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 10/31

Read chapter 10 (pages 267-276) for some more examples of running simulations, then from the exercises on page 277, do 11, 13, 17. For running your simulation in question 17, you should do at least 20 trials, using the random numbers found on Appendix page A-81 in your textbook or elsewhere, or use the random number generator in your calculator (refer to pages 273/274 for instructions).

Fractals & Chaos Lesson Recap for 10/30

We talked some more about hypercubes in class today, including drawing some pictures and building our own models out of gumdrops (take care of them!)

If you’re interested in reading more about the 4th dimension, check out the links below. I especially recommend the short story And He Built a Crooked House.

  • The Adventures of Fred, Bob, and Emily – a detailed look at how the lives of a 2D (Fred), 3D (Bob), and 4D (Emily) creature interact with each other. See especially the “World” section, where the author, Garrett Jones, imagines how wheels, water, and war would work in these universes.
  • And He Built a Crooked House – a short story by “Big Three” science fiction writer Robert A. Heinlein about an eccentric architect who designs a house in the shape of an unfolded hypercube. An earthquake hits, and the house folds back up on itself to concerning results (see also this student film version of the story)
  • Some Notes on the Fourth Dimension – some animations and movies showing the geometry of the fourth dimension, including some of those featured in the Flatland dvd bonus features.

Your homework: You were given a sheet of graph paper at the end of class. This is what you should do with it:

  • Fold it in half lengthwise (“hot dog” style).
  • Unfold and put a dot on the left edge of your crease.
  • Flip a coin (or use some other random procedure). If the coin lands heads, draw a diagonal that goes over and up one square. If the coin lands tails, draw a diagonal that goes over and down one square.
  • Continue with this pattern, creating a zig-zag across your paper, until you reach the other side. Bring that in tomorrow.

Intro to College Algebra Assignment for 10/30

We’ve moved on to the next phase of the rational expressions unit: Complex Fractions. Don’t be intimidated by their name! These are expressions that involve “nested fractions,” meaning the numerator and denominator of the expression are both fractions (or a sum/difference of fractions) themselves.

Remember the basic process for simplifying these beastly expressions:

  1. Combine fractions in the numerator (finding a lowest common denominator and so on) so you have one single rational expression
  2. Combine fractions in the denominator (LCD, etc.)
  3. Keep the top fraction, flip the bottom fraction, and change the operation to multiplication (KFC)
  4. Simplify.

Today’s Files

Cumulative IXL Modules

AP Statistics Assignment for 10/30

Reminder: For Thursday, work on multiple choice question 1-15 from the Unit 2 practice exam, found on pages 264-266 of your textbook. 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.

In addition, you received a playing card today and corresponding instructions about producing 200 coin flips. If the card was black, I’m asking you to actually flip a coin 200 times and record the results. If the card was red, I’m asking you to make up the result of 200 coin flips (not simulate them, totally make them up). Bring in those results tomorrow as well and I will attempt to sort your results based on how they were derived (either legitimate flips or made up ones).

Fractals & Chaos Recap for 10/29

We spent most of today discussing the geometry of hypercubes, and how the patterns of numbers vertices, edges, and faces develop over increasing dimensions.

If you’re interested in reading more about the 4th dimension, check out the links below. I especially recommend the short story And He Built a Crooked House.

  • The Adventures of Fred, Bob, and Emily – a detailed look at how the lives of a 2D (Fred), 3D (Bob), and 4D (Emily) creature interact with each other. See especially the “World” section, where the author, Garrett Jones, imagines how wheels, water, and war would work in these universes.
  • And He Built a Crooked House – a short story by “Big Three” science fiction writer Robert A. Heinlein about an eccentric architect who designs a house in the shape of an unfolded hypercube. An earthquake hits, and the house folds back up on itself to concerning results (see also this student film version of the story)
  • Some Notes on the Fourth Dimension – some animations and movies showing the geometry of the fourth dimension, including some of those featured in the Flatland dvd bonus features.

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

TBD


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)