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# Representing Inequalities

Lesson 11 of 15

## Objective: Students will be able to solve and model one variable inequalities and write them in interval notation.

## Big Idea: Model and solve inequalities both algebraically and graphically as well as introduce interval notation.

*52 minutes*

#### Warm up and Homework Review

*10 min*

I include **Warm ups** with a **Rubric** as part of my daily routine. My goal is to allow students to work on **Math Practice 3** each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. My video narrative specifically explains this lesson’s warm up which asks students to solve a linear inequality.

I also use this time to correct and record the previous day's **Homework**.

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The goals of this lesson are to introduce interval notation, review solving inequalities as well as some modeling with inequalities. This lesson supports the study of inequalities throughout this course.

Modeling with Inequalities

I introduce inequalities by looking briefly at the cake pop story. The students begin by finding the constraints on the number of cake pops that can be produced. I have them think for a minute and then we compile a quick list of ideas. The rest of this problem will focus on the constraint of time.

"*Each batch of 50 cake pops takes 3 hours to make. If there are only 15 hours available each week, write an inequality showing the number of cake pops that can be produced*. " (**Math Practice 4**) I have the students write an inequality for this problem and then share it with their partner. Some scaffolding that may be needed is a review of how to read inequality signs. Often students can say them but not write them. I have had luck explaining to them that we read inequalities from left to right, just like we read words. If we hit the small portion first, it is a less than. If we hit the big portion first, it is a greater than.

Interval notation is now introduced, first using this cake pop example and then generally. Please note that I have purposefully not included real life inequalities using less than. We will look at these in the next lesson when we go over compound inequalities. While interval notation isn't in the Common Core document, it does fall under **Math Practice 2**.

Solving Inequalities

The next goal is for students to gain some conceptual understanding of why the sign is switched when you divide by a negative number. I give the students the problem -2x < 20 and then ask them two questions:

What is this asking?

What numbers could x be to make this true?

This problem is asking us what multiplied to -2 will give us a number that is less than 20. All positive numbers will make this happen since they are multiplied to a negative and will make a negative. Anything between -10 and 0 will also work since they multiply to numbers between -20 and 0. If you graph all of these they will make the inequality x > -10. This means that -2x< 20 is equivalent to x>-10.

I do a guided investigation with the students or allow them to answer these questions as partners and then discuss. This depends on your class.

They then have several practice problems as well as a modeling problem. For the final practice problem, they graph it on the calculator to figure out the answer after they have solved it algebraically which I model with Smartview or an overhead calculator.

I have also included a radical inequality. As of right now, most students don't have the skills to solve this one algebraically but can graphically. This leads to a great discussion on the tools we use to solve equations or inequalities (**Math Practice 5**). One main goal that I have for my students in this course is that they have the ability to find the solutions to problems in different ways and can discuss the strengths and weaknesses of these different methods.

#### Resources

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#### Exit Ticket

*3 min*

I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.

The lesson exit ticket checks that students can solve and graph the solution for a multi-step linear inequality with one variable.

#### Resources

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#### Homework

*2 min*

Homework- Representing Inequalities has several practice problems as well as a variety of modeling and extension problems including the graph of a cubic inequality. These problems move students from basic review towards a rigor expected in the Common Core Algebra 2.

This assignment was created with Kuta Software. I love this software and highly recommend it.

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- UNIT 1: Modeling with Expressions and Equations
- UNIT 2: Modeling with Functions
- UNIT 3: Polynomials
- UNIT 4: Complex Numbers and Quadratic Equations
- UNIT 5: Radical Functions and Equations
- UNIT 6: Polynomial Functions
- UNIT 7: Rational Functions
- UNIT 8: Exponential and Logarithmic Functions
- UNIT 9: Trigonometric Functions
- UNIT 10: Modeling Data with Statistics and Probability
- UNIT 11: Semester 1 Review
- UNIT 12: Semester 2 Review

- LESSON 1: Turn Your Words Into Money
- LESSON 2: Crossing the Snake River
- LESSON 3: Interpreting Algebraic Expressions Day 1
- LESSON 4: Interpreting Algebraic Expressions Day 2
- LESSON 5: Mind Reading with Algebra
- LESSON 6: Selling Cake Pops Day 1 of 5
- LESSON 7: Selling Cake Pops Day 2 of 5
- LESSON 8: Selling Cake Pops Day 3 of 5
- LESSON 9: Selling Cake Pops Day 4 of 5
- LESSON 10: Selling Cake Pops Day 5 of 5
- LESSON 11: Representing Inequalities
- LESSON 12: Modeling Compound Inequalities
- LESSON 13: Modeling Expressions and Equations Review Day 1
- LESSON 14: Modeling Expressions and Equations Review Day 2
- LESSON 15: Modeling Expressions and Equations Test