By William McCallum
In grade 3, as students start to learn about multiplication, they think about products like 6 x 7 in terms of equal groups. 6 x 7 is the number of things when you have 6 groups with 7 things in each group. They might start out calculating that number by drawing a picture of the 6 groups and counting how many things they are. They might use a 6 x 7 array to organize the count. They might then see that the total number is 7 + 7 + 7 + 7 + 7 + 7 and do the additions 7 + 7 = 14, 14 + 7 = 21, etc. From there they might learn to simply write down the multiples, doing the additions mentally:
7, 14, 21, 28, 35, 42
Then, to organize this sort of work, they might make a multiplication table for the multiples of 7:
Of course, by the end of grade 3, the pictures and arrays and tables go away. We expect students to know all these multiples from memory. When I was a child, we achieved that by chanting in unison at the beginning of each class, “one seven is seven, two sevens are 14, three sevens are 21. . .” I used to do that by mentally adding 7 each time. Students also start to solve word problems involving multiplication, for example, how many days are there in 6 weeks?
Later, in grade 6, students start to learn about ratios. They understand that the ratio of days to weeks is 7:1, that the associated rate is 7 days per week, and they make ratio tables, such as
Are these two tables the same thing? Was the first table just a ratio table in disguise? Although they look similar, I think there are some significant differences in the way they are used and conceptualized. The multiplication table is a lookup table, designed ultimately to become obsolete. It is an organized list of individual products. I might look up 6 x 7 for one problem, then look up 8 x 7 for another. It is also a table which, ideally, I would not have to keep producing. I might have the entire multiplication table, not just for the 7s, displayed somewhere in a book or on a classroom wall.
On the other hand, the ratio table is an object of study in its own right, representing a complex of embedded ideas. The rows are ratios. Any two rows represent different but equivalent ratios, related by a scale factor. The columns represent quantities in a context, related by a rate or by a constant of proportionality. The ability to see all these relationships quickly and easily is what the focus on arithmetic in K–5 has been preparing students for.
Studying how the quantities in the columns change in relation to each other is the beginning of a journey that leads to proportional relationships, then linear functions, then functions in general. The 7 in this table is more than just a number, it is a rate, the common rate for all the equivalent ratios in the table. The table is not a throwaway tool for helping to remember products, it is a fundamental representation that will continue until high school and beyond. And although initially the entries and the rate will be whole numbers, we want students to generalize to fractions, then rational numbers, then real numbers. We want them to start using variables to represent the quantities in the columns and write equations for the relationship between them.
Mixing up these two ways of looking at the table has potential for confusion. For example, in the first table I am thinking of the 42 as 6 x 7. If I constructed the entire multiplication table for all single digit products I would notice a symmetry that reflects the commutative property, and see that 6 x 7 = 7 x 6. On the other hand, in the ratio table, where I am thinking of 7 as the rate, I want students to see the 42 as 7 x 6. And the 35 as 7 x 5 and so on. Asking students in grade 3 to switch from seeing 6 weeks as 6 groups of 7 days to seeing the 7 as a rate is a distraction from the main purpose, which is to nail down single-digit products.
There is no harm in elementary students playing around with the multiplication table and seeing patterns that will later stand them in good stead when they dig into ratio tables. But that’s not the same thing as asking them to consider ratio tables as mathematical objects representing an important mathematical concept. That’s a huge conceptual leap, disguised by the smallness of the act of adding headings to the columns.
Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.