4.4 Measures

Learning objectives

4.4A Scales on a range of measuring instrumentsinterpret scales on a range of measuring instruments 4.4B Time intervals in terms of the 24-hour and the 12-hourcalculate time intervals in terms of the 24-hour and the 12-hour clock Use am and pm 4.4C Make sensible estimates of a range of measuresmake sensible estimates of a range of measures 4.4D Angle measureunderstand angle measure including three-figure bearings 4.4E Measure an angle to the nearest degreemeasure an angle to the nearest degree 4.4F Average speed, distance and timeunderstand and use the relationship between average speed, distance and time 4.4G Compound measure such as speed, density and pressureuse compound measure such as speed, density and pressure Formula for pressure will be given

Read the numbered marks first, then count the equal spaces between them. The value of one smallest division is $ext{difference between labels}\div ext{number of spaces}$ .

Step	Check
1	identify the unit and whether the scale increases or decreases
2	find two labelled marks
3	count spaces, not grid lines, between them
4	multiply the number of spaces from a label by one-division value

If 300 and 400 are separated by five equal spaces, each space represents 20. A pointer two spaces after 300 reads $300+2(20)=340$ .

Do not divide by the number of drawn marks between two labels. Four internal marks create five spaces.

In 24-hour time, use four digits: 3:20 pm is 1520 and 12:00 midnight is 0000. In 12-hour time, state am or pm whenever the context does not already fix it.

Situation	Reliable method
same hour	subtract minutes
crosses an hour	count to the next hour, then onward
crosses noon or midnight	split at 1200 or 0000 and add intervals
long interval	convert both times to minutes after midnight, adjusting the next day

From 1635 to 2015: 25 minutes to 1700, then 3 hours 15 minutes to 2015, so the interval is 3 hours 40 minutes.

Subtract times only after using a consistent format. Clock notation is base 60, so 2015 minus 1635 is not ordinary decimal subtraction.

A sensible estimate combines a familiar benchmark, the correct unit and an order of magnitude that fits the object or event.

Measure	Useful benchmark
length	a doorway is about $2 ext{ m}$ high
mass	a bag of sugar is about $1 ext{ kg}$
capacity	a drinking glass holds a few hundred millilitres
time	a short walk is measured in minutes, not seconds or days

Choose the measure type, select a plausible unit, compare with a known benchmark, then reject values that are ten or a hundred times too large or small.

Precision does not make an implausible value sensible. An estimate such as $201.7 ext{ L}$ for a drinking glass has the wrong scale even though it looks precise.

A bearing is an angle measured clockwise from north at the starting point. Write it with three digits from $000^\circ$ to $359^\circ$ , such as $073^\circ$ .

Step	Action
1	draw or identify north at the starting point
2	turn clockwise from north to the direction line
3	measure or calculate the angle
4	write leading zeros when needed

Reverse bearings differ by $180^\circ$ : add $180^\circ$ if the bearing is below $180^\circ$ ; subtract $180^\circ$ if it is at least $180^\circ$ . Thus the reverse of $073^\circ$ is $253^\circ$ .

The north line must be placed at the point the journey starts from. Measuring anticlockwise or from north at the destination gives the wrong bearing.

Place the protractor centre exactly on the angle vertex and align its zero line with one arm. Read where the other arm crosses the correct scale.

Check	Question to ask
centre	is the protractor midpoint on the vertex?
baseline	does the zero line lie on one arm?
scale	does the chosen scale start at $0^\circ$ on that arm?
reasonableness	should the angle be acute, right, obtuse or reflex?

If the ray falls between degree marks, read the closest mark; for example $109.6^\circ$ rounds to $110^\circ$ .

The two printed protractor scales run in opposite directions. Use the scale whose zero is on the aligned arm, not the first number you see.

Average speed is total distance divided by total time: $v=d/t$ . Rearranging gives $d=vt$ and $t=d/v$ .

Desired speed	Match distance with time
km/h	kilometres and hours
m/s	metres and seconds
mph	miles and hours

For 40 km in 2 hours 15 minutes, convert time to $2.25$ hours. Then $v=40/2.25=17.77\ldots$ , so the average speed is $18 ext{ km/h}$ to the nearest whole number.

Average speed uses total distance and total elapsed time, including any stops unless the question explicitly excludes them. Do not average separate speed values without weighting by time or distance.

A compound measure combines two quantities. Use $ext{speed}= ext{distance}/ ext{time}$ , $ext{density}= ext{mass}/ ext{volume}$ and, when given, $ext{pressure}= ext{force}/ ext{area}$ .

Find	Rearrangement
mass	$m= ho V$
volume	$V=m/ ho$
force	$F=pA$
area	$A=F/p$

A $12 ext{ cm} imes8 ext{ cm} imes5 ext{ cm}$ block has volume $480 ext{ cm}^3$ . At density $0.7 ext{ g/cm}^3$ , its mass is $0.7(480)=336 ext{ g}$ .

Convert before substituting: $1 ext{ kg}=1000 ext{ g}$ and $1 ext{ m/s}=3.6 ext{ km/h}$ . The numerator and denominator units determine the compound unit.

Pressure uses contact area, not total surface area; density uses the object's full volume. A correct formula with inconsistent units still gives a wrong answer.

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4.4 Measures

Learning objectives

Interpret scales on measuring instruments

Calculate 12-hour and 24-hour time intervals

Make sensible estimates of measures

Read and calculate three-figure bearings

Measure an angle to the nearest degree

Use average speed, distance and time

Use speed, density and pressure as compound measures

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