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The distributive law

Teaching the distributive law is an interesting prospect. Because really, if students know how to multiply numbers with more than one digit they already use the distributive law, even if they don’t know it. But if you write this on a board:

a•(b + c) = a•b + a•c

and expect kids to understand it, you’ll be met with blank looks. The challenge is not in knowing the law, but in connecting it to prior understanding of numerical multiplication and extending that understanding to algebra.

When Year 9 and I started our unit on Expanding and Factorising (which admittedly was weeks ago, we’re nearly finished the next unit now), I posed a seemingly simple question: what is 29 × 4? The class worked in groups, and I enforced a couple of rules: they couldn’t use calculators and they couldn’t use the standard written algorithm.


Looking back now, I really should’ve created a better graphic than just drawing on the IWB…

Most were clearly happy with themselves for finding the answer, but I ruined that by writing 116 on the board as we came back together as a class. Just giving me the answer was no longer good enough. The groups needed to explain how they got their answers.

As a side note, I find this a constant struggle with my students. How do I get them to unlearn the idea that “getting the answer” is everything, and think more in terms of understanding the problem?

Anyway, it turned out that amongst the groups there were a few different methods used.

I think most readers will know where this is going. Though the methods used were different, they were all dependent on the distributive law. I presented some of their solutions to them in a different form.

This was a good launching point for talking about expanding. After the class did a couple more numerical examples, we started introducing variables into it.

Of course, I was silly enough to do that part on the whiteboard instead of the IWB, so I don’t have an image of it 🙁

Often with algebra, we are teaching rules and patterns that students already know are true, but they are not familiar with the language and structure of algebra. Yes, expanding 2(x+3) is an abstract idea, but 29×4 is itself an abstraction, really. A useful strategy is to help students realise that algebra is often just a layer of abstraction over concepts they already know and understand.

 
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#OzMathsChat

A few weeks ago, @DianeMaths tweeted out the idea of a regular maths chat in a timezone friendly to Australian teachers. At the time, I was the only one to respond, but we were both really keen, so we decided to start one!

#OzMathsChat starts Tuesday (that’s right – Tomorrow!) at 8:30 pm AEST (GMT+10). As long as you’re interested in maths education at any level, we would love to have you there.

Don’t let the name make you think it’s only for Australia. There are many other regular twitter maths chats around the world, but none (that we could find) at a time suitable for us in our isolated part of the planet. But if our weird timezone doesn’t put you off, you’re more than welcome!

And even if you can’t make it, we’d love you to retweet or pass the message on to anyone else you think might be interested.

 
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Discrete Probability Posters

At the start of this term, I said I was going to make more of an effort to make posters and decorations, particularly for the video conferencing room where I teach Year 12 Maths Methods. That hasn’t really happened as much as I’d hoped.

But seeing all the new year classroom posts from the northern hemisphere has reminded me again how bare some of my walls are. Given I’ve just started Probability with Year 12, here’s a set of posters on Discrete Probability Distributions I put together today. I plan to follow these up with a set for Continuous Distributions soon.

Downloads

If you want to edit the document, you’ll want the font Matiz.

 
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Student given factorising questions (or, How to fake being organised)

While my Year 9 class is starting to move on to Area and Volume, I’m still quizzing them on Expanding and Factorising. At the start of Monday’s lesson, I planned to hand out a short set of factorising practice questions to warm up with.

At least, that was the intention. As it turned out, I only remembered that when I walked in the classroom door.

I could’ve moved straight onto the main activity for that lesson, but I still wanted to do some factorising questions first. So I did what any teacher does when they forget something – pretend that’s what the plan was all along. As it turned out, the accidental result was better than my original plan.

I asked students (some volunteers, some I picked on) to give me examples of expressions that can be factorised, which I wrote on the board. I gave everyone a few moments to factorise them, then the class gave me their answers:

(I’m so sorry for my scrawling handwriting. I’ve never really had the patience to write on the IWB neatly.)

The third one was really interesting. Obviously it can’t be factorised, and my reaction at the moment it was suggested was to say that, but fortunately I held my tongue and left it there. That question mark represents a really good discussion the class had about whether this counted as factorising or not. Also, it was awesome that most of the class had already recognised that 1 was the highest common factor of the two terms. They decided it wasn’t factorising, because multiplying by 1 doesn’t change the expression, and it didn’t help simplify the expression.

This is one benefit of getting the class to suggest questions. Had I remembered to organise questions before hand, I would not have used a question like this and not led to that discussion.

I pointed out that all the questions had a numerical common factor. “Is this the only type of factorisation?” I asked. This led to a whole new round of student given questions:

With these, I love how a couple of students recognised that using x2 allowed them to make x the common factor. I also love how the last student refused to allow that pattern to continue, so looked for a different example. There’s much deeper thinking going on here – rather than just giving the answers, students were able to think about and discuss the nature of expressions and how they can be factorised. Which is pretty cool for a warm-up activity I didn’t plan.

 
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Area of a Sector

(Quick confession: this was actually weeks ago. I’m not sure why it took me this long to get around to blogging it.)

Why do textbooks bother trying to explain concepts before each exercise? The kids don’t read it, I don’t refer to it. And the explanations take all the fun of doing mathematics out of it.

In case anyone wants a more pedagogically solid complaint than “takes out all the fun”: The book just gives students formulas and expects them to rote learn their use, rather than using discovery to build understanding.

Anyway, case in point: the area of a sector. The book basically says, “sectors, here’s a diagram, here’s the formula, go do questions.” I thought I could do better.

Enough with the textbook ranting, do some maths already…

Normally this would’ve been something I’d use Geogebra for, but since I’ve become more than a little obsessed with Desmos lately, I went with it instead and put this together:

Desmos area of a sector

[Update 2/2/18: In more recent years, I’ve used another version of this that does both sector area and arc length. You might find that more useful. Link is https://www.desmos.com/calculator/mybwxhjws3.]

I started with the whole circle displayed on my IWB, and we talked about the effect of changing the radius on the area. This was revision, but I wasn’t particularly satisfied about how this discussion went – they got the idea that, say, doubling the radius quadruples the area. but when they gave reasons they only talked about r2. They couldn’t connect it to the circle itself. I moved on at the time, but I think I should’ve spent more time developing the idea.

So I changed the angle to 180°, and asked what the area would be now. “That’s easy, half the area,” came back the response. I showed them a few more angles: 90°, 270°, 45°. They responded with the appropriate fractions. So far so good.

“So what about 213°?” That had them stumped. “Well, maybe you should figure it out in your groups?” (Forgot to mention, I already had them in groups.) I gave them the link to open the Desmos file on their laptops to play around with while they discussed.

What I love is that there were different methods found around the room. Some found the fraction 213/360 quickly, but there were other approaches. Some found the area with 1°, then multiplied by 213. Some recognised that 213° is between 180° and 180° + 45°, and used that fact to determine lower and upper bounds on the solution.

This was a nice activity to differentiate, too. When some individuals found the solution quickly, I had them figure out the formula themselves. When they thought they had it, I made them explain it with their groups, who then discussed whether it worked or not. Others needed more time to find the area in the first place, but they all got there in the end.

I then had each student create three different sectors using the Desmos sliders, which they then printed, stuck in their books and calculated the areas for. (The nice legacy of being in the former computer lab is having a printer in the room!) A few switched on students who had figured out the formula typed it into Desmos so it just calculated the area for them. I never told them how to do this! They thought they were so clever for “cheating” this way 🙂

Bonus features!

A few random tangents I thought of while writing this (I mean discussion tangents, not the … oh, you know what I mean.)

A couple of tech tips for you: To get the images of the of the sectors, I showed students the “Snipping Tool”, a feature of Windows 7 and 8 (and maybe earlier? I can’t remember when it came in.) I’m always surprised by how few people know about this. Everyone knows “Print Screen”, but that’s the old way to get screen images. If you only want part of the screen, Snipping Tool lets you drag a rectangle around only that part! So much easier than having to crop it after.

And my second tip: the easiest way to find it is hit the windows key, type “snip” and hit enter. And this works for any program! You don’t have to waste time searching the start menu/start screen to find that program you want if you know the name of it. Just hit the windows key, start typing the name of the program you want, then hit enter when it comes up. Works in Windows 7 and 8. Again, not enough people know this.

Ed-tech startups: listen, if you want teachers to actually use your product, follow the example of Desmos: make it easy to use, work on pretty much every device (preferably in a browser), easy to share with students, don’t require student logins and don’t make teachers create a new account to use it (use Google, Twitter, etc. to login). If I need to jump through hoops to use your product, I’m not going to bother.

I’ve already shared this on Twitter, but you can’t draw a sector without thinking of Pac-man (or maybe pizza). So, I got distracted for a while and made this. (It’s actually animated of you follow the link!)

Pac-man!