**Take 1: **It all started with a challenge problem that I set them for the topic of Area. Their aim was to calculate the shaded Area of the shape to the right (1/3 of the area outside an equilateral triangle inside a circle, given a side length of a triangle). It took them a while and a few mistakes but eventually they made it to the solution.

**Take 2:** The next lesson I decided to push them further, and asked them to calculate the following shaded area (1/3 of the Area inside an equilateral triangle, but outside the incircle). I hadn’t really thought to much about the solution, but when I went ahead and solved it, I realised that I needed to apply the Sine rule (which they won’t learn for another two years). Much to my surprise, a student came up and showed me the same answer. When I asked him how he did it, he said that he ‘intuitively guessed’ that the height of the incircle would be 2/3 the perpendicular height of the triangle (which it turns out to be!).

**Take 3:** At the start of the next lesson, these students came up to me and said they wanted to combine both questions, and try and get other students to solve it (see left). I was more than happy for them to do it but only if they could produce a full solution for others to see. That was when it all started to get very interesting. One student came up to me and said “it’s really weird; the Area in those sections is almost the same as the Area of the incircle”. I asked him “is it almost the same, or is it exactly the same? Maybe you’ve got a rounding error”. He went back and discovered that in fact **it was exactly the same.** All of a sudden they were getting very excited as they couldn’t find anything about it anywhere on the internet (though there are sites that had info about the ratio of incircles:circumcircles). We chatted about what was going on, and decided it was time to prove it….

**Take 4: **The side length was removed and replaced with pronumeral *l. *Then we went through and produced an expression for both the Area of the incircle, and the shaded Area ((pi**l*^2)/12). Boom! They were exactly the same and the kids were literally jumping in the air at this point.

**Take 5:** I then told them they could prove it another way considering that the radius of the incircle is 1/2 the radius of the circumcircle (something that they came across in their problem solving). Then considering the ratio of Area’s would be 1:4, the annulus would therefore be three times the size of the incircle. Considering symmetry, the shaded Area is 1/3 of the annulus, and therefore equal to the incircle!

**Summary:** These students investigated a problem that they designed (even though it was kind of a conglomeration of my questions), and discovered something that they weren’t expecting or even looking for that was much more interesting. I was able to **use algebra to enhance their solution**, and help them produce their own formula that would work for any given length (thus they learned the beauty of algebra and why it’s so useful!). At the end of the lesson they made me take a photo of them in front of their work and email it to them so they could show their parents and friends. Did anyone say new facebook dp???

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**1)** These guys are passionate about PBL. It’s a nice website and easy to navigate with plenty of stuff to look at (I only saw a small fraction in my 30 minutes).

**2)** There are plenty of great resources out there. BIE has a load of resources which are sorted according to subject/content, making it easy to find something relevant. One of my biggest issues with PBL is the time needed to generate engaging projects; by finding ready-made projects (that can be adapted if necessary) it saves me a lot of time that I don’t really have!

**3)** PBL needs structure. I really liked their article “8 essentials for Project-Based Learning” where they stress the importance of engagement through appropriate content and the right driving question, critical thinking through inquiry and investigation, as well as feedback through the use of rubrics. From my experience with PBL if one of these areas is weak (or non-existent!) the whole experience can fall apart.

PBL can be a really positive learning experience for students and something I would like to more of. BIE is definitely worth the visit if you’re interested in finding out more about PBL or looking for projects (note: they are American and so the content may need to be adjusted for your syllabus).

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**1. Students don’t have to hate maths:** Nearly all of these said they hated maths at the start of the year, for a variety of reasons (see my previous post). I am pleased to say that in recent meetings they have expressed that they have been enjoying maths this year. For all the time and effort I’ve been putting in, it is nice to know that it’s working, even though they sometimes don’t show it….

**2. Just because something worked one lesson, doesn’t mean it will work the next: **This has been my biggest challenge. I have tried different ideas and received comments such as “this is awesome, can we run the lesson like this every day?”, and when I come in the next day and do the same thing I hear “this is stupid. I don’t like it this way”. I’ve had to vary my styles, approaches and methods day to day (I’ll recycle methods at later times).

**3. Everyone in my class can speak: **and they do. If what I’m talking about is less interesting than Masterchef last night, they are going to start chatting. It’s a talent the amount that these guys can talk! So I’ve been trying to use it positively. Heaps of discussion with their friends and as a class. Homework now involves a conversation with a parent or carer about what they have learned today. There’s pretty much no chance of having them sit down and answer some questions, but a chat over dinner works for them. Not only that but it gives them some confidence/excitement to tell their parents something that maybe they didn’t know. Student recall has been much higher since we have started this.

**4. Sometimes you lose: **There are some lessons that just haven’t worked, but it hasn’t stopped me trying new things. Nearly every time I’ve learned something about myself, my students, and my teaching. And on the other occasions, it’s been about dusting myself off, and trying again next time. The beauty is that I get plenty of chances!

**5. Other people have awesome ideas: **If you read blogs then you already know how many great ideas people have out there. I recently got the opportunity to meet Dan Meyer who has long been one of my educational heroes, and it was excellent to chat to him about his methods. I tried out his Pyramid of Pennies lesson with my year 9’s and they loved it. Of course they struggled with some parts, but I allowed them to ask the questions that they wanted answers to. Total buy in right there. Check out other people’s ideas and share them with your colleagues/friends. Most of all share your own!

**6. Challenge them to something they can’t do: **It might sound harsh, but I offered them a $5 prize that I knew they couldn’t win. After an investigation where students discovered that the angle sum of multiple triangles was 180 degrees, I said that the angle sum of every triangle is 180 degrees. “I reckon I could make one that didn’t” said a keen student, to which I pulled $5 out of my wallet and offered it to the first student that could create me a triangle whose 3 angles **didn’t** add up to 180 degrees. The whole class instantly jumped in constructing triangles and measuring angles. “I am so close sir” was one response…. Yes you did. Of all the things I’ve taught them this year, I think this is now the one they are least likely to forget.

So that’s my 6 so far. I’ll keep learning as the year progresses and hopefully so will they! If you’ve had any valuable lessons from a similar situation, I’d love to hear them.

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**1. Prove to them that they can do it.** I speak of our class as a family. We don’t leave a brother or sister behind when it comes to a concept. If someone doesn’t understand something, everyone works with that student until they’ve got it. We succeed as a class, not as individuals.

**2. Bridges Bucks (BB):** I want students to be able to keep account of their achievements as well as motivate them to be involved. For students who complete their homework or something I deem worthy (e.g. having a go at explaining a concept to the class), they earn BB’s. At the end of the term I’ll bring in a bunch of treats that can be exchanged for their BB’s.

**3. No textbooks: **Textbooks have their strengths and weaknesses. Unfortunately they can often turn maths into simply a bunch of numbers, symbols and formula’s. Maths is all about solving meaningful problems, and rarely is anything in a textbook tangible and meaningful (especially for these kids!). It’s taking a lot more effort, but I am working solely off the syllabus, and creating my own stimulus for each concept involving anything from watching partial episodes of “The Biggest Loser” (subtraction of decimals, percentages), to shooting baskets and discovering ways to compare the results (statistics).

The year has started well, and 5 weeks in I have some students seeing me in the playground and being excited to come to maths. There are a few that still think maths is boring (they seem to think what I’m doing isn’t maths?!?!?!), but the good news is I have the rest of the year to get that through to them! My name is Ben, and I love maths!

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**Here’s how it happened:**

1. I showed them a powerpoint that I had previously prepared with 2 Pythagorean examples. My powerpoint is here

2. Then I set them the task of creating their own Pythagorean example using Powerpoint. The problem had to be useful*, and make people laugh.

The winning team would receive a free beverage at the canteen (I also gave prizes to a few others).

**The results:**

Students were focused and excited about their work. 96% of students uploaded a finished powerpoint presentation. Some had errors in their solutions, but all used Pythagoras in an appropriate way. Take a look at some of their work (this is all unedited so expect to see some mistakes). Pretty impressive for a group of students who have struggled in the past.

squeaky the squirrel powerpoint How to choose your shoes powerpoint

Who gets to drink first powerpoint

I have created an online survey for students to fill out next lesson to see how much they enjoyed the experience, and how much their understanding improved. Will update this post once I get the results.

*Useful to a certain situation (i.e. very loose)

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Dan Meyer has a cool basketball ‘strobe’ lesson here that you should check out.

So have a go at some batting or catching practice during your next maths lesson. The cricket coach will love you for it, as will the kids, and most of all they can apply some of that geometry stuff they thought was all theoretical. Who knows, we may even win back the Ashes!

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Why do we set homework? And more importantly, is it having a positive effect on students understanding and enjoyment of maths? These are some of the questions that I have been grappling with over the past few days. My disgruntled year 9 student decided to produce me two graphs to compare the number of homework tasks, and length of task of four of his subjects. The picture tells the story…..

NBCS currently has a 30 mins/5 days maths homework policy for students in years 7-10. For some students this is just what they need, and that’s great. But I think that for far too many of them this is just a total waste of time. Dan Meyer (my favourite Blog) has some great points here.

So after reading plenty of different opinions I thought i’d ask my year 8 class their opinions. The responses were as I expected; some loved being told exactly which questions in the exercise they were to complete each night. Others hated the 30 mins a night as they wanted to spend 60 mins one night to fully understand a concept, and then have a night off. One or two said that they usually understood the concept after one or two questions. One girl said she likes to write up in words how to solve problems as it consolidates her understanding. Another boy said he loved using mathletics because he was able to get check his answers immediately (plus get bonuses for his character to help with his street cred).

So I made a deal. You get to choose your homework. Today I gave them one task for everyone (that the vast majority said they found really useful) and then the choice of 3 options. 1. A set of questions from the text. 2. A set of questions on mathletics with a score > 85% 3. Two questions. One tough, one tougher (took Dan’s idea).

In a couple of weeks I will be setting them a topic test with the first question asking them which homework method they have been using, and the number of hours they have spent on homework each week. I’m interested to see what happens…..

**Update**

So what were the results of this first experiment? 7 chose to complete the textbook exercises, 11 chose the online mathletics, 7 attempted the tough/tougher questions, and 3 chose to do no homework. I was somewhat surprised that only 3 chose to not do anything; either they were fearful that there would be consequences for doing none, or they are more conscientious than I thought!

But here’s the thing that surprised me most.

One student managed to get the correct solution for the ‘tougher’ problem. He said that it took him about half an hr, and multiple pages of working before he was happy with it, but he did in fact get the correct answer. Just take a look at the question again, and remind yourself that the lesson **introduced** equations with pronumerals on both sides. This student isn’t right up the top of the class in results either, but he was totally motivated to stick at it, and apply different things he does know to get the solution.

To be honest I didn’t expect any students to come up with the correct solution to this problem now; my plan was to revisit it later in the year and see if anyone could get a solution then. The other students in the class were blown away that one of their peers could get the solution. Let’s see how many have a real go at the next one.

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