Tuesday, 16 December 2014

Solving Polynomial and Rational Applications

We started class by watching these videos and students asked questions about the videos.

Solving Rational Inequalities

We solved rational inequalities by creating a number line and placing the roots and the vertical asymptotes on the number line. We can then find the sign of one interval using a test point and then figure out the signs in all the other intervals using what we know about odd and even roots and odd and even vertical asymptotes.

Solving Polynomial Inequalities

 
We started by solving some linear inequalities and graphing our solution on a number line.

IROC and AROC

Nth Finite Difference = Leading Coefficient * N!

 
We started by reviewing how to generate the equation and a sketch of the graph of polynomials in factored form.

Saturday, 29 November 2014

Trigonometric Identities

Proving trigonometric identities requires perseverance, a willingness to make mistakes, being able to leave a question and come back to it and a positive mindset.
 
I first wrote out some things we knew.

Wednesday, 26 November 2014

Solving Trigonometric Equations (a third time)

A review of the compound angle formulas, the double angle formulas and the Pythagorean identity.

All the following questions required to find theta (or x or the angle) between 0 radians and 2pi radians.

First a linear example.

Double Angle Formulas

 
Started by using the compound angle formulas and replaced the second angle so it would be equal to the first. By doing this we were able to generate the double angle formulas for sin2A, cos2A, and tan2A. Here is some work from various groups.
 

Saturday, 22 November 2014

Wednesday, 19 November 2014

Solving Exponential Equations by Getting the Same Base

This post is about solving exponential equations by getting the same base on both sides. Since the bases are the same the exponents must be equal. Some examples from class:
 
 

Solving Rational Equations

Solving rational equations is equivalent to finding the y-intercepts of a rational equation.
To make sure you do not lose any roots the best way to do this is to put everything on one side and then solve. Here are examples we worked on in class:
 


 

Solving Polynomial Equations

Solving polynomial equations is equivalent to finding the y-intercepts of a polynomial equation.
If y = x^3-4x^2+8x-32 then to solve 0 = x^3-4x^2+8x-32 is the same as finding the x values that make y = 0. Here are two examples multiple times:

Sunday, 16 November 2014

Logistic Functions



Combining Functions (Adding, Subtracting, Multiplying, Dividing)

 

Solving Trigonometric Equations

After the ferris wheel we formalized solving trigonometric equations. Here is the sequence of questions we worked on.

A trig function equals some number.

Ferris Wheels

Students were asked to write any questions they had on this video of a ferris wheel. As a class we settled on " where does the red bucket finish at the end of the ride?" Then we decided what info we needed. We collected some data from the video. pi on 4 radians every 5 seconds. Students then solved the question for a three minute ride.

Monday, 27 October 2014

Review for test 2

 

Logs and their Laws

Visible random groups and vertical non-permanent surfaces to work on logarithms and log laws.
First evaluating logs.
 

Dominoes - Relationships between the Number of the Dominoe and Various Variables

Students watched this video and then were examined the relationship  between the number of the dominoe and some variable. Here were the variables I assigned.
Once groups had their variable to examine I asked them to create a poster that had the following.
 

Dominoes - What are the Criteria for a Good Question?

Students looked at this picture and were asked to write down any questions or wonderings they had.
In visible random groups they shared their questions with each other and on mini whiteboards they wrote out their three best questions and why they thought they were the best questions. Here are them working on it.

Sunday, 19 October 2014

Rational functions - Oblique Asymptotes

In visible random groups we explored rational functions where the degree of the numerator is one more than the degree of the denominator. For example, a cubic divided by a quadratic.

Before we did this we recalled factoring a polynomial using the area model. My work and some other groups work.


 Then we reviewed a rational function with y = 0 as the horizontal asymptote. Degree of numerator is less than degree of denominator. Two examples.


Then we reviewed a rational function where with a horizontal asymptote of y = some number other than zero. Degree of Numerator = Degree of Denominator. Horizontal asymptote is y = (Leading Coefficient of Numerator) / (Leading Coefficient of the Denominator).
Then we looked at Rational Functions where the degree of the numerator is one more than the degree of the denominator. This creates an oblique asymptote.
Example 1
 Example 2
 Example 3


 Example 4
 
 
Example 5

 Example 6


 


Monday, 13 October 2014

Rational Functions - Looking at Zeros, Vertical Asymptotes and Horizontal Asymptotes

The purpose of this whiteboarding activity was to look at zeros of rational functions - from the numerator - and significance of double, triple etc. roots. We also examined vertical asymptotes of rational functions - from the denominator - and significance of order two, three etc. of the factors in the denominator. Lastly we examined when the horizontal asymptote was y = 0 (degree of numerator < degree of denominator) or when the horizontal asymptote was y = some other number other than zero (degree of numerator = degree of denominator) so horizontal asymptote ended up being y = (Leading coefficient of Numerator / Leading coefficient of the Denominator).
Here are some photos from the whiteboards as we worked our way through this.
We started with something that would have lots of confusion - included a hole