I’m a fan of visual context over simple memorization of formulas. Another pet peeve of mine is the requirement that denominators be rationalized without reason. A good conversation on rationalizing the denominator takes place over at “Why am I teaching this?”. One place I’ve see rationalized denominators lose their context is in a unit circle. I get that it’s a standard. I get that the angle radian measure makes most sense with a radius equal to one.
A friend of mine, Jen Silverman, makes some great protractors that help with this radian angle measure as well. I think my frustration comes from the idea that the unit circle is a beautiful overlay that simplifies so many interactive relationships between the trigonometry ratios. Too often students never come to this realization and instead resort to tricks.
Here’s another post using the same trick.
I used to try to get students to think of the Unit Circle key segments as just a small set of lengths that interact in different ways.
Some applets to help in relating the unit circle to the cartesian form of basic trigonometric graphs:
While others simply go back to memorizing the rationalized denominator forms.
Still, I feel the abstraction away from Trigonometric Ratios loses the relationships within the right triangles created by these points along the unit circle.
So my proposal is: Introduce Circle Trigonometry with a radius of 2. This would double all the segment lengths. Trigonometry ratios would relate more directly, and students could redraw and label the pieces more intuitively [it’s pretty hard to imagine what Sqrt(2)/2 looks like compared to Sqrt(3)/3)]. I haven’t built the applet for this, yet. Expect it to show up in a future revision of this post soon. If you’ve ventured this way before, or you’re interested in encouraging/discouraging me from doing so, please comment below.
It’s Thursday. Yesterday we talked about circles, chords, and kites. Today we asked a very similar question:
Here’s the applet in action: (click the animation to open the applet)
This was similar to Wednesday. We asked the same type of questions. We saw similar relationships. But it stood out enough for a unique post because a student enlightened me with an observation I didn’t articulate. When I repeated yesterday’s question, “What shapes do you see inside the triangle?” one student almost immediately replied, “Is that a kite?” I had to look at it myself. “Yeah, wow, that makes this conversation easy.” My original plan was:
- Focus on the Right angles/right triangles
- Question if certain segments were congruent
- Look at the reflection or congruence theorem that helps confirm the congruence
- then finish off with some color coding.
Instead this students recognizes the 3 kites, then refers to her knowledge of the symmetry in kites. Congruence, simplified.
To help students in transcribing the diagram onto paper to start doing some hand calculations we took a tip from a student in the first class:
- Lay Chromebook on its back
- Increase brightness of screen, turn off some/all light in class
This was another good introduction and discussion with segments in circles. We of course spent the last 1/3 or so of class practicing and becoming fluent with the skill with problems like:
At the end of the week we took a short quiz. 2 questions, same as these. Today/tomorrow I’m going to try something new with how I grade them (thanks to some inspiration from Michael Fenton, Michael Pershan, and Ashli Black). More on that later this weekend.
In the meantime, let’s keep rounding out this circle thing and see what other shenanigans we can come up with.
In geometry we had some fun with kite like shapes for the last couple of days. On Wednesday we asked:
See it in Action (click image to open)
I asked the students to play with the applet. I prompted them to ask for more information. Some noticed the sample questions below the applet and asked those to start the discussion. The key question had to do with decomposing the shape into other shapes that we had more familiar tools to work with. The students were quick to see the right triangles. Once we were able to identify that, the next question was, “How does that help me?” Getting students to connect and then apply relationships they know into a seemingly new context is a constant challenge, but they are getting better. I reminded them to recognize what are we trying to find. Once they narrowed in on the task of finding a length, and the length was a part of a right triangle, some started to see it, “Pythag Thyrem.” (this seems to be a tongue twister for the general high school math population). “Okay, how do you mean?” was my reply. Some didn’t see the given values for the hypotenuse that was also a radius.
Circles seem to be a pain for many geometry teachers, and I feel that it’s because so many people approach circles as a never ending list of formulas. We need to find ways to simplify the overall question and give students an opportunity to fill in the structure(s) needed to respond to the question. I know there are some awesome activities out there dealing with volume and area with round objects. Here I’m trying to put together a series of interactive questions the see the overlaps and relationships with circles, segments and angles.
— Joseph Williams (@jswilliams) May 1, 2014
I enjoy math, thoroughly. I also enjoy design and technology. Recently I’ve had a wildfire like experience in processing and learning material through Geogebra and Desmos. I’ve been learning and experimenting with these much faster than I could possibly archive organize the material. I look forward to the upcoming extended break at summer to truly polish this material. Currently I’m feeling more like
This little presentation I had tonight was a breath of fresh air. It was a mixed crowd and we had a great time. About half mathies in the room, and half techies. It wasn’t a large crowd, so we had a casual, yet productive time. At around 2/3 of the way through the time allotted, one of those in the room inquired about the coding behind some of the applets. Others seconded the question, so then we transitioned from math to tech.
Technology geeks, myself included, often dive into the code and lose some of the social part of the experience. Working on the backend of a program experience too often is a lonely one. Talking about this experience in a real life, social platform was great for myself and them. I was able to reflect and process on my wildfire experience of learning, and from what it seems they were about to start their own versions of something similar.
On the other end of the screen, it all looks so easy.
Geogebra and Desmos both use clean user interfaces that allow for wide audience. Knowing some of the coding and design on behind the scenes still has it’s place though. Recently education has seen growth in the art of coding. It has become more accessible with drag and drops like scratch, tutorials and screencasts shared freely online, and organized movements from large institutions like Khan Academy.
Now I would like to throw another idea into the mix of developing a coding mindset. For those of you that know me, you could probably guess that I’m thinking of Geogebra (and Desmos as well). Tonight we talked about the object oriented code experience it offers, and it’s simplicity with design and interaction. To toggle a picture or make an object move, all you need is a slider control and some checkboxes. I’m not sure where this can go from here, but I like it.
If you haven’t already you should check out these online interactive tools. And when you do, look at them as tools for tech and coding, not just math.
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PS: bonus points if you get the geeky references.
Two-dimensional area starts and ends in pretty much the same place, with base and height. Kids in elementary school calculate space by counting grids. Calculus classrooms do the same thing (on a more complex level of course), but through the short cut of integration. Somewhere in the middle, with geometry and the like, it gets complicated and students lose the conceptual understanding.
Here’s what we did instead.
Start with a few applets:
Then we document our thoughts. Some people call this notes.:
Students watch in amazement as if this were a magical experience. Audible comments of “wow” and “that’s cool” are common.
So then we conclude that there really is just one way to calculate 2 dimensional straight line areas:
base times height (and sometimes half)
Somewhere, I don’t know where for sure, a student learned that one common way to measure a full rotation is 360o. I would even say this generation is more aware of this concept with the popularity of sports that involve rotation(s). I could mention the number 1080 in a class and many students would have at least a minimal intuition that it had something to do with revolutions/rotations. Imagine you are that student and start from there:
The Basics (prior knowledge)
- Full Turn = 360o.
- Half Turn = 180o (call it straight angle)
- Quarter Turn = 90o (call it a right angle)
Number Sense (and decomposition)
“How many ways do you think you can fill in these boxes?”
After 1-3 minutes of getting out their Chromebooks, loading the applet and playing, I give the students a specific angle measure to move to, like angle BAE = 40o. Next a simple task: find all the measures.
“Mr Butler, I don’t know what to do?” “Mr. Butler, but there’s not enough information.” and more of this kind of thing for about 30 seconds.
Then one student elbows the other, “ow, why’d you do that?!”
“Check this out, I can find this one.”
“Oh oh, I can find the next one look here it’s a quarter turn, we just need the other piece and go backwards like we did a few minutes ago”
10 minutes later we share. And there are soooooo many different methods used, a good amount of them correct. In the end it was easy to check though. The total angles made a full revolution, the one measure everyone knew coming in. For those that were incorrect we talk about how the error took place, (was it arithmetic? or reasoning?)
We repeat this process with two or three more different starting measures for the angle controlled by the slider (let the students choose the angle for at least one of those).
Oddly enough, vocabulary seems to stick better once the students have a context to put it in. These quarter turns, that make a CCCCorner are call CCCComplementary, these half turns, that make a SSSSStraight line (or Straight angle) are called SSSSSupplementary. Somewhere in this process students start to notice the relationship in angles across from each other at an intersection are equal. We draw the criss-cross X and get the term Vertical. At this point we are only about 3/4 through 1 class period.
Move the angle to _____ and prove to me (with addition or subtraction) which angles are:
(There’s two for each)
Mr. Butler adds, “Hey class, oh and by the way, we just reviewed Algebra and solving equations.”
“You’re sneaky Mr. Butler”
Then the bell rings…
… to be continued…
The students walk in on Friday to wrap up a roller coaster like week (more posts on that to come). The students heard the music and saw a simple question:
We had been covering a unit on deeper extensions into 2-dimensional area and perimeter. This question fit right in with calculating the the area of a sector in a circle (also known to the layman as slices of pizza).
The students saw the question and almost felt lost. They didn’t even realize it was a question because it wasn’t specific enough. A common response was, “Do you mean which was is bigger?” I replied, “I don’t know, do you want more or less pizza?”
Students then commenced with table talk and less than 10 minutes later were ready with their responses. Before looking at calculations and actual numbers, we took a short poll from all the tables. For each class it the split was nearly even. So we settled the argument with the calculations.
The results were not exactly close. 1 slice had about 10 square inches while the other had 15. I noticed that we could make the decision harder by using a combination of slices to create an equal proportion. 2 of the 15 square inch slices, and 3 of the 10 gave two nearly equal groups. Now the extension.
I had created a #Geogebra applet before school that was originally intended to help visualize the comparison of the two pizzas. This allowed the students to interact with the question. The visualization of numbers is often a challenge, on both sides of the learning experience. These visualizations can be hard to create in a static snapshot, much less something that is interactive and dynamic. Organizations like Geogebra and Desmos have all but eliminated those obstacles.
For homework the students weren’t asked to complete 20 problems practicing this exercise. Instead the students were asked to create 2 more Would You Rather combinations with the following parameters:
- Each pizza needs to have a different diameter
- The difference in the areas must be less than 5 square inches.
- Show process/work for your calculations.
Not every lesson can be awesome like this one. But the format of activity in my classroom has changed significantly with dynamic programs like Geogebra and Desmos. It’s transforming into something much better.