Tuesday, 30 June 2015

My unsolved problem - the magic triangle.

The #mathscpdchat today about perseverance today I suggested an interesting point - share your unsolved problems with them. I was asked for more information and so dutifully supply it here; it is little more than a curiosity in reality but I have been puzzling it through for nearly a decade and have never managed to solve it to my own satisfaction. It centres around completing the magic triangle, like the example below.

For those who haven't come across it before, the point is this - you use the numbers one to nine to complete the three sides of the triangle so that they all have the same total.

Now solving these is not a problem (I probably wouldn't be much of a maths teacher if it were!) but my unsolved part is related to how many solutions. Let me explain in more detail:

It is relatively straightforward to prove that the 3 corner values have to add up to a multiple of 3 (although still a lovely example of a proof for GCSE pupils), but it is also relatively straight forward to find combinations of 3 numbers that do sum to multiples of 3, but cannot be used as the corners of a magic triangle (for example, you cannot complete a magic triangle with 3, 4, and 8 in the corners despite the fact they add to 15). My unsolved problem has been finding a sufficient condition as to which multiples of 3 work. Now I can find them by brute force, but I want an underlying property, I want to understand why certain combinations will and certain combinations won't.

For the longest time I thought it was only sums that lie on straight lines through this grid

1                  2                    3
4                  5                    6
7                  8                    9

(which by the way, can all make magic triangles) but then when I had some Year 6s in a masterclass working on finding them, one of those pupils found a solution that doesn't lie on a straight line through the grid (which I honestly can't remember but which wasn't a problem as I couldn't prove why it would only be lines through the grid that would work in the corners anyway). I have tried rearranging the numbers into a magic square and looking for patterns in that arrangement, but to no avail.

I shared my struggle with my Year 7 pupils today, as we were talking about mathematicians and problem solving, and why not share with your pupils the mathematics that you have puzzled over?

Monday, 29 June 2015

Bearings in the hall - well worth a go!

Well what fun I had with my Year 8s today! The introductory lesson on bearings, with a twist...no writing or drawing at all! Instead we were down in the hall having fun with direction, playing team games and challenges. So much enthusiasm and energy involved, it tired me out just watching them!

On a practical note, the equipment used was:

A ball of wool
Scissors
Tape
Board sized protractors
8 or 9 hula hoops.

Activity 1: North, South, East, West.
You haven't lived if you didn't play this game as a child - someone stands in the middle of a large space and shouts directions, which everyone has to run to. I started by standing in one of the hoops in the middle of the hall and only defining North, and leaving them to figure out where the others are, and then off we went. Following a few of the cardinal directions, I then started to throw in NE, SW etc and ultimately things like NNE. This of course is where it started to get interesting as to how precise I could be, and is what motivated the need for bearings. After a brief discussion about using angle measures, and the need for two lines to create an angle (one, the direction of travel, the other a line pointing due North) and the need to measure clockwise (otherwise two different directions for the same angle), we moved on to Activity 2...

Leaving the hoop in the middle we tied a piece of wool to it and then stretched it out and taped it down to create a North line. I then threw the hoops around the hall and got teams of three to try and measure the bearing from the centre hoop to their assigned hoop. They used the board-sized protractors and more wool (typically each group had one person stood in the hoop in the middle, one person standing on a line to their assigned hoop, and then one person measuring) and had to be accurate enough to satisfy me to score points. This was my checking and consolidation exercise, used to pick up on an early misconceptions (i.e. people not measuring clockwise etc). This finally led on to the fun Activity 3....

Activity 3: Jump in the hoop.

To finish with we went back in to the teams and each team lined up at different points around the room. I put 2 of the board practors down back to back (and overlapping a bit) to create a makeshift 360 degree protractor and we played a game where I would shout a bearing, and the first team to put their circle on the bearing and stand in it scored the point. Each team then had to run back and give the hoop to the next one in line, and join the back of the queue ready to go again. The scramble to get hoops down was a definite sight to behold!

I cant say I have ever enjoyed the start to a topic more; however taxing it was really to manage all of the practical elements (I think I was as exhausted as some of the kids by the end of it), it was definitely worth it!

Thursday, 25 June 2015

Q & A with Nicky Morgan report.

I don't do political blogging; I am driven by developing teaching and learning; however as part of my role with school to school support at with the TELA TSA I was invited to a Q&A today with the SecState and felt honour bound to report back what I had heard, so here goes...

In a quiet drama studio at Thomas Estley Community College a small group of senior leadership and school to school support teachers gathered to hear the gospel of our beloved Secretary of State for Education, Nicky Morgan.

Having submitted our questions for the Q&A we waited eagerly to receive clarity to our questions. My question didn't get past the guard dogs at the DfE but I was able to raise it in the later free for all. It was only part answered: I did manage to find out come things about grade 4/5 but not much. I don't want to go through all the back and forth so I will summarise the outcomes of the discussion:

1) Top five priorities for the DfE were listed as: over-arching - Excellence everywhere, with every child having access to an excellent education. The point was raised that it was deeply unfair for a child, through no fault of their own, to go to a school which wasn't at least good. The numbered priorities were then 1) School places, 2) Teacher recruitment, 3) Funding and making it fairer, 4) schools developing character and resilience rather than just academia and 5) mental health and well being for young people.

2) The plans to make the Ebacc compulsory were discussed for new Year 7 (2015-2016 first year group), with the challenges to meet the recruitment of languages teachers. Apparently there will be incentives for languages teachers, the possibility of overseas recruitment drives, and also ideas around careers education and services used by careers converters to highlight teaching (in general). A point was asked about having government backed schemes like the 'Big Bang project' in science; Ms Morgan suggested that this is something that the Department would love to hear about if teachers have ideas, and she would love for schools to lead themselves in developing this.

3) The question was asked about adopting an always cross-party lead on education (as it apparently is in Finland), to stop education becoming a political plaything for whichever party is in government. The Secretary of State suggested that when discussed this idea was rejected by a majority of stakeholders, including opposition parties, and that the preference was to have that accountability for performance resting with the party in power. The point was repeated about the need for embedding changes and not having new things rushed in.

5) A big one this so I will put it in bold Pupil premium funding to be held at current levels as per the manifesto pledge through the life of the parliament.

6) Talk turned to the life without levels saga, with both primary and secondary colleagues asking about accountability and ensuring that information can be shared between schools. Apparently the 'experts' designing the new curriculum decided that levels weren't fit for purpose; guidance will be forthcoming but schools should develop what they need for their pupils and parents to let them know about the progress their pupils are making, and that schools will likely baseline new entrants for their own system. School collaboration and discussion was said to be important.

7) A question was at least partially ducked about why exam marking was preferred to teacher judgement given the high profile errors with exam marking. The response was basically that there weren't as many mistakes as highlighted and this then segwayed into a brief treatise on the changes to linear exams over two years with no early entry unless best for child blah blah blah...

8) The question about government developing capacity in teaching school alliances given the time and energy involved in administering them; again the DfE welcomes schools and alliances leading on how to develop this capacity (talk about chicken and egg!) Related to this was the question about funding through the NCSL as this seemed to be unclear and was causing anxiety; no answers forthcoming on this one but a promise to find out and report back.

9) Ebacc reared its head again, about the vocational/arts subjects this time. The question was put directly as to whether they were valued, and of course the answer was yes. There will be consultation in september time about groups of pupils that may not have a compulsory ebacc, but still the expectation that most will do. A commitment to 'technical and professional' (the new term for vocational) qualifications was reiterated.

10) I finally got to ask my question about what happens to pupils who get grade 4 post 16 - do they have to resit to grade 5 and are FE colleges expected to change admissions from 5+ A*-C to 5+ 9 to 5 or 9 to 4? The first was expertly dodged with an assertion (ready for this FE teachers...) that FE have a 2 year window before grade 5 becomes their measure as recognition that they need longer to develop (with no mention to what actually happens to the pupils); the second point was partially dodged in that it was clear that colleges set their own admissions policies (true so not really a dodge) but that one would assume in time that the more rigorous standard would be the one to aim for.

11)  A question about inclusion policy and special schools - vague on this one, looking at the area, along with LAC as well, looking at the sometimes slower performance of the services that work with schools. Little titbit, apparently 17% of Free schools set up were special or alternative provision schools. It was said there is a need for more time to see how the whole special needs sector is working and make improvements. Important not to let people slip through the net was how that one ended.

12) Last question, back to funding - It was made clear that the first time we would see changes is the 2017/2018 cohort as the consultation wasn't starting until October. The reason given was the need to get things right first time (we wait with baited breath for this to happen for the first time ever out of the DfE) rather than change quickly and have to change again. There was a final follow up statement rather than question about the government needing to realise that private school pupils are funded at an average 3 to 4 times higher through their parent's payments to schools than state schools are funded by government.

Make of it what you will, Mrs Morgan at least seems to listen a bit more than our last Secretary of State, so that might be a positive thing!

Tuesday, 23 June 2015

Plans and Elevations - Some approaches

Recently I have been teaching the drawing and interpretation of plans and elevations to my set 4 in Year 8; I love teaching this topic as it can be really hands on and can create quite an atmosphere if done well. I love how active the kids are when I set competitions for which teams can build shapes out of multi-link that have the given plan, then watch them have to adapt when I give them the front elevation, or side elevation, or both. So I thought I would take this opportunity to share three of my favourite approaches/activities for this topic.

1) Wall smashing: Remember those cartoons we used to watch when we were young where a character would run off through a wall, leaving a nice character shaped hole; usually with their arms flailing in fear? Welcome to a front elevation! I have yet to find (or get around to making) a compilation with lots of these clips one after the other, but you can find the individual cartoons online and play them if you like, or often the visual images will do. Getting kids to draw these rather than starting with the boring old shapes made of multi-link can be a nice introduction, and if you are brave enough you can create some scared jumpy children as you slam your hand into the table or wall to show you smashing a shape through a wall to create an imaginary elevation or plan.

2) Potato or sponge printing: I don't use this one a lot, but it can be great for certain kids that cant break the need to represent in 3D and understand that these are supposed to be 2D representations. Anyone who has young children will have played the game where you cut a shape into a potato or sponge; dip it into paint and then print onto paper. I prefer sponges as they can often be cut with classroom scissors and get across the idea that if you have depth this impression is destroyed when you print, as the sponge simply flattens.

3) Build and draw around: Pretty much as you would expect really, kids get the opportunity to build the shape and if needs be draw around it. I tend to use the shapes in the plans and elevations lessons at www.MyMaths.co.uk (slides 1 or 3) as they have the added bonus of the animation that changes from the isometric into each of the elevations and plan as well as having the faces from each view coloured. These are all shapes that can be built from multilink, giving pupils a real view of the shape. A lot of pupils won't need to build the shapes (although a lot like to anyway) but for some building the shape is a really useful way to be able to manipulate and view it from the different angles. Those that are really struggling can actually then place the shape flat onto paper and draw around it (note - if you haven't done this before make sure pupils understand they have to hold their pencil perpendicular to the page at all times, or it wont necessarily work). Rarely will you get completely accurate drawings because of the textures of the cubes and some difficulties keeping the pencil perpendicular, but it is usually enough to convey the message and can help pupils then draw a more accurate representation once they know broadly what they are aiming for.

So there you have it, if you feel like your teaching of plans and elevations has gotten a little stale over the years, or you are just starting out and looking for nice introductions to the topic, why not try a little hands on work to freshen up those lessons. There is a link to a prezi here which shows the mymaths link and also the shapes I use for the building competition.

P.S. - Top tip: Did you know that plans and elevations are called orthographic drawing in D&T? Why not talk to your D&T teachers to discuss their approaches and see what resources they use. This could even be tied into a cross-curricular project where they complete orthographic drawings in maths that they use in D&T to build a product (how about a shape sorter where all the shapes go through all the holes when held a certain way!?)

Sunday, 21 June 2015

Ivan the jumping flea and negatives

So yesterday at the excellent National Mathematics Teachers Conference (#mathsconf4) I was discussing with some other teachers approaches to teaching addition and subtraction of negative numbers. I outlined two approaches I have used in the past, the mood cards I outlined in this blog and the other approach which I think underlines more of the conceptual understanding - movements along a number line. To make it a bit more memorable for pupils I invent a nice jumpy character to assist me in the demonstrations - Ivan the jumping flea.

The way Ivan moves is governed by the following rules:

1) Ivan starts at 0 facing the 'positive direction' (i.e. the direction of increasing numbers).
2) Positive numbers cause forward jumps.
3) Negative numbers cause backwards jumps.
4) The operation of addition makes Ivan face the 'positive direction' (i.e. the direction of increasing numbers).
5) The operation of subtraction makes Ivan face the 'negative direction' (i.e. the direction of decreasing numbers).

So lets say Ivan is completing the calculation 3 + (-5). Ivan would start at 0 facing up the number line and the first thing he would do is jump forwards 3 places, as the first part of the calculation is the positive number 3. The next thing Ivan would do is face the 'positive direction' (which would mean he did nothing as he is already facing the positive direction), as the next part of the calculation is the operation addition. The next thing Ivan would do is jump backwards 5, as the last part of the calculation is -5. Once these steps are completed Ivan would be at the value -2, showing 3+(-5) = -2.

Compare this to the calculation 3 - (-5). The first step would be the same, as Ivan still starts at 0 and still jumps to 3 as before. This time the operation is subtraction, so Ivan turns to face the 'negative direction'. Ivan then jumps backwards 5 as before, but because he is facing the 'negative direction' he is actually jumping up the number line, and so ends up at the number 8; showing that 3 - (-5) = 8.

I have a PowerPoint here which shows Ivan solving both of these questions (which I made some changes to in order to have a completely animated sequence, once you click to get going on each slide) which people are free to adapt for other questions as you see fit (provided you can alter the motion paths etc).

Wednesday, 17 June 2015

'Pointless' bar charts

A couple of nights ago some may remember that I put up on the twitter chat #mathstlp that I was teaching bar charts to Year 9 bottom set and was in need of inspiration. I had a couple of contributions (thank you ladies - @missradders I used the challenge you sent me) and then Tuesday morning I had a brainwave - Pointless! I had intended to put a picture of a bar chart on the board and ask pupils questions about it, but then the brainwave I had was - why not just give them the picture and get them to write about it; and from there can they come up with that "pointless" bit of information that no one else can!

Instead of just putting the picture on the board I organised the kids into 11 groups and gave each group a copy of the bar chart stuck into the middle of the paper and told them to write as many bits of factual information from the chart as they could around the outside. After about 10 minutes they had to choose one of the bits of information that they thought was their "best shot" at a pointless answer. They were then given points in the true pointless style - however many groups had the bit of information scored them that many points, or for an incorrect answer the maximum of 10 points (11 teams = maximum of 10 points when counting from 0 to 10).

The kids really enjoyed the competitive element and trying to come up with obscure information, and obviously we got some interesting maths that I wouldn't have thought to ask [how about the bar is 7.5 cm long and 1 cm wide!] We got some great discussion and discord about whether people were right or wrong, and whether two pieces of information were the same. One group said the frequencies add up to 390; and meant the values on the frequency axis rather than the frequencies indicated by the bars; we didn't give that as it was ambiguous.

I can see this working for lots of things; I think putting a straight line graph on the paper and asking for facts here as well, or a two-way table, or any other way of presenting factual information. So if you are looking to get kids answering questions you would never think of asking, try a Pointless Page.

Sunday, 14 June 2015

I can read your mind - now that is what I call an hypothesis test.

A few days ago we were talking on one of the twitter chats about hypothesis testing and I alluded to a fun introduction to the topic I had used before, which I thought I would take the time to flesh out in more detail.

It starts by explaining to pupils that you are going to test if any of them are psychic. Obviously it is nice to ham this up a little bit, give it a bit of dramatic flair etc. Explain to them that the test is that you are going to flip a coin 20 times and keep each result hidden from them. You are then going to concentrate very hard on the result and they have to write down the impression they get from you.

Do the experiment and then see how the kids get on; normally about the maximum you will get is 13 or 14 (any more and you may well actually have a psychic on your hands!) and so the conversation turns to, "well is this enough evidence? What is the probability of someone getting 13/14 by random guessing? How many would be good enough?" This gives me all the tools I need to form a formal hypothesis and discuss things like the significance level (at what point does the probability become so remote we have to agree this is not happening by chance - when it is less than 5%? 1% etc), confidence interval (how many must someone get right before we believe they can read minds) etc.

I find that having this early practical hook to keep coming back to really helps pupils as they navigate what can be quite a tricky topic simply because of the sheer number of different contexts to which it can be applied. So the next time you are teaching kids about hypothesis testing, try giving them a fun hypothesis they can see practically happening in front of them.

P.S. to develop it, ask about what would happen if you switched the coin for a die, and how that would change their views on how many needed to be sure etc...

Wednesday, 10 June 2015

Inspired by JustMaths, looking beyond the basic skill.

Recently I have been planning a lesson on the classic exam situation of completing a partially complete two-way table, such as this one:

Now in my experience pupils grasp the concept of this quite quickly, with most mistakes tending to come in making arithmetic errors rather than mis-understanding the problem. Which of course leads me to the problem of how to stretch the lesson for those pupils who do grasp the concept so quickly.

During an internet trawl for inspiration I came across an excellent resource from the brilliant team over at JustMaths, which has 5 tables to complete (enough to give enough practice at arithmetic) that are all linked together and then provides an interesting activity whereby pupils have to identify which teachers are making mistakes in analysing the resulting tables. It was then that I had one of those nice ideas which occasionally occur to me; here was an ideal moment to link in some prior learning!

Taking the JustMaths tables I then created some statements that go a little beyond their "True or false" on the back, into True/Maybe/False. I brought in these statements:

1) Every student studies English
2) Every student studies Maths
3) More students study Food than Biology
4) A greater proportion of students study Biology than Food
5) The ratio of Boys to Girls studying Art is 2:1
6) More than 80% of the students studying Applied Maths are boys

I won't spoil the surprise of the resource by telling you which are True, which could be true and which are false, but the answers are in the lesson here if you desperately have to know; needless to say there is at least one of each.

Now you may not use the Just Maths resource, you may not use these statements, but if you are looking for a little stretch in your two-way tables lesson, try bringing in some other prior learning number statements, and trying setting some statements that are definitely true, could be true or definitely false.

Tuesday, 9 June 2015

Probability Scales - worth spending time on.

For many years now I have viewed probability scales as something a little beneath the brighter pupils; a Level 4 (in olde English money) skill that bore little practicality for those pupils who would eventually go on to take higher GCSE where such trivial skills would not be tested. Something always nagged at me though - kids would get these questions wrong. Good kids, bright kids, who could use sample spaces and calculate with relative frequency got tripped up when having to describe a situation in words or place it in even broadly the right place on a number line.

So this year I decided to teach it, properly, to my top set Year 7. We looked at describing events in words and looked at where probabilities of events would go on a scale. Boy am I glad I did! Now I am not going to stand here and say every kid made 'rapid progress' nor am I going to stand here and say that every kid benefitted greatly from every activity or every pupil needed to recap every part, but as a whole the lesson was worth doing. Some tips then for making this a success with upper sets:

1) Enforce correct notation [P(...)] alongside correct language.

2) Use alternative language freely and force pupils to describe in different ways.

3) Ensure you hold them to reasonable accuracy when placing values on the number line (I had a lad who told me that a probability that numerically was 1/7 mark an arrow too close to 0 and we challenged this as a class by looking at 1/7 as 14.29% and whether we thought that his arrow was nearly 15% of the way up the line).

4) Use a number line broken into segments (I was nice and went for 10ths) to ensure they have to think about at which point or between which points does it go (and apply point 3 here as well, I was penalising for probabilities of 1/3 being closer to the 4th mark then the 3rd mark)

5) Teach this as a follow on from writing probabilities as fractions, rather than before it (as is often the case), so that you can tie fractional ideas to scales.

I used some really nice stuff from CIMT as well here (in their Maths Enhancement Program, Year 9 book 6) which challenged their thinking a little more and pulled some of the better questions into a nice worksheet.

The link to the full lesson is here, and the link to the worksheet is here, but however you decide to teach it I would heartily recommend making the effort to challenge your upper set pupils on probability scales.

Friday, 5 June 2015

Hannah's sweets and the new GCSE.

Well what a furore the Edexcel Higher paper has caused! I am sure no maths teacher who has any sort of presence online can have escaped noticing that our twitter feeds, facebook pages and even BBC news have been reporting the pupils dissatisfaction with the Higher GCSE paper and in particular the second half. The storm seems to have centred on a couple of questions, one being Hannah and her sweets. The question reads as follows (if you somehow haven't seen it)!
"Hannah has n sweets in a bag. Six of the sweets are orange, the rest of the sweets are yellow.
The probability that Hannah eats two orange sweets is 1/3. Show that n^2 - n - 90 = 0."
Never has a question received such criticism. It has had Edexcel Higher tier pupils across the land calling for lower grade boundaries, calling out examiners for unfair questions etc. Year 11 pupils up and down the country have been raging about it. And one of my Year 9 girls solved it in under 3 minutes.

For me this brings home exactly why the new GCSE has become necessary. This question was a relatively straightforward application of conditional probability and quadratics (or algebraic fractions if you prefer) which required very little in the way of interpretation and yet it stumped so many Year 11 pupils. Never has it been driven home exactly why a renewed focus on problem solving, interpreting and communicating mathematics is necessary for if our top Year 11 pupils (and apparently one rather dense breakfast journalist) are struggling with it then this is a clear indication that our pupils are not being given the skills to understand the mathematics they are doing. To be fair she did solve it by solving the quadratic (giving n = 10) and then showing the tree diagram to find the probability of two orange sweets does equal 1/3, but it seems that was a lot better than a lot of pupils did.

I normally limit my posts to sharing practice, as that is where my passion is, but my passions were enflamed by the furore around this question when it is actually something I would expect most higher tier candidates to at least begin to access. If your pupils are the ones tweeting and shouting about how unfair the paper was, then I would be talking to them about why they found it so difficult; and if you are the journalist who is cannot solve this question then drop me a line and I will give you a few top up lessons!

Wednesday, 3 June 2015

Singing for memory...it really works!

Since moving to an 11-14 school in order to help them become 11-16 I missed teaching A-Level, so when I found out that one of the TAs was resitting the core 4 maths exam and wanted some help I was happy to give up some time to support. We are obviously coming towards the end of the time (the exam is next Tuesday) and only recently discovered she had never learnt the quadratic formula (we were solving some trigonometric equations using double angle formulae). She was having real trouble remembering it so I tried out a technique I had heard about - linking it to a song.

She was a bit embarrassed to sing in front of me, so I sang in front of her instead! I am now seriously considering teaching all of my lessons in song! The reaction I got was brilliant - she couldn't look me in the eye and was laughing her a** off! Talk about engaging them emotionally, the cringe and humour factor is brilliant. I sang Journey's "Don't stop believing" with the quadratic formula as part of the lyrics, basically like this:

"Just a small town girl,
Living in a lonely world;
Minus b plus or minus,
the square root offfff,
B squared,
minus 4ac;
all over 2a
that gives you
the 2 values that solve
the equaaaaaation!"

I explained to her that everyone has a song that they just know the lyrics to, without even thinking about it, so if she can put this sort of information to that tune, she can use her knowledge of the lyrics and tune to support her. Of course the most difficult bit in remembering the lyrics of a song is getting started, so I left the start the same. I guarantee that when I have Year 11 (and I will probably do it with some stuff in Year 7 all the way through), I will be using this for a revision technique!

Monday, 1 June 2015

I have never probability

Ok, so honestly I haven't done this one yet, but it occurred to me whilst doing writing down probability with Year 7 today. Admittedly it would be for relatively low prior attaining but talk about engaging! Count up how many kids in the room and then throw out some "I have never..." statements and get kids to stand up if they have never done something; then get the kids sitting down to put the associated probability on a mini-whiteboard or similar. An alternative might be to test some statements against probabilities sourced on the internet to see how close they are in your class? Here are some possible statements I came up with:

a) I have never flown in an aeroplane.
b) I have never been to a live concert.
c) I have never been to a live sporting event.
d) I have never broken a bone.
e) I have never been abroad.
f) I have never been camping.
g) I have never done the ironing.
h) I have never been on a train
i) I have never ridden a horse.
j) I have never been ice skating.

I am sure I could come up with more if I really thought about it. Think I have my starter for tomorrow!(Or if I can find some way to tie in mutually exclusive/exhaustive events I might use it during the lesson).