## Reflection: Student Ownership A Faster Way to Graph a Parabola - Section 2: Mini-Lesson: Using Roots to Find the Vertex

It's an easy instinct among students to confuse something "new" as being "difficult".  In the current lesson sequence, what's new is the formula for the axis of symmetry, the understanding that the vertex is on the axis of symmetry, and the idea that we can evaluate a quadratic expression for the axis of symmetry to find the y-coordinate of the vertex.  Then, all of that can be used to make our job "easier" as we graph parabolas.

But "easy" is such a relative word.  To a mathematician, everything gets easier as we discover patterns, connections, and shortcuts - in other words, when we gain the requisite knowledge to spot elegance and efficiency that will unify previously unconnected concepts.  In other cases, particularly with a roomful of adolescents, "easy" means being able to shift into cruise-control and repeatedly apply a skill that I've already mastered.

Sometimes, I use that insight into the teenage psyche to my advantage, and I've acknowledged that here on BetterLesson - after days of deep conceptual digging, kids are thrilled to finally have the opportunity to drill some skills on a worksheet.  Structuring that ebb and flow is part of the long game of great classroom management.

But other times, that wish to stick to what they know can impede students from develop the depth of knowledge that I'd like them to gain.  That's what happened today.  A class was certain that for graphing quadratic functions, it was better just to make tables of values and plot points than to go through "all the work" of finding the axis of symmetry and vertex.  When I pressed them to consider the steps, they held fast to that belief, so I challenged them to think about a different scenario.

"Imagine yourself at 25, 28, 33 years old," I said.  "Maybe you've got friends you want to see after work, or maybe you've got kids to go home to.  Would you rather have a job where you're working 20 hours a day at something easy, where you're just doing the same thing over and over again, or 6 hours a day where you have to think really hard, but then you get to leave?"  Now of course, there are plenty of rhetorical holes in that argument, but the kids got my point.  A few even started by saying they'd take the 20 hours of easy work, before passing that off as jest and saying they got my point.

With that in mind, we recommitted to the axis of symmetry and vertex.  That's when we all realized: once you know how to find the axis of symmetry, you just plug that value into the function rule once, as opposed to 10 or 11 times -- and isn't that easier after all?

Which Job Would You Rather Have?
Student Ownership: Which Job Would You Rather Have?

# A Faster Way to Graph a Parabola

Lesson 10 of 21

## Big Idea: A neat but informally defined relationship between the roots and the vertex saves time now and lays foundations for a study of completing the square and the quadratic formula.

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Standards:
Subject(s):
Math, quadratic functions, Quadratic Equations, graphing functions, pure math, Growth Mindset, Algebra 1
43 minutes

### James Dunseith

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