Fair Week and Ferris Wheels

The carnival has arrived in Newport and now is the perfect time to show what you know about applying Sine and Cosine Functions to model periodic behavior.

Who wants to just ride a Ferris Wheel when you now know enough to write and graph a function to model the motion.  You can also describe to your friends when and how often they will be at certain height by just knowing how long it takes to go around once.

That is hands down better.  PERIOD.

There is a wealth of demonstrations of the connections between Ferris Wheels and Trig. class by a simple google search  trigonometry ferris wheel video.  

The only thing you have not learned is how to interpret or use phase shifts to create different types of models.  We will learn that this week and then complete a task.

The visuals displays should really help you put it all together.  I like the second short 60 second video that popped up with my google search.

Tips for El Segundo Task

Your responses to the Arrive at Five task are due Monday.  What you hand in and one final in class summative assessment will be your final opportunties to show what you know for Quarter 3.

I checked my calculations for finding the direct distance between Newport, RI and El Segundo, CA. I used latitude/longitude coordinates, arc length, and the Pythagorean Theorem.  The right triangle and arc length based calculations will not the same as what you get using an online distance calculator (and they shouldn't be).  Do some further reading if you want to know why.

 You are still expected to find the calculations by hand.

I wrote the El Segundo problem to give you an opportunity to do three things:

- Show that you know how to convert coordinates (both Latitude and Longitude) from degree minutes seconds to decimal degrees and vice-versa.

- Make the connection that you can use the difference between any two latitudes (an angle) and the radius of the earth to calculate a curved distance over the surface of the Earth.  You can do a similar set of calculations for the Longitudes.

- THEN APPLY THE PYTHAGOREAN THEOREM.

Kind of like the diagram above.  Basically, get the red and blue then
 a^2 + b^2 = c^2 to get green

Give it a shot and hand in Monday.

Source: 

http://derickrethans.nl/spatial-indexes-calculating-distance.html

BUT:  because of the curvature of the earth and other factors the diagonal distance Between cities will not be precise.  A full explanation is pretty complicated but is explained in more detail using the link above.

 New addition to the Trig. playlist:  A Tribe Called Quest 

FourFive and Knowing Two of Three

FourFive Addition to  Trigonometry Playlist
It still doesn't feel like spring!  In response to a student request to add FourFive Seconds to the Trig. playlist, consider it added.  However, I originally thought I heard "Forty-Five" which connected to 45 seconds per minute loosely relating the lyrics to Trig.   Forty-Five clock seconds representing 3pi/2 radians of a turn of course.  But since Paul McCartney is involved we can add it to the playlist.

Two of Three
If I had to choose one number theme to relate FourFive to what we learned last week to what we will learn this week it is two of three.  In simplest terms, if you can identify Two of Three math facts in a word problem, you typically can connect to the elationship (Formula) you need to apply.    
                   
Typical Problems from Last Week:
                         Know:  Angle in Radians and Radius           Want:  Arc Length      
                         Know:  Rotations and Time                           Want:  Angular Speed    
                         Know:   Angular Speed and Radius               Want:  Linear Velocity

Two of Three Examples you will see this week:

Given a Triangle:                                                        
Know:  An OPPOSITE and HYPOTENUSE
Want:  The SINE of the Angle

SOH CAH TOA.  There is even a two of three mnemonic
to help you make decisions for basic trigonometry. 

Note:  SOH CAH TOA should be familiar from previous classes.   If it is not, I strongly recommend you visit Khan Academy (or another site) for basic trigonometry review.  https://www.khanacademy.org/math/trigonometry/basic-trigonometry  view first video and example.

More Additions to the Playlist
"So What'cha Want" to inspire you to show your work
 https://www.youtube.com/watch?v=RWfqPIyU9Zw

"My Shirona" because it sounds like the mnemonic SOH CAH TOA
https://www.youtube.com/watch?v=g1T71PGd-J0

"The Logical Song"  because you need to know radical form
https://www.youtube.com/watch?v=fBoYZqmcZuc  
and because classic music videos are a lost art form.

What Time is it! Time for My Three Songs.

Happy Spring!

It has been a while since we made musical connections to our class content

(see Missy Elliot in February or of Course Sade’s Smooth Operator in Quarter2).

 Before Pandora, Spotify, and itunes, some radio stations used to play “My Three Songs” and listeners had to guess what they had in common.  These three songs have circles in common!  Here are your three Trig. Songs for the week and 6 extra credit opportunities.

 Note:  You may respond for extra credit one time and be the first commenter/not duplicate someone else’s response to earn credit (read blog comments section before responding).

To clarify who you are and get credit, write your response in the comments section.
e.g. The Artist is Sade! and note that is sounds like “Gaudet” - grs

Unit 6 Trig. Tune	Extra Credit Task
You Spin me Round  (Like a Record) https://www.youtube.com/watch?v=PGNiXGX2nLU	+5% to this week’s quiz. (name this artist in comments below)   +10% to this week’s quiz State 3 unique “circle facts” that relate to learning target 1.
The Spin Doctors What time is it?  https://www.youtube.com/watch?v=eqS5pJDX5j4	+5% State the time/name the album.    +10% State the angle measurement between the hands in radians.  Exact
BHTM Circle https://www.youtube.com/watch?v=TcVIfu_ga8Y	+ 5% Name the Band and state where the lead singer is from.   +10%  Explain what co-terminal means and provide an example of two coterminal angles

This is unconventional way to earn credit, but the time is now Spring and we have to mix things up.  Plus, we need more two-way communication and this is one way to implement a blog.  Also, recall that I did ask you to follow this blog earlier in the year.  I may reward the followers.  My Seniors, hang in there we are not done yet.

 

One New Helpful Resource

I created a google document that will be updated as new material is assigned.  The document can be view here:

https://docs.google.com/document/d/1QDucg-yN_2NcQzTitKsJOk4cHC9XbjpzW8UvkHPRt1U/edit?usp=sharing

 

Note the 6.1d has not been assigned as of Sunday and E-Period has not received the ASMT2 Handouts yet.

The big thaw...

After our cold week off, I want you to assemble and bring in the text homework you have completed so far to show what you know about exponentials.  Next class you will be handed a highlighter and need to choose and highlight about 10 problems that show you know the learning targets listed below.  An even better plan is to highlight (I mean just draw a yellow box around significant problems to draw attention to important problems) before you come to the next class.

The problems you highlight should demonstrate that you know the learning targets described below:

Learning Targets            By the end of this unit you should be able to say and demonstrate that:..

(1)  I understand all the parameters of an exponential function y = Ca^x .   Given a table, I can determine initial values and growth/decay factors to write an exponential function.  Given a graph or situation, I can identify and interpret the appropriate equation using my understanding of exponentials.

(2)  Given a real-life scenario, I can create and evaluate exponential (and later logarithmic) functions to solve problems.  I can interpret different types of compounded interest formulas and y = e^x and use technology to identify solutions to problems.

 

(3)  I recognize a logarithm function as the inverse of an exponential function.  I can solve exponential equations by applying log base 10 and the natural log functions to “undo” exponential functions. 

 There will also be a survey you need to complete next class that rates your understanding of the targets above more specifically.  Be sure you have your work to look at so you can back up what you see about your understanding.

Second week of Exponentials and Missy Elliot

Last Week
We learned exponential growth and decay function key features and  end behavior.

- The y-intercept (0,a) is always a good feature to inspect and 
the end behavior either grows rapidly towards infinity or approaches the x-axis asymptotically.  [Unless the function is transformed of course]

- We learned the basic structure of exponential equations:  y=ab^x and y = a (1+r)^x
and it is all about the base (if "it" means determining growth versus this decay.

Next Step(s):  Apply and interpret more sophisticated exponential models, like models for compounded interest and models to display scientific phenomenon.

- We learned how to interpret patterns (mainly from tables) and determine if it represents an exponential or linear function.  Then, actually write the formula using the appropriate structure of an equation.

Next Step(s):  Contrast exponential functions with Quadratic Functions, and learn about the Exponential Function's Evil Twin Function which can undo it.  If we can undo an operation, we can solve equations.  

Mathematician's call it the Inverse Function, and Missy Elliot refers to it in one of her songs, Work it.  If it is worth it, we can work it and then something about flipping it and reversing it.

Expect a paper pencil quiz this week and one more tenmarks track that will also count as a quiz.

Transitioning from Polynomial Functions to Exponential Functions begins Groundhog's Day!

Monday Task
Last week we did a couple of open-ended practice task problems.  Monday (if we don't get another foot of snow!), you will complete an open response problem that is probably less difficult than the practice problems we did together, but will still challenge you to make sense of a problem and apply what we have learned to solve problems pertaining to volume in a different context.  This will count as the first test grade of the new quarter.  I don't think most of us really need to do a whole lot of studying for this task, but if you want to spend 10 minutes preparing, it would be a good idea to organize your unit 4 polynomial notes and perhaps log on to www.khanacademy.org and search "Dimensions from volume of box".  There is a short 5-minute video that shows a simpler problem than what you will need to complete, but the core problem solving elements are pretty similar and it should build your confidence.
 Guiding Questions for Tenmarks Problems

To prepare for upcoming material, your weekend homework was to complete 4 www.tenmarks.com tracks to refresh  your memory about the key features and behavior of exponential graphs, structure of exponential equations, and patterns to interpret tables of values modeling exponential functions.

Your first homework grade will be your TM scores for the four assigned tracks.  One student finished all four last Monday in about 30 minutes.  I recommend your complete the tracks and as you are working, answer the following guiding questions:

• Representing and Interpreting Exponential Functions [FIF7e] . For polynomial functions we learned about key features and behavior of that family of function.   Given a simple exponential function y = b^x, what key features do we need to recognize?  What is different about the behavior and rates of change of an exponential function compered to other functions we have studied. 

 

• Understanding Exponential Growth and Decay [F-IF.8b] For polynomial functions, we learned what how each of the parameters in y = a (x-h)^3 +k affects the graph of a polynomial.  What can you recognize about the roles of the a and b for a polynomial function y = ab^x?  What happens if b>1?  What happens when 0<b<1?

 

• Calculating Exponential Growth and Decay Algebraically [F-LE.1c] For linear functions, we know all about positive and negative slope and the y-intercept for any y = mx+b.  Exponentials are not characterized by slope and the y-intercept is not a "b".  What can you generalize about the y-intercept of exponential functions?  What is generally true about the x-intercept(s)?

 

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. 

Psyched about sinusoidals

General – You will be assessed on how well you can graph and interpret  sine and cosine functions tomorrow Quiz 4.1-4.2 (or Tuesday if you are in D-period)

Note:  Assignments 4.1 and 4.2 will be handed in tomorrow

Be sure you :

·         know the meaning of the coefficients of y = a sin b (x –c) + d

·         can determine the period, amplitude, and vertical shift

·         can graph sine an cosine functions using a table of values

3 Key Goals for this week

• Show you know the 4.1 – 4.2 material
• Apply trigonometry functions to real-world situations (sinusoidal relationships – see 4.7)
• Complete more challenging word problems/sample tasks to get ready for next week

 

For those of you who did not complete the New Year's Packet, I have created a second problem set.   You must complete the problems handed to you next class AND the New Year's refresher packet. 

 

For those you who did complete the packet.  You will be given the option of checking another students work once it is completed and/or completing less open-ended problems.

Any extra work completed can be used to raise an earlier homework grade from earlier in the year (the review packets really represent material learned eariler this year).  The new multiple choice problems will be posted on my website under the Ch.4 folder.