## Sunday, July 13, 2014

### How should we read 5 × 7?

3.OA.1 makes it clear that 5 × 7 should be interpreted as 5 groups of 7 objects in each group. In other words, 5 is the number of groups and 7 is the group size. But, how should we read this multiplication expression? Interestingly, the CCSS never explains how it should be read. I suspect most people will read “5 × 7” as “five times seven.” However, “times” is not a mathematical term, and another, and perhaps more formal, way of reading a multiplication expression is “__ (is) multiplied by __.” So, is 5 × 7 “five (is) multiplied by seven” or “seven (is) multiplied by five”?

Some people might wonder why we need to worry about this question. In a way, it is a trivial issue. On the other hand, there is at least one instance in the CCSS where this issue is critical. In Grade 4, students are expected to study “multiplying a fraction by a whole number” and in Grade 5, they learn “multiplying a fraction or whole number by a fraction.” So, for example, is 5 × ¾ a Grade 4 topic or Grade 5? How about ¾ × 5? Unless we have an agreement on how to read a multiplication express like “5 × 7,” we can’t answer this question.

Although the CCSS does not discuss explicitly how to read multiplication expressions, there are places in the CCSS and accompanying Progressions documents that suggest what the authors were thinking. For example, 4.NF.1 states: “Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, …” Progressions for the Common Core State Standards in Mathematics: Grades 3-5 Numbers and Operations – Fractions provides this explanation:

“Grade 4 students learn a fundamental property of equivalent fractions: multiplying the numerator and denominator of a fraction by the same non-zero whole number results in a fraction that represents the same number as the original fraction” (p. 6, emphasis added).

Since n in the expression, (n × a)/(n × b), is “the same non-zero whole number,” the numerator, for example, should be read as “a multiplied by n.” Since in the expression, n × a, n is the number of groups and a is the group size, the number following “by” should be the number of groups. In other words, “(group size) multiplied by (number of groups)” is the way to read a multiplication expression. Thus, 5 × 7 should be read as “7 multiplied by 5.”

This interpretation is consistent with the explanation Progressions document provide about Grade 4 “multiplying a fraction by a whole number.” According to their explanation, 5 × ¾ is a Grade 4 topic while ¾ × 5 is a Grade 5 topic. You will also see that the writers of the CCSS tried to pay close attention to this interpretation as you read 5.NF.4.a: “Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b.” In other words, ¾ × 5 means “3 parts of a partition of 5 into 4 equal parts,” or, written in equation, 3 × 5 ÷ 4. In this expression 3 must be written in front of the multiplication symbol because it is the number of groups (or units).

Some of you may be a bit disturbed by the fact that 5 × 7 can be read as “5 times 7” or “7 is multiplied by 5,” reversing the order the factors appear in the expression. Perhaps in the revision of the CCSS, they might choose to use the convention that the first factor is the group size. If 5 in 5 × 7 is the group size, then we can read it as “5 times 7” or “5 multiplied by 7.” The formula for creating equivalent fractions would look like, a/b = (a × n)/(b × n), which might be more familiar. However, we do need to keep in mind that there are multiple ways we describe arithmetic calculations. For example, we can say “7 take away 4” or “7 minus 4,” but we also say “subtract 4 from 7,” again reversing the order of numbers. So, a part of teaching in elementary grades must be to help students become familiar with different ways to describe the same calculation in words. What we do need to avoid is to use different wording without the assumption that it should be obvious to students.