Author: Shaun Carter

So Year 12 are gone and finished their exams a few weeks ago. I always feel a little sad saying goodbye to my Maths Methods students. I have so much fun with them and get to know them pretty well over the year, and this year was no exception. One thing that was different was the fact that I lost the competition we have over the year. Well, if I really wanted to I could claim the win on a few technicalities (drawing on my posters on muck-up day should’ve cost them a few points), but I’ll let it slide.

They decided that if they won, their prize would be getting to write a blog post for me, so this is it. I’m trying to resist the urge to clarify and explain a lot of their comments. Just be aware that some of the things they say might not be 100% accurate… 😉

Next year is not looking good. I’m already down 4-0 after one week of headstart classes…

Mr Carter’s assessment report

Stop

• Looking for your favourite class, you found them this year.
• Giving yourself points for no reason.
• Whinging about hockey injuries, only to find out they don’t exist.
• Giving yourself eighteen points in a lesson as a final attempt to regain dignity in the class competition, no one likes a bad sport.
• Forgetting to bring your laptop charger to class, those minutes of absence, added up might have changed our scores by one or two marks, changing our ATARs, changing the outcome of our future.

Start

• Being fair when awarding points, it’s not “discouraging class participation”, it’s playing strategically.
• Telling students when you are going to be away.
• Preparing students for exams in term one, then they might have a chance.
• Removing sections from the textbook from your study list, particularly: algebraic techniques, functions and graphs, transformations, algebra of functions, differentiation and anti-differentiation, graphs and modelling, discrete and continuous random variables, and the normal distribution. So basically everything.

Keep

• Encouraging students to do maths, but not specialist, that’s just silly.
• Teaching methods so that students don’t have to suffer through distance education.
• Diverging from the maths question you were asked to interesting topics, so that despite students not doing any textbook questions for the lesson, they continue to learn.
• Telling (rubbish) maths jokes, they’re not funny, but at least you tried.
• Telling us how methods can be used in real life, it gave us some hope that the 146 hours we spent in that room this year will be worth it.

Change

• Your lesson plan so there is cake every Wednesday, and so we don’t have a double on Friday afternoon.
• The way you spell chai in the methods probability SAC, it’s not spelt chi.
• The computer for which the license agreement for the CAS software is on, we feel that the five seconds that it takes for you to click on the ‘continue free trial’ button is a major disruption.
• Your career guidance strategies, you’ve successfully convinced us to never become a maths teacher.

Even after all our comments, you are, and always will be our favourite 3/4 maths methods teacher that we ever had, just as we were your favourite methods class. Deep down we did sort of like methods, we just wanted to take this opportunity to whinge about is this one last time, and at least it was better than English (in our reality). So thank you, Mr. Carter, for being our final maths teacher, and for making methods as enjoyable as you could.

Second last week of the year, and my Year Nines have a serious case of the I-don’t-cares. End-of-year activities next week and the fact their reports were written last week have given them the idea that they don’t really need to work this week.

Unfortunately for them, I see this as one last chance to do maths with them this year. So we’re doing maths. The good news is my love of Desmos is starting to rub off on them, so they were happier about working when I told them we were using it.

On Monday we looked at the y = a (x – h)² + k form of quadratic functions, and how to produce its graph by transforming y = x². I could have done this by telling them what each of the transformations were, then have them repeatedly sketch graphs. Instead, I wanted them to actually investigate how to produce the transformations themselves.

I gave the students the following sheet of instructions:

This was written in a bit of a rush, which explains the mistakes (well, that’s my excuse, anyway). In Set Three, I’ve included y = x² twice. When the kids called me out on that, I suggested they make up their own function that they think fits the pattern in the rest of the set. Even though this wasn’t my original plan, I’m thinking now that having students add their own examples to each set might be a good idea.

And Part 2 was just badly written. I found I had to re-explain it numerous times over the lesson. I’m okay with my students being confused because they don’t know how to do something, if it leads to them thinking for themselves to figure a problem out. But it’s not okay that they were confused about what they were being asked to do. (Note to future Shaun: your question is broken, FIX IT before you use it again!) If it’s not clear, the intention was to create something like this:

Part 3 was shown on the IWB rather than on the handout:

Using what they created in Desmos, they needed to find the equation that matched the parabola described. This worked really well – it made students think about how changing the equation transformed the graph. At no point did I tell them that (h, k) is the turning point, or that the y-intercept could be changed by dilating the graph. They had to work this out for themselves. And when they thought they found the equation (and inevitably asked me if they were right), I could tell them to type it into Desmos themselves to make sure it worked as they expected.

The last one worked well, because the students found it much harder than the others – leading to a great discussion about why this form of the equation is not so useful in that situation. Given we just covered the null factor law last week, we were able to discuss how the factorised form was more useful.

Unfortunately, this is as far as we got. My kids can feel the end of the year coming, so they don’t think they need to work. In an ideal world where I didn’t have to stop students from online shopping and playing games instead of opening Desmos, I would’ve continued a bit further with the lesson:

• I wanted students who finished earlier to create their own questions for Part 3, and add them to the list for others to do (which is why this part was on the IWB).
• One nice thing from this lesson was that students were exploring dilations, reflections and translations without me telling them what they are. But that does mean they still don’t know those terms. I wanted to finish the lesson by taking a few notes to define the terminology.

It was more than a little frustrating to not be able to finish this off properly. But I think the work we did get done worked well.

So I ended my last post with the words “I’m excited to see where we end up on Monday”. And now it’s been a month since I actually taught that lesson, and I haven’t blogged since then. Oops. Let’s see how I go remembering what we did…

Quick recap (though I really suggest you read that last post): we were trying to determine the optimum position to kick a rugby ball from to maximise the angle towards the goal the kicker has to aim for. Students worked in groups to figure out a method to calculate the angle, and tried various distances to see what angle resulted.

Diagram I drew on the board at one point, showing some of the angles we were calculating.

The method that each group eventually used was to calculate the angles in the two right-angled triangles, then subtracted the small one from the big one to get our kicking angle.

Students had calculated angles for a few distances by hand, but this was going to be a slow process. So, I suggested they use their computers instead to do the boring manual work. I didn’t specify what software they should use, though I did give them some hints.

The awesome thing about this was that it forced them to think about about the problem abstractly. To get the computer to do the work, they needed to specify how to get the solution generally. So, here we are in a trigonometry unit, and students are inventing and using variables themselves to help solve their problem.

Some groups used Excel:

and others used Desmos:

One group was using Geogebra too, but I didn’t get an image of that.

Each group basically used trial and error, but got the various software to do the calculations for them. I loved the way this showed students the power of thinking algebraically. By using variables, they could generalise the problem. Desmos makes this use of variables explicit. But even using Excel, where there are “Cell References” instead of “Variables”, the same kind of abstract thinking is needed.

As this was a Year Nine class, we didn’t go beyond using trial and error. But afterwards I did play around in Desmos for how you might approach it in a class involving calculus:

In this case, “b” is the distance from the goalpost to where the try was scored, and “x” is the distance from the try line where the ball is kicked.

My major take-away from this lesson: the use of technology should always support getting students to think deeper about problems. Get them to think abstractly. When the boring manual work is removed by the appropriate use of technology, students are freed to experiment, to change the way they approach problems without the cost of losing a large chunk of lesson time to busy work.

What I’ve been doing lately

So, there are a few reasons why I haven’t posted anything for a month. My non-classroom responsibilities have seen me busy with VCE exams, preparing students subject choices for next year, and a few days out of school at various trainings and meetings. Also, my life outside of school suddenly got very busy as well. Don’t worry, everything’s good, but time to blog seems to be less available than it once was. But I’m hoping that as the year starts to wind up, I’ll be finding more free time to clear out my backlog of blog post ideas 🙂

So I need to start this post with an admission: I got this activity from our textbooks. In my own defense, I really only took the initial idea from the book, but then made up the rest of the lesson myself.

So the idea I “borrowed” today was about rugby league and using trigonometry to calculate the angles involved. In particular, trying to work out the optimal position to kick the ball from to make a conversion.

Now, quick aside for those people who know nothing about league. (Sorry to NSW and Queensland readers who are about to see me butcher this explanation. I’m from Victoria, which is an AFL state, not league.) Aim of the game is to score a “try” by crossing the ball over the goal line. If a team scores a try, they get a chance to “convert” it by kicking the ball off a kicking tee through the goals to get a few extra points.

Now here’s the bit that’s relevant to the lesson. The kicker has to place the ball in line with where try was scored, but they can move forwards or backwards wherever they want.

So the textbook basically started by asking something like “Where should the ball be placed to maximise the chance of scoring?”, then proceeded to give a whole heap of smaller questions to step the students through the task. In the process, stripping out all the interesting thinking parts of the investigation and turning it into a set of mechanical steps.

I wanted my students to develop the questions themselves, and I certainly didn’t want them to learn about rugby league by reading the boring description in the book. (Again, my Victorian students don’t really know understand rugby league at all.) So in my preparation, I searched YouTube for “rugby league conversions” and found this video. I didn’t show the entire nine minute thing, just a few of the conversion kicks.

(I also showed part of this video of dreadful conversion attempts at the end of the lesson.)

I gave the kids two minutes to talk in their groups about any questions they had after watching the video. I then asked them to share these with the class. Initially, no-one wanted to answer, but after a little bit of prompting, the floodgates opened:

Sorry for the terrible writing and English. I was trying to get these written on the board as quickly as possible!

Some of their questions were quickly answered by refering to this page showing the dimensions of the field and goals. After a bit more discussion, we set about trying to work out the best position to kick from, to maximise the goal angle to kick through.

I assigned each group a different distance from the goal post to investigate. This divided the labour really well, as within groups different students could investigate different kicking distances to collect a wider range of results.

This worked really well – around the room there were kids discussing angles and distances, drawing diagrams and setting out tables to summarise their results. Groups were discussing what they needed to do to figure out the problem. And in the midst of it all, students were using trigonometry to work out their angles.

The best thing? Beyond the initial prompt, most of this happened without my input. I did have to give help around the room at various times, but students were willing to figure it out together in their groups. They also started using trig without me telling them to!

Though in fairness, it was reasonably obvious given we’re in the middle of our trig unit…

Diagram I drew on the board at one point, trying to explain… something. I’m sure it made sense at the time.

We didn’t quite get to finishing everything off, but that’s mostly because students kept posing new questions and hypotheses to investigate. A few students thought making the distance from the goal line the same as the distance of the try from the post was the best approach, and another said that 45° to the near post was best. It was great to see them slowly realise these were exactly the same thing! It was also great to see them realise their hypothesis wasn’t completely correct, and start searching for more data to figure it out.

I’m excited to see where we end up on Monday 🙂

Update: Follow up to this lesson can be found here: https://www.primefactorisation.com/blog/2014/11/26/kicking-goals-using-technology/

First day of term 4 (which was Monday – I really am running behind at the moment!) saw the start of our Trigonometry unit with Year 9. I wanted a way to get my students to start thinking about how the angles of a right-angled triangle affect its sides, while also defining the Opposite and Adjacent sides. They already all remembered the Hypotenuse from doing Pythagoras’ Theorem 🙂

So I took the class out to the front lawn and had groups of students form right-angled triangles by standing at the corners. (Actually, I didn’t say “stand at the corners” the first time, which was a good reminder that I sometimes need to be clearer with my instructions. One group tried to form the sides of the triangle by lying on the ground.) In each group, one student was given a pink sticky note to indicate they were the right angle, and another had a green sticky note with θ written on it.

Then using different colours of party streamers, the students made the sides of the triangles, defining the Opposite and Adjacent sides as we went.

As they did this, I had the groups check the other groups around them to make sure they all had right-angled triangles. This provoked great conversations amongst the groups, as they evaluated each other’s work and had to communicate clearly their reasons why a triangle was or was not right-angled.

Then I gave this challenge: make the angle at θ bigger.

As they did this, the students needed to work and talk with each other to figure out what they were going to do. Students communicating about maths to figure out a problem together! It also worked well that different groups used different solutions – some shortened the adjacent length, others lengthened the opposite – allowing us to discuss those different solutions.

Once back inside, we then worked on defining the trigonometric ratios, eventually creating this notebook page:

There was more to the lesson than that, but I’m getting sleepy now 🙂 I might elaborate on what other work we did next time.

One more thing: as we were outside, a friend of mine happened to be driving past the school. As I was talking to him that night, he asked me two questions:

• Why did I never get to go outside to do maths at school?
• Why were you making kids stand in rectangles?

I’m a little concerned about how convincing our triangles were now…