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Why we need the same units

Measurement · level 2, lesson 1 of 6

A question to start

In 1999 a spacecraft called the Mars Climate Orbiter flew all the way to Mars, and then it was lost. What went wrong? Two teams of engineers had worked on it. One team's computer gave its numbers in one kind of unit. The other team's computer read those numbers as if they were in a different kind of unit. The numbers looked fine. But they meant different amounts, a bit like "5 of my hand spans" and "5 of your hand spans". The spacecraft's path was worked out wrong, it came much too close to Mars, and it was never heard from again.

A very clever team, a very expensive spacecraft, and the problem was units. In this lesson you will see why everyone in the world agreed to use the same units, and how to measure with a ruler the right way.

The idea

The body measure problem

In level 1 you measured with hands and feet. You found that a child and a grown-up get different numbers for the same table, because their hands are different sizes. A smaller unit gives a bigger number.

That is fine when you are playing. But imagine:

  • You phone a shop and ask for a curtain "8 of my hand spans wide". The shop worker uses her hand span. The curtain arrives far too wide.
  • A builder makes a door "40 of my feet tall" (he has very small feet) and his helper makes the doorway "40 of my feet tall" (big feet). The door does not fit.

The number is not enough. You also need to know which unit, and everyone has to mean the same thing by it.

Standard units

A standard unit is a unit that everybody agrees on. It is the same size everywhere: in your house, in a shop in Japan, and on a spacecraft going to Mars.

People made many standard units over the years. In the 1790s, scientists in France made a new one, the metre. They wanted a unit that belonged to no king and no country, so they based it on the size of the Earth itself. Today almost every country uses the metre. Canada changed to metres (from feet and inches) in the 1970s, although you still hear feet and inches sometimes.

The metre

A metre is about:

  • one big step of a grown-up,
  • the distance from the floor to your tummy or chest, if you are 7 or 8,
  • half the height of a door (a door is about 2 metres tall).

We write m for metres. "The room is 5 m long" means 5 metres long. A metre stick is a long ruler exactly 1 metre long. A tape measure rolls up and can measure many metres.

The centimetre

Metres are too big for small things. A pencil is much shorter than a metre. So the metre is cut into 100 equal parts. Each part is a centimetre. We write cm.

100 centimetres make 1 metre.

A centimetre is small: about the width of your little finger's fingernail. A paper clip is about 3 cm long. A new pencil is about 19 cm long. The school ruler on your desk is probably 30 cm long.

Why is it called a "centi"-metre? "Centi" means "a hundredth part", the same as in "century" (100 years) and "cent" (one hundredth of a dollar). 100 cents make a dollar, and 100 centimetres make a metre.

How to read a ruler

A ruler is a row of centimetres stuck together, with numbers. Here is a pencil measured with a ruler.

0 1 2 3 4 5 6 7 8 9 10 7 cm

The pencil starts at 0 and ends at 7. It is 7 cm long.

What do the numbers on a ruler mean? This is the key idea. The number at a mark tells you how many centimetre spaces there are from 0 up to that mark. From 0 to 1 is one space. From 0 to 7 is seven spaces. The numbers count the spaces (the lengths), not the lines.

That is why the 0 matters. The 0 is where the counting starts: no spaces yet.

Start at 0, not at the end of the ruler

Look closely at a real ruler. On most rulers, the 0 mark is not right at the end. There is a little bit of plastic or wood before the 0. What happens if you line the pencil up with the end of the ruler instead of the 0?

0 1 2 3 4 5 6 7 8 9 10

The same 7 cm pencil now seems to end between 6 and 7! The little bit before the 0 is not counted, so the pencil looks shorter than it is. Always line up the start of the thing with the 0 mark, not the end of the ruler.

A broken ruler still works

What if your ruler is broken, and it starts at 3? You can still measure. Line up one end of the thing with any mark, then count the spaces.

3 4 5 6 7 8 9 10 11 12 13

The pencil starts at 3 and ends at 8. Count the centimetre spaces: 3 to 4, 4 to 5, 5 to 6, 6 to 7, 7 to 8. That is 5 spaces, so the pencil is 5 cm long. It is not 8 cm! The 8 only counts spaces from 0, and this pencil did not start at 0.

You can also think: "from 3 up to 8 is how many?" That is 8 take away 3, which is 5. Measuring from any mark is a take-away sum.

Always write the unit

"The table is 90" does not tell anyone anything. 90 what? Centimetres? Metres? Hand spans? Always write the unit: 90 cm. A number with no unit is like the Mars Climate Orbiter's numbers: easy to read the wrong way.

Worked examples

Example 1. Read the ruler. A crayon is lined up with 0 and its other end is at 9. How long is it?

  1. The crayon starts at 0, so the number at the end tells us how many centimetre spaces it covers.
  2. The end is at 9.
  3. The crayon is 9 cm long.

Example 2. A ribbon on a broken ruler. A ribbon starts at the 4 mark and ends at the 11 mark. How long is it?

  1. It does not start at 0, so we must count spaces.
  2. From 4 to 11: 5, 6, 7, 8, 9, 10, 11. That is 7 spaces.
  3. Or: 11 take away 4 is 7.
  4. The ribbon is 7 cm long.

Example 3. Metres and centimetres. Ana is 1 metre and 25 centimetres tall. How many centimetres tall is she?

  1. 1 metre is 100 centimetres.
  2. 100 cm and 25 more cm is 125 cm.
  3. Ana is 125 cm tall.

Example 4. Which unit? Would you measure a ladybug, a swimming pool and a book in centimetres or metres?

  1. A ladybug is tiny, less than a centimetre or so: centimetres (and even a centimetre is a bit big).
  2. A swimming pool is many big steps long: metres.
  3. A book is shorter than a ruler: centimetres.

Try it

  1. How many centimetres make 1 metre?
  2. A pencil starts at 0 on a ruler and ends at 12. How long is it?
  3. Tom measures a stick. He lines up the stick with the end of the ruler, not the 0, and reads 14 cm. Is the stick really 14 cm, a bit more, or a bit less?
  4. A nail starts at the 2 mark and ends at the 7 mark. How long is the nail?
  5. Would you use centimetres or metres for: a hallway, an eraser, a bus, your hand?
  6. A rope is 2 metres long. How many centimetres is that?
  7. Lea says her desk is "60". Her friend says it is "6". Could they both be right? What is missing?
  8. A broken ruler starts at 5. A paper clip lies from the 5 mark to the 8 mark. Ben says it is 8 cm. What did he do wrong, and what is the real length?
Answers
  1. 100 centimetres make 1 metre.
  2. 12 cm. It starts at 0, so the end number is the length.
  3. A bit more than 14 cm. The bit of ruler before the 0 was not counted, so the reading is too small. The stick is about 14 cm plus that little bit of the ruler's end.
  4. From 2 to 7: 3, 4, 5, 6, 7. That is 5 cm (or 7 take away 2 is 5).
  5. Hallway: metres. Eraser: centimetres. Bus: metres. Your hand: centimetres.
  6. Each metre is 100 cm, so 2 metres is 100 and 100, which is 200 cm.
  7. Yes, they could be! 60 cm and 6 hand spans (or 6 of something else) might be the same desk. What is missing is the unit. Without the unit, the number does not tell us how long the desk is.
  8. Ben read the number at the end, but the clip did not start at 0. Count the spaces from 5 to 8: 6, 7, 8. The clip is 3 cm long (8 take away 5).

Common mistake

Starting at the end of the ruler, or at 1. The length counts spaces from 0. Line up with the 0 mark. If you start at 1, every measurement comes out 1 cm too long, because the first space is never counted. Check: a thing that reaches from 1 to 6 covers only 5 spaces.

Forgetting the unit. Always write cm or m. "5" could be 5 cm (a short crayon) or 5 m (a long car).

Where it leads