Addition & subtraction facts & mental methods
Know from memory all addition facts to 20; commutativity of +; the inverse relationship of + and −; add/subtract within 100 mentally & with informal written methods.
Once you know your number bonds, addition and subtraction stop being counting and start being recall — and you can stretch those known facts to handle numbers up to 100 in your head.
What it means
An addition fact is a small sum you know by heart, without counting: 7 + 5 = 12, 9 + 8 = 17. This concept is about knowing all such facts up to 20 the way you know your own name — instantly. A subtraction fact is the same relationship read backwards: because 7 + 5 = 12, you also know 12 − 5 = 7 and 12 − 7 = 5.
Two ideas make this manageable, so you never have to memorise everything separately.
Commutativity of addition. The order of the two numbers you add never changes the total: 7 + 5 and 5 + 7 both give 12. So learning one fact hands you a second one free. (Subtraction is not commutative — 12 − 5 and 5 − 12 are different, so be careful.)
The inverse relationship of + and −. Adding and subtracting undo each other. If you go up 5 and then down 5, you land back where you started: 8 + 5 − 5 = 8. This means every addition fact carries two subtraction facts with it, and you can check a subtraction by adding back.
Mental methods are the tricks that turn known facts into bigger answers within 100. The main ones:
- Add the tens, then the ones.
34 + 23->30 + 20 = 50, then4 + 3 = 7, so57. - Bridging through ten.
28 + 5->28 + 2 = 30, then+3 = 33. - Counting on / back on a number line for close numbers.
An informal written method just records these jottings on paper when the numbers get too big to hold in your head — not yet the formal column method, only your thinking written down.
Worked examples
Turning one fact into a family of four:
7 + 5 = 12 (a fact you know)
5 + 7 = 12 (commutativity — same total)
12 − 5 = 7 (inverse — subtract one part)
12 − 7 = 5 (inverse — subtract the other part)Adding within 100 by splitting into tens and ones:
45 + 32
40 + 30 = 70
5 + 2 = 7
70 + 7 = 77Subtracting by counting back through a ten:
53 − 6
53 − 3 = 50 (down to the nearest ten)
50 − 3 = 47 (the remaining 3)Checking with the inverse: is 47 + 6 = 53? Yes — so the answer is right.
The generative-art connection
Every fact is a count you can see. In dot-multiplier, a number is a cluster of dots; 7 + 5 is watching five dots join seven, and 12 − 5 is watching five leave — the inverse relationship becomes a literal forwards-and-backwards animation. On a number line (Math Learning Center), addition is a jump to the right and subtraction the same-length jump to the left; the inverse is one arrow drawn in reverse. Commutativity shows up beautifully as symmetry: 7 + 5 and 5 + 7 are two rows of dots that, swapped top-to-bottom, make the identical picture. When children build these visuals themselves — sliding beads on a Number Rack, or arranging dots — the equality isn’t a rule to trust, it’s a shape they can see is the same.
Common misconceptions
- “Subtraction works in any order.” It does not.
5 − 12is not12 − 5. Only addition is commutative; use the number line to see why direction matters. - Counting instead of knowing. Recomputing
8 + 6on fingers every time is slow and error-prone. The goal is instant recall, which is what frees the mind for the mental methods that handle bigger numbers. - Losing a ten while bridging. In
28 + 5, some split the 5 wrongly (say+3then+3). Always make the jump exactly reach the ten first (+2to 30), then add what is left (+3).
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.
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Answer these straight from memory, no working out: `8 + 6 = ?`, `9 + 7 = ?`, `6 + 6 = ?`, and `15 − 8 = ?`
Answer
`8 + 6 = 14`, `9 + 7 = 16`, `6 + 6 = 12`, `15 − 8 = 7`. These are addition and subtraction facts within 20 that should be known by heart — recalled instantly, not counted out on fingers.
Art hook A 'fact-flash' grid: draw a 4x4 grid of dots for each sum. Tap a cell to reveal the two coloured groups (e.g. 8 blue dots + 6 red dots) rearranging into one full block, so the fact is a picture you can check.
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One fact you know is `6 + 8 = 14`. Write the whole fact family: the other addition fact and the two subtraction facts that come with it.
Answer
`8 + 6 = 14` (commutativity), `14 − 6 = 8` and `14 − 8 = 6` (inverse). One known fact gives you four.
Art hook Place 4 dots at the corners of a square labelled 6, 8, 14 and the '+/−' link; draw arrows both ways between them so the family of four facts is a little diamond you can rotate.
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Work these out in your head and say your steps: `47 + 36` and `82 − 27`.
Answer
`47 + 36`: `40 + 30 = 70`, `7 + 6 = 13`, `70 + 13 = 83`. `82 − 27`: `82 − 20 = 62`, `62 − 7 = 55` (or bridge: `62 − 2 = 60`, `60 − 5 = 55`). Both cross a ten, so they test the full mental method.
Art hook A number line from 0 to 100. Type a start number and a jump; a coloured arc hops in tens then ones. Addition arcs curve right, subtraction arcs curve left, so `82 − 27` becomes two backward hops.
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Freya says `9 + 5 = 14`, so `14 − 9 = 6`. Is she right? How can you check without counting?
Answer
No. `14 − 9 = 5`, not 6. Check with the inverse: `5 + 9 = 14` is correct, but `6 + 9 = 15`, not 14, so 6 is wrong. Adding the answer back must return to 14.
Art hook A balance-scale sketch: left pan holds 14 dots, right pan holds a '9-group' plus an unknown group. The scale tips until the unknown group has exactly 5 dots, showing the wrong guess of 6 makes it too heavy.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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Quick recall, no counting: `7 + 7 = ?`, `10 + 4 = ?`, `3 + 9 = ?`
Answer
`14`, `14`, `12`. These are core facts to 20 to say instantly.
Art hook Ten-frame dots: show two ten-frames side by side. Fill 7 dots then 7 more; the overflow into the second frame lands you at 14 without recounting.
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You know `6 + 4 = 10`. Use commutativity: what is `4 + 6`? Draw or picture why the order does not matter.
Answer
`4 + 6 = 10`. Same two groups, just swapped, so the total is the same.
Art hook Two rows of dots: 6 blue over 4 red. Flip the whole picture top-to-bottom to get 4 red over 6 blue. The dot picture is identical, so commutativity is literal symmetry.
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Fill in the missing number: `8 + ? = 15` and `? − 5 = 9`
Answer
`8 + 7 = 15`, so the first is 7. `14 − 5 = 9`, so the second is 14. Use the inverse to find each unknown.
Art hook A number line with a fixed landing spot at 15. Drag an arrow starting at 8 until its tip hits 15; the arrow's length reads off the missing number, 7.
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Because `13 − 8 = 5`, write two more facts (one addition, one subtraction) that belong to the same family.
Answer
`5 + 8 = 13` (or `8 + 5 = 13`) and `13 − 5 = 8`. All four facts share the numbers 5, 8, 13.
Art hook A triangle with 13 at the top vertex and 5 and 8 at the bottom two. The top is the whole, the bottom are the parts; each way you read the triangle gives one of the four facts.
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Add in your head by splitting into tens and ones: `52 + 34`
Answer
`50 + 30 = 80`, `2 + 4 = 6`, `80 + 6 = 86`. No ten is crossed, so it is a gentle warm-up.
Art hook A base-ten dot grid: 5 ten-sticks + 3 ten-sticks stack into a tens column, loose 2 + 4 dots gather into an ones column, and the picture reads 86.
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Bridge through the ten to add: `38 + 7`
Answer
`38 + 2 = 40`, then `+5 = 45`. Split the 7 into 2 and 5 so you land exactly on 40 first.
Art hook A number line hop that snaps to the nearest ten: the first arc jumps 38->40 (glowing at the ten), the second arc finishes 40->45.
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Subtract by counting back through a ten: `61 − 5`
Answer
`61 − 1 = 60`, then `60 − 4 = 56`. Split the 5 into 1 and 4 to reach 60 first.
Art hook A backward number-line hop: first arc 61->60 lands on the ten, second arc 60->56 finishes, both arrows pointing left to show subtraction.
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Spot the mistake: a friend writes `36 + 27` as `30 + 20 = 50`, `6 + 7 = 13`, `50 + 3 = 53`. Where did it go wrong and what is the real answer?
Answer
They forgot to add the whole 13. It should be `50 + 13 = 63`. They dropped the extra ten from `6 + 7`. The answer is 63.
Art hook Base-ten dots where the ones column overflows: 6 + 7 loose dots make more than ten, so ten of them must bundle into a new ten-stick that slides over to the tens column, turning 53 into 63.
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Solve `73 − 28` in your head, then check your answer using addition.
Answer
`73 − 28`: `73 − 20 = 53`, `53 − 8 = 45` (bridge: `53 − 3 = 50`, `50 − 5 = 45`). Check: `45 + 28 = 73`. Correct.
Art hook A number line showing the subtraction hopping left to 45, then a second glowing arrow drawn in reverse from 45 back up to 73 as the inverse check.
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How many different pairs of whole numbers add to make 12? List them (count `3 + 9` and `9 + 3` as the same pair).
Answer
Seven pairs: `0+12, 1+11, 2+10, 3+9, 4+8, 5+7, 6+6`. Working through them all builds the number bonds to 12 into memory.
Art hook Place 13 dots evenly around a circle (0 to 12). For each bond, draw a chord joining the two numbers that sum to 12; the chords form a symmetric fan folding across the 6 axis.
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Reasoning: `48 + 25 = 73`. Without adding again, use this to find `47 + 25` and `48 + 26`. Explain each.
Answer
`47 + 25 = 72` (one less than 48, so one less total). `48 + 26 = 74` (one more than 25, so one more total). Adjusting a part adjusts the total by the same amount.
Art hook A slider that nudges one addend up or down by 1; a dot cluster gains or loses a single dot and the total counter ticks by exactly 1, showing how small changes ripple through.