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The metric family

Measurement · level 3, lesson 1 of 6

A question to start

Three friends measure a race track in three different ways.

  • Ana says: "It is 2 kilometres."
  • Ben says: "It is 1999 metres."
  • Cleo says: "It is 210 000 centimetres."

Who has the longest track? Cleo's number is the biggest by far, and Ana's is the smallest. But the size of a number means nothing until you know the unit. By the end of this lesson you will be able to change all three into the same unit in a few seconds and see the answer. (It is at the end of the worked examples.)

The idea

One family of units

In level 2 you met the metre (m) and the centimetre (cm). They belong to a family called the metric system, which most of the world uses. The family was planned in France in the 1790s. The scientists wanted units that belonged to no king and no country, so they based the metre on the size of the Earth: one ten-millionth of the distance from the North Pole to the equator.

The clever part of the plan is the names. Every length unit is the word "metre" with a little front piece, a prefix, that tells you how big it is.

Prefix What it means Unit How big
kilo- a thousand kilometre (km) 1 km = 1000 m
(no prefix) the main unit metre (m) 1 m
centi- a hundredth centimetre (cm) 100 cm = 1 m
milli- a thousandth millimetre (mm) 1000 mm = 1 m

"Centi" is the same word as in cent (a hundredth of a dollar) and century (a hundred years). "Milli" is like millennium (a thousand years). "Kilo" comes from the Greek word for a thousand. Once you know the prefixes, you know the units.

How big is each one?

It helps to have a picture in your head for every unit. These are called benchmarks.

  • 1 mm: the thickness of a bank card, or of a grain of sugar.
  • 1 cm: the width of your little fingernail.
  • 1 m: a big step for a grown-up, or the height of a door handle from the floor.
  • 1 km: about 10 to 12 minutes of walking, or the length of about 10 football fields end to end.

Look closely at a ruler. The numbered lines are centimetres. Between each pair of numbers there are 10 tiny spaces: those are millimetres.

0123 cm 1 cm = 10 mm

The ladder of units

Put the units in order, from biggest to smallest, and write how many of each fit into the one above.

kmmcmmm × 1000× 100× 10 ÷ 1000÷ 100÷ 10 to a smaller unit: multiply · to a bigger unit: divide

Multiply or divide? Ask one question

When you change units, ask: is the new unit smaller or bigger?

  • Changing to a smaller unit, you need more of them, so the number gets bigger: multiply. A 3 m rope is \(3 \times 100 = 300\) cm, because every metre is 100 centimetres.
  • Changing to a bigger unit, you need fewer of them, so the number gets smaller: divide. A 4000 m run is \(4000 \div 1000 = 4\) km, because every 1000 metres make one kilometre.

Here is a way to remember it. Think of paying for something with coins. If you pay with small coins (pennies) you need lots of them. If you pay with big coins (toonies) you need only a few. Small unit, big number; big unit, small number.

Why the decimal point moves

Our numbers are written in place value: each column is worth 10 times the column to its right.

thousands hundreds tens ones . tenths hundredths
3 . 4 5
3 4 5 .

When you multiply by 10, every digit becomes worth 10 times as much, so every digit slides one column to the left. Multiply by 100 and every digit slides two columns; by 1000, three columns. The table shows 3.45 m changed into centimetres: \(3.45 \times 100 = 345\). The digits 3, 4 and 5 did not change at all. They only slid two places, which looks just like the decimal point hopping two places to the right.

Dividing works the other way: by 10, the digits slide one place to the right; by 1000, three places. So \(2500 \div 1000 = 2.5\): the point hops three places to the left.

This is the whole reason the metric system is so easy. Its units go up in 10s, 100s and 1000s, and so does our way of writing numbers. The two fit together like a key in a lock. (Old units such as feet and inches do not: there are 12 inches in a foot and 5280 feet in a mile, so changing them takes real arithmetic.)

When the point needs to hop past the end of a number, fill the empty places with zeros: \(0.7 \text{ km} = 0.7 \times 1000 = 700\) m, and \(6 \text{ mm} = 6 \div 10 = 0.6\) cm.

Mixed measurements

People often say lengths in two units at once: "1 metre 35 centimetres". To write it as one number of metres, remember that 35 cm is 35 hundredths of a metre: 1 m 35 cm = 1.35 m. In centimetres it is \(100 + 35 = 135\) cm.

Watch out for small parts: 2 m 5 cm is 2.05 m, not 2.5 m. Five centimetres is 5 hundredths, so the 5 goes in the hundredths column, with a 0 holding the tenths place.

Choosing a sensible unit

Any length can be written in any unit, but one choice is usually the most sensible: the one that gives a number that is easy to picture. A bee is about 12 mm long. A book is about 25 cm tall. A classroom is about 9 m long. The drive to the next town is about 40 km. You could say the classroom is 9000 mm, but nobody can picture 9000 of anything.

Worked examples

1. Metres to centimetres. A desk is 1.2 m long. How many centimetres is that?

Centimetres are smaller, so multiply. \(1.2 \times 100 = 120\). The desk is 120 cm long. (The point hops two places to the right; we add a 0 to fill the empty place.)

2. Metres to kilometres. A school run is 3750 m. How many kilometres?

Kilometres are bigger, so divide. \(3750 \div 1000 = 3.75\). The run is 3.75 km. Check with sense: 3750 m is a bit less than 4000 m, which is 4 km. So a bit less than 4 km is right.

3. Millimetres to centimetres. A beetle is 23 mm long. How many centimetres?

Centimetres are bigger, so divide by 10. \(23 \div 10 = 2.3\). The beetle is 2.3 cm long.

4. A mixed measurement. A path is 2 km 50 m long. How many metres?

2 km is \(2 \times 1000 = 2000\) m. Add the 50 m: \(2000 + 50 = 2050\) m. As kilometres it is 2.05 km (50 m is 50 thousandths of a kilometre, so two zeros come first: 0.050).

5. Putting lengths in order. Order from shortest to longest: 0.6 m, 58 cm, 605 mm.

Change all to millimetres: \(0.6 \times 1000 = 600\) mm, and \(58 \times 10 = 580\) mm, and 605 mm stays. In order: 58 cm, 0.6 m, 605 mm.

6. The race track. Change all three to metres.

  • Ana: 2 km is \(2 \times 1000 = 2000\) m.
  • Ben: 1999 m.
  • Cleo: 210 000 cm is \(210\,000 \div 100 = 2100\) m.

Cleo's track is the longest (2100 m), Ana's is next (2000 m), and Ben's is the shortest, just 1 m shorter than Ana's. The biggest number did belong to the longest track this time, but only by luck: Ben's number was bigger than Ana's and his track was shorter.

Try it

  1. Change to centimetres: (a) 4 m; (b) 7.5 m; (c) 90 mm.
  2. Change to metres: (a) 3 km; (b) 0.25 km; (c) 640 cm.
  3. Change to kilometres: (a) 5000 m; (b) 12 500 m; (c) 800 m.
  4. Write 3 m 7 cm as metres, and as centimetres.
  5. Which unit (mm, cm, m or km) would you use for: the length of a pencil; the height of a tree; the thickness of a coin; the distance from Toronto to Montréal?
  6. Order from shortest to longest: 1.5 km, 1450 m, 160 000 cm.
  7. A ribbon is 3 m long. You cut off pieces of 25 cm. How many pieces do you get?
  8. A snail crawls 2 mm every second. How long does it take, in seconds, to crawl 1 m? About how many minutes is that?
Answers
  1. (a) \(4 \times 100 = 400\) cm. (b) \(7.5 \times 100 = 750\) cm. (c) Centimetres are bigger, so divide: \(90 \div 10 = 9\) cm.
  2. (a) \(3 \times 1000 = 3000\) m. (b) \(0.25 \times 1000 = 250\) m. (c) \(640 \div 100 = 6.4\) m.
  3. (a) \(5000 \div 1000 = 5\) km. (b) \(12\,500 \div 1000 = 12.5\) km. (c) \(800 \div 1000 = 0.8\) km.
  4. 7 cm is 7 hundredths of a metre, so 3 m 7 cm = 3.07 m. In centimetres: \(300 + 7 = 307\) cm.
  5. Pencil: centimetres (about 18 cm). Tree: metres. Coin: millimetres (about 2 mm). Toronto to Montréal: kilometres (it is a few hundred kilometres by road).
  6. All in metres: 1.5 km = 1500 m; 1450 m; 160 000 cm = 1600 m. Order: 1450 m, 1.5 km, 160 000 cm.
  7. Use the same unit: 3 m = 300 cm. Then \(300 \div 25 = 12\) pieces.
  8. 1 m = 1000 mm. At 2 mm each second it takes \(1000 \div 2 = 500\) seconds. A minute is 60 seconds, and \(8 \times 60 = 480\), so 500 seconds is 8 minutes 20 seconds: about 8 minutes.

Common mistake

Multiplying when you should divide. "450 cm is \(450 \times 100 = 45\,000\) m" cannot be right: 450 cm is about the length of a car, not 45 km! Always ask "is the new unit bigger or smaller?", and then check the answer with sense. Going to a bigger unit, the number must get smaller: 450 cm = 4.5 m.

Losing the zero. 3 m 7 cm is 3.07 m, not 3.7 m. The digit after the point counts tenths of a metre (10 cm each), and 7 cm is less than one tenth. Writing 3.7 m would add 63 cm by accident.

Where it leads