AFF Factoring to Find Zeros of Polynomials

Week of 12/22/14

Last week we extended our factoring prowess to analyze more sophisticated polynomial functions.

For example, we know that if we analyze p(x) below, we should first factor it (AFF!)  

In factored form, it will be clear that the first graph shown below does not match the given function. But the factored form will tell us the zeros we should expect to see.

In addition, last week we discussed the concepts of multiplicity.  We explored different graphs of polynomials that had repeated binomial factors linear factors raised to a power.  We can now analyze more complex behavior by inspecting the types of zeros (crossing or tangent).

We will discuss more graphing implications this week and after the holiday break.

 After the break we can also apply synthetic division as a solving tool.

Leftover goal which will be our focus this week

• Be able to make sense of word problems that lead to polynomial functions.         Then, persevere in solving them using factoring or other techniques.

Week of December 8th: Revisiting Sade and extending polynomial understanding to solve problems

General  Last week we improved our understanding of graphing polynomial functions and discussed key features of this function family including relative and absolute extrema.  In addition, we learned how to predict the end behavior of their graphs based on the degree and lead coefficient; and the relationship between the degree and number of turns we see on a polynomial graph.  This week we will extend what we learned about quadratic function solving techniques to solve problems involving polynomial functions.    
  

3 Key Learning Goals for the week

• Become and even smoother operator (factorer).  You will be asked to self-assess/rate your current understanding of factoring (GCF, DOT's, Factoring Completely, etc.) and apply similar techniques to rewrite polynomial expressions in a different form to solve problems. 
• Be able to make sense of word problems that lead to polynomial functions.  The persevere in solving them using factoring or other techniques.
• Address some misconceptions that showed up in the last two online assessments. Specifically,  there were some errors related to understanding exponent rules and writing expressions to represent volume algebraically.  These previously learned skills from other classes could be obstacles to doing well with our polynomial modeling and problem solving

Self Regulated Learning

My last post mentioned some of the characteristics of being a self-regulated learner.  One was that a self regulated learner identifies areas of strengths and weaknesses.  My teacher perspective is that graphing is, in general, a strength for our class.  Especially, problems that involve (FIF) interpreting functions, zeros, rates of change, domain and range etc.  However, the mixed tenmarks assessment results indicated that some of the problems reflecting the building function standard (FBF3), mainly transformations, were not as strong.  When I collect your paper homework next class be sure you can communicate any misconceptions related to transformation.  Previous paper pencil quizzes showed pretty strong understanding.  

Also, I need you to know how well you know your rules for exponents and can set up word problems.  If this is not a strength, there are some strategies and follow up activities you can do to improve your understanding.   

And of course, if you are not a smooth operator you need to improve your factoring.you

Self-Regulated Learning and tenmarks

Steppen into Trig is back up and running!  If you are currently in one of my Trig/Precalc. classes you are expected to join the blog (follow my posts).  In the past, I have updated it once each weekend so your responsibility is to mainly check in once each Sunday night to see what is up and coming for the week.  However, we may go further if we see opportunities for student/teacher or better yet student/student communication (about what we are learning in our class!)

You have done and seen a lot of www.tenmarks.com tracks and filled out your own tracking sheets this year.  This quality site allows to you to practice quality problems that can't really be created on a worksheet or copied from a textbook.  That is one great benefit, but I also want you to have and take more ownership in tracking what you know and need to know.   Tenmarks does a nice job of challenging you and giving you that "one more chance" to make corrections.  Those scores provide a consistent way for you to track where you are, and make decisions about what to do next.

The extra work I have put into familiarizing myself with tenmarks and giving your tracking sheets is to add one more tool to your toolbox to help you become a self-regulated learner.  Self-regulated learners take ownership in process of learning.

Self-Regulated Learners

·         Engage in self-observation (monitoring one’s activities), self-judgment (evaluation of one’s performance), and self-reactions (reactions to performance outcomes)

·         Identify academic strengths and weaknesses

·         Attribute their academic successes or failures to factors within their control (e.g. effort expended on a task, effective use of strategies)

  

     Next Steps:  In the upcoming Sunday night post I will talk about self-regulation and how it relates to the rubrics we have been using in class and give a good check-in/preview about how we are doing with Polynomials.

Chapter 7 Polynomial Box Problems Reflection

Need Extra Help or Computer Access for Tenmarks?
I will be in the computer lab room 108 for tutoring help every Tuesday and Thursday until 3 pm if you need help, want to work with someone else, or just use the computer to catch up on tenmarks.  The final exam is just around the corner.

Homework Consistency
About 1/4 of our class has completed all tenmarks assessments and had complete enough homework that I can tell they know the Fundamental Theorem of Algebra (7.4 complex coefficient problems).  But most of you have not.  In addition, there are review problems sprinkled in the homework, like Tangent Graphs, Law of Sines/Cosines, etc.  

Those of us not completing accurate graphs with labeled key features and step-by-step work for old material will not get full HW credit.  More importantly, you are not practicing old material to be ready for the final.

In other words, most of you need to do the most recent HWs and make up any old tenmarks assignments by Tuesday, including those "old" problems.

Analysis of where we are at in Chapter 7
Below is a chart I made for a workshop I am going to tomorrow but it is about our class, so take a look.  I reviewed the recent quizzes and HWs and scored each standard out of 4.  To get more 4's for synthetic division, I need to see how you do with the "complex" coefficient problems.   Word problem box problems and graphing generally looks good.  Numbers would be higher if I had more completed work, especially homework. 

Next Steps include more factoring to solve polynomials and moving on to Rational functions

Take a look.   The left column of the table is essentially a study guide of what you need to know:

:                                        

Essential Learning Outcome:  Model and Solve Real-Life Situations using Polynomials  -- >  “Box” Problems/Volume Applications in Context - Write polynomial, interpret graph, solve
Learning Targets and Relevant Problems                  I can…                                                                           STANDARD	Whole Class Observations
Numerically                                                                      CC9-12.A.APR.2   I can perform polynomial long division and synthetic division / substitution, and understand /can apply the factor and remainder theorems.  7.1 Synthetic Division Assessment (4 problems) and Poly Quiz #1 7.1 Problems Sets (HW)	85% of class consistently performing synthetic division accurately and understand the purpose of the  remainder theorem   Some clarity needed in evaluating using synthetic substitution Only 4 students at level 4 (Homework 7.4 needed to show Fundamental Theorem of Algebra (problems with complex coefficients )
Graphically                                                                       CC9-12.A.APR.3   Sketch the graph of polynomial functions by identifying zeros and understand the behavior near those zeros and as x approaches +/- infinity   7.2 Graphing Assessment 7.2 Problem Sets (HW) and mixed assessment problems	65% of class showing consistent graphical understanding.  Students without graphing calculators (a few non-HW completers) haven’t shown much understanding but seem to have partial understanding   Some students over relying on graphing calculator
Algebraically/Analytically                                                CC9-12.A.SSE.2      Recognize and rewrite polynomials as special products (FACTORS)        to find zeros of polynomial functions.  Examples:  Factor by grouping, Difference of Two Cubes, factor completely 7.3 Polynomials Application Quiz #3, 7.3/7.4 Problem Sets (HW) and mixed assessment problems	50% have shown strong factoring capabilities. Haven’t assigned many multi-step factoring problems but it came up in earlier chapters.   Need to assign more factor by grouping , factoring completely, and difference of squares/cubes.   Many common errors when depressed polynomial is NOT FACTORABLE (need to apply quad formula for irrational or complex solutions)
Making Sense of Problems/Model Using Mathematics    CC9-12.A.SSE.1  Write expressions to represent length, width, and height in a volume function and interpret the meaning of these expressions in a polynomial Apply appropriate tools strategically 7.3 Polynomials Assessment #3    HW 7.3/27-33 word problems	65% of class able to consistently set up/solve  volume problems leading to polynomials.   Box volume problems seem to be understood.  Some students not confident in setting up word problems from other contexts.  Solving process seems strong.

 

Other Comments:  ___________________ ______________________ # Exceeds ___   #Meets ____    #Nearly  ____    #Not Meeting _____      

__________________________________________________________________________________________________________________________

Wrapping up Ch5 and Moving on to Ch6

We are wrapping up our Trig. Applications chapter and this is a good time archive/file away our best work.  We will finish the Chapter 5 test Monday and scan our task work in the computer lab as well. 

Prepared for the test?
Be sure you know when and how to apply the Law of Cosines and how to sketch two distinct possible triangles for the ambiguous case of the Law of Sines.  I have posted some sample student work and worked out solutions for the 5R assignments.  Take a look, especially if you have not submitted those problems yet.  The solutions are on my MHS website at  http://www.mpsri.net/page.cfm?p=1567
Look under Chapter 5 resources. 

C-Period
Be sure to bring your solutions to the 6B applied problems we did a while back (also on my website if you need to see a copy). 

Feeling all set with Chapter 5?
Many of us have masted all the Trig. applications.  Two things you can do are:

(1)  Look over Section 5.5 - There are some other problems we will not cover this year but  you may see in Calculus.  Area with Law of Sines is just another application of the Law of Sines

(2)  Read ahead in Chapter 6.  We will be covering at least sections 6.1 and 6.2 this week.  Historically, Chapter 6 is a little tougher so you may want to read ahead and look over each section before coming to class.





    

The other Trig. Graphs

This week we will continue our graphing of the "other" trig functions and dig a little deeper into different ways to model real-world functions (not just the negative cosine models)

 3 Key  “Graded” Activities

·         HW 4.3b (Due next Class) - 206/11,13,17,22,24,27,29,39-45 AND Ch 4 Practice Quiz (will be assigned on Tuesday) 

·         Chapter 4 Quiz at the end of this week

·         Alternative Assessment (Starting this Week Due Next Monday)– Cool Trig. Graphing Mini-Project.   Those of you that already did it will earn full credit and get some extra feedback.

3 Key Resources

·          Graphs of Trigonometric Functions Yellow Reference Sheet  (use it while doing CW/HW)

·       ~Trig. Graphing Procedure Sine/Cosine Functions~

·       Sample solutions on my webpage (see link above) http://www.mpsri.net/page.cfm?p=1567    

Rule of 4_Real-World Problems

General
At this point we are shifting our angle and radian understanding to actual times and heights.  The purpose of the "leaf bobbing" and "ferris wheel" problems have been to transform your understanding of circular functions to actual situations that can be modeled with trigonometric functions. 

Hopefully, the graphing of the functions has been pretty straightforward  and the recognition of "four to five" different key times and heights connects to your understanding of the unit circle.

Key Products  

• 4.1 and 4.2 text assignments (especially #49) should be handed in with the last assignment sheet (this should show your understanding of the basic trigonmetric graphs).  Due Monday!
• Practice Task Handouts - The sinusoidal functions (see my website for some worked out solutions to one of the practice problems)
• Task (Tuesday or Weds).  You must complete these independently within one class period.  This will count as a test grade.
• Review packet problems (just for general review).  This does not relate to the task or Ch.4.

Understanding Numerically, Algebraically, and Graphically

• We have "solved" a couple of practice equations in class.  This basically involves what you already know solving using inverse operations then applying arcsine and arccosine to undo the function.
• We have not "evaluated" a lot of trig. functions in class but you will need to apply evaluating to our real-world trig. functions
• You do know how to apply unit circle reference angles and will need to apply that understanding to the actual repeating cycles of a trig. function to show proficient understanding on the task.  It all relates back to 180 +/- the reference angle or 360 - reference angle.  But we are not talking about angles anymore.

You are expected to look at the website tonight and see some worked out solutions if you are stuck on the task practice problems.