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Measuring with hands and feet

Measurement · level 1, lesson 4 of 6

A question to start

How long is your table? You do not need a ruler. Stretch your hand out wide, put your thumb at one end of the table, and "walk" your hand along it, span after span, counting as you go.

Now ask a grown-up to do the same with their hand. Did you get the same number? Almost surely not! Maybe you counted 8 and the grown-up counted 5. The table did not change. So who is right?

Here is the surprise: you are both right. In this lesson you will find out why two careful people can measure the same table and get two different numbers, and why that is exactly the reason people later invented rulers.

The idea

From "longer" to "how long"

In Longer and shorter you found out which thing is longer. Now we want to know how long a thing is, with a number. To do that, we pick one small thing to measure with and count how many fit.

The small thing we count is called a unit. Your hand span can be a unit. So can your foot, a paper clip, a block or a crayon.

  • A hand span is from the tip of your thumb to the tip of your little finger, with your hand stretched wide.
  • A foot is the length of your foot, from heel to toe.
  • A pace is one normal step.

How to measure with a unit

Here a pencil is measured with paper clips.

5 clips

The pencil is 5 paper clips long. To make the count honest, follow four rules.

  1. Start exactly at the end. Line up the first clip with the end of the pencil.
  2. No gaps. Each clip starts right where the last one finished.
  3. No overlaps. The clips must not lie on top of each other.
  4. Same unit every time. All the clips must be the same size.

Look what happens when you break the rules.

gaps: 4 overlaps: 7

The strip is really 5 clips long, just like the pencil. With gaps, we only counted 4, because the gaps were part of the length but nobody counted them. That number is too small. With overlaps, we counted 7, because some bits of the strip were counted twice. That number is too big.

"About" and "a bit more than"

Things are not always an exact number of units long. A book might be more than 3 hand spans but less than 4. Then we say:

  • "The book is about 3 hand spans long", if it is just a little more than 3, or
  • "The book is between 3 and 4 hand spans long."

That is honest. Real measuring nearly always has a little bit left over at the end.

Why a grown-up gets a smaller number

Now back to the table. Here is the same table measured with a grown-up's big hand span and with a child's small one.

table big spans: 4 small spans: 6

The table is the same length both times. The big spans are bigger, so fewer of them are needed to cover it. The small spans are smaller, so more of them are needed.

A smaller unit gives a bigger number. A bigger unit gives a smaller number.

Think of filling a jar with sweets. If the sweets are big, only a few fit. If they are tiny, lots fit. The jar did not change, only the size of the sweets.

So you and the grown-up are both right: "8 of my spans" and "5 of your spans" are the same length, said in two different units. The trouble is that the number alone, "8", does not tell anyone how long the table is, unless they know whose hand you used. That is why, later, everyone agreed to use the same units, like the centimetre and the metre. You will meet them in level 2.

Body units long ago

For thousands of years, people really did measure with their bodies. The ancient Egyptians used the cubit, the length from the elbow to the tip of the middle finger, to plan buildings. A foot was once the length of a foot. And even today, the height of a horse is measured in hands. Body units are handy, because you always carry them with you. Their problem is the one you just found: not everyone's hand is the same size.

Worked examples

Example 1. Who has the longer step? Jo walks across a rug and takes 6 steps. Her big brother walks across the same rug and takes 4 steps. Who takes longer steps?

  1. The rug is the same length for both.
  2. The brother needed fewer steps to cover it.
  3. Fewer steps to cover the same length means each step is bigger.
  4. So the brother takes longer steps.

Example 2. Measuring with blocks. Kai lays blocks along a book, end to end, starting at the end of the book. The book is covered by 7 blocks, with a little bit of the last block sticking out past the book.

  1. 7 blocks cover a bit more than the book.
  2. 6 blocks would not quite reach the end.
  3. So the book is between 6 and 7 blocks long, a bit less than 7. We can say "about 7 blocks".

Example 3. Spot the mistake. Lily measures a shelf with crayons and gets 10. She left a gap between each crayon. Is the shelf more than 10 crayons long, or less?

  1. The gaps are part of the shelf, but she did not count them.
  2. So she counted too few crayons.
  3. The shelf is more than 10 crayons long. To fix it, she should push the crayons together with no gaps and count again.

Try it

  1. A desk is 6 of your hand spans long. Your teacher measures it and gets 4 of her hand spans. Whose span is bigger?
  2. Would you measure the length of your classroom in hand spans or in steps? Why?
  3. You lay 5 pencils end to end along a table, but you leave gaps between them. Is "5 pencils long" too big or too small?
  4. You measure a rug with paper clips, but some clips lie on top of each other. Is your number too big or too small?
  5. A bench is 8 blocks long. A table is 11 blocks long (the same blocks). How many blocks longer is the table?
  6. Ben measures his bed with his feet and says "12 feet". His dad measures it with his feet and says "8 feet". Ben says, "My bed must be longer when I measure it." Is he right?
  7. Sam measures a ribbon with 3 big blocks and 2 small blocks and says "the ribbon is 5 blocks long". What is wrong?
Answers
  1. Your teacher's. She needed fewer spans to cover the same desk, so each of her spans must be bigger.
  2. Steps (paces). Hand spans would take a very long time and you might lose count. A step is a bigger unit, so you need fewer of them.
  3. Too small. The gaps are part of the table's length, but you did not count them. The table is really more than 5 pencils long.
  4. Too big. Where clips overlap, some parts of the rug are counted twice. The rug is really fewer clips long than you counted.
  5. Count on from 8 to 11: 9, 10, 11. That is 3 blocks longer.
  6. No. The bed is the same bed. Ben's feet are smaller, so he needs more of them. A smaller unit gives a bigger number.
  7. The blocks are not all the same size. "5 blocks" does not mean anything when the blocks are different. He should measure again with blocks that are all the same.

Common mistake

Leaving gaps or letting the units overlap. Gaps make your number too small; overlaps make it too big. Line each unit up right where the last one ended.

Thinking a bigger number always means a longer thing. "My desk is 10 spans and yours is only 6 steps, so mine is longer" is not fair: the units are different. Compare numbers only when the units are the same.

Where it leads