Item True length Length on model

Settee 2 m 2 cm

Bookcase 1.8 m 1.8 cm

Hall 13.5 m 13.5 cm

Plate 20 cm 2 mm

Living room 5.8 m 5.8 cm

Toothbrush 20 cm 2 mm

Bottle 30 cm 3 mm

5

Item Plan measurement True measurement

Length of patio 4 cm 8 m

Width of patio 2 cm 4 m

Width of vegetable plot 2.5 cm 5 m

Length of vegetable plot 6 cm 12 m

Width of pond 1 cm 2 m

Length of pond 2 cm 4 m

Length of house 6 cm 12 m

Width of house 3.5 cm 7 m

6 a i 8.6 km ii 080.5° or 081° b 260.5° or 261°



Strand 1 Units and scales Unit 9 Scale drawing Band f

Skills practice B1 Own accurate scale drawing 2 a 14 km b 14 km c 14 km3 a 148° b Yes, the distance is 14 km so he will arrive at 5.30 p.m.4 a Own accurate scale drawing b 6.26 m c ‘Max height 5.2 m’ (or 5.1 m)



Wider skills practice1 a Own accurate scale drawing b Own answer c Assuming that each player can search a width of

1 m, they will need to make four sweeps of the length. The total length is 4 × 120 m = 480 m. Assuming a speed of 4 m in a minute, the time taken will be 120 minutes or 2 hours.

2 a 7 mm, actual width 7 m b 12 plots c i 90° ii 2 mm, actual radius 2 m d 18 mm by 7 mm, actual size 18 m by 7 m e Plot: 28 mm by 15 mm, actual size 28 m by

15 m, which gives an area of 420 m2.House: 8 mm by 10 mm, actual size 8 m by 10 m, which gives an area of 80 m2.Percentage covered = 19.0% (correct to 1 d.p.)

f i On plan: 26 mm by 10 mm, which gives an area of 260 mm2.Actual size: 26 m by 10 m, which gives an area of 260 m2.

ii 1 : 1 000 000 (i.e. 1 : 10002) g To convert square metres to square millimetres,

multiply by 1 000 000.The actual area is 1200 m2 or 1 200 000 000 mm2.1 200 000 000 mm2 ÷ 1 000 000 gives 1200 mm2 on the site plan.Treat the playground as rectangular (e.g. 40 m and 30 m) so that the area is 1200 m2.The site plan dimensions of 40 mm and 30 mm give a site plan area of 1200 mm2.

3 a Own accurate scale drawing b i 5 km ii 180° to 190° iii 16 km ii 000° v 21 km c They meet 3.3 km South of the lighthouse

(providing the Swift avoids the lighthouse on its way).



Applying skills1 a Own accurate scale drawing b, c Own answers



Reviewing skills1 a 50 000 cm b 0.5 km c 1.25 km d 20 cm2 a 4.8 m b 73°3 a 3 m b 3 cm


Skills practice A1 a 80 kilometres per hour b i 0.9 metres per minute ii 1.5 cm s−1

c 900 m h−1 or 0.9 km h−1

2 a 150 km/h b 60 km/h c 80 km/h3 a 10 m/s b 36 000 m/h c 36 km/h4 a i 200 m ii 500 m b i 100 m/minute ii 6 km/h c i 500 m/minute i 30 km/h d Nina cycles.5 A 5 g per cm3 or g cm−3 or g/cm3

B 12 000 kg per m3 or kg m−3 or kg/m3

6 Jane £6 per hour, Sarah £5.50 So Jane is better paid.

7 108 km/h



Strand 1 Units and scales Unit 10 Compound units Band g

Skills practice B1 72 mph2 Katherine £1.31 per litre, Elizabeth £1.42 per litre

So Katherine gets the cheaper diesel.3 a 360 beats b 90 bpm c 5 minutes4 a 6.5 g cm−3

b 52 kg5 Peter 56 km/h, Pierre 45 km/h

So Peter is travelling faster.6 The bird‘s speed is roughly 101.25 mph, which is much faster than the speed limit.7 339.9 m/s8 a 40 wpm b 1 hour 12 minutes9 11.25 mph



Wider skills practice1 About 53 km = 1.667 km2 9 460 800 000 000 km3 a John 2, Stephen 2.39, Simon 2.68, Julian 3.21,

Kit 2.43, Lee 3.02 b Julian, Lee, Simon, Kit, Stephen, John c There are no units. The ratio is simply a number.



Applying skills1 £1162 92.5 m/s3 a Miles per gallon b Diesel car c Mileage: 5 × 42 = 210

25235

weekly for work evenings/weekends Total

Consumption: Petrol; 23539 = 6.03 gallons

Diesel; 23543 = 5.47 gallons



Reviewing skills1 a £9 per hour b 15p per minute c £3602 a i 108 km ii 18 km iii 4.5 km iv 1.5 km b 675 mph


Skills practice A1 a p

b qc r

2 e, d, c, a, b3 a Acute: b, f, g, l, k b Obtuse: a, c, e, h c Reflex:d, j4 a 3 right angles

b 4 right angles5 A acute-angled triangle, equilateral triangle

B right-angled triangle, scalene triangle C acute-angled triangle, isosceles triangle D obtuse-angled triangle, isosceles triangle E acute-angled triangle, isosceles triangle F right-angled triangle, isosceles triangle G right-angled triangle, scalene triangle H acute-angled triangle, scalene triangle I acute-angled triangle, isosceles triangle J right-angled triangle, scalene triangle K acute-angled triangle, equilateral triangle L right-angled triangle, scalene triangle

6 a i The four triangles MDA, MBC, MBA and MDC are isosceles.

ii Triangles MDA and MBC are congruent, and triangles MBA and MDC are congruent.

b All of the triangles are right-angled isosceles.7 a i Radius = 1.5 cm, diameter = 3 cm

ii Radius = 2 cm, diameter = 4 cm b Diameter = 6 cm; radius = 3 cm8 a Sphere b Cuboid c Cube9 a False;forexample,anequilateraltrianglehas

three acute angles.b Truec Trued Truee False; the diameter is twice the radius.f True



Strand 2 Properties of shapes Unit 1 Common shapes Band b

Skills practice B1 Grey–reflex

Pink – acuteOrange – right anglePurple–reflexGreen – straight lineYellow – obtuseBlue – acute

2 a PS = PQ = RS = RQ = QS = 4 cmPO = OR = 3.5 cmQO = OS = 2 cm

b i PSQ and RSQ ii QPR and SPR iii OPQ, OPS, OSR and ORQ iv OPQ, OPS, OSR and ORQ

3 2-D shapes 3-D shapes

PentagonSquare

Equilateral triangleHexagonOctagon

Isosceles triangleCircle

RectangleParallelogram

Cuboid Cube

Sphere

4 a Irregularhexagonb Angles a, e and g are acute angles, b is an obtuse

angle, c and farereflexangles,andd is a right angle.

5 a 3 squaresb 6 rectanglesc 9 rectangles

6 a Lucy and Angus are both right.b Arectangleisflat.Itistwo-dimensional(2-D).

Acuboidisasolid.Itisthree-dimensional(3-D).c Circle

7 Height = 6 cm, width = 7 cm8 a i 20 equilateral triangles

ii 12 congruent small equilateral triangles with sides 1 unit each6 congruent larger equilateral triangles with sides of 2 units each2 large congruent equilateral triangles with sides of 3 units each.

b Hexagon

9 a False; parallel lines never meet, so they cannot form two sides of a triangle.

b False; all squares are the same shape, but they may be different sizes and so they are not congruent.

c False; although the letter M is a 2-D shape formed by four straight lines, it is not a closed shape so it is not a quadrilateral.

d Truee False;acircleisflatortwo-dimensional(2-D)but

asphereisasolidorthree-dimensional(3-D).f True

10 a Cuboidsb Rectangle



Wider skills practice1 a Own diagram

There are two answers: ACEG and BDFHb Own diagram

There are four answers: ABEF, BCFG, CDGH and DEHA

2 a i F = 6, V = 8, E = 12 ii F = 6, V = 8, E = 12 iii F = 8, V = 12, E = 18b, c F + V−E = 2 for each solid.d Own answer

3 Cuboids: b, d Cubes: a Spheres: e None of these: c



Applying skills1 a 4 parallelograms each with sides 3 cm and 2.2 cm; 3 isosceles triangles each with sides 2.2 cm, 2.2 cm and 2 cm

b 4 parallelograms and 3 trianglesc No, the parallelograms are congruent to each other and the triangles are also congruent to each other.d No

2 a 6 sidesb i The cells are the same shape. ii The cells are the same size.c Foreverd–f Own tessellations



Reviewing skills1 Obtuse angle – larger than a right angle, smaller than

two right anglesReflexangle–largerthantworightanglesAcute angle – smaller than a right angle

2 Eye – circleEar – isosceles triangleHead – isosceles triangleLegs and body – rectangles

3 a Diameter = 5 cm, radius = 2.5 cmb Squarec 24-sidedpolygon(sometimescalledan

icosikaitetragon)Dodecagon(12sides)OctagonHexagon


Skills practice A1 a i

! ii Danger

b i

ii Keep right

c i

ii No motor vehicles

2 a

b

c

3

A A

A A

4 a i Square ii 4 lines

b i Equilateral triangle ii 3 lines

c i Rectangle ii 2 lines



Strand 2 Properties of shapes Unit 2 Line symmetry Band c

d i Isosceles triangle ii 1 line

e i Regular hexagon ii 6 lines

5 a 1 lineb 12 linesc 1 lined 1 line

6 a 1 lineb 0 lines

7 The statues above entrance, window blinds and roofl ine features (eaves and chimneys)



Skills practice B1 Signs a, c, d and e have line symmetry.2 a, b

p q

c Angles p and q are equal.3 a 3 lines

b 4 lines

c 5 lines

d 6 lines

e 8 lines

4

5 a

b

c

d

6 a



b i A B C D

H

F

F G

ii Shape H (see part i) iii

A B C D

H

F

F G

7 a Diagram A8 a Diagram F (and B)

b Ec Diagram C d D (and E)



Wider skills practice1 a Line AB is a diameter of the circle.

It cuts the circle into two equal parts.It is a line of symmetry.

b Infi nitely manyc Infi nitely many

2 Own answers, e.g.a A rectangleb A parallelogram, kitec An isosceles trapezium



Applying skills1 Own drawings2 a, b

c Own drawings



Reviewing skills1 a

b

c

2 a 1 lineb 2 linesc 2 linesd 1 linee 1 linef 0 linesg 2 lines

3 a

x

b

c

x

d

x

e

x

f

x

4


Note that there may be alternative ways of finding angles so students’ reasons may vary.

Skills practice A1 a = 80°, b = 95°, c = 95°, d = 50° 2 120°3 135°4 A whole turn, 360°, 360°5 a = 110°, b = 60°6 a q = 150°

b r = 30°c s = 150°d 30° + 150° = 180°e p = r and q = s



Strand 2 Properties of shapes Unit 3 Angle facts Band d

Skills practice B1 a = 110°, b = 130°, c = 74°, d = 320°, e = 115°,

f = 55°, g = 155°, h = 60°2 18°3 a 30°

b 120° (or 240° if the refl ex angle is found)4 a = 50°, b = 130°, c = 130°, d = 102°, e = 78°, f = 78°,

g = 65°, h = 115°, i = 115°5 a = 45° (Angles on a straight line add up to 180°)

b = 74° (Angles about a point add up to 360°)c = 90° (Angles on a straight line add up to 180°)d = 120° (Angles about a point add up to 360°)

6 a = 60° (Angles about a point add up to 360°)b = 150° (Angles about a point add up to 360°)c = 93° (Angles on a straight line add up to 180°)d = 60° (Angles about a point add up to 360°)e = 33° (Angles on a straight line add up to 180°)

7 a = 90°, b = c = 135°, d = e = 45°, f = 225°



Wider skills practice1 TURN 155°2 a = 60°, b = 120°, c = 180°



Applying skills1 a 10°

b 50 seconds2 a a = 80°, b = 102°, c = 56°, d = 44°

b Largo, adagio, moderato, presto



Reviewing skills1 p = 100° (Angles about a point add up to 360°)

q = 142° (Angles on a straight line add up to 180°)r = 33° (A right angle is 90°)s = 45° (Angles on a straight line add up to 180°)t = 118° (Angles about a point add up to 360°)u = 73° (Angles on a straight line add up to 180°)v = 230° (Angles about a point add up to 360°)

2 a a = 130°, b = 50°, c = 130°b p = 60°, q = 60°, r = 120°


Skills practice A1 Dominoes c and e have rotational symmetry.2 a Order 2

b Order 3c Order 2d Order 2e Order 1f Order 2g Order 1h Order 1

3 a Order 1b Order 2c Order 2d Order 1e Order 6

4 a i Square ii Order 4 iii

b i Equilateral triangle ii Order 3 iii

c i Regular octagon ii Order 8 iii

d i Regular pentagon ii Order 5 iii

5 a i Order 5 ii

b i Order 4 ii

c i Order 3 ii

6 Sometimes the order of rotational symmetry is the same as the number of lines of reflection symmetry, for example in the case of an equilateral triangle. However, it is possible to have rotation symmetry of order 3 and, say, no lines of symmetry – for example in this shape below.



Strand 2 Properties of shapes Unit 4 Rotational symmetry Band d

Skills practice B1 a No refl ection symmetry, rotational symmetry of

order 2b No refl ection symmetry, rotational symmetry of

order 3c 4 lines of refl ection symmetry, rotational

symmetry of order 4d 1 line of refl ection symmetry, no rotational

symmetry2 a i

ii Order 4b i

ii Order 4c i

ii Order 2d i

ii Order 2

e i

ii Order 43 a i Order 3

ii Order 8 iii Order 6 iv Order 9b i

ii

iii

iv

c The centre of rotational symmetry is where any lines of symmetry cross.

4 a i Other patterns are possible.

ii 2 lines of symmetry, rotational symmetry of order 2



b i Other patterns are possible.

ii 4 lines of symmetry, rotational symmetry of order 4

5 Own answers



Wider skills practice1 a 1 line of symmetry, no rotational symmetry

b i

A

B

1 line of symmetry, no rotational symmetry

ii

A

B C

1 line of symmetry, no rotational symmetry

iii

A

B

D

C

4 lines of symmetry, rotational symmetry of order 4



Applying skills1 Own drawings



Reviewing skills1 a Order 2

b Order 3c Order 1d Order 5

2 a Refl ection symmetry in vertical line, no rotational symmetry

b Refl ection symmetry in horizontal line, no rotational symmetry

c Refl ection symmetry in horizontal and vertical lines, rotational symmetry of order 2

d No symmetrye Refl ection symmetry in horizontal, vertical and

diagonal lines, rotational symmetry of order 4f No refl ection symmetry, rotational symmetry of

order 3


Note that there may be alternative ways of finding angles so students’ reasons may vary from those given here.

Skills practice A1 a = 100°, b = 42°, c = 71°, d = 20°, e = 1° 2 a = 159°, b = 90°, c = 129°, d = 200°, e = 88°3 a = 60°, b = 70°, c = 45°, d = 61.5°, e = 104°4 a a = 45°

b b = 65°c b + 70° = 135° = the exterior angle of triangle

5 a = 27°, b = 153°, c = 148°, d = 97°, e = 147°, f = 265°6 p = 12°, q = 168°, r = 12°, s = 168°7 15°8 70°9 Sometimes true; an equilateral triangle has three acute

angles, but you could have an obtuse-angled triangle with, say, angles of 40°, 40° and 100°.



Strand 2 Properties of shapes Unit 5 Angles in triangles and quadrilaterals Band e

Skills practice B1 Between 15° and 25°2 x = 125°

You have to assume that the walls are vertical and the roof is symmetrical.

3 a p = 50°b s = 65°

4 x = 80°5 106°6 a 180°

b 180°c (a + b + c) + (p + q + r) = 180° + 180° = 360°

So a + (b + p) + q + (r + c) = 360°That is, the sum of the interior angles of a quadrilateral is 360°.

7 25° → a73° → e46° → q53° → s64° → r85° → u

The code word is square.8 x = 107°9 a = 23°, b = 67°



Wider skills practice1 a–c Own diagrams

d p + q = ce AXCZ has rotational symmetry of order 2.

So triangles AXC and CZA are congruent.

A

a

p

X Cp

a Z

f a + p = 90°g ZCYB has rotational symmetry of order 2.

So triangles BZC and CYB are congruent.

B

Yb

q

C

b

q

Z

b + q = 90°h a + p + b + q = 90° + 90° = 180°

a + b + p + q = 180°a + b + c = 180°

2 a Own diagrams b 360°c 540°d i 40 triangles ii 7200°



Applying skills1

30º 30º

30º30º

60º

60º

60º60º

60º 60º

60º

60º

60º60º

60º 60º

60º

60º

120º 120º 120º 120º



Reviewing skills1 a = 35°, b = 55°, c = 53°2 d = 69°, e = 76°, f = 104°, g = 60°, h = 120°3 p = 50°, q = 133°, r = 80°, s = 133°


Skills practice A1 Shape How many in pattern?

Right-angled triangle 6

Other isosceles triangle 4

Square 1

Other parallelogram 2

Other trapezium 1

Kite 2

2 Name of shape How many pairs of parallel sides?

How many pairs of equal sides?

Rectangle Two pairs Two pairs

Square Two pairs Two pairs with same length (All sides equal)

Parallelogram Two pairs Two pairs

Rhombus Two pairs Two pairs with same length

Trapezium One pair None (unless isosceles)

Kite None Two pairs

Arrowhead None Two pairs

3 Square, rectangle, (rhombus or parallelogram)4 Square, rectangle, rhombus or parallelogram5 Own diagrams6 a Square, rectangle, trapezium, (parallelogram,

rhombus)b Arrowhead



Strand 2 Properties of shapes Unit 6 Types of quadrilateral Band e

7 a Trueb False. For example, this trapezium has a right

angle and is not a rectangle:

c Trued False. A rhombus has four equal sides and two

pairs of equal angles. Every rhombus is a kite, but not every kite is a rhombus. For example, here is a kite which is not a rhombus:

e False. For example, this arrowhead has an obtuse angle:



Skills practice B1 Own diagram

a (1, 4)b (3, 2)c (1, 2)

2 Square – HOSMTrapezium – ACJZRhombus – QNUXRectangle – FGWVKite – DEKRParallelogram – BPYL

The letter T is not used.3 ABC – right angled scalene triangle

DEFG – rectangleLMN – equilateral trianglePQRS – irregular quadrilateralABCDE – irregular pentagonXYZ – isosceles triangleJKLM – rectangleDEF – obtuse-angled scalene trianglePQRS – parallelogramTUVW – rhombus

4 Note that there are several possible answers to each part of this question. a Extend lines DE, FG, AH and BC.b Join CE, EG, GA and AC.c Join CD, DE, EF and FC.d Join CD, DE, EH and HC.e Join CH and EH; draw a dotted line DH and mark

a point X on the line so that XH < XD. Join CX and EX.

f Join AB, BE, EF and FA.g Extend lines CD, DE, GH and AH.h Join AB, BC, CE and EA.

5 a Rectangle, squareb Square, rhombusc Rectangle, square, rhombus, parallelogram,

trapeziumd Rectangle, square, rhombus, parallelograme Kite, arrowhead

6 Samir is right: a square is a special rectangle as a rectangle is any quadrilateral which has four right angles.Angus is right: a rectangle is a special parallelogram as a parallelogram is a quadrilateral with two pairs of parallel sides.



Wider skills practice1 Name of shape How many lines of symmetry?

Rectangle 2

Square 4

Parallelogram None

Rhombus 2

Trapezium None

Kite 1

Arrowhead 1

2 a

b To distinguish between: • a kite and an arrowhead, an arrowhead has a

refl ex angle • a rhombus and a parallelogram, a rhombus has

four equal sides • a square and a rectangle, a square has four

equal sides.

AnglesParallel sides

None equal One pair equal Two pairs equal All angles equal

None KiteArrowhead

One pair parallel Trapezium

Two pairs parallel RhombusParallelogram

SquareRectangle



Applying skills1 Own diagram2 Own diagram



Reviewing skills1 a Rectangle

b Squarec Trapeziumd Parallelograme Kitef Rhombus

2 a (3, 3)b (4, 3)c (3, 2)


Skills practice ANote that there may be alternative ways of finding angles so students’ reasons may vary.

1 a a = 70°, b = 125°, c = 64°, d = 64°b e = 109°, f = 109°, g = 115°, h = 115°, i = 115°,

j = 65°, k = 120°, l = 120°, m = 60°c n = 99°, o = 81°, p = 81°, q = 99°, r = 66°,

s = 114°, t = 66°, u = 114°, v = 56°, w = 56°2 a = 125°, b = 55°, c = 55°3 a 67°

b 23°4

40º40º

40º40º

140º

140º

140º

140º

Alice is right.



Strand 2 Properties of shapes Unit 7 Angles and parallel lines Band f

Skills practice B1 a = 70° (Angles at a point add up to 360°)

b = 49° (Vertically opposite angles are equal)c = 70° (Angles on a straight line add up to 180°)d = 73° (Angles in a triangle add up to 180°)e = 64° (Angles on a straight line add up to 180°)f = 65° (Alternate angles are equal)g = 39° (Corresponding angles are equal)h = 141° (Angles on a straight line add up to 180°)i = 45° (Angles at a point add up to 360°)j = 133° (Supplementary angles add up to 180°)k = 47° (Corresponding angles are equal or opposite angles of a parallelogram are equal)l = 133° (Corresponding angles are equal or opposite angles of a parallelogram are equal)m = 72° (Exterior angle equals the sum of the opposite two interior angles; isosceles triangles; alternate angles are equal)n = 94° (Supplementary angles add up to 180°)p = 62° (Supplementary angles add up to 180°)

2 a a = 119°, b = 61°, c = 119°, d = 61°b 360°

3 a a = 94° (Angles on a straight line add up to 180°)b = 33° (Vertically opposite angles are equal)c = 33° (Corresponding angles are equal)d = 33° (Vertically opposite angles are equal or Corresponding angles are equal)e = 53° (Corresponding angles are equal)f = 94° (Vertically opposite angles are equal or Angles on a straight line add up to 180°)

b 180°4 g = 20° (Corresponding angles are equal)

h = 42° (Alternate angles are equal)j = 110° (Corresponding angles are equal)k = 33° (Alternate angles are equal)l = 37° (Angles on a straight line add up to 180° or Angles in a triangle add up to 180°)m = 60° (Corresponding angles are equal)n = 61° (Alternate angles are equal)p = 25° (Corresponding angles are equal)q = 30° (Alternate angles are equal)r = 55° (Corresponding angle to P + Q)

5 In the fi rst diagram, Sam is correct. Angle ABC = angle ADE since corresponding angles are equal. Angle ACB = angle AED since corresponding angles are equal. Hence, all the angles are the same.

In the second diagram, Sam is correct. Angle BAC = angle EAD since vertically opposite angles are equal and angle ACB = angle AED as alternate angles are equal. Hence, all the angles are the same.

6 a = 80° b = 100°



Wider skills practice1 a, b Own diagrams

c Because PSTR is a rectangle with two pairs of parallel sides.

d Angle PQS = xe Angle RQT = zf Angles along a straight line add up to 180°.

So x + y + z = 180°So the angles in a triangle add up to 180°.

2

A

B C

Exx

D

a In triangles ABD and ACD AD is common to both AB = DC Angle BAD = Angle CDA (trapezium is isosceles) Hence triangle ABD is congruent to triangle ACD

(two sides + included angle) Hence angle ABD = angle ACD And angle AEB = angle DEC (vertically opposite) Hence angle BAE = angle CDE (sum of angles in

a triangle) b There is nothing different about the triangles

AEB and CED. c Yes the triangles are congruent.

A

B C

D

10°

80°90° 90°

30°

60°



Applying skills1 Own scale drawing



Reviewing skills1 a = 100° (Corresponding angles are equal)

b = 123° (Opposite angles are equal)c = 72° (Alternate angles are equal)

2 a w = 136° (Vertically opposite angles are equal)x = 44° (Supplementary angles add up to 180°)y = 136° (Supplementary angles add up to 180°)z = 44° (Supplementary angles add up to 180°)

b 360°3 a = 70°, b = 110°, c = 49°, d = 49°, e = 126°, f = 54°,

g = 54°, h = 75°, i = 105°, j = 75°, k = 56°, l = 124°, m = 63°, n = 63°, o = 63°


Skills practice A1 a 72°

b 108°2 a 60°

b 120°3 a = 72°, b = 71°, c = 102°, d = 100°, e = 60° 4 a 900°

b 360°c 900°−360°=540°

5 a Own diagramb 360°

The same is true of all polygons.



Strand 2 Properties of shapes Unit 8 Angles in a polygon Band g

Skills practice B1 a 1440°

b 360°c 1080°d An n sided polygon can be split into n triangles in

the same manner.The sum of the angles in the triangles is n × 180°.The angles at the centre add up to 360°.Hence the interior angles of the polygon add up to n×180°−360°.

e n×180°−360°=n×180°−2×180° = (n−2)× 180°

f 135°2 a 6 triangles

b 1080°c 1080°d 135°

3 a 360°b 18 sidesc 160°d 2880°

4 a 720°b 1260°c 3780°d 360°e 180°f 89 820°

5 a In a regular shape, all the angles are equal.b 111°c Corresponding angles are equal.d 900°e 99.5°

6 a 150°b 30°c 30°d AngleDAJ=angleBAL−angleBAD−angleLAJ

=150°−30°−30° = 90°

By symmetry, ADGJ is a square.

7 a Own tessellationb i Square ii Own tessellation

8 Number of sides n

Name Total of interior angles180(n − 2)°

Each interior angle180(n − 2)°/n

3 Equilateral triangle

180° 60°

4 Square 360° 90°

5 Pentagon 540° 108°

6 Hexagon 720° 120°

8 Octagon 1080° 135°

9 Enneagon(ornonagon)

1260° 140°

10 Decagon 1440° 144°

12 Dodecagon 1800° 150°

20 Icosagon 3240° 162°



Wider skills practice1 Angles in a triangle add up to 180°.

SoangleACB=180°−(angleBAC+angleABC) Angles along a straight line add up to 180°.

SoangleBCD=180°−angleACB =180°−{180°−(angleBAC+

angleABC)} =angleBAC+angleABC as required.2 a Own diagrams

b If all six angles of a hexagon were 90° then the anglesumwouldbe540°.However, the angle sum of a hexagon is 720° so a hexagon cannot have six right angles.



Applying skills1 45°2 20° is the exterior angle associated with the interior

angle of 160°.20° is the angle by which the line of the pipe is changed.



Reviewing skills1 x=47°,y = 30°2 177°3 10 sides


Skills practice A1 a Rectangle Length (cm) Width (cm) Area (cm2)

A 3 2 6

B 2 1 2

C 6 1 6

D 4 2 8

E 3 3 9

F 5 1 5

b The area should be length × width.2 a i 5 cm2

ii 10 cm2

iii 13 cm2

iv 18 cm2

b Own diagrams3 a The right-hand shape has an area of 12, the left-hand one 10. b The right-hand shape has an area of 15, the left-hand one 13.4 a 6 cm2

b 9 cm2

c 23 cm2

d 36 cm2

e Yes f Own answer5 a 12 square units b 8 square units c 9 square units d 6 square units e 10 square units6 A = 28 cm2, B = 16 cm2

7 a A = 4 cm2, B = 5.5 cm2, C = 3.5 cm2, D = 3.5 cm2, E = 7 cm2, F = 2.5 cm2, G = 5 cm2

b E, B, G, A, C and D, F8 a 4 square units b 6 square units c 10 square units d 7.5 square units e 7.5 square units f 10.5 square units g 4.5 square units h 18 square units i 14 square units



Strand 3 Measuring shapes Unit 1 Understanding area Band d

Skills practice B1 a 33 whole squares b 11 half squares c 38.5 cm2 2 a 2100 m2

b 1500 m2

c 22 m2

3 A = 6.5 cm2, B = 5.5 cm2, C = 12 cm2, D = 9 cm2, E = 6 cm2, F = 8.5 cm2

4 a 384 m2

b 120 m2

c 40 m by 12 m d 480 m2

e 984 m2

5 a 84 cm2

b 15 cm2

c 126 cm2

d 9 cm2

6 a 32 m2

b 103 m2



Wider skills practice1 a 21 square units b 20 square units c 33 square units d 30 square units



Applying skills1 a 315 cm2

b 60 sides c 18 900 cm2

2 Own answer



Reviewing skills1 a The right-hand shape has an area of 24,

the left-hand one 25. b The right-hand shape has an area of 18,

the left-hand one 15.2 a i 7 cm, 2 cm ii 14 cm2

b i 5 cm, 3 cm ii 15 cm2

3 5 cm2


Skills practice A1 a i 5 cm, 4 cm ii 20 cm2

b i 6 cm, 2 cm ii 12 cm2

2 700 m3 28 pieces4 a 24 cm2

b 72 cm2

c 96 cm2

5 a 150 m2

b 12 m2

c 138 m2

6 a 38 cm2

b 48 m2

7 a 62 cm2

b 54 cm2

c 65 cm2

8 34 m9 a 300 m2

b 500 mm2

c 1000 cm2

d 600 cm2 or 0.06m2

e 1500 m2

f 1050 mm2

10 12 m2

11 a 1690 mm2

b 10.58 km2

c 40 m2

d 120 m2

e 38 m2

12 a 60 cm2

b 30 cm2

13 24 cm2



Strand 3 Measuring shapes Unit 2 Finding area and perimeter Band e

Skills practice B1 a 35 cm2

b 72 cm2

c 48 cm2

d 1244 cm2

e 110 cm2

f 71 cm2

2 a 55.5 m2

b 74 bags3 34 cm2

4 Accept answers within ± 10% of answers below a 3.1 cm2

b 1.4 cm2

c 3.7 cm2

d 2.16 cm2

e 5.8 cm2

f 2.6 cm2

g 2.7 cm2

h 4.2 cm2

5 a 9.99 m2

b 12.71 m2

c 16.12 m2

d 10.21 m2

e 4.91 m2

6 a 44.5 cm2

b 43.1 cm2

7 2.12 m2

8 a 10 cm b 100 cm2

9 5 tins

10 a Rectangles b 240 cm2, 480 cm2

c 2100 cm2

d 5160 cm2

e 4 m2

11 a 40 cm2

b 6.08 m2

c A = b × h12 7 cm13 AD = BC = 16 cm

14 a Perimeter = 2 × (8 + 7) = 30 m b 3 m, 3 m c 30 m d, e Room Missing lengths Perimeter

Spare 2.5 m, 3 m, 6.5 m 23 m

Bathroom 3 m, 4.5 m 15 m

Kim’s room none missing 20 m

Mum + Dad’s room

5 m 23 m

Landing 1 m, 2 m, 3 m, 4 m, 5 m

16 m

Mum’s offi ce none missing 18 m

f 4 m



Wider skills practice1 a £22.56 b 2 packets

2 24 square units

3 a 3 m2

b 100 cm × 300 cm c 30 000 cm2

d 10 000 cm2

4 Own answer



Applying skills1 4 tins2 a 14.7 m b 2 rolls c Assuming the door is painted also the answer

is 25.44 m2

3 16 m2



Reviewing skills1 a 143 cm2

b 23 cm2

c 19.08 m2

d 4.5 square miles e 8.4 cm2

f 25 cm2

2 a Perimeter = 24.6 m, area = 23 m2

b Perimeter = 25.6 cm, area = 34.48 cm2

c Perimeter = 24 cm, area = 20 cm2

3 a 39.165 m2 or 39.2 m2

b £105.75 or £105.84


Skills practice A1 a 314.2 cm b 12.6 cm c 100.5 cm d 125.7 cm e 157.1 cm f 10.1 cm2 a 31.4 cm b 6.28 cm c 15.7 cm3 a 25.1 cm b 74.1 cm c 23.9 cm4 a 2 m b 1 m5 Radius Diameter Circumference (to 2 d.p.)

4 cm 8 cm 25.12 cm

6 cm 12 cm 37.68 cm

3.5 cm 7 cm 21.98 cm

7.5 cm 15 cm 47.10 cm

12.5 cm 25 cm 78.50 cm

21.6 cm 43.2 cm 135.65 cm

0.20 m 0.40 m 1.26 m

5 km 10 km 31.40 km

6 a 50.3 cm b 603.2 cm c 31.4 cm d 2.6 cm7 a Square 4 cm, hexagon 2 cm b 12 (or 6 × 2), 16 (or 4 × 4) c �The�circumference�of�the�circle�is�4π,�so�12�<�4π�<�16.

Dividing�by�4,�3�<�π�<�4.



Strand 3 Measuring shapes Unit 3 Circumference Band g

Skills practice B1 a 188.5 cm b 10.6 times (or 11 times)2 a 25.1 cm b 69.1 cm c 94.2 cm3 a 175.9 inches b 37.7 inches c 4.7 times d 37.7 inches4 3141.6 cm or 31.4 m5 a 157.1 m b 31.8 turns (or 32 turns)6 63.7 m7 a 157.1 cm b 30.6 times (or 31 times) c 38.2 times (or 38 times or 39 times)8 a 56.5 cm b 3.14 cm c 14.1 cm and 0.79 cm9 47 123.9 km10 19.46 m11 a 7 m b 21.99 m (or 22 m) c 15 lengths d £44.8512 a Own answer, e.g.

12�÷�π�=�3.8197…Dividing�by�2,�radius�=�1.91�m,�which�is�greater�than 1.8 m.

b 10 cm or 11 cm



Wider skills practice1 a 502.65 cm b i 50 265.5 cm ii 502.655 m iii 0.502 655 km2 a He is correct to 3 d.p. or 4 s.f. b i 3.142 857 (recurring) ii A recurring decimal iii 2 d.p. c i 3.162 278 ii� The�value�is�the�same�as�π, correct to 1 d.p.



Applying skills1� Yes,�because�the�circumference�of�table�is�31.415…�m�

and�the�20�politicians�need�20�×�1.5�=�30�m.�

2 4.7 cm

3 a 23.6 cm b 236 cm2



Reviewing skills1 a 15.7 cm b 376.99 m2 a 282.7 cm b 204.2 cm c 157.1 cm3 31.8 cm


Skills practice A1 a 50.3 cm2

b 254.5 cm2

c 201.1 mm2

d 113.1 cm2

2 a 38.5 cm2

b 95.0 m2

c 346.4 cm2

d 0.5 m2

3 a 314.2 cm2

b 78.5 cm2

c 21.2 m2

d 3.14 km2

e 706.9 square feet4 Radius Diameter Area (to 2.d.p.)

4 cm 8 cm 50.24 cm2

6 cm 12 cm 113.04 cm2

2 cm 4 cm 12.56 m2

9 inches 18 inches 254.34 square inches

20 cm 40 cm 1256.00 cm2

2.7 cm 5.4 cm 22.89 cm2

17 mm 34 mm 907.46 mm2

5 a Area = 50.3 square feet, perimeter = 28.6 feet b Area = 76.97 cm2, perimeter = 36.0 cm c Area = 88.3 km2, perimeter = 38.8 km6 a 100 b 314.2 cm2

7 78.5 cm2

8 Radius = 12 m, area = 452.4 m2

9 100.5 cm2



Strand 3 Measuring shapes Unit 4 Area of circles Band g

Skills practice B1 a Area = 28.3 m2, perimeter = 24.8 m b Area = 30.9 square inches, perimeter = 42.8 inches c Area = 503.4 m2, perimeter = 97.1 m d Area = 54.98 square feet, perimeter = 31.99 feet2 36.3 square feet3 a 706.9 cm2

b 353.4 cm2

c 176.7 cm2

4 a 22.48 m2

b 9.19 m2

5 a 25 446.9 cm2

b 314.2 cm2

c 28.3 cm2

d 1369.7 cm2

e 24 077.2 cm2

6 a 452.4 cm2

b 615.8 cm2

c 163.4 cm2

d 102.8 cm2

e 349.6 cm2

7 613 dancers8 a 27.2 m2

b 4 tins



Wider skills practice1 a 706.9 cm2

b 254.5 cm2

c 5770.5 cm2

2 31.04 laps (or 31 laps or 32 laps)3 a Area = 452.4 cm2

b Area = 490.9 cm2

c Area = 465 cm2

d Area = 508.9 cm2

e Area = 484 cm2

f Area = 480 cm2

So the order is a, c, f, e, b, d.4 21.5%



Applying skills1 At least £8667, as the turf can be cut to fi t.2 a 80 m2

b 7.07 m2

c 1.77 m2

d 0.35 m2

e 176.7 cm2 (or 0.02 m2) f 3.80 m2

g 5.11 m2

h 61.89 m2

3 a 4 people b 6 people c 49.5 inches



Reviewing skills1 a 113.1 cm2

b 314.2 square feet c 50.3 cm2

2 17.87 m2 3 Perimeter = 28.3 cm, area = 42.4 cm2


Skills practice A1 601, 24.52 a 17 cm b 29 m c 11.2 m d 20.8 cm3 a 35 cm b 25 cm c 87 cm d 123 cm e 97 cm4 a a = 7.5 cm b b = 13.3 m c c = 13.3 cm d d = 58.8 m e e = 0.7 mm f f = 8.5 cm5 24.7 cm6 a 14.4 yards b 3.5 miles c 146 cm d 11.3 cm e 3000 m f 0.604 m7 40 cm8 a, b Own drawing c 5 cm d–f Own drawings g Shorter sides Hypotenuse Sum of areas of smaller

squaresArea of largest square

AC CB AB

3 cm 4 cm 5 cm 9 cm2 + 16 cm2 = 25 cm2 25 cm2

5 cm 12 cm 13 cm 25 cm2 + 144 cm2 = 169 cm2 169 cm2

8 cm 6 cm 10 cm 64 cm2 + 36 cm2 = 100 cm2 100 cm2



Strand 3 Measuring shapes Unit 5 Pythagoras’ theorem Band g

Skills practice B1 282 = 784, not 3600.

h = 96 cm2 AC is the hypotenuse so a2 + 172 = 222

a = 14.0 cm3 48.2 km (or 48.3 km)4 519.6 miles5 a 27.5 mm b 330.0 mm2

6 a 135° b right angled, isosceles c 141.4 km d 141.4 km7 7.5 cm8 720 m2



Wider skills practice1 a 21, 72, 75

20, 48, 529, 40, 4160, 63, 873, 4, 58, 15, 17

b Own answers; note that any multiple of a Pythagorean triple is also a triple, and that 5, 12, 13 is a triple.

2 a Right angle

a b l Compare … Is x … ?

6 8 10 = Right angle

6 8 12 > Obtuse

6 8 9 < Acute

b, c Own answers d If l2 > a2 + b2 then x is obtuse and if l2 < a2 + b2

then x is acute.



Applying skills1 1.8 m 2 a 21 m b 132 + 202 = 23.92 and not 212.



Reviewing skills1 a a = 7.2 cm b b = 8.9 cm c c = 18.3 cm d d = 14.4 cm2 94.3 miles


Skills practice AFor questions involving measuring or drawing, angles should be accurate to within ±1°.

1 a 2 right angles b 90 degrees c 360 degrees d 3 right angles e 4 right angles2 Angle Type Measurement

a Obtuse 93°

b Acute 17°

c Obtuse 110°

d Reflex 326°

e Obtuse 92°

f Acute 54°

g Straight line 180°

h Acute 13°

i Full turn 360°

j Reflex 232°

3 Angle Size

a About 90°–95°

b About 60°–65°

c About 10°–15°

d About 130°–135°

e About 260°–265°

f About 80°–85°

g About 30°–35°

h About 320°–325°

4 a = 30°, b = 25°, c = 45°5 Own accurate diagram6 a = 132°, b = 48°7 a a = 55°, b = 130°, c = 73° b Own accurate diagrams8 a i 153°

ii, iii Own accurate diagram b Own accurate diagrams

i 360−200=160 ii 360−243=117iii 360−300=60 iv 360−270=90v 360−181=179 vi 360−315=45

9 Own accurate diagrams10 a 90°, 60°, 30° b a about 15°–25°, b about 80°–88°, c about 50°–60°, d about 60°–70°



Strand 4 Geometric construction Unit 1 Angles in degrees Band d

Skills practice B1 9:00 a.m. and 90°

4:00 p.m. and 120°11:00 p.m. and 30°

2 A and H, B and F, C and J, D and G, E and I

3 a 45° b 90°

c 135° d 180°

4 a Own accurate diagram b i

130o

50o

ii

50o

310o

iii

50o230o

5 Angle Estimate Measurement

a 90°–100° 100°

b 55°–65° 61°

c 20°–25° 23°

d 130°–140° 134°

e 30°–40° 38°

f 80°–90° 84°

g 310°–320° 317°

h 245°–255° 250°

6 360°, 180°, 90°, 45°, 22.5°7 a a = 45°, b = 45°, c = 135°, d = 225° b a = b, c=180°−a, d=360°−c8 a a = b = c = 120° b 3 × 120° = 360° so there are no gaps.9 Own answers10 Own answers

11 Turn Estimate Clockwise or anticlockwise Measure

A to B 30°–45° Anticlockwise 38°

B to C 75°–85° Clockwise 82°

C to D 35°–45° Clockwise 42°

D to E 140°–150° Anticlockwise 146°

E to F 40°–50° Anticlockwise 47°

F to G 110°–120° Anticlockwise 117°

G to H 72°–82° Anticlockwise 77°



Wider skills practice1 a i a = 37°, b = 71°, c = 72°

ii p = 78°, q = 45°, r = 105°, s = 132° b i a + b + c = 180°

The angles make a straight line.ii p + q + r + s = 360°

The angles make a full turn.2 The three angles should add to 180°.



Applying skills1 a ii reflex

iii obtuseiv straight linev acutevi straight line

b Own answers, e.g.i 360° = 280° + 80°ii 360° = 110° + 250°iii 360° = 180° + 50° + 130°iv 360° = 90° + 90° + 180°v 360° = 90° + 180° + 30° + 60°vi 360° = 180° + 180°

2 Own answers



Reviewing skills1 a f, b, e, g, a, d, c b, c

Angle Type Measurement

a Obtuse 156°

b Reflex 319°

c Acute 36°

d Right angle 90°

e Reflex 219°

f Reflex 355°

g Straight line 180°

2 a = 30°, b = 90°, c = 135°, d = 210°, e = 330°3 3°, 53°, a right angle, an obtuse angle, a straight line,

240°, 300°4 Own accurate diagrams


Skills practice A1 Own accurate diagrams

Missing angle(s) Missing side(s)

a 60°, 60° 5 cm

b 110° 4.9 cm, 4.9 cm

c 40° 5.8 cm, 5.8 cm

d 82° 5.7 cm, 2.8 cm

e 55° 9.8 cm, 5.6 cm

2 Own accurate diagramsMissing angle(s) Missing side(s)

a 30° 5.1 cm, 7.5 cm

b 77.5°, 77.5° 2.6 cm

c 30°, 30° 5.2 cm

d 75° 5.3 cm, 7.5 cm

e 40° 5.4 cm, 3.5 cm

3 a Own accurate diagram b 4.2 cm, 4.2 cm

It is an isosceles triangle.



Strand 4 Geometric construction Unit 2 Constructions with a ruler and protractor Band e

Skills practice B1 a Own accurate scale drawing b 4.5 m2 a Own accurate scale drawing b 10 km3 a No; although all triangles with base angles

of 40° and 60° will have the third angle 80°, the length of the base is not given, so the triangles are likely to be different sizes.

b In this case the triangles should all be the same.4 Own accurate diagrams a Equilateral triangle b Isosceles triangle c Right-angled triangle d Right-angled isosceles triangle5 Own accurate diagram a 3.8 cm b i 135°

ii 135°iii 90°

c 30.6 cm



Wider skills practice1 a Own accurate diagrams b 36.9° and 53.1° in both triangles c 5 cm and 10 cm

The length in the second triangle is twice the fi rst.



Applying skills1 a Own accurate scale drawings b 135.8 cm



Reviewing skills1 a Own accurate diagrams b i 36.9°

ii 5 cmiii 7.1 cm

2 a Own accurate diagrams b Own answers c All three lengths are equal.

It is an equilateral triangle.


Skills practice A1 Own accurate diagrams 2 Own accurate diagrams3 61.3°, 46.9°, 71.8°4 Own accurate diagram

All three bisectors intersect at one point.5 a, b Own accurate diagram c 60° and 30° d Own accurate diagram6 a Own accurate diagrams

i Equilateral triangleii Isosceles triangle

b It is not a triangle, because the two short sides do not meet.7 Own accurate diagram 8 a, b Own accurate diagram c Rhombus9 a, b Own accurate diagram c The perpendicular bisector of AC passes through B.



Strand 4 Geometric construction Unit 3 Constructions with a pair of compasses Band f

Skills practice B1 Own accurate diagram 2 a, b Own accurate diagram c QS = 6 cm, PS = 8 cm3 Own accurate diagram4 Own accurate diagram5 Own accurate diagram6 a Own accurate diagram b The circle goes through points B and C.7 a, b Own accurate scale drawing c 16.5 m d 13.8 m8 Own accurate diagram9 Own accurate diagrams



Wider skills practice1 a, b Own accurate diagrams c 125.1°, 24.1°, 30.8°

The angles are the same because the triangles are similar.2 Own accurate diagram



Applying skills1 a, b Own accurate diagram c Square d, e Own accurate diagram f Regular octagon2 Own accurate diagram



Reviewing skills1 a Own accurate diagram b Isosceles triangle c Own accurate diagram d BD = DC = 4 cm e BD and DC are the same length.2 Own accurate diagram


Skills practice A1 a Own sketch of locus b Own accurate diagram c Angle bisector2 a Own sketch of locus b Own accurate diagram c Perpendicular bisector3

6 cm3 cm

4 a

locus

b

locus

c

5 a

3 cm

3 cm

b

2 cm

4 cm



Strand 4 Geometric construction Unit 4 Loci Band g

Skills practice B1 a A series of rises and falls, gradually decreasing

in height until the ball eventually rolls along the ground

b A circle of radius the length of the minute hand c A line parallel to the road d The arc of a circle of radius the length of the

wiper arm2 a

45°22.5°67.5°

The locus is the bisector of the angle between the lines, and there are two angles between them.

b 90°3

4 Own accurate drawing, e.g.

A

B

5 a Own accurate drawing b The perpendicular bisector of AB and the

perpendicular bisector of CD c 60° d The centre is at the point where the locus

lines intersect.6 a, b

16 cm

7 cm16 cm

4 cmpost

tree

c No



Wider skills practice1 Own accurate drawing

The locus is the bisector of angle D; this is not the diagonal DB because the diagonal DB does not bisect the angle at D.

2 a Own accurate drawing b Two lines parallel to the original line and 3 cm

away from it, plus the semicircles at the two ends c Area = 70.2 cm2, perimeter = 32.8 cm3

–5–4–3–2–1

12345

–1 1 2 3 4x= −1

y= 3

5–2–3–4–50

4

patio

shed1 cm

2 cm 1cm

3 cm

6 cm

Tree can be planted in area shown in blue.



Applying skills1

5 m

12 m

5 m

5 m

10 m10 m 6 m

5 m9 m10 m16 m

house

plantingarea



Reviewing skills1 a

A

C

B

The point is where the three perpendicular bisectors meet.

b

A

B

C

The point is where the three angle bisectors meet.


Skills practice A1 a Row 2 b C2 c Kim d

Ki S

An Jo

HuCK J

AlJa

L

SoMe

MT H

Mn

PW Mi A

1A B C D E F G H I J K L M

23456789

Key: A Alan, Al Ali, An Andy, C Christina, H Harry, Hu Humza, Ja Jack, J John, Jo Jo, K Karl, Ki Kim, L Lucy, M Mercy, Me Megan, Mn Meena, Mi Michelle, P Pete, S Samir, So Sopie, T Tim, W Wayne

e Own diagram2 a Brown b Working on a laptop c A3, B1, B3, C2, C3, D3 and E2 d A2, D1 and E33 a F2 b A1, A2 and B2 c Cave d F3 and G3 e i Graves ii Swamp iii Haunted forest f A5 g F2, E2, E3, E4, F4, F5, F6, E6, D6, C6, B6,

B5, A54 a Trainers b Chair c A1 and A2 d D2 and D3

5 a i Meet me at the shops. ii Maths is cool! iii Come to my party. iv My dog is called Max. b Own answers6 a A(2, 4), B(5, 4), C(5, 1), D(2, 1) b E(1, 4), F(3, 5), G(5, 4), H(5, 2), I(3, 1), J(1, 2) c K(0, 4), L(4, 0), M(0, 0) d N(0, 3), O(2, 2), P(3, 2), Q(5, 3), R(5, 0), S(3, 1),

T(2, 1), U(0, 0)7 a–c

0

123456789

10

4 5 6 7 8 9 10321

y

x

A fox8 a

0 1 2 3 4 5

y

x

12345

× ×

× ×

b Parallelogram



Strand 5 Transformations Unit 1 Position and Cartesian co-ordinates Band b

Skills practice B1 a 1 Incorrect. The duck pond is in G7. The correct answer is that the parking area is in C1.

2 Correct3 Incorrect. The chicken shed is in J2. The correct answer is that the vegetable garden is in H4.4 Correct5 All the answers are wrong:

F7 is the duck pond, not the tractor.G2 is the farmhouse, not the tree.C7 is the trees, not the duck pond.D4 is the tractor, not the farmhouse.

b B4, C4, C5, D5, E5, F5, G5, G4, G32 a 2J is the grid reference, 67 is the page number. b H13 a 8 b D7 c Because the cell F4 is highlighted. d 63 ice creams e =SUM(E4:E8)4 a i Sports centre ii School iii Ice rink b i (6, 7) ii (8, 5) iii (1, 5) c School d She walks to the end of her road and turns right. She walks to (1, 0) and turns left. She walks to end of this road. e i Tim is wrong. ii You must go along fi rst then up.

5 A(0.25, 1), B(0.75, 0.75), C(1, 0.75), D(1.5, 1), E(1.5, 0.25), F(1, 0.5), G(0.75, 0.5), H(0.25, 0.25)

6 a i (0, 10) ii (20, 15) b i 15 km ii 20 km c i 15 km ii 20 km7 a No b Yes c (3, 3) Yes

(5, 3) Yes(4, 2) Yes(4, 4) No

d (0, 2), (1, 2), (2, 2)(4, 0), (4, 1), (4, 2)(1, 5), (2, 5), (3, 5)(3, 3), (4, 3),(5, 3)



Wider skills practice1 C22 a

0123456

4 5 6 7 8 9 10 11 12 13 14321

y

x

×

×

× ×

×

×

b Hexagon c Yes d No



Applying skills1 a 25 b 34 c A8 d Because cell D8 is highlighted. e It means ‘Find the sum of cells D1 to D6’. f 10 175 g B2, B6, D2, D5, D6, D8, F4 and F82 a i SJ67 ii SJ60 iii SJ08 iv SD33 v SO87 b The letters reference the larger square and the

numbers locate the smaller square within it. c 10 km d 100 small squares e The accuracy is limited to within 5 km east–west

and north–south. f Own grid reference



Reviewing skills1 a 12 seats b 9 seats c 4 rows d 3 seats e 4 seats f G1, G14, F1 and F13 g C1 and C2, D1 and D2, E1 and E2, C11 and

C12, D12 and D13, E13 and E14, H1 and H2, H3 and H4

2 a Kim b Humza c School d Wayne e North-West f Pete

3 A(2, 4), B(5, 4), C(4, 1), D(1, 1)E(2, 5), F(4, 5), G(5, 3), H(4, 1), I(2, 1), J(1, 3)

4 a

0 1 2 3 4 5 6

y

x

10

23456

× ×

××

b (1, 5)


Skills practice A1 a A(−1,3),B(2,4),C(−3,−4),D(1,−2) b C2 a L(−0.5,0.5),M(1,−0.5),N(0.5,−1) b P(−1.4,0.6),Q(0,1.8),R(1.2,0.4),S(−0.2,−0.8)3 A(1.6,1.4),B(−2.4,−1.2),C(−0.8,−1.2),

D(−0.8,−2.8)4 a A(−0.5,0.5),B(0.5,0.5),C(1.5,−0.5),

D(−1.5,−0.5) b P(–1.6,0.6),Q(0.6,0.6),R(–0.5,−0.6)5 a A(−2,3),B(4,3),C(4,−2) b (−2,−2) c 6units d 5units6 a

1

�1

�2

�3

�4

0

2

3

4

1�1 2 3 4 5 6

y

x

b (3,−4)7 A(−2,1),B(3,3),C(1,−2),D(1,1)8 A(5,3),B(4,1),C(6,−4),D(2,−1),E(−5,−5),

F(−8,−8),G(−6,−5),H(−10,−6),I(−6,−4),J(0,1),K(−3,5),L(2,2.5)

9 a U(5,0),V(0,−5),W(−12,0),X(−17,–2),Y(−17,2),Z(0,5)

b (2,2)and(2,−2)orother“symmetricpair”. Theanswerisnotunique.Othersolutionscould

be(3,1),(3,−1),(1,2),(1,−2)10 a (2,−2) b

0 2

2

4

6

8

10

−2−2

−4

−6

−8

−10

−4−6−8−10 4 6 8 10

A

B

C

D

E

x

y

c (−2,2),(−10,0),(−2,2)



Strand 5 Transformations Unit 2 Cartesian co-ordinates in four quadrants Band c

Skills practice B1 a, b

1

�1�2�3�4�5�6

0

234

1�1�2�3�4�5�6�7�8�9�10 2 3 4 5 6 7 8 9 10

y

x

c Akey2 a (1,−2)

–3 –2 –1–1–2–3–4–5

12345

y

x1 2

B

C

D

A

3 4 50–4–5

b (−1.5,3),(2,3.5),(2.5,0),(−1,–0.5)3 a Octagon b A(−6,2),B(−3,5),C(1,5),D(4,2),E(4,−2),F(1,−5),G(−3,−5),H(−6,−2)



4 a (−2,−4)

–3 –2 –1–1–2–3–4–5–6

12345

6y

x1 2

B C

D

EF

A3 4 5 60–4–5–6

b 2lines c No;no5 a (2,4) b i (1,3.5) ii (1,3.5)6 a Own copy of diagram b L(7,7),M(2,4),N(4,2) c–e Own diagram



Wider skills practice1 (5,−2)2 a Own diagram b i ThepointsformaverticallinethroughA. ii ThepointsformahorizontallinethroughB.



Applying skills1 a (−1,1),(2,1),(2,−1),(−1,−1)

No,thepointsformarectangle. b, c Own diagram



Reviewing skills1 a 3rdquadrant b 4thquadrant c 2ndquadrant d 1stquadrant

2 A(1,3),B(−1,1),C(−5,−3),D(3,0),E(0,6),F(0,−4),G(2,−3),H(5,−2),J(−3,−4),K(−6,5)

3 a–c Own diagram d Akite


Skills practice A1 a 4 squares right, no squares up or down b 3 squares right, 1 square up c 3 squares right, 1 square down d 4 squares left, 1 square down e 3 squares left, 1 square up f 1 square right, 1 square up2

x

y

–4–5

a

c

d

1

0 1 2 3 4 5 6

2345

–1–2–3–4–5–6–1–2–3

b


3

–4–5–6

AB

F1

0 1 2 3 4 5 6 7 8 9 10

234567

y

x–1–2–3–4–5–6–7–8–9–10–1–2–3

H

D

EG

C

a i 4 right, 4 up ii 5 right, 1 down iii 5 left, 5 up iv 4 right, 0 up or down v 0 left or right, 4 down vi 4 left, 0 up or down vii 4 left, 4 down b Part vi is the reverse of part iv as D to C is the reverse of C to D.

Part vii is the reverse of part i as B to C is the reverse of C to B.


Strand 5 Transformations Unit 3 Translation Band d

5 a Translation Vector

A → B8

1

A → C5

2−

A → H11

6−

C → B33

C → G−−

7

3

C → D7

1

E → D−

4

2

E → F−−

14

1

G → D14

4

G → A2

5

G → H13

1−

b A(−8, 1), B(0, 2), C(−3, −1), D(4, 0), E(8, −2), F(−6, −3), G(−10, −4), H(3, −5)6 a, b

–4–5–6–7–8–9

1

0 1 2 3 4 5 6 7 8 9

23456789

–1–2–3–4–5–6–7–8–9 –1–2–3

y

x

W

X

c 11 left, 2 down



7 a, c and d

–4–5–6–7–8–9

–10

1

0 1 2 3 4 5 6 7 8 9 10

23456789

10 A B

C

DE

F

–1–2–3–4–5–6–7–8–9–10–1–2–3

y

x

D

C

A B

E

F

B

C

DF

E

A

b 6 left, 6 down c B′(−1, 4), C′(0, 1), D′(−1, 10), E′(−3, −1), F′(−4, 1) d B″(1, −3), C″(2, −4), D″(2, −8), E″(−1, −8), F″(−2, −6) e 4 left, 13 down8 a–c, e

1

�1�2�3

�5�4

2

4

�1�2 3 41 2

5

50

3

�4�5

y

x

PTS

�3Q

R

d Translation by the vector 11−

f Translation by the vector 5

0



Skills practice B1 a i 1 right, 3 down

ii 7 right, 1 upiii 3 right, 2 upiv 1 left, 3 upv 3 left, 2 downvi 7 right, 1 upvii 3 left, 2 downviii 11 left, 0 up or down

b C to B and F to E2 a i 4 left, 1 up ii 3 right, 5 down iii 5 left, 1 down iv 0 left or right, 4 up b A, B, E, D, C3 a 4 left, 10 down b 4 right, 10 down c–d 0

14

12

10

8

6

4

2

0

2 4 6 8 10

2 4 6 8

3 2

10

14

e Own diagram4 a LIMES b Translate L by 1

4−

Translate I by 1

4−

Translate M by 1

4−

Translate E by 1

4−

Translate S by −−

44 c MILES

5 a–c

A

B

C

2

0

4

6

1

3

5

2 4 6 7 81 3 5

y

x

d Translation by the vector 5

2−

6 a i Translation by the vector 7

1

ii Translation by the vector 22

iii Translation by the vector 9

3

b 7

1

+ 22

= 9

3

7 a PA

� ��=

3

2

b AB BC CD

� �� =

=−

, , 3

3

7

1==

−−

, ,

1

4

6

1

11

0

DE

EF

� ��

� ��

−−

−

, , FG GH

� �� 8

7

0

2



Wider skills practice1 a 0

0

b Own sequence of three vectors2 a Own work b VECTOR CODES ARE EASY.



Applying skills1 −

−

−

−

1

0

1

1

1

2

1

3, , , ,

−

1

4

a 0

1

0

2

0

3

0

4

0

5

, , , ,

, , , , 0

6

0

7

0

8

0

9

1

0

1

1

1

2

1

3, , , , , , ,

1

4

1

5

1

6

1

7

2

0

, , , , 2

1

2

2

2

3

2

4

, , , , 2

5

2

6

2

7

2

8

3

0

3

1

3

2

3

3

3, , , ,

44

3

5

3

6

3

7

3

8

, , , ,

b 8 moves



Reviewing skills1 a i 5 left, 4 down ii 4 right, 0 up or down iii 1 left, 4 up iv 1 right, 4 down v 4 right, 1 up vi 1 right, 4 up b A and C are different shapes (not congruent).2

x

y

0–1–1–2–3

–2–3

12345678

1 2 3 4 5 6 7 8

A

B

C

D


Skills practice A1 a

Mirror line b

Mirror line

c

Mirror line

d

Mirror line

e

Mirror line f

Mirror line

2 a

b



Strand 5 Transformations Unit 4 Refl ection Band e

3 a–c

1 2 3�3 �2�1�1

1

0

23

�2�3

y

x

d A rhombus surrounded by squares

4 a Translation by the vector 7

0

b Translation by the vector 0

9−

c Refl ection in the line y = 1 d 4th quadrant5 a–c

–4 –3 –2 –1–1–2

–3

1

2

3

y

x1 2 3 40

P

S

Q

R

d The refl ection of shape S coincides with shape P.

6

7 a–e

23

1

2 31−2 −1−1−4 −3−6 −5−8 −7 4 5 6 7 80

45678

−8

−6−7

−4−5

−2−3

y

x

A

B C

D

e Triangle A is a refl ection of triangle D in the y axis.

8



Skills practice B1 a–c

–3 –2 –1–1

–2

–3

–4

1

2V W

3

4y

x1 2 3 40–4

c Triangle V is refl ection of triangle W in the y axis. d Two refl ections in the same mirror line always bring you back to your starting place.2 a–c

23

1

21−2 −1−4 −3−5 4 50

45

−5

−3−4

−1−2

y

x3

b The letter is back to front. c Although the letter is upside down, it is not back to front.3 a–c

–5 –4 –3 –2 –1–1–2–3–4–5

12345y

x1 2 3 4 50

b This image is still ‘the right way round’. c It is upside down but not back to front. d A is symmetrical but diagram of Y is not.



4 a (0, 0), (0, 5), (3, 5), (3, 4), (1, 4), (1, 3), (2, 3), (2, 2), (1, 2), (1, 1,), (3, 1), (3, 0) b Mirror line 2: x = 7.5

Mirror line 3: x = 11.5 Mirror line 4: x = 15.5 Mirror line 5: x = 19.5 Mirror line 6: x = 23.5

c–f

y

x

123456

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Mirr

or 1

Mirr

or 2

Mirr

or 3

Mirr

or 4

Mirr

or 5

Mirr

or 6

0 g When the letter E is refl ected in an odd-numbered mirror line the image is ‘back to front’; when the resulting

image is refl ected in an even-numbered mirror line the image is the right way round again.

5 a–c

x

y

–2–2–4–6–8–10

–4–6–8

–10

0

246

108

2 4 6 8 10

(7, 7)

d An octagon



6 a, d

M1 M2 M3 M4

A0 A1 A2 A3 A4

6B

4

2

0 2 4 6 8 10 12 14 16 18 20 x

y

b A refl ection in the line x = 8 c i x = 4 ii x = 8 e i A refl ection in the line y = 3, followed by a

refl ection in the line x = 4 ii A refl ection in the line y = 3, followed by a

refl ection in the line x = 87 a y = x b y = −x8 a Own copy of diagram b

c So that car drivers can read the word in their rear-view mirrors.



Wider skills practice1

–4–5–6–7–8–9

–10

1

0 1 2 3 4 5 6 7 8 9 10

23456789

10

–1–2–3–4–5–6–7–8–9–10–1–2–3

y

x

2 a y

ax0

b An isosceles triangle c The triangle would be equilateral. d The triangle would be a right-angled isosceles triangle.



3 a–e

3

30 6 9

–3

–6

6y

x

P

A C D

FEB12

f An irregular hexagon. The hexagon is not regular because the sides are not all the same length and the angles are not all equal. The hexagon is made from six identical right angled, isosceles triangles with side lengths 6, 4.24, 4.24 units.

g The approximate position is P(3, 5.2).



Applying skills1 a–d Own diagram e No it is not possible, because the original shape and the refl ected shape are not the same way round.

2 a Yes, translation by the vector −

8

0 followed by refl ection in m3

b Yes, refl ection in m1 followed by translation by the vector −

8

0

c Yes, refl ection in m2



Reviewing skills1 a–f

–4–5–6–7–8

1

0 1 2 3 4 5 6 7 8

A

BC

D

2345678

–1–2–3–4–5–6–7–8–1–2–3

y

x

d (1, −2), (3, −7), (6, −1) g Translation by the vector 5

1−

2 a The line y = x b The x axis c The line y = −x d The y axis


Skills practice A1 a ii b v c iii d i e vi f iv2 a, b

–3 –2 –1–1–2–3–4–5–6

12345

6

y

x1 2

AB

C

3 4 5 60–4–5–6

c Rotation through half a turn about the origin3 a Own diagram b A′(1, −2), B′(5, −2), C′(1, −4) c A″(−2, −1), B″(−2, −5), C″(−4, −1)4 a–d

–3 –2 –1–1–2–3–4–5–6

12345

6

y

x1 2

A

B

3 4 5 60–4–5–6

e The image is A, so the shape is mapped on to itself.



Strand 5 Transformations Unit 5 Rotation Band e

5 a–c

–3 –2 –1–1–2–3–4–5

12345

x

y

1 2

A

BC

3 4 50–4–5

b (2, −1), (5, −1), (5, −3), (2, −5) c (−1, −2), (−1, −5), (−3, −5), (−5, −2) d Rotation by a quarter turn clockwise about

the origin6 a Refl ection in the y axis b Refl ection in the x axis c Translation by 5 in the x direction and 5 in the

y direction d Rotation through 90° clockwise about the origin e Translation by −5 in the x direction and −3 in the

y direction f Rotation through 90° anticlockwise about

the origin g Refl ection in y = x h Rotation through 180° about the origin i Refl ection in y = 4 j Refl ection in x = 5

7 a i Rotation through 90° clockwise about the origin

ii Rotation through 90° anticlockwise about the origin

iii Rotation through 90° anticlockwise about the origin

b

–3 –2 –1–1–2–3–4

1234y

x1 2

A

D

3 40–4

c i Yes ii Shapes A and D are congruent because they

are exactly the same shape and size.8

c, d

a, b



Skills practice B1 a Rotation through 90° clockwise about the origin b (3, 3), (4, 3), (4, 4), (5, 4), (5, 5), (3, 5) c Emily is right, because shapes A and B are exactly the same shape and size.2 a–c

–3 –2 –1–1–2–3–4–5–6

12345

6y

x1 2

B

C

A

3 4 5 60–4–5–6

d Rotation through 180° about the origin3 a Own diagram b i Clockwise rotation through 30° about C

Anticlockwise rotation through 330° about C ii Clockwise rotation through 200° about C

Anticlockwise rotation through 160° about C iii Clockwise rotation through 240° about C

Anticlockwise rotation through 120° about C iv Clockwise rotation through 160° about C

Anticlockwise rotation through 200° about C4 a The Plough

Midnight 6 a.m.

(10, 12) (−12, 10)

(8, 11) (−11, 8)

(4, 13) (−13, 4)

(3, 15) (−15, 3)

(0, 12) (−12, 0)

(0, 14) (−14, 0)

b Cassiopeia

Midnight 6 a.m.

(−7, −9) (9, −7)

(−7, −11) (11, −7)

(−5, −12) (12, −5)

(−5, −14) (14, −5)

(−3, −14) (14, −3)



5 a–d

–3 –2 –1–1–2–3–4–5–6

12345

6y

x1 2 3 4 5 60–4–5–6

AE

D

C B

e Rotation through 90° clockwise about the origin6 a, b

–4–5–6–7–8

1

0 1 2

A

A’

D’’

C’’

B’’ A’’

B’

C’

D’

D

C

B3 4 5 6 7 8

2345678

–1–2–4–5–6–7–8–1–2–3

y

x–3

c 180° rotation: the co-ordinates are changed by changing their signs (or multiplying by −1).90° rotation: the x and y co-ordinates are interchanged, and the resulting y co-ordinate has its sign changed (or multiplied by −1).



7 a–e

–4–5–6–7–8

1

0 1 2 3 4 5 6 7 8

2345678

–1–2–3–4–5–6–7–8 –1–2–3

y

x

A

BC

D

f Rotation through 180° about the origin g D maps to A.8 a No b The centre, the direction and the angle of rotation c Rotation Angle Direction Centre of rotation

A to B 90° Clockwise (0, 0)

A to C 90° Anticlockwise (0, 0)

B to D 180° Either (4, −3)

C to E 90° Clockwise (−4, 1)

E to F 90° Anticlockwise (−4, −2)

D to G 90° Clockwise (7, −3)



Wider skills practice1 1 Hexagon 2 Order 3 Shape 4 Refl ection 5 Translation 6 Angles 7 Rotation 8 Construct The green squares give the word heptagon.2 a i Refl ection in the y axis or translation by the vector

3

0

ii Rotation through 180° about the origin iii Refl ection in the x axis iv Rotation through 180° about (3, 0)

b Translation by the vector 3

0

or refl ection in the y axis

Both transformations work as the shape has a vertical line of symmetry.3 a Translation by 6 right and 3 up b Translation by 6 right and 3 down c Refl ection in the y axis d Rotation through 180° about the origin e Refl ection in the x axis f Rotation through 90° anticlockwise about the origin g Rotation through 180° about (6, −3) h Translation by 16 left i Refl ection in the y axis j Refl ection in the line x = 1.5 k Refl ection in the line y = x l Translation by 4 down m Translation by 16 left and 4 up



Applying skills1 a

b

2 a It appears the same. b Any three multiples of 60° about C, e.g. 120°, 180°, 240° c Any two multiples of 60° about C, e.g. 60°, 120° d 6 lines of symmetry and rotational symmetry of order 63 a Translation by 1 km due West b Rotation through 180° about P c i Translation by 1 km due East followed by a rotation through 180° about Q ii Rotation through 180° about Q followed by a translation by 1 km due West4 a, b

–12

–16

0 4 8 12 16

4

8

12

16

–4–86 p.m. Pole Star

midnight

–16

–4

y

x

–8

–12

b The co-ordinates have all been multiplied by −1.



Reviewing skills1

b

a, c

2 A translation 6 units to the right


Skills practice AFor questions involving measuring or drawing, lengths should be accurate to within ±0.1 cm and angles should be accurate to within ±1°.

1 a The lengths are doubled. b The angles stay the same.2

1

1

0

23456789

10

a

b

–1–1–2–3–4

–2–3–4–5–6–7 2 3 4 5 6 7 8 9 10

y

x

3 i

PScale factor 3

y

x

ii

PScale factor 2

y

x



Strand 5 Transformations Unit 6 Enlargement Band f

iii

Scale factor 2

y

x

P

4 a

0 1

123456789

101112131415

2 3 4 5 6 7 8 9 10 11 12 13 14 15

A

B C

F

ED

D’

C’B’

E’

F’A’

x

y

b A′(6, 4), B′(6, 14), C′(8, 14), D′(8, 6), E′(14, 6), F′(14, 4)The x and y co-ordinates are all doubled.



5 Shape C

Shape D

Shape E

Shape F



Shape G

Shape H

6 a, b

210 3 5 7 9 11

21

3

5

7

9

11

4

6

8

10

12

4 6 8 10

A

B

12x

y

c The pairs of corresponding sides in the two triangles are all in the same ratio d 3 e (2, 10)



7 a, b

10

123456789

10

2 3 4

P Q R

5 6 7 8 9

y

x

c Translation by the vector 4

0

8 a

2

4

6

8

10

20 4 6 8 10

y

x

D

A

C

B

b

21

3

5

7

4

6

21 30 4

y

x

D

A

C

B

c

21

3

5

78

4

6

21 4 6 70 5

y

x

F

A

E

B

CD



Skills practice B1 a 3 b 2 c 5 d 22 a 4 b Q′(12, 36), R′(32, 28), S′(36, 8) c i P″(10, 25), Q″(15, 45), R″(40, 35), S″(45, 10) ii P″(20, 50), Q″(30, 90), R″(80, 70), S″(90, 20) iii P″(40, 100), Q″(60, 180), R″(160, 140), S″(180, 40) iv P″(200, 500), Q″(300, 900), R″(800, 700), S″(900, 200)3 a, b

Q

Q'

5

5

0

10

15

y

x10 15

c The pairs of corresponding sides in the two quadrilaterals are all in the same ratio. d (3, 13) e 24 a Measurement Red car Blue car

length of car 3.5 cm 7.0 cm

height of car 1.95 cm 3.9 cm

diameter of wheel 0.7 cm 1.4 cm

length of aerial 1.0 cm 2.0 cm

b Each measurement of the blue car is twice the measurement of the red car. c Measurement of angle between Red car Blue car

aerial and roof 30° 30°

bonnet and front windscreen 120° 120°

rear window and boot 100° 100°

bottom and rear wheel arch 110° 110°

d The corresponding angles in the two cars are the same.



5 a Picture 1 – 5 cm Picture 2 – 2.5 cm Picture 3 – 1.25 cm Picture 4 – 0.6 cm b In each case the scale factor of enlargement is approximately 1/2. c Picture 1 – 4.1 cm

Picture 2 – 2.1 cm (1 d.p.) Picture 3 – 1.0 cm (1 d.p.) Picture 4 – 0.5 cm (1 d.p.)

6 a 1 : 50 000 b 4 cm c 20 cm2

d 5 km2

e No7 a, c

10

12345678

2 3 4 5 6 7 8 9 10

T

x

y

b 63°, 63° and 27° (to the nearest degree) d The corresponding angles in T and its image are the same.8 a, b

1

2 310 4 5 7 9 11 13 156 8 10 12 14 16

2345678 P Q

Q”

S”S R

S’

P’ Q’

x

R’R”

P”

y

x

c i (14, 5) ii 1/29 a 4 b (1, 8) c No; although the triangles have the same angles and their sides

are in the same proportion, they are not the same size.



Wider skills practice1 Pictures a and c a i 90% ii 140% iii 75% b i 240 cm2

ii 375 cm2

iii 56.25%



Applying skills1 No; the legs should be twice as wide as the thickness

of the table top and they are wider than twice in the photograph; also, the width of the table should be twice the height of the legs and it is more than twice in the photograph.

2 a The distance from light to fi lm is 0.2 m, and from light to screen is 50.2 m. 50 2

0 2

.

. = 251

b 4.016 m c 4 mm3 a Set of 5 similar rectangles each contained within another b Width Height

1 cm 2 cm

2 cm 4 cm

3 cm 6 cm

4 cm 8 cm

5 cm 10 cm

The scale factor is 5.The centre of enlargement is inside the smallest rectangle. It can be located exactly by extending the black lines until they meet.



Reviewing skills1 a A and E, B and I, C and F, D and H, G and J b A and E are right-angled scalene triangles, B and I are circles, C and F are squares, D and H are parallelograms,

G and J are rectangles. c A to E is 3, I to B is 2 (or B to I is − 1

2), C to F is 4, D to H is 3, G to J is 2.

2 a–c

02 4 6 8 12 14 16

–2

–4

2

4

6

8

10

12

C

X

X

X

x

y

Z Z

Y

Z'

Y

Y

10

d Enlargement with scale factor 1 12

, centre C3 a 1

3

b 1

3

c 1

2

4 Enlargement with scale factor 14

, centre (2 23, 1) or (2.7, 1)


Skills practice A1 a Rectangles E, F, G and H b E – scale factor 2, area scale factor 4

F: scale factor 1.75, area scale factor 3.0625 (1.752) G: scale factor 2.5, area scale factor 6.25 H: scale factor n, area scale factor n2

2 a x = 15 b y = 8 c 3 : 2 d 1 : 13 a 9 cm b 6 cm c 4 cm d 1 : 34 a 14 cm b 6 cm c 6 cm d 3.5 cm



Strand 5 Transformations Unit 7 Similarity Band h

Skills practice B1 a E 20 cm, F 40 cm, G 22.5 cm b 1 : 2 c E 600 cm2, F 2400 cm2, G 337.5 cm2

2 a The angles of both triangles are: 38°, 60°, and 82°; the triangles have the same corresponding angles and so are similar.

b p = 11.2, q = 6.253 a Own accurate diagrams b i 12.0 cm ii 9.3 cm, 4.8 cm iii 4.5 cm iv 11.6 cm, 10.0 cm v 5.6 cm vi 7.0 cm c Triangles ii, iv and vi are similar and triangles i, iii

and v are similar, but none of them are congruent.4 a True b True c False, e.g.

d True

e False, e.g.

f False, e.g.

g False, e.g.

h False; when a fi gure is enlarged, the fi gure and its image are not congruent (though they are similar).

5 Size of paper Longer side (mm)

Shorter side (mm)

Longer ÷ shorter

A4 297 210 1.41

A5 210 148 1.42

A6 148 105 1.41

The ratios are all approximately the same.



Wider skills practice1 a x = 60, y = 84 b i A 156 m, B 260 m, C 312 m ii 3 : 5 : 6 c i A 1512 m2, B 4200 m2, C 6048 m2

ii 9 : 25 : 36 d A 1.8 cm × 2.1 cm, B 3.5 cm × 3 cm, C 4.2 cm × 3.6 cm2 a 1 : n2

b 1 : n3

c Yes, these rules apply to all similar shapes.



Applying skills1 a 2 : 3 b 12 cm

(This could also be found by thinking of D as an enlargement of C by scale factor 1/2.) c 20 cm by 16 cm by 24cm d i 2368 cm2

ii 592 cm2

e i 4 : 1 ii 22 : 12

f Volume of larger box = 7680 cm3, volume of smaller box = 960 cm3

Volume of larger : volume of smaller = 7680 : 960 = 8 : 1 g 23 : 13



Reviewing skills1 a i 2.5 ii 1 : 2 iii 1 : 2 b i 0.5 ii 1 : 3 iii 1 : 3 c i 9 ii 1 : 3 iii 1 : 3 d i 3.8 ii 2 : 1 iii 2 : 1 e i 5.4 ii 3 : 1 iii 3 : 1 f i 5.5 ii 5 : 1 iii 5 : 1


Skills practice A1 a i Own diagram ii H known, O to be found iii sin iv 6.00 m b i Own diagram ii H known, A to be found iii cos iv 4.50 cm c i Own diagram ii A known, O to be found iii tan iv 6.93 mm2 a 6 m b 5.66 m c 20.69 cm3 a 21.82 mm b 21.23 cm c 8.34 m4 a x = 5.67 cm, y = 4.43 cm b x = 0.95 m, y = 1.53 m c x = 6.26 mm, y = 5.45 mm5 a x = 10.15 cm, y = 3.31 cm b x = 9.38 m, y = 6.86 m c x = 15.09 mm, y = 8.00 mm6 a 30° b 45° c 60° d 60° e 90° f 90° g 60° h 29.98° i 26.57°7 a 36.87° b 66.42° c 30.26°8 a 45.92° b 32.78° c 46.88°



Strand 5 Transformations Unit 8 Trigonometry Band h

Skills practice B1 18.13 m2 11.19 m3 8.72°4 6.47 km North and 4.70 km East5 54.7°6 a i AC2 = 12 + 12 = 2 (by Pythagoras’ theorem)

So AC = 2 cm

ii The triangle is a right-angled isosceles triangle, so angle C = angle A = 45°

b sin 45° = opp

hyp=

1

2

c tan 45° = 1, cos 45° = 1

27 a False b True c True d True e True8 3.66 m9 a 14.24 cm b 53.13 cm



Wider skills practice1 a i √2 cm ii √3 cm iii √4 = 2 cm b i 6 triangles ii The 16th triangle does not quite complete 360° so the 17th triangle will be needed. It will overlap the fi rst.

See "Spiral of Theodorus" on the internet.2 a 5 cm b tan θ = 5

12, sin θ = 5

13, cos θ = 12

13

c 22.62°3 a, b Own diagram c 63.43°4 106.26°5 a i PQR is an equilateral triangle, so angle Q = angle P = angle R = 60° ii PM2 = 22 − 12 = 3 (by Pythagoras’ theorem)

So PM = √3 cm

b sin 60° = opp

hyp =

3

2

c cos 60° = 1

2, tan 60° = √3

d M is the midpoint of QR so triangles MPQ and MPR are congruent and angle QPM = angle RPM = 30°

e sin 30° = 1

2, cos 30° = 3

2, tan 30° = 1

3



Applying skills1 a About 037° b 5 km c About 217°2 a tan a = 1

3 so a = 18.435°

tan b = 1

2 so b = 26.565°

tan c = 1 so c = 45° a + b + c = 90° b In the smallest triangle, the other angle is also c since it is an isosceles triangle.

In the largest triangle, the angles are 90°, a and (c + d).So a + c + d = 90° (Angles in a triangle add up to 180°)From part a, a + b + c = 90° also.So b = d, as required.



Reviewing skills1 a x = 9.03 m, y = 11.98 m b x = 2.60 mm, y = 8.41 mm c x = 8.39 cm, y = 12.31 cm2 a 72.8° b 28.6° c 47.8° d 83.2°3 a 36.87° b 200 m


Skills practice A1 a Cube b Triangular prism c Cuboid d Cuboid e Cuboid f Cuboid g Triangular prism h Octagonal prism i Cuboid2 a i

V2

V1

ii 2 vertical planes, 0 horizontal planes b i

V2

V1

ii 2 vertical planes, 0 horizontal planes c i

ii 1 vertical plane, 0 horizontal planes



Strand 6 Three-dimensional shapes Unit 1 Properties of 3-D shapes Band e

d i V1

V2

H

ii 2 vertical planes, 1 horizontal plane

3 a VEG, VFH, VBD, VAC b Because you cannot cut the pyramid in half

horizontally to form two congruent shapes.4 a 5 faces, 9 edges, 6 vertices b V + F = 6 + 5 = 11 = E + 2 �5 a

b

c

6 a Cuboid b Cuboid c On top of the four already placed, to form a

2 × 2 × 2 cube.

7 Shape Number of faces (F) Shape of each face

tetrahedron 4 triangle

cube 6 square

octahedron 8 triangle

dodecahedron 12 pentagon

icosahedron 20 triangle

8 a

b

9 a

b

c



Skills practice B1 a

b

c

d

2 a 7 faces, 15 edges, 10 vertices b V + F = 10 + 7 = 17 = E + 2 �3 a

b

4 a 3 different shaped triangles can be made: 24 like AEG (2 for every edge of the cube), 24 like ADC (2 for every face diagonal of the cube) and 8 like AHF. There are 56 triangles altogether.

b There 56 triangles altogether for a cuboid, but when none of the faces of the cuboid are squares, they are of 12 different shapes.

5 a Vertical: GWEV, FWVH, CVAW, BVDWHorizontal: DCBA

b Square c Rhombus (and square) d GWEV and FWVH, and CVAW and BVDW6 a Yellow b Use a horizontal axis of symmetry through centres

of the yellow faces and rotate the shape vertically upwards through 90°.

7 a Regular octagonal prism

b Regular hexagonal-based pyramid



Wider skills practice1 a, b Own work c Largest surface area:

Smallest surface area:



Applying skills1 a Note that rotations of these arrangements exist but are not shown.

b Note that rotations of these arrangements exist but are not shown.

1 2 3H

v1

v2 v1

v2 v3v4

H

4

v2 v3v4

c i 2 prisms ii 2 cuboids iii 0 cubes



Reviewing skills1 a Isosceles triangular prism b Hexagonal prism c Cylinder d Hexagonal prism e Cuboid f Equilateral triangular prism a i

v1

H1

v2

ii 2 vertical planes, 1 horizontal plane b i

ii 1 vertical plane, 0 horizontal planes c i

ii 1 vertical plane, 1 horizontal plane

d ii 0 vertical planes, 0 horizontal planes3

4 a

b

c

5 a 5 faces b Square-based pyramid or rectangular based pyramid


Skills practice A1 a, b Own design, e.g.

c 7 flaps d Yes2 a 7 cm b 7 cm c 9 cm d 9 cm3 Own design4 a Own designs b i 80 cm2

ii 66 cm2

5 1 and E, 2 and B, 3 and D, 4 and A, 5 and C

6 a Own design b 3 flaps; yes



Strand 6 Three-dimensional shapes Unit 2 Understanding nets Band f

Skills practice B1 a, b Own designs c 80 cm2

2 a M and A b K c J and H3 a B b E c C d 2 fl aps e 7 and 84 a i True ii False iii False (it could be square, but it does not have

to be) iv True v False b Own sketch of the wedge5 Here are the 11 different nets for a cube.

6 Own accurate net7 a 35 shapes b

45 4

4

5 55

6235

5

1

1

1 5 12 4

56 64

21

5

3 336

2 4

46 2

24 1

5 36

2 1 54

36

34

11 1

126

646

3

32

232

36



Wider skills practice1 a Yes b Stella; 32 cm2

2 a QRST b ACEG, DBFH, KIMO, LJNP3 a A prism has constant cross-section. b Tetrahedron: F = 4, E = 6, V = 4

V + F = 4 + 4 = 8 = E + 2 � Octahedron: F = 8, E = 12, V = 6

V + F = 6 + 8 = 14 = E + 2 �



Applying skills1 a Each of its eight faces is an equilateral triangle. b, c Own work2 a Tetrahedron b Yes c Octahedron3 Own design



Reviewing skills1 a Own design b 62 cm2

c 30 cm3

2 a Own design b Yes


Skills practice A1 a–c

Cuboid Number of cubes in one layer

Number of layers Total number of cubes

Volume

A 6 5 30 30 cm3

B 10 3 30 30 cm3

C 4 4 16 16 cm3

D 14 2 28 28 cm3

2 a Both cuboids have the same volume, 48 cm3. b Own answers, e.g.

Length 2 cm, width 2 cm, height 12 cm Length 1 cm, width 2 cm, height 24 cm

3 a 180 cm3

b 40 cm3

c 40 cm3

d 54 cm3

4 a 72 cm2

b 164 cm2

c 150 cm2

d 190 cm2

5 a 214 cm2

b 700 cm2

c 1416 cm2

6 a Volume 200 cm3, surface area 220 cm2

b Volume 96 cm3, surface area 136 cm2

c Volume 72 cm3, surface area 124 cm2

d Volume 64 cm3, surface area 112 cm2

e Volume 2.584 m3, surface area 12.22 cm2

f Volume 4 027 320 mm3, surface area 157 386 mm2

7 a 16 cm2

b 96 cm2

c 64 cm3



Strand 6 Three-dimensional shapes Unit 3 Volume and surface area of cuboids Band f

Skills practice B1 80 matchboxes2 a 60 packs b 15 kg a i 12 sugar cubes ii 2 layers iii 24 sugar cubes iv 24 cm3

b i 24 cm3

ii Both boxes have the same volume. c Own answers and diagrams, e.g.

4 cm × 5 cm × 6cm; this box has the lowest surface area.

4 a 25 cm2

b 5 cm c 125 cm3

5 a 40 cm2

b 40 cm2

c 80 cm3

d The surface area is the same as the area of the net.6 a 1200 cm2

b 320 cm2

c 480 cm2

7 4 m3

8 a 25 m3

b 400 m3

c 425 000 litres d 385 m2

9 a i 200 000 cm3

ii 7000 cm3

iii 255 000 cm3

b 462 000 cm3

c 32 000 cm3

10 No; although both boxes have the same volume, sugar is denser than tea.

11 Pete has multiplied together measurements with inconsistent units (cm and m).



Wider skills practice1 a Cuboid b 400 m3

c 400 000 litres d 320 m2

2 a 1 800 000 cm3

b 1800 litres c 75 000 cm2

3 a Volume (in cm3) = lwh b Surface area (in cm2) = 2lw + 2lh + 2wh4 a 9000 cm3

b 50 m3

c 1500 mm3

d 2500 m3



Applying skills1 a 64 cm3

b 74.07 cm3

2 a–d Own answers e 4 corner pieces f, g Own answers



Reviewing skills1 a Volume 64 cm3, surface area 96 cm2

b Volume 120 mm3, surface area 158 mm2

c Volume 24 m3, surface area 52 m2


b Volume 224 mm3, surface area 336 mm2


Skills practice A1 a 4 cm

4 cm

Plan Front Side b 5 cm

2 cm

1 cm

Plan Front Side2 a

Plan Front Left Right

b

Plan Front Side

3 Triangular prism,4 a i Chair ii Side on b i Washing machine ii In front c i Bath ii Above d i Lamp ii Above5 a Plan b Front elevation c Back elevation d Side elevation



Strand 6 Three-dimensional shapes Unit 4 2-D representations of 3-D shapes Band f

Skills practice B1 Jack is correct.2 a A b C c D d B e

3 a

Plan Front L.Side Right b

Plan Front Side c

Plan Front Side

d


e


f


4 a Square-based pyramid or octahedron b Triangular prism c Hexagonal prism d Tetrahedron e Cuboid f Truncated pyramid5 a

Plan Front Side b

Plan Front Side c

Plan Front Side

6

Plan Front Side



Wider skills practice1 Own design



Applying skills1 a

Plan

0.4 m 0.4 m0.4 m0.4 m

0.4 m0.4 m

SideFront

4 m 3 m 3 m

4 m

4 m 3 m3 m

4 m

10 m

b 40 m2

c 3 tins



Reviewing skills1 a

b

c

d

2 a

Plan Front Side

b

Plan Front Side

c

Plan Front Side


Skills practice A1 a 6 cm3

b 12 cm3

c 18 cm3

d 11 cm3

e 33 cm3

f 8 cm3

g 16 cm3

2 Own sketch of net 132 cm2

3 a 32 cm3

b 16 cm3

c 36 m3

d 30 cubic inches4 a Volume 4.2 m3, surface area 23.4 m2

b Volume 138 000 mm3, surface area 23 100 mm2

5 500 cm3

6 9425 cm2

7 a 48 cm3

b 924 cm3

c 723 cm3

8 a 3519 cm2

b 594 cm2

9 a Areas are ii 360 cm2, iii 320.4 cm2, i 320 cm2

b Volumes are iii 395.8 cm3, i 384 cm3, ii 378 cm3

10 a Own opinion b i 37.7 cm3

ii 18.8 cm3

c No; the second cylinder has three quarters the volume of the first.11 a i 552 cm3

ii 412 cm2

b i 810 cm3

ii 582 cm2



Strand 6 Three-dimensional shapes Unit 5 Prisms Band g

Skills practice B1 a i Cube ii 125 cm3

b i Cylinder ii 75.4 cm3

c i Triangular prism ii 26.8 cm3

d i Cuboid ii 84 cm3

e i Cylinder ii 198 cm3

f i Parallelogram prism ii 70 cm3

g i Semi-circular prism ii 50.3 cm3

h i Trapezium prism ii 48 cm3

2 5 cm3 a Triangular prism b 132 cm3

c 178 cm2

4 a 120 cm3

b 66.0 cm3



6 a 46 cm2

b 112 cm2

7 9000 m3

8 32 tins 9 99.5 m2

10 a 109 000 cm3 (3 sig. fi gs.) b 55 800 cm2 (3 sig. fi gs.)11 7.1 cm12 116 cm3

13 Yes (Area to be covered is 1.7 m2)

14 a i 78.54 m2

ii 113.10 m2

iii 34.56 m2

b 1.73 m3



Wider skills practice1 a 6082 litres b 81%2 96 cups3 a 27 m3

b 54 chickens c 3 tins4 a 1

2 whl

b hw lw hl l h+ + +2 2

+ w5 a Own proof

b π d h2

4

6 5 cm



Applying skills1 a 12 500 cm3 (3 sig. fi gs.) or (3 s.f) b 976 cm3

2 10 cm3 a 1440 cm3

b 2168 cm3

4 Own proof5 Own diagrams



Reviewing skills1 a Volume 16 cm3, surface area 46 cm2


c Volume 108 cm3, surface area 156 cm2

d Volume 875 cm3, surface area 750 cm2

e Volume 96 cm3, surface area 152 cm2

f Volume 36 cm3, surface area 84 cm2

2 a 151 cm2

b 132 cm2

c 24.2 mm2

d 1296 m2

3 Both mugs hold the same amount of tea.4 3520 cm3 (3 sig. fi gs.)

Documents

Skills practice A - Hodder Education