Blogging? Absolutely.

This blog is one month old! Woo hoo! I know that might not seem like much, but I have a bad habit of thinking of great ideas then not really sticking to them. This is something I really want to keep doing, because I can already see it making me a better teacher.

Case in point: I had an idea for an activity on the absolute value function this morning while still in bed. Problem is, on my usual timeline, I cover it in February. So under normal circumstances, I’d think “That’s a good idea, I should remember it.” Then I’d proceed to go back to sleep and immediately forget it (it is school holidays now, after all).

But this time, I actually got out of bed and starting testing the activity. Because now my thoughts and ideas aren’t just dumped in the back of my brain – I can use this blog as a way to process those ideas into something useful.

So I’ve created this thing which I’m dubbing the “Absolutamatron”. (I know that’s a horrible name. Please let me know if you have a better one.) I’ve often described the graph of the absolute value function as “flipping all the negative y-values to positive y-values”, but for some reason it had never occured to me to physically flip the graph. So I made this thing:

Original function

abs of original function

It doesn’t really matter what the function is, but this one is supposed to be f(x) = (x+3)(x+1)(x-2).

So I’ve never used this in class myself (as I said, I only thought of it this morning), but I think learning value for the students would be in making it themselves. I deliberately created the absolute function graph by tracing over the original graph, which is what I would have students do too. So this is how to do it:

You will need:

  • Three identical blank Cartesian planes (which you can download here).
  • Scissors to cut out the planes.
  • Markers that will “bleed” through the paper (I used Sharpies). Use colour, because you don’t want to be boring!
  • A pencil.
  • A glue stick.

Step 1: Using a marker, sketch your first graph onto one of the planes. The function can be anything, but it needs both positive and negative values (polynomials work well).

Step 1

Step 2: Cover the graph with another plane, being careful to line up the axes. Trace over the positive values only using a pencil.

Step 2

Step 3: Flip the original graph so the bottom is at the top. Trace over the negative values (which are now positive).

Step 3

Step 4: Remove the original and draw over the pencil markings using a marker.

Step 4

Step 5: Fold each graph in half along the x-axis. Glue the top half of the original to the top half of the blank graph, and glue the bottom half of absolute graph to the bottom half of the blank graph. (Alternatively, you could just glue each half directly into a notebook.)

Step 5

Step 6: Glue the middle bits together. Now you have your very own Absolutamatron!

So it’s really a simple little idea, but it’s an idea that I can guarantee I would have forgotten about by next year. I’ll put a warning on this in case it wasn’t clear: NOT CLASSROOM TESTED. Yet. But if anyone does give it a go, I’d love to know how it went 🙂

(And seriously, if you think of a better name, please tell me that too!)

Back home from Canberra

We made it home and I’m incredibly tired, but now it’s school holidays 🙂 After re-reading my post about going to Canberra with Year 7 & 8, I’ve realised how pessimistic I sounded. In fact, it was a really good time, and our students were very well behaved. There’s a few things I want to write about in detail, eventually, but for now I’ll just make a note of a few of the things that happened:

  • I really enjoyed the National Portrait Gallery. I know! Not really what I expected either. A lot of this was from our tour guide, who didn’t just show us the artworks, but involved a number of different educational activites and got the students to think about what they meant within a theme of “Australian Identity”. I wouldn’t be at all surprised if she used to be a teacher.
  • The War Memorial continues to be one of the most moving places I’ve been to. I really need to find an opportunity to visit Canberra again so I can spend a whole day there.
  • Questacon is still one of my favourite places in the world. Of course this was always going to be my highlight, but I think it was for a lot of the kids as well. If you don’t know what Questacon is, it’s basically a gallery of a whole range of hands-on science demonstrations and activities (including a decent smattering of maths).
  • Also from Questacon, I probably spent too much in the shop. I don’t get many opportunities to buy books about maths where I live, and my collection consists of mostly uni textbooks. So I picked up some holiday reading. (Admittedly, one of them is about physics).

Books from Questacon

  • Also: pi mug!

PI MUG!!!

  • After I scored better than the kids at ten pin bowling, some of them have taken to calling me “Mr. Skills”. They starting talking about me similarly to how the internet talks about Chuck Norris. When we visited Parliament we met our local member – after they saw me chatting with him, these kids became convinced that I “know him from way back”. They started telling stories about me secretly being friends with the Prime Minister and actually being the one in charge of federal politics. I’m a little worried about where this might lead…
  • Other than the strange rumors being started about me, it was great to get to know these kids and to have them get to know me. I don’t actually teach either of these classes, so a lot of them have previously just been names and faces to me. If I teach them in the future, it should make the start of the year a lot easier.
  • I spent all week limping around Canberra after being hit by a hockey ball on my ankle last Saturday. But it’s OK now – it’s been balanced out by being hit on my other foot today. I think it really is time to have a rest.

Ten pin bowling and Combinatorics

This is just an idea I had tonight while on camp with Year 7 and 8. It doesn’t relate to anything thing I’m doing in class at the moment, but I wanted to get it written down somewhere before I forgot.

So we went ten pin bowling tonight, and I noticed that after each bowl the scoreboard showed a quick video of the pins getting knocked over (I wish I’d gotten a photo of it). Maybe this is pretty standard and you all know exactly what I’m talking about, but we don’t get to see much bowling in a town as small as ours.

Anyway, initially I couldn’t tell if the video was really recorded live with a camera I couldn’t see, or if it was faked by showing a prerecorded video with the appropriate pins being knocked over. (I’m pretty sure the videos were real, but that’s beside the point). It got me thinking – how many videos would need to be prerecorded in order to be able to show every combination of pins falling over?

The simple solution is this: there are ten pins, and each pin has two possible states, standing or knocked over. So the total number of combinations is 210, or 1024 as any addict of “2048” would be able to tell you. Or if we imagine a more general sport called n-pin bowling, then the number of videos needed is 2n.

The thing is, this is not the first solution I tried. Instead, I tried to use combinatorics:

  • There is C(10,0) = 1 combination with no pins knocked over.
  • There are C(10,1) = 10 combinations with 1 pin knocked over.
  • There are C(10,2) = 45 combinations with 2 pins knocked over.

etc. Then we add them together. This is much more work than the other method (particularly when trying to do it in you head while supervising a bunch of excited 12-14 year olds), but should give us our result. We can make this simpler by remember that combinations are contained in Pascal’s triangle, so we can just add the numbers in row 10 to get our result:

1 + 10 + 45 + 120 + 210 + 252 + 210 + 120 + 45 + 10 + 1 = 1024

But this should always work for our hypothetical game of n-pin as well, by adding the entries in row n. Because the number of videos should be the same regardless of which method we use, this leads to the following result:

The sum of the entries in row n of Pascal’s Triangle is 2n.

So that’s kind of cool. A more typical proof of that statement is to consider the binomial expansion of (a + b)n with both a and b set to 1. So now we can relate the problem to algebra as well!

I know this isn’t a lesson yet, but I wanted to get it down while it was still fresh in my mind. Hopefully I’ll remember to look up this post later in the year, when I actually have to teach this stuff! There’s also few possible extensions to this problem I thought of:

  • Are there any combinations we can eliminate and not make videos for because they are impossible to occur?
  • My 2048 reference was really just a joke. But thinking now, 2048 is all about powers of 2. Pascal’s triangle is (in a rather sneaky way) also about powers of 2. Is there some hidden connection between 2048 and Pascal’s Triangle?

For the record, I scored 133 and 126 in our two games of ten pin. Not great scores, but pretty good for me. I did start with two strikes, so now some of the kids think I’m some sort of bowling genius.

Midyear feedback: Year 9

I had a student ask me the other week, “Teachers are always writing reports about us. When do we get to write a report about the teachers?” Well, I thought, why not?

Actually that’s a lie. I’d already decided to gather feedback from my classes weeks ago, but the fact that one student asked this question was a convenient way to start this post :). But seriously, isn’t the whole point of writing student reports so they know what they need to do to improve (well, that’s the theory at least). Though they really aren’t the experts they think they are on what teachers do, students are more than willing to let us know what they think of how we do our jobs. If we want to improve as teachers, it might be a good idea to at least listen to them from time to time.

I decided to use the “Keep, Change, Start, Stop” format and set up a Google docs form to collect responses from students. I thought I might share some of the feedback I got and maybe even respond to it. In each section, I’ve tried to order it from most common to least common feedback. The first class I got responses from was Year 9.

Keep

New Things Thursday. YES! YES! YES! This was overwhelmingly the most popular response. If you missed it, I introduced New Things in this post. I will absolutely continue it in Term 3.

Monday Science Pracs. Something I haven’t explained yet on this blog: our school has a slightly different Year 9 program, the result of which is that I actually teach a combined maths and science subject we call “Exploration” with another teacher. I mostly focus on the maths, but we have one lesson a week with both teachers in the room on Monday, which is when we tend to do science pracs.

Pythagoras’ Therom or however you spell it. Good. I wasn’t planning on getting rid of it. Not that I’d have a choice in the matter anyway. But more significantly, I’ve been trying a few different things in this unit (including New Things), which students seem to be more enthusiastic about.

Your beard. I grew a beard over the last school holidays, which has split student opinion between “It’s awesome” and “It’s scary”. I’m sorry to the jealous boys in my class, I think it’s only going to be a one term thing.

Change

Less book work. I’m pretty sure you meant “less textbook questions” by this. I agree – I’ve been guilty of falling into the too-many-textbook-exercises trap on a few occassions. I’ve been trying to get away from using it as much as possible lately – I promise I’ll keep doing that.

Let us leave class to fill up our water bottles. No. Just no. You know it’s a school rule, and you know you have recess and lunch to fill it up. Don’t waste my time just because you can’t mangage yours properly. (Can you tell I’ve had this conversation a lot over the semester?)

Make questions easier. If you mean “make questions clearer and more approachable”, then yes, I will try to do that. If you mean “let me not have to think in your class”, then no.

Make maths more fun. I’ll keep trying to do this. But my idea of this might be a little different than yours. I believe that maths is fun (because I am a crazy person), and my job is to help you understand it better so you can see how awesome it is. (But if you mean “teach us with different maths activities”, then sure, I’ll make more of an effort to do this).

Start

Most of the feedback here was repeats of the first two questions, but there were some new points.

Short quizzes at the start of the lesson where you time us. I’m pretty sure I did something like that two years ago when I had your class in Year 7, and you guys hated it! But OK, I’ll think about it.

Free time on Friday. No.

Give us more feedback on our work. I agree, this is something I need to work on.

Yes. Ummm, what? (This really was one of the responses. The same student wrote “No” for Stop).

Stop

Stop counting. I assume by this you mean the thing I do to get your attention when you’re all too noisy? (I quietly count by fives, which lets the class know I’m waiting for them. I go up by five because I used to say “five seconds, ten seconds, …” but they kept correctly pointing out that I usually only take three seconds for each count). There’s a simple solution to this – if you’re all quiet when I ask you to be, I won’t have to do it 🙂

Nothing, I guess. Well, that’s good to hear! But I’m under no illusions. Even if I get to the point where my class thinks I’m perfect, I know there will be improvements I can make.

I think that’s a good start for thinking about where I am with my teaching at the moment, but there’s a lot more reflection to do over the two week break. (And anyway, this is only one of my classes). If you’ve never gathered feedback from your students before, it could be a good idea. It’s a little scary giving students permission to examine and comment on your teaching practice, but they know what you do in the classroom better than anyone else. But it can’t stop here. The only feedback of value is that which leads to improved pedagogy – and I think that’ll take harder work than just letting students write a few things about you.

The calculus of glue-sticks

One of the VCE Maths Methods topics I don’t think I’ve covered particularly well in the past has been the estimation of definite integrals using rectangles. By the time I get up to it, we’ve usually had so many interuptions during the first half of the year (this year is no exception) that I’m trying to catch up and often rush through the material with some short board notes and a few textbook questions.

This year I made a conscious decision to improve the way I introduced it. As much as sketched diagrams help a little, I think they still leave the concepts pretty abstract. I wanted to make my examples more concrete. I wanted to set the stage for explaining the definite integral as the limit of a sum. Hopefully this will also later help draw attention to the significance of the Fundamental Theorem of Calculus.

My solution (which I’m starting to realise is the solution to pretty much everything!) was cutting and pasting. Initially my students weren’t terribly keen on it, and thought it was a waste of time when they could just draw it, but I think they came round to the idea – kind of. One of them asked an off-topic question, which they started by saying, “Since we’re not really doing maths today…”. That’s OK, I’ve still got a semester to convince them that cutting and pasting absolutely is maths. (Actually, the question was a really interesting one, but I’ll wait until another post to write about it.)

I gave the students a sheet with the same graph of y = -x2 + 4 repeated four times, on grids where 1 cm = 0.5 units. I also gave them coloured paper to cut into strips 2 cm and 1 cm, to use as the rectangles. The result is below:

Estimating using rectangles

With more time, I would have liked to have also done left and right endpoint rectangles, as well as midpoint rectangles and trapezia (I stuck to the boring smartboard for these). Personally I prefer upper and lower rectangles because you can talk about how they, besides being an estimation, form upper and lower bounds on the actual value, and you can demonstrate that the bounds get closer as the number of rectangles increases. But that could just be my pure maths background – I’m sure I’d prefer a faster method for calculating if I had an applied background.

Thinking ahead to next year, this could be followed up with some sort of technology to quickly demonstrate with even more rectangles – maybe using Excel or Geogebra, or maybe even the CAS has some way of doing that. I’ll need to investigate those options. That way we could experimentally find the limit, and hopefully comfirm it matches the definite integral we find using calculus.

EDIT: I meant to include a link to the worksheets here, I’ll add them once I’m near a computer.

EDIT (23-6-2015): A year later (to the day, apparently), I finally got around to adding those worksheets!

Downloads:

Mr Carter Goes to Canberra

I taught my last lesson for the term last Friday. Reports are written. My year 12s have their holiday study homework. My desk is tidy (well, tidier than it was). My list of paperwork to get through is significantly smaller than it was. There’s only one more thing to get done before I can go on holidays:

Go to Canberra with years 7 & 8 for a week. Hooray?

I have to admit I’m not really as excited as I should be right now. We left school at 6:30 (and I never cope that well with early starts as it is), I just had to growl at a couple of kids who were getting overly excited over a game of Mario Kart, and my leg hurts after I failed to jump out of the way of a hockey ball on Saturday. Plus I know how little sleep I’m going to get over the next week, and I remember how stupidly cold our national capital was last time I went on this excursion – and it’s already been stupidly cold at home.

But I really am trying to focus on the positives. Above all, we put ourselves through things like this to give our kids experiences they can’t possibly get by just staying at school. One of the problems our kids have growing up in a small town it that they sometimes don’t really think about the existence of the wider world – hopefully visiting Canberra gives them a better sense of their place in the nation. But these are some things I’m personally excited about:

  • As crazy as it might sound, I’m looking forward to spending a week with these students. I haven’t taught either class before (aside from a few replacement lessons) so I don’t really know them that well. This is my chance to get to know them before I probably teach them in the next few years.
  • Even though maths is “my thing”, I do tend to nerd out on history a bit. I’ve been to the Australian War Memorial a couple of times, but I’m keen to explore there again. I’m also a bit of a politics nerd (I’m not that political, I just find the process fascinating) so I always get a bit of a kick out of Parliament House and the High Court. (Maybe I should have left this point as “I’m a bit of a nerd” and that would have summarised it all?)
  • Questacon! (aka the National Science and Technology Centre.) Last time I was there, I just wandered around with the kids, which was awesome enough by itself. This time, hopefully I can get some lesson ideas out of it.
  • I’m really looking forward to not going ice skating as we’ve done on previous trips. That’s the kind of stress we really don’t need.
  • I’ve had a coffee since I started writing this, so I’m actually awake now.
  • When I get home, I’m going to sleep for a week.

Should I be worried that the last two points were about how tired I am?

Weirdly, I’m also looking forward to the hours spent on our coach today and at the end of the week. I really haven’t had any spare time for a while now, so I can finally sit down and write a few blog posts that I’ve been thinking about. Stay tuned to either read my reflections of the first semester, or to watch my sanity slowly drain away. Could be either, could be both.

New things follow-up

On Monday I had my first Year 9 lesson since introducing New Things Thursday last week. While I’d had a few positive reactions, I was still a little nervous about how the class felt about the whole thing. As it turned out, they were asking if we could do New Things, and were disappointed that they’re going to be away on Thursday. I was also disappointed, so I’d already decided to New Things on Monday anyway 🙂

As I put my New Things slide up on the IWB, my students were immediately excited about being able to share their new things – well, most of them were. It was only then that I realised two of my students were away last Thursday and had absolutely no idea what was going on. Oops. I told them to just go with it and found a gap in the conversation to explain it all to them.

Proving Pythagoras’ Theorem

Last week explained Pythagoras’ Theorem, and while I mentioned that theorem’s should be proven, I’d only hinted that the proof was still to come. This was Monday’s task. Like last time, we’d do this with a “New Things Page”.

I wanted to make the proof as simple to follow as possible. I also decided to use cutting and pasting of triangles and squares rather than just drawing diagrams – I hoped to help the kinaesthetic learners as well as the visual ones, and make it obvious that all the triangles are the same. The end result (well, my version of it) is below:

Proving Pythagoras' Theorem - new things page

(I tacked on the algebra proof after the class – it’s my personal favourite proof, but we haven’t covered enough algebra yet for me to teach it. Also, I think my arty skills have improved since last time!)

The key point that makes the proof work is realising that the areas not covered by the triangles are the same on both large squares. When I asked the class which area was bigger, they debated with each other for a few moments before declaring it a stupid question because they were the same. (Hooray!)

I’m pretty keen to make New Things Thursday a permanent feature, but I’d like to see if the class stays enthusiastic about it once it’s, well, not a new thing. I asked a few students if they thought it was a good idea or if they thought I’d gone crazy. “Both,” was the answer. I guess that’s good?

Half time score update

One of the features of my Maths Methods classes over the last few years has been the scoreboard, for a game that no-one is entirely sure of the rules for. Well, there are some rules:

  • If I make a mistake on the board and they spot it before I correct it myself, the class gets a point.
  • If they claim I made a mistake when I didn’t, I get a point.
  • I’m the only one who’s allowed to update the scoreboard (otherwise I get a point).
  • Last year they were allowed to claim one ‘cake-point’ per week (that’s exactly what it sounds like: they get a point if they bring a cake to class). Honestly, I don’t mind it when they score this way 🙂

Plus there’s a whole heap of silly rules that tend to get made up on the spot, by both them and myself.

It’s a silly little thing we do which keeps them paying attention and keeps me on my toes. I’d like to say I planned it that way, but the game sort of just developed by itself. At the end of each year the outgoing class seems to tell the next year’s class about the game, so they claim points from me before I’ve even mentioned it.

The first year we did it, I won comprehensively. It was a tie last year (though I may have bent the rules a fair bit to catch up). This year, I think I’m in trouble.

As Unit 3 ended on Friday and Unit 4 starts this week. So it’s half time at the moment. The scoring’s been pretty slow this year – they’re fairly conservative with their challenges and aren’t willing to give up anything easily. Unfortunately for me, that strategy seems to be working:

Half time scoreboard

Oh well. Everyone loves a good comeback.

Girls in maths

I’ve been trying to write this post for a while. I had a student ask me a question one day that’s been troubling me ever since. I’ve put off writing about it until I could come up with an answer I’d be satisfied with, but as that hasn’t happened yet, I’m just going to post what I’ve got so far and hope something meaningful results.

Anyway, the question was this:

Why are 75% of students who do Specialist Maths male?

My answer at the time can basically be paraphrased as “Um, er, yeah, I don’t know?” I’m not sure where she got that statistic from, but according to this document it was about 66% male students in 2013. Certainly less than 75%, but in no way less concerning. This can be compared to Maths Methods with 58% male students, and Further Maths with 52% female students. (If you’re confused right now, I’ve tried to explain the VCE maths subjects below.)

We have a situation where girls are willing to do maths (as evidenced by Further Maths), but are not willing to attempt the more difficult subjects. I personally have seen a number of girls who I’ve believed to be strong maths students decide that it would be too hard to attempt the harder subjects – a decision some later regret when it affects their ability to get into science courses at university.

I just realised that I might be working from an assumption that not everyone agrees with, so let me spell it out clearly so you know where I coming from: This is not a good thing. This tells me there are female students who are missing out on opportunities to suceed in maths and science, not because of their ability but because of what the world around them is telling them they can do. And if students (male or female) are missing out on opportunities they should have, that is a failure in the way we are educating them.

This was where I was hoping to have a lightning bolt of inspiration and have something important to say, but all can think of is more questions.

Where does this stereotype that “girls are bad at maths” come from? There is an undeniable perception in our world that “boys are just better suited to thinking mathematically than girls”, despite the fact that I’ve never seen any real evidence suggesting that. Clearly less female students are choosing to attempt the more challenging maths subjects. The misogynistic view would take this as evidence to support itself, but it’s really a self fulfilling prophesy – if girls are told they can’t suceed in maths, can we expect them to suceed?

Have I, if inadvertently, contributed to this problem at all? Have there been times in the past that I’ve influenced girls away from maths without realising it? I would be horrified if this were true, but I realise that as a male maths teacher I am, in a way, reinforcing the stereotype just by existing. I really think I need to be more aware of the message I give my female students about their ability in maths, and ensure everything I do builds their confidence in their own ability.

What can be done to change this? What can I do in my classroom? What can be done in my school? What can be done in the Victorian education system (and more broadly, the world)?

These are not hypothetical questions. Seriously, if anyone has an answer, please let me and everyone else know.

I’m sorry if you’ve read all that and expected something insightful – I’m still not really sure what point I’m trying to make. But the way I see it, we have a problem at the moment for which there hasn’t been a solution discovered yet. I realise that as a male maths teacher I might not be the most qualified to talk about this (and anyway, y’all don’t know what it’s like, being male, middle class and white). So if someone who actually knows what they’re talking about wants to jump in now, feel free.


xkcd: How it works

For those that don’t know how VCE maths works, we have three subjects in Year 12:

  • Further Mathematics is the easiest of the three – I would hesitate to call it “easy”, but the bar is low enough that any academically minded student should be able to do well in it, even if maths isn’t their strong suit. At my school this subject tends to have the second highest enrollment in Year 12 (behind English, which is compulsory).
  • Mathematical Methods introduces a much more formal treatment of maths, and focusses on algebra and functions, introductory calculus, and probability distributions. This is generally considered a minimum prerequisite for most university STEM type courses.
  • Specialist Mathematics is basically what the name suggests. The enrolments in this subject are much smaller than the others. Most students who do Specialist also tend to do Methods.

New Things Thursday!

So after my last post I started to have a crisis of confidence in what I was planning to do. I felt like I oversold my plans a bit too much. What if this is a complete disaster? I’ve kind of committed to blogging about it now!

Thankfully it seemed to go well, and the feedback I got from the students was really positive.

So anyway, introducing…
(drum roll, drawing out the tension, not really a surprise since it’s in the title…)

New Things Thursday Powerpoint slide I used to introduce ‘New Things Thursday’.

So this is the idea – Thursdays are now all about ‘new things’. As I wrote last time, I want to capture the excitement of discovery that I see in maths but seems to go unnoticed by students. By putting the idea that this is something new front and center, I’m hoping I can sidestep that part of a kid’s brain that tells them each lesson is just the same old boring maths.

So I’m putting these rules on myself for Thursdays. That said, I’m completely prepared to break these rules I feel the need to. And given that New Things Thursday is itself a new thing, it may need to change.

Rules for New Things Thursday

  • Each lesson will be about on something new. A new rule, a new concept, a new perspective, whatever. We will still do some revision, and the new content will build on what the students already know, but the ‘new thing’ will be the focus of the lesson.

  • Rather than explicitly telling students the new thing, I will find a way to let them discover it themselves. This will be a challenge, as it will take a lot longer to introduce each idea, but hopefully it leads to more solid understanding from the start.

  • For each ‘new thing’, students will create a ‘new thing page’: a page in their workbooks that summarises the new content. This page will be their main reference to the concept in future lessons.

To set the tone of New Things Thursday, I’m also allowing students a few minutes where they can share with me and the class any ‘new things’ that they’re excited about, which can be whatever they’re interested in and doesn’t even need to relate to maths. (I realised afterwards that I may have subconciously ripped this idea off from Sarah Hagan’s Good Things Mondays, so thanks goes her. Actually, most of elements of New Things Thursday are probably stolen from other bloggers.)

The first New Thing: Pythagoras’ Theorem

Introducing Pythagoras is always an interesting moment; it’s one of those key points where high school maths transitions from being ‘arithmetic’ to ‘mathematics’, and might even be the first time students hear the word ‘theorem’.

The statement of rule (a2 + b2 = c2) is so simple to students that they don’t always recognise the significance of it, and unfortunately it is often only treated as a tool for calculating lengths. I think our aim should be to helps students realise the surprising elegance of the theorem, the mathematical truth it reveals.

We started the lesson talking about square numbers and I asked why they were called ‘square’ numbers. It took a little prompting, but they realised that they relate to the area of squares. I had them list all of them from 12 to 202, which didn’t take too long. Once they had their lists, I pointed out that 144 + 25 = 169, or 122 + 52 = 132.

I asked if they could find any more combinations like it, and gave them a few minutes in groups to see how many they could find. (I deliberately didn’t use 32 + 42 = 52 as my example so they would have an easier one to find.) A few students asked me whether the ones they’d found were correct. I told them they didn’t need to check with me – they could check themselves using their calculator.

I then listed the combinations on the whiteboard. Some had found up to 82 + 152 = 172. Some had realised that many were multiples of 32 + 42 = 52. (YES!) I had each student choose one equation from the board, and asked them to cut out three squares representing the numbers in their equation from grid paper that I had handed out.

Notice that so far I have not mentioned triangles or right-angles, or even the name Pythagoras. That was the secret truth that I wanted my students to discover.

I asked them to create a triangle by using the edges of their squares. What sort of triangle was it? It turned out they all had right angle triangles!

It was at this point that I finally put up some notes about Pythagoras’ theorem, and had them create their new things page. I’d already prepared one myself in case any students didn’t know what to put on theirs:

My new thing page! Sorry I'm not that artistic...

Thankfully a lot of of the pages the class made looked a lot better than mine. Some students even decided to create a ‘new things’ section in their book so it would be easier to look up in the future.

I pointed out that though we’d discovered this pattern, it’s not really a theorem until we prove it – which hopefully will happen next lesson.

Unfortunately, that was the last Thursday I’ll have them this term – they have an excursion next week and I have one the week after. But I think I’ve built some enthusiasm from the class for ‘new things’, and set a pattern we can start in earnest next term. And anyway, there’s no reason I can’t do this:

New things Friday!