G14, G15 - Measuring, scale drawings and bearings

G14, G15 - Measuring, scale drawings and bearings

Measurements need a number, a unit and a sensible way to read the instrument. In G14 you will use standard measures in geometrical problems. In G15 you will read rulers and protractors, connect measurements to scale drawings, and use compass directions and three-figure bearings to describe a journey precisely.

Choosing and using measures

A rescue boat travels from harbour A to buoy B on a bearing of 064∘064^\circ. What bearing should it follow to come straight back? The same line is involved, but the direction has changed. We will return to the boat after putting its measurements in order.

You already know

Metric conversion factors turn one unit into another. A scale drawing keeps corresponding lengths in the same ratio. Angles on a straight line add to 180∘180^\circ.

Choose the unit for the quantity being measured. A room's floor area is measured in square metres; the paint in a tin is measured by volume, often in litres. A person's mass is measured in kilograms. Metres would not answer either question.

QuantitySuitable standard unitsWhat the unit measures
Lengthmillimetres, centimetres, metres, kilometresdistance along a line
Areasquare centimetres, square metresamount of surface
Volume and capacitycubic centimetres, cubic metres, millilitres, litresspace occupied or space available inside
Massgrams, kilograms, tonnesamount of matter
Timeseconds, minutes, hoursduration
Moneypence, poundscost or value

Capacity is the volume a container can hold. A solid block can have a volume even though it has no capacity. Useful connections are 1 cm3=1 mL1\,\mathrm{cm}^3=1\,\mathrm{mL} and 1000 cm3=1 L1000\,\mathrm{cm}^3=1\,\mathrm{L}. These are not on the formulae sheet: learn them.

Worked example 1: a capacity measurement

Question. A measuring jug contains 750 mL750\,\mathrm{mL} of water. A further 0.4 L0.4\,\mathrm{L} is poured in. Work out the total volume in litres. Non-calculator.

Plan. Express both volumes in litres before adding.

750÷1000=0.75millilitres to litres0.75+0.4=1.15add like units\begin{aligned} 750\div1000&=0.75 &&\text{millilitres to litres}\\ 0.75+0.4&=1.15 &&\text{add like units} \end{aligned}

Why convert first? Adding 750750 and 0.40.4 would combine different-sized units.

Answer. 1.15 L1.15\,\mathrm{L}.

Check. 750+400=1150 mL750+400=1150\,\mathrm{mL}, which is just over one litre.

Change the kind of measure. Time uses groups of 6060, so a time written with hours and minutes is not a decimal number of hours.

Worked example 2: a duration and a charge

Question. A boat leaves at 10:45 and returns at 12:20. Hire costs £6 for each complete 55-minute period. Work out the hire charge. Non-calculator.

Plan. Find the duration in minutes, then count the 55-minute periods.

15+60+20=95minutes from 10:45 to 12:2095÷5=19complete periods19×6=114charge in pounds\begin{aligned} 15+60+20&=95 &&\text{minutes from 10:45 to 12:20}\\ 95\div5&=19 &&\text{complete periods}\\ 19\times6&=114 &&\text{charge in pounds} \end{aligned}

Why work in minutes? The price is for a fixed number of minutes, not for a decimal part of an hour.

Answer. £114.

Check. The trip lasts 11 hour 3535 minutes, which is 60+35=9560+35=95 minutes.

The unit also tells you how to interpret an answer. A hire charge needs money notation; a duration needs a time unit.

Watch out

Wrong: 11 hour 3535 minutes =1.35=1.35 hours. Right: 1+35601+\frac{35}{60} hours. The digits after a decimal point are tenths and hundredths, not minutes.

Keep the units visible when you combine a duration with a rate.

Reading rulers and protractors

An instrument gives a useful measurement only when you know its smallest division and its starting point.

On a millimetre ruler, ten small divisions make one centimetre. Put the ruler's zero mark at the first endpoint, rather than using the physical end of the ruler. Look straight down so the endpoint lines up with the correct mark. If the first endpoint is not at zero, subtract its reading from the final reading.

For an angle, put the protractor's centre at the vertex and its baseline along one arm. Read from the zero on that arm towards the other arm. The two number scales usually add to 180∘180^\circ, so first decide whether the angle should be acute or obtuse.

[DIAGRAM: asset_name: Reading a ruler and a protractor; asset_slug: aqa8300_gd_measuring_scales; caption: Subtract the ruler readings and start the protractor scale from the arm's zero; description: The left panel shows a ruler in centimetres with millimetre divisions. A blue segment begins at 1.4 cm and ends at 6.7 cm, giving length 5.3 cm. The right panel shows an acute 37 degree angle with its initial arm pointing right. A note explains that the paired readings are 37 degrees and 143 degrees and the correct reading starts at zero on the right. The image is a scale-reading illustration and must not be measured on screen.]
Subtract the ruler readings and start the protractor scale from the arm's zero

Worked example 3: endpoints away from zero

Question. A line segment's endpoints align with 1.4 cm1.4\,\mathrm{cm} and 6.7 cm6.7\,\mathrm{cm} on a ruler. Work out its length in millimetres. Non-calculator.

Plan. Subtract the endpoint readings, then convert centimetres to millimetres.

6.7−1.4=5.3length in centimetres5.3×10=53length in millimetres\begin{aligned} 6.7-1.4&=5.3 &&\text{length in centimetres}\\ 5.3\times10&=53 &&\text{length in millimetres} \end{aligned}

Why subtract? The 1.4 cm1.4\,\mathrm{cm} before the segment is not part of its length.

Answer. 53 mm53\,\mathrm{mm}.

Check. The segment is a little longer than 5 cm5\,\mathrm{cm}, so 53 mm53\,\mathrm{mm} is sensible.

Change the instrument from a ruler to a protractor. The starting point still decides which reading matters.

Worked example 4: choosing the angle scale

Question. One arm of an acute angle lies along the protractor's right-hand zero. The other arm crosses paired marks 37∘37^\circ and 143∘143^\circ. Write down the angle. Non-calculator.

Plan. Follow the scale beginning with zero on the right, then check the angle's size.

acute angle<90∘use the stated angle type37∘<90∘select the matching reading\begin{aligned} \text{acute angle}&<90^\circ &&\text{use the stated angle type}\\ 37^\circ&<90^\circ &&\text{select the matching reading} \end{aligned}

Why reject 143∘143^\circ? It is an obtuse angle measured from the opposite baseline direction.

Answer. 37∘37^\circ.

Check. 37+143=18037+143=180, as expected for paired protractor readings.

If an angle to be measured is reflex, first measure the smaller angle between its arms, then subtract from 360∘360^\circ. A small sketch helps you choose the intended region.

Watch out

Wrong: select the first protractor number you see. Right: locate the vertex, align the baseline, and read from the correct zero. An obviously acute angle cannot be 143∘143^\circ.

The next angle uses the same paired-scale decision, with an obtuse angle instead.

Compass and three-figure bearings

A bearing describes a direction from a named starting point, using north as the reference.

Three-figure bearing

An angle measured clockwise from north and written with three figures. For example, east is 090∘090^\circ, south is 180∘180^\circ, and a direction 7∘7^\circ clockwise from north is 007∘007^\circ. “The bearing of B from A” starts at A and looks towards B.

The eight compass points divide one full turn into eight equal angles. Each step is 360∘÷8=45∘360^\circ\div8=45^\circ. North, east, south and west are separated by 90∘90^\circ; north-east sits halfway between north and east.

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