Counting past one hundred
After 120 the count keeps going. What comes next, where are the hard places, and how does the count reach 1{,}000?
Where you are 23 in Counting and cardinality
When you reach for it
A child who can count to 120 can start at any number up to there and keep going. Plenty of counts go much further. A jigsaw puzzle with 1{,}000 pieces, a step counter that adds one for every step on a walk, the pages of a thick book: the count runs on through the hundreds, all the way to one thousand.
This lesson carries counting by ones from 120 up to 1{,}000. The child starts at any number, counts on or counts back, says each number, and writes the numbers as they go. The move is one they already have. Counting on means saying the next number, the one that is one more. Counting back means saying the number before, the one that is one less. Nothing about the move changes past 120.
What needs care are the places where the count rolls over: where a new ten starts, as at 139 then 140, and where a new hundred starts, as at 299 then 300. At those places more than one digit changes at once, and the child has to say a new word.
Counting by ones is the slow, sure way along the count. Jumping a hundred at a time is Counting by hundreds, and naming any three-digit number in words, digits and expanded form is Reading and writing numbers to 1,000. This lesson stays with one step at a time.
The lesson runs over two days.
- Day 1: counting on through the hundreds, the chart page of 101 to 200, and where a ten or a hundred rolls over.
- Day 2: counting back, the last hundred up to 1{,}000, and writing the numbers counted.
The pictures are a number path (a row of numbered squares to touch or step on), the hundred chart’s next page, and a number line. A step counter is a ready-made counting machine: it shows the count as a numeral and adds one for every step.
The procedure
Day 1: counting on through the hundreds
The count past 120 keeps the pattern the child met in Counting, reading and writing numbers to 120: say “one hundred”, then the counting words from one to ninety-nine. One hundred twenty-one, one hundred twenty-two, and on to one hundred ninety-nine. After ninety-nine comes one hundred, so after one hundred ninety-nine comes two hundred, 200. Then the pattern starts over behind “two hundred”: two hundred one, two hundred two, and so on. Each hundreds word (two hundred, three hundred, up to nine hundred) lasts for a hundred numbers before the next one takes over.
The chart’s next page. The hundred chart holds 1 to 100 in ten rows of ten. Turn the page: the next page holds 101 to 200, laid out the same way.
Moving one square to the right is counting on by one. At the end of a row the count goes on at the start of the next row. Each number sits in the same square that its last two digits hold on the hundred chart: 137 where 37 is, 160 where 60 is, and 200 in the last square, where 100 is. The page after holds 201 to 300, and so on. Ten pages hold every number from 1 to 1{,}000. Making a page by hand, on squared paper, is a good task: the child writes each numeral and says it.
Where the count rolls over. Write the numbers while counting on, and watch which digits change.
| Count on | Digits that change | What has started |
|---|---|---|
| 246 to 247 | the last digit only, 6 to 7 | nothing new |
| 139 to 140 | the last digit, 9 back to 0, and the middle digit, 3 up to 4 | a new ten |
| 299 to 300 | the last two digits, 99 back to 00, and the first digit, 2 up to 3 | a new hundred |
Most steps change only the last digit. When the last digit is 9 there is no bigger digit to move to, so it goes back to 0 and the middle digit goes up one: a new ten has started. When the last two digits are both 9, both go back to 0 and the first digit goes up one: a new hundred has started. Why ten ones make a ten and ten tens make a hundred is the business of Tens and ones and Place value. Here the child only needs to see where the change happens and say the new word there.
Count slowly through a roll-over on a number path, touching each square:
The words help at the shaded square. After ninety-nine comes one hundred, so after two hundred ninety-nine comes three hundred.
Day 2: counting back, the last hundred, and writing the count
Counting back says the number before each time, one less. Roll-overs happen going back too. From 140 back one is 139: the last digit goes from 0 to 9 and the middle digit goes down one. From 300 back one is 299: the last two digits go from 00 to 99 and the first digit goes down one. Going back, the places to slow down are the numbers ending in 0, and most of all the numbers ending in 00.
The last hundred. The tenth page, from 901 to 1{,}000, ends the count. After nine hundred ninety-nine comes one thousand, written 1{,}000. Here all three digits roll over at once, and a fourth digit, 1, appears in front. The comma in 1{,}000 is not a digit; it only makes the numeral easier to read. One thousand is the last square of the tenth page.
Writing the numbers counted. Each number in a count is the one before with only the changing digits changed, so the child can keep the front digits and change the back. The zeros are digits like any other and must be written. From 301 to 309 the middle digit is 0, and 305 written without it would be 35. A step counter shows this well: each step it shows the next numeral, and at a roll-over two or three digits change together.
Worked examples
Say every count aloud with the child, and keep the chart page or a number path on the table.
Day 1: Puzzle pieces past three hundred
Sander is counting the pieces of a 1{,}000-piece jigsaw puzzle. He has counted 296 pieces. He counts on 8 more pieces, one at a time. What numbers does he say? How many pieces has he counted now?
SolutionThe 296 pieces are already counted, so Sander does not start again from one. He keeps 296 and says the next number for each new piece, holding up a finger each time so he knows when he has said eight:
One step deserves care. After two hundred ninety-nine the last two digits roll from 99 back to 00 and the first digit goes from 2 to 3: three hundred. Then the small numbers start again behind “three hundred”.
The last number he says is 304. The last number of a count tells how many have been counted (How counting tells how many), so he has counted 304 pieces. Check on Figure 2: start on 296, take eight steps, and land on 304.
Day 1: Along a row and on to the next
On the chart page of 101 to 200 (Figure 1), Kito puts his finger on 176 and counts on to 183. What does he say, and where does his finger go when a row ends?
SolutionHe says each next number in turn:
The square for 180 is the last in its row, the row from 171 to 180, just as 80 ends its row on the hundred chart. So his finger goes to the start of the next row for 181, and then right along it to 182 and 183, the third square of that row.
The row ends where a new ten is about to start. In the digits, from 179 to 180 the last digit rolls from 9 to 0 and the middle digit goes from 7 to 8.
Day 1: A step counter through a ten and a hundred
Mayahuel’s step counter shows 689. She takes 12 more steps. What does the counter show after each step? Which steps change more than one digit?
SolutionThe counter adds one for each step, so it counts on from 689. Twelve steps give twelve numbers:
Two steps change more than one digit.
- The first step, 689 to 690: the last digit rolls from 9 to 0 and the middle digit goes from 8 to 9. A new ten.
- The eleventh step, 699 to 700: the last two digits roll from 99 to 00 and the first digit goes from 6 to 7. A new hundred, seven hundred.
Every other step changes only the last digit. After all twelve steps the counter shows 701.
Day 2: Across six hundred, both ways
Count on from 598 to 602. Then count back from 602 to 598. Where does the count cross a hundred?
SolutionCounting on, each number is one more:
Five hundred ninety-nine is followed by six hundred. Counting back, each number is one less, so the same numbers come in the opposite order:
Both ways the hundred is crossed between 599 and 600. Going on, the last two digits roll from 99 to 00 and the first digit goes up from 5 to 6. Going back, the last two digits roll from 00 to 99 and the first digit goes down from 6 to 5. In words, the number said after “six hundred” when counting back is five hundred ninety-nine: the hundreds word drops by one and the rest of the name jumps up to ninety-nine.
Day 2: The last ten steps
Kito’s step counter reads 990 as he walks through a park gate. He takes 10 more steps. What does the counter show after each one?
SolutionThe counter counts on by one per step:
The first nine steps change only the last digit. The tenth step is the biggest roll-over in the count. In 999 every digit is 9, so every digit goes back to 0, and a new digit 1 appears in front: one thousand, 1{,}000. Figure 3 shows the same ten steps on a number line. Four digits fit the name: one thousand is ten hundreds, and the hundreds digit has no room for ten.
Day 2: Fill in the count
Fill in the boxes. (a) 407, 408, \square, \square, 411 (b) 703, \square, \square, 700, \square
Solution(a) The numbers go up by one, so this is counting on. After 408 comes 409, and after 409 the last digit rolls over: 410. Then 411 follows, as printed. The boxes hold 409 and 410.
(b) From 703 to 700 the numbers go down, so this is counting back. Before 703 is 702, and before that 701, and then 700 as printed. The last box is the number before 700: the last two digits roll from 00 to 99 and the first digit goes from 7 to 6, giving 699. The boxes hold 702, 701 and 699.
Day 2: How many pieces are left?
Sander has counted 993 pieces of his 1{,}000-piece puzzle. How many more pieces does he have to count to reach 1{,}000?
SolutionCount on from 993 to 1{,}000 and keep track of how many numbers are said after 993, one finger for each:
That is seven numbers, so seven fingers are up when he reaches one thousand. Each number said stands for one more piece, so he has 7 more pieces to count.
Boundaries
- Counting by ones reaches every number, but slowly. From 120 to 1{,}000 is 880 steps. For long distances, jumps of ten and of a hundred (Counting by tens to one hundred, Counting by hundreds) are quicker. Counting by ones is for short stretches and for crossing a roll-over carefully.
- 1{,}000 is where this lesson stops, not where counting stops. Every number has a next one. After 1{,}000 comes 1{,}001, and the larger numbers are Numbers to one million.
- Counting back stops at 0. The counting numbers here are the whole numbers 0, 1, 2, 3 and so on, and none of them comes before 0.
- Digits. Every number from 100 to 999 has three digits, and 1{,}000 has four. Counting back from 1{,}000 to 999 drops a digit, as counting back from 100 to 99 does.
- Zeros are written. The 0 in 305 and both zeros in 300 are digits. Without them the numeral names a different number.
- The word “and”. “Three hundred and four” is common in speech and names the same number as “three hundred four”. Written words in this lesson leave it out.
- The chart pages line up because each row holds ten and each page holds a hundred. That is why 137 sits where 37 sits. A chart with rows of another length would not match.
Where this goes next
Saying, reading and writing any three-digit number, and writing it in expanded form, is Reading and writing numbers to 1,000. Why the middle digit counts tens and the first digit counts hundreds is Place value. On a chart page, moving down one square is ten more, and jumping a whole page is a hundred more. Counting tells which of two numbers comes later, which is where the signs for greater and less come in, and every number here has its place on the The number line. Past one thousand, the same roll-overs carry the count on through Numbers to one million.
Exercises
Day 1: counting on through the hundreds
- Count on from 196 to 205.
- Count on from 437 to 444.
- What number comes right after 150? Right after 199? Right after 579?
- Kito has counted 395 pieces of a puzzle. He counts on 7 more pieces, one at a time. What numbers does he say? How many pieces has he counted now?
- Fill in the boxes: 268, 269, \square, \square, 272
- Mayahuel’s step counter shows 849. She takes 3 more steps. What does it show after each step? Which step changes two digits?
- Which numbers are hidden behind the letters A, B, C, D and E?
- On the chart page of 101 to 200, which row is 164 in, counting from the top, and which square of that row?
Day 2: counting back, the last hundred, and writing the count
- Count back from 503 to 497.
- Count back from 811 to 805.
- Count on from 965 to 972.
- Count back from 1{,}000 to 993.
- What number comes right after 699? Right after 909? Right before 800? Right before 1{,}000?
- Fill in the boxes: \square, 899, \square, 901
- Write the five numbers that come right after 606.
- Mayahuel’s step counter shows 897. She takes 6 more steps. What does it show after each step?
- Sander has counted 988 pieces of his 1{,}000-piece puzzle. How many more pieces does he have to count?
- Kito counts on from a number. After the starting number he says five more numbers, and the last one is 403. What was the starting number?
- Kito’s step counter showed 745 at the park gate and 752 at the corner. How many steps did he take between the gate and the corner?
Explain
- Count on from 298 to 301. Which digits change between 299 and 300? Why do they change at the same time?
- Every number from 100 to 999 has three digits. Why does 1{,}000 need four?
- Why does 137 sit in the same square of the chart page 101 to 200 as 37 does on the hundred chart?
- Is it true that the number right after any number ending in 9 ends in 0? Check it with two examples.
- Sander counts on from 950 to 1{,}000. Mayahuel counts back from 1{,}000 to 950. Do they say the same numbers? What is different?
Answers
- 196, 197, 198, 199, 200, 201, 202, 203, 204, 205. After one hundred ninety-nine comes two hundred, a new hundred.
- 437, 438, 439, 440, 441, 442, 443, 444. A new ten starts at 440.
- 151, 200 and 580. After 150 only the last digit changes. After 199 the last two digits roll to 00 and the first digit goes up: two hundred. After 579 the last digit rolls to 0 and the middle digit goes up: five hundred eighty.
- He says 396, 397, 398, 399, 400, 401, 402, and he has counted 402 pieces. He keeps the 395 already counted, says one number for each of the 7 new pieces, and the last number said tells how many.
- 270 and 271. After 269 the last digit rolls over to give 270, then 271, then 272 as printed.
- 850, 851, 852. The first step, 849 to 850, changes two digits: the last digit rolls from 9 to 0 and the middle digit goes from 4 to 5.
- A is 574, B is 580, C is 589, D is 591 and E is 600. A comes after 571, 572, 573; B ends the first row; C comes just before 590; D starts the third row, right after 590; E follows 599 at the end of the page.
- The seventh row, fourth square. The rows go 101 to 110, 111 to 120, and on, so the seventh row is 161 to 170, and 164 is its fourth square, as 64 is on the hundred chart.
- 503, 502, 501, 500, 499, 498, 497. Between 500 and 499 the last two digits roll from 00 to 99 and the first digit goes from 5 to 4.
- 811, 810, 809, 808, 807, 806, 805. Between 810 and 809 the last digit rolls from 0 to 9 and the middle digit goes from 1 to 0.
- 965, 966, 967, 968, 969, 970, 971, 972. A new ten starts at 970: the last digit rolls from 9 to 0 and the middle digit goes from 6 to 7.
- 1{,}000, 999, 998, 997, 996, 995, 994, 993. The first step back drops from four digits to three.
- 700, 910, 799 and 999. After 699 a new hundred starts; after 909 a new ten starts; before 800 the last two digits roll back to 99; before one thousand comes nine hundred ninety-nine.
- 898 and 900. The middle numbers go up by one, so the first box is one less than 899 and the second is one more.
- 607, 608, 609, 610, 611. A new ten starts at 610.
- 898, 899, 900, 901, 902, 903. The third step, 899 to 900, starts a new hundred.
- 12 more pieces. Counting on from 988: 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1{,}000, twelve numbers, one per piece.
- 398. Counting back five numbers from 403 gives 403, 402, 401, 400, 399, so the start is the number before 399. Check: from 398 the five numbers said are 399, 400, 401, 402, 403.
- 7 steps. Counting on from 745: 746, 747, 748, 749, 750, 751, 752, seven numbers, one per step.
- 298, 299, 300, 301. From 299 to 300 all three digits change. The last digit is 9, so it goes back to 0 and the middle digit goes up; but the middle digit is also 9, so it too goes back to 0 and the first digit goes up from 2 to 3. Ten tens make a new hundred.
- In 999 every digit is already 9, the biggest digit, so one more rolls every digit back to 0. The count needs a new place in front to show the new thousand, so one thousand is written 1{,}000, just as one more than 99 needed a third digit and became 100.
- Both charts start in the first square, 101 on one and 1 on the other, and both put ten numbers in a row. So the count on the page runs exactly beside the count on the hundred chart, each number 100 more than the one in the same square. The number 100 more than 37 is 137, so it sits in the same square.
- Yes, it is always true. For example, after 459 comes 460, and after 699 comes 700. A last digit of 9 cannot go up to a bigger digit, so it goes back to 0 and the digit to its left goes up.
- Yes, they say the same numbers, from 950 to 1{,}000. Sander says them from smallest to largest, starting at 950; Mayahuel says them in the opposite order, starting at 1{,}000.