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.

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.