 # K12MATH009: Algebra I

Unit 2: Linear Equations and Inequalities in One Variable   Suppose you are backpacking through Europe this summer. As you’re getting ready to pack, you check the temperatures in the areas you will be traveling and realize those countries use Celsius when displaying temperatures. You are used to seeing temperatures displayed in Fahrenheit, because that is what is commonly used in the United States. You could use a simple formula to determine this conversion.

Imagine you are throwing a fundraising dance. After much planning, you have spent \$1,500 on the DJ, decorations, and some snacks. Tickets are on sale for \$15. Since profit is important from a fundraiser, an equation could be utilized to determine how much money would be collected given a specific amount of tickets sold or an inequality could assist in determining the amount of tickets needed to be sold to produce a profit. By the end of this unit, you will be able to analyze this situation.

This unit addresses basic linear equations and inequalities, expressions, and formulas.

Completing this unit should take you approximately 10.75 hours.

☐    Subunit 2.1: 4.25 hours

☐    Subunit 2.1.1: 30 minutes

☐    Subunit 2.1.2: 1.75 hours

☐    Subunit 2.1.3: 1.5 hours

☐    Subunit 2.1.4: 15 minutes

☐    Subunit 2.1.5: 15 minutes

☐    Subunit 2.2: 3.5 hours

☐    Subunit 2.2.1: 2.25 hours

☐    Subunit 2.2.2: 1 hour

☐    Subunit 2.2.3: 10 minutes

☐    Subunit 2.2.4: 5 minutes

☐    Subunit 2.3: 1.75 hours

☐    Subunit 2.4: 1.25 hours

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Use mathematical properties to justify each step of solving an equation. - Solve linear equations and inequalities in one variable. - Write an equation or inequality to model a situation. - Determine the contextual versus the mathematical solution of a problem. - Evaluate a formula. - Solve for a variable of interest. - Interpret expressions for functions in terms of the situation they model.

2.1 Linear Equations

In order to understand if a problem makes sense, it is important to be able to justify the steps in solving a problem as well as understand what the problem is asking and whether the solution you arrived at is reasonable. If you start off with 3 oranges and you bought 4 more oranges, would it make sense to end up with 1 orange? A lot of times, a solution does not make sense, and it is difficult to determine your mistake, but if steps are justified, the mistake is easier to find. In this unit, you will solve equations using justifiable steps.

Subunit 2.1.1 helps you explain each step in an equation that will allow you to move to the next step of an equation.

2.1.1 Mathematical Properties

Subunit 2.1.1 helps you explain each step in an equation that will allow you to move to the next step of an equation.

• Activity: Illustrative Mathematics’s “How Does the Solution Change?” Link: Illustrative Mathematics’s “How Does the Solution Change?” (HTML)

Instructions: Please click on the link above and complete the task. This task allows you to use algebraic reasoning to explore problems instead of solving them. The solutions to this task are found toward the bottom of the page.

Completing this task should take approximately 10 minutes.

• Activity: Illustrative Mathematics’s “Same Solutions?” Link: Illustrative Mathematics’s “Same Solutions?” (HTML)

Instructions: Please click on the link above and complete the task, which allows you to use algebraic reasoning to explore problems instead of solving them. The solution to this task is found toward the bottom of the page.

Completing this task should take approximately 20 minutes.

2.1.2 Solving

Hot and cold. Up and down. Right and left. What do these combinations have in common? They are opposites. This is the thinking that you use when solving an equation. Subtraction and addition as well as multiplication and division will be some of the opposites you will use when solving an equation. When working with a problem involving a variable, your goal is to isolate the variable from the rest of the terms.

In subunit 2.1.2, think about doing the opposite operations in order to isolate the variable from the rest of the problem. Since this section was covered in a Pre-algebra course, you will go straight to an assessment to see what you remember.

• Checkpoint: Henrico County Public Schools’s “Solving Equations Using Addition or Subtraction” Link: Henrico County Public Schools’s “Solving Equations Using Addition or Subtraction” (HTML)

Taking this quiz should take approximately 15 minutes.

• Checkpoint: Henrico County Public Schools’s “Solving Equations Using Multiplication and Division” Link: Henrico County Public Schools’s “Solving Equations Using Multiplication and Division” (HTML)

Instructions: Please click on the link above and then click on “ExamView Quiz: Solving Equations Using Multiplication or Division.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there are other resources on this site, such as the interactive notes, applet, and flashcards. There is also a PowerPoint presentation of notes about solving equations using multiplication and division as well as a worksheet on solving equations that does not come with an answer key.

Taking this quiz should take approximately 10 minutes.

• Checkpoint: Henrico County Public Schools’s “Solving Multi-Step Equations” Link: Henrico County Public Schools’s “Solving Multi-Step Equations” (HTML)

Instructions: Please click on the link above and then click on “ExamView Quiz: Solving Multi-Step Equations.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there are other resources on this site, such as the interactive notes, applet, and flashcards. There is also a PowerPoint presentation of notes about solving equations using multiplication and division as well as a worksheet on solving equations that does not come with an answer key.

Taking this quiz should take approximately 20 minutes.

• Explanation: Henrico County Public Schools's “Module – Solving Equations: Lesson 4: Solving Equations with Variables on Both Sides” Link: Henrico County Public Schools’s “Module – Solving Equations: Lesson 4: Solving Equations with Variables on Both Sides” (HTML and PPT)

Instructions: Please click on the link above and then view “Notes: Solving Equations with Variables on Both Sides.” This PowerPoint presentation will explain how to solve an equation when there is a variable on both sides of the equal sign. You should work through the guided problems at the beginning of the PowerPoint presentation and then use the PowerPoint presentation to check your work. Each step will be justified so you know what was done and why it was done. At the end, there are problems that you will complete as a self-quiz.

Completing this exercise should take approximately 30 minutes.

• Did I Get This? Activity: Henrico County Public Schools’s “Solving Equations with Variables on Both Sides” Link: Henrico County Public Schools’s “Solving Equations with Variables on Both Sides” (HTML)

Instructions: Please click on the link above and then click on “ExamView Quiz: Solving Equations with Variables on both Sides.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there are other resources on this site, such as the interactive algebra balance scales as well as a worksheet on solving equations that does not come with an answer key.

Taking this quiz should take approximately 30 minutes.

2.1.3 Parameters   - Activity: Mathematics Vision Project’s “Module I – Systems of Equations and Inequalities” Link: Mathematics Vision Project’s “Module I – Systems of Equations and Inequalities” (PDF)

Instructions: Click on the link and scroll to page 4. Read the Pet Sitters task and complete the ‘Ready Set Go’ problems 1-8. Then scroll down to page 37 and read the ‘Food for Fido and Fluffy’ task. Complete practice problems 1-5. Then complete the ‘Ready Set Go’ problems 1-12.

Completing both activities should take approximately 90 minutes.

``````-   [CCSS.Math.Content.HSA-CED.A.2](http://www.corestandards.org/Math/Content/HSA/CED/A/2)
-   [CCSS.Math.Content.HSA-CED.A.3](http://www.corestandards.org/Math/Content/HSA/CED/A/3)

displayed on the webpage above.
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2.1.4 Differences between Mathematical and Contextual Solutions   A lot of times, you solve an equation and you get a solution such as x = 4, but what does that really mean? Applying equations to relevant situations makes the math become real. In subunit 2.1.4, you will work through a problem where the solutions mathematically represent something real, x does not just equal four.

• Activity: Illustrative Mathematics’s “Paying the Rent” Link: Illustrative Mathematics’s “Paying the Rent” (HTML)

Instructions: Please click on the link above and complete the task, which will allow you to understand how solutions relate in a real-world context. The solution is found toward the bottom of the page.

Completing this task should take approximately 15 minutes.

2.1.5 Modeling   In a recipe, each ingredient can usually be used alone. However, when a lot of ingredients come together, a nice meal is created. Mathematics is the same way. Each concept can be learned alone, but used together, a task can be modeled.

In subunit 2.1.5, you will use a variety of concepts you have learned to model a car task.

Instructions: Please click on the link above and complete the task, which will allow you to use a mixture of the concepts you have covered. The solution is found toward the bottom of the page.

Completing this task should take approximately 15 minutes.

2.2 Solving Linear Inequalities   It is tax-free day, so you decide to go shopping. You are going to buy 3 pairs of jeans for \$30 each and you want to buy some shirts that are on sale for \$10. What is the maximum number of shirts you can buy if you are on a budget of \$145? You could solve an inequality to represent this problem such as 3(30) + 10x < 145, where x represents the amount of shirts. You encounter representing these inequalities in the real world even if you are not formally writing them down on paper.

2.2.1 Solving   In subunit 2.2.1, you will solve linear inequalities that are very similar to solving linear equations. Some of these sections only have assessments because they should be review material for you.

• Checkpoint: Henrico County Public Schools’s “Solving Inequalities Using Addition and Subtraction” Link: Henrico County Public Schools’s “Solving Inequalities Using Addition and Subtraction” (HTML)

Instructions: This checkpoint should be used as a knowledge check. Please click on the link above and then click on “ExamView Quiz: Solving Inequalities Using Addition and Subtraction.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there are other resources on this site, such as the interactive notes on solving inequalities as well as a worksheet on solving inequalities that does not come with an answer key.

Completing this quiz should take approximately 30 minutes.

• Checkpoint: Henrico County Public Schools’s “Solving Inequalities Using Multiplication and Division” Link: Henrico County Public Schools’s “Solving Inequalities Using Multiplication and Division” (HTML)

Instructions: This checkpoint should be used as a knowledge check. Please click on the link above and then click on “ExamView Quiz: Solving Inequalities Using Multiplication and Division.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there is a PowerPoint presentation on this same site that will explain and guide you through the steps of solving inequalities using multiplication and division.

Taking this quiz should take approximately 15 minutes.

• Checkpoint: Henrico County Public Schools’s “Solving Multi-Step Inequalities” Link: Henrico County Public Schools’s “Solving Multi-Step Inequalities” (HTML)

Instructions: This checkpoint should be used as a knowledge check. Please click on the link above and then click on “ExamView Quiz: Solving Inequalities Using Multi-Step Inequalities.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there is a PowerPoint presentation on this same site that will explain and guide you through the steps of solving multistep inequalities, and you will also find interactive notes and a worksheet.

Taking this quiz should take approximately 30 minutes.

• Explanation: Henrico County Public Schools’s “Module – Solving Inequalities: Lesson 4: Solving Compound Inequalities” Link: Henrico County Public Schools’s “Module – Solving Inequalities: Lesson 4: Solving Compound Inequalities” (HTML and PPT)

Instructions: Please click on the link above and then click on “Notes: Solving Compound Inequalities.” This PowerPoint presentation will explain how to solve inequalities and graph them on a number line. While viewing this presentation, you should work out the guided problems before viewing the solutions and then at the end of the presentation, there is a self-check quiz.

Completing this exercise should take approximately 30 minutes.

• Checkpoint: Henrico County Public Schools’s “Compound Inequalities” Link: Henrico County Public Schools’s “Compound Inequalities” (HTML)

Instructions: Please click on the link above and then click on “ExamView Quiz: Compound Inequalities.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly. If you struggled with the quiz, there is a compounds inequality worksheet that you can complete; however, it does not come with an answer key.

Taking this quiz itself should take approximately 30 minutes.

2.2.2 Parameters   - Activity: Mathematics Vision Project’s “Module I: Systems of Equations and Inequalities” Link: Mathematics Vision Project’s “Module I: Systems of Equations and Inequalities” (PDF)

Instructions: Click on the link and scroll to page 17. Read the Pampering and Feeding Time task and complete the ‘Ready Set Go’ problems 1-18. This activity lets you practice solving inequalities while working with real world constraints, for example, that a day only has 24 hours.

Completing this activity should take approximately 60 minutes.

``````-   [CCSS.Math.Content.HSA-CED.A.2](http://www.corestandards.org/Math/Content/HSA/CED/A/2)
-   [CCSS.Math.Content.HSA-CED.A.3](http://www.corestandards.org/Math/Content/HSA/CED/A/3)
-   [CCSS.Math.Content.HSA-REI.D.12](http://www.corestandards.org/Math/Content/HSA/REI/D/12)

displayed on the webpage above.
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2.2.3 Differences between Mathematical and Contextual Solutions   - Explanation: MoodleShare’s “Translating Phrases into Mathematical Inequalities” Link: MoodleShare’s “Translating Phrases into Mathematical Inequalities” (Flash)

Instructions: Click on the link and watch the video. This video will help you learn to translate math to everyday English! Pause and write down important phrases and their mathematical “translations.”

Watching this video should take 10 minutes.

``````-   [CCSS.Math.Content.HSN-Q.A.1](http://www.corestandards.org/Math/Content/HSN/Q/A/1)
-   [CCSS.Math.Content.HSN-Q.A.2](http://www.corestandards.org/Math/Content/HSN/Q/A/2)

displayed on the webpage above.
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2.2.4 Modeling   - Explanation: Open High School of Utah’s “Modeling Inequalities” Link: Open High School of Utah’s “Modeling Inequalities” (Flash Activity)

Instructions: Please click on the link and watch the video for a real world example of using inequalities.

Watching this video should take 5 minutes.

``````-   [CCSS.Math.Content.HSN-Q.A.1](http://www.corestandards.org/Math/Content/HSN/Q/A/1)
-   [CCSS.Math.Content.HSN-Q.A.2](http://www.corestandards.org/Math/Content/HSN/Q/A/2)
-   [CCSS.Math.Content.HSA-CED.A.1](http://www.corestandards.org/Math/Content/HSA/CED/A/1)

displayed on the webpage above.
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2.3 Literal Equations   We are so used to equations being in the y = form, but we do not always see them in that format. Sometimes, we have to highlight what letter we are solving for and use all of the methods of solving equations that we have learned to solve for a specific letter. Later this will help us when using formulas to model a real-world situation.

In subunit 2.3, you will solve equations for a specified letter.

• Explanation: Henrico County Public Schools’s “Module – Solving Equations: Lesson 5: Solving Equations and Formula” Link: Henrico County Public Schools’s “Module – Solving Equations: Lesson 5: Solving Equations and Formula” (HTML and PPT)

Instructions: Please click on the link above and then click on “Notes (ver 2): Literal Equations.” This PowerPoint presentation shows how to solve for a specified variable While viewing this presentation, you should work out the guided problems before viewing the solutions and then at the end of the presentation, there is a self-check quiz.

Completing this activity should take approximately 30 minutes.

• Did I Get This? Activity: Oswego City School District’s “Literal Equations” Link: Oswego City School District’s “Literal Equations” (HTML)

Instructions: If you need more examples after viewing the previous PowerPoint presentation, please click on “Lesson: Literal Equations.” If you feel that you are ready to try some problems without viewing any more examples, click on “Practice: Practice with Literal Equations.” There is a drop-down menu for each question to check your solutions.

Completing this activity should take approximately 30 minutes.

• Checkpoint: Henrico County Public Schools’s “Literal Equations” Link: Henrico County Public Schools’s “Literal Equations” (HTML)

Instructions: Please click on the link above and then click on “ExamView Quiz: Literal Equations.” When completed, this can be automatically scored with a click of the “Check Your Work” button. Once scored, the site will also provide the correct answers for any that were wrong. Once the quiz is scored, you should rework the problems by using the “Retake” button or retry the problems using the answers and working backward. Once corrected, you could start the quiz again or just attempt to rework the problems you answered incorrectly.

Taking this quiz should take approximately 20 minutes.

• Activity: Illustrative Mathematics’s “Equations and Formulas” Link: Illustrative Mathematics’s “Equations and Formulas” (HTML)

Instructions: Please click on the link above and complete the task, which reviews solving basic equations as well as literal equations using real-world formulas. The solution is found toward the bottom of the page.

Completing this task should take approximately 25 minutes.

2.4 Formulas   Determining how much wrapping paper it takes to cover a present, how much oil can be poured in a funnel, and the tax and total of a sale can all be calculated with formulas. We use formulas every day, even if we do not realize we are using them.

Subunit 2.4 will allow you to solve problems using formulas.

• Explanation: cK-12’s “Formulas for Problem Solving” Link: cK-12’s “Formulas for Problem Solving” (HTML and YouTube)

Instructions: Please click on the link above and work through all sections: guidance, guided practice, and practice. The practice covers some past concepts. Solutions are provided.

Working through this lesson should take approximately 1 hour.

• Activity: Illustrative Mathematics’s “Planes and Wheat” Link: Illustrative Mathematics’s “Planes and Wheat” (HTML)

Instructions: Please click on the link above and complete the task, which allows you to substitute values into the appropriate places of a given equation.
The solution is found toward the bottom of the page.

Completing this task should take approximately 10 minutes.