"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?





Geoboards in the Car

So I'm driving my daughter to her dance class when she picks up my iPad and asks about the Geoboard app that is on there. I tell her that you can use it to make shapes like the ones she has been learning in class. "Make some trapezoids for example," I tell her. After a while she says "Done! Is this right?". "I can't look now I'm driving!" So we wait until a stop light and then (as she is sitting in the back) she shows me in the rear-view mirror:

"Great. Are they parallelograms too?"
"No... they only have one pair of parallel sides"
"OK. Now make some trapezoids that have a right angle". Now this took longer and a fair amount of "How is that possible?" until a little squeal told me that she got it:

"Right, now make some rhombuses". "Do you mean diamonds?". "No I mean rhombuses!" A little while later she showed me this:

"Hey, what other name can you give that small one in the middle?" A little pause and slight turning of the iPad and then "Oh, a square!!"
"Are all squares rhombuses?"
"Yes! Yes!"
"Are all rhombuses squares?"
"Yes! Wait...NO!!"
"OK. Now it's time to make some pentagons. But make sure that they have at least one right angle"
This is what she made:

I do know some kids who get confused as they think a pentagon will have five 'points' and therefore think that the elastics can only touch five pegs. Thus they won't see these as pentagons as they touch more than five pegs. This is a great opportunity to fine tune what we mean by 'points' and connect it to the number of sides.
Now I know you might be thinking 'Lucky girl, getting to do maths in the car whilst other kids are playing Angry Birds' but it was a neat way to spend 15 minutes. It got me wondering whether or not I prefer this virtual geoboard to the real thing and I think I might be leaning to the virtual side. For a start, the 'elastics' never snap and you never run out of them. Secondly, the vertices look more like they should. For example, look carefully at the 'corners' of this shape below and ask yourself if this really is a rectangle?

That being said, real geoboards are cheaper and I'm sure that some kids will prefer the tactile nature of these as opposed to the virtual geoboard. Either way, geoboards are a great way to get kids really to explore some geometric properties by asking questions such as:

Make a quadrilateral with 3 acute angles.

Make a parallelogram with two right angles.

Are all parallelograms rectangles?

 

For an extra challenge, I sometimes ask to make a shape which I know to be impossible (but the kids don't). This creates huge cognitive dissonance and often gets them reasoning why such a shape is impossible. For example:

Make a triangle with two right angles.

Make a quadrilateral with four acute angles.