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In introducing rotations to my Year 7 class, I had them create a… thing. “Foldable” isn’t really the word I’m looking for here. I think it’s better described as a “spinable”.

Anyway, it looks like this…

…and this.

The main idea is that students aren’t just told what a rotation is, and they aren’t just shown, but they actually create the rotation themselves.

To do this, each student will need:

• An A5-ish sheet of paper (or half a US letter sheet will do).
• An piece of tracing paper half the size.
• One of these pin things. I always called them “split pins” growing up, but I think they’re actually called paper fasteners.

Get students to fold their paper in half, and draw any picture they like (school appropriate, of course) in one half. I only gave them 30 seconds to draw a picture, because I didn’t want them spending the whole lesson on it, and I think a simpler picture is better than this. Of course, many of them took longer than that just to find something to draw with…

Next, unfold the paper and cover the picture with the tracing paper. Attach the two sheets together with the pin. I emphasised the point to students that they could put the pin wherever they wanted, not just the center. I wanted there to be a variety of pin location, so we could see how that affected the image that was produced.

Trace over the picture. I found pencil, rather than pen or marker, works best for this. The colouring is optional.

Finally, glue the back of the paper and fold it in half again. This just stops there being a possibly rogue pin sticking out the back of the sheet.

And we’re done! I also had students write the words “Rotation around a point” on theirs and glue them into their workbooks.

What I like about this is that it emphasises what it means to rotate around an origin. When students were working on questions from their textbooks, it made explaining the origin a whole lot easier: that point doesn’t move, because that is where the pin is. Everything else moves around that.

I also like the way that each student was able to put their own “spin” (get it? Sorry…) on this task. Students could see a whole lot of examples by seeing what other kids did.

Rotations are an interesting topic to try and teach. One battle we have as teachers is finding ways to explain concepts that seem obvious in our own heads, even though they’re not. It turns out that adult-with-mathematics-degree-obvious is not the same as twelve-year-old-obvious. Describing points moving around another point is one of those ideas students can find hard to understand. I think it’s much better to let them create it for themselves.

The gradient of ax + by = c is -a/b.

This is what Year 10 wish I had told them lessons ago.

They’ve been looking at parallel and perpendicular lines lately, which involves finding lots and lots of gradients. They like it when the equation of a line is in gradient-intercept form. The gradient is m from y = mx + b, and everyone is happy.

As an aside, why is m used for gradient/slope? Does anyone know? I’ve had kids ask me that so many times, and my honest answer has been “I haven’t got a clue.”

I was annoyed at the lack of images in this post. So here’s some parallel and perpendicular lines, just because.

Anyway, they aren’t so happy with standard form. They know how to rearrange between the forms, so they’ve been changing them into gradient-intercept form to find the gradient. But they’ve been getting really fed up with it. “This takes too long, Mr Carter! There has to be a quicker way.”

As I mentioned at the start of this post, there is a quicker way. But if I’d just given them that shortcut from the start, they would’ve been able to find gradients a whole lot faster. But they wouldn’t have understood why.

So I hinted that there is a quicker method. I wrote an equation on the board (I don’t remember what, I didn’t save my notes for some silly reason), and asked them to tell what to do to find the gradient. Which they did.

“So what’s the gradient of this?” I asked as I wrote ax + by = c on the board. There was a moment of quiet, before someone called out, “minus a over b!” Someone else added their agreement. Someone else asked “How did you get that?” Someone else disagreed, saying it can’t that simple. “Where did the minus come from?” asked another. And though they wanted me to confirm the answer, I just waited quietly for the class to come to their own agreement about what they’d just discovered.

They weren’t all that impressed with me when they realised I’d known this rule all along…

I feel like I should point out that this is not how the majority of my classes go. This is just an example of one time when everything went really right, and the class responded the way I wanted. My main motivation for writing this post is to remind myself that this type of moment, where I bait the class into their own discussions which lead to their own discoveries, is what I want to happen in my classroom. All the time.

Something I’m realising more and more is that mathematical knowledge has much more value to students when they earn it themselves. Would it have been more efficient to give the class the rule straight away. Well, yes, if by efficient you really mean “students were able to get more questions done”. But I don’t think that’s what efficient learning in class is. Rather than tell my students one specific rule that than can make use of in one specific scenario (and probably completely forget in a week), I want to teach my students to think mathematically and discover mathematics for themselves.

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. 😉