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ALGEBRA II

PURPOSE OF COURSE

The purpose of this course is to familiarize you with the fundamentals of algebraic expressions, including adding, multiplying, factoring, and simplifying; solving equations and inequalities; performing operations on functions; and performing graphing and basic functional analysis. This course is intended to extend your knowledge beyond the foundational information learned in Algebra I and prepare you for more advanced topics, leading toward trigonometry and calculus. Among the benefits that will be gained from learning the material contained here are adding tools for critical thinking, improving skill sets for use in the sciences and improved competitiveness in preparation for college applications. A strong understanding of mathematics is critical toward earning scholarships and gaining admission to many top universities and the knowledge gained here will help in that regard.

COURSE REQUIREMENTS

In order to take this course, you must: Have access to a computer. Have continuous broadband Internet access. Have the ability/permission to install plug-ins or software (e.g., Adobe Reader

or Flash). Have the ability to download and save files and documents to a computer. Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.). Have competency in the English language. Have read the Saylor Student Handbook.

COURSE INFORMATION

Welcome to Algebra II. Below, please find general information on this course and its requirements.

Course Designer: Terrance Harrington

Tips/Suggestions: Focus on understanding what you are doing. The early material creates the tools that you will use in later sections so make sure you don’t skip over it. The answers in the back of the text are a learning tool; occasionally you will get the wrong answer and a tough problem can be cracked by working backwards to understand the process. Be careful of answers—sometimes students look at the answer in the back of a text and see that they made a small mistake—maybe missing a minus sign—and it makes the rest of the problem incorrect. Basically, a student in that situation

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http://www.saylor.org/site/wp-content/uploads/2012/05/Saylor-StudentHandbook.pdf

is almost there, but remember that in a test an incorrect minus sign will get in your way. Keep practicing until you are able to avoid those careless errors. Sometimes a text may be confusing, but with the internet, you have incredible help available. There are many good sources. Often it makes a difference to have a different explanation so type in the name of whatever topic you struggle with, and use the resources to enhance your understanding.

LEARNING OUTCOMES

Upon successful completion of this course, the student will be able to: Solve linear equations and inequalities. Graph linear functions, polynomials, rational functions, radical functions,

exponential functions, and logarithmic functions. Solve systems of linear equations. Solve quadratic, general polynomial, radical, rational, exponential, and

logarithmic equations. Find inverses of functions, if such inverses exist. Find the results of operations on functions. Graph conic sections given their equations. Find the nth term of an arithmetic or geometric sequence.

CONTENT OUTLINE

UNIT 1: LINEAR EQUATIONS AND INEQUALITIESTime Advisory: This unit should take you approximately 9 hours 20 minutes to complete.

Subunit 1.1: 3 hours 55 minutes Sub-subunit 1.1.1: 1 hour 20 minutes

Sub-sub-subunit 1.1.1.1: 45 minutes Sub-sub-subunit 1.1.1.2: 35 minutes

Sub-subunit 1.1.2: 2 hours 35 minutes Sub-sub-subunit 1.1.2.1: 10 minutes Sub-sub-subunit 1.1.2.2: 30 minutes Sub-sub-subunit 1.1.2.3: 1 hour 55 minutes

Subunit 1.2: 3 hours 5 minutes Sub-subunit 1.2.1: 5 minutes Sub-subunit 1.2.2: 40 minutes Sub-subunit 1.2.3: 20 minutes Sub-subunit 1.2.4: 25 minutes Sub-subunit 1.2.5: 65 minutes

Sub-sub-subunit 1.2.5.1: 5 minutes Sub-sub-subunit 1.2.5.2: 15 minutes Sub-sub-subunit 1.2.5.3: 45 minutes

Subunit 1.3: 1 hour 15 minutes Sub-subunit 1.3.1: 30 minutes

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Sub-subunit 1.3.2: 45 minutes Subunit 1.4: 1 hour 5 minutes

Sub-subunit 1.4.1: 30 minutes Sub-subunit 1.4.2: 35 minutes

You will begin Algebra II with an introduction to relations and functions. A relation is any set of ordered pairs. You have encountered ordered pairs in previous math courses and know that (2,3) is a representation of a point on a graph (x-axis coordinate, y-axis coordinate). In this course we will be looking at how relations, with certain rules applied to them, are functions. Functions can be spotted in everyday life. Imagine that you are on a trip to a city that you’ve never been to, you see a coin-pressing machine and decide that the penny in your pocket could be a nice souvenir. You put a coin into the machine, wait while the machine operates, and out comes a flattened piece of metal with an interesting shape on it. This is a function. The penny is the input, the machine is performing an operation (or the whole mechanical process could be called the function), and the flattened souvenir penny is the output. A more specific, algebra-oriented definition for a function would be a set of ordered pairs in which each x-element only has one y-element associated with it. As you delve into the unit, you’ll be able to identify functions, express them fluently in math terms, and even use them to solve more complex problems. Linear equations and inequalities are a good place to start, as they serve as a basis for understanding of more complex functions.

In this section, you will become very familiar with linear expressions, equations, and inequalities. Like a lot of things, equations come in a host of different forms. You will learn about the various forms, why they are important, how and why to use a particular form, and how to move from one form to another. For example, imagine that you are planning a road trip from your hometown to your friend’s town that is 95 miles away and you’d like to calculate how much gas you will use. A simple linear equation will work. Imagine that you are in a situation where you only have $10 in your budget and need to take a cab from a bus stop to a job interview. The cab charges 75 cents per mile, and a flat $4 rate for the ride. You will need to create a linear inequality to help you stay at or under $10. In this section, you will learn some new techniques to graph lines, then how to solve and graph linear inequalities. The unit will introduce graphing based on a comparison test, based on the use of test coordinates, and based on an inequality in slope-intercept form. The importance of the use of test coordinates will become clearer later as you use similar techniques throughout the remainder of your mathematics course... you’ve been warned!

Finally, you will complete this unit by learning how to solve absolute value (the distance from zero) equations and inequalities. Absolute value equations are used frequently. For example, imagine a peanut packing company that makes snack bags with 6 ounces of peanuts. After they are packaged and sealed, they are weighed. There is a .25 ounce range (below or above 6 oz.) within which any bags’ weight must fall—otherwise they are sent back to be repackaged. An absolute value

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equation will tell you the acceptable weight. Absolute value inequalities are solved in a similar way, but take into account that the absolute value could be either a positive number or a negative number. Because of this, they’re solved separately.

Learning Outcomes: Upon successful completion of this unit, the student will be able to:

Write the equation of a line through two points on a coordinate plane. Graph a linear function. Solve linear equations for a given variable. Write a linear equation in three equivalent forms: standard, point-slope,

and slope-intercept. Determine if two lines are parallel, if they simply intersect, or if they are

perpendicular. Graph linear inequalities. Find the solution of linear inequalities by comparison. Find the solution of linear inequalities by using test coordinates.

Standards Addressed (Common Core):● CCSS.Math.Content.HSA-SSE.B.3 ● CCSS.Math.Content.HSA-REI.B.3 ● CCSS.Math.Content.HSA-REI.B.4 ● CCSS.Math.Content.HSA-REI.C.5 ● CCSS.Math.Content.HSA-REI.C.6 ● CCSS.Math.Content.HSA-REI.C.7 ● CCSS.Math.Content.HSA-REI.D.10 ● CCSS.Math.Content.HSA-REI.D.11 ● CCSS.Math.Content.HSA-REI.D.12 ● CCSS.Math.Content.HSF-IF.A.1 ● CCSS.Math.Content.HSF-IF.A.2 ● CCSS.Math.Content.HSF-IF.A.3 ● CCSS.Math.Content.HSF-IF.B.4 ● CCSS.Math.Content.HSF-IF.B.5 ● CCSS.Math.Content.HSF-IF.C.7 ● CCSS.Math.Content.HSF-IF.C.8 ● CCSS.Math.Content.HSF-IF.C.9 ● CCSS.Math.Content.HSF-LE.A.2 ● CCSS.Math.Content.HSF-LE.A.3

1.1 Relations and Functions1.1.1 Introduction to Relations

1.1.1.1 What is a Relation?Instruction: CK-12: Raja Almukahhal et. al’s: “Relations and Functions”

Link: CK-12: Raja Almukahhal et. al’s: “Relations and Functions” (HTML)

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http://www.ck12.org/concept/Relations-and-Functions/?ref=%2Fconcept%2FRelations-and-Functions%2F


http://www.corestandards.org/Math/Content/HSF/LE/A/3


http://www.corestandards.org/Math/Content/HSF/IF/C/9



http://www.corestandards.org/Math/Content/HSF/IF/B/5


http://www.corestandards.org/Math/Content/HSF/IF/A/3



http://www.corestandards.org/Math/Content/HSA/REI/D/12



http://www.corestandards.org/Math/Content/HSA/REI/C/7



http://www.corestandards.org/Math/Content/HSA/REI/B/4


http://www.corestandards.org/Math/Content/HSA/SSE/B/3

Instructions: Please click on the link above. Read the webpage, and watch the videos embedded in the text, stopping at “Practice.” You may also download a PDF, mobi, or ePub version of this text by clicking on the ap-propriate link under “Download.” This resource provides an in-depth discussion on functions and relations. It will discuss what relations and functions are and will give a good explanation on the importance of identifying and distinguishing between them. This page is important as it provides several different ways to identify relations and functions. After reading the article work through the “Guided Practice” examples, the “Practice” examples will be assigned in unit 1.x.x.

Reading and solving the guided practice problems will take approximately 45 minutes to complete.

Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.B.5

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

1.1.1.2 Determining Membership in a RelationInstruction: PurpleMath.com: Modules: “Functions versus Relations”

Link: PurpleMath.com: Modules: “ Functions versus Relations ” (HTML)

Instructions: Please click on the link. There is an in-depth discussion on functions and relations. It will discuss what relations and functions are and will give a good explanation on the importance of identifying and distinguishing between them. This page is important as it provides several different ways to identify relations and functions.

Reading this article should take approximately 30 minutes.

Standards Addressed (Common Core):● CCSS.Math.Content.HSF-IF.A.1 ● CCSS.Math.Content.HSF-IF.A.2

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http://www.purplemath.com/modules/fcns.htm

http://www.purplemath.com/modules/fcns.htm



Instruction: Khan Academy: Math: Algebra: “Domain and Range of a Relation”

Link: Khan Academy: Math: Algebra: “ Domain and Range of a Relation ” (YouTube)

Instructions: Please click on the link. This video (2:43) gives an introduction to domain and range, which are the “input” and “output” of a relation. Basically, these are the sets of values that get matched together by the relation.

Watching this video should take approximately 5 minutes.

Standards Addressed (Common Core):● CCSS.Math.Content.HSF-IF.A.1 ● CCSS.Math.Content.HSF-IF.A.2 ● CCSS.Math.Content.HSF-IF.B.5


1.1.2 Functions1.1.2.1 What is a Function?

Instruction: James L. Sousa’s “Determine if a Relation is a Function”

Link: James L. Sousa’s “Determine if a Relation is a Function” (YouTube)

Instructions: Please click on the link and watch the explanation of the definition of the function introduces the idea of tests to determine if a relation is a function.



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http://www.youtube.com/watch?v=QFOJmevha_Y

http://www.youtube.com/watch?v=QFOJmevha_Y




https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-of-a-relation

https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-of-a-relation


1.1.2.2 How Functions Differ from RelationsInstruction: Monterey Institute: Algebra I Course: “Representing Functions and Relations”

Link: Monterey Institute: Algebra I Course: “ Representing Functions and Relations ” (HTML)

Instructions: Please click on the link “Representing Functions and Relations.” There is an in-depth introduction to functions and relations on this web page. It will discuss what relations and functions are and will give a good explanation on the importance of identifying and distinguishing between them. The pages have a series of self-quizzing questions that appear throughout. The answers to the questions are contained on the page and may be seen by clicking on the “Show/Hide Answer” link for each.

Reading this article, and completing the problem sets should take approximately 30 minutes.



1.1.2.3 Determining Membership in a FunctionInstruction: PurpleMath.com: Modules: “Functions: Domain and Range”

Link: PurpleMath.com: Modules: “ Functions: Domain and Range ” (HTML)

Instructions: Please click on the link. There is an in-depth discussion on how to determine function membership via domains and ranges. This is done by set descriptions, by graphs and by algebraic methods. It is important to have multiple approaches to determining function

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http://www.purplemath.com/modules/fcns2.htm

http://www.purplemath.com/modules/fcns2.htm




http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/index.html

http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/index.html

memberships as some are easier to use than others, depending on the situation.

Reading this article should take approximately 30 minutes.

Standards Addressed (Common Core):● CCSS.Math.Content.HSF-IF.A.1 ● CCSS.Math.Content.HSF-IF.A.2 ● CCSS.Math.Content.HSF-IF.A.3 ● CCSS.Math.Content.HSF-IF.B.5


Instruction: Khan Academy: Math: Algebra: “Domain and Range of a Function Given a Formula”

Link: Khan Academy: Math: Algebra: “ Domain and Range of a Function Given a Formula ” (YouTube)

Instructions: Please click on the link. This video (8:07) provides a discussion on domain and range, which for a function are the sets of “input” and “output” of a function.


Standards Addressed (Common Core):● CCSS.Math.Content.HSF-IF.A.1 ● CCSS.Math.Content.HSF-IF.A.2 ● CCSS.Math.Content.HSF-IF.A.3 ● CCSS.Math.Content.HSF-IF.B.5


Instruction: Khan Academy: Math: Algebra: “Domain and Range 1”

Link: Khan Academy: Math: Algebra: “ Domain and Range 1 ” (YouTube)

Instructions: Please click on the link. This video (3:36)

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https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-1






https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-of-a-function-given-a-formula

https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/v/domain-and-range-of-a-function-given-a-formula





provides a discussion on domain and range, given an example. This walks through the setup of an equation from the information given and then shows how to find domain and range samples from the real numbers.


Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.A.1 CCSS.Math.Content.HSF-IF.A.2 CCSS.Math.Content.HSF-IF.C.7


Instruction: Khan Academy: Math: Algebra: “Domain and Range 2”

Link: Khan Academy: Math: Algebra: “ Domain and Range 2 ” (YouTube)

Instructions: Please click on the link. This video (3:08) provides a discussion on representing domain and range on a coordinate plane, creating ordered pairs and then plotting them.

Watching this video will take approximately 5 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.B.5 CCSS.Math.Content.HSF-IF.C.7


Did I Get This? Activity: Khan Academy: Math: Algebra: “Domain of a Function”

Link: Khan Academy: Math: Algebra: “ Domain of a Function ” (Java)

Instructions: Please click on the link. This page provides a series of practice problems that the student can answer

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https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/e/domain_of_a_function

https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/e/domain_of_a_function








and check online. Each question has solution worked out step by step if the student needs hints along the way.

Solving the problem sets should take approximately 15 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSA-SSE.B.3b CCSS.Math.Content.HSA-REI.B.4b CCSS.Math.Content.HSF-IF.A.2


Did I Get This? Activity: Khan Academy: Math: Algebra: “Range of a Function”

Link: Khan Academy: Math: Algebra: “ Range of a Function ” (Java)

Instructions: Please click on the link. This page provides a series of practice problems that the student can answer and check online. Each question has solution worked out step by step if the student needs hints along the way.

Solving the problem sets should take approximately 15 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.A.1 CCSS.Math.Content.HSF-IF.A.2 CCSS.Math.Content.HSF-IF.B.5


Checkpoint: CK-12: Raja Almukahhal et. al’s: “Relations and Functions”

Link: CK-12: Raja Almukahhal et. al’s: “Relations and Functions” (HTML)

Instructions: Please click on the link above, scroll down to the Practice Problems section, and complete practice

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https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/e/range_of_a_function

https://www.khanacademy.org/math/algebra/algebra-functions/domain_and_range/e/range_of_a_function


http://www.corestandards.org/Math/Content/HSA/REI/B/4/b

http://www.corestandards.org/Math/Content/HSA/SSE/B/3/b

problems 1–16.

Solving the problem set should take approximately 30 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.A.1 CCSS.Math.Content.HSF-IF.A.2 CCSS.Math.Content.HSF-IF.A.3 CCSS.Math.Content.HSF-IF.B.5


1.2 Linear EquationsDid I Get This? Activity: MathIsFun: Algebra: “Linear Equations”

Link: MathIsFun: Algebra: “ Linear Equations ” (HTML)

Instructions: Please click on the link above. The web page contains a great deal of very good review information on linear equations, including the different forms of a line, slope, the identity function and constant functions. Although it's packed with information, it is short and concise, making it an easy read. At the end of the page are a series of self-quizzing questions. Answer the questions to review your knowledge of linear equations.

Reading and solving the problem sets will take approximately 30 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSF-LE.A.2 CCSS.Math.Content.HSF-LE.A.3

Terms of Use: Please respect the copyright and terms of use displayed on the webpage above

1.2.1 Standard Form of a LineInstruction: YouTube.com: Mahalo: “Finding the Equation of a Line in Standard Form”

Link: YouTube.com: Mahalo: “Finding the Equation of a Line in Standard Form ” (YouTube)

Instructions: Please click on the link above. This video (3:46) gives

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http://www.youtube.com/watch?v=fLRkUmE07Jo

http://www.youtube.com/watch?v=fLRkUmE07Jo



http://www.mathsisfun.com/algebra/linear-equations.html





a brief but very useful discussion on the relationship between slope-intercept and standard forms of a line. This video is important for showing how algebraic methods for equations can be used to translate equations from one form to another.


Standards Addressed (Common Core): CCSS.Math.Content.HSF-LE.A.2 CCSS.Math.Content.HSF-IF.C.7 CCSS.Math.Content.HSF-IF.C.9


1.2.2 Slope-Intercept Form of a LineInstruction: James Sousa’s: “Slope-Intercept Form of a Line”

Link: James Sousa’s: “ Slope-Intercept Form of a Line ” (YouTube)

Instructions: Please click on the link above. This video (9:30) gives an excellent discussion on slope-intercept form of a line. Interesting in the instructor's approach is the graphical representation of a line and how the line changes as the slope changes. He also does a very good job of using several different examples to emphasize his points. As a bonus, he discusses how to take two points on a line and find the slope-intercept form.


Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.C.7


Activity: Geogebra: Andy W’s “Transforming Linear Graphs”

Link: Geogebra: Andy W’s “ Transforming Linear Graphs ” (Java)

Instructions: Please click on the link above. This interactive graph allows you to change the slope and y-intercept using slide bars to see how the slope-intercept form of a line changes, as well as the graph. Use the black bar labeled “m =” at the top left hand corner to

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http://www.geogebratube.org/student/m12646


http://www.youtube.com/watch?v=IkHJgg-Xjdk




change the slope of the line. Test out at least three different values and note how the graph changes. Then adjust the blue bar labeled “c =” directly underneath the black bar to adjust the y-intercept. Test out at least three different values and note how the graph changes. Now try adjusting both values and observe the graph. This exercise is a great way to link the form visually with the graph.

Completing this exercise should take approximately 15 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.D.10 CCSS.Math.Content.HSA-REI.D.11


Activity: Geogebra: Matt Huttenlock’s “Exploring Slope and Y-Intercept”

Link: Geogebra: Matt Huttenlock’s “ Exploring Slope and Y- Intercept ” (Java)

Instructions: Please click on the link above. This interactive graph has a wealth of information to it. It allows you to change slope and y-intercept with slide bars to see how other values and the graph change. For an explanation of some of the items seen, “A” is the y-intercept, “B” is the point on the line when x = 2, “a” is the standard form of the line, “f(x)” shows the slope-intercept form and “m” is the slope. Find the blue slope legend in the middle of the graph. Slide the circle left and right, notice how the line changes when the slope is 1 or -1.

Completing this exercise will take approximately 15 minutes.



1.2.3 Point-Slope Form of a LineInstruction: YouTube.com DoctorGenie’s:“Point Slope Form of a Linear Equation”

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Link: YouTube.com DoctorGenie’s: “ Point Slope Form of a Linear Equation ” (YouTube)

Instructions: Please click on the link above. This video (4:02) gives an excellent discussion on the point slope form of a line. The instructor discusses how to take the information given and write the equation in the correct form. This lesson is important not only for the form, but also for showing how to substitute values correctly in an equation by using color coding.




Activity: Geogebra: Geogebra YouTube Team’s “Point-Slope Form”

Link: Geogebra: Geogebra YouTube Team’s “ Point-Slope Form ” (Java)

Completing this activity will take approximately 15 minutes.

Instructions: Please click on the link above. This interactive graph allows you to change the slope using a slide bar or to move a point around to see how the point-slope form of a line changes, as well as the graph. It's a great way to link the form visually with the graph. Please do each of the six exercises underneath the graph.



1.2.4 Translating One Linear Form into AnotherInstruction: YouTube.com: Khan Academy’s “Slope”

Link: YouTube.com: Khan Academy’s “ Slope ” (YouTube)

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http://www.youtube.com/watch?v=hXP1Gv9IMBo






http://www.youtube.com/watch?v=XChGfekxRmE

http://www.youtube.com/watch?v=XChGfekxRmE

Instructions: Please click on the link above. This video (8:28) gives an excellent discussion on the slope of a line and how it does not matter which coordinates one uses first in the formula.




Instruction: YouTube.com: MrCaryMath:“Given Two Points Find the Standard Form Equation of a Line”

Link: YouTube.com: MrCaryMath:“ Given Two Points Find the Standard Form Equation of a Line ” (YouTube)

Instructions: Please click on the link above. This video (13:56) provides us the total picture from two points to point-slope form to standard form. This video is great not only from the perspective of changing forms of a line but, also in showing how to use equation properties to find any equivalent form. Combined with the techniques already learned, the student has a powerful set of algebraic tools that can be used in many ways.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-SSE.B.3 CCSS.Math.Content.HSF-IF.C.7 CCSS.Math.Content.HSF-IF.C.8


1.2.5 Graphing Lines1.2.5.1 Graphing by Finding Two Points on the Line

Instruction: YouTube.com: Mr T’s “Finding the Slope of a Line Through Two Points”

Link: YouTube.com: Mr T’s “ Finding the Slope of a Line Through Two Points ” (YouTube)

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http://www.youtube.com/watch?v=d74I2Zc0rcg

http://www.youtube.com/watch?v=d74I2Zc0rcg




http://www.youtube.com/watch?v=EADbewX7ASU

http://www.youtube.com/watch?v=EADbewX7ASU



Instructions: Please click on the link above. This video (2:49) provides us a way to find the slope of a line through two points and then graph the line. It's a second look at a method we've seen but it never hurts to see another instructor approach a topic.




1.2.5.2 Graphing by Using a Single Point and the SlopeInstruction: YouTube.com: Khan Academy’s “Graphs of Linear Equations”

Link: YouTube.com: Khan Academy’s “ Graphs of Linear Equations ” (YouTube)

Instructions: Please click on the link above. This video (13:09) looks at graphing from several methods, including point-slope form. This is one of the better and most comprehensive lectures on linear equations and reinforces several of the techniques we've already seen.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.B.3 CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.D.10 CCSS.Math.Content.HSA-REI.D.11


1.2.5.3 Graphing by Using x and y InterceptsInstruction: YouTube.com: Khan Academy’s “Graphing Using Intercepts”

Link: YouTube.com: Khan Academy’s “Graphing Using

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http://www.youtube.com/watch?v=6zixwWZ88tk





http://www.youtube.com/watch?v=rgvysb9emcQ

http://www.youtube.com/watch?v=rgvysb9emcQ



Intercepts” (YouTube)

Instructions: Please click on the link above. This video (12:18) looks at graphing lines by finding the x and y intercepts for the line. In addition to being an easy, yet important, method for graphing lines, it is also good practice in using substitutions to solve equations, a technique which is used often in algebra and calculus.


Standards Addressed (Common Core): CCSS.Math.Content.HSF-IF.B.4 CCSS.Math.Content.HSF-IF.C.7a


Activity: Geogebra: David Cox’s “Linear Equation Mixed Forms”

Link: Geogebra: David Cox’s “ Linear Equation Mixed Forms ” (Java)

Instructions: Please click on the link above. This page contains an interactive graph that allows you to plot points for equations of different forms. Once you have the x and y intercepts of the line plotted correctly, by dragging and dropping the two points they provide, the line appears. By clicking on the refresh icon in the top right corner of the graph, you can get a new equation to practice.




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http://www.corestandards.org/Math/Content/HSF/IF/C/7/a


http://www.youtube.com/watch?v=6zixwWZ88tk

Activity: Geogebra: Geogebra YouTube Team’s “Exploring Linear Equations”

Link: Geogebra: Geogebra YouTube Team’s “ Exploring Linear Equations ” (Java)

Instructions: Please click on the link above. This page contains an interactive graph that allows you look at the graphs of linear equations of different forms. You can also change the graph and equations by using the controls in the top left quadrant. By clicking on the refresh icon in the top right corner of the graph, you can reset the graph. This is a good way to learn how to match the equation with the visual graph.




1.3 Linear Inequalities1.3.1 Graphing Linear Inequalities by Using a Comparison

Instruction: YouTube.com: MathPractice2Learn: “The Basics of Graphing Linear Inequalities – Slope-Intercept Form”

Link: YouTube.com: MathPractice2Learn: “ The Basics of Graphing Linear Inequalities – Slope-Intercept Form ” (YouTube)

Instructions: Please click on the link above. This video (9:15) looks at graphing linear inequalities by using a comparison test using slope-intercept form. In addition to being an easy, yet important, method for graphing linear inequalities, it is an easy method to extend for higher power equations. By solving for y, we can make “above or below” comparisons of the line and shade the correct half-plane.


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http://www.youtube.com/watch?v=LBBAfqpqvbQ

http://www.youtube.com/watch?v=LBBAfqpqvbQ









Activity: Geogebra: Terry Lee Lindenmuth’s “Linear Inequalities N1v2 y<=,>= mx + b”

Link: Geogebra: Terry Lee Lindenmuth’s “ Linear Inequalities N1v2 y<=,>= mx + b ” (Java)

Instructions: Please click on the link above. This page contains an interactive graph that allows you to change the y-intercept and slope to adjust the line and change from <= to >= in order to affect the shading. This is good practice for using the comparison method of solving inequalities.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.D.12


1.3.2 Graphing Linear Inequalities by Using Test CoordinatesInstruction: Monterey Institute: “Solving and Graphing Inequalities in Two Variables”

Link: Monterey Institute: “ Solving and Graphing Inequalities in Two Variables ” (HTML)

Instructions: Please click on the link above. This page has a great description of a common mathematical method, which is the use of test coordinates to determine the solution spaces for inequalities. This is not only important for linear inequalities and systems of inequalities, but can be extended to use for any type of inequality.

Reading the article will take approximately 30 minutes.

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http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U05_L2_T1_text_final.html

http://www.montereyinstitute.org/courses/Algebra1/COURSE_TEXT_RESOURCE/U05_L2_T1_text_final.html








Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.C.5 CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.C.7 CCSS.Math.Content.HSA-REI.D.12


Did I Get This? Activity: Geogebra: Math Guru’s “Linear Inequality Solution”

Link: Geogebra: Math Guru’s “ Linear Inequality Solution ” (Java)

Instructions: Please click on the link above and follow the directions. This page contains an interactive graph that allows you move a point around in order to check regions of an inequality. You only need one test point in a region, and if any point in the region works, that's the area that gets shaded with the test-point method. Select the Check Answer box to check your answer. Click New Problem to get a new problem.

Completing the exercise will take approximately 15 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.C.7


1.4 Absolute Value Equations and Inequalities1.4.1 Solving Absolute Value Equations

Instruction: Khan Academy: Math: Algebra: “Absolute Value Equations”

Link: Khan Academy: Math: Algebra: “ Absolute Value Equations ” (YouTube)

Instructions: Please click on the link above. This video (1:47) introduces the student to the concept of absolute value equations. It uses the definition of absolute value with methods for solving linear equations to find the solutions for these equations. This video provides a simple example.

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https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/u02-l2-t2-we1-absolute-value-equations-avi








Watching the video will take approximately 5 minutes.

Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSF-IF.C.7b


Instruction: Khan Academy: Math: Algebra: “Absolute Value Equations Example 2”

Link: Khan Academy: Math: Algebra: “ Absolute Value Equations Example 2 ” (YouTube)

Instructions: Please click on the link above. This video (6:32) contains a more complex example than the first, but continues to use the same techniques as before. These videos are important in that they not only demonstrate how to solve absolute value equations; they demonstrate how mathematical definitions can be used in conjunction with equation properties to solve problems.




Activity: Geogebra: Terry Lee Lindenmuth’s “Absolute Value Function”

Link: Geogebra: Terry Lee Lindenmuth’s “ Absolute Value Function ” (Java)

Instructions: Please click on the link above. This page contains an interactive graph that allows you to change the function shifts (horizontal and vertical) and the graph scaling. This is not only great practice toward absolute value functions but toward understanding shifts and scaling for any basic functions.


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http://www.corestandards.org/Math/Content/HSF/IF/C/7/b


https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/absolute-value-equation-example-2

https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/absolute-value-equation-example-2





1.4.2 Solving Absolute Value InequalitiesInstruction: Khan Academy: Math: Algebra: “Absolute Value Inequalities Example 1”

Link: Khan Academy: Math: Algebra: “ Absolute Value Inequalities Example 1 ” (YouTube)

Instructions: Please click on the link above. This video (3:27) introduces the student to the concept of absolute value inequalities. It uses the definition of absolute value with methods for solving linear inequalities to find the solutions for these problems. This video provides a simple example.




Instruction: Khan Academy: Math: Algebra: “Absolute Value Inequalities Example 2”


Instructions: Please click on the link above. This video (7:30) continues the concept of absolute value inequalities. It uses the definition of absolute value with methods for solving linear inequalities to find the solutions for these problems. This video provides a more complex example.


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https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/absolute-inequalities-2

https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/absolute-inequalities-2



https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/v/absolute-value-inequalities-example-1






Instruction: Khan Academy: Math: Algebra: “Absolute Value Inequalities Example 3”


Instructions: Please click on the link above. This video (2:35) continues the concept of absolute value inequalities. This example is important for it demonstrates that not all equations have a solution. This is the first time we're introduced to the idea of checking the solution. That is, we see only one solution is possible, but that potential solution creates an untrue statement when substituted back for the variable in the original equation. As we move forward it is advisable to begin checking all answers.




Activity: Geogebra: Vitullo’s “Absolute Value Inequality Graphs”

Link: Geogebra: Vitullo’s “ Absolute Value Inequality Graphs ” (Java)

Instructions: Please click on the link above and follow the directions. This page contains an interactive graph that allows you to change several factors to affect the graph of absolute value inequalities. This is not only good practice toward these functions but the same techniques apply to any basic inequality.

Completing the exercise will take approximately 15 minutes to complete.

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Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSA-REI.D.12 CCSS.Math.Content.HSF-IF.C.7b


UNIT 2: LINEAR SYSTEMSTime Advisory: This unit should take you approximately 3.5 hours to complete.

Subunit 2.1: 25 minutes Subunit 2.2: 1 hour 40 minutes Subunit 2.3: 55 minutes Subunit 2.4: 55 minutes Subunit 2.5: 20 minutes Subunit 2.6: 3 hours 15 minutes

Sub-subunit 2.6.1: 3 hours 15 minutes

In Unit 2, you will move from simple linear equations and inequalities to begin exploring more complex systems of equations and inequalities. You will be introduced to various methods to solve these equations and inequalities, and will become familiar with linear programming.

Next, you will be introduced to matrices and methods of solving systems using matrix algebra. Matrices are rows and columns of numbers. They may be used, for example, to tally the party affiliations of a population for political purposes by listing registered voter affiliation by county. For us, the numbers will be the coefficients and constants from linear equations. The methods used to solve these systems are extremely helpful in later courses. You will complete the unit with an introduction to vectors, which are rows or columns from matrices. Learning how to use vectors is very useful in business and science. For example, vectors can be used to calculate how a plane can land safely while there are crosswinds.

Solving linear systems involves finding common points for two or more lines. These are shared solutions among the equations and the techniques learned have important roles to play in solving problems in business and science. For example, when do the costs of two cell phone plans intersect? Solving linear inequalities involves finding all common points among the relations.

Unit 2 ends with an introduction to systems of three equations (which you already know how to solve). The concepts used to solve one equation can be applied when solving three, and will help with scenarios where you are asked to solve for three separate variables within the same situation. Imagine that you have friends in Canada, England, and Japan and you want to figure out how much you paid to

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make phone calls to each of them last month. Calling different countries costs a different amount per minute and you talked to each friend for a different length of time. As a result, you will need to use the different variables (three different per-minute rates, three different call lengths) to calculate how much you spent on international calls.

Learning OutcomesUpon successful completion of this unit, the student will be able to:

Graph systems of two equations in two variables to find a solution to the system.

Use algebraic methods to solve systems of two equations. Use algebraic methods to solve systems of three equations. Graph and solve systems of linear inequalities. Use linear programming methods to solve systems of equations. Use matrix operations to simplify and solve systems of equations. Perform transformations on matrices. Find the determinant of a square matrix. Use the determinant of a square matrix to solve a system of equations via

Cramer’s Rule. Identify a given vector’s graph.

Standards Addressed (Common Core): CCSS.Math.Content.HSN-VM.C.6 CCSS.Math.Content.HSN-VM.C.7 CCSS.Math.Content.HSN-VM.C.8 CCSS.Math.Content.HSN-VM.C.12 CCSS.Math.Content.HSA-SSE.B.3 CCSS.Math.Content.HSA-CED.A.2 CCSS.Math.Content.HSA-CED.A.3 CCSS.Math.Content.HSA-REI.A.2 CCSS.Math.Content.HSA-REI.C.5 CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.C.7 CCSS.Math.Content.HSA-REI.D.10 CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSA-REI.D.12 CCSS.Math.Content.HSF-BF.A.1 CCSS.Math.Content.HSF-LE.B.5 CCSS.Math.Content.HSS-ID.C.7

2.1 Solving Linear Systems by GraphingInstruction: Khan Academy: Math: Algebra: “Solving Systems Graphically”

Link: Khan Academy: Math: Algebra: “ Solving Systems Graphically ” (YouTube)

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-graphically

http://www.corestandards.org/Math/Content/HSS/ID/C/7

http://www.corestandards.org/Math/Content/HSF/LE/B/5

http://www.corestandards.org/Math/Content/HSF/BF/A/1







http://www.corestandards.org/Math/Content/HSA/REI/A/2

http://www.corestandards.org/Math/Content/HSA/CED/A/3



http://www.corestandards.org/Math/Content/HSN/VM/C/12




Instructions: Please click on the link above. This video (2:22) discusses the process of solving systems of equations by graphing. Solving is a method for finding common solutions for two or more equations. Graphing is a great way to begin the topic of systems since it's a visual approach to such solutions, making the concept easy to see. However, when the common solution is not made up of integer coordinates, it becomes harder to find graphically.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.C.7 CCSS.Math.Content.HSA-REI.D.10 CCSS.Math.Content.HSA-REI.D.11


Instruction: Khan Academy: Math: Algebra: “Example 2: Graphically Solving Systems”

Link: Khan Academy: Math: Algebra: “ Example 2: Graphically Solving Systems ” (YouTube)

Instructions: Please click on the link above. This video (3:26) continues discussing the process of solving systems of equations by graphing. In this example, we see that not all systems have a solution. This is easier to see graphically than by algebra, and that is why we approach the topic with graphing first.




Activity: Geogebra: Assnusa’s “Systems of Linear Equations”

Link: Geogebra: Assnusa’s “ Systems of Linear Equations ” (Java)

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-by-graphing-2

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-by-graphing-2





Instructions: Please click on the link above. This page contains an interactive graph that allows you to change several factors to affect the graph of a system of two linear equations. You may use a slider system for both equations to change slope and y-intercepts. Follow the directions and answer the questions underneath the interactive graph.




2.2 Solving Linear Systems AlgebraicallyInstruction: Khan Academy: Math: Algebra: “Example 1: Solving Systems by Elimination”

Link: Khan Academy: Math: Algebra: “ Example 1: Solving Systems by Elimination ” (YouTube)

Instructions: Please click on the link above. This video (5:51) contains an example of solving systems by elimination. This is an important method that can be extended to more than two equations. Pay close attention to the methods used because they serve as a good introduction to matrix methods that we'll see later.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.C.6


Instruction: Khan Academy: Math: Algebra: “Example 2: Solving Systems by Elimination”


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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-by-elimination-2



https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-by-elimination

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/v/solving-systems-by-elimination





Instructions: Please click on the link above. This video (3:45) contains a more complex example of solving systems by elimination. This is an important method that can be extended to more than two equations. Note that the methods involve applying equation properties in new ways.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.A.2 CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.C.7


Instruction: Khan Academy: Math: Algebra: “Example 3: Solving Systems by Elimination”


Instructions: Please click on the link above. This video (4:58) contains a discussion on infinite common solutions. Thus we see that systems may have only one common point, but may also have all points in common.




Did I Get This? Activity: Khan Academy: Math: Algebra: “Systems of Equations with Elimination”

Link: Khan Academy: Math: Algebra: “Systems of Equations with Elimination” (Java)

Instructions: Please click on the link above. This webpage contains practice problems for solving systems of equations via elimination. Each

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/e/systems_of_equations_with_elimination

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/e/systems_of_equations_with_elimination









problem has a set of step by step hints for showing how the problem is solved.

Completing the problem sets will take approximately 15 minutes.



Instruction: Khan Academy: Math: Algebra: “Example 1: Solving Systems by Substitution”

Link: Khan Academy: Math: Algebra: “Example 1: Solving Systems by Substitution” (YouTube)

Instructions: Please click on the link above. This video (3:44) discusses a method for solving systems of equations via substitution. This is another algebraic approach to solving systems and one we'll see again in other units. This particular example shows that two parallel lines have no common solution. Now we've seen the third type of system. We've seen the one common point system (two lines intersecting at one point), the infinite common points system (two lines that are actually the same) and now the no common points system (parallel lines).






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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-with-substitution/v/solving-systems-by-substitution-2










Instructions: Please click on the link above. This video (4:06) continues the approach of solving systems via substitution. This example shows a simple problem that is fairly easy to solve.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.A.2 CCSS.Math.Content.HSA-REI.C.6 CCSS.Math.Content.HSA-REI.C.7 CCSS.Math.Content.HSA-REI.D.11




Instructions: Please click on the link above. This video (6:04) continues the approach of solving systems via substitution. This example shows a simple problem that is fairly easy to solve and then shows how to graph the system.




Did I Get This? Activity: Khan Academy: Math: Algebra: “Systems of Equations with Substitution”

Link: Khan Academy: Math: Algebra: “Systems of Equations with Substitution” (Java)

Instructions: Please click on the link above. This webpage contains

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/e/systems_of_equations_with_substitution

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/systems-of-eq-overview/e/systems_of_equations_with_substitution











practice problems for solving systems of equations via substitution. Each problem has a set of step by step hints for showing how the problem is solved.




Did I Get This? Activity: Geogebra: Mathguru’s “Solving Systems of Equations by Substitution”

Link: Geogebra: Mathguru’s “Solving Systems of Equations by Substitution” (Java)

Instructions: Please click on the link above. This page contains self-quizzing questions. The student is able to check solutions and try new systems. Follow the directions to see how many correct answers you can get in a row.




2.3 Systems of Linear Inequalities

Instruction: Khan Academy: Math: Algebra: “Graphical Systems of Inequalities”

Link: Khan Academy: Math: Algebra: “Graphical Systems of Inequalities” (YouTube)

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/system-of-inequalities/v/graphical-system-of-inequalities











Instructions: Please click on the link above. This video (5:36) shows the best method for solving systems of linear inequalities, which is by graphing. Note that the instructor in the lesson uses both the comparison and the coordinate test method for each inequality. Either can be used. Both need not be.




Instruction: Khan Academy: Math: Algebra: “Graphing Systems of Inequalities 2”

Link: Khan Academy: Math: Algebra: “Graphing Systems of Inequalities 2” (YouTube)

Instructions: Please click on the link above. This video (3:24) demonstrates graphing a solution for multiple constraints. As we will see, this does not make the problem more difficult. Comparison or coordinate tests can be used in this case as well. This particular case shows an example with no common solutions.

Watching the videos will take approximately 5 minutes.



Instruction: Khan Academy: Math: Algebra: “Graphing Systems of Inequalities”

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/system-of-inequalities/v/graphing-systems-of-inequalities-2

https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/system-of-inequalities/v/graphing-systems-of-inequalities-2





Link: Khan Academy: Math: Algebra: “Graphing Systems of Inequalities” (YouTube)

Instructions: Please click on the link above. This video (9:45) demonstrates a system of inequalities from a word problem. Note carefully how the instructor constructs the inequalities.




Activity: Geogebra: Terry Lee Lindenmuth’s “Enter a System of Two Linear Inequalities”

Link: Geogebra: Terry Lee Lindenmuth’s “Enter a System of Two Linear Inequalities” (Java)

Instructions: Please click on the link above. This page contains an interactive graph that can be modified by changing the inequalities or entering new ones. This is great practice for seeing the visual link between the systems and their graphs.




Interactive Lab: Geogebra: Jparete’s “Linear Equations and Inequalities – Systems”

Link: Geogebra: Jparete’s “Linear Equations and Inequalities – Systems”

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/system-of-inequalities/v/u06-l3-t1-we3-graphing-systems-of-inequalities

(Java)

Instructions: Please click on the link above and follow the instructions. This page contains a highly interactive graph that can be modified by changing a variety of values and signs. Follow the instructions in the top left quadrant of the graph. There are many different inputs and possibilities to consider. This is great practice not only for systems of inequalities and equations, but also for linear equations. Also, note that there are three points that may be dragged to different areas to test for solutions.




2.4Linear Programming

Web Media: METAL MathProject’s: Linear Programming: “Introduction to Linear Programming”

Link: METAL MathProject’s: Linear Programming: “Introduction to Linear Programming” (YouTube)

Instructions: Please click on the link above. This video (10:56) introduces the student to linear programming, which is a technique often used to maximize or minimize some value. For instance, in this example, the instructor shows how two businesses maximize profits. The method begins with techniques we've seen for graphing linear inequalities, but instead of considering the shaded areas of a partial plane, linear programming looks at the vertices formed by the polygon created when the boundary lines are graphed.


Standards Addressed (Common Core): CCSS.Math.Content.HSS-ID.C.7 CCSS.Math.Content.HSF-BF.A.1 CCSS.Math.Content.HSF-BF.A.1c CCSS.Math.Content.HSF-LE.B.5

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http://www.corestandards.org/Math/Content/HSF/BF/A/1/c



http://www.youtube.com/watch?v=1CnFjNOewEU&list=PL1CBB58096D12F354

http://www.youtube.com/watch?v=1CnFjNOewEU&list=PL1CBB58096D12F354





CCSS.Math.Content.HSA-SSE.B.3 CCSS.Math.Content.HSA-CED.A.2 CCSS.Math.Content.HSA-CED.A.3 CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSA-REI.D.12


Web Media: METAL MathProject’s: Linear Programming: “Linear Programming-Final 2 Steps”

Link: METAL MathProject’s: Linear Programming: “Linear Programming-Final 2 Steps” (YouTube)

Instructions: Please click on the link above. This video (9:15) concludes the example given in the previous video. The final two steps are using the vertices and checking the final answer to determine whether or not it makes sense (i.e., the recommended checking of the final answer). This is a method commonly used in business and industry for maximizing output or profits and/or minimizing waste or costs.


Standards Addressed (Common Core): CCSS.Math.Content.HSS-ID.C.7 CCSS.Math.Content.HSF-BF.A.1 CCSS.Math.Content.HSF-BF.A.1c CCSS.Math.Content.HSF-LE.B.5 CCSS.Math.Content.HSA-SSE.B.3 CCSS.Math.Content.HSA-CED.A.2 CCSS.Math.Content.HSA-CED.A.3 CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSA-REI.D.12


Activity: Geogebra: Terry Lee Lindenmuth’s “Linear Programming Example: Picturing Pictures”

Link: Geogebra: Terry Lee Lindenmuth’s “Linear Programming Example: Picturing Pictures” (Java)

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http://www.youtube.com/watch?v=qiowP4a7Y_k&list=PL1CBB58096D12F354

http://www.youtube.com/watch?v=qiowP4a7Y_k&list=PL1CBB58096D12F354






Instructions: Please click on the link above. This page contains a linear programming example that shows, graphically, how linear programming works. It is not extensive, but after the videos this may be a good light way to view feasibility and maximization. The program does the work for you, but it would be good practice to use the skills you've already learned to find the equations of the restraints.


Standards Addressed (Common Core): CCSS.Math.Content.HSS-ID.C.7 CCSS.Math.Content.HSF-BF.A.1c CCSS.Math.Content.HSF-LE.B.5 CCSS.Math.Content.HSA-SSE.B.3 CCSS.Math.Content.HSA-CED.A.2 CCSS.Math.Content.HSA-CED.A.3 CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSA-REI.D.12


Activity: Geogebra: LibraryGhost’s “Linear Programming: Problem 1, Car vs. Moped”

Link: Geogebra: LibraryGhost’s “Linear Programming: Problem 1, Car vs. Moped” (Java)

Instructions: Please click on the link above and follow the directions above the graph. This page contains a relatively simple interactive graph that allows the student to move a point within the region of feasibility to find a optimal solution. Again, this doesn't require any computation skills on the part of the student but is good practice nonetheless, as it continues to reinforce the idea that the optimal point is always at a vertex on the region of feasibility.


Standards Addressed (Common Core): CCSS.Math.Content.HSS-ID.C.7 CCSS.Math.Content.HSF-LE.B.5 CCSS.Math.Content.HSA-CED.A.3 CCSS.Math.Content.HSA-REI.D.11

Terms of Use: Please respect the copyright and terms of use displayed on

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the webpage above.

2.5Systems of Equations in Three Variables

Instruction: Khan Academy: Math: Algebra: “Systems of Three Variables”

Link: Khan Academy: Math: Algebra: “Systems of Three Variables” (YouTube)

Instructions: Please click on the link above. This video (8:23) introduces the student to systems of three variables. Techniques for solving are extensions of techniques already learned, with additional steps. There are many real world problems involving many variables that solve with similar methods, but once the basics are mastered any problem of this type can be solved.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.C.5 CCSS.Math.Content.HSA-REI.D.10 CCSS.Math.Content.HSA-REI.D.11


Instruction: Khan Academy: Math: Algebra: “Systems of Three Variables 2”

Link: Khan Academy: Math: Algebra: “Systems of Three Variables 2” (YouTube)

Instructions: Please click on the link above. This video (7:00) contains another example of a system with three variables, with coefficients larger than one. It requires a few more steps of elimination to solve, but the technique only requires patience.


Standards Addressed (Common Core): CCSS.Math.Content.HSA-REI.C.5 CCSS.Math.Content.HSA-REI.D.10 CCSS.Math.Content.HSA-REI.D.11

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https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fancier-systems/v/systems-of-three-variables-2




https://www.khanacademy.org/math/algebra/systems-of-eq-and-ineq/fancier-systems/v/systems-of-three-variables


2.6 Introduction to Matrices2.6.1 Definition of a Matrix and Matrix Operations

Instruction: Khan Academy: Math: Algebra: “Introduction to Matrices”

Link: Khan Academy: Math: Algebra: “Introduction to Matrices” (YouTube)

Instructions: Please click on the link above. This video (11:52) contains a very good introduction to matrices, including notations and basic operations.


Standards Addressed (Common Core): CCSS.Math.Content.HSN-VM.C.6 CCSS.Math.Content.HSN-VM.C.12 CCSS.Math.Content.HSN-VM.C.8


Did I Get This? Activity: Khan Academy: Math: Algebra: “Matrix Dimensions”

Link: Khan Academy: Math: Algebra: “Matrix Dimensions” (Java)

Instructions: Please click on the link above. This page contains a set of practice questions designed to help the student master the concept of matrix dimension. The practice is self-paced and the student can practice with as many questions as he or she wishes. And if the student isn't sure, he or she can get hints step by step.



Terms of Use: Please respect the copyright and terms of use

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https://www.khanacademy.org/math/algebra/algebra-matrices/Basic_matrix_operations/e/matrix_dimensions




https://www.khanacademy.org/math/algebra/algebra-matrices/Basic_matrix_operations/v/introduction-to-matrices

displayed on the webpage above.

Did I Get This? Activity: Khan Academy: Math: Algebra: “Scalar Matrix Multiplication”

Link: Khan Academy: Math: Algebra: “Scalar Matrix Multiplication” (Java)

Instructions: Please click on the link above. This page contains a set of practice questions designed to help the student master the concept of scalar matrix multiplication. The practice is self-paced and the student can practice with as many questions as he or she wishes. And if the student isn't sure, he or she can get hints step by step.




Did I Get This? Activity: Khan Academy: Math: Algebra: “Matrix Addition and Subtraction”

Link: Khan Academy: Math: Algebra: “Matrix Addition and Subtraction” (Java)

Instructions: Please click on the link above. This page contains a set of practice questions designed to help the student master the concept of matrix addition and subtraction. The practice is self-paced and the student can practice with as many questions as he or she wishes. And if the student isn't sure, he or she can get hints step by step.


Standards Addressed (Common Core): CCSS.Math.Content.HSN-VM.C.8

Terms of Use: Please respect the copyright and terms of use

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https://www.khanacademy.org/math/algebra/algebra-matrices/Basic_matrix_operations/e/matrix_addition_and_subtraction

https://www.khanacademy.org/math/algebra/algebra-matrices/Basic_matrix_operations/e/matrix_addition_and_subtraction




https://www.khanacademy.org/math/algebra/algebra-matrices/Basic_matrix_operations/e/scalar_matrix_multiplication

displayed on the webpage above.

Checkpoint: CK-12: Eve Rawley, Anne Gloag, and Andrew Gloag’s “Comparing Methods for Solving Linear Systems”

Link: CK-12: Eve Rawley, Anne Gloag, and Andrew Gloag’s “Comparing Methods for Solving Linear Systems” (HTML)

Instructions: Please click on the link above and scroll to the end of the content until you see the “Practice” heading. Complete practice problems 1-10. If you are not sure how to do any of the problems, review the content before the practice problems. The guidance section may be particularly helpful.

Completing these problems will take approximately 45 minutes.



Checkpoint: Rory Adam’s “Linear Programming”

Link: Rory Adam’s “Linear Programming” (HTML)

Instructions: Please click on the link above. Read the Introduction and Terminology sections. Scroll down and work through the Example of a Problem section and exercises. Finally, complete the End of Chapter Exercises.


Standards Addressed (Common Core): CCSS.Math.Content.HSS-ID.C.7 CCSS.Math.Content.HSF-BF.A.1c CCSS.Math.Content.HSF-LE.B.5 CCSS.Math.Content.HSA-SSE.B.3 CCSS.Math.Content.HSA-CED.A.2 CCSS.Math.Content.HSA-CED.A.3 CCSS.Math.Content.HSA-REI.D.11 CCSS.Math.Content.HSA-REI.D.12

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http://cnx.org/content/m30862/latest/





http://www.ck12.org/concept/Comparing-Methods-for-Solving-Linear-Systems/?ref=%2Fconcept%2FComparing-Methods-for-Solving-Linear-Systems%2F


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Web viewThis lesson is important not only ... when do the costs of two cell phone plans ... linear...

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Transcript of Web viewThis lesson is important not only ... when do the costs of two cell phone plans ... linear...