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:

Harvard Medical School Opportunity, More Linear Programming and Linear Functions

HPREP is a free 9-week program that takes place at the Harvard Medical School campus on Longwood Ave, Boston, on Saturday mornings from 10:00AM to 2:00PM. The program will begin on November 19th, 2011 and end in early February 2012. The application, along with more information, can be found at http://sites.google.com/site/hmshprep/application-deadlines. The application is also attached to this email. Applications are due October 19, 2011 by email to HPREP@hms.harvard.edu.

From http://news.discovery.com/space/nobel-prize-physics-111004.html

The pictures above represent a few images from the work of Saul Perlmutter, Brian P. Schmidt, and Adam G. Riess, the most recent recipients of the Nobel Prize in Physics. If you are interested in their work on the accelerating universe, here is a link to an approachable summary on the topic.  Mathematics is at the heart of their discovery and reiterates the central role of math in understanding our world.


 We have yet another short week ahead of us, but I hope that we can really engage the material and deepen our understanding of mathematics.

Algebra II will be examining linear equations within the context of functions and addressing any and all misconceptions about linear relationships.  We will spend these four days solidifying linear equations as a basis for a brief but rigorous investigation of systems of equations as well as the basis of our exploration of polynomials.
Homework is due in two weeks, October 21st.

Slides:	Classwork (from eMathInstruction):	Homework (from XYZ Custom Plus):	Linear Equations Reference Sheet

________________________________________________________________________________________ Precalculus is zeroing in on linear programming. We started our investigation with a modified version of the National Council for Teacher's of Mathematics "Dirt Bike Dilemma" and will continue to explore it over the next six days. Homework is due in two more Fridays!  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 to linear programming here as well and some exemplar problems to help guide you on your way.

Slides:	Text Examplars:	Homework	Reference Material

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
________________________________________________________________________________________ Remember, you all have ongoing Khan Academy assignments with the first installment due November 1st, and the ability to practice and get immediate feedback at TenMarks. If you have forgotten your password for TenMarks just e-mail me.

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
_______________________________________________________________________

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!