This week, my algebra class has (finally) gotten around to completing the square. I like to do this using algebra tiles, but I was thinking about desmos and how to connect the concepts of finding the vertex using the formula -b/2a (which they derived as a class using desmos) and completing the square to convert a quadratic equation from standard to vertex forms.
It's funny how sometimes these things seem to be staring back at us when we finally go looking for them. So this is what we did:
1. I had them multiply out some perfect square trinomials and think about the pattern (again...)
2. We graphed the resulting quadratics on desmos and they conjectured a pattern. i.e. they all bounce off of the x axis. We talked about the question "why is that?"
3. I gave them some problems that were not perfect squares and asked them to figure out how to change the equations to make them into perfect squares by graphing the results. For example, on the one in the picture, they figured out you needed to add 14 because the y-coordinate of the vertex is -14.
4. We then talked about how these aren't the same parabolas anymore, so how what number would we need to add or subtract to/from the perfect square so that the two parabolas matched again. This gave us the structure of y = perfect square trinomial +- number.
5. Then we factored the perfect square and, voila!, vertex form.
It sounds so simple when you say voila, doesn't it?
They struggled with it though. I'm inclined to think that this is okay because they always struggle with completing the square. I want to be sure the process makes sense to them before we move on. About half the class had "aha" moments at some point today, so maybe? I'm thinking that tomorrow, we will go through some of the same problems using the algebra tiles to show that "look, you're making a square!"
It will be interesting to see where this goes. I am hoping they make a much better connection than they usually do when we show the quadratic formula in a couple of days.
A quick exit ticket last 5 minutes of class will give you a more accurate feel of who got it. Then you can launch the next day with it and even say what percent were successful, and possibly turn it into a "My Favorite No" opportunity where you can all analyze the most common misconception (usually not adding the c term to both expressions or not writing the perfect square trinomial as (x+b/2)^2 rather than (x-b/2)^2
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