Common Core: Possible revisions 4 – multiplication as scaling

The idea of multiplication as scaling appears in 5.NF.5:

However, this is the only place the idea (multiplication as scaling) is explicitly mentioned. Moreover, the relationship between the main standard and sub-standards a and b isn’t really clear to me. The standard says “by” doing sub-standards a and b, but does that mean doing sub-standards a and b means interpreting multiplication as scaling? Or, are students to develop the interpretation of multiplication as scaling by tackling problem situations related to those sub-standards?

So, how does the CCSS develop the meanings (interpretations?) of multiplication? As most of you know, multiplication (and division) is introduced in Grade 3 in the CCSS. 3.OA.1 suggests that students initially understand multiplication in the equal group situations – “interpret products of whole numbers, e.g., interpret 5 x 7 as the total number of objects in 5 groups of 7 objects in each. Then, in 3.OA.3, students are expected to “solve word problems in situations involving equal groups, arrays, and measurement quantities.”

Solving array problems then plays a role in considering another interpretation of multiplication. In the Measurement and Data domain, the CCSS expects Grade 3 students to “relate area to the operations of multiplication and addition” (3.MD.7). Specifically, students are expected to “represent whole-number product as rectangular areas in mathematical reasoning.”

Then in Grade 4, the CCSS expects students to “interpret a multiplication equation as a comparison, e.g., interpret 35 = 5×7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5.” I assume the idea of multiplication as scaling is really an extension of this multiplication as comparison idea. One way to make connection between multiplicative comparison and scaling is to put comparison situations in measurement situations involving continuous quantities. So, instead of just thinking of 5×7 as 5 times as many as 7 objects, think about the length if 7 cm is stretched 5 times as long. Thinking about stretching (and shrinking) seems to be easier to make connection to the idea of scaling. So, perhaps the idea of multiplication as scaling should be developed in Grade 4, in conjunction with the idea of multiplication as comparison. Then, in Grade 5, students can use their understanding of multiplication as scaling to make sense of multiplication by fractions/decimals, as well as the ideas discussed in the sub-standards of 5.NF.5.

Another idea that the CCSS does not explicitly discuss is how division is to be interpreted when the interpretations of multiplication change. In Grade 3, students are expected to understand “the relationship between multiplication and division.” Although 3.OA.2 provides 2 different interpretations of division – how many in each share or how many shares – there is no corresponding interpretations of division in other situations. For example, in a multiplicative comparison (or scaling) situation, division may be interpreted as what is the base of comparison or how many times as much. These interpretations parallel how many in each share and how many shares interpretations in equal groups. I think in mathematics education research, the former is referred to as partitive division and the latter quotitive division.

In an array/area situation, though, there is really one interpretation of division – what is the missing dimension – because unlike the other two situations (equal groups & comparison/scaling), the two factors in a multiplication equation are interchangeable – some consider it a ‘symmetrical’ situation. I suppose all of these ideas can be included in 3.OA.6, which states, “understand division as an unknown-factor problem.” But I wonder if that’s too abstract for students in Grades 3 through 5.