Dreaming about Maths

I've only been back at work three weeks and I've already started to dream about maths again. And not in a good way. I mark a set of homeworks and dream about the answers - they swim around my head all night long, tormenting me. Perhaps I shouldn't mark in the evenings.

I also lie awake at night thinking about lessons that didn't go well. I know, I know - don't dwell on them - after all, I learn from my mistakes and they make me a better teacher.

Today I want to focus on the good lessons. This post is about three things that went well this week. As well as being therapeutic for me, hopefully there'll also be some ideas here to inspire you.

1. Angles
At the end of a sequence of lessons on angles with Year 7 I gave out this vocabulary check. I told my students to try to fill it in without referring to their notes. I was pleased that some high quality work was submitted:

It turned out to be a really worthwhile exercise for revealing misconceptions. The most common misconception was that co-interior angles are equal. There were also a lot of sketches missing the correct parallel line notation:

Just forgetting the arrows or unaware the lines must be parallel?

A few more surprising misconceptions came up too:

Worrying misunderstanding of the definition of parallel

Confusion over vertically opposite angles

Normally I'd avoid an activity that doesn't actually involve solving maths problems, but I think I'll use this one again because it was so revealing.

I took a first look at my Year 7's books this weekend. Some were a big mess so I need to do more work on improving their presentation. I think it's really important to establish good working practices in Year 7. The picture below shows one of the best books in the class. This is why I want to use iVisualiser in my classroom - I want to show this student's book to the class as an example of how classwork should look (margins, sketches in pencil, clear workings, underlined answers etc).

2. Problems
At the end of a sequence of lessons I often make an A3 sheet of problems relating to that topic. The questions come from a variety of sources including Median Don Steward and Brilliant.org, and sometimes I include one or two past GCSE questions. My students work in pairs answering the questions in any order - sometimes this is for a whole lesson, sometimes half a lesson. The example below is one I used in a Year 10 quadratics lesson, after they'd completed a card sorting activity.

The radius problem (bottom left) originated from Chris Smith's newsletter and I've mentioned it in a couple of blog posts before. I told my Year 10s I'd be very impressed if anyone managed to solve it. The next lesson, two girls brought me the correct solution and talked me through their method. I love it when students do maths 'voluntarily' in their own time, and I love it when they surprise me by figuring out the solution to a challenging problem.

Solving the radius problem

I gave the same problem to my Year 9s. A few girls came up the solution 36 - although it was incorrect, their thinking was interesting. They correctly argued that the circle must be in a square - they thought that the top must be a multiple of 9 and the side must be a multiple of 8, so the sides of the square must be a multiple of both 9 and 8 (ie 72). I like their reasoning and they explained it well, but there's no reason why the sides of the square have to be a multiple of 8 and 9. Actually the radius is 29, so the square has sides of length 58.

I've been doing a lot of problem-based lessons this year (using up my school's stocks of A3 paper!) and I've been very impressed by my students' efforts so far. They seem to enjoy these lessons, so I'll keep doing them every now and then.

Lovely problem for a surds problem solving lesson

3. #LikeAGirl
This isn't maths, but it does have a link to maths so bear with me.

I've taken on a Year 11 tutor group this term. They're lovely girls, if a little loud at times. I have three 20 minute afternoon registration slots to fill each week. This week I showed them the brilliant #thisgirlcan video, but they didn't seem particularly inspired. I think it has more of an impact on women my age than teenage girls. The Head of PE then recommended the #likeagirl video and this one got a better reaction. If you haven't seen it, do have a look.


It struck me that this video has parallels with gender issues in maths and science. I work at a girls' school where these gender issues are non-existent. We have over 200 sixth form students taking maths - it's the most popular subject in the school - and there is absolutely no perception of it being a 'male' subject. I admit that there are some disadvantages of single-sex education, but this freedom from the influence of gender stereotypes is one of the clearest advantages I've seen. If I was to tell my students that maths and science are male-dominated, they'd be surprised and perplexed. But in many schools (and in society in general), doing maths 'like a girl' is considered an insult, like throwing like a girl or running like a girl.

The video really got me thinking about the maths gender debate so I'm going to put 'discovering this video' down as my third success of the week.

The rest
Please don't think that every lesson I teach goes well... far from it! Many lessons aren't worth commenting on, and some could actually be classified as disastrous... I tried Plickers for the first time with Year 7 on Friday. At the last minute my iPad app randomly froze so I had to use my iPhone instead. Scanning the room took far too long. In fact, it was painfully slow. We gave up after four questions, which was a shame because I'd spent a while setting it all up. I still really like the idea though, and my students seemed to like it too. So if I can get it working again on my iPad, I'll definitely try again another day.

Speaking of trying new technology, I've set up a multiple choice test for my Year 10s on Monday which I intend to mark using Quick Key. I'll write about how it goes next week.

Polygraph was the absolute highlight of the week for me but I haven't featured it here because I wrote a separate post about it (as soon as I got home from school on Wednesday, because I was so excited!).

That's it from me. I hope there's been some helpful ideas in this post. I think all teachers would find it really worthwhile to reflect on a few things that went well each week. If you want to sleep soundly, don't dwell on the bad stuff, but pat yourself on the back for the good stuff. And don't mark after 9pm!

Polygraph Rocks

Just a short post from me today to sing the praises of Polygraph from Desmos, which I first featured in Gems 19.

My Year 10s are very smart girls and we've been working on quadratics for a couple of weeks - it's been a bit 'death by algebra' to be honest. We've practised factorising, solving, using the formula, sketching and completing the square. Today I got them in an IT room and we had a go at Polygraph: Parabolas. It was superb.

One of the huge advantages of this lesson is that it requires absolutely zero preparation. Once you've registered with Desmos, just log in and start the activity. Your students go to student.desmos.com and enter a code and their name, then off they go. You don't really need to give any instructions because Desmos does it all for you.

The majority of the lesson is like a game of 'Guess Who' but for parabolas. Each student chooses a parabola from a selection and Desmos randomly pairs them up with another student. Their partner then has to type yes/no questions to figure out which parabola they've chosen.

Behind the scenes the teacher is able to monitor all the conversations. The interface is fantastic. 

Here's some examples of the questions asked by my students today:

These examples were from quite early in the lesson. Throughout the lesson I occasionally picked out really good questions and shared them with the whole class. At one point I wrote six words on the board and encouraged students to start using those words in their questions:

• Roots
• Quadrant
• Intercept
• Vertex
• Origin
• Symmetry

I was really impressed by how quickly their mathematical language developed. They started using the new words (roots, vertex and quadrant) fluently. It was a pleasure to watch. Even when they got a bit silly, they were using sophisticated terminology:

The lesson includes other tasks which reveal misconceptions, like the question below.

Overall it was a fantastic lesson. I really saw my students' mathematical vocabulary develop. I also saw progress in their understanding of quadratic graphs. The lesson was easy to plan and utterly engaging. 

At the end I let them play around with Polygraph: Hexagons for 10 minutes. Wow, they really don't have a clue how to describe polygons! We'll tackle that another day though.

Well done Desmos, Polygraph rocks. 

5 Maths Gems #22

Hello and welcome to my 22^nd gems post - this is where I share five teaching ideas I've seen on Twitter.

I've discovered at school this week that change is hard. It's easy to say 'I'm not going to teach it like that anymore' or 'I have a fantastic new idea for teaching this topic' but when it comes to actually delivering lessons it's easy to stick with tried and tested methods and resources. We want our students to take risks but how often do we take risks as teachers? This has been on my mind a lot this week - I'll write a post about this soon.

1. 2015

Every year Chris Smith features the Math Forum Year Game in his newsletter. The idea is to use the digits in the year 2015 and the operations +, -, x, ÷, ^ (raised to a power), sqrt (square root), and ! (factorial), to write expressions for the integers 1 to 100. Answers can be submitted through the Math Forum website, or perhaps you could make a classroom/corridor display like these examples:

2015 Challenge classroom display by @Jeremy_Denton

Challenge 2015 display by @c0mplexnumber

Clarissa Grandi has very helpfully loaded her display resources onto TES. Paul Collins' post about the 2014 challenge is also worth a read if you're going to try this.

2. The Constant Character
Teaching indefinite integration? I loved this tweet from @inFinnityPi.

This lovely idea originated from @edenspresence who shared the student-decorated constant characters below. When students forget the 'plus C' they have to decorate a character - a good way to make it stick!

3. 'I helped'
A nice idea from @GiftedBA - 'I helped' stickers to encourage students to offer each other meaningful help and to recognise those who do so. Download a sticker template here.

4. What's the question?
A really nice idea from Marcus Fleet is to write an answer on the board and ask students for possible questions.

It's nice that the class have chosen a winning question!

I'm quite surprised to see the kind of algebra students are doing in Year 6 - it's more advanced than I realised.

In the conversation that followed this tweet, Miss_Ren said that she likes to write a number on the board and ask what the question could have been. It links to all areas of maths and requires no printed resources. Paul Godding shared some similar ideas - such as writing 5, 9 and 10 on the board and asking for the odd one out (students have to come up with two reasons for each answer). There's load more fantastic problems like this on Paul's website 7puzzleblog.com.

5. What the examiner sees
If you're teaching a Year 11 class I think it's well worth showing them this picture from @TheMathsMagpie to ensure they are aware that they shouldn't work outside the allocated spaces in their exams.

This post gives more information about the picture above. Use it alongside this very helpful set of slides GCSE Maths - Easy Ways to Make Sure You Don't Lose Marks, also from @TheMathsMagpie.

WWW/EBI

I'm resisting the urge to write a weekly post reflecting on what went well (and what didn't go well) at school. Instead I'll just feature one personal reflection in each of my gems posts.

I'm teaching angles to Year 7 at the moment. I gave them a homework in which they were required to give reasons for their answers. Some students gave brilliant reasons and I was pleased to see the correct use of mathematical terms I'd taught them such as 'supplementary'. But many students just wrote out their calculations in words instead of stating the angle facts they'd used.

In their defence, we hadn't practised 'giving reasons' in class, so I decided to address this by setting some angle problems and asking for full reasons with each answer. Conveniently Don Steward has just published a set of angle problems that were the right level of challenge.

It seemed like a good idea but I wouldn't say the lesson went well. What happened was that my students were writing such detailed reasons in full sentences that it became really time-consuming (for example solving question 8 above involves using three different angle facts, hence three sentences). They were doing so much writing that they weren't doing enough maths for my liking. Plus you could hear a pin drop in my classroom, and that makes me uncomfortable. In the end I told them to stop writing full sentences and write one-word bullet points instead (eg isoscoles/triangle/straight line/complementary etc). They were really relieved! I'd love to hear my readers' thoughts on this - should I have persevered with the dull but important sentence-writing practice? I'm not sure. Please tweet me your opinion!

A Week in the Classroom

My first week back at work has been challenging to say the least. Eight new classes, a new Year 11 tutor group, over 200 names to learn, changes to school policies and practices, new nursery routines for my toddler and baby... the list goes on. I also celebrated a Fibonacci birthday this week, so I’ve been really busy both at work and at home. I know it will get easier when I get back into the swing of things.

I received my timetable by email during the Christmas holidays so there was no opportunity for a handover with the previous teacher. This week I've been busy trying to figure out what topics to teach and, for my exam classes, how I'm going to fit everything in before study leave commences in May. I'm out of my comfort zone without detailed plans! I only work three days a week but this year I'm teaching eight different classes under a job share arrangement (double the number of classes I had last year!). I'll have to be very mindful of the experience of my students, who all have at least two maths teachers now - switching between teachers and topics can be difficult for them.

Job sharing has its challenges but there are some advantages, the main one being that I now have my own classroom (most part-timers are school nomads). I love not having to move from room to room all the time - it really does make my days less stressful.

When I left work to have a baby I was Key Stage 5 Coordinator, a role that I loved. I don't have a TLR anymore so am focusing all my efforts on teaching. Since starting to write this blog and entering the Twittersphere I've become a much more reflective teacher. In this post I want to share a couple of things that have gone well this week.

Angles with Year 7
The thing I find hardest about teaching Year 7 is that I don't really know what maths they've done in previous years. I need to visit some local primary schools and observe some Year 6 maths lessons. This week I found out that most of my students were aware that the interior angles of a triangle sum to 180^o, but not aware that vertically opposite angles are equal. I found some questions (example below) in which they had to combine their existing knowledge with their newly acquired knowledge. They found these quite challenging - I don't think they're used to multi-step questions like this.

Brilliant.org is an excellent source of extension questions for lessons on angles, like the example shown below.

I'll be moving onto angles in parallel lines with Year 7 next week and I'm looking forward to using 'Spot the Angle' from mathspad.co.uk which I mentioned in my post Angle Facts.

I did quite a lot on vocabulary with Year 7 this week because there's lots of new words in this topic. When I was defining 'vertically opposite angles', I spotted the fact below on mathsisfun.com. This was new to me - I probably should have known this! It was helpful to share this with my students because I'd just taught them the meaning of the word vertex.

Finally, I recommend the short video All the different types of Triangle - Euclid by Shoo Rayner (featured in Gems 19). I showed it in my lesson on triangles and it went down well.

Problem Solving with Year 11
It's important that I quickly establish a good relationship with my Year 11 class. I've become their teacher at a late stage in their GCSE course, but I still have time to make a difference. I need this class to trust me. I see them five times a fortnight and that gives me 36 lessons to finish the syllabus and prepare for their exams. This week I decided to spend a lesson getting to know them. I handed out a problem solving activity then went round and met each student individually, chatting to them about their predicted grades and the topics they want me to review. While I circulated, they worked on eight problems of varying levels of difficulty. In the end I ran out of time to talk to everyone but I still think the lesson went well - they were really engaged in solving the problems and I heard lots of discussions about mathematics.

These problems are from a variety of sources (thanks to all originators!). At the end of the lesson the only problem that remained unanswered was the 'find the radius of the circle' problem from Chris Smith's newsletter - most students hadn't even attempted it but I promised them we'd look at it another day.

I did a similar lesson with Year 8 (top set), except all their problems involved Pythagoras' Theorem (I told them to use Pythagoras, meaning they had a starting point for each question. Perhaps I shouldn't have told them). Again, my students were really engaged in this activity. They particularly liked the 'Spider Box' question. The only question that no-one answered was Ed Southall's semi-circle problem (I suggested they challenge parents or older siblings to have a go). Also, the question below was answered incorrectly by almost every student (they squared the height to get 4w² instead of 16w²).

Support
Another change I've made this year is to set up a Twitter account for my students. Last year I ran a couple of blogs for student support and they worked well (my Year 11 class said their blog was really helpful, so it was well worth doing, albeit a little time-consuming for me. Read about it in my post Student Communication using Blogs). I'm trying Twitter as an alternative this year, the idea being that students (or their parents) follow me for useful links (eg for exam revision), interesting maths, reminders (eg 'you have a test next week') and so on. I've already started tweeting links to homeworks so students can print a copy if they lose their sheet. I've also said that I will offer maths support through Twitter, so they can ask for help anytime they get stuck on a homework or when revising.

I'm not sure how successful this will be. Other schools are much more Twitter-friendly than mine. We don't have a school Twitter account to communicate with parents, and I think I'm the first teacher at my school to try to use Twitter for educational purposes. So I got nervous giggles from my classes when I told them about it. I doubt many of them have a Twitter account so it might not work, but we'll see. I'll report back on this in a few months!

What's Next?
Well it's only been a week so I've still got loads to try. I downloaded iDoceo but haven't had a chance to look at it yet. I'm also increasingly keen on setting up a visualiser (via my iPad), to show examples of student work on the IWB (amongst other things). Funnily enough one of the most useful things I've done this week is use my own website, resourceaholic.com, to find resources! My resource libraries have saved me a lot of time. I've already noticed gaps in them though - nothing on angles in triangles for example - so I'll work on boosting my libraries over the next few months.

That's it from me. I've found it quite therapeutic to reflect on my first week back at school and I hope there's been something helpful for you here too. There'll be no gems post this week because I've not had much time to get on Twitter, but I'll write one next week. Please don't forget to get a free ticket to La Salle's National Mathematics Teacher Conference which is on Pi Day in Birmingham (I'm delivering a session). I hope to see you there. Also, White Rose Math Hub's Celebration of Maths is definitely worth attending. I'm gutted I can't go (I can't afford the train fare to Leeds!) but it looks awesome.

Selfie with my girls on my last day of maternity leave

5 Maths Gems #21

Happy New Year and welcome to my 21st gems post - this is where I share five teaching ideas that I've seen on Twitter in the past week. Now I'm back at work from maternity leave, I expect these posts will become fortnightly rather than weekly. Sadly, marking and lesson planning will mean I have less time to gather ideas from Twitter.

My aim for 2015 is to continue to work towards a post for every topic, so that when you're planning lessons you can refer to these posts and there's everything you need - concepts, ideas, resources, enrichment and links. I really hope these posts reduce the amount of time you have to spend searching the internet. My latest post was on Teaching Indices, and you can see the full range here.

1. Brilliant
I've been enjoying the questions from brilliant.org this week. It has a big range of mathematical problems that you could use in lessons (algebra, geometry, calculus, mechanics etc). Plenty of the problems are available for free - full solutions, extra problems and a bunch of other stuff is available for a small subscription.

Here's a couple of examples of problems for your students:

Some topics also have explanations and interesting worked examples - check out the chapter on the volume of a pyramid. The associated quiz has questions like the one shown below.

So if you're looking for interesting questions to use when teaching a specific topic then you might find Brilliant helpful.

Also, if you're looking for some challenging problems to have a go at yourself, I recommend you like Brilliant.org on Facebook - you'll get a nice flow of problems in your newsfeed. Here's what Facebook presented me with this morning - this is a level 2 question (levels go from 1 to 5, with 5 being the most difficult).
2. FOIL

Whether or not you teach 'FOIL' as a method for expanding brackets, watch the video below.

By the way, I don't think there's necessarily anything terribly wrong with teaching 'FOIL' as long as students know that the order doesn't matter. When I was at school I was just told to 'multiply everything in the first bracket by everything in the second bracket' and as a result I use a range of FOIL, FIOL, FOLI and FILO on the board... sometimes my students who've been taught FOIL think I'm doing it wrong. It's really important that we emphasise that there is no set order. I suppose FOIL just helps students keep track of their workings.

3. BODMAS
Sticking with the excellent videos from @minutephysics, if you haven't seen this one about PEDMAS (that's BODMAS in the UK), it's well worth a watch (thanks to @JustinAion for sharing). The order of operations turns humans into robots!


4. Furbles
I may be the last maths teacher to hear about Furbles, but in case anyone else hasn't heard of them, they deserve a mention here. Thanks to Hannah (@missradders) for bringing this cute statistics teaching tool to my attention.

Furbles can be used to teach statistical graphs and probability - if you've not seen them before then have a play around - there's two web-based versions on the website, plus a downloadable version here. Examples of Furbles-related questions are shown below, and there's a more comprehensive list here (thanks to @clwatson999 for sharing this link).

• Which is the best graph to show you which is the most and the least? 
• What would the key for this pie chart look like? 
• What is the probability the next starred Furble will be … a purple triangle with 4 eyes?

You can also ask students to make a set of Furbles to give particular chart properties or a specified probability distribution.

5. Euclid the Game
A few people have recommended Euclid the Game to me and I finally got round to having a good look at it. I see what all the fuss is about now. I'm teaching loci and constructions soon so I think I'll book an IT room to let my students have a go at this game. After a short tutorial in which they learn how to use the basics of GeoGebra, students are asked to do a series of constructions.

What I like about it is that students who normally just learn a procedure for these constructions have to think about how the constructions actually work (ie they're not just drawing arcs). It's a great game so if you've not seen it before, do have a look.

Well that's it from me today. Hope you all have a good first week back at school!