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Counting things

Numbers · level 1, lesson 1 of 6

A question to start

There is a bowl of strawberries on the table. Mia counts them and says "seven". Sam counts the same bowl and says "eight". Then their dad counts and says "seven".

Nobody ate a strawberry. Nobody added one. So how can two people count the same strawberries and get different answers? And who is right?

Counting looks easy. You just say "one, two, three". But real counting has a few secret rules. If you break one of them, you get a wrong answer, even if you know all the number words. In this lesson you will learn the rules, and then you will be able to say exactly what went wrong for Sam.

The idea

The counting words

First we need the counting words, and we need them in the same order every time:

one, two, three, four, five, six, seven, eight, nine, ten, ...

The order never changes. "Two" always comes after "one". "Five" always comes after "four". If you said "one, two, four, three", the count would go wrong, just like a song goes wrong if you mix up the words.

We also write the counting words as numerals, which are the short signs for numbers:

Word one two three four five six seven eight nine ten
Numeral 1 2 3 4 5 6 7 8 9 10

"Five" and "5" mean the same number. One is a word and one is a sign.

Rule 1: one number for each thing

When you count, you touch each thing (or point at it) and say one number. One touch, one number. Then move on to the next thing.

123 45 one touch, one number: the last number is 5, so there are 5

Two things can go wrong here:

  • Skipping: you jump over a strawberry and never say a number for it. Then your answer is too small.
  • Counting twice: you touch one strawberry, say "three", and then touch it again and say "four". Then your answer is too big.

That is what happened to Sam. He touched one strawberry twice, so he said "eight" when there were only seven.

Rule 2: the last number tells how many

When you have touched every thing, stop. The last number you said is the answer to "how many?"

If you counted "one, two, three, four, five" and then there was nothing left to touch, there are five strawberries. You do not need to count again. The "five" does not belong to the last strawberry only. It tells you about the whole group. Five is how many there are all together.

This is a big idea, even if it sounds small. Young children often count "one, two, three, four, five" perfectly, and then, when you ask "so how many are there?", they start counting again! The last number is the answer. That is what counting is for.

Rule 3: the order does not matter

You can start with any strawberry. You can go left to right, right to left, or jump around. If you follow Rule 1 (one number for each, none skipped, none counted twice), you always get the same answer.

315 24 a different order, the same answer: 5

Why is this true? Think of the counting words as stickers: a "1" sticker, a "2" sticker, a "3" sticker, and so on. Counting means putting one sticker on each strawberry, in order. Whichever strawberry you start with, you use up the stickers "1", "2", "3", ... one at a time. You run out of strawberries at the same moment, because there are the same number of strawberries every time. So the last sticker you use is always the same one: "5".

The things can also be moved. Spread the strawberries out, bunch them up, put them in a line or in a circle. As long as you do not add any or take any away, the number stays the same.

Rule 4: you can count anything

You can count strawberries, chairs and cars. But you can also count things you cannot touch:

  • sounds: clap your hands and let a friend count the claps;
  • jumps: count how many times you can hop on one foot;
  • days: count the sleeps until a birthday;
  • ideas: count the colours of the rainbow you can name.

Counting does not care what the things are. A group of 5 elephants and a group of 5 ants are different, but they have the same number: 5. Numbers are one of the first big ideas in math, because the same number fits so many different things.

Tricks for keeping track

When there are many things, or they are all mixed up, it is easy to skip one or count one twice. Here are tricks that help:

  1. Move each thing as you count it. Push it to the other side of the table. Things on the left are "not counted yet", things on the right are "counted".
  2. Line them up first. Things in a straight row are easy to count from one end to the other.
  3. Mark each thing. On paper, put a small tick or dot on each picture as you count it.
  4. Count twice, in different orders. If you get the same answer twice, you can be sure. If you get two different answers, one count has a mistake. Count a third time, slowly.
not counted yet counted: 1, 2, 3, 4

Zero: when there is nothing to count

What if the bowl is empty? Then you do not say any counting word at all. The number for "none at all" is zero, and we write it 0. Zero is a real number. If someone asks "how many dinosaurs are in your kitchen?", the right answer is "zero".

Worked examples

1. Counting a row. How many stars? ★ ★ ★ ★ ★ ★

Touch each star from the left: one, two, three, four, five, six. There are no more stars. The last number was six, so there are 6 stars.

2. The same things, a new order. Leo counts 9 crayons from left to right. Then he mixes them up in a pile and counts again, starting with a green one. What should he get?

He should get 9 again. Mixing up the crayons does not add any crayons or take any away. If he gets 8 or 10, he skipped one or counted one twice, and he should count once more, slowly, moving each crayon as he counts it.

3. Finding the mistake. Ana counts these balls: ● ● ● ●. She says "one, two, three, four, five". What went wrong?

There are only 4 balls, but she said 5 numbers. She must have said two numbers for one ball (perhaps she said "two, three" while touching just one). One touch must get one number. Counting again carefully gives 4.

4. Counting what you cannot touch. Grandpa claps: clap, clap, clap, clap, clap, clap, clap. How many claps?

Hold up one finger for each clap: 1, 2, 3, 4, 5, 6, 7. Seven fingers are up, so there were 7 claps. Your fingers "remember" the claps after the sounds are gone.

5. The strawberries. Mia and Dad both got 7. Sam got 8. Two careful counts agree, so the answer is 7, and Sam counted one strawberry twice. (Here is something sneaky. If someone skips one strawberry and counts another one twice, they get 7 too! Skipping takes one away and counting twice adds one, so two mistakes can hide each other. That is why counting twice, in different orders, is a good habit.)

Try it

  1. Count the hearts: ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥. How many?
  2. Say the counting words from one to ten. Which word comes just after six? Which comes just before nine?
  3. Tom counts 6 blocks. He builds them into a tower. How many blocks are in the tower? Do you need to count again?
  4. Mars has two moons, called Phobos and Deimos. How many moons does the Earth have? How many moons is that for the two planets all together?
  5. Raj counts the cars in a toy box and says "one, two, three, four". Then he says "there are one". What did he forget?
  6. A rover on Mars has 6 wheels. Hold up one finger for each wheel. How many fingers? How many fingers are still down?
  7. Jo counted the same marbles three times and got 12, then 13, then 12. What would you do? What is probably the answer?
Answers
  1. Touch each heart once: 1, 2, 3, 4, 5, 6, 7, 8. There are 8 hearts.
  2. Seven comes just after six. Eight comes just before nine.
  3. 6 blocks. Building a tower moves the blocks but does not add or take any away, so you do not need to count again.
  4. The Earth has 1 moon. Two moons and one moon make 3 moons all together. You can check by touching three fingers: Phobos, Deimos, our Moon.
  5. He forgot Rule 2: the last number tells how many. He said "four" last, so there are 4 cars.
  6. 6 fingers up. You have 10 fingers, so 4 are still down. (Count the ones that are down: 1, 2, 3, 4.)
  7. Count once more, slowly, and move each marble as you count it. Two of the three counts said 12, so the answer is probably 12, and the count of 13 had one marble counted twice. But only a careful count can be sure.

Common mistake

Touching and talking at different speeds. Sometimes your mouth goes faster than your finger: "one-two-three" while your finger touches only one thing. Or your finger goes faster than your mouth, and you skip things. Slow down so that each touch has exactly one word. Tapping each thing firmly, like pressing a button, helps a lot.

Counting again to answer "how many?". You do not need to count again. The last number you said is the answer.

Where it leads