The Prime Factorisation of Me

Only a week left of school holidays. I’d better get started on the huge list of things I mean to do over the break!

I spent some time at school today working on an idea I had last week. My next topic with year nine (once we properly finish off Pythagoras) will be Expanding and Factorising. I plan to include some work on simplifying expressions, but I’ve noticed that some students struggle with recoginising ‘like’ terms. Some get mixed up between terms like x and x² terms, and some don’t realise the differences between terms at all and try to do things like 4x + 5y = 9xy. I wanted an activity that let students practise recognising when terms are like or not.

So I came up with “Algebra Snap”:

It’s basically what it sounds like – “Snap”, but with matching like terms instead of numbers. (Everyone knows how to play Snap, don’t they? EDIT: Sorry, I shouldn’t make assumptions like that! I’ve tried to explain it below.) I haven’t had a chance to test it out yet, but I’ll post here again once I’ve used it in class. The cards are simply laminated coloured paper, with different colours so I don’t get the sets mixed up.

An extension to this activity could be to play the game the same way, except that students have to state the sum of the cards they’re snapping before they get it.

Even though I’ve printed “Algebra Snap” on the back of the cards, I hope there will be chances to reuse them in other activities too. One idea I had was something around factorising – draw two cards, and have students decide whether they can be factorised or not (and have them do the factorisation). I wish I’d planned the coefficients a bit better – if I’d used 6 more often instead of 5 and 7, then there would be more opportunities to factorise. I guess that’s what permanent markers are for.

If you have any other ideas for these cards, or any improvements, let me know!

Downloads (both files are the same thing, but last page of the Word document looks weird unless you fix up the font):

• algebra snap.pdf
• algebra snap.docx

EDIT: How to play Snap:

• Divide the cards between each player, who holds them face down.
• Players take turns to reveal the top card and place it in a stack in front of them.
• If at any point two stacks have matching cards (in this case, cards with like terms), the first player to shout “Snap!” wins both stacks and places them at the bottom of their hand. (Don’t shout if you don’t want to disturb the rest of the school. But it’s more fun if you shout.)
• The winner is the player to get all the cards.

There are rules for dealing with wrong and tied calls, but there seem to be many different versions around – google it to find one you like. The version I played growing up involved everyone putting their cards into one stack, which would be won by yelling “Snap!” and slapping your hand onto the stack if the last two cards match. It used to get a little violent, so it probably isn’t a good way to play with rowdy 15 year olds.

Here’s a few things I’ve been meaning to share, but I can’t really justify a post for each of them. So here we go, in no particular order:

My little sister has a job!

After finishing training to become a primary teacher last year, my sister just got her first teaching job! I travelled with her on Wednesday to go visit her new school. I didn’t see much of it because I went shopping while she had important stuff to do, but she was positive about it! She has a lot to do before she starts in a couple of weeks, including finding somewhere to live and setting up her classroom (which is completely empty at the moment).

She’s nervous about it, but I’m really excited for her. Teaching comes to her much more naturally than me, and she’s always known she would be a primary teacher, since she was in primary school herself (as oppossed to me who decided halfway through second year of uni). Also, she’s going to teach maths really well, partly because she’s good at maths herself, and partly because I’m not really going to give her a choice.

Maths shopping

As I mentioned above, I went shopping on Wednesday, and it was an opportunity to do something I call a “maths shop”. That is, I wander through shops looking at stuff, seeing if anything inspires any teaching ideas. I didn’t really really get any specific ideas this time, but I did buy some stuff under the vague notion that something might be useful eventually:

• A couple of buckets. (Because really, when do you ever have too many buckets?)
• A whole bunch of coloured electrical tape.
• A roll of EFTPOS machine paper. (I have no idea why, but a really long strip of paper seems like something that might be useful for… something.)
• A bunch of stationery supplies – my town’s only newsagency closed down a while back, so it’s often hard to get stuff. (I get a little crazy when I get the chance to buy stationery. It’s probably a good thing I don’t live anywhere near an Officeworks.)

I did also buy some non-teaching related stuff, but who really cares about that?

About page updated

So if you visited the about page for this blog before today, you would’ve only found three short bullet points (and the last one was just a stupid comment about how I couldn’t be bothered writing anything else). But now, it might actually be worth reading!

I’m slightly more important according to Google!

OK, I know I blog so I can improve as a teacher, not to become famous. But it would at least be a little nice if my blog could at least be found by googling it?

Well, if you google the prime factorisation of me now, this blog has reached the fourth page! It was on the seventh, so progress!

Honestly, I’ve always felt a little hard done by from Google. If you search for my name, you’ll find someone a lot more famous than me in the top spot (and his name isn’t even spelt the same as mine). At least Google doesn’t insist my name is spelt wrong like it used to.

I’m starting to ramble, so I’d better leave it here.

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:

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

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

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

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 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!)

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).

• Also: 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.

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.