I keep trying to do mental multiplication by rounding one number to something friendly and then “fixing it,” but my brain does a little detour and I forget which way the fix goes. For example, with things like 52×19 or 198×6, I’ll nudge a number to something nicer and then I’m not sure if I’m supposed to compensate by changing the other number or by adding/subtracting something at the end. Sometimes I accidentally do both (oops). Is there a simple rule of thumb for when to adjust the other factor versus when to correct at the end? And is there a neat mental checklist so I don’t double-count or adjust in the wrong direction? I feel like I can almost see the pattern but then it slips away mid-sum.
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3 Responses
Short answer first: pick exactly one move and remember one sign. Either
– Add/subtract a small k to one factor and fix it once at the end, or
– Scale one factor by c and immediately unscale the other by c, with no end correction.
If you mix those, you’ll get lost.
Two reliable “flavours”
1) Additive round-and-correct (change by a small k)
– Pattern: a × b = a × (b ± k) = a×b_easy ± a×k.
– What to do: keep the untouched factor as your “anchor.” Do the easy product with the rounded number, then correct by anchor × k.
– Sign memory: round up, subtract; round down, add. I keep re-checking that: if I made b bigger, I created an extra strip of area a×k, so I must subtract it.
Examples
– 52 × 19
– Round 19 up to 20 (k = +1): 52×20 = 1040; subtract 52×1 = 52; answer 988.
– Or round 52 down to 50 (k = −2): 50×19 = 950; add 19×2 = 38; answer 988.
– 198 × 6
– Round 198 up to 200 (k = +2): 200×6 = 1200; subtract 6×2 = 12; answer 1188.
2) Multiplicative scale-and-swap (change by a small factor c)
– Pattern: a × b = (a ÷ c) × (b × c) or (a × c) × (b ÷ c).
– What to do: apply times c to one factor and divide the other by c immediately. No end correction.
– When it shines: when 2, 4, 5, 8, 10, 25, 50, 100 are in play or you see an easy halve/double, third/triple, quarter/quadruple, etc.
Examples
– 198 × 6 → halve/double: 99 × 12 = 1188.
– 48 × 25 → quarter/quadruple: 12 × 100 = 1200.
– 297 × 34 → divide/multiply by 3: 99 × 102 = (100 − 1)×102 = 10200 − 102 = 10098.
A tiny picture to lock the sign
Think of a rectangle of width b and height a. If you widen b to b + k, you’ve added a thin strip of area a×k stuck to the side. So:
– If you rounded up (added the strip), subtract that strip once.
– If you rounded down (removed a strip), add it back once.
That “strip” is anchored by the untouched side, which is why the correction uses the other factor.
A simple mental checklist to avoid double-counting
1) Choose the flavour first:
– Additive? Then you’ll do one correction at the end.
– Multiplicative? Then you’ll do no correction at the end.
2) Name your anchor (the factor you did not change). Say it in your head.
3) Track the direction:
– Up → subtract the anchor×k at the end.
– Down → add the anchor×k at the end.
4) Do not also adjust the other factor when you’re using the additive flavour. That’s the usual place people double-count.
Quick practice on your examples
– 52 × 19
– Additive with 19 → 20: 52×20 − 52 = 1040 − 52 = 988.
– Additive with 52 → 50: 50×19 + 2×19 = 950 + 38 = 988.
– 198 × 6
– Additive with 198 → 200: 200×6 − 2×6 = 1200 − 12 = 1188.
– Multiplicative halve/double: 99 × 12 = 1188.
Rule-of-thumb for choosing which flavour
– If you’re within 1–3 of a round number (10, 100, 1000), additive is usually fastest: one easy product, one small correction.
– If you can exploit 2/4/5/8/10/25/50/100 relationships, multiplicative is often cleaner: rebalance immediately, then finish.
Common pitfalls and how to dodge them
– Mixing flavours: e.g., bump 198 to 200 (additive) and also double 6 (multiplicative). That changes the product twice. Pick one move.
– Correcting with the wrong factor: the correction uses the anchor, not the rounded number. I sometimes say “anchor times k” out loud in my head to keep it straight.
– Rounding both numbers: if you push both toward friendlier values, you must repair three pieces: the two side strips and the little overlap corner. People usually forget the overlap.
– For rough mental estimates, some ignore that tiny overlap uv when the nudges u and v are small; it’s “close enough.” For exact answers, though, tracking that overlap is fussy, so I avoid double-rounding.
Mini-analogy
It’s like moving a bookshelf against a wall:
– Additive: you slide it 1 inch to the right; you now have a thin gap you must fill or shave off. One strip to fix.
– Multiplicative: you shrink the width by 1/2 and stack two copies; the total wall coverage stays the same automatically. No strip to fix.
If you want one sticky slogan to remember the sign: round up, subtract the strip; round down, add the strip. And only one strip unless you deliberately chose to scale-and-swap, in which case there are no strips at all.
I find it much less slippery if I separate “two different tricks” in my head. Trick A is add/subtract rounding: fix one factor as your base, round the other to a friendly number, and then correct at the end using the base. The formula in words is: base × (friendly ± offset) = base × friendly ± base × offset, with the same sign inside becoming the opposite effect outside (if you rounded up, subtract; if you rounded down, add). Say 52×19: keep 52 as the base and see 19 = 20−1, so 52×19 = 52×20 − 52 = 1040 − 52 = 988. The correction always uses the factor you did not round (here, 52), which stops the double-counting wobble. Trick B is scaling: if you multiply one factor and divide the other by the same number (double/halve, ×10/÷10), the product stays exactly the same and there’s no end correction at all. A quick mental checklist I use: 1) Pick your base and stick to it; 2) Decide the kind of move-add/subtract or scale-don’t mix them; 3) If you rounded up, subtract one “base × offset”; if you rounded down, add it. Example: 198×6-round 198 up by 2 to 200, compute 6×200 = 1200, then subtract 6×2 = 12 to undo the overcount, giving 1188. Once you keep those two buckets separate-“round and correct” versus “scale and compensate”-the direction choices stop tripping you up.
Use one change, one correction: round exactly one factor by d, leave the other untouched, do the easy product, then fix once at the end-round up ⇒ subtract d×(other factor); round down ⇒ add d×(other factor) (e.g., 198×6 = 200×6 − 2×6, 52×19 = 52×20 − 52). Mental checklist: choose which factor to round, note “up or down,” compute the friendly product, then apply ± d×(other factor) with the opposite sign of your arrow-no other adjustments. Hope this helps!