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Teens and beyond

Numbers · level 1, lesson 6 of 6

A question to start

We say twenty-one, twenty-two, twenty-three. We say thirty-one, thirty-two, thirty-three. So why don't we say "ten-one, ten-two, ten-three"? Why do we say eleven and twelve instead?

And here is another puzzle. You know nine numbers that have just one digit: 1, 2, 3, 4, 5, 6, 7, 8, 9. Plus 0, that is ten digits. How can only ten little signs write every number, even a number as big as the number of stars?

The answer to both puzzles is the same idea: groups of ten. Once you see it, numbers up to 100 (and far beyond) stop being a long list to remember and become a simple pattern.

The idea

A ten and some ones

In the last lesson you made ten. Now keep going past it.

13 is a ten and three ones. On ten frames, it is one full frame and three more counters. With blocks, it is one long ten stick (ten little cubes stuck together) and three single cubes.

13: 1 ten and 3 ones 31: 3 tens and 1 one

Look at how we write thirteen: 13. The 1 on the left tells how many tens. The 3 on the right tells how many ones. Every two-digit number works like this:

  • 17 is 1 ten and 7 ones.
  • 20 is 2 tens and 0 ones.
  • 45 is 4 tens and 5 ones.
  • 31 is 3 tens and 1 one.

The place where a digit sits tells you what it is worth. That is why ten digits are enough to write every number: the same digit can mean "ones" or "tens" (or, in level 2, "hundreds") depending on where it is. A digit is one of the ten signs 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A number like 45 is one number made of two digits.

The teen numbers

The numbers from 13 to 19 are called the teens, because their names end in "teen". "Teen" is an old way of saying "ten".

Number Name Means
11 eleven 1 ten and 1 one
12 twelve 1 ten and 2 ones
13 thirteen three and ten
14 fourteen four and ten
15 fifteen five and ten
16 sixteen six and ten
17 seventeen seven and ten
18 eighteen eight and ten
19 nineteen nine and ten

Notice something odd: we say the ones first ("four-teen") but we write the ten first (14). That is why children often write 41 when they mean fourteen. The name and the numeral go in opposite orders. It is not your fault; it is the language's fault!

And eleven and twelve? Their names are very old. People who study the history of words say that "eleven" comes from old words meaning "one left over" and "twelve" from "two left over": one or two left over after counting ten on your fingers. Some languages are more sensible. In Chinese, eleven is said "ten-one" and twelve "ten-two".

Counting by tens

Ten ones make a ten. What about many tens?

Tens 1 ten 2 tens 3 tens 4 tens 5 tens 6 tens 7 tens 8 tens 9 tens 10 tens
Number 10 20 30 40 50 60 70 80 90 100
Name ten twenty thirty forty fifty sixty seventy eighty ninety one hundred

The "-ty" at the end of twenty, thirty, forty means "tens". Sixty is "six tens". Ninety is "nine tens".

Counting by tens sounds just like counting by ones, with "-ty" stuck on: sixty, seventy, eighty. Only a few names are a little bent: twenty (two), thirty (three), fifty (five).

And ten tens is a new, bigger group: one hundred, written 100.

The hundred chart

Here are all the numbers from 1 to 100, in rows of ten.

123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100

The chart is full of patterns:

  • Going right one square is one more (like the number line).
  • Going down one square is ten more. Look down any column: 4, 14, 24, 34, 44, ... The ones digit stays the same, and the tens digit goes up by one.
  • The last column (green) is counting by tens: 10, 20, 30, ..., 100.
  • Every row (except the first) starts with a number ending in 1 and ends with a number ending in 0.

What comes after 39? After 99?

Count on from 35: thirty-five, thirty-six, thirty-seven, thirty-eight, thirty-nine, ... and then?

39 is 3 tens and 9 ones. One more makes 10 ones. But 10 ones are a new ten! So bundle them into a ten stick. Now there are 4 tens and 0 ones. That is 40, forty.

3 tens, 9 ones + 1 one 4 tens, 0 ones 39→40

That is why the chart jumps from the end of one row to the start of the next: after a 9 in the ones place, the ones go back to 0 and the tens go up by one. 29 → 30. 59 → 60. 89 → 90.

Now 99: 9 tens and 9 ones. One more makes 10 ones, which make a new ten. Now there are 10 tens. And 10 tens make a hundred. So after 99 comes 100: 1 hundred, 0 tens, 0 ones. The number needs a third digit, and a new place, the hundreds place. You will explore it in level 2.

This is exactly like a car's distance counter (or the counter on a Mars rover that shows how far it has driven): when a wheel reaches 9 and turns once more, it goes back to 0 and pushes the next wheel up by one.

Worked examples

1. Tens and ones. How many tens and ones are in 58?

The 5 is in the tens place and the 8 is in the ones place. 58 is 5 tens and 8 ones. Check: 5 tens is 50, and 50 and 8 make 58.

2. Writing a number from tens and ones. Write the number with 7 tens and 2 ones.

7 tens is 70. And 2 ones. That is seventy-two: 72. (Not 27; 27 is 2 tens and 7 ones.)

3. Counting by tens from a number that is not a ten. Start at 6 and count on by tens.

Each ten more goes one row down the hundred chart, and the ones digit stays 6: 6, 16, 26, 36, 46, 56, 66, 76, 86, 96.

4. Crossing a ten. What comes just after 69? Just before 70?

69 is 6 tens and 9 ones. One more makes 10 ones, a new ten: 70. Just before 70 is one less than 70: 69.

5. Ten more and ten less. What is ten more than 47? Ten less than 47?

Ten more adds one ten: 4 tens become 5 tens, and the 7 ones stay. 57. Ten less takes one ten: 37.

Try it

  1. How many tens and ones are in 64? In 90?
  2. Write the number that has 3 tens and 8 ones.
  3. Which is bigger, 41 or 14? Explain with tens and ones.
  4. What number comes after 79? After 59? After 99?
  5. Count by tens from 3 to 93. How many numbers did you say?
  6. What is ten more than 82? Ten less than 82?
  7. Ben has 2 bags of 10 sweets and 6 loose sweets. How many sweets? Ava has 26 sweets. Who has more?
  8. On the hundred chart, you start at 23, go down 2 squares and right 4 squares. Where are you?
Answers
  1. 64 is 6 tens and 4 ones. 90 is 9 tens and 0 ones.
  2. 3 tens is 30, and 8 ones: 38.
  3. 41 is bigger. 41 has 4 tens; 14 has only 1 ten. Tens are worth more than ones, so the number with more tens is bigger (when both have two digits).
  4. After 79 comes 80 (9 ones and one more make a new ten). After 59 comes 60. After 99 comes 100.
  5. 3, 13, 23, 33, 43, 53, 63, 73, 83, 93: 10 numbers.
  6. Ten more than 82 is 92. Ten less is 72.
  7. 2 bags of 10 is 20, and 6 more is 26. Ava also has 26. They have the same.
  8. Down 2 squares is 20 more: 23 → 33 → 43. Right 4 squares is 4 more: 43 → 44, 45, 46, 47. You are on 47.

Common mistake

Writing teen numbers backwards. You hear "seventeen" and write 71. But seventeen is 1 ten and 7 ones, so the 1 (for the ten) goes first: 17. Say to yourself "seventeen is ten and seven" before you write it.

After 39, saying "thirty-ten". When the ones reach ten, they turn into a new ten. 39 is followed by 40, never by "thirty-ten".

Where it leads