Unit 3: Expressions and Equations

Have you ever noticed that patterns are found everywhere? We use expressions and equations to put those patterns into mathematical sentences. Comparing the equations allows people to recognize how one pattern compares to another. For example, if you have a cell phone, you or a parent might have done some “comparison shopping” before signing up for the plan. Without even knowing it, you used an equation to compare. If you and your friends want to rent bicycles while on vacation, you might use an equation to figure out how much it would cost for eight friends to rent bikes. You’d use the same equation if only five friends plan to rent. Equations and expressions involve variables, which helps open lots of doors into algebra. While working through this unit, you will be constructing the basic building blocks of algebra, which is important to your mathematical future.

Completing this unit should take you approximately 13 hours.

☐ Subunit 3.1: 15 minutes

☐ Subunit 3.2: 2 hours and 30 minutes

☐ Subunit 3.3: 3 hours and 15 minutes

☐ Subunit 3.4: 4 hours and 15 minutes

☐ Subunit 3.5: 2 hours and 45 minutes

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to:

• Evaluate whole number expressions using exponents.

• Use variables to represent numbers.

• Write expressions for specific situations using variables.

• Use mathematical vocabulary to identify parts of an expression.

• Know and apply the rules of order of operations in expressions.

• Evaluate expressions for the given variable value.

• Apply and explain mathematical properties to compare equivalent expressions.

• Write and solve an equation for a real-world situation.

• Compare and contrast the difference between an equation and an inequality.

• Explain that solving equations or inequalities can be used within the process of answering questions about making the equation/inequality true.

• Write an inequality to represent a constraint or condition for a real-world situation.

• Define dependent and independent variables within a real-world situation.

• Write an equation using dependent and independent variables.

• Analyze the relationship between dependent and independent variables within a table and graph.

3.1 Exponents

When students first learn about multiplication, teachers often relate it to addition. Perhaps you have heard the phrase “multiplication is repeated addition.” An exponent is similar because an exponent helps mathematicians represent repeated multiplication. This subunit will teach you about how to solve for and write exponents.

• Explanation: YouTube: Mathispower4u: James Sousa’s “Exponential Notation”

Instructions: Watch the video about exponential notation. Take notes on the 2nd slide (0:20). There is important vocabulary in red text that you will want to take notes on. As you watch, pay attention to how the instructor uses those vocabulary words throughout the video. The slide starting at 1:19 is also very valuable; notice that multiple ways to verbally say x2 and x3 exist. You want to be familiar with both “x squared” and “x to the second power.”

Starting at 3:13, the instructor begins evaluating some exponents. At this time, you only need to be able to do and understand #1. Feel free to watch him solve the other three examples; do not worry if the negative numbers or the multiplying with a variable are confusing. You will learn those concepts throughout your math career.

Watching this video and taking notes should take you approximately 15 minutes.

3.2 Order of Operations

Mathematicians have a set of rules called order of operations; the rules help everyone know what steps to take when solving a set of numbers. As a mathematician you need to know these rules and how to apply them because you will find that you need them for both simple and complex problems you will encounter.

• Explanation: CK-12: “Order of Operations”

Link: CK-12: “Order of Operations” (HTML)

Instructions: Read about Keisha and counting birds. Take notes about the difference between an equation and an expression. You will continue to use those two words throughout this unit. Take notes about order of operations. This will be an important concept to remember whenever you are solving number sentences.

It’s often helpful to work the problems step-by-step to help organize your information. A benefit to organizing your work step-by-step is if you make a mistake, it’s easy to go back and figure out where the mistake was made. If you try to do mental math, you might end up with an incorrect answer. If you have not shown all your work, finding your error will be hard.

Look at the example problem below. The student first wrote out the entire problem, then solved the exponent operation and re-wrote the entire problem with just the 42 solved. The next step was to see that multiplication comes next, so the third line shows everything written out with the 5 x 2 solved. Now the student is ready to add/subtract from left to right. The fourth line shows the first addition part solved with the remaining number sentence staying the same. The final two lines show the last two operations solved as they show up left to right. The number sentence gets shorter with each line because the student solves one operation and then re-writes everything else just as it was given in the problem.

3 +  42 – 5 x 2 + 9

3 + 16 – 5 x 2 + 9

3 + 16 – 10 + 9

19 – 10 + 9

9 + 9

18

Continue doing the examples. In section III, you will get to be a math detective. Take time to check each of Joaquin’s problems and make sure he followed the proper steps. Do all example problems. Write down any vocabulary words that you have not already added to your notebook.

Watch the videos. Notice how the instructors show each of their steps as they work through the problem. Try to solve the expression before the instructor so you can check your work. Pause the video while you are working.

Work the “Time to Practice” problems for additional review.

Taking notes, watching the videos, and working through the examples should take you approximately 1 hour.

• Did I Get This? Activity: Wade Ellis and Denny Burzynski’s “Exponents, Roots, Factorization of Whole Numbers: Grouping Symbols and the Order of Operations”

Link: Wade Ellis and Denny Burzynski’s “Exponents, Roots, Factorization of Whole Numbers: Grouping Symbols and the Order of Operations” (HTML)

Instructions: Do “Practice Set A,” “Practice Set B,” and “Practice Set C” (exercise 19 in this last set is a challenge question). Check your solution with the “Show Solution” button next to each problem. Remember to show each step of your work. If you get a problem incorrect, go back through each of your steps and find the error.

Solving these problems should take you approximately 45 minutes.

3.3 Expressions

Remember in subunit 3.2 when you learned about the different between an equation and an expression? Now you are going to focus on expressions and how to evaluate expressions that involve variables (a variable is a letter representing a number). Expressions can represent situations that happen in your daily life. For example, as you will see in the first resource, an expression can be used to figure out a weekly paycheck based on the number of hours someone works.

3.3.1 Solving Expressions   - Explanation: CK-12: “Expression Evaluation with Powers and Grouping Symbols” Link: CK-12: “Expression Evaluation with Powers and Grouping Symbols” (HTML)

``````Instructions: Read about Lydia and Bart working part time jobs.
Think about the steps that would go into solving this initial
question. Continue reading and taking notes under the “Guidance”
section. Solve the examples step-by-step on paper (not just
mentally); the process of substituting numbers for variables and
remembering to follow the order of operations shouldn’t be something
that is rushed. Continue doing all the guided practice, watch the
short video, and complete the practice problems.

Reading, taking notes, and practicing this concept should take you
approximately 30 minutes.

-   [CCSS.Math.Content.6.EE.A.1](http://www.corestandards.org/Math/Content/6/EE/A/1)
-   [CCSS.Math.Content.6.EE.A.2](http://www.corestandards.org/Math/Content/6/EE/A/2)

attributed to CK-12.
``````

3.3.2 Writing Expressions for a Given Situation   - Explanation: CK-12: “Patterns and Expressions”

``````Link: CK-12: [“Patterns and
Expressions”](http://www.ck12.org/algebra/Patterns-and-Expressions/lesson/Patterns-and-Expressions/) (HTML)

the examples. Watch the first video about patterns.

Extra Challenge: The second video takes these concepts to a
higher-than-sixth-grade level. Feel free to watch it to “see where
you are going” in math. However, don’t get overwhelmed or
intimidated. The video uses equations instead of expressions. The
rules and ideas are the same, but since the instructor is solving
for a specific day, we use an equal sign.

Reading and watching the first video take should take you
approximately 15 minutes.

-   [CCSS.Math.Content.6.EE.A.2a](http://www.corestandards.org/Math/Content/6/EE/A/2/a)
-   [CCSS.Math.Content.6.EE.A.2b](http://www.corestandards.org/Math/Content/6/EE/A/2/b)

attributed to CK-12.
``````
• Did I Get This? Activity: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 3, Section 1: Mathematical Expressions - Exercises”

Link: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 3, Section 1: Mathematical Expressions - Exercises” (PDF)

Instructions: From the table of contents, click on “Exercises” under “3.1 Mathematical Expressions.” Do exercises 1-19 ODD. Scroll down to the next page and check your work.

Completing these exercises should take you approximately 30 minutes.

3.3.3 Identifying Parts of an Expression   - Explanation: CK-12: “Single Variable Expressions”

``````Link: CK-12: [“Single Variable
Expressions”](http://www.ck12.org/algebra/Single-Variable-Expressions/lesson/Single-Variable-Expressions/) (HTML)

Notice in the introduction there is the expression “20x.” Continue
specifically how to write variables and apply them to multiplication
and division problems. Work through the examples and guided
practice. Write down the vocabulary words that are not already in

Skip the first video. Watch the second and third video. Pause the
videos and solve each of the expressions before the instructor.
Remember to follow the order of operations and show each of your
steps.

Work through the practice problems.

Taking notes, watching the videos, and completing the practice
problems should take you approximately 30 minutes.

-   [CCSS.Math.Content.6.EE.A.1](http://www.corestandards.org/Math/Content/6/EE/A/1)
-   [CCSS.Math.Content.6.EE.A.2b](http://www.corestandards.org/Math/Content/6/EE/A/2/b)

attributed to CK-12.
``````

3.3.4 Evaluating Expressions within Real-Life Situations   - Explanation: CK-12: “Expressions for Real-Life Situations”

``````Link: CK-12: [“Expressions for Real-Life
Situations”](http://www.ck12.org/algebra/Expressions-for-Real-Life-Situations/lesson/Expressions-for-Real-Life-Situations/) (HTML)

Instructions: Read through the introduction. Pay close attention as
you read the first paragraph under the “Guidance” section. Take each
problem slowly; think about what type of operation would be used to
write an expression for each situation. Continue working on the
examples and guided practice. Skip the video as it covers
higher-level expressions with negative integers. Complete the ODD
practice exercises.

Taking notes and completing the practice problems should take you
approximately 30 minutes.

-   [CCSS.Math.Content.6.EE.A.1](http://www.corestandards.org/Math/Content/6/EE/A/1)
-   [CCSS.Math.Content.6.EE.A.2a](http://www.corestandards.org/Math/Content/6/EE/A/2/a)
-   [CCSS.Math.Content.6.EE.A.2b](http://www.corestandards.org/Math/Content/6/EE/A/2/b)
-   [CCSS.Math.Content.6.EE.A.2c](http://www.corestandards.org/Math/Content/6/EE/A/2/c)

attributed to CK-12.
``````
• Did I Get This? Activity: Howard County Public School System’s “Word Problems”

Link: Howard County Public School System’s “Word Problems” (PDF)

Instructions: This activity has eight problems. Two problems, #5 and #8, involve inequalities, which you will learn later in this unit. Consider how you might solve them, but don’t get too caught up in the exact expression. Complete all other word problems.

When you are finished check your answers here (PDF). If your variable is a different letter than the one on the answer key, that is fine. Sometimes mathematicians always use the same variable, like x; other times mathematicians use a variable that best suits the situation.

Completing these exercises should take you approximately 30 minutes.

3.3.5 Distributive Property   - Explanation: YouTube: Mathispower4u: James Sousa’s “Introduction to the Distributive Property”

``````Link: YouTube: Mathispower4u: James Sousa’s [“Introduction to the
Distributive

Instructions: As you watch the video, write down the number
sentence. It also helps to draw the arrows (as seen in the video) to
guide you to multiply the outside number by each of the inside
numbers. Take notes while the instructor does each of the examples,
including drawing the rectangle to show how the distributive
property can be used to find area.

Taking notes and watching the video should take you approximately 15
minutes.

-   [CCSS.Math.Content.6.EE.A.3](http://www.corestandards.org/Math/Content/6/EE/A/3)

attributed to James Sousa.
``````
• Explanation: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 3, Section 3: Simplifying Algebraic Expressions - The Distributive Property”

Link: College of the Redwoods Mathematics: Prealgebra Textbook: “Chapter 3, Section 3: Simplifying Algebraic Expressions - The Distributive Property” (PDF)

Instructions: From the table of contents, click on “The Distributive Property” found beneath “3.3 Simplifying Algebraic Expressions.” Read this section, take notes, and do the sample problems on pages 189 - 190. Understand that the number outside the parentheses needs to be multiplied by each of the numbers inside the parentheses. As you continue to work through higher-level algebra, you will extend your knowledge of the distributive property.

Reading this section, taking notes, and working the sample problems should take you approximately 15 minutes.

3.4 Equations

Solving an equation is like answering a question. Remember, an equation is going to have an equal sign. Think of the equal sign as a balance beam. Both sides are equivalent, and sometimes you have to work backwards, or undo, one operation to figure out the unknown value (the variable). In this subunit you will use substitution, like you did with expressions - the concepts are the same.

3.4.1 Solving Equations   - Explanation: CK-12: “Equations that Describe Patterns”

``````Link: CK-12: [“Equations that Describe
Patterns”](http://www.ck12.org/algebra/Equations-that-Describe-Patterns/lesson/Equations-that-Describe-Patterns/) (HTML)

Instructions: Read the introduction and “Guidance” section. Take
notes on the bold words and information as you read. The question in
example A is a different type of question than in examples B and C.
You should feel comfortable being able to write equations as well as
using the substitution method to check if a specific value for a
variable results in a solution.

Listen to the first 30 seconds of the video. Pause it, and try to
write and solve the equation before continuing with the video. Watch
the remainder of the video so see if your equation and solving
method matched the solution. As the second example begins playing,
stop the video at the 2:15 mark. See if you can write and solve an
equation for the verbal expression before continuing the video.

Complete the exercises under the “Guided Practice” section, and
check your work with the given solutions. Stop there. Do not
continue with the next video or practice problems.

Taking notes, completing the practice problems, and watching the
video should take you approximately 30 minutes.

-   [CCSS.Math.Content.6.EE.A.2](http://www.corestandards.org/Math/Content/6/EE/A/2)
-   [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
-   [CCSS.Math.Content.6.EE.B.6](http://www.corestandards.org/Math/Content/6/EE/B/6)
-   [CCSS.Math.Content.6.EE.B.7](http://www.corestandards.org/Math/Content/6/EE/B/7)

attributed to CK-12.
``````
• Explanation: CK-12: “Sentences as Single Variable Equations”

Link: CK-12: “Sentences as Single Variable Equations” (HTML)

Instructions: Read through the amusement park problem and the dilemma about the cost. Read through the “Guidance” section and example problems. Take note of the key words that help hint toward the operation or task. Complete the exercise under the “Guided Practice” section.

Watch the first video. The instructor does a good job of really explaining what 7x = 14 means. Take notes as you watch the video. Listen at the 3:53 mark for the word “coefficient” - this is an important vocabulary word to know. Write down the definition as you hear it. Continue watching the video and take notes as you go. Combining “like variables” is a skill the instructor shows you. This concept will come up again as you continue studying algebra in the future.

Complete the practice problems.

Taking notes, completing the practice problems, and watching the videos should take you approximately 45 minutes.

• Did I Get This? Activity: Khan Academy’s “One Step Equations”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice the one-step equations until you feel confident that you understand how to solve for the unknown variable (recommended a minimum of 10 minutes of practice).

Practicing these one-step equations should take you approximately 15 minutes.

• Checkpoint: The Mathematics Common Core Toolbox: “Inches and Centimeters”

Link: The Mathematics Common Core Toolbox: “Inches and Centimeters” (HTML)

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

3.4.2 Writing Equations in Real-Life Situations   - Explanation: CK-12: “Applications of One-Step Equations”

``````Link: CK-12: [“Applications of One-Step
Equations”](http://www.ck12.org/algebra/Applications-of-One-Step-Equations/lesson/Applications-of-One-Step-Equations/) (HTML)

different the equations would look for the two situations about your
“Guidance” section and solve the problems in the “Examples” and
“Guided Practice” sections. Write out each equation and show the
steps to find the solution. Don’t just use mental math. As you move
into higher-level math, the examples will become more challenging,
but the steps will stay the same. Now is the time to really
understand how to write and solve equations.

Watch the video and solve each of the problems. The instructor goes
into helpful detail by using number lines, models, and number
sentences.

Solve questions 1 - 3 under the video for additional practice.

Reading this lesson, watching the video, and completing the practice
problems should take you approximately 45 minutes.

-   [CCSS.Math.Content.6.EE.A.2](http://www.corestandards.org/Math/Content/6/EE/A/2)
-   [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
-   [CCSS.Math.Content.6.EE.B.6](http://www.corestandards.org/Math/Content/6/EE/B/6)
-   [CCSS.Math.Content.6.EE.B.7](http://www.corestandards.org/Math/Content/6/EE/B/7)

attributed to CK-12.
``````
• Checkpoint: Illustrative Mathematics: “Firefighter Allocation”

Link: Illustrative Mathematics: “Firefighter Allocation” (PDF)

Instructions: Write and solve an equation for the town’s firefighters. Check your solution on the second page.

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

• Explanation: National Center for Education Statistics: “What Are Independent and Dependent Variables”

Link: National Center for Education Statistics: “What Are Independent and Dependent Variables?” (HTML)

Instructions: Read and take notes on the definitions for both independent and dependent variables. Read how they relate to each other.

Taking notes and reading the explanation should take you approximately 15 minutes.

• Checkpoint: Illustrative Mathematics: “Chocolate Bar Sales”

Link: Illustrative Mathematics: “Chocolate Bar Sales” (PDF)

Instructions: Remember in the previous resource you learned about independent and dependent variables. When you graph information from a table to a graph the independent variable goes on the x-axis and the dependent variable goes on the y-axis.

Complete the questions on the first page. Check your solutions and read the commentary on pages 2 - 4.

Completing this checkpoint and checking your solution should take you approximately 45 minutes.

• Checkpoint: The Mathematics Common Core Toolbox: “Gasoline Consumption”

Link: The Mathematics Common Core Toolbox: “Gasoline Consumption” (HTML)

Instructions: This checkpoint calls on you to apply your previous learning about ratios to your current learning of equations. There are three parts (a, b, and c) to this resource. Read the introduction in the light gray box about the cars featured in the magazine.

Part a: Fill in the blanks to complete the rate table. Check your answer with the Submit Answer button. Do not move on until you have the correct answer.

Part b: Click on “Part b” in the upper right corner. In case you are not familiar with miles per gallon - a consumer wants to own a car that has the highest number of miles per gallon. This allows the person to travel a farther distance before needing to fill up the gas tank (saves money!). Use the information in the gray box to answer the question. Check your answer with the Submit Answer button. Do not move on until you have the correct answer.

Part c: Click on “Part c” in the upper right corner. Use the information in the gray box to answer the question. Check your answer with the Submit Answer button. Do not move on until you have the correct answer.

Completing the questions and checking your work should take you approximately 30 minutes.

3.5 Inequality

Have you ever solved a math problem where there was more than one correct answer? Perhaps you were considering the minimum number of lawns you needed to mow to buy a new skateboard. You might conclude that mowing more than nine lawns would get you an adequate amount of money. This means the correct answer for how many lawns to mow would be anything larger than nine. This is where you would use an inequality. An inequality is like an equation, but the equal sign is replaced with a greater-than symbol or a less-than symbol. You probably have experience with greater-than (>) and less-than symbols (<). There are often infinite possible solutions for some inequalities (and sometimes there are no solutions at all!). The rules of substitution and variables remain generally the same.

3.5.1 Representing Inequalities on a Number Line   - Explanation: K12 Handhelds, Inc.: Inequalities: “Overview” and “Graphing Inequalities”

``````Link: K12 Handhelds, Inc.:
*[Inequalities](http://k12opened.com/ebooks/math/ebook-inequalities/index.html)*:
“Overview” and “Graphing Inequalities” (HTML)

Instructions: This page will give you a really good summary of what
an inequality is and how to apply it. Read the “Overview” section;
take notes of each of the words in blue font. Read through the
examples, taking additional notes as you go. Do practice problems
1 - 11 (you can check answers 8 - 11).

In the “Graphing Inequalities” section, read and take notes.
the outlined boxes. This is very important to graphing inequalities
on a number line. Read through the examples, and complete practice
problems 12 - 19.

Reading this selection, taking notes, and completing the practice
problems should take you approximately 30 minutes.

-   [CCSS.Math.Content.6.EE.B.8](http://www.corestandards.org/Math/Content/6/EE/B/8)

United States
``````
• Explanation: Khan Academy’s “Inequalities on a Number Line”

Instructions: Watch the video. Draw a number line and graph the inequality on a number line. Notice how the instructor makes a circle around the dash representing the number 4. This is because the less-than sign represents the inequality.

Watching this video should take you approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Inequalities on a Number Line”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice graphing inequalities and writing inequalities until you feel confident that you understand how to graph and identify an inequality (recommended a minimum of 10 minutes of practice).

Practicing graphing inequalities on a number line should take you approximately 15 minutes.

3.5.2 Solving Inequalities   - Explanation: K12 Handhelds, Inc.: Inequalities: “1-Step Addition and Subtraction Inequalities”

``````Link: K12 Handhelds, Inc.:
*[Inequalities](http://k12opened.com/ebooks/math/ebook-inequalities/index.html)*:
“1-Step Addition and Subtraction Inequalities” (HTML)

Instructions: Scroll down to the section titled “1-Step Addition and
addition and subtraction inequalities. You will notice that solving
an inequality is similar to solving an equation. An important step
to remember is to always check your solution once you are complete.
This is a learning objective for sixth-grade students.

Reading and taking notes should take you approximately 15 minutes.

-   [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
-   [CCSS.Math.Content.6.EE.B.8](http://www.corestandards.org/Math/Content/6/EE/B/8)

``````

Instructions: Watch the video and work the problems out with the instructor. As a sixth-grade student, your learning objective is to be able to use substitution to determine whether a given number will make an inequality true.

Reflection: Write a paragraph comparing and contrasting the difference between expressions, equations, and inequalities. How are they the same? How are they different? Give some examples in your own life where you might use one of these algebraic formats over another.

Watching the video, taking notes, and writing the paragraph should take you approximately 30 minutes

3.5.3 Writing Inequalities in Real-Life Situations   - Did I Get This? Activity: Khan Academy’s “Inequalities”

``````Link: Khan Academy’s

Instructions: Read the word problem at the start of the video. Stop
the video at the 0:30 mark and try to solve it on your own. Use the
instructors as a guide if you need help. Notice that two questions
for how many tiles are needed to build the stone patio.

Watching this video and solving the problem should take you
approximately 15 minutes

-   [CCSS.Math.Content.6.EE.B.5](http://www.corestandards.org/Math/Content/6/EE/B/5)
-   [CCSS.Math.Content.6.EE.B.8](http://www.corestandards.org/Math/Content/6/EE/B/8)

``````
• Checkpoint: Illustrative Mathematics: “Fishing Adventures 1”

Instructions: Solve questions a and b about the fishing boat. Scroll down to the second page to check your answers.

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

• Checkpoint: Illustrative Mathematics: “Log Ride”

Link: Illustrative Mathematics: “Log Ride” (HTML)

Instructions: Read through the situation about the theme park. Notice how the inequality is already written for you. Your job is to figure out which group of children can safely ride the log ride.

The solution on the second page shows you two ways to solve the problem. Compare the way you solved it with both ways. It is a good strategy to know both ways to solve an inequality like this.

Completing this checkpoint, checking your solution, and reading about the different methods should take you approximately 30 minutes.

`````` Instructions: Test your knowledge by completing this checkpoint.