Right now the Big Birds are studying fractions in math and I've been working with my group to multiply and divide with fractions. For multiplication, I had the students reflect on what multiplication meant before we began. We then applied those definitions to make sense of multiplying by fractions and mixed numerals. 3x1/2 is much easier to understand if you think of it as "3 groups of a 1/2" or "3 halves" or, using the commutative property, "1/2 of 3" After a series of examples, the students developed their own algorithm as a shortcut. One student noticed, "You get the same answer as we got if you just multiply the numerators and then multiply the denomenators...is that always true?" We set out to find out (and, of course, discovered that she was right.)
That sense of discovery is an important piece...but I'll save it for another blog entry.
When it came time for division with fractions, I shared with the students the article that Cathy had found: How Einstein Started Solving its Math Problem. The article mentions the mnemonic the students used to remember how to divide fractions, "Don't ask why...invert and multiply." The Big Birds were wise to me, "Uh oh, you're going to make us ask 'why,' aren't you?"
We began by the students describing division. "It's repeated subtraction." "It's asking how many of the divisor fit into the dividend." "It's splitting into groups." We broke out the playdough to model the number sentences. 3 divided by 1/2 was our first challenge. We made three wholes with the playdough and we split them into halves. We got six groups. We did more examples of increasing complexity, working through confusion and disagreements together. For each, I wrote a picture and the number sentence on the board. Soon, kids started trying out the flip and multiply algorithm to see if it matched the results we were getting. It did! But why?
The next day, we worked with an expanded set of division descriptions and tackled some more complex problems. Each time students responded, I asked them to share which descriptions they used to prove their answer to themselves. Lots of things started to fall into place and students were very excited to feel like they were understanding. On day three, we played a game in which you had to be able to envision the impact of different numbers in the numerator and denominator of the divisor in order to create a winning strategy.
By the end of that day, students felt they really did understand dividing by fractions and how it was related to multiplying by a reciprocal. Even more important, I know they'll be able to understand situations in which this approach will help them find a solution. As one student said, "Once you know why...then flip and multiply!"
It was a fun challenge for me, too. I'd learned the shortcut as a kid and I certainly understand dividing by fractions, but could I really explain why multiplying by a reciprocal produced the same answer as dividing by the original fraction? The students enjoyed watching me struggling to hold my understanding solid as the examples became more complex (whole number dividends were no problem...but while I could see why it still worked for fractional dividends, it was much more difficult to get words around.)
How about you? Can you explain why you flip and multiply? How about why cross multiplication works to find a fraction of a set? By asking the Big Birds to struggle with the "why" they're using the kind of mathematical logic they need to be effective problem solvers and, eventually, do the proofs of higher mathematics.
Here's a video from earlier in our fraction work -- we were creating equivalent fractions (eventually making 100ths to see the relationship between fractions, decimals and percentages.