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Measuring and estimating length

Measurement · level 2, lesson 2 of 6

A question to start

Stand up and stretch your arms out wide to the sides, as wide as they go. The distance from the tip of one middle finger to the tip of the other is your arm span.

Now guess: is your arm span longer than you are tall, shorter, or about the same?

For most people, the answer is a surprise: your arm span is about the same as your height. Not exactly, but close. Ask someone to check with a tape measure. If you are 125 cm tall, your arm span is probably somewhere near 125 cm too.

Now you carry a "ruler" for your own height wherever you go. That is the idea of this lesson: knowing a few lengths by heart lets you make good guesses about everything else.

The idea

What is an estimate?

An estimate is a sensible guess, made by thinking. It is not a wild guess. "The bus is 3 cm long" is a wild guess (that is smaller than your thumb!). "The bus is about 12 m long, because it is about as long as 12 big steps" is an estimate.

We estimate all the time. Builders estimate before they buy wood. Cooks estimate before they cut a cake. Scientists estimate first, so that when they measure carefully they can tell at once if the careful answer is silly. Estimating is not cheating. It is a skill, and you get better with practice.

Benchmarks

A benchmark is a thing whose length you already know, which you use to help you estimate. Here are some good ones.

Benchmark About
The width of your little finger's nail 1 cm
A paper clip 3 cm
A new, unsharpened pencil 19 cm
A school ruler 30 cm
A grown-up's big step 1 m
A door, from the floor to the top 2 m

The best benchmarks are on your own body, because you always have them with you. Measure these on yourself with a ruler and remember them:

  • the width of your little fingernail (about 1 cm),
  • your hand span (thumb tip to little finger tip, stretched),
  • the length of your foot,
  • your arm span (about your height).

How to use a benchmark

To estimate a length, picture the benchmark fitting along it, again and again, and count. This is just like measuring with paper clips in level 1, except you do it in your head.

  • A book is about as long as one ruler, so it is about 30 cm.
  • A window is about 3 rulers wide, so it is about 30, 60, 90: about 90 cm.
  • A classroom is about 8 big steps long, so it is about 8 m.

Why does this work? Because each benchmark has a length you know. If one ruler is 30 cm, then three rulers end to end are 30 cm and 30 cm and 30 cm, which is 90 cm.

Estimate first, then measure

Here is a good habit.

  1. Look and estimate. Write down your estimate.
  2. Measure carefully with a ruler or tape measure.
  3. Compare: how far off were you? Was your estimate too big or too small?

The next estimate will be better, because your brain learns from each one. This is how carpenters get so good at "eyeballing" a length.

Measuring to the nearest centimetre

Most things are not exactly a whole number of centimetres long. Look at this crayon.

0 1 2 3 4 5 6 7 8 9 10

Its end is between 6 and 7. Which is it closer to? It is past the little half-way mark, so it is closer to 7. We say the crayon is "about 7 cm" or "7 cm to the nearest centimetre".

Now this key.

0 1 2 3 4 5 6 7 8 9 10

Its end is between 4 and 5, but before the half-way mark. It is closer to 4. The key is about 4 cm long.

The half-way mark between two numbers is very helpful. Past it, go up to the next number. Before it, stay at the smaller number.

Things longer than your ruler

What if the desk is longer than your 30 cm ruler? Measure in pieces.

30 cm30 cm12 desk 30 + 30 + 12 = 72 cm
  1. Lay the ruler down from the end of the desk. Put a small pencil mark where the ruler ends (30 cm).
  2. Move the ruler so its 0 is exactly on the mark. No gap, no overlap. That is another 30 cm.
  3. The last bit is shorter than a ruler. Read it: 12 cm.
  4. Add the pieces: 30 and 30 is 60, and 12 more is 72 cm.

The same no-gaps, no-overlaps rule from level 1 still applies. A gap or an overlap at a mark makes the whole answer wrong.

Centimetres or metres?

Choose the unit that makes a sensible number.

  • Small things you can hold: centimetres. A spoon is about 15 cm.
  • Big things, rooms and distances you walk: metres. A room is about 4 m wide.

You could say the room is 400 cm wide, and that is correct, but "4 m" is easier to picture. And you could say a spoon is "a tiny bit of a metre", but "15 cm" is much clearer.

Worked examples

Example 1. Estimate with a benchmark. About how tall is a doorway in your house, and about how tall is a bookcase that reaches halfway up it?

  1. Benchmark: a door is about 2 m tall.
  2. The bookcase reaches halfway up the door.
  3. Half of 2 m is 1 m. So the bookcase is about 1 m tall.

Example 2. Estimate, then measure. Zoe estimates her lunch box is 20 cm long. She measures it: the end is between 23 and 24, past the half-way mark.

  1. Past the half-way mark means go up: the lunch box is about 24 cm.
  2. Her estimate was 20 cm. From 20 to 24 is 4.
  3. She was 4 cm too small. Next time she might picture the ruler: her lunch box is a bit less than one ruler (30 cm), so a guess like 25 cm would be closer.

Example 3. Longer than the ruler. A rug is measured with a 30 cm ruler: 3 whole rulers, then 15 cm more. How long is the rug?

  1. 3 rulers: 30, 60, 90 cm.
  2. 15 cm more: 90 and 15 is 105 cm.
  3. The rug is 105 cm, which is 1 metre and 5 centimetres.

Example 4. Is that sensible? Max measures his little brother and writes "his height is 9 cm". Is that sensible?

  1. 9 cm is about the length of a finger. No brother is that short!
  2. Max probably meant 90 cm (about up to an adult's hip), or he misread the tape.
  3. An estimate (about 1 m, a bit less) would have caught the mistake straight away.

Try it

  1. Which benchmark would you use to estimate the length of a spoon: your fingernail, a ruler, or a big step?
  2. Estimate in centimetres: a paper clip, a pencil, your hand span. (Then measure, if you can.)
  3. A hallway is about 6 big steps long. About how many metres is that?
  4. A crayon's end is between 8 and 9, before the half-way mark. About how long is it, to the nearest centimetre?
  5. A ribbon's end is between 11 and 12, past the half-way mark. About how long is it?
  6. A table is measured with a 30 cm ruler: 2 whole rulers and then 20 cm. How long is the table?
  7. Sara estimated a book was 30 cm. It was really 26 cm. Was her estimate too big or too small? By how much?
  8. Which of these is sensible for the height of a classroom door: 2 cm, 20 cm, 2 m, 20 m?
Answers
  1. A ruler, or your hand span. A spoon is shorter than a ruler, so "about half a ruler" works. A fingernail (1 cm) is too small to count easily, and a step is far too big.
  2. Paper clip about 3 cm. New pencil about 19 cm (a used one is shorter). Hand span: a child's is often about 12 to 16 cm, but measure your own.
  3. A big step is about 1 m, so 6 steps is about 6 m.
  4. Before the half-way mark, so stay at the smaller number: about 8 cm.
  5. Past the half-way mark, so go up: about 12 cm.
  6. 2 rulers is 30 and 30, which is 60 cm. 60 and 20 more is 80 cm.
  7. Too big, by 4 cm. From 26 up to 30 is 27, 28, 29, 30: 4 cm.
  8. 2 m. A door is about 2 m tall. 2 cm is smaller than your thumb, 20 cm is shorter than a ruler, and 20 m is taller than a house.

Common mistake

Wild guessing instead of estimating. An estimate uses a benchmark. Ask yourself: "How many rulers? How many big steps?" Then count.

Leaving a gap or an overlap when you move the ruler. When a thing is longer than your ruler, make a small mark where the ruler ends, and start the next ruler's 0 exactly on that mark.

Where it leads