## Reflection: Developing a Conceptual Understanding The Axis of Symmetry and Vertex Form - Section 3: Guided Notes: What Do You Notice?

We've made a lot of tables this year, and I'm happy to report that students recognize tables as a useful tool for making sense of problems.  Although I hope that you recognize my references to Mathematical Practices #1 and #5 in the previous sentence, what I'd really like to reflect on here are the last two:

- MP7: Look for and make use of structure.

- MP8: Look for and express regularity in repeated reasoning.

We've made tables in all sorts of different contexts.  Of course there have been the customary x/y tables that are one of the three canonical ways to represent functions.  There have also been frequency tables, guess and check tables, and the kind that students see today: tables that just help us organize what we've seen in a way that reveals new insights.  That's a pretty big skill, both in terms of learning something new in an algebra class, but also in any field that accesses and attempts to use the ever-growing trove of data produced by our world.

When students see the list of functions with their roots, axis of symmetry, and vertex, my hope is that I'll barely need to prompt them for structure to emerge.  That repeated values in different parts of the table will get kids asking their own questions.

In conversations with colleagues, we often struggle to distinguish between Mathematical Practices #7 and #8, but I think this is ok, because they're related!  When I notice repetition in a table, I should try to figure out what that structure reveals to me.  When I perform a repetitive task, hopefully I'm able to recycle the structure of the task until I come up with some new shortcut.

Which brings me to these "student friendly" versions of MP7 and MP8 that I found bouncing around the internet a while ago, and have co-opted for use in my classroom.  I'm not even sure where to attribute these, because they seem to pop up all over, and they're worth considering here:

- MP7: I can use what I know to solve new problems.

- MP8: I can solve problems by looking for rules and patterns.

The questions are: (a) do these capture the essence of what the MP's, as originally written, are going for? and (b) if we want students to be able to do both, which I believe we do, how much should we worry about the words used to express the idea?

Ideally, today's activity achieves both of these goals: students will use what they know about making tables, about finding roots, and about noticing patterns to come up with new generalizations that they'll be able to apply to more efficient problem solving later.  If students leave my 9th grade classroom able to do that in any context, I'll be pretty stoked.

When In Doubt, Make a Table!
Developing a Conceptual Understanding: When In Doubt, Make a Table!

# The Axis of Symmetry and Vertex Form

Lesson 12 of 21

## Big Idea: As teachers, it's so important that we pay attention to what our kids can actually do, and make small adjustments accordingly. Today is a day to practice that!

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Subject(s):
Math, completing the square, quadratic functions, Quadratic Equations, graphing functions, axis of symmetry, vertex form, Growth Mindset, Algebra 1
38 minutes

### James Dunseith

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