On the NCTM discussion site myNCTM, there’s currently an extended discussion about “Division and multiplication of fractions.” As the discussion has continued, I’ve grown concerned with what I see as a fundamental problem with the way we often introduce multiplication as repeated addition: “Multiplying 4 by 5 means we’re combining five groups of four items. How many do we have altogether?”
This approach sets our students up for problems when they start to consider multiplication and division with fractions. What could it possibly mean to “combine two and two-thirds groups with four-fifths of an item in each group?”
Keith Devlin wrote a series of columns on this for the MAA, the last of which is here: What Exactly Is Multiplication? I am convinced that Devlin and many others who’ve thought deeply about this are correct: teaching kids something that’s wrong and that they’ll have to relearn as soon as they go beyond whole-number arithmetic is a mistake. Even young students understand the ideas of stretching and shrinking, and exploiting these intuitions when we teach multiplication will serve them much better in the long run.
The myNCTM discussion was kicked off with this question: “Is there a way that is internally consistent to model division of fractions for both a whole number divided by a unit fraction and a unit fraction divided by a whole number?” The post challenged us to provide similar explanation for how to compute 4 ÷ 1/3 and 1/3 ÷ 4. Fortunately the discussion didn’t devolve into a debate about memorizing “Ours is not to reason why; just invert and multiply”. (Nix the Tricks does a good job of explaining why this is a bad idea.) Still, the discussion didn’t move in the direction of describing how a scaling approach provides students with a way to relate multiplication and division, to understand the role of multiplicative inverses, and to use one simple model to compute both compute 4 ÷ 1/3 and 1/3 ÷ 4.
Showing is better than telling, so here it is: Below (and here) is a Web Sketchpad double number line model for multiplying and dividing whole numbers and fractions:
Task 1: Play with the number lines: Set different scale factors (n = 2, d = 1 is a good start). Show the control points and experiment to find out what they do.
Task 2: Use these number lines to illustrate two multiplication problems: (a) multiply two whole numbers, and (b) multiply a whole number by a fraction.
Task 3: Illustrate two division problems: (a) divide two whole numbers, and (b) divide a whole number by a fraction. Notice that you can drag the red and green arrows onto each other to show a corresponding problem pair (one multiplication and the other division).
Task 4: Illustrate these two problems from the myNCTM discussion: (a) 4 ÷ 1/3 and (b) 1/3 ÷ 4.
I’d love to receive your comments about this model. How well does it connect multiplication and division? How can it help students understand multiplicative inverses? How hard is it for students to figure out how to use the denominator points? How can the model be improved, or how would you like to see it extended?