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