Saturday, 23 November 2013

How Many Each?

After a great day at EdCampOttawa, I came home to a roast chicken dinner (yes, my husband is a great cook) along with roasted veggies.  While helping himself to seconds, my 8-year old, Noah, complained that he hadn't gotten enough potatoes.  My husband said that he had cut 4 potatoes into 4.  The kids quickly figured out that there had been 16 potato slices.  I then asked how many we would each get if they were equally distributed among us (us being 6 people).  They tried 2 and counted up to 12, then tried 3 and counted up to 18 (in 3s) so they knew it was more than 2 but fewer than 3 pieces each.  Noah then said that it wasn't 2 and a half.  I asked him how he knew.  He got mad at me, screamed that he wasn't doing this anymore and stomped away from the table.  So I turned to Jacob, who is 6, and asked him the same question as I had seen him making gestures with his hands that looked like he was figuring it out.  He told us that if they each got 2 and a half pieces, that would be 15 pieces.  At some point my 10-year old said she had figured it out, but actually managed to not yell out the answer (good job, Isabelle!).  So back to Jacob.  We went back to knowing it was more than 2 pieces each which he told me would be 12 pieces. I asked how many pieces were left to be divided among us.  He said 4.  He wanted to give each kid 1 and the adults none, but I said we each had to get the same amount.  Then he thought Noah shouldn't get any since he had left the table.  With a little encouragement he said we could "chop, chop" the pieces into 3.  So how many pieces would that make?  12, he counted. And then he figured that once around the table was 6 so he could around the table another time and we would each get 2 of the cut-up pieces.  So 2 and two-thirds each.  Just as we were saying this Noah yelled from the next room - 2 and a half and one sixth each!  Yes, Noah.  Yes :)  Then he sat back down and finished eating.

Sunday, 17 November 2013

Completing the Square with Algebra Tiles

There seems to be some interest in how to use algebra tiles in the MTBoS so I thought I would attempt a blog post.  I apologize now if I confuse anyone - I don't claim to be an expert, but I do find they help kids make connections.

For years I avoided algebra tiles. It didn't help that one of my previous colleagues told the story of how one of her students managed to choke on a tile! Thankfully, none of my students have done that.  Our tiles are red and blue so they all feel the need to look through them as if they are 3D glasses (sigh). The kids get to choose which should be positive and which should be negative.  They invariably say red is hot so positive and blue is cold so negative so that is what I am using here.  
We introduce the tiles in grade 9 when we simplify polynomials.  They use them to multiply binomials (it makes a rectangle) and to factor trinomials (find the length and width of the rectangle).  So my students are very familiar with algebra tiles by the time they get to completing the square.

I start with a warm-up to make sure they remember what is special about perfect square trinomials. They work in groups of 4 - each student does one question then they add up their answers.  If the sum is correct, we can move on, if not, they have to find the error.  I stole this from someone at TMC13 (who stole it from someone)^n, who stole it from Kate Nowak.

Then we tackle our first example:

Students understand that instead of making a rectangle, they need to make a square with their tiles.  They each have their own set to work with.  We will place the 7 unit tiles off to the side and work with the rest.

They see that they need to add 1 unit tile to make a square.  In order to do that we have to add a zero pair so a -1 unit tile goes with the 7 off to the side.
We can then write the area of the square a its side length squared and simplify the unit tiles.  And just like that we have vertex form!

We work through a couple more examples in the same way, with students working with their tiles then consolidating with the whole class.  Each time they have to divide the x-tiles - half for the length and half for the width so that they make a square.  Each time they have to add unit tiles.  I get them to notice patterns in what they are doing.   

Next, we use a chart to connect the algebra tiles to the algebra.  The cool thing is that they actually understand why we are dividing 'b' by 2 and squaring it because they have done it with the tiles.

The next day we extend to quadratics where 'a' is not equal to 1.  Again we start with tiles and connect to algebra.  You need identical tile diagrams for each x^2 you have, but it's the same process.  It connects factoring out the 'a' value to the tiles.

I find algebra tiles really help explain why we are doing what we are doing. It helps that my students are asking how to change a quadratic from standard form to vertex form. Well, some of them are anyway!

My SMART Notebook file is here. I left off the extra practice for day 2 as I am not really happy with it, but am not sure how to change it.

If you are looking for on-line algebra tiles, The National Library of Virtual Manipulatives is a great resource.