I’m trying to get faster at mental multiplication for everyday stuff (like rough totals while shopping), but when I try two-digit-by-two-digit, my brain drops a gear and starts juggling jelly. For example, with 38 × 27 I try the distributive thing: (40 − 2)(30 − 3) = 40×30 − 40×3 − 2×30 + 2×3. I can start: 40×30 = 1200, then I subtract 120, and then I get wobbly-do I subtract 60 or 80 for the next part? And then I add 6, but by then I’ve lost track and my tens and ones are doing a conga line.
I also tried the “round then fix” idea: 38 × 27 = 38 × (30 − 3), which I rewrite as 38×30 minus 38×3. I can hold 38×30 in my head, but I keep mis-doing 38×3 (I say 90-something and then forget exactly what). Another attempt: halve/double to 19 × 54-felt clever, still got tangled.
What’s a reliable mental strategy to do this without dropping the cross-terms or place values? Is there a preferred order (like always do tens×tens, then tens×ones, etc.) or a neat way to keep a running total so I don’t mix up −60 and −80? If you can show how you’d step through something like 38 × 27 or 47 × 58 in your head, that’d be amazing.
Any help appreciated!
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3 Responses
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I like to think of two-digit-by-two-digit as stacking boxes on a shelf from left to right so nothing topples: split each number into tens and ones, then keep a running total that you say out loud after every box you “place.” For 38 × 27, go (30 + 8)(20 + 7): start with tens×tens = 30×20 = 600, park “six hundred”; add 30×7 = 210 to get 810; add 8×20 = 160 to get 970; add 8×7 = 56 to land on 1026. Saying the totals out loud-six hundred, eight-ten, nine-seventy, one-thousand-twenty-six-keeps the middle bits from dancing away. I used to try (40 − 2)(30 − 3) and get tangled because I’d think the two minus cross-terms sort of cancel so I’d add 120 and then take off 60, which isn’t right, and my place values went wobbly. The left-to-right build avoids that because you only ever add the next chunk to the running pile. One more example: 47 × 58 → 40×50 = 2000, plus 40×8 = 320 → 2320, plus 7×50 = 350 (I sometimes mistakenly say 300 and then remember to tack on the extra 50) → 2670, plus 7×8 = 56 → 2726. If you like a visual backup, this is basically the area model done mentally; Khan Academy’s 2-digit multiplication page shows the same idea with boxes: https://www.khanacademy.org/math/arithmetic/arith-review-multiply-divide/arith-review-multi-digit-multiplication/a/multiplying-2-digit-numbers