Mental Strategies with Large Numbers

When one of the numbers is nearly round, treat it as round and fix it afterwards. Three thousand nine hundred and ninety-eight is two short of four thousand, and four thousand plus two thousand seven hundred and forty-five is easy: six thousand seven hundred and forty-five. But you added two too many, so take two off: six thousand seven hundred and forty-three. Subtraction works the same way in reverse. Eight thousand two hundred and fifty minus one thousand nine hundred and ninety-five: take away two thousand to get six thousand two hundred and fifty, and because you took away five too many, give five back: six thousand two hundred and fifty-five.

When neither number is near anything round, split them both into their place-value parts and add the parts. Four thousand three hundred and fifty-six plus two thousand eight hundred and thirty: the thousands make six thousand, the hundreds make eleven hundred, the tens make eighty, the ones make six. Then add those from the biggest down, saying the running total: six thousand, seven thousand one hundred, seven thousand one hundred and eighty, seven thousand one hundred and eighty-six. Eleven hundred being more than a thousand is not a problem — it simply carries into the thousands when you recombine.

A subtraction is the distance between two numbers, and you can measure that distance by counting up from the smaller one through round landmarks. Five thousand minus three thousand seven hundred and eighty: from three thousand seven hundred and eighty up to four thousand is two hundred and twenty, and from four thousand up to five thousand is another thousand, so the distance is one thousand two hundred and twenty. Giving change is the same move: from thirty-seven dollars forty-five, five cents reaches thirty-seven fifty, fifty cents reaches thirty-eight, twelve dollars reaches fifty. No borrowing, no crossing out.

None of these strategies is best; each fits certain numbers. Look before you start. If one number is within a few of a round number, compensate. If you are subtracting from a round number, bridge up. If neither is true, partition. And if the numbers are genuinely awkward — six digits, lots of carrying — the written column method from the next lesson is the right tool, and there is no prize for doing it in your head. On a test, choosing well is what buys you the time for the hard questions.

Test questions chain two or three of these together. A shop starts with twelve thousand tickets and sells three thousand nine hundred and ninety-five on Friday and four thousand two hundred and sixty on Saturday. Take the first sale off by compensating: twelve thousand minus four thousand, plus five, is eight thousand and five. Take the second off by partitioning: eight thousand and five minus four thousand is four thousand and five, minus two hundred and sixty is three thousand seven hundred and forty-five. Write each result down as you go; the numbers that get swapped are the ones you tried to hold in your head. And estimate: twelve thousand minus about eight thousand is about four thousand, so three thousand seven hundred and forty-five is believable. Quiz and exercises now.