of 62 KS3 Mathematics S7 Measures Boardworks Ltd 2006 Contents of 62 S7 Measures A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles

A S7.5 Bearings Boardworks Ltd 2006 Metric units of 62 The metric system of measurement is based on powers of ten and uses the following prefixes: KiloCentiMilliMicro- meaning 1000 meaning one hundredth meaning one thousandth meaning one millionth

These prefixes are then followed by a base unit. The base unit for length is metre. The base unit for mass is gram. The base unit for capacity is litre. Boardworks Ltd 2006 Metric units of length of 62 Metric units used for length are kilometres, metres, centimetres and millimetres. 1 kilometre (km) = 1000 metres (m) 1 metre (m) = 100 centimetres (cm) 1 metre (m) = 1000 millimetres (mm)

1 centimetre (cm) = 10 millimetres (mm) Boardworks Ltd 2006 Metric units of length of 62 A race track measures 400 m. An athlete runs 2.6 km around the track. How many laps is this? 400 m = 0.4 km Number of laps = 2.6 0.4 = 6.5 laps The following day the athlete completes 8 laps. How many km is this? 8 laps = 8 0.4 km

= 3.2 km Boardworks Ltd 2006 Metric units of mass of 62 Metric units used for mass are tonnes, kilograms and grams and milligrams. 1 tonne = 1000 kilograms (kg) 1 kilogram (kg) = 1000 grams (g) 1 gram (g) = 1000 milligrams (mg) Boardworks Ltd 2006

Metric units of mass of 62 60 tea bags weigh 150 g. How much would 2000 tea bags weigh in kg? We can solve this problem using a unitary method. 60 tea bags weigh 150 g So: 1 tea bag weighs (150 60) g = 2.5 g Therefore, 2000 tea bags weigh (2.5 2000) g = 5000 g = 5 kg Boardworks Ltd 2006

Metric units of capacity of 62 Capacity is a measure of the amount of liquid that a 3-D object (for example a glass) can hold. Metric units of capacity are litres (l), centilitres (cl) and millilitres (ml). 1 litre (l) = 100 centilitres (cl) 1 litre (l) = 1000 millilitres (ml) 1 centilitre (cl) = 10 millilitres (ml) Boardworks Ltd 2006

Metric units of capacity of 62 A bottle contains 750 ml of orange squash. The label says: Dilute 1 part squash with 4 parts water. How many of litres of drink can be made with one bottle? If the whole bottle was made up we would have: 750 ml of squash + (4 750) ml of water = 750 ml of squash + 3000 ml of water = 3750 ml of drink = 3.75 l of drink Boardworks Ltd 2006 Converting metric units

0 of 62 To convert from a larger metric unit to a smaller one we need to _______ multiply by 10, 100, or 1000. Complete the following: 34 cm = 340 mm 0.0471 km = 47.1 m 0.4 l = 400 ml 0.3428 m = 342.8 mm 7.3 kg = 7300 g

23.51 g = 23 510 mg 54.8 cl = 548 ml 0.085 m = 85 mm Boardworks Ltd 2006 Converting metric units 1 of 62 To convert from a smaller metric unit to a larger one we need to ______ divide by 10, 100, or 1000.

Complete the following 920 mm = 92 cm 65800 m = 65.8 km 530 g = 0.53 kg 526 mg = 0.526 g 3460 ml = 3.46 l 4539 cl = 45.39 l 43.1 cm = 0.431 m 87 kg = 0.087 tonnes

Boardworks Ltd 2006 Converting metric units 2 of 62 Boardworks Ltd 2006 Imperial units of length 3 of 62 Commonly used imperial units for length include inches, feet, yards and miles. 1 foot (ft) = 12 inches (in)

1 yard = 3 feet (ft) 1 mile = 1760 yards Although these units can be difficult to convert between they are still in common use, for example, for road traffic signs and related measurements of speed. Boardworks Ltd 2006 Imperial units of length We can convert between metric and imperial units of length using the following approximations: 4 of 62 1 inch 2.5 cm 1 foot (or 12 inches) 30 cm

1 yard (or 3 feet) just under 1 metre 5 miles 8 km (or 1 mile 1.6 km) Boardworks Ltd 2006 Imperial units of length 5 of 62 The following road sign was seen from the side of the motorway in France. What are the distances in miles? Paris Calais Dieppe

264 km 36 km 144 km 8 km 5 miles 1 km 0.625 miles Distance to Paris (264 0.625) miles = 165 miles Distance to Calais (36 0.625) miles = 22.5 miles Distance to Dieppe (144 0.625) miles = 90 miles Boardworks Ltd 2006 Imperial units of mass

6 of 62 Commonly used imperial units for mass include ounces, pounds, and stones. 1 pound = 16 ounces 1 stone = 14 pounds Although these units can be difficult to convert between they are still in common use, for example, most people weigh themselves in stones. Boardworks Ltd 2006 Imperial units of mass We can convert between metric and imperial units of mass

using the following approximations: 7 of 62 1 ounce (oz) 30 grams 1 pound is just under kilogram (or 1 kilogram 2.2 pounds) 1 stone is just over 6 kg Boardworks Ltd 2006 Imperial units of mass 8 of 62 Which is cheaper: apples costing 53p per kilo

or apples costing 30p per pound? 1 kilogram 2.2 pounds So: 30p per pound (2.2 30)p per kilo = 66p per kilo The apples priced at 53p per kilo are cheaper. Boardworks Ltd 2006 Imperial units of capacity 9 of 62 Commonly used imperial units for capacity include fluid ounces, pints, and gallons.

1 gallon = 8 pints 1 pint = 20 fluid ounces Although these units can be difficult to convert between they are still in common use, as an example beer and milk are still commonly measured in pints. Boardworks Ltd 2006 Imperial units of capacity We can convert between metric and imperial units of capacity using the following approximations: 0 of 62 1 gallon 4.5 litres

1 pint just over litre (or 1 litre 1.75 pints) Boardworks Ltd 2006 Imperial units of capacity 1 of 62 A litre of petrol costs 84.7 p. Approximately, how much would 1 gallon cost? 1 gallon 4.5 litres Therefore: 1 gallon of petrol costs 4.5 84.7 p = 381.15 p 3.81

Boardworks Ltd 2006 Imperial to metric conversions 2 of 62 Boardworks Ltd 2006 Spider diagram 3 of 62 Boardworks Ltd 2006 Units of area

4 of 62 Area is measured in square units. Here is a square centimetre or 1 cm2. How many mm2 are there in a cm2? 1 cm = 10 mm 1 cm = 10 mm 1 cm 1 cm = 1 cm2 10 mm 10 mm = 100 mm2 So:

1 cm2 = 100 mm2 Boardworks Ltd 2006 Units of area 5 of 62 Area is measured in square units. Here is a square metre or 1 m2. How many cm2 are there in a m2? 1 m 1 m = 1 m2 1m = 100 cm 2 2 100 cm 100 cm = 10

10000 000m cm 1m = 100 cm So: 1 m2 = 10 000 cm2 Boardworks Ltd 2006 Units of area 6 of 62 We can use the following to convert between units of area.

1 km2 = 1 000 000 m2 1 hectare = 10 000 m2 1 m2 = 10 000 cm2 1 m2 = 1 000 000 mm2 1 cm2 = 100 mm2 Boardworks Ltd 2006 Units of area 7 of 62 A rectangular field measures 150 m by 250m. What is the area of the field in hectares? The area of the field is

150 m 250 m = 37 500 m2 250 m 1 hectare = 100 m 100 m = 10 000 m2 37 500 m2 = 3.75 hectares 150 m Boardworks Ltd 2006 Units of volume 8 of 62 Volume is measured in cubic units. Here is a cubic centimetre or 1 cm3.

1 cm = 10 mm 1 cm = 10 mm 1 cm = 10 mm How many mm3 are there in a cm3? 1 cm 1 cm 1 cm = 1 cm3 10 mm 10 mm 10 mm = 1000 mm3 So:

1 cm3 = 1000 mm3 Boardworks Ltd 2006 Units of volume 9 of 62 Volume is measured in cubic units. Here is a cubic metre or 1 m3. 1m = 100 cm 1m

= 100 cm 1m = 100 cm How many cm3 are there in a m3? 1 m 1 m 1 m = 1 m3 100 cm 100 cm 100 cm = 1 000 000 cm3 So: 1 m3 = 1 000 000 cm3 Boardworks Ltd 2006 Units of volume

0 of 62 We can use the following to convert between units of volume. 1 km3 = 1 000 000 000 m 3 1 m3 = 1 000 000 cm3 1 m3 = 1 000 000 000 mm 3 1 cm3 = 1000 mm3 Boardworks Ltd 2006 Units of volume 1 of 62 Dice are packed into boxes measuring 20 cm

by 12 cm by 10 cm. If the dice are 2 cm cubes, how many of them fit into a box? The volume of the box = (20 12 10) cm3 = 2400 cm3 The volume of one dice = (2 2 2) cm3 = 8 cm3 Number of dice that fit in the box = 2400 8 = 300 dice Boardworks Ltd 2006 Volume and capacity 2 of 62 Capacity is a measure of the amount of liquid that a 3-D object can hold.

A litre of water, for example, would fill a container measuring 10 cm by 10 cm by 10 cm (or 1000 cm3). 1 l = 1000 cm3 1 ml = 1 cm3 1000 l = 1 m3 Boardworks Ltd 2006 Volume and capacity 3 of 62 Boardworks Ltd 2006 Volume and capacity

4 of 62 Which holds more juice when full; a litre bottle or a carton measuring 6 cm by 10 cm by 20 cm? The volume of the carton is (6 10 20) cm3 = 1200 cm3 1 litre = 1000 cm3 The carton holds more juice. Boardworks Ltd 2006 Converting units of area, volume and capacity 5 of 62

Complete the following 3 ha = 30 000 m2 4000 m2 = 0.4 ha 2.8 m3 = 2800 l 6 200 cm2 = 0.62 m2 4.35 cm2 = 435 mm2 9.6 cl = 96 cm3 0.07 cm3 = 70 mm3

38 000 cm3 = 0.038 m3 0.72 l = 720 cm3 5630 cm3 = 5.63 l Boardworks Ltd 2006 Units of time Time does not use the metric system. Units of time include years, months, weeks, days, hours (h), minutes (min) and seconds (s). 6 of 62 1 minute (min) = 60 seconds (s)

1 hour (h) = 60 minutes (min) 1 day = 24 hours (h) 1 week = 7 days 1 year = 365 days = 52 weeks 1 leap year = 366 days Boardworks Ltd 2006 Units of time 7 of 62 A machine takes 4 minutes and 10 seconds to make a toy car. How long would it take to make 18 toy cars? 4 minutes 18 = 72 minutes 10 seconds 18 = 180 seconds = 3 minutes

72 minutes + 3 minutes = 75 minutes = 1 hour 15 minutes Boardworks Ltd 2006 Contents 8 of 62 S7 Measures A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings Boardworks Ltd 2006

Choosing units 9 of 62 What units would you use to measure the following: The mass of a child Kilograms The length of a finger nail Millimetres The area of a field

Hectares The mass of an ant Milligrams The distance between two cities Kilometres The capacity of a pool Litres The volume of a room

Cubic metres The distance between two stars Light years Boardworks Ltd 2006 Estimating measurements 0 of 62 When we estimate measurements we usually compare known measurements to find unknown measurements. Some useful measurements to know are: The height of a door is about 2 m.

The mass of a large bag of sugar is 1 kg. A teaspoon holds 5 ml of liquid. Most adults are between 1.5 and 1.8 m tall. A small car weighs about 1 tonne. The area of a football pitch is 7500 m2. The capacity of a can of drink is 330ml. It takes about 20 minutes to walk one mile. Boardworks Ltd 2006 Estimating measurements 1 of 62 Boardworks Ltd 2006 Contents

2 of 62 S7 Measures A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings Boardworks Ltd 2006 Reading scales 3 of 62 What numbers are the arrows pointing

to on the following scale? 2.8 C 3.8 A 3 4.4 B 4 5

Each small division is worth 1 5 = 0.2 A is pointing at 3.8 B is pointing at 4.4 C is pointing at 2.8 Boardworks Ltd 2006 Reading scales 4 of 62 What numbers are the arrows pointing to on the following scale? C 57.5 B

72.5 65 A 60 70 80 Each small division is worth 10 4 = 2.25 A is pointing at 65 B is pointing at 72.5 C is pointing at 57.5 Boardworks Ltd 2006

Reading scales 5 of 62 What numbers are the arrows pointing to on the following scale? 1.96 C 2.03 A 2.0 2.165

B 2.1 2.2 Each small division is worth 0.1 10 = 0.01 A is pointing at 2.03 B is pointing at 2.165 C is pointing at 1.96 Boardworks Ltd 2006 Reading scales 6 of 62

Boardworks Ltd 2006 Contents 7 of 62 S7 Measures A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings Boardworks Ltd 2006 Measuring angles

8 of 62 An angle is a measure of turn and is usually measured in degrees. 360 A full turn measures 360. Boardworks Ltd 2006 Measuring angles 9 of 62

An angle is a measure of turn and is usually measured in degrees. A quarter turn measures 90. 90 It is called a right angle. We label a right angle with a small square. Boardworks Ltd 2006 Measuring angles

0 of 62 An angle is a measure of turn and is usually measured in degrees. A half turn measures 180. 180 This is a straight line. Boardworks Ltd 2006 Measuring angles 1 of 62

An angle is a measure of turn and is usually measured in degrees. A three-quarter turn measures 270. 270 Boardworks Ltd 2006 Acute, obtuse and reflex angles 2 of 62 All angles are acute, obtuse or reflex. An acute angle is between 0

and 90. An obtuse angle is between 90 and 180. An reflex angle is between 180 and 360. Boardworks Ltd 2006 Acute, obtuse and reflex angles 3 of 62

Look at each interior angle in this shape. Is it acute, obtuse or reflex? B D C A I F E H G

Boardworks Ltd 2006 Estimating angles 4 of 62 Boardworks Ltd 2006 Using a protractor 5 of 62 We measure angles with a protractor. Notice that the protractor has two scales.

Before you measure an angle decide whether it is acute or obtuse. Boardworks Ltd 2006 Measuring angles with a protractor 6 of 62 Boardworks Ltd 2006 Contents 7 of 62 S7 Measures A S7.1 Converting units

A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings Boardworks Ltd 2006 Bearings 8 of 62 Bearings are a measure of direction taken from North. If you were travelling North you would be travelling on a bearing of 000. If you were travelling from the point P in the direction shown by the arrow then you would be travelling on a bearing of 000.

075. N 75 Bearings are always measured clockwise from North and are written as three figures. P Boardworks Ltd 2006 Compass points 9 of 62

000 N 315 NW 045 NE 270 W E 090 SW 225

SE S 180 135 Boardworks Ltd 2006 Virtual snooker 0 of 62 Boardworks Ltd 2006 Bearings

1 of 62 Boardworks Ltd 2006 Bearings 2 of 62 The bearing from point A to point B is 105. What is the bearing from point B to point A? N The angle from B to A is N A

105 + 180 = 285 105 ? 105 This is called a back bearing. B 180 Boardworks Ltd 2006