## Wednesday, January 17, 2018

### First attempt at stations in the live online class

I've been trying to find an alternative to the usual group work, and also, I've long been wanting to try out "stations" in my live, virtual classroom. The trick was how to simulate easy movement, make clear goals at each station so I don't get run ragged, keep things moving, and have time to process the math at the end. Today I finally did it.

In our environment, the closest thing to a station is what we call a breakout room, or bor for short. They are really just other websites, of course, that are somehow attached to the main classroom. Here is how they usually look from the dashboard:

Usually, I put students into whichever room I want by dragging and dropping their name into it, much like moving a file into a folder, and in fact it ends up looking that way too (pretend I'm a student):

But I can create as many bors as I want, and call them what I want, and make them any colour I want, so today, here is how my stations looked:

Why did I set them up this way? Because this was the activity:

The task was to shade equivalent boxes in matching colours, using the properties of exponents, and the exponential situations we've studied so far, including rational exponents, negative exponents, and the property (a^n)^m = a^(nm).

Here's how it went down, along with how it would look in a brick and mortar situation, in case it's hard to get your head around what the heck I'm talking about:

Step 1: Simulate easy movement:

First I gave them all moderator status so THEY could move themselves into and out of any bor they wanted. This way they all get to go to each room at their own pace, plus I found it was a nice break from me picking who is going to work with whom.  Also it makes it feel more like they're moving around in a classroom station situation. We first practised that a bit, so they could get used to the controls.

Brick and mortar equivalence: Kids can usually move around in a classroom.

Step 2: Clear goals at stations

Upon entering a bor, they all saw the same slide, which was the one above. Then, if they were in the turquoise bor, they'd colour in (using the virtual highlighter pen) all the things equivalent to y = (1/2)^x, if they were in the yellow bor, they'd colour all the things equivalent to y = 3^x, etc. Students can see what bor they're in, so again, it's all as obvious as if they were in a brick and mortar classroom.

Brick and Mortar equivalence to this: Having 4 big tables, each with a giant place-mat like the above, preferably with a covering that makes it easy to erase, with each table equipped with highlighter pens in only one of the four colours, so there's a bunch of turquoise pens at one table, and a bunch of pink pens at another etc. It's important that the highlighter pens be erasable though, and I don't know if such a thing exists. So also you'd need erasers.

Step 3: Keep things moving

After a few minutes, I said "Change rooms!" and watched their names jump around. I had already told them that if they agreed with the colouring that had already been done, leave it there, and see if there were any other things that should be coloured. If they disagreed with what was already coloured, they could change it. The idea was that this would filter out the easy egs from the harder ones, by a loosely formed consensus.

Brick and mortar equivalence: Saying change stations, whereupon students move to another table. They can change what's already coloured and add as needed. I'd probably have to make sure the right coloured pens would stay at the right table!

Step 4: Keep things moving and varied

Repeat step 3 twice, so that everyone has had a chance to go to each station.

Step 5: Process the math

I took snapshots of each of the final bor screens as everyone moved them selves back to the main classroom.

Brick and mortar:  Telling everyone to go back to wherever they usually sit while I gather the place mats.

Step 5: I displayed each snapshot, we discussed, I showed the answers, and we discussed some more. (Brick & Mortar I'd do the same.) Here are a few of the actual ones, compared to the answer slides:
 The turquoise were all correctly coloured

 The greens were mostly correct

 Missing a few pinks and one incorrect
 And missing a few yellows

Here's where my evil plan started really playing out:

• Each colour had some really obvious ones that were not in dispute, like 3^(-x) and (1/3)^x, so we didn't have to spend much time at all on those.
• Each colour also had some that weren't so obvious, like why is 27^(-x/3) the same as (1/3)^x? Also the one about why losing 2/3 gave a base of 1/3. This was a chance to add a layer to their understanding of exponential properties. I asked for volunteers to explain these.
• Each colour had something that I anticipated no one would colour, and bingo, I was right! These examples involved e, and for most this was the first time they'd seen it. So this was a nice way to introduce it.
I usually try to think of what I'd do to improve, but all I can think of is additional examples to add. Since there was no dispute over the turquoise, that could use something in it to uncover a gap in understanding. Otherwise, this went well, and it didn't take a lot of set up, since there was only one slide for all the rooms. Also, I didn't have to chase after any actual pens!

## Friday, November 10, 2017

### GeoGebra Group Hug

I've been getting my students to create their own GeoGebra applets for a while now. If you're interested in the big picture of what and why, you might want to read this first. I'm doing the same overall thing this year, but this time, I'm incorporating GeoGebra groups. And this post is a big GeoGebra Group hug. Groups have been around for a good two years now, but it's only this year I really dove in.

Every year when it's time to get my students using GeoGebra, I start with the linear function, with which they are theoretically already familiar. This way we spend most of our time getting familiar with the platform as opposed to the math, first time around.

How-to:

First, I created a group for us. Then, I had them create accounts at GeoGebra, and join our group. This is all pretty easy - here are some instructions if you need them. Here's my first post to the group, which was their first "task", visible as soon as they'd joined the group:

As you can see, there is a place for them to comment right there under the post, which for this task, I didn't ask them to do (but some are just naturally friendly!). Comments posted here are visible to everyone in the group, and we can all reply to comments. Anyway, once they clicked where it says Linear1, they saw this:

How it looks to me, the owner of the group, is almost the same except instead of the turn in button, I have, for obvious reasons, this:

Hugs right off the bat:

My first favourite thing is right there in the instructions - that they don't have to each make their own copy - it's all done. This alone saves them, and me, SO MANY CLICKS.

I gave them a simple task, which, once they were done, they just turned in by clicking the button. If they wanted to, here was another opportunity for them to make a comment, but the difference between doing it here and on the previous space is here only I will see it. Nice opportunity for intervention.

I also love love that I can reply here, and we can have a conversation if necessary. This is the first mathy edtech tool I know of that has this continuous two-way conversation capability.

Next lovely thing is as they were all working, I refreshed every so often so that I could see where they all were via a "feedback" table, which shows me, in one glance, who has started, who hasn't, who has left me a comment. Right now it doesn't look at all like it did live but it was pretty cool. An updated table appears later in this post.

Eventually, all were turned in, so the next phase was me looking at them all.

After turn-in, more hugs:

Another enormous saving of clicks. All I had to do was open up one of their sheets, comment if I wanted (and I did), then click "complete" or "incomplete" and the next student's work is automatically opened. I think I screamed when that happened. I was like - you mean I don't have to save anything, then go somewhere else to open up the next one, wait for some program to start.......nope.

So this is now a cycle that repeated as needed. If anyone needed to fix something, they did, if they have a question, they put it in the comment, then turned it in etc etc.

Bigger picture, bigger hugs:

After a few of these tasks, this is what my feedback table looks like:

In a glance, I can see who has started, who hasn't, who has turned it in, whose is all good, whose needs fixing, who was late, who has left me a comment. I can open up anyone's work in a click. It's just lovely.

I love the flow. It's not quite live yet like a Desmos activity, but it's pretty close.

Next post will be about my next foray in which I used the public comments section.

## Friday, February 3, 2017

### I Wonder Why I Stopped Blogging?

I know, I know, I'm blogging about why I've stopped blogging, I get the irony.

But it's been on my mind. I don't blog anymore, at least not like I used to. I used to compose posts in my head during the week, or even during the day, and spend hours making it look and sound just right.

Did I stop being inspired? No.

Did I stop learning and growing as a teacher? No.

Did I stop wanting to share my ideas? Even to my future self? No.

So what happened? Did I just get tired? Maybe. I am tired.

I still read my own posts though, so that's something. I read last year's one about introducing log properties, just in time to repeat it today, which is what made me wonder.

Why did I stop blogging?

## Wednesday, December 21, 2016

### Another First in My 30th Year of Teaching

Welcome to the continuing saga of how I, in my 30th year of teaching, am doing things that I should have been doing for all these 30 years. Today's practice that I'm embarrassed to admit I'm only starting to do now is having students present to the class.  I'm not sure why I avoided it up till now, but I definitely know why I wanted to start very much this year. I was inspired by my colleague Peggy Drolet, whose classroom management skills I learn from every day. She and I both teach online, and so constantly have to work hard just to create a sense of community, establish our presence, and get the kids to do the same. A couple of weeks ago, she had her students do class presentations, with great success, and so I jumped on it. The presentations were, at least on the surface, about math, but I had lots of other ulterior motives - mostly social ones, as did Peggy.

Topic for Presentation:

We're studying optimization, so for a presentation topic, I chose the polygon of constraints. I always compare the pgoc to a sculpture, so I thought this might inspire some creativity. The constraints do the same thing as a sculptor's knife - they remove something so as to let the shape emerge little by little. I asked my students to design their own pgoc from their own constraints. The requirements for the pgoc were that:
• it be in the first quadrant
• it be composed of at least 3 constraints
• one constraint be in the standard linear form
• one constraint be in general linear form
• the slides be designed so that the shape of the polygon emerges one step at a time, slide by slide, with the resulting pgoc clearly outlined in the last slide, like this:
 Each constraint takes another slice off
The Medium:

Also inspired by Peggy, I had them create their presentations using Google Slides, so that I could watch them working live, and so that they could work outside of class time without having to actually be in the same place, which would have been impossible considering they're scattered all over the province of Quebec. I've had my students do activities using google slides before, but it was always something I had created, and they were to edit. Also, in the past they were able to access the slides without signing in, so I didn't know exactly who was in which document. This time it was all done through their GAFE accounts. No more anonymous bobcats or chupacabras.

Day 0: The Set-Up

I created a bunch of blank google slide documents, one for each team. I shared each doc according to who was on that team,via their emails. I had considered getting them to do this part themselves, which would have probably taken a whole day in and of itself, but I caved to save time. Plus, this way, since I was the owner, I also had immediate access to their slides, which I wanted because I want to be able to watch them work.

Day 1: Meeting in the Cloud

I described the project to the whole class, told them they had 3 days, then they just clicked on the links I'd shared via email, and boom, they were all in their respective presentation work spaces, with their team members, and I was sitting there waiting for them all. I had all of the files open at the same time on my screen, so I could just click one tab after another, and go from team to team to watch. This is what mission control, aka my screen, looked like:

At the top you can see all the tabs, with the names of team members visible. The tabs are just the file names, which, since I created them, I chose a title that made my life easy.

If I clicked on any one of the tabs, here's what I'd see:

I could see who was in the doc, which slide they were currently looking at, their live edits, and the live chat. It was really fun to watch them creating their pgocs while they were chatting using Google's live chat tool, which is right there in the same window. It was such an eye-opener to see how much they enjoyed collaborating, socially and mathematically. I feel like I benefited from knowing this at least as much as they did experiencing it.

But it wasn't only there that things immediately felt different. There was more communicating happening everywhere. That first day: "Bye guys, have a great day!" to the whole class, from kids who'd NEVER said anything like that before. Kids who'd never tweeted suddenly started to. I definitely noticed a difference in the social atmosphere - in and out of class - right away.

Days 2-3: Riches:

The math:
• Some groups needed help getting their constraints to match the pgoc they wanted, and some needed intervention in their work due to mistakes, so that was a classic just-in-time teaching opportunity. This felt like productive struggle.
• I got to deepen one group's understanding of solution sets: They had their pgoc ready and looking the way they wanted, and their constraints matched the pgoc, but they asked me to help them rewrite one of their constraints. It was in the form  y > -x + 7, but they wanted to put it in general form because that was one of my requirements (I'll make sure to include that next year too). They wanted it written in a different form, but they didn't want it to change how the pgoc looked. What a teachable moment! These are kids who've solved a ton of linear inequations before, but never realized that the algebraic steps they'd been doing actually didn't change the solution set. They were surprised, and relieved, to learn that x + y > 7 had the exact same graph as y > -x + 7.
• The usual benefit of teaching something - it helps you understand it better. I could literally see it happening before my eyes as they would find a mistake and fix it. Also, the task of organizing a large amount of info, for example all the systems of equations that define the vertices, gave them a bigger picture of the problem.
• The sheer variety of the inequations - not only did they go with standard and general, but also symmetric! Also things that don't really fit into any category, like 3x + 1 > 10, or 12 ≥ 3x + 4y, with the variables on the right side. I plan to use their actual constraints as a jumping off point for even more complex ones.
So many other cools:
• One group asked if they could include a quadratic inequation, which technically doesn't make a polygon, but which was definitely thinking outside the box. So I said yes!
• Creativity! One group decided to make their pgoc perfectly symmetric.
• One group organized themselves as follows: X designed the entire pgoc, Y found all the vertices, and Z created the slides for the presentation. That seemed like a pretty fair distribution of labour.
• Total GeoGebra and Desmos fluidity. I didn't have to help anyone with these tools, they just ran with it. The tools have become transparent.
Presentations: The PGOCS

I'm just going to let the pics speak for themselves. (I know there's a way to get blogger to put images side by side, but I can't even right now.)

Such style, such pizzazz!

As for the actual live presentations? So much fun to watch - how they handled it, and how the others reacted. Voice is a huge presence for anyone who teaches or learns in an online class. So much personality is revealed in the tones of a person's voice. Not to mention the little touches some added, like getting the others to participate, and putting little jokes in their slides. Everybody got to know everybody else a bit better, which for me, was the whole point.

My only problem was that I tried to record them all and chop the video files into individual ones, and in doing so, inadvertently deleted some. I could definitely use a digital assistant. This seems to be happening a lot lately. Must also have something to do with being in my 30th year of teaching....

But for sure, I'll be doing this in my 31st year too!

## Tuesday, November 29, 2016

### The Classic Homographic

Why I love rational applications:

Rational applications are worth spending more time on than absolute value or square root ones, I find, because they can harness so much algebra AND reality in one single problem. I spent two unhappy years working as a cost accountant, in which this type of problem came up a lot, and it was the only fun part of the job. So I'm giving my kids an assignment that's only on rational application problems, and in order to prepare them, I showed them 3 type of situations that they could expect:

1. The Constant Product situation:

This is the simplest one, in which the two variables involved always multiply to give the same result. They may notice this from inspecting a table of values:

Or by reading between the lines:

In either case, the relationship between the variables can be expressed as xy = a, or y = a/x, in which a is the constant product.

2. The cost-per-person situation:

In which there are two types of costs involved - a fixed and a variable (which I happen to know is how they're referred to in the cost-accounting world):
The fixed is the \$3000, and the variable is the \$750. The calculation that naturally occurs to students to make is the cost per person, and coming up with a rule for cost per person is fairly easy for most.

The fact that no person can ever pay exactly \$750, but always slightly more, makes the concept of asymptote very real. This is a really nice intersection between their intuition and the very abstract concept of asymptotes.

Next level up would be a situation that involves something per something other than cost and people, but which has both a fixed and a variable quantity.

3. The Class Homographic

The name for which only occurred to me moments before I went into class - in which two linear quantities are being divided, so that the resulting function is rational, but its asymptotes would need to be discovered via long division:

Now this is where some really beautiful intersection between algebra and reality happens. They do the long division, which reveals the asymptotes, one of which is at 1000. So is that truly the maximum? Time to discuss. Not to mention that this is a great argument for doing long division in the first place, because it reveals so much about the nature of the function.

Next level up is a situation in which the homographic rule is not given, but two linear relations are, and they need to be divided, for example, the concentration of a solution in which the amount of solute and the amount of solution are both changing linearly.

How will this help them?

So often I hear - Mrs. - I don't even know where to start. Well, we spent the rest of the time looking at other problems and identifying which of the three types each was - which can help point the way.. Which I hope will help them with their assignment - we'll see on Friday when it's due.

At any rate, I just love how Classic Homographic sounds!

## Monday, October 31, 2016

### How to Get Organized in Your 30th Year of Teaching

Many years ago, as a young, disorganized, overwhelmed, but still enthusiastic teacher, I fantasized about opening up my plan book in September and having every lesson on every page already filled in. A stress-free year, in which I'd never have to wing it. I was so wrong about many things, not the least of which was that that would be a truly dull and cognitively dead classroom where nothing surprising or unexpected ever happened. But I was right about one thing - I did need to get organized, so that I could handle those surprises.

I tried a few things - colourful filing systems, getting advice from my mentors, just plain disciplining myself, with some small success, but everything seemed to require that I practically change my personality. Just stay up-to-date and keep everything neatly filed - it's as easy as that! It was meant to help but it didn't, not in any long term way.

Turning Point #1:

Sometime around 2002, my school got Smart Boarded, to my great delight. And say what you will about Smart Boards, but they helped me get organized.  I didn't expect that to happen, and it didn't happen right away - it took a year from the time I started using one. During that first SB year, I wrote on that fancy board much the same way I'd written on the chalkboard, but the biggest difference was that I could save all of that writing by date, topic, whatever I chose. That meant that I began the next year already possessing a real timeline for each unit, not to mention a starting point for planning better lessons for this year. In one stroke, my organization and pedagogy were improved and I hadn't had to suddenly change anything I was already doing in order to achieve it. I used what I did, and built on it. I think that's the key to getting organized - be who you are, watch what you do, keep a record of what you do, & build on that.

This year, I used some of what I'd learned from the SB years, plus I intentionally made one very small but powerful change.

Turning Point #2: The Small But Powerful Change:

I took attendance every single day for every single class. Absolutely non-negotiable.

I didn't do it at the beginning of class, mind you, because that was always the reason I'd skipped it in the past, that it took away class time. So this year I did it right after class, while it was still fresh in my mind who'd been there. Unexpected bonus - if I couldn't remember whether or not a certain student had been there, that meant I hadn't interacted with that child. Not good, but good to know.

How did this become so powerful?

• I got addicted to how good that one simple change felt - always knowing that my attendance was up to date. The data alone didn't make the huge difference, not right away anyway, but it made me realize how badly I'd been feeling all those years when it wasn't up to date, and especially when it invariably got so far away from me that I was literally pulling numbers out of the air at report card time.
• I also got addicted to having a daily routine. I'm embarrassed to admit that I reached my 30th year of teaching without any kind of daily routine. Not good, but good to know.
• Both of the above gave me energy - or more specifically removed a ton of negative energy. No guilt, no cringing when I see my attendance book, no time wasted on "OMG where do I even start today?"
• It lead to what happened next.

Turning Point #3: Documenting my Organizational Routines

From the first day of this school year, I paid attention to myself. Every time I noticed myself doing something along the lines of management/routine/organization (corrections, time sheets, reading emails, etc), I jotted it down on a post it and stuck that within reaching distance of my chair. Very quickly I noticed categories emerging - things I did every day, once a week, end of a unit, so I transferred this info onto colour coded post-its:
This took time - I was finding time to do something I'd never done before. Which meant that my own behaviour was changing, not because I or anyone else was forcing it, but as a natural consequence of simply watching myself.

Next level up

These post-its started turning into checklists, like the ones the pilots in Sully used, that helped them stay calm and think clearly while the plane was headed for the Hudson river with 155 people on board. Nothing on the checklist directly changed the outcome, but indirectly it gave the pilots the mental energy they needed to focus on solving a problem.

And that's how I think my checklists, routines, and post-its are helping me this year.

Next levels up

I refer to the daily checklist once or twice a day. I've added things to it, and moved some onto the weekly.

Every Monday, and again every Friday afternoon, I go through my weekly checklist. I've added things to it, too, and moved some around.

These routines, which aren't static by the way,  make me feel stronger, less overwhelmed, and less stressed. They help me in the short term and in the long term. They're not the same as those endless to-do lists I used to make. These are much more long term, and their benefits keep on multiplying.

So is my life perfect now?

I can't honestly say I'm getting everything done exactly on time, but I can say that I know how far away I am from that goal. I can make the adjustments I need to on the fly, weigh the costs, and use written words and facts as a basis for my decisions rather than full-out panic.

I can say that I'm organizing my class time better - fitting in those wonderful WODB activities, even a couple of mini-contemplate-then calculates!

I can say that I'm more energetic, more likely to remember to deal with all those unexpected things, having new ideas more often, trying out more things, doing a much better job of rolling out GeoGebra (more about that in a future post) ...it just feels a heck of a lot better.

I can also honestly say that for the first time in my career, two months into the year, instead of feeling the ropes slipping away from me, and in spite of the fact that I've been sick since October 12, I feel, in my bones, that I'm still at the helm, and doing a good job of keeping us all moving ahead. Good AND good to know.

## Friday, May 6, 2016

### Crowd-Sourcing Brings Personality

Since I teach online, I can have a lot of students writing on the eboard at the same time. I've been using this to my, and their, advantage. Instead of getting everyone to do the exact same thing, I make sure to ask something in which there is potential for a variety of answers. I also get them to own their answers by putting their initials or name next to it on the board. Then we talk about the differences or similarities, or classify them, etc just try to use the variety to get a bigger picture much faster than they could get with one example at a time. Not to mention that attaching students' names to each answer, and then addressing them by those names, lends personality and life to the whole activity. Variety, big picture, personality: Win-win-win!

We did this the day after a GeoGebra lab, in which students used Jennifer Larson Silverman's Drawing an Ellipse GeoGebra to experiment with drawing ellipses using virtual string. I had them keep string length constant while moving the endpoints, then vice versa. It was concluded that:
• When you keep the string length constant, the distance between the ends of the string influences the shape but only in the length of the smaller axis - the longer axis remains the same.
• When you keep the string ends constant, changing the length of the string influences the shape of the ellipse in both directions
• the length of the longer axis is always equal to the length of the string.
Last night, their homework was to watch a voicethread on the parts and vocabulary of the ellipse. At this point, they didn't know any formulas or rules about the ellipse, only the geometry of it. Then today I had them do this all together:

1. (My voice in bold) Draw an ellipse with major axis length 6 and minor axis length 4.

 (After some minor corrections, and with the names changed of course)
Do all of these have the right major and minor axis length? Yes How are they the same, how are they different? All are same shape but oriented differently. If you had had to draw these yesterday, would you all have had the same string length? Yes. And end points? No. Julie and Bridget's ends would be vertical. What would be the same for all endpoints? distance between them. If you had to find their rules, would they all be the same? Horizontal ones will all have same rule and it will be different from the vertical ones.

2. Draw the foci in their approximate locations.

What part of yesterday's lab do the foci represent? ends of string What do you notice about the foci? Lots here: horizontal ones have foci on x axis, vert on y, all hor foci should be in exact same locations and at same distance from each other. Vert ones will be at same dist from each other as the hor but on y axis (based on yesterday that moving the string ends changed the ellipse). Everyone's foci have origin as midpoint. Personality bonus - Julie and Bridget think a bit differently than most - and that's cool.

3. Draw any point on your ellipse in purple.

What from yesterday's lab does your purple point represent? pencil tip What do we call Heather's point? Covertex. Anyone else picked a covertex? Julie. And who picked a vertex? Bridget and Bob. (In both classes, someone picked those key points, fortunately. If they hadn't I would have had to draw my own.)