Category Archives: Transformulas.org
I’ve been off. For a long time. But I’m coming back with lots to tell. Firstly let’s do a then & now to summarize a fews items.
I wanted to get back to working with people, in classrooms, so I applied for and was chosen to work at Heritage High School as an instructional technology teacher. It feels good to be a regular on a campus. I get to hear, “hi Mr. Butler” again. I missed that. I’m working in depth with a handful of teachers, running some side projects with digital citizenship and social media, and showing teachers ninja moves with ed tech. A couple of those teachers I work closely with have been starting to use Desmos, particularly the activity builder content.
My wife and I are expecting a child on valentine’s day. We don’t know if it’s a boy or a girl, and honestly I’m not sure if I’m biased one way or the other. People keep asking us if we’ve at least got a name. Some have encouraged to follow the J-trend from our families. My name is Jed, and I have an older sister (Julia) and a younger brother (Jake). Both of my brothers-in-law are J’s (Justin and James), and my nephews are Jace and Jayden. I think we’ve about exhausted it, so we’re aiming for something from the other 25 letters from the english alphabet.
For those that don’t recognize it, the image is from Son of Flubber, the follow-up of Disney’s Absent Minded Professor. Friends and family sometimes listen to what I say and they imagine this guy in the picture, conjuring up crazy experiments for the classroom. In my position as a coach, I’ve lost the opportunity to use my own classroom as a lab, but the alternative is actually turning out to be awesome. After some convincing, the teachers that I work with volunteer to host my experiments. I’ve been able to see students use Google drawings and slideshows to improve vocabulary in a Spanish classroom by personalizing the content, 3 ELA teachers are piloting a new internal blogging system that utilizes the open platform from wordpress.org, help support the video production course in establishing a daily news show, desmos activities in math classrooms, and building a digital citizenship program for the freshmen foundations courses. The assistant director in my district now shares a workflow spreadsheet with me entitled “Jed’s Hair-Brained Schemes”.
With all this excitement, I still want to do some old favorites – so I have plans for two big math + tech series, both housed over at transformulas.org.
- I love transformations, and I see how it builds a backbone for secondary math in today’s classroom. I need to share this conversation with others. So I’m writing about it over the next while (let “while” be somewhere between 6 months and a year; I really have no idea of the time line)
- Desmos Activity builder is awesome. I need to push myself to use it more. (Others should too). I want to dive into lesson (re)design playing in the desmos platform. No set goal here, but more a desire to build.
Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments. This post will look at using visual information from central angles and inscribed angles.
Students sometimes lack intuition for the measure of something. Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises. Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.
At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees. Built with this intuition in mind, we look at a triangle created by inscribed angles.
The next day we get to see the formula that collapses 3 ideas down to 1.
Dynamic Angles in…
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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.
End of Line
PS: bonus points if you get the geeky references.
There is a little bit of mystery and magic to these relationships, if you don’t believe me just ask Mr. Vaudrey. Students trust that a triangle is simple, yet if you asked them to communicate anything beyond the magical balance of 3 angles, and 3 sides, most wouldn’t know what else is true. Sometimes students see triangles as snowflakes, each one of them unique. Little do they know how much all these triangles are alike.
- Get quarter sheets of graph paper
- Draw your own unique triangle
- Color in the angles in each corner
- Cut Out the triangle
- Tear off each corner
- Piece together the puzzle, and what do you see?
Describe in your own words what’s happening.
This is HUGE. Students need time to digest this transformation. If it feels like the engine is stalling, change gears:
Start practicing lo-tech with some paper examples.
Constantly bring back the triangle with the transforming corners. Have the students take some
Then go back again to the applet. Get the class to a point where students are articulating what is happening in the triangle. Have them say it in multiple ways. These angle relationships have patterns and consistencies, but often get lost in the multiple perspectives (what about this corner, or that one, or the inside, or the out, or what does a parallel line have anything to do with it). If students can transform a triangle and its angles, then adding in the relationships with parallel lines is only a half step away.
Don’t forget to check out the other gems over at Transformulas.
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…