Students Making Sense of Quadratics

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I realize some folks will bash me for sharing this from an Algebra 2 class, but based on benchmarks, most of my students have major gaps in quadratics. 

I began with reviewing multiplying 2 binomials on our whiteboards.  I shared the box/area model and several smiles celebrated because they “saw it” and were doing it correctly!

Last week, I pulled out a box of Algebra Tiles.  We literally explored building squares.  I wish I had taken pictures because some of their squares were like a grandmother’s beautiful quilt blocks.  I began tying it back to our box/area models -I’d rather think of it as leading (not forcing) their thinking – but they were quickly picking up the patterns. 

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We then began looking at the algebraic equivalents, again, with a sketch along side allowing them to “see” the process.

Our next step was to find the missing value without tiles/picture models…and then I asked them to review their multuplying with 5 expressions alongside.

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“What? You think they’re the same thing?!?” I asked,  “Prove it to me. Well, by-golly-jee. You are on to something!”

The following day in class, I made a HUGE ordeal of different ways to write zero.

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I explained our next few minutes were a process. But we talked about it, step by step, completing the square, adding ‘that zero’ in our expression, the separating the trinomial and 2 constants.  Rewriting our trinomial as a binomial squared.

Ok. Why in the world would anyone want to do this?  I told them we were finding hidden information.

As they arrived at this form (x+4)^2 – 9, I paused, reminding them to think back on our function transformations before Christmas break.  How would this function y= (x+4)^2 – 9 move on our graph from this one y=x^2?  Quiet. “Move left 4 and down 9!” Someone exclaimed.  Really? Are you sure? We graphed the two and yes, it did just that.  So what does this tell me about my parabola?  They didn’t say vertex. Or minimum.  They said it shows us how the graph was transformed. 

I will take that.

I then asked them to move left 4 and down 9 from the origin.  What have you found? The lowest point.  The vertex. The minimum. All their responses, not my statements.

We set our expression equal to zero and solved the equation, using our inverse operations.  They made the connections with the x-value of the vertex being the “center line” of the parabola.  They realized the +- 5 were steps in either direction from the center line.

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I most appreciated the questions they asked on #3, 4 and 7.  Several chose #7 thinking it was shorter, thus less work. Snafoo. No middle term. What happens? 

I suggested they look at it from a transformations point of view.  Someone shared-It doesn’t slide left or right, only down.  Another student said-well, that’s the easiest equation to solve! (Yep.)

Why did #4 bother some? The middle term had an odd coefficient.  But once they shared their thinking, ok. Got that one too!

#3 was what we math folks recognize as perfect square trinomial.  But for the students, it was an a-ha.  Again, using the transformations context, we moved right 5, but not up or down.

L: But I thought all quadratics intersected x-axis twice?  I asked – did this one? No.

What about y=x^2 + 3?  It moves up 3. Ok. How many times did it intersect the x-axis? It doesn’t.   Hmmm.

A student who is rarely engaged then asked, if you can make a parabola that doesn’t intersect the x-axis, can you find one that doesn’t intersect either axis?  Me: Can we? What would it look like? S: Noooo. As its going up, increasing, it would be increasing outward, too!  More discussion, between them. Me not included. I was smiling.

And their questions were what drove our lesson today.   And I was so excited, telling them their questions make me think! And when they’re asking questions, their brains are processing the information – making it their own.

It was a good day.

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2 responses »

  1. Been seeing a lot of discussion back and forth in blogs about quadratics: why we spend so much time on them – we should spend less time – we should spend more time – what value is factoring – etc. etc. i think you’ve made some beautiful points about the value of examining quadratics. They are puzzles, they require making connections between seemingly random events, and while most quadratics will never be perfect trinomials – they do allow us to utilize their patterns for business, computing, astronomy, science… So many things. I just like solving them, seeing the way their graphs change, shrink, grow, move… Like watching a kaliedescope as you turn the wheel. Sounds like your kids had a great day exploring!

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