Author: Shaun Carter

So, I think a lot of people will be familiar with this puzzle. Start with an arrangement like this:

The aim is to swap the positions of the green and blue counters. The counters can move either one or two spaces forward, and each space can contain only one counter.

I was given the idea to use this in class at a PD a couple of years ago. The puzzle still works with any number of counters on each side, it just takes more steps to complete it. In fact, the number of steps forms a very nice algebraic pattern.

We did this activity on Monday. I added a silly story about two families of kangaroos trying to get past each other on a small path. (Now that I think about it, frogs would’ve made more sense with the colours we used. It probably wasn’t necessary at all anyway.) Initially, the class struggled to work out how to solve the puzzle. A few students excitedly called me over because they’d “worked out how to do it!” But when I asked them to show me, they’d forgotten. But that’s OK, this helps reinforce the idea that getting the answer is not the same as understanding the problem.

After a little while, I suggested they try the puzzle with only one counter in each group. “That’s easy!” they all said. OK then, try it with two.

(Sorry for the ugly black blob, but that student had written their name on their hand in black marker, for some reason.)

Slowly, kids around the room started to figure it out. Once they knew how to solve the puzzle, I had them count how many steps it takes to solve. They then added more counters and counted those steps.

“How many steps will it take to solve with 56 in each group?” I asked. They thought I was crazy (maybe past tense isn’t right for this sentence…). Their first thought was that I wanted them to physically solve the puzzle with that many. “Well, maybe there’s a pattern we can find, instead.”

So they drew up tables comparing n (the number in each group) to the number of steps. Two different patterns were spotted: multiply n by the number two bigger than it, or multiply n by itself and add double n. Of course, when they wrote the patterns as algebraic expressions, the following result emerged:

n (n + 2) = n² + 2n

To check our results worked, we completed this table on the whiteboard:

(And now I’ve revealed to the world just how horrible my handwriting is…)

We never did it, but the next step would be explaining why this pattern works. The nice thing about this is that while n (n + 2) is the easier pattern to see, n² + 2n is easier to explain (each blue counter passes each green counter once, plus the extra small step each counter makes).

What I like about this lesson: Expanding and factorising are fairly abstract concepts for high school students. This activity makes them more concrete, without trying to make up a “real world” application that doesn’t really exist. I’m enjoying that so far this term, I haven’t been asked the dreaded question, “When are we going to use this?”

Some of the class still wanted to do the puzzle with 56. I should’ve seen that coming – this same class in Year 7 wanted to confirm that 1 m³ = 1 000 000 cm³ by collecting every 1 cm³ block from the primary classrooms and stacking them into a giant cube. They still want me to ask if we can drain the town pool to measure the volume of it. I love this class – always curious and creative but no sense of practicality.

Update (April 2016): I’ve coded an online version of this puzzle. See my blog post about it at https://www.primefactorisation.com/blog/2016/04/14/interactive-jumping-puzzle/

Today was a good day.

Anyone who takes the time to read a blog like this probably doesn’t need me to tell them that teaching can be a frustrating job. It often feels as though you’re fighting against pressures on all sides, and every time a student doesn’t progress as far as you’d hope, you question your own ability and wonder if the effort is really worth.

But for me, today was not like that. Despite Monday being the busiest day of my week, I had many things go right. Not earth-shattering moments, just little moments of success for me and my students. I think it’s important to note these small victories and celebrate them. If we really think we’re in this profession to make a difference, it’s a good idea to have evidence of that difference for the times you can’t see it.

So these were my small victories today:

• I’m continuing to experiment with SBG. Today I handed back a quiz from Friday. One student had not achieved a 4 (max score) yet and didn’t expect to, but he did this time He was so excited – and so was I!
• One student who had been struggling with expanding, and was disappointed with her quiz results, asked for extra help to understand it. She re-took the quiz, which she checked herself using my answers, and was able to identify why she made the few mistakes she did. She then asked for extra questions she could practice for homework. She would never had done that in the past. The way SBG focusses on improvement over scores is already making a difference.
• “I think this topic is my favourite thing we’ve done this year,” said a student while doing algebra.
• “It’s Monday New Things – better get my coloured textas out!” and “Mr. Carter, you and [the English teacher] have the best markers!” (Many of my lecturers insisted maths looks best on chalkboard. I disagree. Maths looks best in Sharpie.)
• The problem solving task I gave led students to move around the room to discuss how they were doing it without me having to tell them to. I think they’re starting to get how to do this type of lesson. I’ll write a post about the activity some other time (edit: here. Here’s a sneak peek:

• My class can still expand binomial products. They still don’t know what FOIL is.
• Today I introduced factorising. I had students make up their own expansion problems, for which they worked out the answers. I had some students give me answers, which I wrote on the whiteboard. Then the rest of the class worked out what the question possibly was. When faced with 10x + 2x², there were a few different, yet correct, questions given. But after a discussion (with next to no involvement from me) they all decided 2x(5+x) was the best because “it makes the inside of the brackets the simplest”. I hadn’t even told them what factorising was at that point!
• One of my classes was away playing basketball, so I took PE extras with Year 2/3 and Prep. And I survived. 🙂

I think it’s a tough balancing act to find the point where students can investigate mathematical ideas independently. Not enough guidance, and the class will stare at me, wondering what they’re supposed to even be doing. Too much help, and they just start guessing at the answer that will get me to move on from this prompt. The sweet spot is when they learn a new idea that they’ve discovered themselves, which is “coincidently” the content I was trying to teach all along.

Today I wanted to give my year nine students the chance to “be mathematicians”: to pose a conjecture, then either prove or disprove it. We’ve been working on expanding, and we’re at the point of discussing expanding binomial products and preparing to introduce factorising. I thought letting students discover that

(x + a)(x + b) = x² + (a+b)•x + ab

was a good opportunity to prove a conjecture, as well as practice expanding.

I produced the following template. You’ll notice that there’s nothing about expanding in particular in it, so hopefully I’ll be able to reuse this in other lessons.

Download: conjecture proof theorem template.docx

I broke the students into groups of four. They each had to create their own example of a binomial product, then they had to expand the examples of each person in their group. As they did this, I wrote some of their answers on the whiteboard.

From there, they had to see if they could notice any pattern, which became their conjecture. Some needed a little prompting, but someone in most groups noticed, leading to discussions within the groups, about whether it really did fit the examples. Hooray!

Next, they had to see if they could find any counter-examples. Of course, the rule that we were leading to is true, but they didn’t know that yet. Also there was still a bit of refining to do to the conjectures, as most hadn’t taken non-monic cases into account (though not in those words). Luckily, one student in the class had used (2x + 4)(x + 3) as their original example, and as it was on the whiteboard many students had noticed it. To some students, I suggested they explore this further: could they find any other examples that didn’t work?

Next step was the proof. Most needed a little help at the start – the idea of substituting the numbers with pronumerals was a step of abstraction they weren’t quite ready for yet. But they did pretty well from that point on.

A couple of students finished early, so I managed to snap pictures of their work. They were in the same group, so their examples are the same:

(Not sure why the second one is backwards.)

By the end of the lesson, I was even able to get some students started on the same process, but this time with perfect squares.

Most of the class took to this activity really well. Some did not – I think they struggle a little with staying focussed when tasks are slightly more open-ended (not that this task was particularly open-ended, but the kids didn’t know that). Is this the sort of thing they’ll get better at with practice, or do I need to make the activity more structured for them? As I said at the start of this post, that balance can be difficult.

But it was still fantastic to get the class thinking through the properties of expansions themselves. And not a “FOIL” in sight!

While shopping the other day, I saw some of those decorative garden windmill things for sale at 50 cents each. Normally they would be the type of thing I would just walk past and ignore (not really one for gardening), but this time I had an light-bulb moment. So I bought a handful and came up with this creation:

Before and after: the original “Windmill” and my Unit Circle Spinner.

Close up view.

If you look at it from side on it gives a visual demonstration of sin (or cos from above), represented by the amount of yellow paper you can see and the direction it’s pointing in. For example, this is roughly showing that sin(π/6)=0.5:

(Trying to balance the spinner in the right place while simultaneously taking a photo was easier said than done.)

I’m not completely sure destroying a garden decoration was necessary to make this, but sometimes it’s good to take inspiration from wherever you find it.

I hope this will help my students understand the definitions of sin and cos (and tan) better than they have in the past. A lot of my previous students have seemed pretty happy to rely on mnemonics like “CAST” to remember where the different functions are positive or negative. Sometimes I wish I could tear that page out of their textbooks.

Yesterday I wrotes about the reasons I’m experimenting with Standards Based Grading this term. Today I want to show how I’m planning to implement SBG in my classroom.

Because I was away in the last week of last term, we weren’t able to finish our unit on Pythagoras’ Theorem before the holidays. Because I was already planning to introduce SBG this term, I had already produced a list of skills and given them to the students before I left on the Year 7 & 8 camp.

Whenever I plan a unit (even before SBG), I like to copy the relevant standards and content descriptions from AusVELS, so I can clearly understand what I’m working towards:

I normally like to include content from levels above and below our year level, so I can cater for that range of student experience and ability. However, Pythagoras only starts at Level 9 in the Australian Curriculum, so I don’t have any previous levels this time.

From there I decided on my SBG skill list:

This week we did our first quiz, which I’m planning to do regularly on Thursdays. I’m enjoying being able to write questions that directly test each of the skills. I also like being able to set less questions while being able to demand higher quality responses from my students. Here’s one of the questions on that quiz, including a box for marking against the relevant skill:

I have finished “content teaching” for Pythagoras’ Theorem, so we are just starting a new unit on Expanding and Factorising. I like the scope that SBG gives to retesting skills, so I plan to include skills from both units in this week’s quiz. For the new unit I also have my AusVELS standards:

And my skill list:

So there we are, my first attempts at SBG. I’m sure this will need a lot of refinement over time. If anyone with more experience with SBG wants to give me any pointers or ideas, I would really appreciate it!