As elementary teachers, we rarely have the opportunity to explore division of a fraction by a fraction. When we do, it’s normally accompanied with Keep-Change-Flip or the saying “Yours is not the reason why, just invert and multiply.”
Both are conceptual cripplers.
I’ve been drafting the 4th installment of the Making Sense Series involving fractions and I’m sharing this post as more of a personal reference should K-C-F make its way round these parts again…and I’m sure it will.
Let’s start with a model for 3/5 ÷ 1/4.
At some point along the way it becomes inefficient for students to draw models once the conceptual understanding is established. As students represent measurement division of fractions they should be formally recording their thinking.
From here students generate their own algorithm (shortcut). They begin to recognize that they will always get a denominator of “1 whole” so they begin to purposefully leave it out. In doing so, they become more efficient in the procedure of dividing fractions.
Some students begin to eliminate the green and red steps from the above equation because they’re seen as repetitive. We’ve even had one student that “invented” and generalized cross multiplication for division of fractions as they searched for ways to record fewer numbers and symbols.
It looked something like this…
I keep reminding myself that if fractions are the gatekeeper to algebraic reasoning then I need to slow the process down and conceptually understand what’s happening. This includes K-C-F.
As students understand the power of creating a whole number denominator they begin to search for more efficient ways to get 1 whole. It’s here that they being to explore with equivalent fraction and the idea of using the reciprocal to create a whole.
What I’ve realized is that aside from complex fractions, the underpinnings of this equation are developed in the elementary grades.
- 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a÷b = a/b).
- 4.NF.1-Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b).
- 3.NF.3.c- Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers.
Just like before, students will look to generalize the equation above and find shortcuts. They’ll do so by eliminating repetitive steps which would leave them with…