How can you use repeated reasoning to find multiplication facts?

A multiplication fact-practice exercise might, for example, ask children to choose three numbers in a row (e.g., 5, 6, and 7) and compare the middle number times itself to the product of the two outer numbers. In this example, the two products are 36 and 35. After they do this for several triplets of numbers, they are likely to conjecture a pattern that allows them to multiply 29 × 31 mentally because they expect it to be one less than 30 × 30, which they can do easily in their heads. Seeing that regularity is typically easy for fourth graders; expressing it clearly is much harder. Initial attempts are generally inarticulate until students are given the idea of naming the numbers. A simple non-algebraic “naming” scheme was used above to describe the pattern: the numbers were named “middle” and “outer” and that was sufficient. A slightly more sophisticated scheme would distinguish the outer numbers as something like “middle plus 1” and “middle minus 1.” Then children can state (middle – 1) × (middle + 1) = middle2 – 1. The step from this statement to standard algebra is just a matter of adopting algebraic conventions: naming numbers with a single letter like m instead of a whole word like “middle,” and omitting the × sign.

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