Written Methods to Six Digits, Estimating to Check

Before any six-digit sum, estimate. Round each number to its front two digits: three hundred and eighty thousand plus two hundred and fifty thousand is about six hundred and thirty thousand. Write that in the margin. Now do the exact working and get six hundred and thirty-four thousand one hundred and twenty-three. It is close to the estimate, so you can trust it. If your working had produced five hundred and thirty-four thousand, or seven hundred and thirty-four thousand, the estimate would tell you immediately that a digit slipped a column. That is the mistake that costs the most marks with big numbers, and ten seconds of estimating is the cure.

Column addition with six digits is the method you know, only longer. Line the numbers up so that ones sit under ones, and work from the right. Six plus five is eleven: write the one, carry the one. Zero plus nine plus the carried one is ten: write zero, carry one. Eight plus one plus one is ten again: write zero, carry one. Five plus eight plus one is fourteen: write four, carry one. And so on to the left. Three carries in a row is nothing unusual with big numbers; the mistake is to hold a carry in your head instead of writing it small above the next column. Estimate at the end: two hundred and fifty thousand plus one hundred and forty thousand is about three hundred and ninety thousand, and three hundred and eighty-four thousand and one is close.

Subtraction across a run of zeros is where the marks go, so here it is slowly. Five hundred and three thousand minus one hundred and seventy-six thousand four hundred and twenty-eight. In the ones, zero take eight cannot be done, so you need to trade — but the tens are zero and the hundreds are zero, and a zero has nothing to lend. Go left until you find a digit that can lend: the three in the thousands. It becomes two, and every zero you passed becomes nine, and the ones column becomes ten. Now the columns all work: ten take eight is two, nine take two is seven, nine take four is five, and in the thousands, two take six needs another trade from the hundred thousands. Write every new digit above the old one. Estimate to check: five hundred thousand minus one hundred and eighty thousand is about three hundred and twenty thousand.

Test questions dress these sums in a story, and the job is to see which operation the story wants. An odometer reads one hundred and twenty-eight thousand four hundred and seventy-six before a trip and one hundred and thirty thousand and fifty-two after. How long was the trip? The trip is the change, and a change is the difference: after minus before. One thousand five hundred and seventy-six kilometres. Estimate: one hundred and thirty thousand minus one hundred and twenty-eight thousand is about two thousand, which fits. Words like altogether, total and combined mean add; how much more, how much left, the change, the difference mean subtract. The size of the numbers has nothing to do with it.

An estimate does more than say yes or no; it points at the mistake. Sam adds forty-seven thousand eight hundred and twelve and thirty-eight thousand two hundred and ninety-five and gets seventy-six thousand one hundred and seven. The estimate is forty-eight thousand plus thirty-eight thousand, about eighty-six thousand. Sam's answer is ten thousand short, so the ten-thousands column is one out — and sure enough, seven plus eight in the thousands made fifteen, and the carry never arrived. Eighty-six thousand one hundred and seven. When the estimate disagrees by a round amount, look for a lost carry or a digit in the wrong column. Quiz and exercises now.