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 10/29

You had you Unit 2 test in class today, and you’ll do the test redo in groups tomorrow.

For Thursday, work on the Unit 2 practice exam, found on pages 264-266 of your textbook, specifically the multiple choice questions 1-15. We’ll start things off on Thursday with a brief discussion of these answers.

Note: Unlike other assignments from your textbook, this assignment is required and will be checked for completeness.

Fractals & Chaos Recap for 10/28

Report your findings for our final box-counting project here.

After talking so much these past few weeks about fractional dimension—dimension values that fill in the gaps between 1, 2, and 3 dimensions, we turn our perspective in the other direction on the number line: towards the 4th dimension. It stands to reason that the trends we’ve observed could be extended past the 3rd dimension, but considering fractals, or even Euclidean shapes, is immensely challenging for us as 3-dimensional creatures.

What helps is to consider the perspective by analogy: we can better understand the fourth dimension by putting ourselves in the mindset of a 2-dimensional creature considering the third dimension. Fortunately, this is territory that has been well-covered.

In 1884, British teacher and theologian Edwin Abbott Abbott published Flatland: A Romance of Many Dimensions. It is told from the perspective of A. Square, a denizen of the titular 2-dimensional world, and starts off explaining aspects of their universe in great detail. The second part describes his first encounter with the 3-dimensional Spaceland, and the changes to his world-view as a result.

Abbott wrote the book partly as an exercise in geometry, but also partly as a satire on the regimented Victorian-era social hierarchy. As a result, there are some rather uncomfortable characterizations of women as lower-class citizens, among other shocking commentary.

You can read the full book here (I have paper copies if you’d prefer that), but for tomorrow please at least read this excerpt.

Intro to College Algebra Assignment for 10/28

Your first quiz of Unit 2, covering all of our work with Operations with Rational Expressions, is tomorrow, October 29.

Today’s Links

Extra Practice Links (IXL Review)

Greetings IHS Parents/Caregivers!

Hello and welcome!  As I mentioned during the open house, I use this site to communicate with students, parents, and whomever else might be interested outside of class.  On this site, you can find assignments, announcements, links to instructional videos and online review, copies of handouts I distribute in class, and the occasional mathematical musing.

Please take some time to check out this website.  If you’re looking for a course syllabus or other information, you can find it under Course Information above.  If you’re wondering about how to obtain some extra help for your student, I recommend checking the Contacting me and Getting Help link above.  For more general information about what you can find on this site, check out the welcome post from the beginning of the year.

If you’re interested in learning a little bit more about my philosophy of teaching, please read Jo Boaler’s Fluency without Fear. I base much of my concept about my role as a teacher around what Dr. Boaler says in this article (her recent LA Times Op-Ed with “Freakonomics” professor Steven D. Levitt is also worth a read!)

Thanks for stopping by!

Fractals & Chaos Lesson Recap for 10/24

Today, used the Box Count method to find again the dimension of Great Britain (report your findings here) then completed one last project to find calculate the dimension of one of the spiral fractal seen on the last dimension calculation sheet (this took most of the remainder of the period).

For Monday, read pages 83-92 in Fractals: The Patterns of Chaos (about fractal math limitations)

If you’d like to rewatch Adam Neely’s Coltrane Fractal video we saw in class (or check out some of his related videos), click the link.

Intro to College Algebra Assignment for 10/24

We have a formal quiz on operations with rational expressions planned for Tuesday, October 29. We did some more review in class today.

Today’s Links

Extra Practice Links (IXL Review)

Intro to College Algebra Assignment for 10/23

We have a formal quiz on operations with rational expressions planned for Tuesday, October 29. Today in class we worked on a mastery mini-quiz.

Today’s Links

Extra Practice Links (IXL Review)

AP Statistics Assignment for 10/23

We discussed a bit of re-expression in class today, and tonight you should read pages 232-236 and 243-246 about an overview of why we re-express and some general strategies in doing so. From pages 251-252, do 18, 19, 20, 21, 22 for a neat little story about evidence for why Pluto isn’t a planet anymore.

Your next PPC (personal progress check) has been assigned: Unit 2 MCQ Part B. It will be due by the start of class on Monday, October 28. As with the previous PPC, there is a 35 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.

Finally: Your Unit 2 test will be on Tuesday, October 29.

Fractals & Chaos Recap for 10/23

After discussing the reading from the text and the answer to yesterday’s question of the border between Spain and Portugal, we moved on to the last method of finding dimension, the Box Count method.

This method of finding dimension produces the same table of values and log-log plot that we made with the Richardson plot, but the values of S and C are found using a different method. Imagine overlaying a grid on top of a fractal image. We then count (C) the number of boxes of that grid that contain some portion of the fractal. We then repeat this process using a grid with smaller boxes, the sizes of which relative to the original give us S.

The YouTube channel 3Blue1Brown has a great video summarizing all of this.

After enough counts are collected at different scales of boxes, we can create a log(c) vs log(s) plot and find the dimension using the slope as we did before. Your homework is to make the necessary counts with the coastline of Great Britain.

Also homework: For Monday, read pages 83-92 in Fractals: The Patterns of Chaos (about fractal math limitations)

Fractals & Chaos Recap for 10/22

We continued yesterday’s applications of the Richardson Plot to the Koch Curve and finally to the coastline of Great Britain, largely confirming Richardson’s findings as included in Mandelbrot’s article. The results of these can be found here.

Also, at the end of class today, we discussed the border between Spain and Portugal and looked at three maps.  Take the data below and answer the following questions:

  1. What is the dimension of the border between the two countries?
  2. One country has historically given the length of the border as 987 km, while the other has given a length of 1214 km.  Which country is which, and why might this difference have a logical basis (in other words, why might the countries have truly measured the borders in this way? The answer isn’t political!)
Step Size S C Distance measured
100 km 1 7.3 730 km
50 km 2 16.2 810 km
25 km 4 35.4 885 km
10 km 10 93.2 932 km
5 km 20 200.6 1003 km

Remember: for Friday please read pages 61-73 of your new book Fractals: The Patterns of Chaos

Intro to College Algebra Assignments for 10/22

For Period 3

We did some more work with adding/subtracting rational expressions today, this time with polynomial denominators. The key thing to remember with these is that we can only obtain a common denominator through multiplying. We cannot add/subtract denominators. See the notes and the video below for examples.

Period 3 Links

For Period 8


Extra Practice Links (IXL Review)

Fractals & Chaos Recap for 10/21

In class today, we derived a new method for finding a dimension of a fractal, the method proposed by Mandelbrot in his article and originally conceived of by mathematician and meteorologist Lewis Fry Richardson.  We observed that if we make a “log-log” plot (a so-called “Richardson plot”) of the step sizes and counts of steps that “fit” in a curve, the distribution of points comes out to a roughly linear association, the slope of which is the dimension of the fractal.

We concluded class today by testing this theory to find the dimension of a circle, the results of which can be found here.

We also got a new book, Fractals: The Patterns of Chaos, and our first reading assignment: pages 61-73 (on fractal dimension)

Intro to College Algebra Assignment for 10/21


I have some ideas of how homework accountability might change for our class. Please complete this Google Survey about homework. The survey is completely anonymous. Please be honest!

For Period 3

We spent today practicing adding, subtracting, multiplying, and dividing with an IXL assignment. Your homework is one of two:

For Period 8

We did some more work with adding/subtracting rational expressions today, this time with polynomial denominators. The key thing to remember with these is that we can only obtain a common denominator through multiplying. We cannot add/subtract denominators. See the notes and the video below for examples.

Period 8 Links

Extra Practice Links (IXL Review)

Fractals & Chaos Recap for 10/18

If you have a compass (the circle-drawing kind) please bring it with you to class over the next few days.

We discussed Mandelbrot’s Article, then used it to segue into a further discussion of dimension. As we are aware, we still have problems with the Hausdorff dimension formula for calculating dimension of fractals. It can’t handle fractals with stems (i.e., non-iterating segments that never disappear) and with fractals that are not exactly self-symmetric.

Today, we considered a football field, a circle, and a Koch Curve, and looked at how the size of the measuring stick we use to measure the length or perimeter of such things has an impact on the total amount of length we actually calculate. For a football field, the size of the stick makes no difference. We’ll be obtaining 100 yards worth of length even if we use a foot (S = 3) or an inch (S = 36) as our step size.

For a circle, this isn’t the case. Use a measuring stick the length of the diameter, and we can only make two steps before we end where we started. Use a stick the size of the radius (S = 2) and we can make 6 such steps (resulting in a measure of three diameters). Use a half-radius (S = 4), and we wind up with a total length of slightly more than 3 diameters. There is a limit to this, of course: pi*d, which is precisely the formula for the circumference of a circle.

For the Koch Curve, the story is very different. Use a step size the length of the original baseline, and we can make one step. Use a step size of 1/3 the baseline (S = 3), we can make 4 steps, giving a length of 4/3 the base. Use a step size of 1/9 the baseline (S = 9), and we can make 16 steps, for a total length of 16/9 the base. As we shrink the length of the ruler we use, the number of steps increases more quickly, and so the total length increases without bound.

We’ve seen suggestions at this idea before. In the second article we read (The Diversity of Life), we saw that reducing the scale of our perspective dramatically increases the amount of living space we can find. This idea is also found at the center of the coastline paradox, hinted at in the Ants in Labyrinths article (see also this blog post from UK Urban Planner Alasdair Rae)

We will be expanding on this in class tomorrow, including a discussion on what all this has to do with the dimension of what we’re measuring.

Intro to College Algebra Assignment for 10/18

For Period 3

We worked with adding and subtracting rational expressions today, modifying the usual “lowest common denominator” process to accommodate monomial denominators. We’ll do some more practice with this on Monday.

Period 3 Links

For Period 8

We spent today practicing adding, subtracting, multiplying, and dividing with an IXL assignment. Your homework is one of two:

Extra Practice Links (IXL Review)

Fractals and Chaos Lesson Recap for 10/17

We had our official Fractal Art Show Gallery Opening today!

For tomorrow, please read Mandelbrot’s revolutionary paper that sparked the recognition of fractals and fractal geometry How Long is the Coast of Britain? You might also want to read this version, where Mandelbrot himself explains how he originally wrote this paper as a “Trojan Horse” to introduce his vision of fractal dimension into the scientific community conversation.