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.
4 × 3 = 122 × 3 = 61 × 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.
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.
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.
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