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As my students and I are starting to fall into the rhythm of SBG, I’m seeing benefits I didn’t predict. Though they found it a little uncomfortable at first, my students are getting their heads around the idea of seeing the quizzes as chances to show what they know, and aren’t afraid of questions they can’t do yet. Because they know there’ll be opportunities to demonstrate those skills in the furture.

Which has this side-effect: I can include questions for skills we haven’t actually covered yet. If students can’t do it, that’s fine – I explain it’s a skill we still need to learn, and they’re not bothered by it. But if it turns out they can figure it out, well, hooray!

I don’t think this is appropriate all the time – don’t worry, I’m not going to start putting calculus questions on my Year 9 quizzes. But in certain situations…

We’re still working through expanding and factorising, and I was hoping to have had covered Perfect Squares (and maybe even Differences of Squares) by the quiz we did last week. But alas, we hadn’t. I really wanted to include a Perfect Squares question on that quiz. Most of the class already had a really good understanding of using the Distributive Law, so I thought if I rewrote the question a bit to both teach and test Perfect Squares. This is what I ended up with:

I have to admit, I thought one or two students might get it if I was lucky.

But it turns out, 25% of the class had a go at it and they all pretty much got it.

I’ll repeat that in case it wasn’t clear: I set a question for a skill I hadn’t taught yet, and a quarter of my students were still able to do it. If you give students opportunities to surprise you, they’ll probably go ahead and do just that.

Now I realise that doesn’t mean I can skip Perfect Squares for these students. For starters, they haven’t shown they can recognise them apart from other quadratic expressions.

One other interesting point – all the students who got it were girls. I’m not sure what to make of that. I really think my female students are thriving under the learn-test-improve-repeat mindset that I’m trying to develop with the class. So, yay! But I’m wondering why some of my higher achieving boys didn’t even attempt the question. Maybe removing test scores has reduced the competitive motivation for some of them. As I said, I have no idea why, I’m just guessing. But it’s something I need to pay attention to.

If you get the chance, find a primary teacher and talk about maths.

A friend of mine, a primary teacher in another school, asked for some help around fractions and decimals. So one night last week we spent an evening doing just that. It’s been a couple of years since I taught fractions to Year 7s, so I didn’t have many resources close to hand, but with a stack of plain paper, a handful of markers, scissors and glue, we were able to do a lot. I wish I had a photo of all the notes we made.

So we did maths. And it was awesome. We covered a lot of different concepts in a short amount of time. But not only that, but every time I explained an idea, drew a picture or cut out strips of paper (there was a lot of that), I also explained why I would teach those concepts that way. Which meant I was going through the teaching-reflection cycle at an incredibly rapid rate. It’s not that I don’t reflect on my regular teaching (this blog is part of that), but the reflection is never as immediate as this.

Given that I teach in a P-12 school, it’s fairly ridiculous that it’s taken this long for me to realise this is a good idea. But I’m trying to rectify that. I spent some time in the Year 4 classroom the other day which I’m hoping to do more regularly.

One resource I think is essential to teaching fractions is “fraction strips” sheets. I used to have a better template which I can’t find, so here’s one I quickly threw together instead:

Download: fraction strips.docx

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!