##
* *Reflection: Developing a Conceptual Understanding
Features of a Parabola - Section 2: Complete Your Gallery Walk, With One New Instruction

As we continue to shift our emphasis from the algebraic manipulation of quadratic expressions to a study of the graphs of quadratic functions, I still haven't named "f(x) = (x - h)^2 + k" as "vertex form". In fact, I haven't even hurried to replace the "+" sign that was on "Quadratic Functions in Three Forms" to the "-" sign that is traditionally used in the vertex form of a quadratic function.

But we're getting there.

When students started their work on Quadratic Functions in Three forms at the beginning of the week, I made sure to name the first two columns of that assignment as "standard form" and "factored form". For the last column, however, I told students that we'd figure out what to call this eventually, and that I was open to suggestions. In the next lesson, students recognized that this form was easiest to manipulate on an interactive graph. In last few days, as we've gotten rolling on the gallery walk, students have learned to recognize and label the vertex of a parabola.

As they finish their gallery walks, I'll often assign the additional extension of writing each function in vertex form, even though we still don't have those words yet. Once students rewrite their functions that way, and they look at the coordinates of each vertex on the graph, they are thrilled to discover this connection for themselves, and instead of being told, "hey, this is called vertex form," they find those words on their own - and that feels as natural as can be!

*What should we call f(x) = (x + h)^2 + k?*

*Developing a Conceptual Understanding: What should we call f(x) = (x + h)^2 + k?*

# Features of a Parabola

Lesson 9 of 21

## Objective: SWBAT identify the roots and the vertex on the graph of a quadratic function.

## Big Idea: Students finish up the Quadratic Functions Gallery Walk and begin to spot connections between where the roots and where the vertex are located.

*35 minutes*

Today's opener is about the three "green functions" from the Quadratic Functions Gallery Walk that kicked off yesterday. By referencing work that kids have already done, I hope to give everyone a chance to use what they've got, to share this knowledge with the class, and to help our community build an understanding of quadratic functions.

The instruction asks, "What is the 'lowest point' for each function?" I expect that students will be able to look at their work from yesterday and to identify these points. On the side board, I make a table in which we can record the lowest point and roots for each function, and I ask for students to share what they've got. When someone says that the lowest point for Function A is at (-2, -4), I ask if anyone else agrees or disagrees, and we reach consensus for the next two functions in the same manner.

With the next slide, I ask if everyone can identify the roots of each function, and we follow the same steps. At minimum, I want students to be able to identify roots by looking at a graph, and I watch closely to see who can and cannot. Depending on how confidently my students can approach this task, I might ask them to factor each expression to see that the roots are what we expect them to be. If that doesn't happen now, we'll get to it soon; tomorrow's lesson is about using what we know about the features of a parabola to more efficiently produce a graph.

There is a lot to notice about the relationships between a quadratic function, its vertex, and its roots. I don't rush into naming any of these connections, but I do leave room for students to say what they notice, and that helps us build upon what we know.

*expand content*

The opener leads neatly into continued work time on the gallery walk. Please see yesterday's lesson for details on how I get students started on this activity. I tell students to pick up where they left off and I add one new instruction: everyone should label the roots and the lowest point, "which is called the vertex," I say, on each of their four graphs. To get everyone started, I provide this example for Function A. For some students, this is a nice confidence boost because they see they've already got it right. For struggling students, it always helps to see another example. Either way, this is a chance for kids to see exactly what I mean by labeling the vertex and roots, and they're able to do the same.

On today's lesson slides are the green functions (these were the focus of today's opener) and the blue functions, just in case we have any reason to discuss them as a class. With most of my classes, students are ready to get to work, and we won't touch these slides, but they're there just in case everyone gets stuck.

The value in this activity is that students can really get a feel for the shape and behavior of a parabola. As I wrote yesterday, the amount of arithmetic required may make for some slow going. It's my job here to make sure that students are building enough of an appreciation for how these graphs work that they're able to embrace the shortcuts that we'll develop tomorrow.

I give students until there are about five minutes left in class to finish their work. When students finish early, they can choose between two extensions, which are outlined in the next section of this lesson.

*expand content*

If students finish the Gallery Walk with time to spare, I collect their work and tell them they have two choices of what to do next.

**Option #1: Add Your Work to the Gallery**

There are stickies notes posted on each function in the gallery walk, and I invite students to take a sticky and make a table of values or a graph to post as an exemplar for each each function. I always keep a supply of 11x17 inch ledger paper and colored pencils in my room, and there's plenty of graph paper available. Students can use these supplies to complete the gallery, which will serve as a great reference for the rest of the unit.

**Option #2: Complete Quadratic Functions in Three Forms**

Many students still have to finish up the Quadratic Functions in Three Forms activity that preceded the Gallery Walk. When a student turns in today's work, I ask them where they stand on that activity. I ask if this assignment makes more sense now that the gallery walk is done, and suggest that they sketch a graph of each functions, again labeling the key features of each graph.

#### Resources

*expand content*

With five minutes left in class, I call everyone to attention and post the exit task, which is on the last slide of the lesson notes. Students are given the equation for a quadratic function, and asked to "make a quick sketch" and label the roots, the vertex and the y-intercept.

I expect to see a wide spread among what students know and can do with this task. No matter what kids have done on their gallery walk, here's a chance for them to show what they understand. Everyone has done a different amount of work over the last two days, but here's the bar that I expect everyone to reach.

The short amount of time students have to complete this exit task means that it's all about how efficiently they can work. If a student has developed some shortcuts of their own, they will use them here. If they must make a table of values, then they'll have to work quickly. Even when students can't produce a perfect graph, they're often able to accurately label the roots and vertex. Every partial solution I get gives me insight into the next steps I'll take with each student.

We'll end class like this again in about a week, which is exciting because it will make it clear to students how much they've learned by that point.

#### Resources

*expand content*

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- UNIT 1: Number Tricks, Patterns, and Abstractions
- UNIT 2: The Number Line Project
- UNIT 3: Solving Linear Equations
- UNIT 4: Creating Linear Equations
- UNIT 5: Statistics
- UNIT 6: Mini Unit: Patterns, Programs, and Math Without Words
- UNIT 7: Lines
- UNIT 8: Linear and Exponential Functions
- UNIT 9: Systems of Equations
- UNIT 10: Quadratic Functions
- UNIT 11: Functions and Modeling

- LESSON 1: Rectangular Gardens
- LESSON 2: Wait, Can You Solve That?
- LESSON 3: Area Models for Multiplying Polynomials and Factoring Quadratic Expressions
- LESSON 4: Can You Factor It?
- LESSON 5: Finding Roots of All Sorts
- LESSON 6: Quadratic Functions in Three Forms
- LESSON 7: Different Forms and Different Parameters
- LESSON 8: Graphing Quadratic Functions (Gallery Walk)
- LESSON 9: Features of a Parabola
- LESSON 10: A Faster Way to Graph a Parabola
- LESSON 11: How Does a Parabola Grow?
- LESSON 12: The Axis of Symmetry and Vertex Form
- LESSON 13: What if We Start With the Axis of Symmetry?
- LESSON 14: Finishing Five Point Graphs
- LESSON 15: What Does "a" Do?
- LESSON 16: Solving Quadratic Equations (Delta Math)
- LESSON 17: Moving Toward Mastery: Completing the Square (Day 1)
- LESSON 18: Moving Toward Mastery: Completing the Square (Day 2)
- LESSON 19: Review or Move On (to the Quadratic Formula)
- LESSON 20: A Review Day, In Its Own Way
- LESSON 21: Unit 6, Two Day Exam