Study for tomorrow’s Unit 4 test. Refer to the review materials posted yesterday, or do some additional review questions from the textbook (I can provide answers to those on request).

# Author Archives: Benjamin Kirk

# Intermediate Algebra Assignment for 1/28

We’ve started a new unit, this one on rules of exponents. Today was a review of pre-existing shortcuts, along with an introduction of some new ones.

Finally, because I want you to be able to work through this chapter without relying on the calculator, we will have a daily Powers Quiz. I will choose 10 powers from the powers reference sheet you make in today’s homework. I will ask you to evaluate them **without using the calculator** in a brief, timed quiz. We will do this **every day** through the rest of the unit, and only your **highest grade** on any one quiz will be the one used in your Unit 5 grade.

## Today’s Files

- HW 5.1 – Review of Exponent Shortcuts
- Notes
- Video Archive
- Review of existing exponent shortcuts (From Algebra 1)
- Negative Exponents
- Rational Exponents

## Cumulative IXL Modules

# AP Statistics Assignment for 1/28

Our test on Unit 4 will be on **Wednesday, January 30**.

You got a comprehensive review packet in class today, and I’ll ask that you work on finishing that tonight for homework.

Included in this packet (but not in the link above) is this sheet with some more practice with Chapter 16 specifically, my answers to which can be found here.

# Fractals & Chaos: Final Lesson Recap

If you are here to submit a course evaluation, please do so here, and thank you. If you enjoyed this class, please mention it to any of your friends who may be considering signing up in the 2019/2020 school year. Word of mouth is the best advertising for the class!

Read on for the recap of the last two days

# Intermediate Algebra Assignment for 1/18

No homework over exam week. Good luck on your tests!

# AP Statistics Assignment for 1/18, plus Test Warning

The applet for today’s activity can be found here. Using this simulator, find:

- Two
**very different**values of*p*and*n*that create a distribution that you would confidently identify as “normal” - Two
**very different**values of*p*and*n*that create a distribution that you would confidently identify as**“not**normal” - At least one value of
*p*and*n*that creates a distribution that*might*be normal, but you’re on the fence about.

For homework this weekend/next week, please finish reading Chapter 16, pages 423-429, then work on exercises 32, 33, and 34 from page 432.

Out Unit 4 Test will be on **Wednesday, January 30**. We’ll spend Monday and Tuesday after we get back from midterm exams reviewing the unit. Over next week, I recommend you work on the following practice problems from the Unit 4 review, pages 434-439: 5, 16, 20, 21, 23, 25, 27, 28, 37.

# AP Statistics Assignment for 1/17

First, some more about a big topic that came up at the end of class: The Difference Between an Assumption and a Condition

In order for a *model* (e.g., a normal model or binomial probability model) to be valid for a scenario, we must be able to make certain *assumptions* about that scenario. These assumptions can include the results of individual trials being independent from each other, the distribution of results being sufficiently unimodal and symmetric, etc.

If it is appropriate to make these assumptions, then go ahead and do it. But if it is **not** appropriate to make these assumptions, you can still proceed provided certain *conditions* are satisfied. These conditions mean that the scenario is “close enough” to allow the model to be valid.

For example, coin flips are independent. There is no finite population of coin flips that you are “drawing” a sample from, and so each coin flip’s outcome is independent of the next. Drawing cards from a deck are **not** independent, as the deck is finite and the probability of a certain outcome changes with each card that is removed. However, if the population is large enough, or more specifically *if the sample is small enough in comparison to the population*, then that probability change is very small, small enough to be ignored.

In general, as long as the **sample size is less than 10% of the overall population**, the probability change isn’t big enough to be worrisome. The reason why the magic number is 10% has to do with something called the Finite Population Correction Factor, and a thorough description of where it comes from and how it affects probabilities can be found here.

**Your homework tonight**

From pages 431-432, please do 19, 21, 26, and 28.

# Intermediate Algebra Assignment for 1/17

Your Unit 4 test is tomorrow. To prepare, consider the following:

- Work on HW 4.14 – Test Review 2
- Review my answers to both review assignments here
- Post here, or email me with questions you might have
- Read over your notes from the unit, and rewatch some of the videos I’ve posted through the unit
- Adding/subtracting radicals
- Multiplying/distributing radicals
- Rationalizing denominators
- Evaluating nth roots
- Simplifying radicals with variables
- Simplifying radicals with negative radicands
- Multiplying complex numbers
- Powers of
*i* - Rationalizing imaginary denominators
- Solving quadratics with non-real roots

- Work on some IXL practice modules
- H.4 Multiply complex numbers
- H.5 Divide complex numbers
- H.8 Powers of i
- J.6 Solve a quadratic equation by factoring
- J.9Solve a quadratic equation using the quadratic formula
- L.4 Simplify radical expressions with variables I
- L.6 Nth roots
- L.7 Multiply radical expressions
- L.8 Divide radical expressions
- L.9 Add and subtract radical expressions
- L.10 Simplify radical expressions using the distributive property

# Fractals & Chaos Recap for 1/16

We discussed an open problem related to the **connectedness** of the Mandelbrot Set today. It seems silly, but “connected” is actually a tough idea to define mathematically. We discussed the following ideas:

**Not disconnected**: That is, the set is not divided into pieces. We can’t draw a line “between” pieces of the set without crossing the set. The Mandelbrot set is not disconnected**“Simply” connected:**Both the set and its complement (everything*not*a member of the set) are not disconnected. A circle is not simply connected, because it divides the plane into regions inside and outside of the circle, which are disconnected from each other by the circle. A disk — a circle that includes its interior —*is*simply connected, as anything not in the disk is outside the disk. The Mandelbrot Set is simply connected**“Path” connected:**Any two points of the set can be joined by a path that stay in the set. We used the Topologist’s Sine Curve to define an example that is not “path” connected. The Mandelbrot Set, though,*is*path connected.**“Locally” connected:**Around any point of the set, we can draw a circle small enough such that the set is not connected inside of it. We used the idea of a comb defined by the line segments y = 1/2^x from x: [0,1] as an example of a space not locally connected. Mathematicians are still unsure whether the Mandelbrot is Locally Connected or not, an open problem known as the*MLC*.

The last topic, which we will get into tomorrow, is a brief history lesson of the Julia Set that starts in Newton’s Method for Approximation.

For tomorrow, please read this article from the November, 1991 issue of *Science News*, published just after an important discovery was made about the border of the Mandelbrot Set, as well as this follow-up of sorts from 2017 in *Scientific American*

# Intermediate Algebra Assignment for 1/16

Our test is on **Friday, January 18th** and today we started reviewing.

## Today’s Files

- HW 4.13 – Unit 4 Test Review 1
- Video Archive
- Adding/subtracting radicals
- Multiplying/distributing radicals
- Rationalizing denominators
- Evaluating nth roots
- Simplifying radicals with variables
- Simplifying radicals with negative radicands
- Multiplying complex numbers
- Powers of
*i* - Rationalizing imaginary denominators
- Solving quadratics with non-real roots

## Cumulative IXL Modules

- H.4 Multiply complex numbers
- H.5 Divide complex numbers
- H.8 Powers of i
- J.6 Solve a quadratic equation by factoring
- J.9Solve a quadratic equation using the quadratic formula
- L.4 Simplify radical expressions with variables I
- L.6 Nth roots
- L.7 Multiply radical expressions
- L.8 Divide radical expressions
- L.9 Add and subtract radical expressions
- L.10 Simplify radical expressions using the distributive property

# AP Statistics Assignment for 1/16

From chapter 16, read pages 418-423, up to the section titled “The Normal Model to the Rescue!”

From the exercises on pages 430-432, do 7, 10, 12, 17

# Fractals & Chaos Recap for 1/15

We wrapped up yesterday’s game of “Find the Julia Set” and reviewed some other interesting properties of the Mandelbrot Set:

- “Area” Julia Sets come from “areas” in the Mandelbrot Set, and “string” Julia Sets come from “strings” in the Mandelbrot Set
- There are “veins” through the main cardioid of the Mandelbrot Set where we can watch the orbit of a single fixed point die towards a 2-, 3-, 4-, or any magnitude cycle. Along these veins, the fixed point forms spokes. Offset slightly from these veins and they become spirals
- Relatedly, there are balls off of the main cardioid for every possible cycle, and every variant of every cycle (1/5-cycles, 2/5-cycles, etc.). Imagine a circle centered on the cusp of the cardioid. Any angle rotation around that circle points straight at a ball on the edge of the cardioid, whose orbit-cycle magnitude exactly matches the fraction used to find that angle. A 120 degree rotation, representing 1/3 of a full rotation, points straight at the 3-cycle ball on the top of the Mandelbrot set, and a 240 degree rotation, 2/3 a full rotation, points straight at the 3-cycle ball on the bottom. A 72-degree rotation, 1/5 of a full, points straight at the first 5-ball. Rotations of 144, 216, or 288 degrees point straight at the second, third, and fourth 5-balls.

Our last two topics of the course: an open problem related to the connectedness of the Mandelbrot Set, and a discussion of Newton’s Method of Approximation, and its surprising connection to the Mandelbrot Set

# Intermediate Algebra Assignment for 1/15

We revisited solving quadratic equations today, now with the capability of evaluating negative radicands!

## Today’s Files

- Notes
- HW 4.12 – Solving Quadratics with Non-real Roots
- Lesson Video (solving quadratics with non-real roots)
- Video Archive

## Cumulative IXL Modules

- H.4 Multiply complex numbers
- H.5 Divide complex numbers
- H.8 Powers of i
**J.6 Solve a quadratic equation by factoring****J.9 Solve a quadratic equation using the quadratic formula**- L.4 Simplify radical expressions with variables I
- L.6 Nth roots
- L.7 Multiply radical expressions
- L.8 Divide radical expressions
- L.9 Add and subtract radical expressions
- L.10 Simplify radical expressions using the distributive property

# AP Statistics Assignment for 1/15

From page 430, do exercises 1, 2, 4, 9. That’s all!

# Fractals & Chaos Recap for 1/14

The central question I have asked of you is **In what way ****are the shape of the Julia Set, the size of the cycle with it, and its location in the Mandelbrot Set related to each other? **Today, we started to answer this question.

First, we observed that orbits along the horizontal *x*-axis of the Mandelbrot Set bifurcate in the same way that they do in the Feigenbaum plot, and for good reason. Furthermore, we noticed that elsewhere along the edge of the central cardioid of the Mandelbrot Set, we can find balls that yield cycles of any magnitude, and in general **the size of the ball is inversely related to the magnitude of its cycle** (bigger ball = smaller cycle).

When we look at the Julia Sets we get in each of these balls, we realize that each of them have a central area, centered at the origin, and each with an infinity of “pinch points” found along its edge. The number of areas pinched together at this point **exactly matches** the size of the orbit contained! Given a mystery Julia Set, then, we can determine the size of its contained orbit just by its shape!

Finally, we observed that every ball connected to the central cardioid has an infinity of balls attached to it, also of decreasing size. The fractal nature of the Mandelbrot Set manifests itself here: the largest sub-ball corresponds to a bifurcation, the next largest to trifurcations, and so on. Thus, the 2-cycle found in the San Marco Julia Set can bifurcate into a 4-cycle, trifurcate into a 6-cycle, or any other multiple of two. The 3-cycle found in the Douady Rabbit can bifurcate to a 6-cycle, trifurcate to a 9-cycle, and so on for any multiple of 3. And this pattern extends to the Julia Sets themselves: the 3-ball off of the 2-ball creates a Julia Set that is a San Marco of Douady Rabbits:

And the 2-ball off of the 3-ball creates a Julia Set that is a Douady Rabbit of San Marcos:

With this in mind, we can play the game: Find the Julia Set. For each Julia Set in the attached file, use the shape/structure of the Julia Set to predict where in the Mandelbrot Set it can be found, and as a result the magnitude of the cycle found within it. Then verify your predictions using Fractal Zoomer.

# Intermediate Algebra Assignment for 1/14, plus Test Warning

As we continue to explore the use of *i* to represent the square root of -1, and the consequences that such a definition involves, we revisited the topic of rationalizing today, and noted that since *i* is a square root, the rules of Simplest Radical Form require that we not leave *i* in the denominator of a fraction.

Our test on Unit 4 will be on **Friday, January 18th**. Please do not be absent on that day! If you know for some reason that you will be absent on that day, please arrange a time **before the 18th** to take the test! Otherwise, we will have to schedule a time for you to take the test during exam week.

## Today’s Files

- Notes
- HW 4.11 – Rationalizing Imaginary Denominators
- Lesson Video (you can stop watching this video at the 8-minute mark. The examples he does beyond that point are beyond what we will do in this class)
- Video Archive

## Cumulative IXL Modules

# AP Statistics Assignment for 1/14

Tonight, after today’s quiz, please start reading Chapter 16: pages 413-418, up to the section “The Binomial Mode: Counting Successes”

# Fractals & Chaos Recap for 1/11

We spent the period playing around with Fractal Zoomer, and have made a few observations:

- The central “bulb” of the Mandelbrot Set is a cardioid, and each other bulb off of that central body is a perfect circle.
- Each bulb has its own cycle of orbits. Some bulbs have the same size orbit as others, but with a different “order” (1,2,3,4,5,6 vs 1,3,5,2,4,6 for example)
- If you tile the plane with Julia Sets of a sufficient density, the collection of Julia Sets make a photo-mosaic of the Mandelbrot Set. This further reinforces that there is something important between the location of C within the Mandelbrot Set and the shape of the corresponding Julia Set.
- Along the central “spike” on the left side of the Mandelbrot Set, we can find baby Mandelbrot Sets strung along that string. The main body of these baby Mandelbrots show cycles instead of fixed points. Realizing that the horizontal axis on which the Mandelbrot set is centered is the
**real number**axis, this suggests that there may be a meaningful connection between the Mandelbrot Set and Feigenbaum Plot…

Keep working with Fractal Zoomer and considering the questions posted yesterday. Please also read the section on Mandelbrot Set from pages 74-81 in your copy of Fractals: The Patterns of Chaos (and bring that book back on Monday).

# Intermediate Algebra Assignment for 1/11

More work with imaginary numbers today, in particular simplifying expressions that involve multiplying them or with higher powers. We saw in the videos we watched yesterday that powers of *i* don’t increase like other real numbers, but instead cycle through the numbers 1, *i*, -1, *-i*, and then back to 1 again.

## Today’s Files

- Notes
- HW 4.10 – Multiplying Complex Numbers
- Lesson Video 1 (multiplying complex numbers)
- Lesson Video 2 (powers of
*i*) - Video Archive

## Cumulative IXL Modules

# AP Statistics Assignment for 1/11

Your quiz on Chapter 15 will be on **Monday, January 14.** There is no special review sheet for this quiz, this time I advise you work on the following problems from Chapter 15’s exercises: 19, 20, 21, 36, 38, 41, 42, 52. You can find my solutions here.