Over the last two days, I've had a few important breakthroughs with discouraged Algebra 1 learners about quadratic functions and their graphs. I wanted to document this for myself before I have a chance to forget about it for next year.
If you have rigid beliefs about the only ways for students to approach quadratic functions, analysis, and graphs, then this post is definitely not for you.
Consider yourselves warned.
My most discouraged Algebra 1 learners are extremely gifted kids, but this year's crop are definitely dreamers, not stalkers. They need to really marinate in something for a long time before it takes root in their minds. They are factoring warriors, but the quadratic formula and complicated answers can be really daunting for them.
The connection I have wanted all of my students to make between quadratic functions and their graphs is this: even when a quadratic doesn't have neat, simple, integer answers, it still has a number of neat, simple aspects that they can grab hold of. There will always be an axis of symmetry. There will always be a vertex.
What I have discovered — or rather, what they have been teaching me all this week — is that if you approach a quadratic function from the understanding that you already know how to find the AOS and the vertex, you can make a TON of important discoveries and understandings about its graph and the many properties of the function that are important to understand.
What they taught me today is that they know how to use the axis of symmetry formula like a key in a lock. They can identify a, b, and c in a quadratic, and they understand that they can find and graph the AOS equation quickly and fearlessly. Then they can plug in the value they found for the axis of symmetry to identify the vertex of the parabola.
From there, it is a simple matter to find some more points for your sketch and to identify their mirror reflections across the axis of symmetry.
The beauty of their method is that it makes it easy for them to develop a meaningful conjecture about whether or not a quadratic function has any zeros.
If the parabola is floating above the x-axis, then they can use the QF to confirm their hunch that there are no real-number zeros to the function. Likewise if the parabola is submerged below the x-axis.
I love that they naturally figured out today what it means for the vertex to be a maximum or a minimum.
And I especially love the fact that they made these discoveries themselves.
They still don't understand how to complete the square or use the quadratic formula to blast through problem after problem to find complicated zeros of non-trivial quadratics.
But this feels less important to me than the fact that they have made important connections and developed their own methods for investigating quadratic functions. And it has been an important reminder to me to design learning experiences that empower them to make these connections and discoveries for themselves.