Reasoning About Decimal Multiplication and Division

Multiplying and dividing decimals can be tricky for students, but it’s often not for the reason we think. Once they know the algorithm, the computation is really just like dividing and multiplying whole numbers. The part that trips them up is deciding where the decimal point belongs in the answer.

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Don’t believe me? Try multiplying 12.1 × 34, 121 × 34, and 121 × 0.34. You’ll find that all three products contain the same digits—4114. The only difference is the placement of the decimal point.

We often teach students to count the decimal places in the factors and then move that many places from the right in the product. While that procedure works, many students don’t understand why they’re doing it. If they forget the procedure—or simply get mixed up while using it—they often end up with an unreasonable answer and have no way of recognizing that it’s incorrect.

That’s why estimation is such a powerful tool. In his excellent book Teaching Student-Centered Mathematics, John Van de Walle suggests that instruction on computation with decimals should begin with estimating. If students can accurately estimate products and quotients, they are much more likely to place the decimal point correctly. Before students ever put pencil to paper, they should have a pretty good idea of what a reasonable answer will be. Is the product about 6? Is the quotient a little less than 1? Having that estimate gives students a benchmark for deciding whether they’ve placed the decimal point correctly instead of relying solely on a memorized procedure.

Ready to give it a try? The card below shows the factors and the digits of the product have already been calculated for you. Your challenge is to estimate first and then decide where the decimal point belongs. Resist the temptation to multiply—you don’t need to! Instead, use your estimate to justify your answer.

If you estimated first, the answer probably jumped right out at you. Since 12 × 2 is about 24, the product should be somewhere around 28. That immediately tells us the answer has to be 28.032. No counting decimal places required.

Ready for a bigger challenge? This time, the product has already been given, but you’ll need to place the decimal points in the factors instead. It’s a little trickier because there may be more than one correct answer (don’t you just love that!). Once again, don’t start multiplying. Estimate first, then use your reasoning to decide where the decimal points belong.

What solution—or solutions—did you come up with? 18 × 14.5 works, but so does 1.8 × 145. Can you explain how estimation supports both answers?

For 18 × 14.5, we can estimate 20 × 15, which is about 300, so a product of 261 is reasonable. For 1.8 × 145, we can estimate 2 × 145, which is about 290, so 261 is reasonable for that expression as well. In both cases, the estimate tells us that the product should be in the hundreds, helping us place the decimal points correctly.

Not surprisingly, the same reasoning works with division. Take a look at the card below. Before you do any computation, estimate the quotient. Then decide where the decimal point belongs and be prepared to justify your answer.

Did you place the decimal after the 2? How did you know?

A quick estimate tells us that 235 ÷ 100 is a little more than 2, so the quotient should be somewhere around 2.4. That makes 2.4 a reasonable answer, while 0.24, 24, or 240 clearly don’t make sense. Notice that we never had to perform the division to know where the decimal belonged.

Helping students reason about decimal operations gives meaning to the traditional procedure. When students estimate first, they have a benchmark for deciding whether their answers make sense and where the decimal point belongs. Download the free Place the Decimal…Justify cards and let your students practice estimating, justifying, and reasoning about decimal multiplication and division.