Systems of Linear Equations and Inequalities

The first week and a half we reviewed linear functions as tools to model the relationship between two quantities with an emphasis on distance vs. time relationships.  This week we are moving into a brief review systems of linear equations, which will culminate in exploration of linear programming.  The picture at left is an example of what we will eventually will be building, interpreting, and using to solve real-world problems.

Please find a video on systems of equations in three variables below as well as one on systems of inequalities immediately below.  I encourage you to take a moment and critically view them, pausing and re-watching when needed.The documents for our first unit are embedded beneath that, including the next two homework assignments.  

Systems of Three Variables:

Graphical Systems Application Problem:

Linear Programming Example:

Documents from our first unit:

More Transformations and Systems of Inequalities

 

Transformations will continue in Precalculus with even more difficult problems coming your way!  I encourage you to review your notes nightly until you can, without hesitation, describe a single transformation and the order of transformations.  These are simple tasks if you take it one transformation at a time.  At the end of the week will refocus on our attention on common characteristics of functions such as symmetry, end behavior, etc.  The homework is posted with odd solutions to help you navigate this, at times challenging, homework assignment. Beneath the table is a link to a new set of videos/animations you may find more engaging or interesting if you need to review the material.

Pearson Powerpoints	Text Examplars:	Classwork:	Homework (from University of Houston)	Reference Sheet (from fortress math)

Precalculus animations/review from Cengage Learning:
Transformations of Graphs and Functions
Nonrigid Transformations of Functions
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Algebra 2 will apply their problem solving abilities to three-variable systems as well as revisit some of the classic simultaneous equation problems.  Given that I will be working outside of the building on Wednesday and  we have no school on Thursday, this will be a short work with a great deal of perfecting of our basic skills.  We will return to systems when we add polynomial functions to the mix so master the skills now!

Please find some relevant resources below:

Solving by Graphing Powerpoint	Homework	Reference Material	Reference Sheet 1	Reference Sheet 2

Problems for homework with one student's action plan:

Brightstorm Videos:
These videos are aligned to the sections from your textbook, so as you are completing homework from your textbook you can easily locate the most relevant videos for you!

Also check out the videos recently posted by LearnZillion. They provide a nice basic overview of the underlying skills.

Transformations of Functions and Applications of Systems of Equations

Algebra 2 will be moving into applications of systems of equations that will range from profit, cost, and revenue, to mixtures, to numerical relationships, to planes, trains, and automobiles. It should be a good solid honing of your equation writing skills as well as your ability to manipulate equations. I look forward to lively debate, self-reflection, and solid revisions of you personal approaches as we go through the problem-solving process together. Please find some relevant resources below:

Solving by Graphing Powerpoint	Homework (from the University of Houston)	Reference Material	Reference Sheet 1	Reference Sheet 2

Brightstorm Videos:
These videos are aligned to the sections from your textbook, so as you are completing homework from your textbook you can easily locate the most relevant videos for you! _______________________________________________________________________
Please find the upcoming material for Precalculus the next week or so!
We have also started reading short articles at the beginning of class and hope that connecting math to the everyday newsfeed has been interesting and revealing. Here is what we have been reading and will read!
CBO Blog-Trends in the Distribution of Income
UPI News-7 billion people is a 'serious challenge'

PHYSORG-Fighting violent gang crime with math
Washington Post: Want to earn more money? Study STEM.
Google Blog: The bright side of sitting in traffic: Crowdsourcing road congestion data

Our work on transformations thus far!
Gallery Walk Conclusions:


Our formalizations of our gallery walk:

Pearson Powerpoints	Text Examplars:	Classwork:	Homework	Reference Material

Parent function "analysis" files with template (modified from Pete Falzone)



Videos on Transformations: 
Brightstorm Videos

Graphs of Functions and Systems of Equations

Precalculus will be moving beyond their introduction to functions and function notation from Algebra II and into a in-depth development of transformations. We will be applying transformations (translations, reflections, dilations, etc.) to many of the parent functions included in the image above (we will save trigonometry for the Spring!). Understanding transformations is a crucial link between the algebraic form of a function and its graphical behavior. We will spend a considerable amount of time exploring this topic and working toward mastery. You all will continue to complete the lengthy linear programming problem set and for that reason I have a great deal of information to post--I will post the homework, reference materials, and a YouTube video that goes over the basics of linear programming from the previous week on linear programming as well as the upcoming information on transformations! Enjoy.

Text Examplars:	Homework	Reference Material	Wolfram Demonstration

Upcoming Material for the next two weeks.

Text Examplars:	Homework	Reference Material

Videos on Transformations: 
Brightstorm Videos

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Algebra 2 will be moving into systems of equations through the lens of coordinate geometry.  Below is a student work example of some of the resources I will be using and what we will be exploring.  The purpose is for students to have a concrete link between their algebraic worlds and geometric worlds as well as to ground the solving of systems in the graphical world and slowly move to more abstract manipulation.

Check out this 3D Lathe that will help you visualize the creation of 3D geometric figures from the two-dimensions of the coordinate plane.
http://www.fi.uu.nl/toepassingen/00182/toepassing_wisweb.en.html

Homework                   (from the University of Houston)	Reference Material	Reference Sheet 1	Reference Sheet 2

Brightstorm Videos:
These videos are aligned to the sections from your textbook, so as you are completing homework from your textbook you can easily locate the most relevant videos for you!

Moving Toward Linear Programming and Simultaneous Equations

MIT SEED applications are now being accepted!  All interested students should come see me.  An amazing opportunity and one that should not be passed up!





Precalculus will be completing their application of inverse matrices to solve "traditional" and "non-traditional" three variable system word problems and then move onto an eight day exploration of linear programming.  The image to the left shows a system of inequalities that represents the constraints on two variables.  The region that is darkest represents all the possible, feasible, values that "work" given the five different constraints.  The vertices of the polygon of the "feasible region" give the maximum and minimum values given the constraints.  We will primarily use these maximum and minimum values to solve business related problems that require us to determine what will maximize profit and/or minimize cost.



Here is an example problem from the Transition Mathematics Project:

Your great aunt left you a trust fund of $18,000 that you may now have to invest with some restrictions.  You have three different funds from which to choose.  The municipal bond fund has a 5.26% return, U.S. Bank has certificate of deposits (CDs) with a 2.25% return and the high-risk fund has an expected (but not guaranteed) return of 9%.  To minimize risk, you decide to invest a maximum of $3,000 in the high-risk account.  A stipulation of the trust indicates that you must invest at least three times as much in the bank CDs as in the municipal bonds.  Assuming the year-end returns are as expected, what are your best investment choices to maximize your return?  Be sure to state how much you will invest in each fund. 

You will be able to solve this!

Homework is due in two weeks!  A great deal of graphing and interpreting is involved so work steady and check-in with me regularly.Here is the link to the chapter from which the problem set came.
Please find the slides related linear programming here as well and some exemplar problems to help guide you on your way.

Slides:	Text Examplars:	Homework

Video resources:
Brightstorm Videos on solving and graphing inequalities
Khan Academy on graphing inequalities
Youtube Video 1 on Linear Programming
Youtube Video 2 on Linear Programming
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Algebra II will quickly cover lines and move directly into classic system of equations problems.  The aim here will be twofold:  review of all things linear and translation of word problems.  We will continue to keep midpoint and distance formulas as well as exponents and radicals in play as much as possible by reviewing with Tarsia triangles, hexagons, and dominoes (see image of completed Tarsia triangle at left).

Slides:	Homework (from University of Houston):	Linear Equations Reference Sheet

Brightstorm Videos:
These videos are aligned to the sections from your textbook, so as you are completing homework from your textbook you can easily locate the most relevant videos for you!

Solving Systems with Matrices and All Things Linear

     From Pearson Education

Algebra II will be putting algebra basics behind them and moving on to linear equations with a coordinate geometry twist and an introduction to the concept function.  The goal of this week is to recall all you have learned about linear equations and really push your understanding of linearity to the limit.  Please find the Pearson PowerPoint slides below, a collection of videos on lines, the homework, and a linear equations reference sheet!

Slides:	Homework (from University of Houston):	Linear Equations Reference Sheet

Brightstorm Videos:
These videos are aligned to the sections from your textbook, so as you are completing homework from your textbook you can easily locate the most relevant videos for you!
_______________________________________________________________________________ Precalculus will continue to develop the properties of matrices and apply them to solving systems of equations. We will complete the final investigation of this unit on mixtures and solidify our ability to organize and manipulate information using matrices and apply them in problem-solving contexts.

Slides:	Text Examplars:	TI-83 Guide (see ch. 10-pp.167-182)

Homework: Handout from Finite Mathematics and Calculus with Applications Khan Academy Video:
Matrices to solve a system of equations


Brightstorm Video:
Solving linear systems using matrices  (this is a link to four videos that start with previous topics and build to the final topic of solving systems with matrices)

Solving Systems with Matrices and Factoring Expressions

                                         Factoring Flowchart Image Link




Precalculus will begin applying matrices to the solution of systems of equations.  It is an exciting week!  Please find the slides for the upcoming week below (starting with ppt5), a keystroke guide for matrices, as well as the homework.  We will have another investigation from the College Board's Springboard materials.  We completed "Gems in the Matrix" and "How Much Wood Do You Need," and will end with an investigation of mixtures.  Continue to focus on argument and explaining how matrices can help us manage and manipulate large sets of information.

Slides:	Text Examplars:	TI-83 Guide (see ch. 10-pp.167-182)

Khan Academy Video:
Matrices to solve a system of equations


Brightstorm Video:
Solving linear systems using matrices  (this is a link to four videos that start with previous topics and build to the final topic of solving systems with matrices)


Homework:
pp. 103-104 #27-52, pp. 119-122 #1-48, pp. 124-125 #1-10
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Algebra II will hone their factoring skills and will be introduced to the basics of rational expressions.  After next week algebraic basics is over and we will be moving onto linear functions and piecewise functions.  Please find the notes from the week below (factoring is ppt2), factoring reference sheets, and the homework.

Slides:	Factoring Reference Sheet	Factoring Flowchart

Brightstorm Videos:
These videos are aligned to the sections from your textbook, so as you are completing homework from your textbook you can easily locate the most relevant videos for you!


Homework:

Handouts taken from XYZ Custom+.  Completes the multiples of three, please.