 Unit 2: Expressions and Equations   Suppose your parents recently told you they would help you purchase your first cell phone. Your job is to figure out which company to use. Call for Less Phone Company offers a low cost up front, but they charge you per text message. Text for Less Phone Company is costly to enroll, but they offer free text messaging. Which company is better for you? You might compare two different cell phone companies using a system of equations. In this unit, you will learn about proportional relationships. You can express proportional relationships in tables, graphs, and equations. Throughout the unit, you will focus on finding slope, finding y-intercept, finding patterns, and comparing graphs to equations.

Completing this unit should take approximately 11 hours.

☐    Subunit 2.1: 3 hours and 15 minutes

☐    Subunit 2.2: 1 hour

☐    Subunit 2.3: 3 hours

☐    Subunit 2.4: 3 hours and 45 minutes

Unit2 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Graph proportional relationships. - Interpret the slope of a graph. - Compare different proportional relationships that are represented in different forms. - Explain how graphs and equations use slope and y-intercept to compare proportional relationships. - Use similar triangles to explain slope on a coordinate plane. - Solve linear equations involving expanding expressions using the distributive property. - Solve linear equations involving combining like terms. - Solve linear equations with rational number coefficients. - Interpret the solution of two linear equations using a coordinate graph. - Solve simultaneous linear equations algebraically. - Estimate simultaneous linear equation solutions. - Recognize and solve real-world examples of two linear equations in two variables.

Standards Addressed (Common Core): - CCSS.Math.Content.8.EE.B.5 - CCSS.Math.Content.8.EE.B.6 - CCSS.Math.Content.8.EE.C.7 - CCSS.Math.Content.8.EE.C.8 - CCSS.ELA-Literacy.WHST.6-8.2

2.1 Proportional Relationships   A proportional relationship is when two quantities vary directly with each other. This means that when one quantity is doubled, the other quantity is doubled. Proportional relationships can be written as equations, tables, or graphs. In this subunit you will use proportional relationships to compare different situations, and you will graph proportional relationships and interpret the unit rates.

2.1.1 Slope   - Explanation: Monterey Institute for Technology and Education: HippoCampus: “Graphing Linear Equations” Link: Monterey Institute for Technology and Education: HippoCampus: “Graphing Linear Equations” (Flash)

Instructions: On the left side of the screen, select Developmental Math – Beginning Algebra, which you will find under the “Presentations” heading. On the pop out menu, scroll down to the “Graphing” heading, select Graphing Linear Equations, and play the video. While you watch, copy the example, including the information, table, and graph. Understand what linear relationship and linear equation mean. You should understand how to find information by using the table, graph, and equation.

Watching the video and taking notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.EE.B.5](http://www.corestandards.org/Math/Content/8/EE/B/5)

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• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Finding the Slope of a Line” Link: Monterey Institute for Technology and Education: HippoCampus: “Finding the Slope of a Line” (Flash)

Instructions: On the left side of the screen, select Developmental Math – Beginning Algebra under the “Presentations” heading. On the pop out menu, scroll down to the “Graphing” heading and select Finding the Slope of a Line. While you watch the video, make sure you are taking notes and writing down examples. Focus on the following questions while you watch the video:

• What is slope?
• How do I find slope?
• How does the slope change as the steepness of the line changes?
• What is the formula for slope?
• What is the slope for horizontal and vertical lines?
• What is the difference between a positive slope and a negative slope?

Stop the video while you watch if you need extra time to take notes.

Watching the video and taking notes should take approximately 15 minutes.

Standards Addressed (Common Core): - CCSS.Math.Content.8.EE.B.5

• Did I Get This? Activity: Illustrative Mathematics: “Peaches and Plums” Link: Illustrative Mathematics: “Peaches and Plums” (PDF)

Instructions: Solve questions a-b on page 1. Read the commentary and check your solution on page 2.

Completing this activity and checking your solution should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Coffee by the Pound” Link: Illustrative Mathematics: “Coffee by the Pound” (PDF)

Instructions: Solve questions a–d on page 1. Read the commentary and check your solution on page 2. Understand the two different ways you could have graphed the situation as explained in the solution for parts c and d.

Completing this activity and checking your solution should take approximately 15 minutes.

2.1.2 Comparing Graphs and Equations   - Explanation: Monterey Institute for Technology and Education: HippoCampus: “Writing the Equation of a Line” Link: Monterey Institute for Technology and Education: HippoCampus: “Writing the Equation of a Line” (Flash)

Instructions: On the left side of the screen, under the “Presentations” heading, select Developmental Math – Beginning Algebra. On the pop out menu, scroll down to the “Graphing” heading and select Writing the Equation of a Line. While you watch the video, make sure you are taking notes and writing down examples. The video will show and explain the formula y = mx + b. You will use the information from the previous subunit to help you find m. Make sure you understand how to find the y-intercept and slope, and how to write a linear equation. Stop the video while you watch if you need extra time to take notes.

Watching the video and taking notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.EE.B.5](http://www.corestandards.org/Math/Content/8/EE/B/5)

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• Explanation: CK-12: “Writing Equations Given a Slope and a Point” Link: CK-12: “Writing Equations Given a Slope and a Point” (HTML and YouTube)

Instructions: This lesson will build on the concepts from the previous video regarding how to write an equation. Read the introduction and take notes. Complete examples A, B, and C and understand the solutions. Complete the “Guided Practice” and understand the solution. Watch the video in the “Practice” section. In the video the instructor shows a problem before solving it. You should pause the video each time you see a new problem, answer the problem on your own, and then watch the video to check your solution.

Reading this lesson, taking notes, completing practice problems, and checking solutions should take approximately 45 minutes.

• Did I Get This? Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Drops in a Bucket Math Task” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Drops in a Bucket Math Task” (PDF)

Instructions: Read the situation about the rain in the bucket. Solve the problem using at least two strategies. Your strategies may include graphing, making a table, finding the slope, using an equation, writing a proportion, and working backwards. Check your answer here. Answers are on pages 2-5.

Completing this task and checking your solutions should take approximately 30 minutes.

2.1.3 Comparing Different Proportional Relationships   - Did I Get This? Activity: Illustrative Mathematics: “Proportional Relationships, Lines, and Linear Equations” Link: Illustrative Mathematics: “Proportional Relationships, Lines, and Linear Equations” (PDF)

Instructions: Use the graph to solve the question on page 1. Make sure to explain your response. Read the commentary and check your answer on page 2.

Completing this task and checking your solution should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.EE.B.5](http://www.corestandards.org/Math/Content/8/EE/B/5)

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• Did I Get This? Activity: Illustrative Mathematics: “Who Has the Best Job?” Link: Illustrative Mathematics: “Who Has the Best Job?” (PDF)

Instructions: The table explains Kell’s situation, and the statement explains Mariko’s job situation. Using each of these, answer questions a-d on page 1. Read the commentary and check your answers on page 2.

Completing this activity and checking your solutions should take approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Comparing Speeds in Graphs and Equations” Link: Illustrative Mathematics: “Comparing Speeds in Graphs and Equations” (PDF)

Instructions: Use the graph to solve the question on page 1. Make sure to explain your response. Read the commentary and check your answer on page 2.

Completing this task and checking your solutions should take approximately 15 minutes.

2.2 Using Triangles on the Coordinate Plane   You learned in the previous subunit how to find slope using any two points on a line. In this subunit you will use similar right triangles to explain why slope is the same between any two points on a non-vertical line in the coordinate plane. Similar triangles have the same shape, but can be different sizes. Triangles can be similar even if one is rotated or one is a mirror image of the other. Similar triangles will have the same angles, and their side lengths will be proportional. Congruent means the same shape and same size, so congruent triangles will have the same angles and same side lengths. You may encounter these vocabulary words throughout this subunit.

• Activity: Howard County Public School System’s Grade 8 Common Core Mathematics: “Similar Triangles and Slope” Link: Howard County Public School System’s Grade 8 Common Core Mathematics: “Similar Triangles and Slope” (PDF)

Instructions: Start on page 3 looking at the downhill skier. Answer the three questions beneath the picture to help develop ideas for what slope really means and how you can think of it in your own life. Answer questions 1-8 on page 4. You might need to zoom in on the graph to see the grid lines. Skip page 5, for now. On page 6, complete questions 1-7. On page 7, complete questions 1-3.

Challenge (optional) – complete the slide and ski slope activity on page 5. The activity gives you a lot of numbers. Sketch a picture and label the information. Answer the questions on page 5 as well. Finally, write one equation each for the slide and the ski slope. Completing the challenge should take approximately 15 minutes.

Check all answers on pages 8-10.

Completing this activity and checking your solutions should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Slopes Between Points on a Line” Link: Illustrative Mathematics: “Slopes Between Points on a Line” (PDF)

Instructions: This resource gives you the opportunity to use similar triangles to explain the relationship of the triangles to the slope on a line. Read through the information on page 1 and answer questions a-b. Read the commentary and check your solutions on pages 2-4.

Completing this task and checking your solutions should take approximately 30 minutes.

2.3 Solving Linear Equations   In sixth and seventh grade, you worked with solving expressions and inequalities. This subunit will review some of the same skills you worked on in the past. The subunit will continue to use rational numbers and integers while you learn about the distributive property and collecting like terms.

• Explanation: Monterey Institute for Technology and Education: HippoCampus: “Solving One-Step Equations Using Properties of Equality” Link: Monterey Institute for Technology and Education: HippoCampus: “Solving One-Step Equations Using Properties of Equality” (Flash)

Instructions: On the left side of the screen, select Developmental Math – Beginning Algebra from beneath the “Presentations” heading. On the pop out menu, scroll down to the “Solving Equations and Inequalities” heading and select Solving One-Step Equations Using Properties of Equality. While you watch the video, take notes and write down examples. Focus on the vocabulary words at the beginning of the video. Although the words might be review, copy the definitions in your notebook for reference. Stop the video while you watch if you need extra time to take notes. Make sure to copy the notes about how to solve equations found in the last 0:35 of the video.

Watching the video and taking notes should take approximately 15 minutes.

• Did I Get This? Activity: HS Tutorials: “One Step Equation Practice” Link: HS Tutorials: “One Step Equation Practice” (HTML)

Instructions: This interactive activity will help you practice the steps to solve simple linear equations. There is a hint button to use if necessary. Practice for at least 10 minutes.

Completing this activity should take approximately 15 minutes.

2.3.1 Distributive Property and Collecting Like Terms   - Explanation: James Sousa’s Mathispower4u: “Combining Like Terms” Link: James Sousa’s Mathispower4u: “Combining Like Terms” (YouTube)

Instructions: Watch the video and take notes. Make sure you understand what terms can be combined. Pause the video at the 1:25 mark to take notes on the slide about similar terms. The next slide introduces the distributive law, which is also called the distributive property. Take notes on how you can change the way expressions are written by rearranging the values in specific ways. Listen closely at the 2:35 mark. The instructor mentions the commutative property as a reason for why a can be placed either before or after the set of parenthesis. This is another good vocabulary word to know. Watch and practice the examples with the instructor.

Watching the video and working through the examples should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.EE.C.7](http://www.corestandards.org/Math/Content/8/EE/C/7)

attributed to James Sousa.
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• Explanation: James Sousa’s Mathispower4u: “Simplifying Algebraic Expressions” Link: James Sousa’s Mathispower4u: “Simplifying Algebraic Expressions” (YouTube)

Instructions: The video starts with another review of the distributive property. Add to your notes as necessary. Pause the video at the 0:53 mark and try to solve the problem. Watch the video and listen to the instructor as you check your work. Pause the video again at the 2:41 mark and try to solve the problem. Watch the video and listen to the instructor as you check your work. Pause the video once more at the 4:04 mark and try to solve the problem. Watch the video and listen to the instructor as you check your work. Don’t try to skip steps to speed up the process. Write down each step, as it helps you remember what you are doing and also allows other mathematicians to follow your thinking.

Watching the video and working through the examples should take approximately 15 minutes.

• Did I Get This? Activity: Department of Mathematics, College of the Redwoods: Prealgebra Textbook: “Chapter 3, Section 3: Simplifying Algebraic Expressions” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook: “Chapter 3, Section 3: Simplifying Algebraic Expressions” (PDF)

Instructions: Start at page 187. Read about and take notes on the gray box that explains the commutative and associative properties. Look at Examples 1, 2 and 3 on pages 187-188. Be sure you understand the solutions. Go to page 195 and complete problems 1-35 ODD. Check your answers on page 196. Make sure you understand the solutions before you move on.

Completing the practice problems and checking your solutions should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Coupon versus Discount” Link: Illustrative Mathematics: “Coupon versus Discount” (PDF)

Instructions: Answer the question on page 1. Make sure you use a strategy that includes combining like terms. Do not just guess and check. Read the commentary and check your solution on page 2.

Completing the problem and checking your solution should take approximately 15 minutes.

2.3.2 Solving Multi-Step Equations   - Explanation: Monterey Institute for Technology and Education: HippoCampus: “Solving Multi-Step Equations” Link: Monterey Institute for Technology and Education: HippoCampus: “Solving Multi-Step Equations” (Flash)

`````` Instructions: On the left side of the screen, select *Developmental
Math – Beginning Algebra* from beneath the “Presentations” heading.
On the pop out menu, scroll down to the “Solving Equations and
Inequalities” heading and select “Solving Multi-Step Equations.” At
the beginning of the video, the instructor explains the five steps
for solving multi-step equations. Make sure you take notes and
understand these steps. Stop the video while you watch if you need
extra time to take notes.

Watching the video and taking notes should take approximately 15
minutes.

-   [CCSS.Math.Content.8.EE.C.7](http://www.corestandards.org/Math/Content/8/EE/C/7)

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• Explanation: Department of Mathematics, College of the Redwoods: Prealgebra Textbook: “Chapter 3, Section 5: Solving Equations Involving Integers II” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook“Chapter 3, Section 5: Solving Equations Involving Integers II” (PDF)

Instructions: On pages 208-212, you will work through examples and take notes on solving equations. For each example, solve the equation for the given variable. In seventh grade, you learned about operations with integers. You will be using those skills as you solve each equation throughout this section. You should also solve the problems in the “You Try It!”sections. The “You Try It!”answers are along the edges of the page.

Taking notes and working through examples should take approximately 30 minutes.

• Did I Get This? Activity: Department of Mathematics, College of the Redwoods: Prealgebra Textbook: “Chapter 3, Section 5: Solving Equations Involving Integers II” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook“Chapter 3, Section 5: Solving Equations Involving Integers II” (PDF)

Instructions: Go to pages 213-214 in Chapter 3, Section 5. Complete exercises 1-9 ODD, 17-25 ODD, 35-39 ODD, and 53-57 ODD. Check your answers on pages 214-215. If you get an answer wrong, go back through the five steps for solving multi-step equations and figure out how to solve it correctly.

Completing the practice problems and checking solutions should take approximately 30 minutes.

2.4 Simultaneous Linear Equations   Often when working with linear equations, you are comparing two or more situations. Perhaps, for example, you are trying to figure out which cell phone plan is the cheapest for you based on the data plan you want. In this subunit, you will learn how to solve simultaneous linear equations using the coordinate plane and substitution with the equations. You will also have the opportunity to work with real-life situations. It is important that you know how to solve simultaneous linear equations a few different ways because some situations lend themselves to graphing, and some situations are more efficient when you use the substitution method.

2.4.1 Solving Using the Coordinate Plane   - Explanation: Monterey Institute for Technology and Education: HippoCampus: “Graphing Systems of Linear Equations” Link: Monterey Institute for Technology and Education: HippoCampus: “Graphing Systems of Linear Equations” (Flash)

Instructions: On the left side of the screen, select Developmental Math – Beginning Algebra from beneath the “Presentations” heading. On the pop out menu, scroll down to the “Systems of Equations and Inequalities” heading and select “Graphing Systems of Linear Equations.” While you watch the video, take notes and write down examples. Pay attention to what it means when lines intersect. Understand the difference between linear equations that have one solution, infinite solutions, and no solutions. The instructor explains what a system of equations is when the timer reads 1:00. Make sure to take notes on this concept. Stop the video while you watch if you need extra time to take notes.

Watching the video and taking notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.EE.C.8](http://www.corestandards.org/Math/Content/8/EE/C/8)

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2.4.2 Solving Algebraically and through Estimation   - Explanation: Monterey Institute for Technology and Education: HippoCampus: “The Substitution Method” Link: Monterey Institute for Technology and Education: HippoCampus: “The Substitution Method” (Flash)

Instructions: Instructions: On the left side of the screen, select Developmental Math – Beginning Algebra under the “Presentations” heading. On the pop out menu, scroll down to the “Systems of Equations and Inequalities” heading and select “The Substitution Method.” While you watch the video, take notes and write down examples. The instructor explains why graphing is not always the most efficient way to solve a system of equations. Understand the difference between linear equations that have one solution, infinite solutions, and no solutions. When there is 0:35 left in the video, take notes and be sure you understand the steps for using the substitution method to solve a system of equations. Stop the video while you watch if you need extra time to take notes.

Watching the video and taking notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.8.EE.C.8](http://www.corestandards.org/Math/Content/8/EE/C/8)

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• Did I Get This? Activity: Illustrative Mathematics: “Fixing the Furnace” Link: Illustrative Mathematics: “Fixing the Furnace” (PDF)

Instructions: Read and solve the problem on page 1. Consider using strategies from the graphing subunit and/or the substitution subunit. Read the commentary and check your solution on pages 2-3.

Completing this task and checking your solution should take approximately 30 minutes.

2.4.3 Real-World Examples of Systems of Equations   So far in this unit, you have completed a variety of problems that involve solving systems of equations. Your task in this subunit is to relate these skills to real-world applications. As you work, think about what the problem is asking, and solve each problem step-by-step. There are often multi-step solutions that need to be solved. Don’t guess and check – solve the problems algebraically.

• Did I Get This? Activity: Illustrative Mathematics: “Cell Phone Plans” Link: Illustrative Mathematics: “Cell Phone Plans” (PDF)

Instructions: Write an equation and graph the cell phone plans as described in part a of the question. Answer question b. Read the commentary and check your solutions on pages 2-3.

Completing this task and checking your solution should take approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Summer Swimming” Link: Illustrative Mathematics: “Summer Swimming” (PDF)

Instructions: Solve questions a-e on the first page. Read the commentary and check your solutions on pages 2-3.

Completing this task and checking your solution should take approximately 30 minutes.

• Checkpoint: Illustrative Mathematics: “How Many Solutions?” Link: Illustrative Mathematics: “How Many Solutions?” (PDF)

Instructions: This resource requires you to use skills from all of Unit 2. Answer the question on page 1. Try to solve the bonus question for the applicable cases. Read the commentary and check your solutions on pages 2-3.

Completing this task and checking your solution should take approximately 30 minutes.