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# Rate of Change & Comparing Representations of Quadratic Functions

Lesson 3 of 13

## Objective: SWBAT calculate the average rate of change for a function over a specified interval SWBAT compare properties of two functions represented in different ways. SWBAT provide examples and evidence to support their ideas/arguments.

## Big Idea: Students interpret the rate of change in context for quadratics AND compare and contrast features of functions presented in different ways!

*90 minutes*

For today's Entry Ticket (also first slide of PowerPoint), I ask students to review the concept of rate of change by interpreting the slope of three linear functions. It is a quick warm up. During this opener I expect students to be conversing with each other. I will make my way around the class checking on student understanding. Some of my students continue to struggle with this idea. Others, who have the concept down, can assist their peers, in the process improving their skill at academic conversation. This activity provides an opportunity to assess the progress of the class with respect to **HSF-IF.B.6**.

After students complete the Entry Ticket, I will turn to the Agenda and review the learning and language objectives, plan, and homework for today. I typically ask students if they have anything else they want to include on the agenda. I find that this simple invitation, whether accepted or not, provides an opportunity to give students increased agency and ownership for their own learning. If, in the case a student brings up an inappropriate idea for the agenda I ask the class for their input on whether or not that should be included – we use the class objectives and essential question as a decision making guide as to whether or not the proposed agenda item is relevant or not.

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After we discuss the agenda, I plan to introduce new concepts using **Rate of Change & Comparing Functions** PowerPoint slides. As always, I ask students to actively take notes on the material, preferably in two-column form. As I present, I check-in with students to observe whether or not notes are being taken.

To begin, I review a problem with a constant rate of change over specific intervals. The class has reviewed this concept during the warm up. I present this problem to activate their background knowledge further as we prepare to apply the idea of rate of change to quadratic functions. namely.

The lesson then turns to the topic of comparing functions.

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In this section of the lesson, I have students work as a class on the **Collaborative Work: Interpreting Rates of Change** to compare two quadratic functions, one that is represented as an equation and the other as a graph. Both functions model a relationship between hours studied and score on a test. This section was designed intentionally to meet the standard **HSF-IF.B.9, **and provides more practice for students to show understanding of standard**HSF-IF.B.**7a as graphing the quadratics helps in the solution to this problem. The problem provides an opportunity for students to engage with the math practice **MP.1** because students are asked to make sense and persevere in problem solve, **MP.2** as quantitative and abstract reasoning are necessary to engage deeply in the problem, **MP.3** as again students are creating their own arguments and critiquing those of others, and **MP.4** as students have to model the situation to create their own unique response to the prompt.

During this section, the class is working together to solve the **Collaborative Work: Interpreting Rates of Change**. I am in front of the class helping facilitate the discussion. For example I may ask a question like, “well, what would the function from the history class look like as a graph? How could we go about translating the function to a graph?” to help the group stay on track and give hints to the characteristics and skills that could be helpful in solving the problem.

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#### Exit Ticket and Homework

*15 min*

To close this lesson, I add a third function in the form of a graph to the hours studied and test score problem. I have students work in pairs to work on the **Homework: Comparing Quadratic Functions**. At this point I am decreasing the level of support as the class as a whole has already modeled what types of questions to ask when comparing functions in different representations. I added the third function as a table to help students understand a table is in fact another representation of a function.

The homework provides the information from the three tables and asks students to synthesize their classwork and compare the three functions through the completion of an Idea Organizer. As an alternative, I always allow students to write a 1-2 paragraph response in lieu of the Idea Organizer because that adds another step of taking their organized ideas and putting it into standard text convention.

#### Resources

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- UNIT 1: Thinking Like a Mathematician: Modeling with Functions
- UNIT 2: Its Not Always a Straight Answer: Linear Equations and Inequalities in 1 Variable
- UNIT 3: Everything is Relative: Linear Functions
- UNIT 4: Making Informed Decisions with Systems of Equations
- UNIT 5: Exponential Functions
- UNIT 6: Operations on Polynomials
- UNIT 7: Interpret and Build Quadratic Functions and Equations
- UNIT 8: Our City Statistics: Who We Are and Where We are Going

- LESSON 1: Introduction to Quadratic Functions
- LESSON 2: Interpreting and Graphing Quadratic Functions
- LESSON 3: Rate of Change & Comparing Representations of Quadratic Functions
- LESSON 4: Rearranging and Graphing Quadratics
- LESSON 5: Graphing Functions: Lines, Quadratics, Square and Cube Roots (and Absolute Values)
- LESSON 6: Building Quadratic Functions: f(x), kf(x) and f(kx)
- LESSON 7: Factoring and Completing the Square to Find Zeros
- LESSON 8: Forming Quadratics: Math Assessment Project Classroom Challenge
- LESSON 9: The Three Musketeers: Simplifying the Quadratic Formula
- LESSON 10: Quadratic Quandaries: Modeling with Quadratic Functions
- LESSON 11: Performance Task: Pulling It Together with Quadratics
- LESSON 12: Study Session for Unit Test on Quadratics
- LESSON 13: Unit Assessment: Quadratic Functions and Equations