Number bonds & +/− within 20
Number bonds within 20; add/subtract within 20 (concrete/pictorial); meaning of +, −, =; missing-number problems.
A number bond is a pair of numbers that fit together to make a total — like 7 and 3 making 10. Once you know the bonds, adding and subtracting within 20 becomes a matter of remembering, not counting.
What it means
A number bond splits a whole into two parts. If the whole is 10, then 6 + 4, 7 + 3 and 8 + 2 are all bonds of 10 — different ways to break the same total. Written as a fact family, one bond gives four true statements:
6 + 4 = 10, 4 + 6 = 10, 10 − 6 = 4, 10 − 4 = 6.
The three symbols do the work. + means put together — combine two parts into one whole. − means take away or find the difference — start with the whole and remove a part to see what is left. = means is the same amount as; the left and right sides balance, like a see-saw that hangs level. = is not an instruction to “write the answer next”; 7 = 3 + 4 is just as true as 3 + 4 = 7.
Within 20 means every number in the problem, and the answer, stays between 0 and 20. The powerful move here is making ten: to work out 8 + 5, split the 5 into 2 + 3, give the 2 to the 8 to make 10, then add the leftover 3 → 13. Bonds of 10 are the stepping-stones for everything larger.
A missing-number problem hides one part and asks you to find it: 9 + ▢ = 14. Because addition and subtraction are two views of the same bond, you can solve it by subtracting: 14 − 9 = 5, so ▢ = 5.
Worked examples
1 — Making ten to add. 7 + 6:
7 + 6
7 + (3 + 3) split 6 into 3 and 3
(7 + 3) + 3 give 3 to the 7
10 + 3 = 132 — A fact family from one bond. Starting from 5 + 8 = 13:
| Addition | Subtraction |
|---|---|
5 + 8 = 13 | 13 − 8 = 5 |
8 + 5 = 13 | 13 − 5 = 8 |
3 — Missing number. ▢ + 4 = 11. The whole is 11, one part is 4, so the other part is 11 − 4 = 7. Check: 7 + 4 = 11. ✓
4 — Subtracting through ten. 15 − 7: take 5 to land on 10 (15 − 5 = 10), then take the remaining 2 → 10 − 2 = 8.
The generative-art connection
Every bond is a partition of a set of dots, and a partition is something you can see. Lay out 10 dots and slide a divider along the row: at each position you read off a different bond — 1 and 9, 2 and 8, 3 and 7 — the whole staying constant while the two parts trade. The Math Learning Center Number Rack is exactly this: ten beads a row, pushed left or right, so a bond becomes a gesture.
In mathartcademy, dots are the raw material of the art. The dot-multiplier tool grows a single dot into a symmetric burst — and the count in each arm, plus the count across arms, is a live bond: arm + arm = total. Because the arrangement is symmetric, splitting the total into equal or unequal parts changes the shape you see, so a child adjusting a number is really composing a picture. Making ten stops being a rule to memorise and becomes a satisfying visual click: two groups snapping into one full ten-frame.
Common misconceptions
- “
=means the answer comes next.” No —=means balances. Practise reading13 = 8 + 5aloud so the symbol means sameness, not “do it now”. - Counting on from 1 every time. If a child recounts all the dots, they haven’t yet trusted the bond. Cover one part and ask them to know the other — that is the leap from counting to fluency.
−is only “take away”. Subtraction also means difference:12 − 9can be found by asking “how far from 9 up to 12?” — sometimes far quicker than removing nine things.
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.
-
Add these within 20, using number bonds so you don't have to count on your fingers: `8 + 7`, `6 + 9`, and `5 + 8`.
Answer
`8 + 7 = 15` (split 7 into 2+5: 8+2=10, then 10+5=15). `6 + 9 = 15` (9+1=10, then +5). `5 + 8 = 13` (8+2=10, then +3). All cross ten by making ten first.
Art hook A ten-frame filler: draw two 5x2 grids of empty circles side by side. For `8+7`, light 8 dots in the first frame in one colour, then fill its 2 empty holes with the second colour and spill the remaining 5 into the next frame — the moment of 'making ten' is the frame turning fully solid.
-
Find the missing part in each: `7 + ▢ = 16`, `▢ + 9 = 15`, and `13 − ▢ = 5`.
Answer
`▢ = 9` (16 − 7 = 9). `▢ = 6` (15 − 9 = 6). `▢ = 8` (13 − 5 = 8, since you take 8 away to leave 5). Each uses the bond: whole minus known part = missing part.
Art hook A balance-beam animation: a horizontal bar pivoting on a centre point. Left pan holds `7` dots plus an empty box; right pan holds `16` dots. Dragging dots into the box tilts the beam until it hangs level exactly when the box holds 9 — the child feels `=` as balance.
-
Here is one bond: `9 + 5 = 14`. Write the whole fact family — all the addition and subtraction facts that use these three numbers.
Answer
`9 + 5 = 14`, `5 + 9 = 14`, `14 − 9 = 5`, `14 − 5 = 9`. One bond of three numbers always gives these four true statements.
Art hook A triangle bond diagram: place the whole `14` at the top vertex and the parts `9` and `5` at the two bottom vertices, drawn as clusters of that many dots. Tapping any vertex to 'hide' it reveals the four facts as arrows sweeping around the triangle.
-
Is each statement true or false? `6 + 8 = 14`, `13 = 4 + 9`, and `20 − 5 = 5 − 20`. Explain the tricky one.
Answer
`6 + 8 = 14` true. `13 = 4 + 9` true (`=` means 'is the same amount as', so the total can sit on the left). `20 − 5 = 5 − 20` false: `20 − 5 = 15` but `5 − 20` is not the same, so the two sides don't balance.
Art hook A see-saw checker: two stacks of dots, one per side of the `=`. If the counts match, the see-saw hangs level and glows green; if not, it tips down on the heavier side and glows red — a true/false statement becomes a physical tilt.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
Say the pairs that make `10`. Fill each blank: `10 = 6 + ▢`, `10 = 3 + ▢`, `10 = 8 + ▢`.
Answer
`4`, `7`, `2`. These are bonds of ten — the stepping-stones for everything bigger.
Art hook Ten dots evenly spaced on a circle. A sweeping line splits the circle into two arcs; wherever it lands you read the bond of ten (6 dots on one side, 4 on the other). Rotate the line to see every pair 1+9, 2+8, 3+7...
-
Add without crossing ten (the answer stays under 20): `12 + 4`, `13 + 5`, `11 + 6`.
Answer
`16`, `18`, `17`. The tens stay the same; just add the ones.
Art hook A number line from 0 to 20 drawn as 20 dots in a row. A frog starts on dot 12 and hops 4 dots to the right, landing on 16 — each hop leaves a coloured footprint so the jump is visible.
-
Subtract without crossing ten: `18 − 5`, `16 − 4`, `19 − 6`.
Answer
`13`, `12`, `13`. Take the ones away from the ones.
Art hook Same 0–20 dot line, but the frog hops left. Start on 18, hop back 5, land on 13 — footprints fade behind it to show 'taking away'.
-
Make ten to add: `9 + 4`. Show how you split the `4`.
Answer
`13`. Split `4` into `1 + 3`; give the `1` to the `9` to make `10`; `10 + 3 = 13`.
Art hook One ten-frame with 9 dots already lit and a single empty hole. Watch 1 dot slide across from a waiting pile to fill it — the frame flashes 'full ten!' — then the remaining 3 dots drop into a second frame below, so the split of `4` into `1 + 3` is a single visible slide.
-
Add these that cross ten: `7 + 5`, `8 + 6`, `9 + 3`.
Answer
`12`, `14`, `12`. Each time, make ten first, then add what's left. `9 + 3`: 9+1=10, then +2 = 12.
Art hook For each sum, place the two numbers as dot-clusters on a colour wheel; when they merge past 10 the hue jumps to a new colour band, so 'crossing ten' becomes a visible colour change.
-
Find the missing part: `▢ + 6 = 13` and `9 + ▢ = 17`.
Answer
`7` (13 − 6 = 7) and `8` (17 − 9 = 8). Whole minus the known part gives the missing part.
Art hook A bar split into two coloured segments totalling 13 units long. One segment is fixed at 6; drag the divider until the second segment length reads 7 and the whole bar exactly matches the 13-mark.
-
Spot the mistake: a friend writes `8 + 5 = 12`. Is that right? If not, fix it and explain.
Answer
Wrong — `8 + 5 = 13`, not 12. Make ten: 8 + 2 = 10, then + 3 = 13. The friend probably lost one on the way over ten.
Art hook Show 8 dots then 5 dots dropping into a ten-frame set; a running counter ticks each dot, catching the off-by-one — the miscount lights up red as the 13th dot lands.
-
Subtract by crossing ten: `14 − 6` and `13 − 8`.
Answer
`8` and `5`. For `14 − 6`: take 4 to reach 10, then 2 more → 8. For `13 − 8`: take 3 to reach 10, then 5 more → 5.
Art hook Number-line frog again: from 14, first hop back to the nearest ten (10), pause, then hop the rest — the two-part jump makes 'bridging ten' obvious.
-
Which pairs are bonds of `20`? Choose from `15 + 5`, `12 + 6`, `14 + 6`, `11 + 9`.
Answer
`15 + 5`, `14 + 6`, and `11 + 9` all make 20. `12 + 6 = 18`, so it is not.
Art hook Twenty dots on a circle; a divider line splits them into two arcs. Only when the two arc-counts add to the full 20 does the whole ring glow gold — test each pair by placing the line.
-
True or false, and explain: `7 + 6 = 6 + 7`.
Answer
True. Swapping the order of the two parts gives the same total (both are `13`). Addition doesn't care which part comes first.
Art hook Two dot-groups (7 and 6) sitting on a mirror line. Reflect them to swap sides; the combined count stays 13 either way, so the picture is symmetric — order-swap as a reflection.
-
A missing number in the middle: `15 − ▢ = 9`. What was taken away?
Answer
`6`. Since 15 − 6 = 9 (check: 9 + 6 = 15). Ask 'how far from 9 up to 15?' — that gap is 6.
Art hook A bar of 15 units; slide a shrinking cover from the right until only 9 units show. The length now hidden reads 6 — subtraction seen as covering up the difference.
-
Write your own bond of `16` two different ways, then write a subtraction fact from one of them.
Answer
Example: `10 + 6 = 16` and `9 + 7 = 16`; from the first, `16 − 6 = 10`. Any correct pair summing to 16 and a matching take-away fact is fine.
Art hook A 'bond builder' triangle with 16 at the top; drag dots into the two lower corners in any split — the tool instantly draws the matching + and − facts underneath, so each split you invent generates its own fact family.