Unit 3: Expressions and Equations   Imagine you are keeping track of the growth of bacteria in a science lab. You would like to make predictions based on the patterns you are starting to see, but extending a table of values would take way too long. One of the many goals of this unit will be to represent this real-life data as an equation and then solve it.

In this unit, you will develop your algebraic skills by writing and simplifying expressions using variables, numbers, and symbols. You will extend these skills to equations with multiple steps, including combining like terms and using the distributive property. Finally, you will take all of these skills and use them to help represent and solve real-life problems using equations and inequalities.

Completing this unit should take approximately 14 hours and 15 minutes.

☐    Subunit 3.1: 1 hour and 15 minutes

☐    Subunit 3.2: 1 hour and 15 minutes

☐    Subunit 3.3: 1 hour

☐    Subunit 3.4: 3 hours and 15 minutes

☐    Subunit 3.5: 3 hours and 15 minutes

☐    Subunit 3.6: 4 hours and 15 minutes

Unit3 Learning Outcomes
Upon successful completion of this course, you will be able to: - Write, simplify, and evaluate algebraic expressions.
- Solve and check multistep algebraic equations.
- Solve and graph inequalities with one variable.
- Model real-life situations using equations and inequalities.

3.1 Writing Expressions   Equations are created by piecing together different expressions. In this subunit, you will focus on changing descriptions of different situations into mathematical expressions by converting words into numbers, variables, and symbols.

• Activity: Mathematic Vision Project and Utah State Office of Education: “Serving Up Symbols: A Develop Understanding Task” Link: Mathematic Vision Project and Utah State Office of Education: “Serving Up Symbols: A Develop Understanding Task” (PDF)

Instructions: Complete this activity using the items and variables given. The important point of this task is to make sense of the expressions that you write. Be sure to think of how the different expressions might be used to give real-life data.

Completing this activity should take approximately 30 minutes.

• Explanation: CK-12: “Writing Expressions and Equations” Link: CK-12: “Writing Expressions and Equations” (PDF)

Instructions: Read the lesson up to the “Time to Practice” section. While you are reading through the examples, make a table with four columns in your notes. The columns will be labeled “Adding,” “Subtracting,” “Multiplying,” and “Dividing.” In each column, make a list of vocabulary words that let you know which operation to use in the expression. Then, solve the 26 practice problems at the bottom of the page.

Completing this activity should take approximately 45 minutes.

3.2 Simplifying Expressions   As we begin to model different real-life situations using mathematical expressions, we will need to show the expressions using multiple representations. Oftentimes a different, but equivalent, representation of a mathematical expression will help shed light on different features of the expression. For example, we might use the expression 20x + 36 to represent the area of a rectangle. However, if we had written it as 4(5x + 9), we could then see two possible factors that could represent the length (4) and width (5x + 9) of the rectangles. Both expressions are equal to each other, but the second expression gives us a little more information.

• Explanation: CK-12: “Simplify Variable Expressions Involving Multiple Operations” Link: CK-12: “Simplify Variable Expressions Involving Multiple Operations” (PDF)

Instructions: Read through the examples to find out what is meant by “like terms” and the steps you can use to combine them together. Then watch the six-minute video to see some practice questions in action. Finally, it is your turn to solve the 15 practice problems at the bottom of the page.

Completing this activity should take approximately 45 minutes.

• Explanation: CK-12: “Simplify Products or Quotients of Single Variable Expressions” Link: CK-12: “Simplify Products or Quotients of Single Variable Expressions” (PDF)

Instructions: Read through the passage to see how simplifying products and quotients differs from the sums and differences. Watch the review video and then solve the 15 practice problems at the end of the passage.

Completing this activity should take approximately 45 minutes.

3.3 Evaluating Expressions   To better understand how and what an algebraic expression represents, it is helpful to look at the value of the expression depending on the value of the variable. For example, we are given the expression 3x + 5 to represent the total cost of visiting a fair, where the variable x represents the number of rides or activities. If we know someone took part in 11 rides or activities, we can now calculate the total cost.

3x + 5
3(11) + 5
33 + 5
38

This expression is equal to 38 when x = 11. That means that this person spent \$38 to go to the fair and participate in 11 rides or activities. The value of the expression will change as the value of x changes.

In this subunit, you will practice evaluating different expressions and formulas and seeing how the value changes as the variables change.

• Explanation: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 3, Section 2: Evaluating Algebraic Expressions” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 3, Section 2: Evaluating Algebraic Expressions” (PDF)

Instructions: Read through the examples on pages 179 - 182, beginning with example 1. Focus on the red sections that share the steps to follow when evaluating. Then go to page 183 and complete problems 1 - 12 and 29 - 38. The answers can be found on pages 185 -

1. Use them to check your work.

Completing this activity should take approximately 30 minutes.

2. CCSS.Math.Content.7.EE.A.1

• Activity: Space Math @ NASA: “Kelvin Temperatures and Very Cold Things!” Link: Space Math @ NASA: “Kelvin Temperatures and Very Cold Things!” (PDF)

Instructions: Learn about how the Kelvin temperature relates to Celsius and Fahrenheit by evaluating formulas to convert temperatures.

Completing this activity should take approximately 30 minutes.

Terms of Use: This resource is in the public domain. It is attributed to Space Math @ NASA. The original version can be found here.

3.4 Solving Equations   In this subunit, we will grow from expressions to equations. Instead of being able to plug in multiple values for the variable in expressions, we will now be able to solve for a certain solution in an equation. This subunit starts with solving two-step equations and then builds to equations with multiple steps, including steps that involve combining like terms and the distributive property.

3.4.1 Two-Step Equations   - Explanation: Wikispaces: Math Made Easy for Middle School: “Two-Step Equations” Link: Wikispaces: Math Made Easy for Middle School: “Two-Step Equations” (HTML) (YouTube)

Instructions: While watching the video, copy the equations, practice showing the steps, and then check algebraically. For example, if you were solving the equation 3x – 5 = 25, you need to show the algebraic steps to prove that when x = 10, the equation will be true. Then, when you check your work, you will replace the x with the answer and evaluate to ensure that it is correct. Here is an example of checking algebraically.

3x – 5 = 25
3(10) – 5 = 25
30 – 5 = 25
25 = 25

Watching this video and practicing the equations should take approximately 15 minutes.

``````-   [CCSS.Math.Content.7.EE.B.3](http://www.corestandards.org/Math/Content/7/EE/B/3)

``````

3.4.2 Multistep Equations   - Explanation: CK-12: Barbara Greer's “Two-Step Equations” Link: CK-12: Barbara Greer's “Two-Step Equations” (PDF)

Instructions: Read the examples of how to change equations into two-step equations. Then solve number 1, a - p in the “Review Questions” section.

Reading the examples and solving the problems should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.EE.B.3](http://www.corestandards.org/Math/Content/7/EE/B/3)
-   [CCSS.Math.Content.7.EE.4a](http://www.corestandards.org/Math/Content/7/EE/B/4/a)

attributed to CK-12 Foundation and Barbara Greer. The original
version can be found
[here](http://www.ck12.org/user:YmdyZWVyQG1pdGFjYWRlbXkub3Jn/section/Two-Step-Equations/).
``````
• Activity: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 4, Section 8: Solving Equations with Fractions - Exercises” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 4, Section 8: Solving Equations with Fractions - Exercises” (PDF)

Instructions: Scroll down to page 337 and complete odd problems 49 - 59, using the same skills you used to solve the integer equations. The answers can be found on pages 339 - 340. Use them to check your work.

Completing this activity should take approximately 30 minutes.

• Activity: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 3, Section 5: Solving Equations Involving Integers II - Exercises” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 3, Section 5: Solving Equations Involving Integers II - Exercises”(PDF)

Instructions: Scroll down to page 213 and complete odd problems 17 - 51. Be sure to show an algebraic check as well. You can find examples of algebraic checks on pages 210 through 212, labeled “Check,” as well as in Subunit 3.4.1 of this course. The answers can be found on page 214 and page 215. Use them to check your work.

Completing this activity should take approximately 1 hour.

• Activity: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 5, Section 6: Equations with Decimals - Exercises” Link: Department of Mathematics, College of the Redwoods: Prealgebra Textbook, 2nd edition: “Chapter 5, Section 6: Equations with Decimals - Exercises” (PDF)

Instructions: Scroll down to page 421, and complete and check algebraically odd problems 17 - 33. The answers can be found on page

1. Use them to check your work.

Completing this activity should take approximately 30 minutes.

2. CCSS.Math.Content.7.EE.B.3

3.5 Solving Word Problems and Writing Equations   Using all of the skills we have developed to write expressions and solve equations, we will build equations from word problems and solve and discuss what the solutions mean in terms of the problem. A wide variety of problems can be modeled using equations. In this subunit, you will discover several different types and build skills for writing and solving any type of equation involving one variable.

• Explanation: Cheryl Wilcox’s “Perimeter Problems with Related Variables” Link: Cheryl Wilcox’s “Perimeter Problems with Related Variables” (PDF)

Instructions: Read this lesson to discover how the perimeter formula can be used to create algebraic equations to solve. Complete questions 14 - 16 of “Homework 55A” on page 7 after the reading. The answers follow on page 9.

Completing this activity should take approximately 30 minutes.

Instructions: Watch the video on writing let statements to use when building algebraic equations. While watching the video, be sure to answer the three questions given for each example.

The three questions are as follows:

1. Who do we know the least about?
2. How can I write the verbal model using words?
3. How can I write the algebraic model using the variables?

Completing this activity should take approximately 30 minutes.

• Explanation: CK-12: SCC Mathematics: “Writing Equations” Link: CK-12: SCC Mathematics: “Writing Equations” (PDF)

Instructions: In this section, you will write equations and define variables in terms of the story problems given. After reading the concession stand example, write equations to represent the six practice problems. You do not have to solve the equations in this section.

Reading this example and writing the equations should take approximately 30 minutes.

• Explanation: CK-12: “Interpreting Word Problems” Link: CK-12: “Interpreting Word Problems” (PDF)

Instructions: Skip the “Unit Conversion” section at the beginning of this webpage. Begin reading when you see the title “Using Algebraic Models.” While reading the remainder of the text, take note of key vocabulary terms that help convert words into equations. Then, create equations and solve the 10 problems under “Problem Set.”

Completing this activity should take approximately 1 hour.

• Explanation: Cheryl Wilcox’s “Writing Equations for Word Problems” Link: Cheryl Wilcox’s “Writing Equations for Word Problems” (PDF)

Instructions: Read this lesson to see how the way you solve problems can help you write equations. Then, solve the word problems that follow on pages 6, 7, and 9.

Reading this lesson and solving the word problems should take approximately 45 minutes.

3.6 Inequalities   Imagine that you have \$300 to go shopping for school clothes. The total of all the items you want to purchase must be less than or equal to \$300. It would be OK to get to the register and have a total of \$238 or \$287 or even \$199. As long as the total is not more than \$300, you have enough money to cover the costs. This is an example of an inequality. This subunit will use your equation-solving skills to now solve inequalities and graph the solutions.

3.6.1 Introduction to Notation and Graphing on a Number Line   - Explanation: OpenAlgebra.com: James Moore’s “Introduction to Inequalities and Interval Notation” Link: OpenAlgebra.com: James Moore’s “Introduction to Inequalities and Interval Notation” (PDF)

Instructions: Explore the examples in the reading to see how multiplying and dividing inequalities by negative numbers affects the outcome of the solution. Then, solve the two example problems before viewing the video solutions.

Reading this explanation, solving the problems, and watching the videos should take approximately 30 minutes.

``````-   [CCSS.Math.Content.7.EE.B.4b](http://www.corestandards.org/Math/Content/7/EE/B/4/b)

It is attributed to [John Redden](http://www.openalgebra.com/). The
original version can be found
[here](http://www.openalgebra.com/2012/11/introduction-to-inequalities-and.html).
``````
• Explanation: CK-12: “Inequality Expressions” Link: CK-12: “Inequality Expressions” (PDF)

Instructions: This reading will introduce you to the notation and graphing of inequalities. Be sure to note how the symbols are connected with the open and closed circles for graphing. Solve the 17 practice problems at the bottom of the page.

Reading this selection and solving the practice problems should take approximately 30 minutes.

• Web Media: XP Math: T. L. Hui’s “Inequality Wars” Link: XP Math: T. L. Hui’s “Inequality Wars” (Flash)

Instructions: As the game begins, you will be given an inequality. Your job is to hit asteroids with numbers that would be a solution to the inequality. To play this game, you will use the arrow keys and the space bar. You will use the space bar to hit the target.

Playing this game should take approximately 15 minutes.

3.6.1.1 Less Than or Greater Than   - Activity: CK-12: “Inequalities” Link: CK-12: “Inequalities” (PDF)

Instructions: This is a series of practice problems for working with inequalities. Complete all of the problems until you come to the section entitled “Absolute Value Equations.” Note that this activity covers the material you need to know for Subunit 3.6.1.1 and Subunit 3.6.1.2.

Completing this activity should take approximately 45 minutes.

``````-   [CCSS.Math.Content.7.EE.B.4b](http://www.corestandards.org/Math/Content/7/EE/B/4/b)

attributed to CK-12 Foundation. The original version can be found
[here](http://www.ck12.org/section/Inequalities/).
``````
• Explanation: CK-12: “Inequalities on a Number Line” Link CK-12: “Inequalities on a Number Line” (PDF)

Instructions: After reading the lesson, write in complete sentences why you might need to graph the solution to an inequality. Then watch the Khan Academy video. Finally, solve the 15 practice problems at the end of the passage. Note that this explanation covers the material you need to know for Subunit 3.6.1.1 and Subunit 3.6.1.2.

Reading this lesson, taking notes, watching the video, and solving the problems should take approximately 45 minutes.

3.6.1.2 Less Than or Equal To and Greater Than or Equal To   Note: This subunit is covered by the activity assigned beneath Subunit 3.6.1.1. Focus specifically on sections that include the symbols ≤ or ≥ and how graphing inequalities is different when these symbols are included.

3.6.2 Solving Inequalities   - Explanation: CK-12: Jack Hatert’s “Solving Inequalities” Link: CK-12: Jack Hatert’s “Solving Inequalities” (PDF)

Instructions: Read through the examples to see how solving inequalities is similar to solving equations. Then, solve the 22 practice problems at the bottom of the page.

Reading these examples and solving the problems should take approximately 1 hour.

Note to Educators: A planned unit with lessons and activities can be found at the Howard County Public School System Wiki.

``````-   [CCSS.Math.Content.7.EE.B.4b](http://www.corestandards.org/Math/Content/7/EE/B/4/b)

attributed to CK-12 Foundation and Jack Hatert. The original version
can be found
[here](http://www.ck12.org/user:amhhdGVydEB5c3NjaG9vbHMub3Jn/section/Solving-Inequalities/).
``````
• Checkpoint: Illustrative Mathematics: “Writing Expressions” Link: Illustrative Mathematics: “Writing Expressions” (PDF)

Instructions: Write an expression to represent the descriptions for parts a and b. Then scroll to page 2 to check your work.

Completing this checkpoint and checking your work should take approximately 15 minutes.