##
* *Reflection: Routines and Procedures
Graphing and Solving Systems of Inequalities - Section 2: Entry Ticket

The Entry Ticket is a routine that I implement in my classroom everyday. I utilize the routine as a way to support students to have a smooth transition at the beginning of class. AS soon as students enter the class they are expected to take out their materials and get to work on the Entry Ticket (I have a white board designated for entry tickets so students always know what to work on).

In this particular entry ticket, I wanted to assess student understanding of solving systems of equations from a previous lesson. I appreciated how many students **labeled the solution **to the system and also paid attention to detail, including **defining the variables** in the system.

From this entry ticket I was able to assess understanding of previously covered topics, and also (through the graph) get a rough baseline on the level of student understanding of solving systems by graphing that I can compare and contrast to a similar Exit Ticket on solving systems of inequalities.

*Routines and Procedures: Entry Ticket as Routine and a Check for Understanding*

# Graphing and Solving Systems of Inequalities

Lesson 8 of 12

## Objective: SWBAT define variables and create and rearrange equations to model systems of inequalities. SWBAT interpret and solve systems of linear inequalities using algebra and graphing.

I like making systems of inequalities using the TI NSpire or CAS software. It is relatively simple and generates really nice looking inequalities that can be used for powerpoint slides, homework or any practice problems.

This tutorial is also a useful tool to teach students how to graph inequalities as a technology skill:

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#### Entry Ticket

*15 min*

For this **Entry Ticket: Graphing Systems of Inequalities **I have students practice graphing a system of linear functions as an entry point to graphing systems of inequalities. For this lesson I like to have students use colored pencils or markers to really "see" the differences in functions and to be able to shade areas modeled by the inequalities.

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After completing the entry ticket, the class shifts to an Active Note taking session where students are taking **2-Column Notes**.

I begin this section by revising the entry ticket problem so that it is better modeled by a system of inequalities rather than equations. As we review the classnotes, the students learn about how to represent systems of inequalities graphically and how to interpret the solutions based on the graph and the inequalities.

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#### Guided Practice

*20 min*

I start this section by having students working in small groups creating graphs of linear inequalities. I like to have students use colored pencils to add some color and clarity to the process. I like to use the **Kuta Software worksheet on Solving Systems of Inequalities** and assign each group 1-2 of the problems. Alternatively, the teacher can cut up the worksheet to help groups focus on only the problems assigned to them.

After having a chance to persevere and work through the process of graphing by hand, I then have each group graph their assigned inequality using technology. I was lucky enough to obtain a grant from the **Salem Education Foundation **to purchase copies of teacher and student software for TI NSpire and secure grant money for class ipad minis from the **Hardscrabble Education Fund**

Before diving in, I show students this quick tutorial video from TI on how to create graphs of linear inequalities:

There are a number of other free graphing calculator resources for teachers to implement technology in the classroom. Khan Academy, for example, has an interactive website for graphing inequalities that allows students to make choices about the graphs (solid or dotted line, etc.) when creating the graphs. The link is here: **Khan Academy Graphing Linear Inequalities**

In order to capture student work using technology I find that having students utilize the screen capture (ipad, iphone, laptop, etc.)on their device is a relatively quick and easy way to capture and consolidate student work that is online.

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Students continue to work in small groups on the **Collaborative Work **for the next section, but now we shift to a higher order thinking skill of creating for the lesson. At this time I want to see the majority of students getting comfortable graphing and shading systems of inequalities. If this is not the case I would reteach and review an additional example problem or two to provide further support to either the whole class or differentiate the instruction based on the needs of individual students.

With that being said, moving forward each group has the task of creating a scenario that can be modeled by a system. I only require at least one of the functions in the system to be an inequality because I want to balance the need to practice the skill with allowing creativity and giving students a chance to think about whether a certain rule or parameter of a scenario is better modeled with an equation or an inequality.

The task can be completed with paper and pencil, with technology or a balance between the two. I personally like to provide each group with two ipads - one to focus on creating the graph and capturing the system that they graph, and a second to write out explanations and the group's reasoning. Using a technology like **Google Drive **can be helpful in making the process of sharing work between group members and the teacher more efficient and manageable.

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The **Exit Ticket (Solving and Graphing Systems of Inequalities) **for today serves as a formative assessment as it is a good point in the unit to take a temperature check of student understanding so that I can adjust my practice accordingly.

I assign two systems of inequalities, one set where the inequalities are set up in slope-intercept form and the second that is not. The reasoning for this is I know I stress slope-intercept form a lot in my instruction, and I have to be careful that students are flexible in their thinking to be able to solve the inequalities.

The second problem also gives me valuable information on where students are in terms of their ability to manipulate and rearrange equations for a variable of interest.

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For homework students complete at leat 5 practice problems on **Graphing systems of linear inequalities on Khan Academy**. I like this interactive applet because it provides a lot of nice support for students to practice the concepts.

For example, the site provides a coordinate plane and makes it easy for students to move and manipulate the graphs. The site also has a simple choice of whether to shade above or below the line, giving students a reminder of one of the important decisions to make when graphing a system of inequalities. This is an excellent example of Universal Design for Learning that not only supports students with fine motor difficulties, but a wide range of students who struggle setting up graphs.

In addition the website provides hints to help students initiate the process of graphing and solving systems of linear inequalities. And as always, Khan Academy has an excellent repertoire of instructional videos for students to review when/if they get stuck on a particular problem. An example video from the system of inequalities (one in which there is no intersection/solution) library at Khan Academy is included below:

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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