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!

Matrices and more algebra basics

Precalculus will venture from algebraic manipulation of three variable linear systems using "elimination" and "substitution" and venture into the properties, characteristics, and application of matrices to multivariable linear systems. This adventure will be facilitated by a good dose of reasoning and technology. The basic key strokes associated with entering and solving matrices can be found here. We will go over instructions for this in class in greater detail. Please find both the PowerPoint slide shows for the coming week as well as relevant fact sheets and homework.

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

Brightstorm Videos:
Systems of Linear Equations and Matrices


Homework:

pp. 76-77 #8-22; pp. 82-85, #1-54; pp. 102, #14-26

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Algebra II will continue their review of basic algebra concepts by steaming on into the concept of factoring. We will loop back to concepts of number, especially as it pertains to exponents and radicals, and apply these concepts to sums and differences of terms as they emerge in polynomials. The image above is the factorization of a fifth degree polynomial, which shows that the expression has three distinct factors with one factor having a multiplicity of two. The irreducible factorization of the polynomial will not be covered in this basics review, but we will emerge in our study of polynomials over the coming months. Please find the PowerPoint slides as well as reference sheets and homework below:

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:

pp. 260-261 #1-63; pp.242-243 #1-45; pp. 247-248#1-59.

Graphical Solutions and Algebra Basics

In Precalculus we will finish our brief overview of lines and modeling and move onto a conceptual analysis of solutions in all of their many forms.  As shown in the pictures above we will be leaning on technology to conceptualize solutions to equations primarily in their tabular and graphical forms.

We will be finishing some of the remaining PowerPoint slides* from last week and moving onto a new set.  Please find both of them below (start with the tab ppt1 for both) as well as two reference sheets on systems of linear equations:

Video Resources at Brightstorm:
Graphs of Linear Equations
Systems of Equations

Exemplars from Textbook (start with Exemplar 1)
Homework (due September 9, 2011): 
pp. 41-44, #1-19 evens; pp. 54-56, #1-22; pp. 70-72, #1-45. 

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In Algebra II we will continue with our algebra basic review* and move onto a review of factoring, rational expressions, and radicals. I have included the set of powerpoints below as well as resources on the "rules" around these different basic algebra concepts. The algebra cheat sheet is from Paul's Online Math Notes.

Video Resources at Brightstorm:
Factoring Methods
Rational Exponents
Exponent Rules
More on Exponent Rules

Homework (due September 9, 2011): 
 pp. 226-228, #1-65 all; pp. 37-39, #1-58 all

 *Powerpoints are modified from 2007 Pearson text.