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Seeing small amounts at once

Numbers · level 1, lesson 2 of 6

A question to start

Roll a dice and look at the top. Do you count the dots, "one, two, three, four, five"? Or do you just know it is a five?

Most people just know. They see the shape of the dots, like a face they recognise, and the number pops into their head. Grown-ups who play board games never count the dots on a dice.

Now try this. Someone flashes a pile of 13 beans at you for one second and hides them. Can you say how many? Almost nobody can. So what is the difference? Why can you "just see" five dots, but not thirteen beans? This lesson is about that difference, and about a trick that lets you "just see" bigger numbers too.

The idea

Seeing without counting

Look quickly at each group and say how many. Do not count.

Two, three, one, four. You probably knew each one in a flash. Mathematicians have a long word for this: subitizing (say "SOO-bi-tie-zing"). It comes from an old word that means "suddenly". Subitizing means seeing how many things there are, all at once, without counting.

Most people can do this for up to about four things scattered anyhow. After that, the eyes get muddled and we start counting. But there is a way around this: patterns and groups.

Dice patterns

The dots on a dice are not scattered anyhow. They sit in a neat pattern, the same on every dice in the world.

123 456

Look closely, and you can see that the bigger patterns are made of smaller ones:

  • Four is a square: a dot in each corner.
  • Five is the square of four, and one more in the middle.
  • Six is two rows of three, or three and three.

So when you "just see" five, your eyes are really seeing "four and one". This is the secret. You cannot see 13 beans at once, but you can see small groups, and then put the groups together.

Dominoes

A domino is a flat tile split into two halves, with a dice pattern on each half. To find how many dots a domino has, you see each half at once and put them together.

4 and 3 is 7

The domino has 4 on one side and 3 on the other. Four and three make seven. You know it is 7 without touching every dot.

Fingers: five on each hand

You carry a counting tool everywhere: your hands. Each hand has five fingers. Two hands have ten.

Show seven on your fingers. Most people hold up one whole hand (that is five, you do not need to count it) and two more fingers. Five and two more is seven. Show nine: one whole hand and four more, or two whole hands with one finger down. Your hands teach you two very important groups: five and ten.

The five frame and the ten frame

A five frame is a row of 5 boxes. We put one counter in each box, starting from the left.

3 counters, 2 empty boxes

The empty boxes help as much as the full ones. You can see 3 counters, and you can also see that 2 more would fill the frame. So 3 and 2 make 5.

A ten frame is two five frames, one on top of the other: 2 rows of 5 boxes. We fill the top row first, from the left, then the bottom row.

five and two more is seven

Look how fast you can read it. The top row is full, so that is five. The bottom row has two. Five and two more is seven. And there are 3 empty boxes, so seven needs three more to make ten.

Why does the frame make it so easy? Because it turns a big number into small groups that your eyes can see at once: a full row (always 5), a few more (small enough to subitize), and some empty boxes (small enough too). You never have to count one by one.

Two ways to see the same number

There is often more than one way to see a number. Here are ways to see eight:

  • On a ten frame: a full row of 5 and 3 more. Five and three.
  • On a ten frame, looking at the gaps: 2 empty boxes. Two less than ten.
  • On a dice and a domino: two squares of four. Four and four.

All three are right. Being able to see a number in many ways is what makes adding easy later. When you meet 8 + 2 or 10 − 2 or 4 + 4, you will already know the answer is 8, because you have seen it.

Worked examples

1. A dice and one more. You roll a five on one dice and a one on the other. How many dots?

See the five pattern (no counting). Then one more is six. So there are 6 dots. A grown-up would say "five, six", starting from the five, not from one.

2. A ten frame showing 9. A ten frame has a full top row and four counters in the bottom row, with one empty box. How many counters?

Way 1: a full row is 5, and 4 more. Five and four is 9. Way 2: only one box is empty, so it is one less than ten. One less than ten is 9. Both ways agree.

3. A domino with six and six. Each half has two rows of three. Three and three is six on each half. Six and six is 12. (This is the "double six", the biggest domino in a normal set.)

4. Fingers for eight. Show eight on your fingers without counting. Put up one whole hand: that is five. Then put up three fingers on the other hand. Five and three is 8. Check: two fingers are still down, and ten take away two is eight.

Try it

Here are three ten frames. Call them A, B and C.

ABC
  1. How many counters are in frame A? Say it as "five and ...".
  2. How many counters are in frame B? How many more would make ten?
  3. How many counters are in frame C? How many empty boxes does it have?
  4. On a dice, which number looks like the four pattern with one more dot in the middle?
  5. A domino has 5 on one half and 2 on the other. How many dots does it have?
  6. Show six on your fingers in two different ways.
  7. Draw a ten frame for 10. How many empty boxes are there? Why is ten frame "full" a special number?
Answers
  1. A full top row (5) and 1 more: five and one is 6.
  2. A full top row (5) and 4 more: 9. Only 1 box is empty, so 1 more would make ten.
  3. 4 counters, all in the top row. There are 6 empty boxes: 1 in the top row and 5 in the bottom row. Check: 4 and 6 make 10, the whole frame.
  4. Five. The four pattern is a dot in each corner; five adds one dot in the middle.
  5. Five and two is 7. See the five pattern, then say "six, seven" for the two.
  6. For example: one whole hand (5) and one finger on the other hand; or three fingers on each hand (three and three). There are other ways too, like four and two.
  7. Every box has a counter, so there are 0 empty boxes. Ten is special because it is "one full frame", or "two full hands". In the last lesson of this level you will see that we write all big numbers using groups of ten.

Common mistake

Starting from one every time. If you see a full row of five on a ten frame and then count "one, two, three, four, five, six, seven", you have thrown away the pattern. Trust the row: say "five", then count on, "six, seven". This is faster, and it makes fewer mistakes.

Filling the frame anyhow. Counters on a ten frame go in the top row first, from the left. If they are scattered, nobody can read the frame at a glance, and it is just a box of beans again.

Where it leads