**Unit 3: Linear Equations and Inequalities in Two Variables**
*Suppose you know the age and weight of your dog and like to feed her a
combination of wet and dry food. Based on the information provided on
the packaging of the food, you have to determine what combination of
amounts of wet and dry food you could feed the dog. An inequality would
assist you in this matter.*

*Suppose you go to the candy store with a budgeted amount of money. You
want to know how many of each type of two candies you could purchase to
stay within that budget. Writing an equation and/or an inequality or
viewing the graph could quickly tell you the different combinations you
could buy to spend exactly the amount of money you have or stay under
budget.*

*This unit covers equations and equalities in two variables. *

**Unit 3 Time Advisory**

Completing this unit should take you approximately 8.25 hours.

☐ Subunit 3.1: 25 minutes

☐ Subunit 3.2: 1.25 hours

☐ Subunit 3.2.1: 1.25 hours

☐ Subunit 3.3: 4.25 hours

☐ Subunit 3.3.1: 3 hours

☐ Subunit 3.3.1.1: 1 hour

☐ Subunit 3.3.1.2: 30 minutes

☐ Subunit 3.3.1.3 55 minutes

☐ Subunit 3.3.1.4: 35 minutes

☐ Subunit 3.3.2: 30 minutes

☐ Subunit 3.3.3: 20 minutes

☐ Subunit 3.3.4: 30 minutes

☐ Subunit 3.4: 2 hours 20 minutes

☐ Subunit 3.4.1.: 55 minutes

☐ Subunit 3.4.1.1: 10 minutes

☐ Subunit 3.4.1.2: 45 minutes

☐ Subunit 3.4.2: 20 minutes

☐ Subunit 3.4.3: 10 minutes

**Unit3 Learning Outcomes**

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

- calculate and interpret the average rate of change.
- graph an equation and inequality in slope-intercept and standard
forms using a variety of methods.
- model a situation using an equation or inequality.
- determine the parameters of the solution within the context of the
problem.

Standards Addressed (*Common Core*):

- CCSS.Math.Content.HSF-IF.A.1
- CCSS.Math.Content.HSF-IF.B.4
- CCSS.Math.Content.HSF-IF.B.6
- CCSS.Math.Content.HSF-IF.C.7
- CCSS.Math.Content.HSN-Q.A.1
- CCSS.Math.Content.HSN-Q.A.2
- CCSS.Math.Content.HSA-CED.A.2
- CCSS.Math.Content.HSA-CED.A.3
- CCSS.Math.Content.HSF-LE.A.2
- CCSS.Math.Content.HSF-LE.B.5
- CCSS.Math.Content.HSA-REI.D.10
- CCSS.Math.Content.HSA-REI.D.11
- CCSS.Math.Content.HSA-SSE.B.3

**3.1 Discrete and Continuous Linear Functions**
*If you wanted to track your speed versus your time on a trip, or if you
wanted to collect data on people’s shoe sizes versus their age, you
would have to know what type of graph would be needed.*

*In subunit 3.1, you will explore when to use a discrete graph and when
to use a continuous graph.*

**Explanation: YouTube: STPAMrGill’s “Algebra Discrete vs. Continuous Graphs”**Link: YouTube STPAMrGill’s “Algebra Discrete vs. Continuous Graphs” (YouTube)

Instructions: Please click on the link above watch the video. While watching the video, pause to work out the examples and then push play to check your work.

Watching this video and working through the problems should take approximately 25 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.2 Average Rate of Change**
*Most people think of skiing when they hear the term slope. If you are a
beginning skier, then you want a less steep slope, but the experienced
skier wants a steep hill. This is the concept of slope, steepness.*

*We encounter other slopes every day, such as when we climb up a hill
and walk on a ramp. *

**3.2.1 Types of Rates of Change and Determining Rates of Change**
*In subunit 3.2.1, you will learn about the different rates of changes
that we call slopes and learn how to calculate rate of change.*

**Explanation: HippoCampus’s “Algebra I – An Open Course”**Link: HippoCampus’s “Algebra I – An Open Course” (HTML)

Instructions: Please click on the link above and then click on “Algebra 1 – An Open Course (2011) under the “Presentations” tab. Then click on “Rates of Change and Slope” under “Analyze and Graph Linear Equations, Functions, and Relations.” Then play the video, which explains the different types of rates of change and how to calculate slope.

Watching this video should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Explanation: HippoCampus’s “Algebra I – An Open Course"**Link: HippoCampus’s “Algebra I – An Open Course” (HTML)

Instructions: Please click on the link above and then click on “Algebra 1 – An Open Course (2011) under the “Worked Examples” tab. Then click on “Rates of Change and Slope 1” under “Analyze and Graph Linear Equations, Functions, and Relations.” Then play the video, which shows an example of calculating slope from a graph.

Watching this video should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Explanation: HippoCampus’s “Algebra I – An Open Course”**Link: HippoCampus’s “Algebra I – An Open Course” (HTML)

Instructions: Please click on the link above and then click on “Algebra 1 – An Open Course (2011) under the “Worked Examples” tab. Then click on “Rates of Change and Slope 2” under “Analyze and Graph Linear Equations, Functions, and Relations.” Then play the video, which shows an example of calculating slope from a table.

Watch this video should take approximately 5 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Explanation: Henrico County Public Schools’s “Module – Linear Equations: Lesson 1: Slope”**Link: Henrico County Public Schools’s “Module – Linear Equations: Lesson 1: Slope” (HTML and PPT)

Instructions: Please click on the link above and then click on “Notes: Finding Slope.” This PowerPoint presentation will explain how to determine slope from a graph as well as a formula. 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.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Checkpoint: Henrico County Public Schools’s “Slope”**Link: Henrico County Public Schools’s “Slope” (HTML)

Instructions: Please click on the link above and then take the quiz. 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 an interactive slope activity and a slope worksheet. The worksheet does not include answers, but you could check your answers using a calculator.

Taking this quiz should take approximately 20 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.3 Equations in Slope-Intercept and Standard Form**
**3.3.1 Graphing an Equation**
*When some people get up, the first thing they do is brush their teeth;
others take a shower, and some people might eat breakfast, but they are
all getting ready to start their day. There are multiple ways of
accomplishing a task, just as there are multiple methods to graphing an
equation. In this unit, you will explore this multitude of graphing
methods.*

**3.3.1.1 Tables**
*It would be very helpful for you to keep all of your notes together in
one place, such as in a binder or folder for each subject. It would also
be helpful for you to keep information for each equation in a nice neat
table. The table method is an organized manner of keeping the x and the
y categorized to make it easier to graph the equation. In subunit
3.3.1.1, you will graph lines by making tables.*

**Explanation: BrozanskaAlgebra I’s “Graphing Using Table of Values”**Link: BrozanskaAlgebraI’s “Graphing Using Table of Values” (HTML)

Instructions: Please click on the link and read the explanation for how to use a table of values to create a graph. Then complete the practice problem.

Completing this activity should take you about 15 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Activity: Math Help’s “Graphing Lines”**Link: Math Help’s “Graphing Lines” (PDF)

Instructions: Please click on the link above and complete the worksheet using the table method to graph lines. There are some problems where you have to solve for*y*first before using the table method. When complete, there is an answer sheet you can use to check your solutions.

Completing this exercise should take approximately 45 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.3.1.2 Intercepts**
*Since only two points are needed to graph a line, the x- and
y-intercepts are two points that would accurately make a graph if the
graph had intercepts. In subunit 3.3.1.2, you will graph the equation of
a line using the intercept method.*

**Explanation: HippoCampus’s “Algebra I – An Open Course”**Link: HippoCampus’s “Algebra I – An Open Course” (HTML)

Instructions: Please click on the link above and then click on “Algebra 1 – An Open Course (2011)” under the “Presentations” tab. Then click on “Intercepts of Linear Equations” under “Analyze and Graph Linear Equations, Functions, and Relations,” Then play the video, which models graphing steps using the intercept method.

Watching this video should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Explanation: HippoCampus’s “Algebra I – An Open Course”**Link: HippoCampus’s “Algebra I – An Open Course” (HTML)

Instructions: Please click on the link above and then click on “Algebra 1 – An Open Course (2011)” under the “Worked Examples” tab. Then click on “Intercepts of Linear Equations 1” under “Analyze and Graph Linear Equations, Functions, and Relations,” Then play the video, which provides practice for graphing with intercepts. Once the problem is given, pause the video and attempt the practice. Once you are done, play the video to check your work.

Completing this exercise should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Explanation: HippoCampus’s “Algebra I – An Open Course”**Link: HippoCampus’s “Algebra I – An Open Course” (HTML)

Instructions: Please click on the link above and then click on “Algebra 1 – An Open Course (2011)” under the “Worked Examples” tab. Then click on “Intercepts of Linear Equations 2” under “Analyze and Graph Linear Equations, Functions, and Relations,” Then play the video, which provides practice for graphing with intercepts. Once the problem is given, pause the video and attempt the practice. Once you are done, play the video to check your work.

Completing this exercise should take approximately 10 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.3.1.3 Slope and y-intercept**
*Now that you are familiar with slope and intercepts, you are going to
learn a method of graphing that incorporates both concepts. In subunit
3.3.1.3, you will graph a line using the slope-intercept method.*

**Explanation: Math Help’s “Graphing Slope Intercept Form”**Link: Math Help’s “Graphing Slope Intercept Form” (HTML)

Instructions: Please click on the link above and watch the video, which shows an example of graphing using the slope and the*y*-intercept.

Watching this video should take approximately 5 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Checkpoint: Math Help’s “Graphing Slope Intercept Form”**Link: Math Help’s “Graphing Slope Intercept Form” (PDF)

Instructions: Please click on the link above and complete the worksheet using the slope intercept method to graph lines. When complete, there is an answer sheet you can use to check your solutions. If you need more practice, you could use the worksheet from 3.3.1.1 and re-graph those using this method. The lines should turn out exactly the same as using the table method.

Completing this exercise should take approximately 50 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.3.1.4 Technology**
- **Explanation: NASA.gov’s “Suit Yourself: Fitted Through Space”**
Link: NASA.gov’s “Suit Yourself: Fitted Through
Space” (PDF),
“Answer
Key” (PDF),
and Optional Video “Interview with Amy Ross - Spacesuit
Engineer” (WMP)

Instructions: Read the Background, Instructional Objectives, and
the Problem. Solve problems 1-6. Check your answers using the Answer
Key.

Reading the background, learning objectives and problem should take
approximately 30 minutes. Completing the problems and checking the
answers should take approximately one hour. Watching the optional
video should take approximately 5 minutes.

Standards Addressed (*Common Core*):

```
- [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)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.3.2 Writing an Equation**
- **Explanation: One Mathematical Cat’s “Finding Equations of Lines”**
Link: One Mathematical Cat’s “Finding Equations of
Lines” (HTML)

Instructions: Click on the link and read the explanation of how to
find the equation of line. Then do at least 4 of the practice
problems until you feel confident about the concepts.

Reading this explanation and doing these problems should take about
30 minutes.

Standards Addressed (*Common Core*):

```
- [CCSS.Math.Content.HSA-REI.D.10](http://www.corestandards.org/Math/Content/HSA/REI/D/10)
- [CCSS.Math.Content.HSA-REI.D.11](http://www.corestandards.org/Math/Content/HSA/REI/D/11)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.3.3 Modeling**
- **Explanation: Khan Academy’s “Equation of a Line: Word Problem
Solving 4”**
Link: Khan Academy’s “Equation of a Line: Word Problem Solving
4” (YouTube)

Instructions: Watch the “Equation of a Line: Word Problem Solving
4” and take notes about to translate real-world situations into
slope-intercept form.

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

Standards Addressed (*Common Core*):

```
- [CCSS.Math.Content.HSA-CED.A.2](http://www.corestandards.org/Math/Content/HSA/CED/A/2)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.3.4 Parameters of Solutions**
- **Did I Get This? Activity: One Mathematical Cat’s “Point Slope
Form”**
Link: One Mathematical Cat’s “Point Slope
Form” (HTML)

Instructions: Click on the link and read the explanation of point
slope form. Some of this content will be review, but make sure you
understand what the parameters of an equation are. Then do at least
4 of the practice problems until you feel confident about the
concepts.

Doing this reading and these practice problems should take about 30
minutes.

Standards Addressed (*Common Core*):

```
- [CCSS.Math.Content.HSF-LE.B.5](http://www.corestandards.org/Math/Content/HSF/LE/B/5)
- [CCSS.Math.Content.HSS-ID.C.7](http://www.corestandards.org/Math/Content/HSS/ID/C/7)
- [CCSS.Math.Content.HSA-REI.D.11](http://www.corestandards.org/Math/Content/HSA/REI/D/11)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.4 Inequalities in Slope-Intercept Form and Standard Form**
*The difference between an inequality and an equation is that an
equation has to be exactly equal to something, whereas an inequality can
be equal to an amount, but it could also be less than or greater than an
amount as well. You will be using many of the concepts you previously
learned for solving equations, with a few extra steps added in. In this
unit, you will be working with graphing, writing, and modeling with
inequalities.*

**3.4.1 Graphing an Inequality**
*If I am told that the number of people attending an event is less than
60, there are many options for how many people might come. In planning
for the event, you would consider that there might be 20 people, or
there might be 40 people. When graphing an inequality, you have to keep
all those potential options in mind. In subunit 3.4.1, you will learn
how to graph an inequality.*

**3.4.1.1 Graphing an Inequality**
- **Explanation: James Sousa’s “Graphing Linear Inequalities in Two
Variables (Slope Intercept Form)”**
Link: James Sousa’s “Graphing Linear Inequalities in Two Variables
(Slope Intercept
Form)” (YouTube)

Instructions: Please click on the link above and watch the video
which shows how to use slope intercept form to graph linear
inequalities with two variables. As you watch, pause the video to
take notes.

Watching the video and taking notes should take about 10 minutes.

Standards Addressed (*Common* *Core*):

```
- [CCSS.Math.Content.HSA-REI.C.6](http://www.corestandards.org/Math/Content/HSA/REI/C/6)
- [CCSS.Math.Content.HSA-REI.D.12](http://www.corestandards.org/Math/Content/HSA/REI/D/12)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.4.1.2 Technology**
- **Explanation: Khan Academy’s “Graphing Linear Inequalities:
Graphing Inequalities”**
Link: Khan Academy’s “Graphing Linear Inequalities: Graphing
Inequalities” and
“Graphing Linear
Inequalities” (YouTube)

Instructions: Watch Khan Academy’s video, “Graphing Inequalities,”
and complete the checkpoint at “Graphing Linear Inequalities.” The
site will check your answers.

Watching the video and taking notes should take approximately 15
minutes. Completing the exercises in the checkpoint should take
approximately 30 minutes.

Standards Addressed (*Common Core*):

```
- [CCSS.Math.Content.HSA-REI.B.3](http://www.corestandards.org/Math/Content/HSA/REI/B/3)
- [CCSS.Math.Content.HSA-REI.C.6](http://www.corestandards.org/Math/Content/HSA/REI/C/6)
- [CCSS.Math.Content.HSA-CED.A.2](http://www.corestandards.org/Math/Content/HSA/CED/A/2)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```

**3.4.2 Writing an Inequality**
*In subunit 3.4.2, you will build upon what you learned in the last
section. However, this time, instead of being given an inequality and
graphing it, you will be given the graph and you will then write the
inequality.*

**Explanation: Khan Academy’s Graphing Linear Inequalities in Two Variables 3”**Link: Khan Academy’s “Graphing Linear Inequalities in Two Variables 3” (YouTube)

Instructions: Please click on the link above and then play the video, which shows how to write an inequality given a graph.

Watching this video should take approximately 5 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**Did I Get This? Activity: Khan Academy’s “Graphs of Inequalities”**Link: Khan Academy’s “Graphs of Inequalities” (HTML)

Instructions: Please click on the link above and then answer the problems provided. This site will automatically inform you after each problem if you are correct with the click of the “Check Answer” button. Try problems until you get five correct answers in a row without using the “I’d Like a Hint” button.

Completing this activity should take approximately 15 minutes.

Standards Addressed (*Common Core*):Terms of Use: Please respect the copyright and terms of use displayed on the webpage above.

**3.4.3 Modeling**
- **Explanation: Sophia’s “Equations in Standard Form”**
Link: Sophia’s “Equations in Standard
Form” (Flash)

Instructions: Scroll down to the third video, and watch “Equations
in Standard Form,” and take notes.

Watching the video should take approximately 10 minutes.

Standards Addressed (*Common Core*):

```
- [CCSS.Math.Content.HSA-SSE.B.3](http://www.corestandards.org/Math/Content/HSA/SSE/B/3)
Terms of Use: Please respect the copyright and terms of use
displayed on the webpage above.
```