Multiplication makes things bigger and division makes things smaller, right? Well, not so fast...
Students often think that multiplication makes numbers bigger, and division makes them smaller. This notion is understandable, and it’s part of a healthy number sense—when you’re using whole numbers.
Alas, when students multiply and divide by fractions, this sensible pattern turns out to be wrong.
What does this mean for a math teacher?
Be alert for these misconceptions. When you see a student doing something you can’t account for, consider whether the student might think that multiplication must make numbers larger. For example, suppose Aloysius is using a calculator to multiply 3 times 0.35. His display shows 1.05. If he thinks that result must be too small, he might assume that he misplaced a decimal point and report 10.5 as the answer.
Then, develop multiple strategies to support students' understanding of multiplication and division, and to help them develop number sense that extends beyond whole numbers. Here are several approaches for multiplication.
- Help students see that multiplication often means “of,” especially with fractions. So 0.35 × 3 means, “0.35 of three”—and that’s smaller than three. This works (albeit a bit more awkwardly) with whole numbers, e.g., three of five really is 15.
- Help students reason from number patterns. For example,
4 × 3 = 12
2 × 3 = 6
1 × 3 = 3
½ × 3 = ?
- The first number—4, 2, 1, ½—gets divided by 2 each time. The number after the equals sign does as well: 12, 6, 3, what’s next? Half of three, or 1½.
- If you go one more step, you get ¼ × 3, which is half of 1½, or ¾.
- Use a calculator, not as the sole source of answers, but for confirmation.
- Address the issue head-on and ask students to generalize: “If you multiply two numbers, how can you tell whether the product is larger or smaller than the two factors?” (This way of stating the task lets students discover and resolve an interesting question: when is a product between the two factors?)
Division
The same principles hold with division: be alert for the misconception (in this case, that division always makes things smaller), and develop strategies to confront that misconception. With division of fractions, however, both the concepts and the computations are harder than with multiplication.
- Check out these short videos about division to see the different meanings of division for integers.
- Check out the division-of-fractions tune-up for several ways to visualize division of fractions using a “measurement” perspective and for examples of the other perspectives on division.
- “Sharing” division with a fractional denominator is weird. How do you share 2⅔ pizzas among ⅝ of a person? It might be easier to phrase a question in terms of a group. For example, if ⅝ of the group ate 2⅔ pizzas, that makes some sense even if the context sounds fake. But we can avoid explicitly “sharing” while keeping the underlying math the same by “spreading” something equally. For example, Penelope put 2½ ounces of gas in her go-kart. It ran ⅝ of a mile. How much gas does it need to go a whole mile? (2½ ÷ ⅝ , or 40/10 ounces: 4 ounces, a half-cup.) We’re spreading the gas over the distance to get that quotient. And we intuitively understand that the answer has to be bigger than 2½.
- For students who are good with arrays and area, it might help to cast a problem as a missing-factor situation, perhaps using a rectangle or array for which they know the area and the length of one side.
- As with multiplication, use the calculator as a check.
- And as with multiplication, have students generalize.
One danger with fractional division is that students rely too much on calculators or the “invert and multiply” algorithm to find answers without actually understanding the process. Part of that process is recognizing that division is what you need to do. So don’t assume that a student who can reliably calculate answers to disembodied problems understands the topic. Translating situations into mathematical expressions is essential.
Check out tune-ups about delaying "invert and multiply" and about ratios for more about calculating fractional quotients without the algorithm.