Friday, May 27, 2011

Unit Circle and Wave Functions Project - Part 2

(This is a continuation of the previous post, talking about a previous GeoGebra project we did in Precalc.) Here are some samples of circles and associated waves that students did from scratch in GeoGebra: Part 1, where kids had to create a slowly counterclockwise rotating point along the unit circle; Part 2, where they had to make the circular path bigger; and Part 3, where they had to make the circle smaller and make the point rotate clockwise.

Here are some excerpts of student explanations as they reflect on how coefficients affect wave graphs and circular rotations. I pasted a different student response here for each question so you can read a variety of student "voices" as they attempt to articulate the concepts.

Student A - Question #1 asks the student to analyze the effects of coefficients on cosine wave and on the circle.

1. For the function f(x)=A*cos(Bx), A determines the radius of the circle and it therefore changes the amplitude of the cosine graph. In the circle, if A is greater than 1, for example 2, then the radius would be bigger because it would be 2 and the wave would be taller because it would go from 2 to -2. If it is less than 1, like 0.5, then the radius would be 0.5 and the wave amplitude would be smaller because it would go from 0.5 to -0.5. If the A happens to be negative then the graph would flip across the x-axis. When you flip the wave graph vertically along the x-axis the graph would start at -1 instead of 1. A horizontal flip across the y-axis wouldn’t matter for cosine.

If B is greater than 1 then the point takes less time to make a period because it is moving faster. For example, if B would be 2 then you would complete 2 periods in the time it normally takes to make 1. It B is less than 1, i.e.: 0.5, you are slower, thus it takes you twice the time to make a period. For B in the cosine wave, it doesn’t matter if it is negative or positive because it will be the same x-value regardless if the point in the circle is moving clockwise or counterclockwise.

Student B - Question #2 asks the student to analyze the effects of coefficients on sine wave and on the circle. Here, some of the explanations may duplicate part of their answer to #1, but I'm looking for the student to articulate that the sine wave flips if you make B negative, because it matters greatly whether your point rotating around the circle is going clockwise or counterclockwise, if you're tracking its height over time. This kid's response isn't perfect, but she's getting most of the ideas.

2. g(x)=A*sin(Bx)

A affects the amplitude of the graph and makes the circle smaller or larger. Say A=4 will make the graph have its y-maximum at 4 and y-minimum at -4; the circle will have a radius of 4. Say A=0.5 will make the graph have its y-maximum at 0.5 and y-minimum at -0.5; the circle will have a radius of 0.5.
This means that the point along the circle will have more or less height over a certain amount of time.

B affects the wavelength distance and the speed a point moves along a circle. Say B=4 will make the wavelength shorter because it will travel more distance over a certain amount of time. B=4 will make the point in a graph move faster. B=0.5 will make the graph have a longer wavelength. B=0.5 will make a point in a graph travel much slower.
This means that the point along the circle will travel in a slower or faster pace the height in a certain amount of time.

Sine is the height of the point along the circle over time.
*Signs of B do affect the shape of the sine graph because it is not symmetrical. This means that a point in the opposite side of another point, won’t have the same height.

Student C - The next two questions, #3 and #4, are fairly straight forward. The kids need to explain how you could manipulate sine and cosine functions in order to make the circle bigger, smaller, or to turn the rotational direction around.
3. In order for you to create either a larger or smaller circle you have to change the number in front of the sine or cosine formula. If you want the circle to be bigger the number in front has to be greater than 1 and if you want the circle to be smaller the number in front has to be less than 1. For example for a circle to be smaller the number has to be less than 1 such as C=(0.25cos(t), 0.25sin(t)). This will make the circle be smaller and have a radius of 0.25. On the other hand if you want the circle to become larger the value has to be replaced by a number greater than 1 such as C=(2cos(t), 2sin(t)). This will make the circle larger by augmenting its radius to 2.

4. In order for the point of the circle to rotate in a clockwise rotation you have to change the formula by adding a (-) sign before the (t) value. An example to make the rotation clockwise is C=(cos(-t),sin(-t)).

Student D - Questions 5 and 6 ask the student to analyze a given sine or cosine equation to tell me about the circular motion that is (partially) described. I'm looking for them to describe the size of the circle, its speed, the direction of rotation along the circle, and the (x, y) point along the circle where the point starts rotating at 0 seconds.

5. m(x) = 2sin(-5x) models a circle with a radius of 2. Point C is moving in a clockwise direction, meaning that it first goes down. It’s going 5 times faster than normal, which would be 5 radians/second. Point C starts at (2,0).

Student E - See above. This kid should have shown more work or explained how they got to their starting point (x, y).
6. What behavior is modeled by n(x)= 0.5 cos(x+3)? Include in your description the size of the circle, the speed, the direction and the starting location of the point that is moving along the circle.

Size: diameter of 1 (radius of 0.5)
Speed: Starts 3 seconds “before”- has a “head start”
Direction: counterclockwise
Starting point on the circle: (0.5, 0.7056)

Student F - For #7, the students needed to try to create a point that's rotating around a circle NOT centered at (0, 0).
7. [In order to create a circle whose center is not at (0, 0)...] I plugged in the formula, C = (cos(t)+3, sin(t)+3) and the center of the circle is (3,3). Sine would be f(x) = sin(x)+3 and cosine would be f(x) = cos(x)+3

This activity was very helpful to our later discussions on how to graph sine and cosine graphs by hand and how to set up and solve for simple trig equations, and what it all means. Stay tuned for more follow-up worksheets and discussions! :)

Wednesday, May 25, 2011

Unit Circle and Wave Functions Project - Part 1

A short while back, I set out to create a GeoGebra project that would help my juniors in Precalculus understand sine and cosine functions. I remembered that over the summer when I was playing with GeoGebra, I had discovered that if you define a point like C = (1, 0), GeoGebra sets the point down statically, but if you first define a slider variable (such as t) and then you define point C using simple trig as C = (cos(t), sin(t)), then GeoGebra creates a point that rotates in a circular path around the origin as you drag to change the value of t along your slider. I remember feeling amazed by this, even though obviously it made total mathematical sense.

So, I sat down in May to create a multi-day project for my Precalc kids, to help the meaning of sine and cosine sink in. It turned out to be pretty darn cool as a learning tool. In the project, the kids had to create several circular points of different radii and rotational speeds (we assumed t meant seconds, and C=(cos(t), sin(t)) thus meant that your point is rotating along a circle at "regular" speed, 1 radian/sec) and directions (clockwise vs. counterclockwise). For each rotating point, they had to create corresponding sine and cosine waves each with a point that traces along the wave as you drag t. This allowed them to verify visually/kinesthetically that that sine is always positive when the circular point is above the x-axis, and cosine is always positive when the circular point is to the right of the y-axis. And then they had to answer a bunch of questions to fully show me that they understand what circular motion is being modeled by
f(x) = 3sin(-0.5x), or g(x) = 0.5cos(x + 3), and WHY and HOW that circular motion impacts the shape of the wave graph.

Anyway. I've got to finish writing about this later. So, consider this the Episode 1 of the unit circle project. You can read about the project specs here. The really cool thing is that my kids submitted their writeup and GGB project files digitally, and so I will get to really show you the results of their understanding.

But, that will come. A bit later. That's Episode 2. Maybe Episode 3 I'll go into then how this feeds into their understanding of solving trig equations. Stay tuned.

Monday, May 23, 2011

Hot Off the Press

Our Honors Geometry magazines Issue 1 and Issue 2!

Stay tuned for the Precalc unit circle GeoGebra projects.

In other news, we sold our car today. It feels... glorious. I've got about 3 weeks left in San Salvador. OMG I love the feeling on the eve of a big move. It makes having only intermittent water supply seem almost bearable. :)

Wednesday, May 18, 2011

3-D project photos!

Check out the 3-D objects my regular Geometry kids designed, drew 2-D nets for, calculated surface area/volumes for, and then finally built:

I love them! :)

The year's winding down. I feel GREAT. We're definitely ending on a good note. Kids are doing some cool things across all of my classes these days. Life's been busy, but when I get a chance to catch up I'll write more about other projects besides this one. Ciao!

Saturday, May 14, 2011


The 3-D project turned out to be the coolest project I've done all year with my regular kids. The constructions looked amazing and I think I actually looked over ALL of their volume and surface area results before they submitted their work on Friday, so their calculation grades should be fairly high as well. It was really amazing -- the kids were super motivated because they thought designing/measuring nets and then building things out of a net was ridiculously fun, and they really understood when I said that surface area does not include any two surfaces that are glued together (ie. the bottom of a pyramid, if the pyramid is attached to another prism, and you'd also have to subtract that pyramid base area from the top surface of the prism as well). And they had to really understand slant height versus real height of a pyramid, in order to correctly calculate both surface area and volume of the pyramids. Some groups chose to make hexagonal prisms and hexagonal pyramids, which made their calculations even more exciting/challenging! A bunch of groups also built 3-D concavities into their solid, which added to their overall surface area and took away from their overall volumes.

It was really, REALLY great. On Friday in class I let the kids just have a fun day decorating their composite solids. Meanwhile, I went around and did a final check of their volumes and surface areas to make sure they understood all the big concepts of the project.

After school, some 11th-graders who came by to turn stuff in were amazed that the projects piled on my desk were made by regular 9th-graders. :) Yeah!!!

For sure, pictures will come. I can't decide if I should be taking pictures of their beautifully decorated solids or scanning in their beautiful perspective drawings and 2-D nets and calculations. This week has been CRAZY but oh-so-worth-it!!

PS. The geometric magazine is also coming along smoothly. I think in about a week I'll have a polished PDF I can post to share with you all! This entire week has been crazy from reading their gazillion drafts and offering detailed feedback daily, while managing other projects across every class. I haven't touched Google Reader all week; that's how bad it was. But I think I am seeing the light at the end of the tunnel, and their final drafts are quite lovely! :)

Tuesday, May 10, 2011

3-D Object Project

Last year, when I taught 3-D topics at the end of the year, we: watched Flatland (which we did again this year... the kids LOVED the movie and were bubbling with opinions about the existence of a 4th dimension); did some awesome computer investigations (which we are still going to do this year); and we also did a bunch of textbook practices involving volumes and surface areas of composite 3-D solids. This year, I scrapped the in-class algebra practices, am assigning just a few problems as Do Nows and daily homework, and instead we are doing an in-class project where kids design and build their own composite solids, for which they'd have to calculate the volume, surface area, and draw related nets.

My regular kids have already: learned how to keep track of surfaces (in computing surface areas); figured out how to find the height of a cone or pyramid given its slant height; reviewed how to calculate volumes and surface areas of a prism/cylinder or a pyramid/cone. In terms of their projects, most of the groups in my regular Geometry classes have gotten their 3-D designs and dimensions and nets with dimensions approved by me, and some groups are starting to cut out the nets to create the 3-D shapes.

I am excited! The regular projects are on track to be finished by Friday. (Honors kids will have a few extra days, since I want their efforts this week to be focused on polishing their Geometry magazine articles.) Their designs look pretty awesome and are all very different; the kids are practically tripping over themselves with excitement over this last project. Pictures to come!

Friday, May 6, 2011

Some Old Action Shots

My friend, the head librarian at the school, emailed me some photos today from a while ago when my kids were testing out their centroids in the library by trying to balance their triangles on top of a sharpened pencil. Cute!! (We went to the library because there we could sit in the coolness of the AC and not have the ceiling fans blow the kids' triangles all over the place. Plus, the library has flat tables, which are good for trying to balance things on.)

Here are a couple of good pictures. The kids were so intent on making their little triangles balance! They have to be absolutely serious and have still hands and their centroid has to be absolutely PERFECT for it to happen. It's a really fun activity, last year and this year. (Last year, as a fluke, I was able to balance my own triangle on a sharpened pencil held in my own shaky hand, while talking to the class, for about 15 full seconds! At some point I even took a couple of steps, and the triangle was still hanging on. They thought I was their hero.)

Monday, May 2, 2011

Square Root Fun

Today and tomorrow, we're spending Geometry class time exploring the geometric nature of square roots. Well, that's a loose description, anyway. It's a transition topic and something I loved from last year, so we're doing it.

Today my kids constructed using protractors and rulers a Spiral of Theodorus (see above). They started with the smallest right triangle measuring 1cm in each leg. In the margins they showed all the Pythagorean math work, which helped them practice a gazillion times how square root and square cancel each other out, and helped them see why $\sqrt{5}^2 + 1^2 = c^2$ means that $c = \sqrt{6}$, for example.

Then, we simplified things like $\sqrt{9}$ and $\sqrt{16}$ to verify with a ruler that our integer lengths matched what we saw on the ruler. And we simplified expressions like $\sqrt{12} = 2\sqrt{3}$ to make sure that we were able to compare the two lengths (as measured with rulers) and verify one length to be double the other. All in all, it was a productive, visual, and relatively pressure-free practice of their arithmetic skills without using the calculator.

Then, I took the opportunity to quickly explain visually what square root means and why it's not correct to simplify as such: $\sqrt{a^2 + b^2} = a + b$. I told the kids that it's one of those common mistakes in algebra that you can avoid if you have a geometric understanding of what's going on.

I drew this on the board and we discussed it briefly.*
It was immediately evident to the kids that you can't simplify as described above, because the area you have is "too small" to be broken up nicely into two sides of lengths $a + b$ each. I re-emphasized that if you have a geometric understanding, it helps you out a lot of times with algebraic concepts!

I'm excited about tomorrow! I will be doing more square root investigation with my kids... this time on Geoboards! My worksheet is based off of this lovely activity, but I re-worded/scaffolded it a bit more and added a few questions. I am, in fact, getting videotaped for this lesson, which makes me OH SUPER NERVOUS!!

Anyway, I am cherishing these last weeks with my kiddies. June is coming way too fast. I am running out of time to play Geometry with them! :(

*PS. I guess I wasn't being totally accurate in the second diagram. Should have used abs(a) as the square root value. Oops.