Friday, October 2, 2015

If Only I'd Used a Hinge Question 3 Weeks Ago

After correcting this week's assignments, I discovered many students are still not able to find a vector's direction, given its components. This is something that I supposedly taught 3 weeks ago, and thought I'd checked for understanding, but....oh well. A hinge question 3 weeks ago would have been awesome. I would have known who, what, why, and how bad things were, and come up with a way to straighten out their vector issues.

Haha! Get it? Straighten out the vectors? It's been a long week.

As usual, it's only when it's too late for this year's students that I have clarity on what to do, but in my defense, seeing so many possible wrong ways to do it today was what guided me to writing this hinge question.

To generate the wrong answers, I used today's mistakes. The 3 big ones I saw today: not taking absolute value of components, wrong order of ratio, wrong quadrant formula. I saw one person using the y-axis as a reference instead of the x-axis, so I'll put that in just a few answers.

Find the direction of the vector <9, -20>.

If they get 294 degrees, they're right.
If they get 426 degrees, they did 360 - arctan (-20/9), ie didn't take abs value of 9
If they get 335 degrees, they did 360 - arctan (9/20), ie wrong order of ratio, OR they did 270 + arctan (20/9)
If they get 66 degrees, they did arctan (20/9), ie wrong quadrant
If they get 384 degrees, they did 360 - arctan (9/(-20)), ie didn't take abs value of 9 AND wrong order
If they get -65 degrees, they did arctan (-20/9), ie didn't take abs value of 9 AND wrong quadrant
If they get 24 degrees they did arctan (9/20) wrong order and wrong quadrant
If they get -24 degrees, they did arctan (9/(-20)), ie wrong ord, wrong quad, no abs val
If they get 204 degrees, they did 270 + arctan(-20/9) ie used y-axis as reference AND no abs val.

Thursday, September 17, 2015

Vector Addition Goodness - and Quirkiness

This went well.

It turned out to be a quick way to generate lots of examples - simple ones and quirky ones, and to target specific examples so that we could look back and look ahead in the vector unit.

We've just started vector addition. Here's the activity I put together for my online class today:

We're all online, so anyone can easily move these arrows around by clicking and dragging them.

I split the kids into groups of 2-3, and told them to each pick one blue and one green vector, add them, then draw the resultant in red (these colours matched the ones from the previous evening's voicethread.). Since there are 6 blue and 6 green, there are many possible combinations of vectors - 36 in fact (an opportunity to talk about math that's not usually part of the vectors unit!). They didn't have to do them all, of course, just do 3 pairs, or enough to use up all the vectors.

Because they were dragging instead of drawing, it only took about 5 mins for them all to finish. I took snapshots of each group's work, then we reconvened in the main classroom space. Here were some of the results
Group 1:
First we together looked at all the examples group 1 made and decided if there were any mistakes - which happily, there weren't. Then I saw the first part of my evil plan unfold. (I had only a 1/36 chance of that combo happening, yet it happened!) The combo at the very lower left was two horizontal vectors of opposite directions being added. A quirky one that I wanted everyone to see, and since only one group did that combo, it kind of was an example that had a personality to it - it belonged to a person!

Group 2:
We repeated for group 2 - checked the answers first, then more evil plans unfolded. (Another long shot happened, geez I should buy a lottery ticket today.) In the lower left corner, someone had added a green and blue that were perpendicular to each other. I asked them to think about where they'd seen something like that before - and after a few seconds I heard - the blue and green are almost like the components of the red. Almost? OR EXACTLY?!? We talked about how all along, the components of a vector (with which they're already familiar) actually add up to that vector. When they'd been drawing a vector's components all last week, it was kind of like they started with the answer, and drew a question to go with it.

Group 3:
Weird how all my evil combos ended up in the lower left corner...anyway here we saw the same component-style example except that they're added in a different order, but still giving the same resultant.

I then showed them a combo that no one had picked, but that I wanted to address:
I had, of course, deliberately put in a blue and green that were identical to each other. We talked about what the resultant for this would look like, and it was a nice intro to something we haven't done yet - multiplication of a scalar and a vector.

What I liked about this activity:
  • the possibility of everyone seeing so many examples in a relatively short time (this all took a total of 20 minutes)
  • the possibility of interesting and unusual examples for discussion
  • the personalization of the examples - I didn't make them up, not really anyway, so that I could refer to "Susie's example", instead of a cold "number 3"
  • It gave kids who think outside of the box a chance to try something outside of the box, and feel like it's ok to do that. In fact, it was fantastic to do that!
  • It was a great intro to the geogebra that I then had them do individually, in which they were pretty much doing the same thing, manipulating vectors to add them, with a few twists, like find what vector I have to add to this one to get that one (intro to subtraction)
Next time:
  • I'll make sure to keep WAY more copies of the original arrows - easier to show combos that no one did that way
  • Find cases where two people added the exact same vectors and got the exact same resultant
  • I'll include the potential for more quirkiness, like opposite vectors adding to zero. You can never have too much quirkiness.

Thursday, April 16, 2015

Always-Sometimes-Never Intro to Identities

Today, I tricked my students into writing identities, without saying the word identity even once. And I think I got them to appreciate how awesome truth is.

The progression during this class was that my students:

1. classified statements as Always true, Sometimes true, or Never true

2. filled in the blanks in a statement so that it was Always true

3. wrote their own statements that are Always true

Part 1: Classifying:
Here are the statements I wrote on the eboard, each of which they then labeled with A, S, or N, and after each of which we discussed why (that's the italics):

Θ = Θ + 2π     
NThe notion that = means "is the exact same number as".  Coterminal is another thing entirely.

sinΘ = sin(Θ + 2π) 
AA lot of confidence about this being an A, due to the previous discussion. The seed was planted that finding an A is kind of a big deal.

cos Θ = 1
S - I insisted on hearing some Θ's for which it was true.

cos Θ = sin Θ
S - Same as before, except this time, tell me all of the Θ's for which this is true

Here I had to pause and get them comfortable with locating angles like Θ + π, π - Θ, etc on the Unit Circle, so that they could visualize the next statements. I had them drawing random angles for Θ, then the corresponding Θ + π etc.  Once they were ready, I asked them to classify this one:

cos (Θ) = cos (π - Θ)
N - Lots of lovely arguing, many said well they're equal but opposite. How to say that algebraically ..... and let's convince ourselves with a few angles on the calculator (in degrees though!) it was time to segue to part 2.

Part 2: filling in blanks to make an A:

sin Θ ______ sin (π - Θ)
cos Θ _______ cos (π + Θ)
sin Θ _______ sin (π + Θ)

This part was done in groups. I witnessed some fantastic discussion - which I was unable to copy and paste due to a tech glitchy thing, but there were drawings being done, there were angles being tested on the calculators, there was correct vocabulary being was truly exciting to see the strategies they were using to decide and then convince. I didn't have to say much. I shut up really well.

Part 3: writing A statements from scratch:

This time all I said was "Tell me about the relationship between cos Θ and cos (-Θ)."

Back into their groups they went. (There was no need to say, oh by the way, I want the truth.) They figured out where -Θ was in relation to Θ, they looked at their x-coordinates, they wrote a statement, they tested it out on random angles, and then ALL groups proclaimed:

cos Θ = cos (-Θ).

We repeated this to get (in a lot less time btw):
sin Θ = -sin (-Θ)

and talked about whether or not we could also say that
-sin Θ = sin (-Θ)

Tonight's assignment:

Tell me about cos Θ and cos (Θ + π/2). And I want the truth. I can handle it.

Monday, March 2, 2015

Off the Wall Flashblog post

What an awesome idea, the flashblog post. Topic- what is the most off-the-wall lesson you've ever done?

The only thing I can think of is the voicethread I did last year called "What Would the Teacher Say?" in which I showed a step-by-step procedure (mathematical of course) and I asked them to supply the explanatory comments for each step as if they were the teacher. It was a procedure made up of things they already knew about, but applied to a new function. I think I tried to start a new hashtag #wwtts but it didn't take off.

The idea came to me in a flash, I can't even tell you from where. But some kids took me at my word and tried to mimic me, while others went too far into the back ground knowledge and over-explained things. But I'm going to try it again!

Friday, January 30, 2015

Teaching Logarithms Using Suspense (okay and GeoGebra)

Every year, when I start the logarithms unit, I brace myself. I know I'm going to lose a lot of kids. I've tried all kinds of ways to make logs clear. I've written about it here, in fact, and each time I think "This is gonna be great!" but then it's not. But this year, I think I hit on something. I really mean it this time. No really!

And guess why!?!? It's because of the geogebras I've been getting them to create. (If you have no idea what I mean by that, this will give you some idea.) Another unexpected benefit that just kind of fell into my lap. I'm using their exponential geogebras to create suspense about logarithms!

Where we are now

We're currently studying the exponential function, and this time I had them start creating their geogebra explorers very early in the unit, so that they could use them in parallel to the lessons, for pinning things down, validation, exploring, whatever. (Note to self: Do that next year for every function. It's too overwhelming for them to do it all at the end of the unit.)

Anyway, I knew that they'd be able to get their sliders, asymptote, domain etc etc all done, but that when they'd get to the zero, specifically figuring out a formula for it, they'd be at a loss. When I first realized this, I thought, ooh, that might be frustrating or confusing, but then I realized that it would be an opportunity to motivate the need for logs. Just the act of asking me "How do we solve for the x in this?", ie an equation like   indicates that they are aware that this is a thing. A new thing. Which let's face it, logs are.

The plot thickens

They haven't done logs, so this kind of makes it suspenseful for them! I'm hoping that in a week or so, when I reveal logs to them for the first time, instead of the usual confusion and horror and OMG THAT'S IT I'M GOING INTO ART, I'll get "Oh! So that's how you solve that equation!" or something like that.

To add to the suspense, I had them spend some time struggling with the question: What can I do to both sides of this equation: , that is the opposite? I wanted to use that kind of language because that's what they are used to - again to motivate that this calls for something totally different. Trying to find that opposite operation, and failing to do so (which they did spectacularly), points them away from that old familiar safe language, which is a good thing. Kids, you're so not in Kansas anymore.

Questionable pedagogy?

It's not completely true that they haven't seen logs, actually, because I have shown them that to solve , they can get the value of x by either trial and error, or by using the log button on their calculator and punching in:

But they have no idea why that works, or what the log button does yet. Normally I don't encourage my students to do something without understanding it, but logs are different. I find the word itself is intimidating and doesn't sound at all like what it is - an exponent. So I get them used to hearing it for a while before actually explaining it. I'm not sure about the pedagogical appropriateness of that....but emotionally I think it helps. And it also lets them know that logs have something to do with solving exponential equations.

My evil plan's results so far 
Dr. Eeeevillllll!!!

So far, it has all worked exactly according to plan, at least for a few students. Several students got everything done in their geogebra except for the zero, and asked me to help them figure out the formula for the zero. I of course refused. Nicely though!

Another asked "Are we allowed to use logs to do the zero in our geogebra?" I said sure, in fact, you'll have to, there's no other way!  Now I think I should have just batted my eyes, all innocently, and said, "Well, sure, if you think it'll help..."

So the suspense is building, for them and for me! If only one kid figures out how a formula for their zero, all by themselves, I will be thrilled. That'll be better than all preceding years.

In about a week, maybe two, I'll write an update. Watch this space!

Wednesday, October 15, 2014

Recursive Learning Using Geogebra.

Last year was the first year I had my students making geogebra applets. Now that I look back, I think I went too fast at first, because their first function assignment was this. I think it was too much too soon. I may have frightened a few of this year,

I'm slowing down now so I can speed up later.

This time around, they did that same linear function geogebra, but in several layers. I devoted a whole week to letting them get to know geogebra, using the linear function, with which they are already familiar from grade 10. I wanted to start with the linear so that this time around, they're learning about geogebra, as opposed to the math. Although frankly, the two never happen in isolation, but I digress. I had them watch my "Learning Geogebra" video, do the practices that went with them, and then create a new geogebra every day of the week, each one a copy of the previous with more information added to it. I gave feedback on every single version, and helped individuals so that everyone was good before going on to next version. Here's what we did:

Day 1: Create a linear function controlled by sliders for slope and initial value & make sure the graph matches with your own knowledge of the linear function (ie increases for a > 0, flat for a = 0, etc)
Day 1: Sliders for a and k, rule y = ax + k
Day 2: Add a t-slider and a point P whose position is controlled by the slider, and which slides along the line, no matter how much a and k are. This was really hard for them, partly because they thought that sliders were only for parameters, which was my fault. Will have to do that better next year.
Day 2: t-slider and point P = (t, at + k)
Day 3: Add the initial value and the zero. Developed formula by: Got them to pick their own a and k, calculate I and Z, then use their own ggb to check. Survey everyone's calculations to get pattern, develop formula for I and Z, type in point to geogebra.
Day 3: I = (0, k) and Z = (-k/a, 0)
Day 4: special stuff, like conditional colours depending on whether the function increases or decreases (or is constant), displaying rule/coordinates using the "show value", or by using text boxes with objects in them.
Day 4: Cool fun pretty stuff
Bonus teachable moment

Some students didn't seem familiar with good computing practices such as saving subsequent versions with a new name, or file naming practices. The version created on day 1 was linear1, day 2 was linear2, etc. I discovered this when one student sighed and said it was tiring having to start all over each time....

And on that note, I made this for them:

And now, for the rest of this year, they're going to make geogebras AND USE THEM!!! To make more!!!!

I want them to do this for EVERY function we're studying. And not only make geogebras, but USE them. And not just because I tell them to, but because they are compelled to, in order to move their own learning forward, in whatever direction they choose. I'm seeing a cyclical formation, in which they use their own paper graphs, their own calculations, and their own instincts to create, use, then improve their geogebras, which then feed the next one...

The recursive learning: Create - Check - Use - Repeat

This week we're starting the absolute value function. Here's what I'm planning:

Tuesday: (Yesterday) Graph, on graph paper, many graphs of absolute value functions, and worked their way up until they could quickly graph and describe y = a |b(x - h)| + k. So now they knew what a graph should look like and why it looks that way.
Wednesday: (Today) Create version 1 with sliders for a, b, h, and k, and verify they're doing what they should by comparing to paper graphs. Also put in a vertex with (h, k) and make sure it's where it should be.
Thursday: Math lab! Use version 1 to explore relationships between parameters, to develop point P.
Friday: Create version 2 with time slider and point P, verify it using own calculations
Next week: To add initial value, domain, range, interval of increase etc

New Rule: Trust yourself first, geogebra second

What I really, really want to EMPHASIZE is that they verify, as much as possible, their own geogebra, using their own calculations, and not vice versa. Most importantly they verify NOT by showing it to ME and asking ME if it's right.

If this works, by the time they're done this, they will know the absolute value function like a boss!

And by the end of the year, they'll be total geogebrainiacs like me!

Sunday, October 5, 2014

Audrey Learns to Code

I’m an online teacher for LearnQuebec, and I recently became a student in a classroom again, which hasn't happened in a long time. In my development as a teacher, I tend to spend a lot of time online, learning new things independently in a just-in-time fashion, but this post is about an instance in which that didn't work out, and I needed to be face-to-face with an instructor and peers. As usual, I learned way more than just what I set out to learn...

Audrey code

Until very recently, the only code I knew was Audrey code. For example, the first time I asked someone what “html” was, they answered me by saying “hyper text markup language.” I responded by blinking and saying thank you, which is Audrey code for “Now I have four more questions in addition to the one I just asked you.”

Probing further did not help. Every explanation seemed to make things worse, and intimidate me even more. Brow-furrowing, sighing, and wincing became part of my code. Nevertheless, I had a vague notion that it had something to do with the internet.

Coding? What is this coding?

Sometime later, I started seeing hashtags about coding on twitter, like #kidscancode, #codingforkids, and #coding. There was a lot of enthusiastic buzz from teachers about the many benefits of coding. Not only is it fun, addictive, & creative, but it improves understanding in math and languages as well. It was the creative part that interested me most!  I just wasn't sure of what type of coding everyone was talking about, or what exactly was being created. But I knew that before I tried to get my students to code, I needed to know how to do it myself - teaching usually works out better that way.

I decided to join and try to learn coding on my own.  I started with JavaScript, because I had heard it referenced while using my favourite software, geogebra. The lessons were easy enough to follow, and I made “progress” according to the site, but I still felt like I was in the dark as far as what I was creating. Where would I use this JavaScript interactive thingy? I was missing the big picture, and I just couldn't keep at it without that. I felt constantly distracted, even agitated by that.

Google Apps scripts

Another effort that seemed, at the time, to be unrelated to html and coding was that I tried to learn how to write google apps script. I use google forms a lot, and there were specific things I wanted to be able to do with the data that my students were entering on those google forms. Off I went to google, and entered their google apps script “tutorials”. The problem here was that each link lead to so many other links that I lost my way very quickly. Unlike my codeacademy experience,  I was clear on what I wanted to create, but the tutorials didn't seem organized in a user-friendly way. In fact, one of the links lead back to codeacademy, specifically to their JavaScript course, which I’d already tried. What this had to do with google apps script I didn't know, which added to my confusion.

Convergence at Ladies Learning Code

These mysteries were finally solved for me on Sept 27 at a workshop in Ottawa called Ladies Learning Code. A friend had happened to mention to me that Sept 27 was National Ladies Learn to Code Day all across Canada. LLC (@llcodedotcom) is a not-for-profit Canadian organization devoted to teaching code to anyone who wants to learn in a comfortable, friendly, collaborative environment. They were having an introductory one-day workshop in many cities across Canada on Sept 27, so off I went to register. Unfortunately, the Montreal one was already full, so I decided to go to the one in Ottawa. I was persistent, because I was really interested in not only the coding, but the people who were organizing this amazing event, for free, on their weekend. People are endlessly fascinating to me, especially people who are passionate and creative.

I was not disappointed, in any way! Everyone working at the LLC session was a volunteer – our instructor, Jessica Eldredge (@jessabean),  the mentors (satellite teachers, one for every 4 participants), and the students from U of O. And everyone was friendly. You could tell right away that they were there to have fun and to help people. My favourite kind of people! I had a very strong sense that web developers are highly creative people who love doing what they do. And they love teaching other people how to do it! As for the participants, most were young, but there were a few my age, one of which sat at my table – coincidence? Probably not.

I'm sitting just right of centre. Coding!
At last - the Big Picture

Within the first few minutes of the session, a lot of my previous confusion was cleared up by our instructor, Jessica Eldredge. She said that html was what created webpages, and that you could think of webpages as being in three layers, each one in a different type of code:
  1. The first is the content (text, pictures, links etc) which is created by the html.
  2. The second is the CSS, which is another language altogether, and which makes the content have a certain colour or style or placement on the webpage. In other words, it makes it look pretty.
  3. The third is the interactive elements, such as a gizmo on That’s where code like javascript comes in, and that’s where I had unwittingly started on my unsuccessful learning-to-code journey prior to this workshop. No wonder I had been confused – I had started with the last thing – javascript! Suddenly all the pieces fell into place for me. It felt like my mind was now truly open.

I really liked the way that the workshop was organized. It was kind of a mix of the flipped class and direct instruction. Jessica would spend a few minutes explaining something, then we would work for a while to complete the accompanying set of instructions, while getting lots of support from our “mentor.” Each group of four people had their own mentor. Ours was Gavin (@GavinNL), who was wonderful.  And he happens to be a math and science teacher! He was there in a heartbeat when we needed him, which was tremendously reassuring, but we also had the ability to move forward at our own pace as well, because we had already downloaded, prior to the workshop, all kinds of software and files, including all of Jessica’s slides and instructions. Hence the flipped element. I feel validated, because I use the flip in my own classes.

Audrey learned to code!

By lunchtime, I had made this:

Incredibly, I had written some html and css, and it had worked! We didn't get to the interactive stuff, but at least now I know what it is, what it's for, and where to go to continue to learn.

What else did I learn?
  • Learning really is social. It means so much to be able to turn to someone, for a reaction, for help, for reassurance, and to offer it to them. Humans need humans.
  • I like having the option to move ahead or go back as I wish. And at different times during the day, I did both. Although at around 2:30, my saturated mind ground to a complete halt.
  • That option to move at one's own pace is only truly available if the material given is well organized, easy to find, and contains good visuals and examples, which Jessica's did.
  • Hearing someone say something is way more powerful than reading it to yourself.
  • A webpage is a file! That blew my mind. To see my webpage, I double-clicked on a file with .html at the end. I don't know why that was so eye-opening for me, maybe because it made it all seem a lot less like magic and more like logic.
  • I need to have the big picture to learn some things. Otherwise, I'm constantly distracted and agitated.
  • Web developers are highly creative people who are passionate and love to teach other people how to do the same! I'm encouraging my own kids to learn, because they are very creative people too. So far no luck, after all, I'm their mom.
  • Finally, there are an awful lot of people out there who love to teach, and are really good at it, but very few of them do it for a living like I do. I'm lucky like that.
What's next?

So what am I going to do with this? Not sure yet - I had a vague notion that I would rebuild my own blog from scratch, but that seems like it might be a bit too much to start off with. I remember feeling this way when I started to learn geogebra  - I had no idea what to make with it, I just knew that it was really really cool. That's where I am now - any suggestions would be more than welcome! And that's not Audrey code for anything!