Box 6–4 Thinking Strategies for Single-Digit Multiplication
In single-digit arithmetic, there are 100 multiplication combinations that students must learn. Commutativity reduces that number by about half. Multiplication by 0 and by 1 may quickly be deduced from the meaning of multiplication. Multiplication by 2 consists of the “doubles” from addition. Single-digit multiplication by 9 is simplified by a pattern: in the product, the sum of the digits is 9. (For example, 9×7=63 and 6+3=9.) Multiplication by 5 may also be deduced through patterns or by first multiplying by 10 and then dividing by 2, since 5 is half of 10.
The remaining 15 multiplication combinations (and their commutative counterparts) may be computed by skip counting or by building on known combinations. For example, 3×6 must be 6 more than 2×6, which is 12. So 3×6 is 18. Similarly, 4×7 must be twice 2×7, which is 14. So 4×7 is 28. (Note that these strategies require proficiency with addition.) To compute multiples of 6, one can build on the multiples of 5. So, for example, 6×8 must be 8 more than 5×8, which is 40. So 6×8 is 48. If students are comfortable with such strategies for multiplication by 3, 4, and 6, only three multiplication combinations remain: 7×7, 7×8, and 8×8. These can be derived from known combinations in many creative ways.
patterns are a hallmark of mathematics. Thus, treating multiplication learning as pattern finding both simplifies the task and uses a core mathematical idea.
After children identify patterns, they still need much experience to produce skip-count lists and individual products rapidly. Little is known about how children acquire this fluency or what experiences might be of most help. A good deal of research remains to be done, in the United States and in other countries, to understand more about this process.
Division arises from the two splitting situations described above. A collection is split into groups of a specified size or into a specified number of groups. Just as subtraction can be thought of using a part-part-whole relation, division can be thought of as splitting a number into two factors. Hence, divisions can also be approached as finding a missing factor in multiplication. For example, 72÷9=? can be thought of as 9÷?=72. But there is little