I’m trying to get faster at multiplying numbers in my head by using the “close to a nice number” idea, but I keep tripping over the adjustment step. For example, with 35 × 19, I think “19 is 20 minus 1,” and then I somehow do 35 × 20 = 700 and subtract 1 to get 699. I know that feels off, but I can’t seem to stop doing it. Then my brain goes in circles: if I think of 100 × 99, I end up thinking 10000 − 1 = 9999, which also seems wrong but I can’t articulate why in the moment.
A similar thing happens with 48 × 25. I try to be clever: 50 × 25 = 1250, so I just subtract 2 to “fix” it and say 1248, which I’m pretty sure is not the right adjustment. I suspect I’m misusing the distributive property – I’m subtracting the 1 (or the 2) instead of subtracting the 1 (or 2) times the other number, but when I’m doing it mentally, I lose track of which thing I’m supposed to adjust and by how much.
What’s a simple, reliable way to keep the compensation straight in my head for cases like a × (b ± 1) or when I replace a number with something round like 20, 50, or 100? Is there a quick mental cue that stops the “subtract 1” mistake without having to write it down each time?
Any help appreciated!
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3 Responses
Mental cue: when you nudge a factor, you must “pay” by the other factor times the nudge-formally a·(b ± k) = ab ± a·k, so 35·19 = 35·(20−1) = 700−35 = 665, 100·99 = 100·(100−1) = 10000−100 = 9900, and (50−2)·25 = 1250−2·25 = 1200 (also 48·25 = 12·100 = 1200-nice!).
Does thinking “correction = change × the other number” lock it in for you, or do you like the halving–doubling route better?
I trip over that same wire if I don’t give myself one tiny reminder: change a factor by k, and the product changes by k times the other factor. Think in “groups.” If 19 means “19 groups of 35,” then pretending it’s 20 means you’ve added one extra group of 35, so you must subtract one 35, not one. In symbols that’s just the distributive property: a·(b ± k) = ab ± k·a. If you round b to a nearby round number r, the correction is (r − b) times a. If you round a instead, the correction is (r − a) times b.
Quick examples you can say to yourself:
– 35 × 19 = 35 × (20 − 1) = 700 − 35 = 665. “Twenty groups of 35 is one group too many; take away one 35.”
– 100 × 99 = 100 × (100 − 1) = 10,000 − 100 = 9,900. “One group of 100 too many; subtract 100.”
– 48 × 25 = (50 − 2) × 25 = 50 × 25 − 2 × 25 = 1,250 − 50 = 1,200. “Two groups of 25 too many; subtract 50.” (Or use the neat 25 × 4 = 100 trick: 48 × 25 = (48 ÷ 4) × 100 = 12 × 100 = 1,200.) My mental cue is to say “groups” in my head: “I added k groups, so I must remove k of the other number.” That phrasing keeps me from doing the “subtract 1” slip.
A simple rule keeps the compensation straight: when you change one factor by k, you must change the product by k copies of the other factor. In symbols, a×(b±k) = ab ± a×k and (a±k)×b = ab ± k×b. A quick mental cue is: “What I add or subtract from one factor, I add or subtract that many groups of the other factor.” This stops the “subtract 1” mistake because you remember to subtract 1 group of the other number, not just 1.
Now apply it to your examples. 35×19 = 35×(20−1) = 35×20 − 35 = 700 − 35 = 665. Notice the check: since 19 is one less than 20, the product should be about 35 less than 700, not just 1 less. Similarly, 100×99 = 100×(100−1) = 10000 − 100 = 9900; being one less in a factor knocks off one group of the other factor (here, 100). For 48×25, think (50−2)×25 = 50×25 − 2×25 = 1250 − 50 = 1200, not 1248. The same cue works for any k: if you rounded up by k, later subtract k times the other number; if you rounded down by k, later add k times the other number.