About Benjamin Kirk

I am a New York State Master Teacher in Mathematics at Ithaca High School. I have been teaching at Ithaca High School since 2007.

AP Statistics Assignment, plus Test Warning

Your homework for tonight is to read pages 114-121 (up to “z in reverse”), then do exercises 15, 18, 20, 22, 23, 26 from pages 132-133.

Since Chapter 5 is the last of the first unit, we will be having our first unit test on Friday, September 28. We’ll spend some time Thursday reviewing, but you may want to start the review assignment early: from pages 134-146, exercises 5, 6, 7, 8, 18, 19, 21, 31, 32f, 33

Fractals & Chaos 9/21 Recap/Assignment

We practiced finding dimension with our new definition for a few additional fractals, including seeing a surprising result with the Dragon Curve, then spent the remaining time working on our fractal designs in lab.

This weekend, please read the passage Ants in Labyrinths, from Ivars Peterson’s The Mathematical Tourist. As usual, make a note of questions you have and passages you think are significant.

Fractals & Chaos 9/20 Recap/Assignment

We wrapped up yesterday’s lesson with an explanation of the formula for the Hausdorff Dimension of a fractal and a few examples of finding the dimension for the Sierpinski Triangle, Koch Curve, and a newly designed fractal called the Sierpinski Carpet (essentially the same design as the Sierpinski Triangle, but with a square as the starting shape). We spent the rest of the time in class working with FractaSketch.

AP Statistics Assignment for 9/20

Tonight, complete your response to the Chapter 4 Investigative Task. As with the previous task, your response should be completed in a Google Document and submitted to me (bkirk@icsd.k12.ny.us) by the beginning of class tomorrow. Please name your file appropriately.  The file name should have the format “LastName.FirstName.Ch4InvTask”. For example, mine would be “Kirk.Benjamin.Ch4InvTask“. Remember also that you may use no resources except for your own notes (including classwork sheets)

In order to create your graph, I recommend you use the online tools found at Stapplet, specifically the 1 Quantitative Variable, Multiple Groups tool. You can export your graph from the online tool, but you may want to take a screenshot instead. Instructions can be found here (I recommend the “partial screenshot” option).

Finally, we discussed in class how boxplots are great for comparing distributions of data, but maybe aren’t the best option for assessing a distributions shape. We saw this with some interesting visualizations here, which you’re welcome to check out on your own.

Fractals & Chaos Recap/Assignment for 9/19

Today was an important day.

We started with a conversation about dimension, acknowledging the difference between intrinsic dimension—the dimension of an object itself—and extrinsic dimension—the dimension of the space the object occupies. For example, the edge of a circle is one dimensional, since there is only one axis of movement around the circle, but the circle exists in two dimensional space. We further observed that the extrinsic/containing dimension of an object must necessarily be greater than or equal to the intrinsic dimension of the space.

We also discussed how the dimension of an object affects the dimension with which we can measure it. We cannot find the volume of a square, and we can’t find the area of a cube. Put another way, if we tried to use a cubic centimeter measuring device to measure the area of a circle, we’d end with a measure of zero. And if we tried to use a square centimeter measuring device to measure the volume of a cube, we’d get a measure of infinity. In general, too small a dimension of measure results in a measure of infinity, and too large a dimension results in zero.

Finally, we considered the implications of this to the Sierpinski Triangle. When we attempted to measure the area of the Sierpinski triangle, we resulted in a measure of zero (an infinity of triangles each with area zero results in a total area of zero). When we attempted to measure its length, we resulted in a measure of infinity (an infinity of segments each with length zero results in a total length of infinity). Thus, 2 is too high a dimension to describe the Sierpinski Triangle and 1 is too low. It must therefore mean that the dimension of the Sierpinski Triangle is strictly between 1 and 2, and is therefore fractional.


We ended class by deriving a formula that we could use to measure such dimension, called the Hausdorff Dimension, and given by the formula S^d = N, where S is the scale factor by which the fractal is shrunk when it is repeated and N is the number of times it is repeated at that scale.

For homework: Complete the Sierpinski Carpet drawing we started in class!

AP Statistics Assignment for 9/19

From chapter 4, read pages 83-90 and 93-95. Skip the sections on re-expression (or go ahead and read them, idc)

From pages 100-102, do exercises 20, 26, 27

Finally, read over the Chapter 4 Investigative Task. As we did last week, you’ll have some time in class tomorrow to start on this task, and it will be due by the start of class on Friday, September 21st (for what it’s worth, we do not plan on having one of these every week; it just works out that way at the start).

Fractals & Chaos Assignment for 9/19

We had a day working with FractaSketch today, which included an announcement of the 2018 Fractal Art Show.

The exact date of the art show will be determined later, but I will expect one entry from each student in three of falling into the following categories:

  • Fern
  • Tree (or shrubs, bushes, weeds, etc.)
  • Spiral
  • Realistic (other natural phenomena)
  • Artistic (patterns, designs, etc.)

Again, each student will be submitting three entries, each falling in a separate category.

Please feel free to work on your designs outside of class. We’re still trying to get the PC mobile lab open, so until then we will continue to work with the MacBook lab.

For tomorrow: continue to think about what the term “dimension” really means, per yesterday’s conversation.

Fractals & Chaos Assignment for 9/17

We saw the Sierpinski Triangle pop up in a surprising place today, arising from bending a single line segment in the same manner as FractaSketch does. How can it be, then, that the same image can arise from an obviously 1D structure as it did from an obviously 2D structure, as we saw before? What is the dimension of the Sierpinski Triangle?

To help us answer this question, we need a clear idea on what we mean by dimension. When we say something is “two dimensional,” what does that mean, exactly? Think about this and try to have some ideas to share on Wednesday.

Intermediate Algebra Assignment for 9/14

Due to the complexity of solving absolute value equations, we took today to do some more practice with this particular skill. So there are no new notes, but there is a new homework assignment: HW 1.5 – More Solving Absolute Value Equations.

Cumulative IXL Modules


Intermediate Algebra Assignment for 9/13

Our first day of new content! Today we looked at solving absolute value equations, and noticing how they always have two solutions. Check the links below for the notes, today’s homework assignment, and a video with a few more examples.

Today’s Links

Cumulative IXL Modules

AP Statistics Assignment for 9/13

Complete the Chapter 2 Investigative Task. Your response should be completed in a Google Document and submitted to me (bkirk@icsd.k12.ny.us) by the beginning of class tomorrow. Please name your file appropriately.  The file name should have the format “LastName.FirstName.Ch2InvTask”.  For example, mine would be “Kirk.Benjamin.Ch2InvTask“.

A reminder: This is essentially an open-notes, open-book take-home quiz. You may use whatever notes you have taken, and you may refer to your textbook, but you may not use any other resources than that. Please do not go online looking for more information, even supplemental background information about the cases cited. And definitely no working with other students on this assignment.

Intermediate Algebra Assignment for 9/10

Our first lesson of the first unit was on the Order of Operations, and how it can go wrong if you’re not careful! See below for links to today’s notes and homework assignments, as well as a short video discussing the proper order of operations

Today’s Links

Cumulative IXL Modules