Fractals & Chaos Recap/Assignment for 9/25

We discussed the article Ants in Labyrinths, then did some more practice with finding dimension of a variety of fractals. Some of the designs were templates, but some of them were “completed” fractals, which prompted us to discuss how to recreate the template from these designs. We have also already observed that the S^d = N definition for dimension may not be sufficient!

We’ll have some more time to work in the lab tomorrow, and for Thursday you should read On Being the Right Size, an essay written by biologist JBS Haldane in 1926.

Intermediate Algebra Assignment for 9/25

Today in class we did our Quiz 2 Review since we will be having our second quiz tomorrow.  The quiz will cover polynomial vocabulary, polynomial operations, translating expressions or equations, and word problems including perimeter word problems.  Attached below is a copy of the review sheet which you are to complete for homework as well as a cumulative list of IXL links that you may use for additional practice.  Tomorrow is the last day to hand in quiz 1 corrections as well.

HW 1.11- Quiz 2 Review

IXL: Additional Practice

Z.1- Polynomial vocab

Z.4- Add and subtract polynomials

Z.6- Multiply a monomial by a polynomial

Z.8- Multiply two binomials

F.1- Perimeter problems

F.3- Perimeter and area word problems

Intermediate Algebra Assignment for 9/24

Today we we did some more practice with perimeter word problems, and other word problems.  A copy of the class notes is attached below as well as the homework and additional practice.

We will also be having our second quiz on Wednesday.  The quiz will cover polynomial vocabulary, polynomial operations (addition, subtraction, multiplication), and translating expression including perimeter problems.

Class notes: More practice word problems and perimeter problems

HW 1.10: More Translating Expressions/ Perimeter problems

IXL: Additional Practice

Z.1- Polynomial vocab

Z.4- Add and subtract polynomials

Z.6- Multiply a monomial by a polynomial

Z.8- Multiply two binomials

F.1- Perimeter problems

F.3- Perimeter and area word problems

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.

Intermediate Algebra Assignment for 9/21

In class today we did an activity where in groups you found the area and perimeter of different classroom items.  You then created your own word problems and added them to a google doc. Your homework is to answer the questions created by your classmates.  Please make sure you do the right set of questions for your period.

Period 2 HW 1.9- Perimeter Word Problems

Period 8 HW 1.9- Perimeter Word Problems

Intermediate Algebra Assignment for 9/20

We did a fun activity today on creating list of words that imply the different operations, addition, subtraction, multiplication, and division.  We later discussed translating expressions and equations to and from written sentences and algebraic representations.  Attached below is a copy of the class notes from today, and tonights homework.

Operations vocabulary list

Copy of class notes

HW 1.8- Translating Expressions/Equations

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.

Intermediate Algebra Assignment for 9/19

Today we covered a lot of material during class.  We went over polynomial vocabulary, adding polynomials, subtracting polynomials, and multiplying polynomials.  Attached below is a copy of todays lessons notes as well as a copy of the homework assignment and additional practice problems from ixl.

Copy of notes: Polynomial vocab and operations

HW 1.7- Polynomial operations

IXL: Additional Practice

Z.1- Polynomial vocab

Z.4- Add and subtract polynomials

Z.6- Multiply a monomial by a polynomial

Z.8- Multiply two binomials

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.

Wow.

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 9/17

Today we reviewed for quiz 1, which will be tomorrow Tuesday 9/18.  Quiz 1 will include the following topics: Order of operations, solving linear equations, solving linear inequalities, evaluating and solving absolute value expressions and equations. The homework assignment is to complete HW 1.6- Quiz 1 Review. You can find the answers to this review here!

Cumulative IXL Models