KS3 Mathematics S7 Measures The aim of this unit is to teach pupils to: Use units of measurement to measure, estimate, calculate and solve problems in a range of contexts; convert between metric units and know rough metric equivalents of common imperial measures Extend the range of measures used to measure angles and bearings Material in this unit is linked to the Key Stage 3 Framework supplement of examples pp 228-233. S7 Measures

S7 Measures Contents A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings

Metric units The metric system of measurement is based on powers of ten and uses the following prefixes: Kilo- meaning 1000 Centi- meaning one hundredth Milli- meaning one thousandth Micro- meaning one millionth These prefixes are then followed by a base unit. Ask pupils why they think powers of ten are used for the metric system of measurement. Discuss the fact that it is easy to convert between units by multiplying or dividing by a power of ten. Link: N1 Place value ordering and rounding – powers of ten. The base unit for length is metre. The base unit for mass is gram. The base unit for capacity is litre.

Metric units of length 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 (cm) Pupils should know these conversions and the abbreviation for each unit. Ask pupils to give you an example of something that measures about 1 mm. For example, the width of a grain of rice. Ask pupils to give you an example of something that measures about 1 cm. For example, the width of a little finger. Ask pupils to give you an example of something that measures about 1 m. For example, the width of a desk. Ask pupils to give you an example of a distance of about 1 km. For example, the distance from the school to a well-known local landmark. 1 centimetre (cm) = 10 millimetres (cm)

Metric units of length 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? For the first part of the problem we can either convert both measurements to m or both measurements to km before dividing. Link: N8 Ratio and proportion – direct proportion. 8 laps = 8 × 0.4 km = 3.2 km

Metric units of mass Metric units used for mass are tonnes, kilograms and grams and milligrams. 1 tonne = 1000 kilograms (kg) 1 kilogram (kg) = 1000 grams (g) Pupils should know these conversions and the abbreviation for each unit. Distinguish between mass and weight. We often use the word weight to mean mass. However, weight is actually a force due to gravity and can change depending on gravity. Mass, on the other hand, is a measure of the amount of matter contained in an object and is constant throughout space. (The imperial system refers to weights; the metric system refers to mass. So to convert between them using the common conversion formulae we assume them to refer to objects here on earth!) The tonne is also used in the imperial system to describe a similar weight spelt ton. Point out that when the metric system system of measurement was devised in France more than 200 years ago, 1 gram was defined as the weight of 1 cm3 of pure water. 1000 cm3 (that’s a cube 10 cm by 10 cm by 10 cm) of water weighs 1 kg, and 1 m3 (or 1 000 000 cm³) weighs 1 tonne. A mm3 of water weighs 1 mg. Ask pupils to give you an example of something that has a mass of about 1 tonne. For example, the mass of a small car. Ask pupils to give you an example of something that has a mass of about 1 kg. For example, a bag of sugar. Ask pupils to give you an example of something that has a mass of about 1 g. For example, an aspirin tablet (a 1p coin weighs about 3.5 g). Ask pupils to give you an example of something that has a mass of about 1 mg. For example, the mass of 1 mm3 of water. 1 gram (g) = 1000 milligrams (mg)

Metric units of mass 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 Link: N8 Ratio and proportion – direct proportion. Therefore, 2000 tea bags weigh (2.5 × 2000) g = 5000 g = 5 kg

Metric units of capacity 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)

Metric units of capacity 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 Link: N8 Ratio and proportion – direct proportion. = 3750 ml of drink = 3.75 l of drink

Converting metric units To convert from a larger metric unit to a smaller one we need to _______ by 10, 100, or 1000. multiply Complete the following: 34 cm = mm 340 0.0471 km = m 47.1 0.4 l = ml 400 0.3428 m = mm 342.8 Pupils often believe that to convert from a larger unit to a smaller unit we should divide to make it smaller. Explain that we actually have to multiply because there are more smaller units in each larger unit. For each problem ask pupils what we have to multiply by to complete the conversion before asking for the solution. Change the numbers to make a new set of problems. Link: N1 Place value, ordering and rounding – Multiplying by 10, 100 and 1000. 7.3 kg = g 7300 23.51 g = mg 23 510 54.8 cl = ml 548 0.085 m = mm 85

Converting metric units To convert from a smaller metric unit to a larger one we need to ______ by 10, 100, or 1000. divide Complete the following 920 mm = cm 92 65800 m = km 65.8 530 g = kg 0.53 526 mg = g 0.526 Pupils often believe that to convert from a smaller unit to a larger unit we should multiply to make it larger. Stress that we actually have to divide because there are fewer large units for each smaller unit. For each problem ask pupils what we have to divide by to complete the conversion before asking for the solution. Link: N1 Place value, ordering and rounding – Dividing by 10, 100 and 1000. 3460 ml = l 3.46 4539 cl = l 45.39 43.1 cm = m 0.431 87 kg = tonnes 0.087

How many mm2 are there in a cm2? Units of area Area is measured in square units. Here is a square centimetre or 1 cm2. How many mm2 are there in a cm2? 1 cm 1 cm × 1 cm = 1 cm2 = 10 mm 10 mm × 10 mm = 100 mm2 Stress that area has two dimensions, length and width. A square centimetre is 1 cm × 1 cm. 1 cm is equivalent to 10 mm, so 1 cm2 is equivalent to 10 mm × 10 mm = 100 mm2. So, = 10 mm 1 cm2 = 100 mm2

How many cm2 are there in a m2? Units of area 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 m = 1 m2 = 100 cm 100 cm × 100 cm = 10000 m2 10 000 cm2 A square metre is 1 m × 1 m. 1 m is equivalent to 100 cm. Ask pupils to imagine that the square is divided into 100 cm across the length and 100 cm across the width. We can conclude that 1 m2 is equivalent to 100 cm × 100 cm = 10000 cm2. So, = 100 cm 1 m2 = 10 000 cm2

Units of area We can use the following to convert between units of area. 1 km2 = m2 1 000 000 1 hectare = m2 10 000 1 m2 = cm2 10 000 As each unit appears remind pupils verbally that: 1 km2 = 1000 m × 1000 m = 1 000 000 m2 A hectare = 100 m × 100 m = 10 000 m2 1 m2 = 100 cm × 100 cm = 10 000 cm2 1 m2 = 1000 mm × 1000 mm = 1 000 000 mm2 1 cm2 = 10 mm × 10 mm = 100 mm2 1 m2 = mm2 1 000 000 1 cm2 = mm2 100

Units of area 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 Link: S8 Perimeter, area and volume – area. = 10 000 m2 37 500 m2 = 3.75 hectares 150 m

How many mm3 are there in a cm3? Units of volume Volume is measured in cubic units. Here is a cubic centimetre or 1 cm3. How many mm3 are there in a cm3? 1 cm = 10 mm 1 cm × 1 cm × 1 cm = 1 cm3 10 mm × 10 mm × 10 mm Ask pupils to imagine filling a cubic centimetre with cubic millimetres. = 1000 mm3 = 10 mm So, 1 cm3 = 1000 mm3 = 10 mm

How many cm3 are there in a m3? Units of volume Volume is measured in cubic units. Here is a cubic metre or 1 m3. How many cm3 are there in a m3? 1 m = 100 cm 1 m × 1 m × 1 m = 1 m3 100 cm × 100 cm × 100 cm Ask pupils to imagine filling a cubic metre with cubic centimetres. = 1 000 000 cm3 = 100 cm So, 1 m3 = 1 000 000 cm3 = 100 cm

Units of volume We can use the following to convert between units of volume. 1 km3 = m3 1 000 000 000 1 m3 = cm3 1 000 000 1 m3 = mm3 1 000 000 000 As each unit appears remind pupils verbally that: 1 km³ = 1000 m × 1000 m × 1000 m = 1 000 000 000 m³ 1 m³ = 100 cm × 100 cm × 100 cm = 1 000 000 cm³ 1 m³ = 1000 mm × 1000 mm × 1000 mm = 1 000 000 000 mm³ 1 cm³ = 10 mm × 10 mm × 10 mm = 1000 mm³ 1 cm3 = mm3 1000

Units of volume 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 An alternative method would be to work out how many dice would fit along each length of the box and multiply these together. This would be 10 × 6 × 5 = 300. Number of dice that fit in the box = 2400 ÷ 8 = 300 dice

Volume and capacity 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

Volume and capacity 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.

Converting units of area, volume and capacity Complete the following 3 ha = m2 30 000 4000 m2 = ha 0.4 2.8 m3 = l 2800 6 200 cm2 = m2 0.62 4.35 cm2 = mm2 435 9.6 cl = cm3 96 The problems on the left involve converting larger units to smaller units. The problems on the right involve converting smaller units to larger units. 0.07 cm3 = mm3 70 38 000 cm3 = m3 0.038 0.72 l = cm3 720 5630 cm3 = l 5.63

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). 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

Units of time 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 = Discuss any alternative methods pupils suggest. 75 minutes = 1 hour 15 minutes

S7.2 Estimating measurements Contents 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

Choosing units 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

Estimating measurements 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 tea spoon holds 5 ml of liquid. Most adults are between 1.5 and 1.8 m tall. A small car weighs about 1 tonne. Ask pupils to use the height of the classroom door to estimate your height in cm. Ask for estimates of other lengths or weights in the classroom. 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. The mass of a large bag of sugar is 1 kg.

S7 Measures Contents A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings

What numbers are the arrows pointing to on the following scale? Reading scales What numbers are the arrows pointing to on the following scale? 2.8 C 3.8 A 4.4 B 3 4 5 Each small division is worth 1 ÷ 5 = 0.2 Explain that when reading a scale it is important to start by working out the value of each small division. We do this by taking two consecutive numbered divisions, finding the difference between them and dividing this by the number of small divisions. Stress to pupils that the number of divisions is actually the number of gaps between the lines and not the number of lines themselves. In this example, we have five divisions between each whole unit. This means that one small division is worth 0.2 units. A is pointing at 3.8 B is pointing at 4.4 C is pointing at 2.8

What numbers are the arrows pointing to on the following scale? Reading scales What numbers are the arrows pointing to on the following scale? C C 57.5 65 A 72.5 B B 60 70 80 Each small division is worth 10 ÷ 4 = 2.25 In this example, we have four divisions between ten units. This means that one small division is worth 2.25 units. A is pointing at 65 B is pointing at 72.5 C is pointing at 57.5

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

S7 Measures Contents A S7.1 Converting units A S7.2 Estimating measurements A S7.3 Reading scales A S7.4 Measuring angles A S7.5 Bearings

Bearings 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 Bearings are always measured clockwise from North and are written as three figures. 75° P

Compass points 000° N 315° 045° NW NE 270° W E 090° SW SE 225° 135° S Revise the basic compass points. These can be remembered using a mnemonic such as ‘Naughty Elephants Squirt Water’. Introduce the points NE, SE, SW and NW, pointing out that N or S always comes before E and W. Ask pupils to tell you how many degrees there are between each compass point (45°). Ask questions such as, A ship is sailing due southwest. What bearing is it sailing on? Reveal the directions of the compass using bearings in orange. You may like to mention that in addition to these points we can also have NNE (north by north east), ENE (east by north east), ESE (east by south east), SSE (south by south east), SSW (south by south west), WSW (west by south west), WNW (west by north west) and NNW (north by north west). As an extension exercise ask pupils to draw these compass points using a ruler and a protractor and give the bearing of each one. There will be 22.5º between each compass point. SW SE 225° 135° S 180°

Bearings Use this activity to demonstrate how to find the bearing from one numbered place on the map to another. Also, ask pupils to use the scale given on the map to estimate the distance between two points. Ask, for example, What is the approximate bearing from 6 to 5? Approximately how far in metres is it between 1 and 3? What point is on a bearing of 255° from 6? Find the bearing from 7 to 9. Ask pupils if they can give you the bearing from 9 to 7 without measuring. Link: S5 Coordinates and transformations 2 – Scale drawings.

Bearings The bearing from point A to point B is 105º. What is the bearing from point B to point A? N The angle from B to A is 105º + 180º = N 285º 105º A This is called a back bearing. 105º ? Explain that the north lines are parallel and so we can work out he corresponding angle of 105°. The angle from B to A is therefore 105° + 180° = 285°. If a given bearing is less than 180º, we find the back bearing by adding 180°. If a given bearing is more than 180º, we find the back bearing by subtracting 180°. B 180°