Fractals & Chaos Lesson Recap for 12/7

We closed out our work with the Feigenbaum Plot today and took a moment to reflect. With our work in the real numbers, we started with linear functions (mx+b) and quickly discovered nearly everything there was to find. Linear functions are easy and non-chaotic. When we moved to non-linear functions, things started getting interesting. We looked at a few examples of the form x^2 + c, but did a deep dive with the logistic map of ax(1-x). For a single parameter, we could make a cobweb diagram to show the behavior of that specific function, but the really interesting things happened when we made our catalog of behavior for all parameters, creating the Feigenbaum Plot.

We will follow the same path through the forest of Complex Numbers. We will start with linear functions of the form Az+B, and understand what we can find there. We’ll then move on to non-linear functions, specifically of the form z^2 + C, and look at the behavior of single, specific values of C. Eventually, we will move to a catalog view there and see what we find.

Today was our first step towards that goal, with a discussion of how arithmetic on complex numbers can mimic geometric transformations. Complex numbers have two coordinates, a real part and an imaginary part, so their operations provide a convenient way of movie around the coordinate plane. Specifically, adding complex numbers produces translations and multiplying complex numbers produces a dilation/rotation. Exactly how the dilation/rotation is understood requires another way of referring to these complex numbers: Polar Form. We will discuss this on Monday.

Intermediate Algebra Assignment for 12/7

The third method of solving quadratics is to simply use the square root, but there’s a danger here: you still have two solutions. The + is built into the quadratic formula, but it still needs to be added here. And that falls to you, the solver!

Today’s lesson also included a brief revisit to word problems, this time resulting in a quadratic equation.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 12/6

You had your solving quadratic equations quiz in class today. I want you to do some more for homework. Tonight, work on IXL Module J.9 Solve a quadratic equation using the quadratic formula to a score of 80 or higher. Remember: any score above 80 will count as extra credit!

Cumulative IXL Modules

Intermediate Algebra Assignment for 12/5

You will have a quiz tomorrow on solving quadratic equations. Today you were given a review sheet (see below) with some more practice. Work on this review and the IXL modules linked below in preparation for tomorrow’s quiz!

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 12/5

We did some more work with the quadratic formula today, specifically with simplifying the radical expressions we sometimes get when we apply it.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 12/3

More work with solving quadratic equations today, this time with equations that did not factor. Our solution? The quadratic formula.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 11/30

Part 1 of the unit was on graphing quadratic functions and identifying the key features of the parabolas we create. Part 2 is more algebraic, focused on solving quadratic equations by hand (and eventually finding those key features without the graph). Today was a review of factoring and how that method of algebraic manipulation allows us to solve these equations.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 11/28

Your first quiz in chapter 3 is tomorrow, and in class we spent time reviewing.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 11/27

We did some work with our graphing calculators today, using them to find the vertex, x-intercepts, and line of symmetry for parabolas.

Today’s Files

Cumulative IXL Modules

Intermediate Algebra Assignment for 11/26

I met with many of you in class today to discuss your first marking period grade and what you should do in the 2nd marking period to either improve or continue the good work you’ve been doing.

Your homework is on IXL. Your assignment is as follows:

Use your graphing calculator or Desmos.com/calculator to make graphs. If you exceed the minimum requirements of this assignment, your work will be recorded as extra credit.

Fractals & Chaos Lesson Recap for 11/20

We reviewed results from 4b through 4f on the Iterated Functions sheet, making a brief detour to discuss the Golden Ratio which became surprisingly relevant for 4d. Every example had a single attracting fixed point until 4f, which had two repellers. But what did these repellers repel to? It wasn’t infinity, it was a cycle of values between 0 and -1. This is our first example of a limit cycle, a surprising result in algebraic iterations where a pattern of steps is pushed into a repetitive cycle.

Then we pushed the function a little further, from x^2 – 1 to x^2 – 2. Still with two repellers, this time the system of iterations dissolved into complete chaos, careening randomly within the interval -2 to 2, never settling into one spot. Changing the value of the seed by only one ten-thousandth produced a completely different pattern of iterations, demonstrating the sensitivity to initial conditions that are characteristic of chaos.

We will see plenty more examples of weird behavior in our iterations, but for now please finish question 4, parts h, i, and j. We’ll be discussing these results on Monday when we return.

Intermediate Algebra Assignment for 11/20

Today was Day 3 of the Desmos-based activities we’ve been using in class to re-introduce ourselves to parabolas. In class yesterday, many of you had some difficulty in properly describing parabolas or in identifying features of parabolas to ask about in order to guess which one your partner had chosen. This shows us that there is a need for proper vocabulary when describing these entities. Today’s lesson reviewed that vocabulary.

Today’s Files

Cumulative IXL Modules

 

Intermediate Algebra Assignment for 11/19

We continued our re-introduction to Parabolas and Quadratic Equations today by playing a game similar to the classic game Guess Who, where you had to ask a series of questions to guess which of 25 parabolas your randomly matched partner selected. This taught us the value of having clearly defined vocabulary to describe the key features of such shapes, as several questions that you asked each other led to confusion!

For tonight’s homework, please complete the Parabola Polygraph Debrief assignment. You will also need to bring in your Chromebook once more tomorrow for the third and final introductory activity to this unit.

Snow Day Items of Note

Happy snow day, all! I hope you are able to stay warm, dry, and safe today. It was a mess out there this morning, and it still is in many places, so if you can avoid going out you probably should!

There will be no make up assignments over the weekend for any of my classes, with the exception of AP Statistics. For you, I would like you to watch a pair of videos that will give you a solid introduction to our next chapter about observational studies and experiments. They are

  • The Question of Causation – A historical story describing how researchers untangled the relationship between smoking and lung cancer
  • Designing Experiments – A discussion of the difference between observational studies and experiments, with examples about marine life on the remote Line Islands and a medical study about osteoarthritis treatments.

Have a good weekend!

Intermediate Algebra Assignment for 11/14

We had our Unit 2 test today, and at the end you were given this extra puzzle sheet. You must complete one side (your choice which!) for a homework credit. If you complete both sides (and you must show your work!), you will get extra credit on today’s test.

Also, bring your Chromebook tomorrow, charged and ready to go!

Intermediate Algebra Assignment for 11/13

Your Unit 2 Test is tomorrow. Here are some ideas on how to prepare:

Cumulative IXL Modules

Intermediate Algebra Assignment for 11/9

Your Unit 2 test will be on Wednesday, November 14. We played a game of survivor to review much of the unit so far, and your homework tonight is the first of two test review sheets.

Today’s Links

Cumulative IXL Modules

Intermediate Algebra Assignment for 11/8

We did some more review with solving rational equations today, focusing especially on dealing with equations that require us to factor denominators to find a common denominator.

Today’s Links

Cumulative IXL Modules