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.

AP Statistics Assignment for 12/3

Today, you formed your groups for the Midterm Project and started brainstorming ideas for projects. By the end of the period, I want you to submit a Google Document to me with 2-3 “We wonder” statements that show me what you are considering for your research question.

Each statement statement must start with the phrase “We wonder…” Where you go after that is up to you, for example “We wonder if…” “We wonder why…” “We wonder whether…” For now, it should be fairly broad, you’ll make them more specific later. See the pre-break post for ideas.

What you should avoid is “We wonder what will happen when…” That is a method-forward, not question-forward.

Fractals & Chaos Recap for 11/26

We did some more research in the logistic map and the catalog of behavior we’ve been using for the past few days. We’ve answered the first question from yesterday’s list: the first split happens at precisely a = 3.0. With a proof in class, we showed that this is because for a = 3.0, the slope of the curve at the value of the fixed point (which happens to precisely equal 2/3) is exactly -1. Any value of a less than 3.0 will have a slope at its fixed point that is shallower than -1, and as a result the fixed point is an attractor. Any value of a greater than 3.0 will have a slope at its fixed point that is steeper than -1, and as a result the fixed point is a repeller. So when we notice that there’s a 2-cycle at a = 3.1, this is because the fixed point is a repeller, pushing the orbit to the 2-cycle. This 2-cycle exists for all values of 3.0 < a < 3.1.

We also have seen some progress on question 3, finding a 5-cycle at a = 3.906, a 7-cycle at a = 3.702, and our very first 3-cycle at a = 3.83. A follow-up question we could ask here: Can a cycle of any length be found for a > 3.6?

We still haven’t answered question 2: whether or not both points of the 2-cycle at a = 3.4 split simultaneously to form the four cycle found at a = 3.5, or if, for the briefest of moments, one point splits before the other and we can find another 3-cycle somewhere between 3.4 << 3.5. Your homework this break is to investigate this some more.

InCA Assignment for 11/26

We finished yesterday’s Desmos activity with some curve sketching (drawing smooth polynomial curves that met certain criteria). There are a few examples of this in the homework.

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for 11/25

Complete the Unit 3 practice exam multiple choice questions (1-20) and free response question 1 from pages 336-341. This will be due on Monday, December 2 when we return from Thanksgiving break.

Also, we will be starting on our midterm projects after we get back. Over the next few days of break, you should read some blogs, listen to some podcasts, or watch/listen to the news to think of interesting ideas that you’d like to explore. You’ll be asked to come up with a research question and a design of a survey, study, or experiment that you could use to ask it. Some ideas of successful past projects include:

  • Are Double Stuf Oreos actually double the stuffing of regular Oreos?
  • Do various brands of fishing line actually hold the weight they claim?
  • Do students who play sports have better hand-eye coordination that students who do not?
  • Is there an association between students’ academic background and their knowledge of US geography?
  • Does salted water boil faster?

You can’t use any of these ideas, but these are the types of questions you could ask. Also valid would be to take a noted social experiment that you’ve learned about previously and redo it in the high school.

Notice that none of the questions above are “What will happen when we do x to people?” or “What is the effect of x on y?” Your project should be question forward, not method forward. A lot of the time, groups start by thinking about something fun they’d like to do, then struggle to think of a question they’d like to ask about it. This is backwards. You should come up with a question, then come up with a method of how to answer it.

On Monday, you’ll form your groups and start brainstorming ideas. A formal proposal will be due on Wednesday, December 4. Full information about the project can be found online here.

Fractals & Chaos Assignment for 11/25

We did some more work collecting information for the Catalog of Behavior for the Logistic Map function Mr. Drix introduced to you last week, getting a clearer idea on the changes in behavior as a increases towards 4.0. The patterns we observed raised a few questions:

  1. Somewhere within the interval 3.0 < a < 3.1, the single fixed point attractor “bifurcates” (i.e., splits) into a 2 cycle. Where exactly does this happen?
  2. Later, the 2-cycle becomes a 4-cycle. Do all points of the cycle bifurcate simultaneously, or can one split before another? In other words, is there a 3-cycle within the interval 3.4 < a < 3.5, or does the 2-cycle split directly to a 4-cycle?
  3. Once chaos appears past a = 3.6, is that it? Or is there anything more to be found there?

Continue your explorations of this function to try and answer these questions. Use the applets posted on the left side of the page. I have a special prize for the first person who can find a 3-cycle…


InCA Assignment for 11/25

We did another Desmos activity today about expanding our ability to discuss key features of any polynomial graph, not just parabolas. We observed that terms like “y-intercept” and “zero” still apply, but terms like “vertex” and “axis of symmetry” no longer work when there are multiple turning points on a polynomial graph. So instead we now have terms like “local maximum,” “local minimum,” and “inflection point.”

We haven’t finished our exploration yet, so the homework linked below is not due just yet. Take a look at it, though!

Today’s Files

Cumulative IXL Modules

Fractals & Chaos Recap for 11/22

We continued our discussion of the logistic map from yesterday by iterating the function for various values of the growth parameter a (which we identified as being bound between 0 and 4). We observed that if a is too low (a < 1), the population will die out and the destination of the orbit is zero. Once a passes 1, the population will eventually settle at some proportion of the maximum population; for example at a = 2.6, the orbit of iterations settles on a value of approximately 0.6154 (precisely, this is 8/13 of the possible maximum population). For a = 3.2, we observed a two-cycle of {0.5130,0.7995}, suggesting that the population here will year by year fluctuate between roughly 51% and 80% of its possible maximum population. For a = 3.6, we observed two bands of chaos bound within (0.32,0.6) and (0.79,0.9), suggesting that the population never dies out, but never settles at a stable value (or set of values).

This weekend, please continue to explore values of a and the destinations of orbits within this graph. Use the apps at to help, as well as this Logistic Function Cobweb Diagram I made in Desmos. Keep track of your observations on the Catalog of Behavior you got in class. By Monday, you should at least have an observation for every tenth value of a between 2.2 and 4.0.

InCA Assignment for 11/22

We did one more Desmos Activity in class today to practice our identification and communication of a parabola’s key features. On Monday, we will have a homework quiz about Wednesday’s HW 3.3 and today’s HW 3.5.

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for 11/22

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 spent today reviewing the unit, including looking at the Unit 3 Review materials posted here. If you have any questions between now and the test, please post here or send me an email!

If you missed the video about the Placebo Effect we watched in class, or want to take another look at it, you can find it here. This TED-Ed video is very good as well.

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!

You spent the next portion of class finishing 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. Using the ideas noted above, and confirming the few unclear examples with the Geogebra apps on, you classified the differing behaviors of each of those functions.

From there, we moved to a different family of functions defined by the graph y = ax(1-x). This function is called a logistic map, and it represents the growth of a capped population where x represents the ratio of the current population to a maximum sustainable population (as defined by the carrying capacity), and a is known as a fecundity rate. Naturally, both x and y are bound within the interval (0,1), as a population that reaches 0 is extinct and a population that reaches 1 will necessarily exceed its carrying capacity and become doomed. Your homework tonight is to consider this equation and those limits and to derive the boundaries for a.

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 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