Instead of calling it factoring and expanding, I like to refer to the two different equations the students are responsible for as this.
y = ax^2 + bx + c as y intercept form or area form.
y = a (x-p) (x-q) as x intercept form or length width form
The curriculum requires that students can work with these equations when a=1
To get at changing from area form to length width form or vice versa, I use rectangles and algebra tiles.
The first thing I like to do is to give the students an equation in area form. They are already familiar with it as we have modelled quadratics with the graphing calculator.
So here is an example. y = x^2 + 4x +3. I will use this example for the rest of the blog post.
Then I ask them to get three different colored piles of cube a links. One for the x^2 term, one for the x term, and one for the number term.
The curriculum requires that students can work with these equations when a=1
To get at changing from area form to length width form or vice versa, I use rectangles and algebra tiles.
The first thing I like to do is to give the students an equation in area form. They are already familiar with it as we have modelled quadratics with the graphing calculator.
So here is an example. y = x^2 + 4x +3. I will use this example for the rest of the blog post.
Then I ask them to get three different colored piles of cube a links. One for the x^2 term, one for the x term, and one for the number term.
Once the students have the three piles I ask them to create a pile of the correct number of tiles for stage 1 (when x=1), stage 2 (when x=2), stage 3 (when x=3) and stage 4 (when x=4). Here are what the piles look like for our example.
Then we create a stack for each stage. Numbers on the bottom, then the x cubes in the middle followed by the x^2 cubes on the top. Nice discussion at this point about the graph of y = 3, y = 4x + 3 and y = x^2 + 4x + 3. Here is what it looks like.
Students then deconstruct their towers and I ask them to create a rectangle for each stage. x^2 blocks need to be in a square and the ones blocks touch the x^2 blocks at a corner. We also create the rectangles with algebra tiles. I particularly like doing this as students can see x increase (be variable). Eventually students will get to this.
And a little poster action demonstrating some algebra and some key characteristics of quadratics.
This is just brilliant!
ReplyDeleteI should have commented before, but this is how students experience multiple representations.
Thanks for all you do Alex!
Alfonso Garcia