Author: Shaun Carter

There was sport on today, so a large portion of my Year 7s were away. As a result, I had an exchange with some of the remaining students at lunch time that went a little like this:

Student 1: We’ve only got 6 students in our class today! [Stretching the truth a fair bit, it was more like 15.]

Me: Okay.

Student 1: So we don’t have to do maths, right?

Student 2: Or if we have to, you can only make us do easy maths.

Unfortunately for them, I had a different idea. Sometimes when a lot of kids are away, it’s necessary to not continue with normal plans so those kids don’t get left behind. However, I still think that having the students that are left with me for 50 minutes is still an opportunity for them to learn.

So today, we looked at intercepts of linear graphs. This doesn’t really arise as a topic until Year 9 under the Australian Curriculum, but given that we’ve been doing a lot of work on solving equations and plotting graphs lately, it seemed like a good idea.

I’m a fan of giving students opportunities to see some of the maths they’ll use in future years, as long as it’s presented in a way that’s not too overwhelming. It allows them to see why we do some of the work we’re doing now, and gives stronger students a chance to achieve at that higher level. And it’s nice to be able to tell parents that their Year 7 child has been working on Year 9 work. 🙂

(I also may have told the Year 10s that they should get working, because the Year 7s were catching up to them…)

This was the process I came up with to let students discover intercepts:

1. Define the x-intercept and y-intercept.
2. Use Desmos to find intercepts.
3. Notice the pattern in all the coordinates (i.e. there’s a lot of zeroes).
4. Have students find x- and y-intercepts for equations using algebra.
5. Sketch graphs for those equations. (This is their first time sketching, because it’s the first time using intercepts.)
6. Use Desmos to check their graphs.

I created this worksheet to for the lesson:

The sheet is purposely vague in places, because I wanted students to figure things out for themselves as much as possible, and where they couldn’t, I could tailor my assistance as needed. Or preferably, they could help each other.

This task seemed to differentiate well, too. Some students needed very clear instructions on how to enter the equations into Desmos and how to read off the intercepts, and had to be reminded which axis was which. But once they did, they found the task fairly easy to follow.

On the other hand, some students recognised what was going on very quickly. I even had one student say, “Won’t all of them have a coordinate with zero?” before she’d even started the task. For these students, the process of finding the intercepts with Desmos was a way to verify the results they predicted.

You can download the worksheet here:

• PDF: Linear graph intercepts.pdf
• Word document: Linear graph intercepts.docx

The other day I posted this problem that one of my students discovered. We played around with it for a while and came up with this solution. I don’t know if this is the easiest or most elegant solution, but it’s what I have.

Quick recap, we’re trying to solve the equation 6a + b = ab, where a and b are non-negative integers. In Australian rules football, if a is the number of goals (worth 6 points each) and b is the number of behinds (worth 1 point each), the solutions to this equation are the scores where the total score is equal to the product of the goals and behinds.

Solutions can be found by trial and error, but how can we be sure we’ve found all of them? How do we know the solutions don’t just continue on forever? Well, it turns out that a little algebra and an understanding of asymptotes makes it clear that only a handful of solutions exist.

The original equation can be rearranged as so:

When graphed, this produces a hyperbola with asymptotes at a = 1 and b = 6 (see below). Which means that for a > 1, b is strictly decreasing and b > 6.

Since both a and b have to be integers, this puts an upper bound on a. Once b is 7, a can get no larger; if it did, b would not be an integer. As it turns out, b = 7 when a = 7.

We already have a lower bound: a = 0, as we only accept non-negative solutions.

So, we can simply use trial and error within this domain, and find these solutions:

Or, reported as football scores, 0.0.0, 2.12.14, 3.9.27, 4.8.32 and 7.7.49.

If we allow negative scores (not possible in football, but let’s do it anyway) we can get a few more solutions.

An interesting result is that b is always a multiple of a in these solutions. We actually noticed this result before finding all the solutions. a and a – 1 are coprime, so they share no prime factors. Since 6•a = (a – 1)•b, the prime factors of b must contain all the prime factors of a. A result that, as it turns out, didn’t end up helping us solve the problem, but is relevant to the name of this blog. 😉

The real question I have now is this: how do I go about turning this problem into a lesson?

As a maths teacher, one of my aims is to get students to think about the world mathematically. So there aren’t many things more exciting than having a student come to me with a problem they noticed and are trying to solve themselves. Just for the fun of it. This is the story of one of those moments.

The other day I had a student stay back after school and told me of a problem he was going to figure out. He had noticed a pattern in the football scores he’d seen over the weekend, and wanted to know how many different ways that pattern was possible.

Now, unless you are from Australia, this going to take some explaining. In this part of the world, “football” refers to Australian rules football (which is not rugby, despite the fact that I’ve blogged about that before).

Credit: Tom Reynolds. Sourced from Wikipedia.

There are two ways to score in “Aussie Rules”:

• A “goal”, which is worth 6 points.
• A “behind”, which is worth 1 point.

For example, a team with 3 goals and 4 behinds has 22 points, which is usually reported as “3.4.22”.

My student had noticed that it is possible for the total score to be the product of the goals and behinds. For instance, 7.7.49 is a possible score, and 7 × 7 = 49.

His question was: how many scores like this are possible?

He’d already made some progress on the problem when he told me about it. He defined the problem as being the solution to 6a + b = ab, where a and b are both non-negative integers.

How awesome is that?

Now, this particular student is the type of kid who’ll go looking for problems like this, who just naturally love maths. But I’m wondering how I would go about using this in a whole class setting. How would I structure a lesson around this idea? What curriculum could it be fit into? This type of equation that only allows integer solutions was something I studied as an undergrad, but this seems simple enough for high school kids to get – one of them did pose the problem, after all.

We did manage to solve the problem. But I think I’m going to leave that for another blog post. I’ll give you a hint: 0.0.0 is also a solution. 😉

To be clear, this is the strangest post I’m ever going to write. Completely unlike any other you’ll see on this blog. This is about what’s been happening in my life personally. But also totally about teaching math. And blogging. And the MTBoS. And the fact that I dropped the ‘s’ off the word maths just now.

This kangaroo is also related to the story. Sort of.

I’m not exactly sure how to explain what I’m about to share. I’ve been thinking about this for a while, but haven’t really gotten anywhere. But I promise what I’m about to share is totally worth hanging around to read.

Last year I started dating this really amazing girl. Someone who really inspires me. She is so wonderfully talented and passionate about what she does. She is so incredibly cute. A couple of months ago, I got down on one knee and asked this girl to marry me. And with a single word, she made me the happiest guy in the world.

(That word was ‘Yes!’, if that wasn’t immediately obvious.)

Now, if I just left this post here, that would be exciting enough. It’s certainly the most exciting thing that’s ever happened to me. But there’s a twist to this story. Because if you’re the type of person to read my blog (i.e. a mathematics teacher who reads mathematics teaching blogs), there’s a good chance you’re already familiar with this person.

My fiancée is Sarah Hagan, who writes Math = Love.

Yeah, that really happened. I really started dating a girl who lives on the other side of the world to me. I really fell in love with another math teacher who goes by the name @mathequalslove. I feel like I need to apologise to everyone who wanted to catch up with Sarah at TMC15, because she spent her summer in Australia with me.

If this seems like a story that’s too ridiculous to be true, it seems like that to us, too. But here we are. I discovered the MTBoS at some point in the first half of last year, and Sarah quickly became one of my favourite bloggers. One day I left a comment on Sarah’s blog, not realising that by doing so I’d caught her attention as well. Over time, the messages between us increased, and we discovered that we have a lot in common, even aside from being teachers who blog about mathematics. Eventually, our messages led to us talking over Skype, when we finally realised we were both really into each other.

If you want to read Sarah’s take on this same story, click on through to find that. Though given her blog’s popularity, I’m guessing most people reading this have already seen it. 🙂

The other day I wrote a post about having students build their own equations. I decided to use this idea as a starting point for solving equations by backtracking.

The class has already solved equations in a number of ways, and even used flow charts to solve them with backtracking. But now I’m trying to introduce them to more formal algebraic notation to show their backtracking. As it turns out, they’ve already written their working out like this before, last lesson. Now they need to learn to write each step working backwards.

I wrote the following on the interactive whiteboard:

I made it clear that I was making this equation up as I went, and that I was following the exact same process as they did when they constructed their own equations. The only difference being that I haven’t written down all the steps for them to see.

Their job is to figure out what those steps were.

I asked them what I would’ve done first. “Chosen x as a variable, and a value.”

Of course, we don’t know what that value is yet. But we can work out what operations I did to it. So what did I do first? “Timesed the x by 2.” One of these days they’ll learn to say “multiplied” instead of “timesed”.

Then, it was subtract 6, then divide by 5. And, of course, our final result was 2.

Now, we need to work backwards through these steps. Here’s the cool bit, and the advantage of using an IWB for this lesson: I can drag the steps around and reorder them:

So in the last step, I divided by 5 and got 2. What number did I divide by 5?

And what number did I subtract 6 from to get 10? What number did I multiply by 2 to get 16?

And now we have our solution. 🙂

Now, I’m not expecting students to write these steps out forwards, just to write them out again to do them backwards. But I’m hoping that seeing this highlights for students that these steps of working that we write down are not arbitrary, but actually result from somewhere.

I can imagine using blank cards as a way to have students write down the steps, then reorder them themselves. Unfortunately, this was an idea I only had part way through the lesson.

Unsurprisingly, the expectation that students should show each step of working (and particularly my warning that I would mark any quiz questions missing working as wrong) annoyed some of them, particularly the stronger students. “Why should we show our working if we already know the answer.” Of course, I don’t particularly care that their answer is correct if they can’t convince me that their answer is correct.