Monday, January 30, 2012

First Full Week of Quarter Three "3 Key Things"

As we approach February and launch the Third Quarter - Here are 3 sets of 3 things to consider
 3 Key  “Graded” Activities
·         HW 4.3b (Due next Class) - 206/11,13,17,22,24,27,29,39-45 AND Ch 4 Practice Test
        Practice Test Answer Key will be posted at  http://www.mpsri.net/page.cfm?p=1567 
·         Chapter 4 Test on Thursday (Day 3)
·         Alternative Assessment (Starting this Week Due Next Tuesday)– Cool Trig. Graphing Mini-Project (Preview of the Project Expectations are posted at my Middletown Webpage (see link above). 
3 Key Resources
·         Graphs of Trigonometric Functions Reference Sheet  (use it while doing CW/HW)
·       ~Trig. Graphing Procedure Sine/Cosine Functions~
·         A new Google document assignment sheet will be added and linked to this page (but is not yet.  Our current assignment sheet can be found at ….(stay tuned) This Live Document can be accessed any time to see what you  should be working on.

3 Key Takeaways from Today’s class (Monday, January 30, 2012 – Day 3)
·         Our graphs have to show 6 ” key features” and  there should be a table of values showing our exact abscissa values (pi/4, pi/2, etc.) and important ordinates (fancy or mathematically precise words for (x,y) coordinates. 
·         We can use the graphing calculator to “check” the accuracy of our key features but must know how to adjust the window, table, tableset, etc.  This systematic approach to graphing Trig. functions will take practice but will pay off in the long run
·         There are two Algebraic forms of Trigonometric equations y = asinb(x-c) + d (when the b is “factored out”) and y = a sin(bx + c) which matches standard formats on our new reference sheet.  The “factored out” form  is better for identifying phase shifts and relating to earlier non-Trigonometry functions, shifts, etc.
  

Wednesday, January 25, 2012

Midyear Exam Reflection

Below are some thoughts about the mid term results.  Stop by and see me tomorrow if you want to see the test.  The actual exams will not be handed back, but you may take a peek.

Observations:

The good – Scores were generally high, mostly B’s and C’s and only a few A’s.  Pretty much the entire class was solid on inverses, graphing aspects of Trigonometric Functions, questions relating to the Unit Circle, and Domain, Range, Asymptotes.

The bad – Some of the open-ended answers and recognition of concepts learned earlier in the year.  Specifically, when analyzing an angle (like 188-degrees) we use a Reference Angle.  That is important vocabulary and matches the little yellow triangle we often had on the TV as a visual. Even functions are symmetrical about the y-axis.  Many of us missed this question.  Think about a parabola y = x^2. 

Do you recall what an odd function is symmetrical about?    And what is a good way of remembering that characteristic?  First new commenter who responds to those questions correctly starts of the new term with extra credit.

The Ugly – Solving polynomial equations (material from earlier in the year – try factoring first and check for zeros with your graphing calculators). 

Co-functions and the Domain/Range of Inverse Trig Functions– This is a pretty straightforward concept, we just need more practice.  

Expect some more HW problem coming soon.  We also need more practice with phase shifts.

See some of you Friday!  Good luck with any remaining exams.

Saturday, January 21, 2012

The Final Push Before the Midyear Exam

If you are reading this, you are most likely already prepared for the midyear exam.  However, you are probably likely to do just a little more to build some confidence. 

I always found that for math and science classes (especially in Engineering school) it just made me feel more confident to do problems and see that I am getting them right.  Also, you may just uncover one old problem that might force you to look up and old formula or just page through your old notes one more time (like say, an axis of symmetry problem for a quadratic).  Then, when you hit that problem on the actual exam you will feel happy (oh yeah, I remember x=-b/2a) instead of feeling anxious and losing confidence during the test.

So, remember that I did provide you with a page of answers to multiple choice problems (the day I gave out the pink topic sheet). 

If I were studying, I would open my textbook and put myself in "practice test mode".  Take out 4-5 pieces of paper, set aside an hour, and then just do each of those problems as if it were the final (hey it is cold and snowy out anyways).   I would pass over any of the problems I did not feel as confident about (good test taking skill) and then go back to them after finishing aobut 50  problems I do feel confident about.  I would then spend 15 minutes really focusing on those and then check all of my answers.  Then score your test and make any decisions about what else to study.

Good luck!  And if you don't have your Trig Midterm Review Book Problems answer sheets, I am sure one of your classmates will.  If not, email me and I will scan it and get you a copy.

ps.  This can be part of your final 50 HW problems as long as you show some evidence of calculations.

Tuesday, January 17, 2012

Even and Odd Intervals

The following are a few student observations about graphing the secant and cosecent functions:

Re:  Secant/Cosecant Graphs-

The graphs of co-secant and secant graphs are both similar and different. 
1.       A co-secant graph is a shift of 90 degrees or pi/2 to the right of a secant graph. 
2.       The asymptotes for a secant graph cut through the center of the parabolic-structure of co-secant graphs. 
3.       The asymptotes for co-secant graphs cut through the center of a secant graph.
4.       When a sine graph's x-value is zero, the x-value for a secant graph is also zero. 
5.       When the x-value for a cosine graph is zero, the x-value of a co-secant graph is also zero. 
6.       All odd intervals of pi have the secant graphs opening downward. 
7.       All odd intervals of 3pi/2 have co-secant graphs opening downward. 
8.       All even intervals of pi have secant graphs opening upwards. 
9.       All even intervals of  3pi/2 have co-secant graphs opening upwards
Question:  Is there an algebraic way of modeling

"all odd intervals of"

"all even intervals of"

Hint:  It may involve k's and Z's

First commenters may receive extra credit.  Post a comment!  and now the wikipedia is going dark at midnight!

Monday, January 16, 2012

Ferris Wheel_Intial Post

We have had a great year in Trigonometry so far and are getting very close to the midyear exam.  As the year has developed, I see more and more how important it is to use visuals to describe certain concepts and how great it is to see how you (my students) describe your understanding in writing.  I plan to use this blog to provide roughly a weekly update (T = one week, probably each weekend)  about what we learned, any updates about assignments etc. and to preview what is on the horizon.  Ideally, it can also be a place to share good links and also hear your thoughts and descriptions about what you have learned.  In fact those of you that did not e-mail me last week will be our first commenters!  Stay tuned.

To borrow a Trigonometry model, much of what we learn in mathematics is cyclical (it just gets more sophisticated), and as we go through each week we can learn more and more how all the ideas are related.

Last Week
 We got back our graded HEP5 tasks.  The Ferris Wheel problem really helped me see who understands the relationships between circles, sine/cosine graphs, and how time and height are modeled using a trigonometry function. 

Everyone seemed to master the "vertical shift" considering the Ferris Wheel was on top of a building and we had to choose a "middle" height and think about whether cosine or sine could be used to model the changing height over time.

Sine versus Cosine.    After we all completed our tasks, there was  a learning opportunity to see how either sine or cosine model could be used to model the height at any time ( negative cosine was the most used model because it started at "the bottom").  The graded task transitioned nicely into the idea of a "cofunction", realizing that sine and cosine are the same function, just shifted by a quarter turn (or 90-degrees or pi/2 for you radian purists).

Problem Solving - The most unfamiliar step of the task was using the graph to see how often the ferris wheel was at 40-meters above the ground.  This was a great question to recognize the value of our graphical knowledge to estimate how often and when that height occurs and how to use the graphing calculator to find the "actuals".  That solving step is just like a systems of equations problem with a horizontal line.  The intervals between the 40's ended up happening about every 3 minutes (lower parts of the wheel) and 11 minutes (the longer part of the rotation).  Not just a coincidence that 11+3 = 14 minutes, the period of rotation for the Ferris Wheel.

This Week
We are in exam preparation mode.  Using our pink topic sheet as a guide, we can review mixed problems to prepare for the exam.  Making sure you hit each topic at least 4 times (to follow a topic mastery "rule of 4").  Most of what we learned 2nd quarter can be found in Section 6 of the Algebra2/Trig Page. of http://www.regentsprep.org/    It is up to you to do problems and check yourself.  Keep enough evidence of showing your work to be able to show me and another student you have fully prepared yourself.  Your completed work should be organized and placed into your folder by the Monday of exam week.  (This Friday, is even better!) 

50 questions is a minimal expectation (and your final HW assignment).  Those of you who are struggling should do more and seek out help for anything you don't understand.

Inverse Functions
One new concept that we will hit this week is the graphing inverse functions.  Since you will be at the regentsprep website, take a look at the lesson page for that topic.  One reason I assign this website as a review is so you can re-learn or teach yourself any concepts that you may be struggling with.  That is what you will be doing in college!

Ok - stop reading now and (1) Scan/upload your tasks  (2) attempt the mixed multiple choice Trigonometry problems at the end of Section 6 then (3) check your solutions on the website. 

If you finish that, page through each of the topics in Section 6 and attempt the practice pages highlighted in the classroom.

Good luck!