S8 Measures

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Transcript S8 Measures

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KS4 Mathematics

S8 Measures

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S8.1 Converting units

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S8.2 Accuracy in measurement

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S8.3 Calculations involving bounds

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S8.4 Compound measures

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S8.5 Bearings 2 of 51

Converting units

It is important to be able to convert between a variety of units quickly and accurately.

When using a formula

For example, if we are using a formula to find the volume of an object we must make sure that all the lengths are written using the same units before using the formula.

When comparing measurements

For example, suppose one mother gives the weight of her baby in pounds and another mother gives the weight of her baby in kilograms.

We usually use

metric units

for calculations.

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Metric units

The metric system of measurement is based on powers of ten and uses the following prefixes:

Kilo Centi Milli Micro-

meaning one thousand 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.

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Metric units of length, mass and capacity

You should know the following metric conversions for length, mass and capacity (liquid volume):

Mass Length 1 km = 1000 m 1 m = 100 cm 1 m = 1000 mm 1 cm = 10 mm 1 tonne = 1000 kg 1 kg = 1000 g 1 g = 1000 mg Capacity and Volume 1 litre = 1000 ml 1 cl = 10 ml 1 m 3 = 1000 litres 1 cm 3 = 1 ml 5 of 51

Metric units of area and volume

The metric units for area are mm 2 , cm 2 , m 2 and km 2 .

1 m 2 = 100 cm × 100 cm 1 m 2 = 10,000 cm 2 1 cm 2 = 10 mm 1 cm 2 × 10 mm = 100 mm 2 The metric units for volume are mm 3 , cm 3 , m 3 and km 3 .

1 m 3 = 100 cm × 100 cm × 100 cm 1 m 3 = 1,000,000 cm 3 1 cm 3 = 10 mm × 10 mm × 10 mm 1 cm 3 = 1000 mm 3

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Converting metric units

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To convert from a larger metric unit to a smaller one we need to

_______

by 10, 100, or 1000.

For example, Convert 0.43 kg to grams 0.43 kg = 0.43 × 1000 g = 430 g To convert from a smaller metric unit to a larger one we need to

_______

by 10, 100, or 1000.

For example, Convert 7.6 cm to metres 7.6 cm = 7.6 ÷ 100 m = 0.076 m © Boardworks Ltd 2005

Converting metric units

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Ordering units

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Imperial units

Imperial units are still frequently used and you should be aware of the following imperial conversions:

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Converting imperial units

We can convert between metric and imperial units using the following approximate conversions:

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Converting imperial units

1 kilogram is about 2.2 pounds.

About how many kilograms are there in 1 stone?

1 stone = 14 pounds Every 2.2 pounds is worth about 1 kilogram. This means that we have to divide 14 pounds by 2.2 to find the equivalent number of kilograms.

14 pounds ≈ 14 ÷ 2.2 kilograms = 6.4 kilograms 1 stone is about 6.4 kilograms

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Metric and imperial conversions

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Spider diagram

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S8 Measures

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S8.1 Converting units

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S8.2 Accuracy in measurement

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S8.3 Calculations involving bounds

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S8.4 Compound measures

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S8.5 Bearings 15 of 51

Continuous measurements

The continuous nature of measurements means that they can never be exact. There is always an element of rounding involved.

If we were measuring the length of a pencil, for example, we would measure to the nearest cm or mm. These three pencils all measure 14 cm to the nearest cm:

13 14 15 13 14 15 13 14 15

The length of a pencil given as 14 cm to the nearest cm can be any length between 13.5 cm and 14.5 cm.

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Accuracy in measurement

Remember, any measurement given to the nearest whole unit could be up to half a unit longer or shorter.

Suppose we are told that a pencil measures 14.2 cm.

Can we assume that this measurement is exact?

No, although this measurement has been given to a higher degree of accuracy it is not exact.

What is the shortest and longest length it could be?

The length

of a pencil given as 14.2 cm, to the nearest 0.1 cm, could be anywhere in the range 14.15 cm ≤

< 14.25 cm

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Upper and lower bounds

When we give a range for a measurement as in: 14.15 cm ≤ length < 14.25 cm This is an

inequality

this value is called the lower bound … … and this value is called the upper bound.

The length could be equal to 14.15 cm so we use a greater than or equal to symbol.

If the length was equal to 14.25 cm however, it would have been rounded up to 14.3 cm. The length is therefore “

strictly less than

” 14.25 cm and so we use the < symbol.

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Upper and lower bounds

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S8 Measures

A

S8.1 Converting units

A

S8.2 Accuracy in measurement

A

S8.3 Calculations involving bounds

A

S8.4 Compound measures

A

S8.5 Bearings 20 of 51

Adding measures

The following triangle has sides of length 3 cm, 4 cm and 5 cm.

5 cm 3 cm What is the range of possible lengths for the perimeter?

4 cm The least the lengths could be is 2.5 cm, 3.5 cm and 4.5 cm.

The smallest possible perimeter = 2.5 + 3.5 + 4.5 = 10.5 cm The most the lengths could be is 3.5 cm, 4.5 cm and 5.5 cm.

The largest possible perimeter = 3.5 + 4.5 + 5.5 = 13.5 cm

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Adding measures

The following triangle has sides of length 3 cm, 4 cm and 5 cm.

5 cm 3 cm What is the range of possible lengths for the perimeter?

4 cm The range of possible values for the perimeter is 10.5 cm ≤ perimeter < 13.5 cm Notice that the more lengths that are added together the greater the error on either side.

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Adding measures

When a calculation involves

adding

two or more measurements together: The lower bound is found by adding the lower bounds together The upper bound is found by adding the upper bounds together

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Calculations involving bounds

A piece of wood measuring 170 cm has a piece of length 50 cm cut off of it. These are given to the nearest 10 cm.

What is the range of possible sizes for the remaining piece?

The original piece of wood could be between 165 cm and 175 cm.

The piece of wood that is cut off could be between 45 cm and 55 cm.

The smallest possible size of the remaining piece is: 165 – 55 = 110 cm

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Calculations involving bounds

A piece of wood measuring 170 cm has a piece of length 50 cm cut off of it. These are given to the nearest 10 cm.

What is the range of possible sizes for the remaining piece?

The original piece of wood could be between 165 cm and 175 cm.

The piece of wood that is cut off could be between 45 cm and 55 cm.

The largest possible size of the remaining piece is: 175 – 45 = 130 cm

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Calculations involving bounds

A piece of wood measuring 170 cm has a piece of length 50 cm cut off of it. These are given to the nearest 10 cm.

What is the range of possible sizes for the remaining piece?

The original piece of wood could be between 165 cm and 175 cm.

The piece of wood that is cut off could be between 45 cm and 55 cm.

The range of possible sizes for the remaining piece is: 110 cm ≤ length < 130 cm

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Calculations involving bounds

The angles in the following diagram are rounded to the nearest degree: This angle could be between 34.5

° and 35.5

°.

35 °

78 ° This angle could be between 77.5

° and 78.5

°.

What is the range of possible values for the angle

The smallest

could be is 180 ° – (35.5° + 78.5°) = 66 ° The largest

could be is 180 ° – (34.5° + 77.5°) = 68° The range of possible values for

is therefore, 66 ° ≤

< 68 °

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Subtracting measures

When a calculation involves

subtracting

two measurements: The lower bound is found by: subtracting the upper bound from the lower bound The upper bound is found by: subtracting the lower bound from the upper bound

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Calculations involving bounds

The dimensions of a small tile are given as 15 mm by 18 mm.

18 mm What is the smallest possible area the tile could have?

15 mm The smallest values for the length and the width of the tile are 14.5 mm by 17.5 mm.

To calculate the smallest possible area we multiply these values together.

Smallest possible area = 14.5 × 17.5 = 253.75 mm 2

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Calculations involving bounds

The dimensions of a small tile are given as 15 mm by 18 mm.

18 mm What is the largest possible area the tile could have?

15 mm The largest values for the length and the width of the tile are 15.5 mm by 18.5 mm.

To calculate the largest possible area we multiply these values together.

Largest possible area = 15.5 × 18.5 = 286.75 mm 2

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Calculations involving bounds

The dimensions of a small tile are given as 15 mm by 18 mm.

18 mm What is the range of possible values for the area?

15 mm The smallest value for the area is 253.75 mm 2 and the largest value for the area is 286.75 mm 2 The range of possible values for the area is therefore, 253.75 mm 2 ≤ area < 286.75 mm 2

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Multiplying measures

When a calculation involves

multiplying

two or more measurements together: The lower bound is found by: multiplying the lower bounds together The upper bound is found by: multiplying the upper bounds together

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Calculations involving bounds

A boy runs 200 metres in 27.8 seconds.

The distance is given to the nearest metre and the time is given to the nearest tenth of a second.

What is his greatest possible average speed to 2 decimal places?

The distance could be between 195.5 m and 200.5 m.

The time taken could be between 27.75 s and 27.85 s.

greatest possible = average speed greatest distance shortest time = 200.5 m 27.75 s = 7.23 m/s

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Dividing measures

When a calculation involves

dividing

two measurements: The lower bound is found by: dividing the lower bound by the upper bound The upper bound is found by: dividing the upper bound by the lower bound

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S8 Measures

A

S8.1 Converting units

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S8.2 Accuracy in measurement

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S8.3 Calculations involving bounds

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S8.4 Compound measures

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S8.5 Bearings 35 of 51

Compound measures

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Which is heavier: a kilogram of cotton wool or a kilogram of rocks?

Compound measures

Of course, a kilogram of rocks weighs the same as a kilogram of cotton wool – they both weigh one kilogram!

If you thought that the rocks were heavier, you were probably thinking of their density rather than their weight.

Density is an example of a

compound measure

It is a measure of the mass of an object per unit volume.

Density is usually measured in g/cm 3 be measured in kg/

or kg/m 3 . It can also A density of 1.2 g/cm 3 means that every 1 cm 3 of the material has a mass of 1.2 grams.

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Compound measures

Which is heavier: 1 cm 3 of cotton wool or 1 cm 3 of rock?

This is a more sensible question. Rock is denser than cotton wool and so 1 cm 3 of rock is heavier than 1 cm 3 of cotton wool.

We can find the density of a given material using the following formula: Density = mass volume For example, if a block of metal has a mass of 760 g and a volume of 80 cm 3 : Density = 760 80 = 9.5 g/cm 3

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Compound measures

We use compound measures when we are comparing how one measurement changes with another.

When one measurement changes in direct proportion with another it is said to change at a

constant rate

For example, suppose a man is running around a race track.

The total distance he has run changes with time.

The rate at which he runs is called his speed.

Speed = distance travelled time taken Speed is usually measured in km/h, m/s or mph.

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Average speed

In many situations the speed is not constant.

For example, the man running around the track will probably speed up or slow down as he runs.

We can still calculate his average speed using the following formula: Average speed = total distance travelled total time taken For example, if the man runs 1560 metres in 300 seconds Average speed = 1560 300 = 5.2 m/s

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Common compound measures

Commonly used compound measures include: Density Speed Pressure Fuel consumption mass volume distance time force surface area distance volume Measured in g/cm kg/m 3 or kg/

3 , Measured in m/s, km/h or mph.

Measured in N/m 2 N/cm 2 .

or Measured in km/

or mpg.

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Pairs – compound measures

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Calculating densities

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Converting compound units

What is 45 mph in m/s?

When we convert compound units we usually have to do it in several steps, especially if both units are being changed.

45 mph stands for 45 miles per hour.

We have to change the 45 miles into metres and 1 hour into seconds.

5 miles = 8 km 45 miles ≈ 45 × 1.6 km So 1 mile = 1.6 km = 72 km = 72,000 m 1 hour = 60 minutes = 3600 seconds

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Converting compound units

What is 45 mph in m/s?

Travelling 45 miles in 1 hour is equivalent to travelling 72 000 m in 3600 seconds.

Therefore, 45 mph ≈ 72,000 3600 m/s = 20 m/s How could we use this answer to convert any speed given in mph to a speed in m/s?

45 mph ≈ 20 m/s Dividing by 5 we have, 9 mph ≈ 4 m/s We can divide the speed in mph by 9 and multiply it by 4.

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S8 Measures

A

S8.1 Converting units

A

S8.2 Accuracy in measurement

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S8.3 Calculations involving bounds

A

S8.4 Compound measures

A

S8.5 Bearings

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Bearings

Bearings

are a measure of direction taken from North.

If you were travelling North you would be travelling on a bearing of 000 °.

bearing of

000 °.

N 75 ° P

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

Compass points

315 °

000 °

N NE

045 ° 270 °

W E

090 °

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225 °

SW S

180 °

Measuring bearings

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Back bearings

The bearing from point A to point B is 105 º.

What is the bearing from point B to point A?

N A 105 º N B 180 ° ?

º The angle from B to A is 105 º + 180º =

285 º

This is called a

back bearing

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Investigating back bearings

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S8 Measures

Transcript S8 Measures

KS4 Mathematics

S8 Measures

Contents

S8 Measures

A

S8.1 Converting units

A

A

A

A

Converting units

Metric units

metre.

gram.

litre.

Metric units of length, mass and capacity

Metric units of area and volume

Converting metric units

Converting metric units

Ordering units

Imperial units

Converting imperial units

Converting imperial units

Metric and imperial conversions

Spider diagram

Contents

S8 Measures

A

A

S8.2 Accuracy in measurement

A

A

A

Continuous measurements

Accuracy in measurement

Upper and lower bounds

Upper and lower bounds

Contents

S8 Measures

A

A

A

S8.3 Calculations involving bounds

A

A

Adding measures

Adding measures

Adding measures

Calculations involving bounds

Calculations involving bounds

Calculations involving bounds

Calculations involving bounds

Subtracting measures

Calculations involving bounds

Calculations involving bounds

Calculations involving bounds

Multiplying measures

Calculations involving bounds

Dividing measures

Contents

S8 Measures

A

A

A

A

S8.4 Compound measures

A

Compound measures

Compound measures

Compound measures

Compound measures

Average speed

Common compound measures

Pairs – compound measures

Calculating densities

Converting compound units

Converting compound units

Contents