Polar Graphing Activities

Happy #mtbosblaugust once again!! I was thinking about some of the activities I did in Honors Precalculus and AP Calc last year to reinforce some new or tough ideas, thinking about what worked and what to use again this year. I’ll briefly try to describe something I learned last summer at an APSI from Dr. Joe Brandell, our AP Calc guru!

Graphing polars is a topic discussed in my honors Precalculus class since we now have a majority students who move onto AP Calculus BC. I do everything from plotting points by hand, plotting on graphing calculators and on Desmos. All seem abstract to my students but they are fun to play with, especially with sliders on Desmos. I don’t want to even think of a time before Desmos existed!

Back to the activity from Dr. Joe. He introduced our group to human polar graphing!! You will need a long rope and a lot of space, oh, and a class of students! Depending on how new or quick students are at computing the radius from given theta, you may want to give them an equation or two to create a table. Pick a point to be the pole (origin in polar) and how far out a radius of 1 or 2, etc. will be.

The first student will hold onto the end of the rope and find their first point in the polar space you created. Then the next volunteer will hold onto the rope at the point of the second coordinate, etc. Eventually, a circle or a cardioid or a rose…will be in the space with kids as the coordinates and the rope as the curve! Human Polar Graph!!

Another polar graphing activity that I do thanks to Infinite Sums can be found here. Now, his goes on and on and I strive to do that with my group! We started small this year.

It was very fun, and who doesn’t love to create with sidewalk chalk?! This is a great activity to get the students to practice plotting points and graphs. It usually shows up at a nice time of year where we need to get out of the class.

AP Calculus MPACs

Nothing major to share tonight, but did create some classroom decorations I wanted to put out there. AP Calculus has Math Practices similar to the common core. I made some displays to remind me and my classes (AP and Honors Precalculus) why we do the things we do.

The first is a link to the actual MPAC banners. This second link is to the subtopics banners. These are editable PowerPoint files. I will try to edit this post with PDF files if I think of it later.


A Few Changes

I have taught Honors Precalculus for years…though I try to switch some things up, mostly to keep things fresh for me. This semester I thought that the first chapter needed a couple changes, since the things at the end tended to be the things students missed (if anything) on the test. Rather than doing the Piecewise Functions graph right after the Parent Functions (as the book does) I thought to teach Operations on Functions with Domain and then go back to graphing after. It gives the students some extra time to work with composition of functions, and it also spreads graphing out, leaving time for students to process that. I think it was a good choice. Students did well on their test, even on the composition questions, something that in the past they may have gotten wrong because not enough days were spent thinking about the topic and how to work with the domain. Here are some pictures from my Interactive Notebook on these topics.





I also changed some of the notes from past years…with more emphasis on composition of functions.

Polynomials Chapter

This chapter did not turn out as well as it could have…we had state testing in the middle of it, and the distractions and days off from class did not help a lot of my students. It’s nothing I can control, so it is what it is. I changed my organization of the chapter in my notebook a little bit, and tried my best to adjust for all of the interruptions.

I start this chapter with a factoring review since this chapter expects the students to be able to factor any factorable polynomial. I found a great flow chart to include in my notebook on a precalculus blog. I used a few more of his ideas later in the chapter! I love finding resources out there, so hopefully someone else will be able to benefit from my posts as well. I also gave students practice problems to complete after their last test.

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The chapter starts with quadratics, a quick review. I’m finding that no matter how much time is spent on these functions, students struggle.  It’s a lot to sort out, when to set it equal to zero, when to look for the vertex, etc.

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Next, we talked about division and the theorems that utilize the operation. And then some vocabulary terms.

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This part of the chapter, students are creating polynomials given some “zeros”. I showed them how to more easily multiply polynomials, or to organize their work a little more using charts. These are honors students, but they appreciated not having to distribute as usual and to try to add up like terms that someone go all over their paper. It’s good that they have as much trouble reading their own work as I do sometimes!

We then go through a few tests that help students to make a challenging polynomial more factorable, and then graph these polynomials. I tried to teach end behaviors using limit notation, to introduce the students to this topic. We will work more with this notation later in the class, it is just meant to be an introduction. During this time, I have students explore the shape of the graphs around the zeros on this  WS_Exploring_Polynomial_Functions_with_repeated_linear_fac.

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At the end of the chapter, I gave students a worksheet I borrowed from Sam Shah. It gets them thinking about polynomials in reverse, writing equations from the graph. They have to pay attention to the end behaviors, zeros, even y-intercepts. equations from graphs ws

My project for this chapter is related to this idea as well…Birthday Polynomial Project. I found the idea for this project on Miss Rudolph’s site. I can’t wait to see what they come up with!

Function Chapter – Honors Precalculus

I made a few changes to my first honors precalculus chapter this semester. Having gone through a whole course with my INB once, I decided I had a lot more room and didn’t need to cram a lot of information on one page. The unit starts out discussing functions and all that is related, domain, range, notation… I got the vocabulary sheet from Math=Love blog, just retyped it myself to have 2 per page rather than 6.

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The inside of this notesheet is modified a bit from last term…I think students made the transition from inequality notation to interval notation a lot easier by starting with that, then discussing domain and range from the graphs. I’m trying to emphasize the importance of color a lot more this term, and students seem to be taking to that better than last semester as well. Some were even using color on the short answer portion of their test at the end of the chapter.

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This even/odd functions page is new…Students weren’t sure about taking notes without a notesheet in their INBs, but mine were planned out enough that I could tell them to divide up the page into three sections to be sure they had enough room.

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So is the brochure-like page about Analyzing Functions. I wanted it to make more sense to me, hopefully then making sense to the students a little better. The Left Hand Side activities and problems are also causing me to assign less homework problems, because there is time set aside in class to do some practice that most students seem to get done because they know they will get immediate feedback on it.

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My last INB did not have a Difference Quotient/Rate of Change page…I changed that here.

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DQs are still a work-in-progress. I have to remember to keep assigning one of these here and there. Next, graphing topics and operations on functions.

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I think my students appreciated the little graph squares on the piecewise functions pages. It made it easier to budget space as well as to keep the graphs neat. I think as far as the composition of functions go, I need to put more emphasis on determining functions that were composed together (breaking down the composition) because that is how they will be used the most in Calculus. This unit, I did a few group activities too. One is a translations assignment: Translations_and_the_Coordinate_a_b_GW1. The others are saved at school, and I will try to remember to post them later!

Ice Day, Christmas Break, 2 Snow Days and Maybe More?!

It has been a while! The title of this post pretty much sums up what I’ve been doing, a whole lot of nothing (in regards to school!). I also noticed that there are a few things I missed in my posts from last year, you know, a couple of weeks ago!

Here are some things from my INB in Honors Precalculus…

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We worked on the binomial theorem, and Pascal’s triangle before break. Then I had 2 days to talk about rational functions, which leads us into limits with Honors Precalculus (Pre-AP) so I searched the blogs and this is why the MTBoS is so fantastic…of course Rebecka had exactly what I was looking for! This note sheet worked perfectly to review rational functions with my students.

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We were then supposed to have a test the Friday before break, but my title tells what is going on with that! I think my notebook is going to come in handy when we get back to school (whenever that may be). Students will be able to look through that to remind them of what we were learning before the extended vacation. A lot of modifications are going to have to be made to the rest of my curriculum though…exams are supposed to be the end of next week! What do you do when your semester is suddenly cut short? We are on block scheduling, and are done with these classes at the end of next week…

180 – ??? Honors Precalculus Edition

I’m still working on catching up with my blog. In Honors Precalculus we are in the middle of our chapter on Sequences and Series. My students had fun with the notes we took the first day of the chapter:

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I think they liked the colored paper. I’ll have to track down some more of that for future use. Anything to mix things up. The Vi Hart videos on the Fibonacci Sequence are also fun to watch this chapter. Just go to YouTube and search Vi Hart Fibonacci. It’s a 3 part series, and there’s a lot to learn on the brief videos.

More with arithmetic and geometrics:

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This is mainly review from Algebra Two, but we reinforce the main idea, and focus a lot of attention on the applications.

Thursday and Friday this week involved Mathematical Induction, a little proof to brighten their days!

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