##
* *Reflection: Rigor
Solving Non-linear Systems of Equations - Section 1: Entry Ticket: Comparing Stocks (Linear and Exponential Growth)

In this Entry Ticket I ask students to not only create a **graph **involving a system comprised of a linear and an exponential function, but also compose a **written response **to interpret the scenario.

I purposively put in two different types of functions for this Entry ticket because I want students to persevere in problem solving. At this point in the school year, I have covered the general characteristics of exponential functions, but only briefly in the opening unit on functions. I want students to utilize the tools they have to be able to work on a complex task like today's Entry Ticket.

This Entry Ticket went well in class, partially because I was cognizant to give students more time than usual to process and work on the problem in groups. The more I teach, the more I see the benefit to providing students with real life and relevant problems that help frame the lesson as an entry ticket/opener. When I implement high quality and engaging entry ticket, the class tends to go smoother and student motivation and engagement tend to be higher.

*Rigor: Comparing and Contrasting to Develop Critical Thinking Skills*

# Solving Non-linear Systems of Equations

Lesson 7 of 12

## Objective: SWBAT solve systems of equations involving a linear function and a quadratic function both algebraically and graphically. SWBAT understand the process of factoring polynomials as a tool to solve non-linear systems

## Big Idea: Students create and interpret non-linear systems of equations and use technology to find and understand the solution(s)!

*80 minutes*

In this **Entry Ticket** I give students practice solving a system of equations with graphing. I also present them with a new situation: **What happens if one of our functions in a system is not linear? **I strongly favor this HSA-REI Common Core standard because it sets a high bar of expectations for learners to make connections, comparisons, and explanations.

Today's Entry Ticket takes a bit longer than the typical 10-15 minutes. 20 minutes is usually enough time for students to engage fully with the activity. The task encourages students to work on a relevant problem with enough scaffolding to practice higher-level thinking skills in a collaborative environment. The added bonus to this word problem is it gives a preview of a coming attraction, Comparing Linear and Exponential Functions, a topic we will cover later in the year!

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To continue the discussion around non-linear systems I show one or two Khan Academy video examples dealing with the topic. During this time students are actively taking **Two-Column notes **and I am pausing the video and making annotations to the video to check in on student understanding. Each video presents examples of a linear function intersecting with a quadratic function.

**Video 1**

For most of my classes, I like to show a second example video. The reason for showing two videos addressing the same topic is that many of my students need to see a particular type of problem more than once in order to understand the big idea. For my advanced and honors sections, I typically only show the first video.

**Video 2**

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After the video note-taking and discussion, I allow students to use an iPad from the classroom set that I have available to practice solving examples of non-linear systems. Today, I plan for students to work on the section of the Khan Academy website focusing on **Solving Non Linear Systems**. I encourage my students to complete the practice problems on the site while working in small groups.

My students find these problems challenging at this point in the course. As a result, I am actively supporting students throughout this activity. I also encourage students to seek help from their group members. Some students like to refer back to their notes or to the example videos.

In order to enable the sharing of reasoning, I ask each group take a screen shot of at least one solution and annotate it for a presentation.

**Instructor's Note**: There is an option on Khan Academy for a sketchpad where students can take notes on the screen.

At an appropriately timed moment, I plan to project a group's work and ask the group to give a short presentation to the class about the solution that they chose to capture. I generally ask several groups to present in sequence, but it is possible to break them up a little more.

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For the **Exit Ticket** I have students complete a problem-based activity comparing the flight trajectory of two planes (a picture of the graphical solution is a resource in this section:**Graphical Solution for Exit Ticket: Solving Non-Linear Systems**).

The intent of this problem is to have students think about different applications of the concepts and skills they have used to solve systems with non-linear functions.

Students can also complete the Exit Ticket for homework and/or elaborate more on their thinking by completing a written response to their work to the problem as a homework assignment.

*expand content*

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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: Creating and Interpreting Equations to Solve Systems
- LESSON 2: Bring in the Sub!: Solving Systems of Equations in 2 Variables
- LESSON 3: The Whole is Greater than the Sum of Its Parts
- LESSON 4: Practice Session on Solving Systems
- LESSON 5: Graphing Systems of Equations
- LESSON 6: Solving Linear Equations in Two Variables
- LESSON 7: Solving Non-linear Systems of Equations
- LESSON 8: Graphing and Solving Systems of Inequalities
- LESSON 9: Job Offers and Cell Phone Plans: Making Informed Decisions with Mathematics
- LESSON 10: Study Session for Unit Test on Systems of Equations and Inequalities in 2 Variables
- LESSON 11: Unit Test: Systems of Equations and Inequalities in 2 Variables
- LESSON 12: Writing in Math Classroom, Part 2: Solving Equations & Inequalities in 2 Variables