Intermediate Algebra Assignment for 1/2

We expanded on the specifics of Simplest Radical Form today by adding the requirement that expression written in SRF have no radical expressions in their denominator. The process of tweaking expressions to accomplish this is called rationalizing.

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Fractals & Chaos Recap for 12/21

After the weird results we observed yesterday with Phi and the golden ratio, and its ubiquity in nature, we doubled-down by showing that we can use Phi to derive an explicit formula for the Fibonacci Sequence. Yes, the most famous recursively defined set of integers can be derived explicitly using an irrational number. Wild. If you missed it, check out this video from Mathologer.

I also demonstrated another Fractal-generating software: IFS Construction Kit. Selecting “rotation” mode in this software allows you to define a piecewise-linear function in complex numbers, one that allows a greater amount of flexibility by using different dilation or rotation factors for the real and imaginary components of the complex number you are iterating. Use the default settings when the software opens up, and you get a very familiar image…

This break, please think about the back of Complex Paint Worksheet 1. It is a summary of the work we’ve been doing with complex linear functions, and completing this will allow us to move on to the final stage of the course: iterations of complex quadratics.

Intermediate Algebra Assignment for 12/20

Yesterday we discussed adding/subtracting radicals (and a bit of dividing). Today is multiplying

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Fractals & Chaos Recap for 12/19

We’ve been reflecting on the question of why the pattern of attraction for some values of R for the same value of T will produce spiral patterns and why others (specifically slower values of R closer to 1) will produce spokes. A clue was found in discussing T = 0.32, with the note that 0.32 is close to, but not quite equal to, 1/3. So every rotation of 0.32 is close to a perfect 1/3 rotation. But since 0.32 < 1/3, every 3rd step is slightly short of a full 1/3 rotation, so we wind up with a clockwise spiral pattern (T = 0.35 on the other hand would be slightly ahead of 1/3 with every three steps, so it creates a counterclockwise spiral pattern).

But why 3 specifically? 319/1000 is closer to 0.32 than 1/3 is. So is 6/19. Why don’t we see 1000 spiral arms, or 19? The answer to this has to do with a novel way of breaking down real numbers into a sort of skeletal structure: a continued fraction. Each step of a continued fraction produces rational approximations of a real number of progressively improved accuracy. It allows us to claim that some rational numbers are actually “more rational” than others and, even more surprisingly, some irrational numbers are “more irrational” than others. But most immediately, the denominators of these approximations correspond exactly to the numbers of spirals we see.

We confirmed this with T = 15/62 and T = 5/17. The continued fraction sequence for T = 15/62 is [0;4,7,2], producing a first rational approximation of 1/4 and then 7/29. The number 15/62 < 1/4, producing the four CW spirals we see in Complex Paint, and the number 15/62 > 7/29, producing the 29 CCW spirals. The sequence for T = 5/17 = [0;3,2,2], giving approximations of 1/3 (too big -> 3 CW spirals) and 2/7 (too small -> 7 spirals).

Not only does the continued fraction give us rational approximations, it gives us a way of assessing how “good” these approximations are. We notice in the complex paint that the 4 spiral arms of 15/62 last for several steps of R (it’s not until R = 0.99 that we even see the 29 spiral arms), whereas T = 5/17 only shows 3 spiral arms for R = 0.9, and then only barely (we can also somewhat see the 7 spiral arms at this stage). This tells us that 1/4 is a better approximation of 15/62 than 1/3 is of 5/17 (which makes sense, since 15/62 is a lot closer to 0.25 than 5/17 is to 0.3333….).

So, a “good” rational approximation can be illustrated by the following characteristics in Complex Paint:

  • It is easy to see, and not hidden by another pattern
  • It is straight or spoke-like
  • It lasts through several magnifications (R = 0.9->0.95->0.99->0.995->…)

We can see this numerically as well: the third term of the sequence for 15/62 (a_2 = 7) is bigger than the second term (a_1 = 3), suggesting that we’re not adding a lot when we add on the third term of the continued fraction. On the other hand, the third term of 5/17 (a_2 = 2) is less than the second term (a_1 = 3), suggesting that we’ve left off a lot by stopping at just two terms.

All of this leads to why we see 7 “spokes” which very slowly turn into spirals for T = pi. The rational approximation 22/7 is a very common, and accurate to three digits, approximation of pi. The 7 “spokes” we see at low values of R slowly tighten as we move R closer to 1, until we get 113 “spokes.” This suggests there is another, even more accurate rational approximation of pi with a denominator of 113. Your homework tonight is to complete the continued fraction sequence for pi and find that approximation.

Intermediate Algebra Assignment for 12/19

We have started Unit 4, a unit working extensively with radical expressions. We can treat these expressions as a class of numbers in their own way, with their own “rules” for adding, subtracting, multiplying, and dividing them. Today we started with adding/subtracting.

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Intermediate Algebra Assignment for 12/17

Your Unit 3 test is tomorrow! See below of a list of resources you should use to prepare.

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Intermediate Algebra Assignment for 12/14

Your Unit 3 test is on Tuesday, December 18. We started our two days of review by making a study guide, which you should finish for homework if you did not.

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Intermediate Algebra Assignment for 12/13, plus Test Warning

The last bit of new material for this unit takes us back to the connection between the roots/solutions of a quadratic and the x-intercepts of its graph. This relationship exists even with polynomials of higher degrees, which means we can use that relationship to factor using its roots! We introduced two tasks today:

  • Using the known roots of a polynomial to write the polynomial, and
  • Using the graphing calculator to find the zeroes, and by extension the factors, of the polynomial.

These are the last two new items, in addition to everything else you’ve learned this unit, that will appear on your Unit 3 test on Tuesday, December 18.

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Intermediate Algebra Assignment for 12/12

We reviewed the key features of parabolas today, but this time from an algebraic perspective. We’ve already seen that the roots of the parabola (aka the x-intercepts or the zeroes) are the solutions we get from the quadratic equation. We saw today that the equation for the Axis of Symmetry can be obtained from the standard form of a quadratic equation using the formula x = –b/(2a). And since the vertex is on the axis of symmetry, the x-coordinate of the vertex is that same value (and the y-value is obtained by plugging that x-value into the equation).

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Fractals & Chaos Recap for 12/11

We spent some time reviewing answers to the Complex Numbers Transformations sheet from yesterday, and used the principles to find the precise value of the fixed point you were finding graphically last week with the transformations Dilation x1/2, Rotation 45°, Translation Up 4.

We then introduced our next Very Important Program: Complex Paint. This software was created by former F&C students Devon Loehr and Connor Simpson, and will be invaluable to us going forward. We’ll discuss the features of the program as they become relevant, but after downloading/unzipping the folder listed in the Google Drive folder linked, read the README file for help on how to use the software.

Your homework tonight is to finish the back of the Complex Numbers Transformations sheet.

Fractals & Chaos Lesson Recap for 12/10

We reviewed that the values of a and b of the Rectangular Form tell us the horizontal and vertical components, respectively, of the translation achieved by adding a complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.

To better understand this requires practice, and so we spend a considerable amount of time in class today converting between rectangular and polar forms for a single complex number and identifying the transformations they refer to. This work was on the Complex Numbers Transformations sheet, and your homework is to finish the front as well as questions 3, 4, and 5 on the back.

Intermediate Algebra Assignment for 12/10

We took a moment today to pause and look back on the three methods of solving quadratic equations that we have discussed: factoring, the quadratic formula, and the square root method. We discussed when each method might be most efficient, but reinforced that point that the quadratic formula is always a valid solution method.

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AP Statistics Assignment for Week of 12/10

You will spend this week working on your projects. You should strive to be done with data collection by Thursday the 13th, and you’ll have up to Monday the 17th to work on your presentations and written report. Refer to the contents of the

We’ll spend the 18th-21st presenting on your findings, and your reports (one per group!) will be due on the 21st.

Intermediate Algebra Assignment for 12/7

The third method of solving quadratics is to simply use the square root, but there’s a danger here: you still have two solutions. The + is built into the quadratic formula, but it still needs to be added here. And that falls to you, the solver!

Today’s lesson also included a brief revisit to word problems, this time resulting in a quadratic equation.

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Intermediate Algebra Assignment for 12/6

You had your solving quadratic equations quiz in class today. I want you to do some more for homework. Tonight, work on IXL Module J.9 Solve a quadratic equation using the quadratic formula to a score of 80 or higher. Remember: any score above 80 will count as extra credit!

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Fractals & Chaos Recap for 11/5

We started in class today watching a Numberphile video about the Feigenbaum Plot and an interesting number that can be found in it that appears to have some surprising universality. The video does a great job of recapping what we’ve done over the past few days, so watch it if you’ve missed anything. You were also given an article to read about the plot and this constant.

The video also makes a point that the pattern we see in the Feigenbaum Plot is not unique to the Logistic function we’ve been iterating. In fact, any function that creates a bound area with the x-axis can exhibit such behavior. With the rest of the period, play around with Paul Fischer-York’s Bifurcation Diagram. Use the dropdown in the upper-right corner to examine diagrams for other functions. Some other functionality:

  • Use the Darkness slider to make the image darker and easier to see.
  • Left-click and drag a rectangle around any portion of the diagram to zoom in on that portion.
  • Right-click anywhere in the diagram to run a series of iterations at that value of a along the x-axis. The software will identify the magnitude of the cycle you’re looking at (though sometimes it makes a mistake, so look at the list as well!)

Use this map to explore the Sharkovskii Ordering mentioned in the article above. Why does that ordering make sense given this picture?

Intermediate Algebra Assignment for 12/5

You will have a quiz tomorrow on solving quadratic equations. Today you were given a review sheet (see below) with some more practice. Work on this review and the IXL modules linked below in preparation for tomorrow’s quiz!

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