"Whatever you do to the top,..."

I wonder how many people read the title of this post and automatically completed it by saying "... you must do to the bottom." It's a phrase I was drilled in when learning about equivalent fractions, a phrase I used myself when I started teaching. It was only after seeing student work like this one that I began to wonder on the wisdom of using such tips:

So when I think of some of the standard procedures for adding, subtracting, multiplying and dividing fractions, I wonder if as teachers we are guilty of rushing in too quickly to computational strategies before students have a solid enough understanding of the quantity of fractions. I read a quote by Jon Allen Paulos that made me ponder on this even more:

"Mathematics is no more computation than typing is literature."

Recently I was at the Ontario Mathematics Coordinators Association's annual conference. The keynote speaker was Christine Suurtamm from the University of Ottawa. Among the many great ideas and activities that she led us through was this one: 

You can see my solutions to the first two questions. What I love about the questions is that I can see how they will expose and challenge many misconceptions that students have about the quantity of fractions. This gives us an opportunity to fix these misconceptions which in turn will put students in a better position to understand any computational procedures they will need to learn.

Of course, we took up the challenge to describe another structure and then have a colleague build it. My challenge was to build a hexagon that is 3/5 yellow, 1/5 green and 1/5 blue; a simple enough question to state but it provoked a lot of thinking. Chad's challenge to me was to build a hexagon that is 1/6 green, 1/2 red and 1/3 blue. After I came up with one answer, I wondered if others were possible and indeed there was. Are there others?



What I really liked about this activity was its openness: there are so many points of entry and it truly is a 'low floor, high ceiling' question.

This was followed by a 'Fractions War' game. If you are unfamiliar with 'War' games, two players have a pack of cards and both turn over one card. The player with the higher card wins. As Sean and I ran through this game, we faced this situation: 



It reminded me of a misconception that I've often seen where students compare the numerators (and see 3>2) then the denominators (and see 12>4) and then conclude that 3/12 must definitely be bigger than 2/4. These students have not had enough hands-on experience to understand the quantity of fractions like those shown above or in a previous post on Fraction Flags .

And if students really do think that 3/12>2/4, how would teaching them to add these two fractions be beneficial for them?