Group the ones into tens
Take 23 cubes. Count out ten of them and join them into one stick. Count out ten more and make a second stick. Now 3 cubes are left, too few for another stick. So 23 is 2 groups of ten and 3 ones.
Counting in groups makes a big number quick to read. Two sticks are 10, 20, and then count on the loose cubes: 21, 22, 23.
2 sticks of ten and 3 loose cubes: 23.
What each digit counts
The two digits of 23 count different things. The digit on the left, 2, counts the groups of ten, so it is called the tens digit. Its 2 tens are worth 20.
The digit on the right, 3, counts the ones that did not fill a group of ten, so it is called the ones digit. Put the two parts together: 23 = 20 + 3.
The tens digit, 2, counts the 2 sticks of ten.
The ones digit, 3, counts the 3 loose cubes.
Why ten is written 1 and 0
The ones digit can only be 0, 1, 2, and up to 9. Add one more cube to 9 loose cubes and there are ten of them, enough to fill a group. The ten ones become one ten.
Then there is 1 ten and no ones left, and that is written 10: the 1 counts the ten, and the 0 says there are no loose ones. The 0 matters, because it keeps the 1 in the tens place. Without it, the number would read 1.
fewer than ten ones: the ones digit counts them, the tens digit is 0, and 10 = 1 ten is not yet reached
Drop ten ones, then drag the bundle into the tens column
Add ones until there are ten of them, then move the ten across into the tens column as one bundle.
The place changes the value
23 and 32 use the same two digits, but they are different numbers. 23 is 2 tens and 3 ones, and 32 is 3 tens and 2 ones, so 32 is the bigger one.
One more stick of ten makes the tens digit go up by 1 and leaves the ones digit alone: 38 and one more ten is 48. The usual mistake is to add the digits, 3 + 8 = 11, but the 3 counts 3 tens, which is 30, so the number is 30 + 8 = 38.
Worked example: Sticks in Bundles of Ten with Some Loose Sticks
Question Ali has 4 bundles of sticks. Each bundle has 10 sticks. He also has 6 loose sticks. (a) How many sticks does Ali have? (b) His teacher gives him 1 more bundle. How many sticks does he have now?
1.Draw the 4 bundles and the 6 loose sticks. Each bundle is 1 ten. Each loose stick is 1 one.
Each bundle is 1 ten. Each loose stick is 1 one. 2.Count the bundles in tens: 10, 20, 30, 40. Then count on the loose sticks in ones: 41, 42, 43, 44, 45, 46.
Count the bundles in tens: 10, 20, 30, 40. Then count on the loose sticks: 41, 42, 43, 44, 45, 46. 3.(a) 4 tens and 6 ones is 40 + 6 = 46. Ali has 46 sticks.
(a) 4 tens and 6 ones is 40 + 6 = 46 sticks. 4.(b) One more bundle is 1 more ten. Now there are 5 tens and 6 ones: 50 + 6 = 56. Ali has 56 sticks. Check: 46 + 10 = 56.
(b) One more bundle is 1 more ten: 50 + 6 = 56 sticks.
Answer: (a) 46 sticks; (b) 56 sticks
Common mistakes
- Adding the two numbers: 4 + 6 = 10. The 4 is the number of bundles, and each bundle has 10 sticks, so the 4 stands for 40 sticks.
- Adding 1 in part (b) to get 47. One more bundle is 10 more sticks, so the tens digit goes up by 1, not the ones digit.
Worked example: A Mystery Number from Clues
Question Lina thinks of a number. It is less than 100. It has 5 tens. (a) How many numbers fit these two clues? (b) The ones digit of her number is 2 more than its tens digit. What is Lina's number?
1.A number less than 100 that has 5 tens is a two-digit number. Its tens digit is 5.
The number is less than 100 and has 5 tens, so it has two digits and its tens digit is 5. 2.List the numbers with 5 tens: 50, 51, 52, 53, 54, 55, 56, 57, 58, 59.
These are the numbers with 5 tens: 50, 51, 52, 53, 54, 55, 56, 57, 58, 59. 3.(a) The ones digit can be any digit from 0 to 9, so 10 numbers fit the two clues.
(a) The ones digit can be any digit from 0 to 9, so 10 numbers fit. 4.The ones digit is 2 more than the tens digit, so the ones digit is 5 + 2 = 7.
The ones digit is 2 more than the tens digit: 5 + 2 = 7. 5.(b) Lina's number is 57. Check: 57 is less than 100, it has 5 tens, and 7 is 2 more than 5.
(b) The number is 57. It is less than 100, it has 5 tens, and 7 is 2 more than 5.
Answer: (a) 10 numbers; (b) 57
Common mistakes
- Writing 75. That number has 7 tens and 5 ones, so it does not fit the clue that the number has 5 tens.
- Writing 52. The clue does not say that the ones digit is 2. It says that the ones digit is 2 more than the tens digit, which is 5 + 2 = 7.