Tuesday, September 16, 2014

A Math Teacher Teaching Science?!?

Last year I started to wonder what it would be like to teach a new course, because I've taught the same courses for a while now. I guess I made the fates laugh heartily, because guess what happened?!?

This year I'm teaching grade 10 science for the first time. I had taught science and physics in the very distant past, but I am a math teacher by trade and by comfort. New course? Check.

But the science isn't the only new thing. The students in my class happen to be teen moms, who are in a program and a school designed for their particular situation. Which means that they aren't able to do much, if any, homework between classes. So no flipped-class videos either! Whatever happens, has to happen during class. New student situation? Check. New strategy? Check.

AND there is a provincial exam at the end of the year, which they have to pass in order to graduate from high school. Just so long as there isn't too much pressure!

Week one: In which I go to an actual school

Last week was my first week with the girls, and since their school isn't far from here, I drove downtown to meet them face-to-face. (If you're wondering why a teacher would even mention this, just know that I teach online, and don't usually get to meet my students.) We had a friendly chat, and I got to meet their babies, because the nursery is right there in the school. Which one girl said made it hard sometimes, because when you hear your baby cry and you can't go to them, it hurts. I certainly get that.

It was especially important for me to meet these students right away, because if anything is going to get them through this year, it'll be relationships. Sometimes when you're tired and fed up, the only thing that makes you show up is that you like the person or people you're going to see. I hope that's how they'll feel.

Week two: Atoms & baby germs

This week so far has been good in the sense that I've gotten a lot done during class - so far, we've done the different models of the atom, electron configuration, and the periodic table. But there have been a lot of absences already - you know how it is when you're exposed to baby germs, the strongest germs known to man or woman. I really have no idea how to deal with that.

Behold all the science things!

For the last two weeks, it's been Christmas for me. I got to open the Science folder in Smart Notebook's Gallery - woo hoo there's gold in there! And now all those gorgeous geogebras that people made for science? They're mine to covet! That part's been fun - I always felt like science people just had more stuff!

An Act of Love

The greatest impact of all is that I met with the ladies who run this program. They are so passionate, devoted, no-nonsense, and team-oriented that I feel like I've been hit with a lucky stick just to work with them. I've met only a few people in my life that I could honestly say this about, but what these ladies do every day is an act of love. It's hard work, it doesn't pay a lot, it's not glamorous, and sometimes the people you're trying to help only get that long after they've exited the building. But they do it because they love it and they know it's important and they own the job.

Kind of like what moms do for their kids.

Sunday, August 17, 2014

Less Paper, Not Paperless

This past week's #flipclass chat exit ticket was to write a post about our workflow, which I have to admit I didn't even know where to start with. Then Brad Holderbaum helped me out by asking me this:
This is in response to my answer for Q2, which was about how paperless our classroom is. Now that I am actually thinking about it, I'm not sure mine's as paperless as I thought. Since I teach online, everything that flies between me and my students has to be, at some point, in digital form, but as it turns out, there are wildly varying degrees of paperless-ness during the year. To measure the degree, I'm looking at how much paper is involved in each task at each of these stages:
  1. The actual task
  2. How I deliver that to them
  3. What they do
  4. How they deliver that to me
  5. How I assess
  6. How I deliver that to them
I looked at five different types of tasks, and filled out this table with green = no paper, red = paper. It made it easy for me to see that the most recent things I've been giving them are, or can be at least, 100% paperless.


A little more detail about those tasks:

Tests:

When I give a test, I email it to each of the schools' secretaries, who then prints them up. The students then write on it, and it then gets either scanned or faxed to me. If it's faxed, obviously, there's more paper used. Either way, from then on, it's paperless. I don't put my corrections on any paper. By now their tests are all in digital form, either as an image or a pdf, so I send them to Smart Notebook, where I can mark them up with digital ink, stickers, whatever. Here's how that actually looks in real life:



Assignments: 

These might be a worksheet, or a set of problems, or one big multi-step problem (called a situational problem here in Quebec). I deliver it via our CMS (Sakai) but I know most of the students print it up right away. I always give the option to do it on paper, but unlike the tests, they can also do it using some digital tool, such as voicethread, or geogebra, or whatever they choose. I usually offer bonus points for that - questionable I suppose, to motivate using marks, but hey. At any rate, as long as it's presented so that I can follow their reasoning, it's all good. If they go with the paper, then it's exactly the same degree of paperlessness as the test, but if they do it digitally, for example, like this, I create a rubric in Smart Notebook, fill it in with digital ink as I do the tests, export to pdf, then upload that to their dropbox on Sakai. Here's what that looks like to the student:


Blogpost / Geogebra / Portfolio entries:

My students all have their own blogs, as well as digital portfolios. The blogs are of course viewable by anyone, and they are linked from the classblog, but the portfolios are only viewable in our immediate community. Usually the post or the portfolio entry is about an applet they're creating using Geogebra, my favourite dynamic geometry software. Regardless of which of these three they're doing, the degree of paperlessness is the same - 100%!

Toward the end of last year, I gave them a task involving all three of these things. I wanted them to use their portfolios to track their own progress and to get feedback from me, then use geogebra to confirm and organize their learning, and once they were ready to commit, share it with the world on their blogs.

Here's how that looked: assignment description, assessment descriptionstudent's post, and a filled-in rubric:


Note that all of these points came directly from the assessment description that I linked above. I didn't want my students to be surprised at how I would be assessing their work! Next year, I plan to get them involved in creating the rubric itself.

I can't show any students' actual portfolios of course, but I did do a post that showed snippets from their reflections for another assignment, this one involving just creating a geogebra. (I got their permission to share, and they're anonymous anyway.) You can read about that here if you're interested to know what I mean by portfolio entries.

Good on paper:

Full disclosure - several times a week, my students write notes, on paper, based on the voicethreads that I create for them to watch. If I put that into the table above, IT WOULD LOOK LIKE THIS. I know it's a lot of paper, but I also find that writing on paper can be a worthwhile thing to do, so I don't know if I'll ever want to change that. I'll reduce, but I don't see myself ever being 100% paper-free. Besides, like my mother, I love books too much!

Wednesday, August 13, 2014

After the Ten Stages of Twitter

This post has been in draft mode for about 3 months. I've never stared at that publish button for as long as I have for this one. Then this summer, at Twitter Math Camp, in Jenks, Oklahoma, this happened:

Photo credit: Greg Taylor @mathtans
What is this? This is one of the slides presented by none other than Dan Meyer. His presentation was about who the Math-Twitter-Blogosphere (#MTBoS) is and, among other things, how the members of it use twitter. He showed us some very interesting stats, some of which are in this picture. Under #FOLLOWING are the top three people in the MTBoS in terms of how many people they follow. If you look closely, you'll see my twitter handle. Yup. If Greg had taken a picture of me at this moment it would have looked like this:

♫ Psycho shower scene music ♪

Because it means out of all the people in the MTBoS, almost nobody follows as many people as I do. Dan mentioned that he'd like to hear about how one would manage this many followees. Well, here it is. How do I manage it? I don't. Which is why I was writing this for so long. Nothing like being a statistic in a Dan Meyer presentation to motivate finishing that 3-month-old blogpost! Here it is, folks:

I remember when I read The Ten Stages of Twitter, by Daniel Edwards. I recognized every single one of those. But now I'd like to add a few more stages to his list, based on my own recent experience, maybe yours too.

Stage 11: Vexation.

I'm not really sure when this stage started, all I know is that at some point, my twitter experience started to sour, and to distract me from my growth as a teacher. Reading my twitter feed used to make me feel stimulated and inspired, but suddenly, I was getting vexed instead.

Part of the vexation was sheer quantity. I was following too many people. Checking the general feed felt like drowning, getting pulled in too many different ways. I had tried to filter by creating all kinds of lists, but they also got too big. My "top ten" list had 27 people on it. And I wasn't very good at remembering to check each list anyway.How did I even end up following so many people? Some people I followed simply because they were nice enough to follow me, and after all, I am Canadian. But whether or not I ever saw any of their tweets afterward, or connected at all...for the most part, no. I couldn't honestly call twitter my PLN, because everyone can't be in your PLN. If yours includes everyone, then it really includes no one.

But that wasn't the main reason for the vexation.

Some tweets were actually upsetting me.

I'd see an unbelievably sarcastic, condescending tweet about someone or something, and wonder how an educator could behave that way - especially toward another educator. I often thought - How would this teacher feel if they suddenly realized that their students had witnessed this whole exchange? Would they be proud of themselves? Would they feel they had modeled respectful debate? I know no one's perfect, but shouldn't we try to move through this world the same way we want our students to? That means we treat each other the way we want to be treated, even on twitter.

I have to admit, some of the aforementioned tweets were also very funny, but does that make it okay? I don't think so, but based on how many retweets these funny retorts got, it seemed that on Twitter, as on the playground, sometimes mean was masquerading as clever.

And then there was the twunning.

In real life, when you speak directly to someone, and they don't answer you, it's rude. If they then turn to answer someone else instead, it's downright insulting. And if that happens over a sustained period of time, it's a form of harassment called shunning. I don't know what it's called on twitter, twunning maybe? But it sucks.

I wondered if it was just me - maybe I'm not supposed to have the nerve to be addressing a tweet to someone who has thousands of followers. And of course I get that some people don't have enough time to answer all the tweets they receive. But then someone like Darren Kuropatwa or Pernille Ripp would have that excuse, right? But they DO answer. And besides, isn't that what twitter is for? The chance to connect with ANYONE?

Stage 12: Self-analysis

Instead of getting more and more upset, I had to ask myself what I really wanted out of twitter. Because clearly, either I was looking for the wrong thing, or looking in the wrong place.

I realized that to a certain extent, I was looking for attention. I saw my melancholy self sitting by a stream with a dozen fishing lines going, resting my chin in my hand, just waiting for a bite. Not appealing. So I had to face up to that childish need. This was really hard to admit. *pats self on back*

But I was also looking for authentic and meaningful interaction - with people who are interesting and interested in the same things I am, who challenge me, or who just make me smile. But interaction is by definition two-way, not one-way. So people who tweet some deep thought, then get a zillion responses but never engage beyond that, (or only do so with a select few), have at it, but that's not what I was looking for at all. At all.

Most important of all though -  I needed to take care of the solid connections I had made already. I knew that the twitter fire hose was making me miss out on the good stuff - not only the info that I could truly use, but tweets from people who really mattered to me, with whom I did interact. Those people need interaction and responses too, just like me.

Stage 13: Re-twittering

It was about destroying in order to rebuild:

First of all, I got rid of a lot of the lists that I had.

Then I unfollowed a lot of people. Unfollowing someone is hard, because I'd never want to hurt someone's feelings. I didn't do it to be mean though, or as a tit-for-tat thing, I just felt I had to trim it down. It's simple logic to me - if interaction is what I want, and it's not what's been happening with @PersonX, I unfollowed. But that started to be really tedious and it felt really negative too.

So I took another tack - a more positive one. I decided to take some responsibility toward my actual, real, honest-to-goodness PLN. I realized that a PLN isn't just about what you get, it's about what you give.

I made a new list. A really short one. These are people who already consistently interacted with me. Some on a daily basis, some less frequently, but at least consistently. Or who did any of the following:
  • get me
  • are nice
  • answer me
  • have some common interests to mine, but that's not an absolute necessity
This list has become my PLN. There are teachers on it, there are gardeners on it, there are relatives on it...I can't even characterize any commonalities other than that I know I'll get just as much as I give to these people. I'm sure that being on this list confers no special honour, because it's only a confirmation of a relationship that already existed anyway. By the same token, I'm sure that not being on it won't upset or insult anyone. I don't assign such importance to myself. But I do to my feelings, and to my PLN.

My New Twitter Routine

Now the first thing I do when I get onto twitter is check my daily list. It currently has 46 members. Totally manageable. It doesn't take long for me to scroll through that feed, I see everything these blessed people are saying, I respond because I want to and I can! I feel like I'm in the company of old friends.

One last stage: when online becomes face-to-face:

I hope everyone gets to this stage eventually. When you get to meet these people face to face, after being twitter friends for years. I had the opportunity to do that at a tweet-up in Ottawa, then at Flipcon14, in Mars, Pennsylvania, and also at Twitter Math Camp, in Jenks, Oklahoma. This is a whole other-nother level. (apologies to English teachers everywhere.) Now when I read the tweets of these people, I hear their voices, I see their smiles, I remember moments we shared. The line between online and f2f just got blurred.


Twitter, like life, is a double-edged sword. There's just as much potential for good experiences as for bad ones. But I do have way more control over my twitter experience than my life one, and this change of tack has made a HUGE difference. No more vexation, WAY more growth, and wonderful friendships. I just hope my Daily list doesn't get out of control.......cue Psycho shower scene music again......

Wednesday, July 2, 2014

Big Ideas at Flipcon14

Flipcon14 took place in Mars, Pennsylvania June 23-25, and I'm happy to say I was there. I had previously attended Flipcon12 in Chicago. My perception may be a little skewed, but it felt like this conference had a Big Idea feel to it that I don't remember from the Chicago one, maybe because I was still a relative beginner then. Where did this feeling come from?

First from the keynote speaker, Molly Schroeder. She said these three words:

Think - Make - Improve.

That's what all these teachers have been doing with their own work, with other teachers' work, and it's what we want our kids to do.

Later, Brian Gervase said this, which I just had to tweet:
Throughout the conference, I found that even though most of the sessions I attended were not specific to math, what I heard was nevertheless applicable to any subject. Big concrete ideas that are making their way through the Think - Make - Improve cycle, taking on new colours as they move into different subject areas, then branching out further, in a sort of learning fractal.

A rundown of what went down, and my takeaways:

Day 1: Session A: Andrew Thomasson & Cheryl Morris:

Creativity: Biggest takeaway for me was that routine is a significant factor to creativity. Counter intuitive for me! I always thought it happens when it happens, you can't schedule it. Blogging, for example - whenever I've heard people say they stick to a blogging schedule, I've thought, well not me, I wait to be inspired. But it turns out there is evidence that routine really does help people be more creative.

Gradual release of responsibility: The year begins with bootcamp, and ends with work for which the student assumes full responsibility. During bootcamp, the essential skills are covered, like how to watch a video, how to talk to peers, how to take notes. For math class, it will be about those exact same things, plus using Geogebra and Desmos. 

Opportunities to practice: no grades for these, simply practice on the essential skills but building in variety, such as taking notes from a variety of media, video, text, or a website. Perfect for what I'm planning for next year - giving my students more geogebra and less direct instruction, so they'll need to take notes from their own explorations with geogebra.

Grammarly: Cheryl showed how she uses this tool with her students. She feeds the students' text to grammarly, which it then scans it for grammatical or spelling errors, then it reports how many errors there are. It's up to the kids to find them and fix them! I'm going to do the same with math examples, maybe even create a geogebra with mistakes in it, and have them find and fix them. I can train them to use geogebra at the same time, kill two birds with one stone if you will.

Networking by subject
This was a great idea - one entire slot of time devoted to informal chatting amongst people who teach the same subject. Of course, I headed to one of the math rooms. We organized ourselves into groups by level, and I ended up in a circle of about 10 people who teach senior high. We had a great time just sharing our questions, ideas, general thoughts. There was a teacher there who had come all the way from China (if I recall correctly)!

This was just the right amount of subject-specific stuff for me. And even this discussion yielded some generalizations - via Steve Kelly, for example, about how kids organize themselves into groups according to their ability, which usually turns out to be about 4 different levels.

Soon after this, I started to feel overwhelmed, even though it was still day 1. I expressed this to Steve Kelly, who was kind enough to tweet it out:
Session B: Book panel:
Next it was time for all the coauthors of this wonderful book:


to join Jason Bretzmann, our publisher and tireless supporter, for the panel discussion he organized. Jason gave a wonderful presentation about the book, and had some questions for us all to take turns answering. It was an honour to be included in this group of talented people.

Does it kind of look like Jason has a halo? Just saying.

That evening we were all bused to the beautiful Carnegie Science Center in downtown Pittsburgh. I have to say, I had no idea that Pittsburgh was such a beautiful city! Our view from the science centre cafeteria was stunning - we were sitting in a valley, where two rivers become one, surrounded by an astonishing amount of greenery. Lots of huge yellow bridges, too! We had a great time eating, learning, and DANCING. Michelle Karpovich said it best:


Day 2: Session C: Jonathan Thomas-Palmer
Videos: Jonathan makes physics videos full-time, so he's pretty good at it. He used to teach, and flip his class using videos, but found too many students did not watch them, so decided to make them so good that they would WANT to watch them. He gave us tips on making engaging videos.

Audio is of prime importance, get a really good mic, check your audio level, don't film outside
Talking head should be doing more than just talking - pick up stuff, point to parts of presentation, anything to vary. Don't film with light behind you.
Frequent visual changes ie text popping up that paraphrases what speaker has just said, arrows, callouts

Jon cautioned against trying to make something like this, with one person playing multiple characters:



....unless you have a professional grade software. Darn. That looks really fun.

Session D: Brian Gervase
This was one of those sessions that got me so worked up that I didn't take any notes. I spent this session either tweeting what I was hearing Brian say or just picking my jaw up from the floor, being stunned that someone else could speak my thoughts so eloquently and passionately. Brian's session was called Flipped Assessments. He uses mastery with his classes. Here are some of the tweets I managed to make, apart from the one at the top of this post:

Anyone who knows me at all knows what my favourite math edtech tool is, and at one point during the session I became afraid I might lose what little professional composure I still had:

As it turned out, Brian ran out of time in his session, which is probably for the best, because he later replied:
Session E: Crystal Kirch
Crystal explained her WSQ system, which she uses in her math class.  I had read about it, and had even tried a form of it before, but it's always way better to hear things explained f2f and see it for yourself. Crystal's talk, like all of her writing, isn't just about the details of what and how she does the WSQ, or the TWIRL, it's much deeper and further-reaching. Brian said it best:
Crystal also spoke of routine in her class, just as Andrew and Cheryl did. Bigger ideas included Organization, Accountability, Processing, Feedback, and Discussion, all features of her classroom, which we can all use, regardless of what we teach.

Session F: Stacy Lovdahl and Eric Marcos
Stacy gets her students to create videos as projects, and Eric's students create videos to help their peers' understanding. In either case, the underlying idea is purpose. Kids learn best when they're making something that they see the purpose of. Think - Make- Improve must be happening constantly when kids make videos. There are other benefits, though, such as kid-friendly language, both in the video and in the feedback other kids can then give:

One example of the benefits of student-created videos was Stacy's - you get 16 kids to each create a video showing an example of a chemical change, and boom, not only did they engage in their learning, have a purpose, but they also now have 16 examples! My favourite rationale for kids to present their learning this way instead of in front of the class is here:

Biggest takeaway, biggest idea from Flipcon14:

There is so much variety in what teachers who "flip" are doing, that the word really doesn't mean anything anymore. But I will still use it because:

Friday, May 23, 2014

There's knowing and then there's KNOWing

Knowing:

My students did this activity yesterday, in which they unwittingly drew a parabola using geometry instead of algebra. So they "knew" about the focus and directrix of a parabola, and that all the points on a parabola are the same distance from the focus as they are from the directrix. And they "knew" the formula c = 1/(4a).

But I had a sneaking feeling they didn't really KNOW, you know?

KNowing:

So today I showed them this:


...and asked "Which point is the focus of this parabola?" There were guesses for each of the colours. Some said they all could be the focus. Bingo. They know what it is but not what it isn't. That's not KNOWing.

"If the pink one is the focus, then which line has to be the directrix?"

Everyone went with the pink line of course, but their reasons were varied:  because the colour matches, because it's the farthest away, because it's the same distance from the vertex as the pink dot.

Right. We've just established that as soon as you have a focus, you also have a directrix - they work as a team. That's a slightly bigger picture. Also I need to knock it off with the colour coding.

KNOwing: 

"Is it possible that any one of these could be the focus, as long as we pair it with the correct directrix?" Some said yes, some said no. 

Time to test out their hypotheses. I had everyone make a dot somewhere on the parabola with their initials.

"Draw L1 and L2 for your point using the green focus/directrix."

The board looked something like this:


It was very fortunate that K picked the vertex for her point!

"Does anyone's point have L1 = L2?"

K's did, no one else's though. 

"Well does that mean the green point is the focus or isn't the focus?"

Great discussion on why it isn't - it's not good enough to have one point on the parabola with L1 = L2, they ALL have to have it. They knew that yesterday, but this was knowing on a different level. I think the fact that each person "owned" a different point reinforces the all or nothing idea here.

Then each person picked a new points, and we tested out the blue pair. As soon as ONE person's point didn't work, the reaction was immediate - it can't be the right pair. 

Now they knew that there is only ONE possible location for the focus and directrix of a parabola. Move anything and it doesn't have the L1 = L2 property.

KNOWing:

We finally tested the pink pair and found it to be the actual focus and directrix. Now they knew where it was and where it wasn't, and there's only one possibility for the former.

Way to drop the ball McSquared:

The final point I wanted to make was the connection between what they just did and the formula c = 1/4a. Not the algebraic connection, but the bigger, deeper one:
The numbers are connected like this: Change the value of a, and you'll change the value of c, and vice versa.
The things are connected like this: Change the parabola, and you change the focus and vice versa.

Unfortunately, that part was just me talking, which I'll replace next time with them doing something, not sure what. Something involving matching parabolas with c values and focus/directrix pairs.....it was time to zoom out here and I dropped the ball, but I'll pick it up next year.

But I do think I helped them to KNOW, you know?

Friday, May 9, 2014

Developing the standard rule of an ellipse

This activity was inspired by two people: Teresa Ryan, a fabulous math teacher tweep, and Amanda R., one of my students. A few days ago, Theresa tweeted this
and that question started the cogs turning. Around the same time, I had my students playing around with circles on desmos. Amanda happened to type in the equation 2x² + 2y² = 1, and notice that it had a smaller radius than our unit circle. That lead to a nice discussion as to why the radius was less than one, and then why it was equal to the square root of 1/2.

So today, again, all of this kind of gel-ed on my way into class. Here are my guiding questions, and their collective answers:

Open a desmos or ggb, and get the unit circle to show up.

Now type in 2x² + 2y² = 1. Tell me what you get, (smaller circle), what's the approximate radius? (0.7)

Type in another equation like this, which = 1, but make an even smaller circle appear, and write your equation on the eboard, plus the approximate radius.


Find pattern: as coeffs get bigger, circle gets smaller.

And what's the relation between the coefficient and the radius? Radius is the square root of one over the coefficient.

Okay, if bigger coefficients make smaller circles, what coefficients will make bigger circles? (1/2 or 1/3)

Type those in, measure the approximate radius, and write on eboard:


Is it okay if we write these equations this way instead? Are they equivalent?


Now how can we calculate the radius from the rule? It's always the square root of the number in the denominator.

Which denominator? Well, it doesn't matter. Doh. They're the same.

Oh right. I didn't notice that. Well type one in that doesn't have the same number under each term, what do you get? An ellipse!

Tell me your equations and the dimensions of your ellipses:

Unfortunately I didn't take a snip of this, but the variety was wonderful, some ellipses were horizontal, some vertical, it was absolutely no big deal for them to see that the number under the x always governed the width and the one under the y governed the height, plus that a square rooting was involved.

From there it was a piece of cake to generalize to the standard form of the ellipse! We did a bit of practice where I gave them the rule and they graphed, and vice-versa. It felt like I'd covered 2-3 days' worth of concepts just today.

Thanks Teresa and Amanda!

Tuesday, May 6, 2014

New Intro to Locus and Circles

All of this actually happened today, although, well, maybe not all during the same class. So it's piece-wise true...

Part 1: Locus intro:

This was the first day of our last chapter, conics. I wanted to begin with the idea of the locus of a point. But I didn't want to actually tell them what a locus is, I wanted to show them, then get them to tell me.

I got this idea on my way into class, which by the way there has to be something to why that happens so often at that exact time. Anyway, I thought of a use for one of my geogebras that was not at all what I had intended it for. This video explains what I had intended it for, and what I ended up doing instead:




By the way, if you're interested, here's that geogebra. Next I asked my students what they thought a locus was. Here are a few samples, word for word:
  • The path of a point followed by a specific function
  • a locus is the path a point takes
  • The path of a point of a function
  • The trace of a moving point
Their words, not mine. Which, collectively, touched on all of the key points - that it's a path, that it's created by a point, that the point is moving, that as it moves, the point is following some kind of rule.

Part 2: The circle as a locus:

I then wove all of these locus ideas into this geogebra, made by the brilliant Jennifer Silverman:
How beautiful is this?

I let them play with it a bit, to draw a few circles, then identify which of these virtual things was the locus, which was the moving point P, and what rule that point was following as it moved. Here are their answers, again collectively:

What is the locus? The circle is the locus! (Just that right there was huge. All these years I've been the one saying that, and approximately no one was really seeing the circle any differently than they had always seen it - as a static thing.)

Which point traced this locus? The point at the tip of the pencil.

What rule did the point follow as it moved? It stayed the same distance from the red pin.

Then we formalized that into the locus definition of the circle, which for the first time since I've ever taught it, I didn't have to dictate or get them to fill in the blanks on pre-made notes. Okay, I did give them the word equidistant.

Part 3: The rule of the circle

Next I wanted to move onto the Cartesian coordinate system, so we reviewed that:
  • the rule for the unit circle is x² + y² = 1
  • where that rule came from (right triangle inside circle)
  • that really the 1 in the rule was 1².
 I gave them this desmos:  and had them work on that in groups. Just like when they're using geogebra, there is no need for me to tell them if they're right or not. If it is, they'll see a circle with the right radius. Again, there were no notes, no me telling them what the rule is. It took some trial and error, but eventually everyone noticed that the radius has to be squared in the rule. After a bit we regrouped, discussed, even a few things that I hadn't expected would come up:
  • Why is the radius squared in the rule? Why isn't it just x² + y² = r?
  • Is it possible to get a circle that's even smaller than the unit circle?
  • One student noticed that  2x² + 2y² = 1 gave a smaller circle than the unit circle.  Why would that be?
On the way out, I had another idea. I need to write this down so I'll remember it all next year. That was 9 hours ago!