Place value to two digits
Place value of a two-digit number (tens & ones); compose/decompose (30 + 5); use =, <, >.
A two-digit number is written with two figures, and the place each figure sits in tells you how much it is worth. In 35, the 3 is not just “three” — it means three tens.
What it means
We count in groups of ten. When you have ten single things, you bundle them into one ten. A two-digit number records how many of those tens you have, and how many loose ones are left over.
The number has two places. The right-hand digit is the ones place; the left-hand digit is the tens place. So in 35:
- the
5is in the ones place and means5ones, - the
3is in the tens place and means3tens, which is30.
Put together, 35 is 30 + 5. Splitting a number into its tens and ones like this is called decomposing it; joining 30 + 5 back into 35 is composing it. The same digit means different amounts depending on its place: in 53 the 5 now means 50, because it has moved into the tens place.
Because place tells you size, you can compare two numbers by their tens first, then their ones. We record the comparison with three signs: = means equal to, < means less than, and > means greater than. The open mouth of the sign always faces the bigger number, so 4 < 9 and 62 > 60.
Worked examples
Decompose 47. Four tens and seven ones: 47 = 40 + 7.
Compose 8 tens and 2 ones. 80 + 2 = 82.
Compare 63 and 68. Same tens (6 each), so look at the ones: 3 ones is fewer than 8 ones, so 63 < 68.
Compare 71 and 39. 7 tens beats 3 tens straight away, so 71 > 39 — you never even need the ones.
| Number | Tens | Ones | As a sum |
|---|---|---|---|
| 20 | 2 | 0 | 20 + 0 |
| 56 | 5 | 6 | 50 + 6 |
| 90 | 9 | 0 | 90 + 0 |
The generative-art connection
Place value is an arrangement, so you can draw it. Take a scattering of dots and snap them into full rows of ten: the number of complete rows is the tens digit, and the leftover dots in the last, unfinished row are the ones. 35 becomes three tidy rows of ten with a short row of five underneath — the picture and the number carry exactly the same information. The dot-multiplier tool lets children multiply and array dots, so a two-digit count turns into a grid you can literally see and rearrange.
Colour makes the two places visible: paint every complete ten one hue and the loose ones another, and 47 reads at a glance as “four blue bars and seven red dots”. Number Pieces (Math Learning Center) does this with virtual base-ten blocks — long ten-rods and single ones-cubes you drag, bundle and break apart, watching the written number update as the picture changes.
Common misconceptions
- Reading a number as loose digits. A child may see
35as “a three and a five” and think it is3 + 5 = 8. Bundling ten dots into a single rod shows why the3is worth30. - Face value over place value. In comparing
19and21, the digit9looks big, so19seems larger. Counting tens first fixes this: two tens beat one ten, so21 > 19. - Pointing the sign the wrong way. Remind them the wide open end faces the larger number, the narrow point faces the smaller —
<points its point at the little one.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
-
Split `74` into its tens and its ones, and write it as a sum. Then say how many tens and how many ones there are.
Answer
`74 = 70 + 4`, which is `7` tens and `4` ones. The `7` sits in the tens place so it is worth `70`; the `4` is loose ones.
Art hook Draw 74 dots that snap into 7 full rows of ten stacked vertically, with a short row of 4 underneath. Colour the full tens rows blue and the 4 leftover dots red so the split reads at a glance.
-
I have `6` tens and `9` ones. What number is that? Now swap the two digits around — what number do the same figures make then?
Answer
`60 + 9 = 69`. Swapped, `9` tens and `6` ones make `90 + 6 = 96`. Moving a digit into the tens place makes it worth ten times as much.
Art hook Show two dot-grids side by side: one with 6 rows of ten + 9 loose dots (69), one with 9 rows of ten + 6 loose dots (96). A button swaps the digits between the tens and ones boxes so children watch the same two figures land in different places and the grids resize live.
-
Put the right sign (`<`, `>`, or `=`) between each pair: `48` and `52`; `37` and `30 + 7`; `81` and `79`.
Answer
`48 < 52` (4 tens is fewer than 5 tens); `37 = 30 + 7` (both are 3 tens 7 ones); `81 > 79` (8 tens beats 7 tens).
Art hook A number-line ribbon of dots where two markers slide to the two numbers; the gap between them lights up green for the bigger, and a big `<`/`>`/`=` sign draws itself with its open end facing the larger marker.
-
Put these four numbers in order from smallest to largest: `63`, `36`, `60`, `30 + 3`. Explain how you decided.
Answer
`30 + 3 = 33`, so the numbers are 33, 36, 60, 63. Order: `33 < 36 < 60 < 63`. Compare tens first (3 tens before 6 tens), then ones to break ties.
Art hook Four dot-grids that animate into place along a shelf, sorting themselves shortest-to-tallest by number of full ten-rows; ties among equal tens then settle by their leftover ones.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
Here are 3 full rows of ten dots and then 2 loose dots. How many dots are there altogether?
Answer
`3` tens and `2` ones is `30 + 2 = 32` dots.
Art hook Canvas of 3 rows of ten dots plus 2 extra; a running counter shows the number building as each ten-row bundles into a single glowing bar.
-
Write `50 + 6` as a single two-digit number.
Answer
`56` — 5 tens and 6 ones.
Art hook Two stacks of coloured dots (five ten-bars and six singles) slide together and the number 56 fades in above them.
-
In the number `29`, which digit is in the tens place, and what is it worth?
Answer
The `2` is in the tens place; it is worth `20` (two tens). The `9` is 9 ones.
Art hook A two-column place-value chart where tapping a digit highlights its whole column of dots (20 dots for the tens, 9 for the ones).
-
Fill the gap: `40 + ___ = 48`.
Answer
`8`. There are 4 tens already, so the ones must be 8 to make `48`.
Art hook A grid with 4 full ten-rows drawn and one empty row; dragging dots into the empty row stops when the total reads 48.
-
Split `85` into tens and ones and write it as a sum.
Answer
`85 = 80 + 5`, which is 8 tens and 5 ones.
Art hook 85 dots pour in and auto-snap into 8 full ten-rows plus a short row of 5, tens in one hue and ones in another.
-
Choose the right sign for `55` ___ `45`: is it `<`, `>`, or `=`?
Answer
`55 > 45`. The ones are equal (5 each), so the tens decide: 5 tens beats 4 tens.
Art hook Two dot-towers grow to 55 and 45; the taller one glows and a `>` sign snaps in with its open mouth facing it.
-
Spot the mistake: a friend says '`19` is bigger than `21` because 9 is bigger than 1.' Are they right? Explain.
Answer
No. You must compare tens first. `21` has 2 tens, `19` has only 1 ten, so `21 > 19`. The lone `9` in the ones place is worth just 9, not more than a ten.
Art hook Side-by-side dot-grids for 19 (1 row + 9) and 21 (2 rows + 1); the second ten-row on 21 flashes to show why it wins, ignoring the tall-looking 9.
-
The digit `7` is worth `70` in one of these numbers: `17`, `70`, `27`. Which one?
Answer
`70` — there the `7` is in the tens place. In `17` and `27` the `7` is in the ones place, worth just 7.
Art hook Three numbers on cards; tapping each lights the tens column, and only the card where 7 fills a full 70-dot block sparkles.
-
Make the biggest two-digit number you can using the digits `4` and `8`, each once. Then make the smallest. Write both.
Answer
Biggest `84` (put the larger digit, 8, in the tens place). Smallest `48`. So `48 < 84`.
Art hook Two draggable digit tiles that snap into a tens box and a ones box; the resulting dot-grid resizes live, and a leaderboard marks the biggest and smallest arrangements.
-
Count on in ones from `68`: what is the next number? Now count back one from `68`: what comes before it?
Answer
After `68` comes `69`; before `68` comes `67`. Each step of one adds or removes a single loose dot.
Art hook A number line of ten-bundles with a single dot that hops one step right (+1) or left (−1); crossing a ten-boundary (like 69→70) triggers 10 loose dots bundling into a fresh ten-bar.
-
Put these in order from smallest to largest: `72`, `27`, `70`, `77`.
Answer
`27 < 70 < 72 < 77`. Compare tens first (2 tens comes before the three 7-tens), then break the 7-tens ties by ones: 70, 72, 77.
Art hook Four dot-towers of these heights auto-sort left-to-right along a baseline, tens rows locking in first and leftover ones nudging the tied towers into final order.
-
A number is 5 tens and some ones. It is bigger than `54` but smaller than `56`. What number is it?
Answer
`55`. It has 5 tens, and to sit between 54 and 56 the ones must be 5.
Art hook A slider fixed at 5 full ten-rows with a ones-dial from 0–9; two fences at 54 and 56 glow, and only when the dial hits 5 does the number fit snugly between them.