A hundred is a flat square
Ten small cubes in a row make a rod of ten. Ten rods side by side make a flat square, 10 cubes across and 10 cubes down, and that flat is one hundred: 10 × 10 = 100 small cubes.
One flat is one hundred small cubes, 10 across and 10 down.
Stack ten hundreds
Now put a second flat on top of the first, then a third, and keep going until there are 10 flats in the stack. The stack is a solid cube: 10 small cubes long, 10 wide and 10 high.
It is made of 10 hundreds, so it holds 10 × 100 = 1000 small cubes. This block is one thousand.
Ten hundreds stacked up make one solid thousand: 10 × 100 = 1000.
7 hundreds = 700
Stack flats until ten hundreds make a thousand
Each flat is a hundred. The tenth flat closes the stack into one thousand cube, and a flat after that starts a new stack.
The same ten flats laid side by side hold 10 × 100 = 1000 small cubes, as many as the thousand cube.
The same step every time
Ten ones make a ten. Ten tens make a hundred. Ten hundreds make a thousand. Each block is ten of the block before it, and that is why each place in a number is worth ten times the place on its right.
Four places, four digits
A number with four digits has a thousands place. In 1234, the 1 counts the thousand cubes, the 2 counts the hundred flats, the 3 counts the rods of ten and the 4 counts the single cubes. Each digit counts its own kind of block: 1234 = 1000 + 200 + 30 + 4.
1 thousand, 2 hundreds, 3 tens and 4 ones make 1234.
When the hundreds fill up
Add 100 to 3980. The 9 hundreds and 1 more hundred make 10 hundreds, and 10 hundreds stack into one thousand. So the thousands digit goes up from 3 to 4, the hundreds digit goes back to 0, and 3980 + 100 = 4080.
The usual mistake is to count a thousand cube as if it were a flat. It is one block, but it is worth 1000, not 100 and not 1. With 1 cube, 2 flats, 3 rods and 4 single cubes, the number is 1234, not 334.
Worked example: Place-Value Blocks, Then One Flat More and One Cube Fewer
Question Mei has place-value blocks. She has 2 thousand cubes, 5 hundred flats, 3 ten rods and 8 ones. (a) What number do her blocks show? (b) Mei puts in 1 more hundred flat and takes away 1 thousand cube. What number do her blocks show now?
1.Write the value of each kind of block. 2 thousands are 2000. 5 hundreds are 500. 3 tens are 30. 8 ones are 8.
2 thousands are 2000, 5 hundreds are 500, 3 tens are 30 and 8 ones are 8. 2.(a) Add the values: 2000 + 500 + 30 + 8 = 2538. The blocks show 2538.
(a) 2000 + 500 + 30 + 8 = 2538. 3.One more hundred flat is 100 more. Only the hundreds digit changes, from 5 to 6: 2538 + 100 = 2638.
One more hundred flat: the hundreds digit goes from 5 to 6, so the number is 2638. 4.One thousand cube fewer is 1000 less. Only the thousands digit changes, from 2 to 1: 2638 − 1000 = 1638.
One thousand cube fewer: the thousands digit goes from 2 to 1. 5.(b) The blocks now show 1638. Check: there are now 1 thousand, 6 hundreds, 3 tens and 8 ones.
(b) The blocks now show 1638.
Answer: (a) 2538; (b) 1638
Common mistakes
- Adding the numbers of blocks, 2 + 5 + 3 + 8 = 18. Each kind of block has its own value: a thousand cube is worth 1000, not 1.
- Changing the wrong digit for 100 more, for example writing 2548. That is only 10 more. 100 more changes the hundreds digit, which is the 5 in 2538.
More thousands, angles and time problems, worked step by step →
Worked example: A Number Pattern That Crosses a Thousand
Question Five lockers in a row are numbered in a pattern. The numbers are 3780, 3880, 3980, a number that has rubbed off, and 4180. (a) By how much does the number go up from one locker to the next? (b) What is the number that has rubbed off?
1.Subtract the neighbors that you can see: 3880 − 3780 = 100 and 3980 − 3880 = 100.
3880 − 3780 = 100 and 3980 − 3880 = 100. 2.(a) The number goes up by 100 from one locker to the next.
(a) The number goes up by 100 from one locker to the next. 3.Add 100 to 3980. 9 hundreds and 1 more hundred make 10 hundreds, which is 1 more thousand: 3980 + 100 = 4080.
9 hundreds and 1 more hundred make 1 more thousand: 3980 + 100 = 4080. 4.(b) The number that has rubbed off is 4080. Check: 4080 + 100 = 4180, which is the last locker.
(b) The missing number is 4080. Check: 4080 + 100 = 4180.
Answer: (a) 100; (b) 4080
Common mistakes
- Writing 3990 for the missing number. That adds only 10, but the numbers go up by 100 each time.
- Writing 3080 when the hundreds digit passes 9. Ten hundreds are exchanged for one thousand, so the thousands digit goes up from 3 to 4 and the hundreds digit becomes 0.
More thousands, angles and time problems, worked step by step →