All posts by Mr. Kirk

I have been Mathematics Teacher at Ithaca High School since 2007. In that time, I have taught Algebra 1, Intermediate Algebra, Honors Algebra 2, Honors Pre Calculus, Advanced Placement Statistics, Math AIS, and a one-semester elective course in Fractal Geometry and Chaos Theory.

Fractals & Chaos Recap for 11/21

Mr. Drix was in again today and reminded you of a point he made when he was with you last time: the graph of a function “near” a fixed point will look linear if you magnify it enough, even if the function itself is not linear. As a result, we can use the rules we derived for classifying fixed points based on the slope of a linear function to help us classify fixed points now. For example, if the “nearby” slope around a fixed point is positive and greater than that of y=x (i.e., is greater than 1), then the fixed point will be a “direct attractor.” This raised the question: What if the nearby slope is exactly equal to 1? This occurs when the graph of f(x) is tangent to the graph of y = x at the location of the fixed point.

It turns out that the fixed point has no unique classification, as behavior of seeds to the left of the fixed point differs from the behavior of seeds to the right. A fixed point could be directly attracting on one side, but directly repelling on the other!

Your homework tonight is to finish the Iterated Functions sheet by exploring the equations of part 5. All five pairs of functions have a fixed point at 0, all of which having a nearby slope of 1. Use the Geogebra apps on MathInsight.org to make a graph and test the behavior of seeds on both sides of the fixed point and write down your observations on the Iterated Functions Supplement. All five examples of two equations each: the positive and negative version. Make sure you look at both!

InCA Assignment for 11/21

Today we took some time to review the vocabulary we introduced yesterday. Your homework is on IXL, and 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.

AP Statistics Assignment for 11/21

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 3 MCQ Part B PPC by Monday, November 25

On Monday, we will have our Unit 3 Test. We will spend Friday reviewing, and in preparation for that you should take a look at the review problems for Unit 3: pages 331-336, exercises 1, 3, 7, 9, 11, 15, 19, 20, 23, 29, 32, 40, 41. You’ll get answers and supplemental review tomorrow.

Fractals & Chaos Recap for 11/20

After a discussion of the recently-assigned article, we finished #4 from the Iterated Functions sheet, observing that the two fixed points for 4i seem to attract much more rapidly than some of the other examples we’ve seen (the ancient divide and average method of approximating irrational square roots explains why; see if you can figure out the connection!), and seeing in 4j our first example of a 4-cycle (a limit cycle of length 4).

InCA Assignment for 11/20

See yesterday’s post for the link to the debrief from the Polygraph activity

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 a shared vocabulary when describing these entities. Today’s lesson reviewed that vocabulary.

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for 11/20, plus Test Warning

Finish reading Chapter 12. Pay particular attention to the sections on matching, blinding, and placebos, as these were important topics we didn’t enough time to discuss in class.

Your next Personal Progress Check has been assigned. Log into AP Classroom and complete the Unit 3 MCQ Part B PPC by Monday, November 25

On Monday, we will have our Unit 3 Test. We will spend Friday reviewing, and in preparation for that you should take a look at the review problems for Unit 3: pages 331-336, exercises 1, 3, 7, 9, 11, 15, 19, 20, 23, 29, 32, 40, 41. You’ll get answers and supplemental review on Friday.

Tomorrow, Thursday the 21st, you’ll have an in class investigative task for Chapter 12, where you will be asked to design an experiment for a scenario. Unlike other tasks, this will be closed notes/closed book, and your hand-written response will be due at the end of the period.

Fractals & Chaos Recap for 11/19

We reviewed results from 4b through 4h on the Iterated Functions sheet. Every example had a single attracting fixed point (and possibly an additional repelling one) 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.

Day 53 - Iterating Functions 4f

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.

Day 53 - Iterating Functions 4g.png

We will see plenty more examples of weird behavior in our iterations in the days to come. For tomorrow, be sure to read the article posted yesterday.

InCA 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 reflection. You will also need to bring in your Chromebook once more tomorrow for the third and final introductory activity to this unit.

AP Statistics Assignment for 11/19

Read pages 313-319 (up to the section on Confounding), then from the exercises do 7, 17, 30

For some additional reading about the ongoing Replication Crisis in science, see this article from The Atlantic, or this Crash Course video. Check out this New York Times article for more about Brain Wansink, the Cornell food science researcher who resigned amid this scandal.

Fractals & Chaos Recap for 11/18

We discussed the last of the exercises from #3 of the Iterated Functions sheet, then debriefed all of the observations of Ax+B again, deciding once and for all that Ax+B is solved. A fixed point can be found by the expression x = B/(1-A), but only the value of A impacts the behavior of the fixed point:

  • If |A| < 1, the fixed point is attracting
  • If |A| > 1, the fixed point is repelling
  • If A > 0, the pattern is direct
  • If A < 0, the pattern is alternating
  • If A = 1, there is no fixed point
  • If A = -1, the fixed point is neutral
  • If A = 0, the fixed point is “super-attracting”

We also observed that as |A| gets nearer to 1, the attracting/repelling behavior becomes slower.

We looked at another hypothetical non-linear system, then set to work on exploring more of #4 from the IF sheet, using the Iterated Functions Supplement as a guide. Your homework is to finish up through #4h (by tomorrow), and read Chaos and Fractals in Human Physiology, from the February, 1990 edition of Scientific American

 

InCA Assignment for 11/18

Today in class, we did a (hopefully fun!) activity introducing our next unit on Parabolas and Quadratic Equations. Your homework tonight is to reflect on two questions:

  1. Describe one thing that you learned from the Will It Hit the Hoop Desmos activity from class.
  2. What else do you think you could use a parabola to model the shape/path of?

Write your reflection on a piece of paper or record them in Socrative (room code G102KIRK)

Bring your Chromebook in tomorrow again for Part 2 of this introduction!

Fractals & Chaos Recap for 11/15

With Mr. Drix, you observed some patterns that emerge in cobweb diagrams for certain types of fixed points. For an attractor, the pattern of steps in the cobweb diagram will be drawn towards the fixed point; for a repeller they move away. Furthermore, if the fixed point is direct, the pattern of steps will look like actual steps as they move towards or away from the fixed point, trapped between the two lines defined by the function we are iterating and y = x. If the fixed point is alternating, on the other hand, the pattern will spiral around the fixed point with each step, producing a picture that most clearly gives the cobweb diagram its name.

From there, we took our first look at non-linear functions, realizing quickly that in addition to having more than one fixed point, such functions can have different types of fixed points. One could be a direct repeller, while another an alternating attractor. The cobweb diagram remains our best way to observe these differences, and we used the Geogebra-based applications found here to create them.

We looked at 4a (y = x^2) together, classifying the fixed point at 1 as a direct repeller and the fixed point at 0 as a direct attractor. We furthermore observed that -1 is a “pre-image” to the fixed point at 1, and therefore we have different behavior for different seeds. Your homework tonight is to continue and look at 4b-4f, using the cobweb diagram app linked above and the Iterated Functions Supplement to keep track of your results.

InCA Assignment for 11/15

In class today, you were given a preview of Unit 3, which will be a review of Quadratic Functions.

Complete the table using whatever methods you have available to you. Some of it you might remember from Algebra 1 or Geometry. Some you might need to look up online. If you need to go to the HW Help Room in K22, that’s fine too. You have plenty of resources available: use them!

I’ll expect this to be completed on Monday. We’ll review your answers in class.

AP Statistics Assignment for 11/15

After swapping responses to the AP review problems, see the scoring guidelines here:

For each, read over the sample solution, then look at the scoring guidelines, scoring each section of the response as an E, P, or I. At the end, convert the E/P/I breakdown to a numerical score (out of 4). Give the scored document back to its owner.


If you missed the videos shown in class, you can watch them here and here. The notes sheet we started in class can be found here.

For homework this weekend, in addition to the PPC below, please read pages 305-313 (stop at “Experiments & Samples) and do exercises 1, 2, 3, 9, 13.

Reminder: Your next PPC (personal progress check) has been assigned: Unit 3 MCQ Part A. It will be due by the start of class on Monday, November 18. As with the previous PPC, there is a 25 minute timer. You are not required to work by the timer, and its expiration will not lock you out of the PPC. It is there only as a guide for how long I expect this PPC should take.

Fractals & Chaos Recap for 11/14

With Mr. Drix, you continued your work with the Iterated Functions sheet, making some observations about how we might predict the classification of fixed point we get based on the parameters of the function we are iterating (see a friend in class for the exact notes on this if you missed them!)

We also introduced a new way of visualizing the behavior of functions: the cobweb diagram. In this style of graph, we draw the function we are iterating on the same plot as the line y=x. We pick a seed and move vertically to the graph of y=f(x), then horizontally to the line y=x, resulting in the output we just got becoming the input for the next iteration. We then move vertically again to y=f(x) and horizontally again to y=x, continuing until we get a view of the behavior of the function.

Your homework tonight then is to finish question 1 (parts f and g) and to do parts a, b, and c of question 3 of the Iterated Functions sheet. Draw both the time diagram and cobweb diagram for each of these functions, and try to make some observations about how the patterns of the two representations align with each other.

AP Statistics Assignment for 11/14

Click here to review my answers to the hospital drug testing examples you did in class.

In class today, you were given two free response questions from old AP exams. Finish your responses tonight. You’ll be trading them with a partner in class tomorrow and scoring them according to the official AP scoring guidelines.

Reminder: Your next PPC (personal progress check) has been assigned: Unit 3 MCQ Part A. It will be due by the start of class on Monday, November 18. As with the previous PPC, there is a 25 minute timer. You are not required to work by the timer, and its expiration will not lock you out of the PPC. It is there only as a guide for how long I expect this PPC should take.

Fractals & Chaos Recap for 11/13

We’ve started down the road towards understanding the mathematics of chaos by iterating linear functions. Imagine a recursively-defined sequence, where each term of the sequence is defined based on the previous value. That’s what we’re doing here: we start with a seed, then plug that seed into a formula. Each output becomes the next step’s input, and we seek to understand the long-term behavior of formulas.

We are primarily working with this classwork sheet: Iterated Functions. We’ve done parts a, b, and c for part 1, looking at the behavior of seeds both numerically and graphically, creating a plot of steps vs. value called a Time Diagram. In all three examples, we found a fixed point: a value of a seed that is constant through the formula (plugging that value in gets that same value out). Some of the fixed points were attractors, and some were repellers. We discussed how iterating the function repeatedly is a fine way to find an attracting fixed point, but obviously won’t work to find a repeller. The only option there is to solve the equation f(x) = x.

Tonight for homework, please do parts d and e of question 1, then all of question 2. You should work on identifying and classifying the fixed point (as an attractor or repeller).

InCA Assignment for 11/13

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

Cumulative IXL Modules