That's definitely a question my maths colleagues and I have asked each other after an afternoon double period... But in this case I'm talking about beating them at chess.
Now that all of my students understand the rules of chess, I have been able to spend lesson time playing against my students instead of walking around the classroom reminding little Billy (not actual name) that Queens can't turn corners and that Pawns should not be used as false fingernails. My first attempt at duelling a student was actually in my Pokemon/Yugioh/and now Chess club after they had had their first three chess lessons. He beat me. He reminds me that he beat me every time he sees me in the corridor.
My next attempt was in their fourth lesson, when they were just playing friendly matches against each other in preparation for the tournament starting from lesson 6. I was very evenly matched with this particular student (let's just try to forget, for the moment, that I have a degree in mathematics and his highest qualification is an "expected standard" pass in his year 6 SATs.). We ended up with just three pieces left each. I was struggling. I asked the university student who is helping me out with this project to advise me. This does not count as cheating, because I taught him for five years and was his form tutor, so his knowledge is actually my knowledge, if you think about it. Unfortunately, he got called away at a crucial moment and I managed to give away one of my pieces. After that I wouldn't let him leave me until I had won, which I managed to do with just my King and a Bishop. I'll admit, it wasn't the sweetest victory I have ever tasted.
In the next lesson I played against a student who, in the past, I might have described as "not all there", perhaps accompanied with a circular gesture with my hand near my ear. He destroyed me.
This week in Pokemon/Yugioh/socially awkward club, I re-matched the student I had played against last week. He beat me even more embarrassingly than last week. Then, students who are not in my class but who have a vague understanding of chess said they wanted to play me. I got beaten three times in quick succession. In fact, I was so rubbish that my older students simply took over my games, because my moves were so bad it was painful for them to watch. I pointed out to my students that I never took their exercise books off them and solved their equations for them because their algebra was too painful to watch. They then pointed out that actually I do do that. They were making fun of me and my pathetic strategies (or lack thereof). I pointed out that I don't laugh at them when they fail their Maths mock exams. They pointed out that actually I do do that too. I should probably stop doing that.
All of this leads me to ask the question: why can't I beat my students at chess?
I am usually good at stuff. It's pretty rare for me to try something and not be good at it straight away (obviously this does not extend to stupid things like map reading, long jump, and the bus stop method of division). So I'm really frustrated about this. The only positive spin I can put on this is that I must have taught my students really well. I'm not even convincing myself with that line.
So I've made an account at chess.com and I've started working my way through their online tutorials. So far (two hours into my course), I've learnt how to do my first move. If I'm White, I know exactly what the (statistically speaking) best move is, and if I'm Black, I know six possible first moves I could do. I also learnt how to do the four-move checkmate and how to fianchetto. I refuse to be beaten by 11 year olds!
Anyway, back to my Chess lessons at school. In lesson 6 we started the tournament. I paired students up systematically and separated all of the desks. The students found their opponents and started playing. Originally, the plan was to play two half-hour matches per lesson, but then I realised that an hour was a more appropriate length of time. Students who finished their matches were told to just play friendly matches for the rest of the lesson. Most matches seemed to take around 40 minutes.
Challenges I've Faced
Noise! My students are not quiet chess players. They celebrate every capture with a loud noise and sometimes even a dance. The only silver lining is that at least they seem to have finally stopped dabbing. I may have to impose some volume rules, although I can't really ask for silence as students do often need to talk to each other during their matches.
The concepts of Check and Checkmate. For many students, this is difficult to grasp. Many students are claiming they won by killing their opponent's King. They are not telling each other when they are in Check and often they don't even realise they're in Check. This is why it's important to have adults walking around and checking for checks and getting the students to move themselves out of check if needed. I have three wonderful assistants with my class now (a trainee plus two of my favourite ex-students who have finished university for the summer). I could probably just about manage with just myself, but I'm glad I don't have to. Especially because I still find myself occasionally wondering whether it's the Bishop or the Rook that moves diagonally.
Missing pieces. Every lesson starts with having to re-distribute pieces that have ended up in the wrong boxes. Despite only having bought them a week ago, some chess sets are now using multi-link cubes as stand-ins for pawns. I need a better system for keeping track of all the pieces. Any suggestions?
Moving Forward
For the last two weeks of term, my students will just be playing their tournament matches. They will get three points for a win, two for a draw, and one for a loss. They get zero if they're absent from the lesson or if they display unsporting behaviour or if they misbehave. If there isn't a clear first, second and third by the end of the two weeks, we'll do some deciding matches between the top few.
Now for the really exciting part: my boss read my previous blog post and decided she really liked what I was doing so has offered to buy us more chess sets so that my colleagues can teach chess to their year seven classes too! And we're going to teach chess to our new year sevens next year too, although I'm not sure exactly when in the year that will be.
Have any of you tried it yet? Let me know if you have any questions or comments!
Emma x x x
Showing posts with label Teaching Ideas. Show all posts
Showing posts with label Teaching Ideas. Show all posts
Friday, 7 July 2017
Saturday, 1 July 2017
How and Why I taught my Year 7s to Play Chess
Last Monday I stumbled upon an article in the Guardian titled "Schools teach chess to help "difficult" children concentrate". Whilst I wouldn't describe my year 7 class as "difficult" (at least not on a public and non-anonymous blog), the article did strike a chord with me when it talked about how "as schools grapple with screen addiction and short attention spans, chess is also seen as a way to encourage 'digital detox'." All of the year 7s in my school were given the opportunity at the start of the year to buy an ipad mini at a very very very reduced price. Almost all of them have one, and it appears to me that an alarmingly high percentage of them are addicted to it. I see them walking through the corridors with their eyes glued to the screens. They sit in the canteen at lunch time staring at their devices, not talking to anyone, not noticing anything that's going on around them. In my lessons, where I have a strict no-ipads-on-the-desks rule, I see my students reach for their ipads every few minutes, just flick the screen on and off, as if to reassure themselves that it's still there. I was frustrated when I would give my students an open-ended task or a puzzle to solve and they would look at it for five seconds before declaring it "impossible!" and giving up, and turning their attention to their fidget spinners instead. The Guardian article gave me hope that chess might be the solution to all of my problems. And so began the ---- ---- Academy Chess for Success Project.
I recruited a former chess club champion SCITT trainee and a chess-loving ex-student who owed me a favour, both of whom have been invaluable in setting up this project because, well, did I mention I had no idea how to actually play chess? I wasn't too worried about that though, as in the timeless words of Marge Simpson, "I've just gotta stay one lesson ahead of the kid!" So during a year 13 Further Maths lesson I got a particularly nerdy student to give me my first and last lesson in chess. I learnt how the pieces move. I learnt about castling. I learnt that to win, I needed to trap my opponent's King. But I had absolutely no idea how to go about doing that.
I decided that I needed to think carefully about how to teach my students how to play chess. I knew from my experience of learning chess that knowing how the pieces move isn't enough. I had an idea of how to do it and after doing some research I found a WikiHow article (method 2) that was similar to the idea I had had, so I decided to go with that.
Here's exactly how I did it:
At this point I only had one chess board for every 6 students (scrounged from the library). I started the lesson by explaining the benefits of chess and telling my students why I had decided we were going to abandon our scheme of learning for the next few lessons and play a game instead. Then I gave them the pawns. I told them where to place them. I explained the movement of the pawns. And then I told them their objective was to get one of their pawns to their opponent's end of the board. (I feel it is my duty at this point to warn any teachers planning to do something similar that the word "pawn" will be very amusing to some of your students, and I encourage you to develop a semi-Scottish accent if possible). The students played 3 v 3 and were immediately engaged in this game, which does sound rather basic but is actually still quite a rich source of critical thinking and strategising. The disadvantage of only have 5 chess boards became an advantage because it forced students to communicate with each other and verbalise their thought processes.
About half way through the lesson I introduced the Bishop. I explained the Bishop's movements, and told them their aim was still the same. They learnt how to use their Bishops to defend their pawns.
The lesson was a big success, with students that usually spend more time in the corridor than in the lesson actually engaged and concentrating for long periods of time. To my great surprise, absolutely no flicking/throwing/inserting into orifices of chess pieces occurred during the entire lesson. The students left already looking forward to their next chess lesson. And I was too!
By this point I now had 9 chess boards (thank you Amazon Prime) and a ratio of one board to every 3 students. Interestingly, some students still chose to play in bigger groups. I started by introducing the Rook, and letting them play for a while. A third of the way through, I introduced the Knight, and then at two thirds I introduced the Queen. The aim was still to get a pawn to your opponent's edge of the board. The students were still really enjoying it. Several students told me they had gone home the previous day and asked their parents for a chess set. A few had installed a chess app on their ipads (sort of defeating the purpose, but oh well). A few students had even come up to the maths area at break time and lunchtime asking if they could play.
I told the class that the following week we would be having two tournaments taking place at the same time: a competitive one with prizes for first, second and third place, and a friendly one for those who don't like to compete. Several students got really excited about this, and were determined to win.
As soon as the students came in they grabbed their boards, set up, and started playing. They could hardly wait! I had to interrupt them almost straight away though, as I had a new addition to make. I introduced the King, and crucially, introduced the new objective: to kill your opponent's King! I did not at this point mention check or check mate (concepts which I think are a little bit challenging), and one of my students who had already known how to play chess did give me a slightly incredulous look when I talked about killing the King (and he turned to the trainee teacher and they exchanged a look that basicaly said "pah, n00b"). After playing for around thirty minutes I approached individual games and explained check and check mate to them. In the last ten minutes or so I mentioned promotion, and with that I had pretty much finished teaching the rules of chess. I will casually drop castling into lesson 4, and some advanced players will be taught en passant too (once I have learnt it myself).
Lesson 4 is on Monday, so I'll have to let you know later how that works out. But I should have 5 more boards arriving, which means finally the students can play one on one.
Everyone in my class seems to be enjoying playing chess. This surprised me, I thought that at least one or two would hate it, or would resist learning it because they perceive it as boring or difficult.
That student that I've described as "incapable of concentrating on anything for more than 5 seconds" is actually on his way to becoming a grand master and can sit in a zen-like state of deep intellectual thought for a good ten minutes as he considers his next move. So maybe it's my maths lessons that are the problem, not him?
Chess brings people together. I have never seen so much collaboration and communication in my classroom before. They are all talking about the game. There is next to no off-topic chatting and the students are talking to the people on their assigned table, who are not necessarily their friends.
The students actually feel like they're becoming cleverer. My students are proudly telling me that their memory is improving and they are becoming better at concentrating. This is probably the placebo effect but that's fine with me!
Moving Forwards
I will keep you updated with how the project goes. I'm sort of planning on spending all of my year 7 lessons until the end of term on this project. I'm lucky in that my academy is allowing me to ignore my scheme of learning and do this. I really hope some of you reading this are inspired to try the same thing in your school. If you do, let me know how it goes!
Emma x x x
I recruited a former chess club champion SCITT trainee and a chess-loving ex-student who owed me a favour, both of whom have been invaluable in setting up this project because, well, did I mention I had no idea how to actually play chess? I wasn't too worried about that though, as in the timeless words of Marge Simpson, "I've just gotta stay one lesson ahead of the kid!" So during a year 13 Further Maths lesson I got a particularly nerdy student to give me my first and last lesson in chess. I learnt how the pieces move. I learnt about castling. I learnt that to win, I needed to trap my opponent's King. But I had absolutely no idea how to go about doing that.
I decided that I needed to think carefully about how to teach my students how to play chess. I knew from my experience of learning chess that knowing how the pieces move isn't enough. I had an idea of how to do it and after doing some research I found a WikiHow article (method 2) that was similar to the idea I had had, so I decided to go with that.
Here's exactly how I did it:
Lesson 1
At this point I only had one chess board for every 6 students (scrounged from the library). I started the lesson by explaining the benefits of chess and telling my students why I had decided we were going to abandon our scheme of learning for the next few lessons and play a game instead. Then I gave them the pawns. I told them where to place them. I explained the movement of the pawns. And then I told them their objective was to get one of their pawns to their opponent's end of the board. (I feel it is my duty at this point to warn any teachers planning to do something similar that the word "pawn" will be very amusing to some of your students, and I encourage you to develop a semi-Scottish accent if possible). The students played 3 v 3 and were immediately engaged in this game, which does sound rather basic but is actually still quite a rich source of critical thinking and strategising. The disadvantage of only have 5 chess boards became an advantage because it forced students to communicate with each other and verbalise their thought processes.
About half way through the lesson I introduced the Bishop. I explained the Bishop's movements, and told them their aim was still the same. They learnt how to use their Bishops to defend their pawns.
The lesson was a big success, with students that usually spend more time in the corridor than in the lesson actually engaged and concentrating for long periods of time. To my great surprise, absolutely no flicking/throwing/inserting into orifices of chess pieces occurred during the entire lesson. The students left already looking forward to their next chess lesson. And I was too!
Lesson 2
By this point I now had 9 chess boards (thank you Amazon Prime) and a ratio of one board to every 3 students. Interestingly, some students still chose to play in bigger groups. I started by introducing the Rook, and letting them play for a while. A third of the way through, I introduced the Knight, and then at two thirds I introduced the Queen. The aim was still to get a pawn to your opponent's edge of the board. The students were still really enjoying it. Several students told me they had gone home the previous day and asked their parents for a chess set. A few had installed a chess app on their ipads (sort of defeating the purpose, but oh well). A few students had even come up to the maths area at break time and lunchtime asking if they could play.
I told the class that the following week we would be having two tournaments taking place at the same time: a competitive one with prizes for first, second and third place, and a friendly one for those who don't like to compete. Several students got really excited about this, and were determined to win.
Lesson 3
As soon as the students came in they grabbed their boards, set up, and started playing. They could hardly wait! I had to interrupt them almost straight away though, as I had a new addition to make. I introduced the King, and crucially, introduced the new objective: to kill your opponent's King! I did not at this point mention check or check mate (concepts which I think are a little bit challenging), and one of my students who had already known how to play chess did give me a slightly incredulous look when I talked about killing the King (and he turned to the trainee teacher and they exchanged a look that basicaly said "pah, n00b"). After playing for around thirty minutes I approached individual games and explained check and check mate to them. In the last ten minutes or so I mentioned promotion, and with that I had pretty much finished teaching the rules of chess. I will casually drop castling into lesson 4, and some advanced players will be taught en passant too (once I have learnt it myself).
Lesson 4 is on Monday, so I'll have to let you know later how that works out. But I should have 5 more boards arriving, which means finally the students can play one on one.
What I've Discovered so Far
Everyone in my class seems to be enjoying playing chess. This surprised me, I thought that at least one or two would hate it, or would resist learning it because they perceive it as boring or difficult.
That student that I've described as "incapable of concentrating on anything for more than 5 seconds" is actually on his way to becoming a grand master and can sit in a zen-like state of deep intellectual thought for a good ten minutes as he considers his next move. So maybe it's my maths lessons that are the problem, not him?
Chess brings people together. I have never seen so much collaboration and communication in my classroom before. They are all talking about the game. There is next to no off-topic chatting and the students are talking to the people on their assigned table, who are not necessarily their friends.
The students actually feel like they're becoming cleverer. My students are proudly telling me that their memory is improving and they are becoming better at concentrating. This is probably the placebo effect but that's fine with me!
Moving Forwards
I will keep you updated with how the project goes. I'm sort of planning on spending all of my year 7 lessons until the end of term on this project. I'm lucky in that my academy is allowing me to ignore my scheme of learning and do this. I really hope some of you reading this are inspired to try the same thing in your school. If you do, let me know how it goes!
Emma x x x
Thursday, 17 November 2016
Draw Your Brain: a Growth Mindset Activity
Today I shared an activity with my colleagues and they seemed to like it so I thought I'd share it here too.
I told my colleagues (Maths, Science and Learning Support teachers) that they needed to get into the mindset of 15 year olds so that they could get the most out of the demonstration. Of course then one of my colleagues got a little bit too in character and drew an item of male anatomy on his piece of paper. Actually, knowing this particular colleague's personality, he would have done this even if he wasn't pretending to be a fifteen year old.(Typical Physics teacher).
But anyway, I had given everyone a piece of A5 paper and a felt tip. I told them to pretend to write on their piece of paper, and pretend they were answering a really difficult maths problem. They had to pretend they had worked on it for ages and they were really struggling. Finally, they come to an answer. However, a minute later, their teacher tells them their answer is wrong. I asked my colleagues to think about the emotions they would experience then. They gave me some examples: anger, frustration, embarrassment, disappointment. Then I told them to take all of those emotions, and channel those emotions into crumpling their piece of paper up into a ball. I told them to really make sure that all of their anger, frustration, and feelings of failure were screwed up into that ball of paper. And then I told them to throw their piece of paper as hard as they could, and with it release all of those negative emotions.
Then I asked them to retrieve their piece of paper and uncrumple it, and smooth it out so it's nice and flat. I told them that the piece of paper represented their brains. All of the tiny crease marks on the paper are the synapses, or pathways, inside their brain. I got them to draw over the crease marks with a felt tip pen, and as they did that, I asked them to think about the knowledge flowing through their brains thanks to these pathways. Every time you make a mistake, your brain gains an extra synapse. The only way to gain extra synapses is through making mistakes. So all of the connections and pathways in your brain are due to making mistakes. The piece of paper representing their brain would not have any synapses at all if it hadn't been crumpled up. Those pathways are only there because they made a mistake.
With students, I then tell them to keep this picture of their brain in their folder, and every time they make a mistake in Maths, they should look at it and think about how their brain has just gained another synapse. They can even draw on another synapse each time they make a mistake, and by the end of the year they will be able to see how much their brain has grown, and how much progress they've made.
This is a great activity to do with a class that lacks confidence or is stuck in a fixed mindset. I did this with my year 11 intervention class last year, and I think it was a real turning point for them.
Please try this with a class and let me know what impact it has!
Emma x x x
PS I got this idea, plus many others, from the book Mathematical Mindsets by Jo Boaler.
I told my colleagues (Maths, Science and Learning Support teachers) that they needed to get into the mindset of 15 year olds so that they could get the most out of the demonstration. Of course then one of my colleagues got a little bit too in character and drew an item of male anatomy on his piece of paper. Actually, knowing this particular colleague's personality, he would have done this even if he wasn't pretending to be a fifteen year old.
But anyway, I had given everyone a piece of A5 paper and a felt tip. I told them to pretend to write on their piece of paper, and pretend they were answering a really difficult maths problem. They had to pretend they had worked on it for ages and they were really struggling. Finally, they come to an answer. However, a minute later, their teacher tells them their answer is wrong. I asked my colleagues to think about the emotions they would experience then. They gave me some examples: anger, frustration, embarrassment, disappointment. Then I told them to take all of those emotions, and channel those emotions into crumpling their piece of paper up into a ball. I told them to really make sure that all of their anger, frustration, and feelings of failure were screwed up into that ball of paper. And then I told them to throw their piece of paper as hard as they could, and with it release all of those negative emotions.
Then I asked them to retrieve their piece of paper and uncrumple it, and smooth it out so it's nice and flat. I told them that the piece of paper represented their brains. All of the tiny crease marks on the paper are the synapses, or pathways, inside their brain. I got them to draw over the crease marks with a felt tip pen, and as they did that, I asked them to think about the knowledge flowing through their brains thanks to these pathways. Every time you make a mistake, your brain gains an extra synapse. The only way to gain extra synapses is through making mistakes. So all of the connections and pathways in your brain are due to making mistakes. The piece of paper representing their brain would not have any synapses at all if it hadn't been crumpled up. Those pathways are only there because they made a mistake.
With students, I then tell them to keep this picture of their brain in their folder, and every time they make a mistake in Maths, they should look at it and think about how their brain has just gained another synapse. They can even draw on another synapse each time they make a mistake, and by the end of the year they will be able to see how much their brain has grown, and how much progress they've made.
This is a great activity to do with a class that lacks confidence or is stuck in a fixed mindset. I did this with my year 11 intervention class last year, and I think it was a real turning point for them.
Please try this with a class and let me know what impact it has!
Emma x x x
PS I got this idea, plus many others, from the book Mathematical Mindsets by Jo Boaler.
Thursday, 14 May 2015
A Maths Game with a Psychological Twist
Today I'm going to tell you about one of my favourite ways to kill a spare fifteen minutes in a maths lesson. It's great because it requires absolutely no preparation, no prior knowledge, and no differentiation. All you need is a class set of mini whiteboards and pens.
Tell students that you're going to ask them to write down a number. It must be a whole number (no decimals, fractions or surds) and it must be zero or greater. They have to write this secretly and not show anyone their number. Tell them you are going to count down from 5. At 2, everyone's pens need to be down. At 1, everyone needs to show their boards. The winner, and this is the important bit, is the person who has the smallest number that no one else has.
This should be simple enough to understand, and if a student doesn't quite get it, they can at least write down a number and participate until it clicks.
When students hold up their boards, first look for zeros. If there's just one, the zero wins. If there's more than one, tell all the zeros to put their boards down and then look for ones. Repeat until you have found a number that only one person has written. They are the winner. This doesn't take very long at all, even with 32 students like my year 7 class.
I have found that this game intrigues students immediately. They have to try to predict what everyone else is thinking. I have found that I can usually predict who in the class will write zero. Zero rarely wins, and I've never seen it win in the first round. I've found that the winning numbers vary massively from class to class. The results are also very different when I've played this with adults. I'm sure there's a lot that can be explored here in terms of psychology (and economics, actually) but I'll leave that for you to think about.
I like to try to encourage strategic thinking and annoying my students at the same time by saying things like "oh, so three won that time. That means you are probably thinking about choosing three this time. But if other people think that too, you won't win, so you'd better choose four instead. But now that I've said that, you can't choose four, it's too obvious, so..." The students usually interrupt me at this point and beg me to shut up because I'm ruining their strategy.
The beauty of this game is it can be done with year 7s, year 13s, and even the Maths faculty as a pre-meeting warm-up. I have never met a class (or group of adults even) that doesn't enjoy this game.
A more mathematical version of this game is to have the same rules but this time the winner is the student whose number is the closest to the mean of all of the numbers. This is too difficult to do as a whole class, so I do this in table groups. The four students each choose a number, they calculate the mean, then the winner is whoever's closest. They keep a tally of how many times each person wins so they can declare a winner at the end of the fifteen minutes.
What's really nice about this variation is that students will be calculating means with much more enthusiasm and motivation than if they were meaningless numbers on a worksheet. Also, by playing this game and trying out different strategies, students begin to appreciate the nature of the mean. There will always be a student who will write down a million, thinking it will skew the mean towards them. However, if the other three numbers are low they still won't win.
You can also play this game with the median instead of the mean. It doesn't quite work with mode! Actually, maybe one of my amazing readers could come up with a way of making a mode variation. Comment below!
Try one of these games out when you have ten minutes to spare. Let me know how it goes by commenting below.
Emma x x x
Tell students that you're going to ask them to write down a number. It must be a whole number (no decimals, fractions or surds) and it must be zero or greater. They have to write this secretly and not show anyone their number. Tell them you are going to count down from 5. At 2, everyone's pens need to be down. At 1, everyone needs to show their boards. The winner, and this is the important bit, is the person who has the smallest number that no one else has.
This should be simple enough to understand, and if a student doesn't quite get it, they can at least write down a number and participate until it clicks.
When students hold up their boards, first look for zeros. If there's just one, the zero wins. If there's more than one, tell all the zeros to put their boards down and then look for ones. Repeat until you have found a number that only one person has written. They are the winner. This doesn't take very long at all, even with 32 students like my year 7 class.
I have found that this game intrigues students immediately. They have to try to predict what everyone else is thinking. I have found that I can usually predict who in the class will write zero. Zero rarely wins, and I've never seen it win in the first round. I've found that the winning numbers vary massively from class to class. The results are also very different when I've played this with adults. I'm sure there's a lot that can be explored here in terms of psychology (and economics, actually) but I'll leave that for you to think about.
I like to try to encourage strategic thinking and annoying my students at the same time by saying things like "oh, so three won that time. That means you are probably thinking about choosing three this time. But if other people think that too, you won't win, so you'd better choose four instead. But now that I've said that, you can't choose four, it's too obvious, so..." The students usually interrupt me at this point and beg me to shut up because I'm ruining their strategy.
The beauty of this game is it can be done with year 7s, year 13s, and even the Maths faculty as a pre-meeting warm-up. I have never met a class (or group of adults even) that doesn't enjoy this game.
A more mathematical version of this game is to have the same rules but this time the winner is the student whose number is the closest to the mean of all of the numbers. This is too difficult to do as a whole class, so I do this in table groups. The four students each choose a number, they calculate the mean, then the winner is whoever's closest. They keep a tally of how many times each person wins so they can declare a winner at the end of the fifteen minutes.
What's really nice about this variation is that students will be calculating means with much more enthusiasm and motivation than if they were meaningless numbers on a worksheet. Also, by playing this game and trying out different strategies, students begin to appreciate the nature of the mean. There will always be a student who will write down a million, thinking it will skew the mean towards them. However, if the other three numbers are low they still won't win.
You can also play this game with the median instead of the mean. It doesn't quite work with mode! Actually, maybe one of my amazing readers could come up with a way of making a mode variation. Comment below!
Try one of these games out when you have ten minutes to spare. Let me know how it goes by commenting below.
Emma x x x
Wednesday, 22 April 2015
Show Me Your Knowledge - An Alternative to Tests
I had become fed up of marking my year 10s' fortnightly assessments. They are aiming for a grade C, and following the Higher syllabus, which means the assessments are quite challenging for them. Marking takes a lot longer when the student has got almost everything wrong.
One day I decided to do something different. I offered my class the choice. They could either do the assessment as normal, or they could Show Me Their Knowledge (Patent Pending). I explained what I meant by this. I wanted them to pour out the contents of their brains on that particular topic (it was averages) onto an A3 sheet of paper. I told them they needed to make up their own examples to demonstrate all of the skills. I gave them a list of skills so they wouldn't forget anything: mean, mode and median from a list, mmm from a frequency table, mmm from a grouped frequency table.
What was quite interesting was all of the boys chose to take the normal test, whereas all of the girls chose to Show Me Their Knowledge. And the girls loved it, they were begging me to do this every assessment. This gender difference is worth exploring in more detail, so I will save that for a separate post.
The pages that were created were impressive. Most girls did an example for each of the three sections, and heavily annotated the solution with explanations. For example, "You find the midpoint here because you don't know the actual number" then an arrow from this led to, "To get the midpoint add them together and half it or just work it out by looking at it".
Some girls put up their hands at various points and said they didn't know how to explain a certain bit. I told them that means they maybe didn't understand that bit. I told them to leave a bubble or box where they wanted to write the explanation, so that they could see there was a gap in their understanding. I told them I would help them fill that gap in the follow-up lesson.
When it came to marking, I read through the explanations and tweaked some of the wording if it wasn't quite right. I did this very neatly because girls don't like you defacing their work (male teachers please take note of this). If I thought some bits needed further explanations, I drew bubbles or boxes (copying their style) but left them blank, or started the sentence for them. For example, I might have written, "If there are two numbers in the middle..." or "The question will usually tell you to r____ your answer...".
When I gave these back, they filled in the bits they could by themselves, then I talked to them on each table to explain any bits they didn't understand. They also talked to each other to fill in the gaps. By the end, they all had a poster explaining the whole chapter. That is now a very useful resource for their revision. The best looking one I could have copied and had laminated for future classes to use.
I will continue to use this method of assessment, although I will still do some normal tests too, as at the end of the day, they will have to sit a traditional exam, and they need to be prepared for this. I can't help but think the fact that girls prefer this method suggests that maybe they are disadvantaged by our exam system and that girls could do much better if we changed this. But again, that's a post for another day.
Emma x x x
One day I decided to do something different. I offered my class the choice. They could either do the assessment as normal, or they could Show Me Their Knowledge (Patent Pending). I explained what I meant by this. I wanted them to pour out the contents of their brains on that particular topic (it was averages) onto an A3 sheet of paper. I told them they needed to make up their own examples to demonstrate all of the skills. I gave them a list of skills so they wouldn't forget anything: mean, mode and median from a list, mmm from a frequency table, mmm from a grouped frequency table.
What was quite interesting was all of the boys chose to take the normal test, whereas all of the girls chose to Show Me Their Knowledge. And the girls loved it, they were begging me to do this every assessment. This gender difference is worth exploring in more detail, so I will save that for a separate post.
The pages that were created were impressive. Most girls did an example for each of the three sections, and heavily annotated the solution with explanations. For example, "You find the midpoint here because you don't know the actual number" then an arrow from this led to, "To get the midpoint add them together and half it or just work it out by looking at it".
Some girls put up their hands at various points and said they didn't know how to explain a certain bit. I told them that means they maybe didn't understand that bit. I told them to leave a bubble or box where they wanted to write the explanation, so that they could see there was a gap in their understanding. I told them I would help them fill that gap in the follow-up lesson.
When it came to marking, I read through the explanations and tweaked some of the wording if it wasn't quite right. I did this very neatly because girls don't like you defacing their work (male teachers please take note of this). If I thought some bits needed further explanations, I drew bubbles or boxes (copying their style) but left them blank, or started the sentence for them. For example, I might have written, "If there are two numbers in the middle..." or "The question will usually tell you to r____ your answer...".
When I gave these back, they filled in the bits they could by themselves, then I talked to them on each table to explain any bits they didn't understand. They also talked to each other to fill in the gaps. By the end, they all had a poster explaining the whole chapter. That is now a very useful resource for their revision. The best looking one I could have copied and had laminated for future classes to use.
I will continue to use this method of assessment, although I will still do some normal tests too, as at the end of the day, they will have to sit a traditional exam, and they need to be prepared for this. I can't help but think the fact that girls prefer this method suggests that maybe they are disadvantaged by our exam system and that girls could do much better if we changed this. But again, that's a post for another day.
Emma x x x
Wednesday, 15 October 2014
Another New Way to Teach Dividing Fractions
Remember my post from June 2013 about a new way to teach dividing fractions? Well the other day I came up with another new method!
You might be wondering why I need a new method anyway, when the normal method (flip it and times it) works so well. Here's why: because that method is not intuitive. Well, it is if you understand reciprocals properly, and that the multiplicative inverse of a is 1/a. We don't normally go into the axioms of fields though, when we teach year 8 fractions.
Here's the way my method works. Say you want to do 3/4 divided by 2/3.
The way I usually think of integer division in my head is to make it into a multiplication. So 20 divided by 4 becomes 4 times something is 20. And then I think of what the something is. I think this is the way many students think about division.
So applying that to my question:
But 2 times something makes 3 and 4 times something makes 4 is quite difficult. So what we'll do is find an equivalent fraction for 3/4 so that 2 goes into the numerator and 3 goes into the denominator.
Then we just have to work out 2 times something is 18 and 3 times something is 24. Easy!
I thought of this method because there was a question in the year 9 MEP textbook that my students came across that was something like 3/4 x = 5/7 and you had to solve for x. But not having done algebra recently, my year 9s didn't think to divide both sides by 3/4. This led to them trying to find the answer by the method above. Interestingly, this year 9 class is the same class (then in year 7) that provided inspiration for the previous blog post on this topic!
What do you think, a waste of time, or a nice way in?
Emma x x x
Saturday, 11 October 2014
Challenging Gifted and Talented Students Accidentally
This post is about challenging our most able young mathematicians.
I always thought I was good at giving students challenging learning activities. But on Friday I learnt something surprising. Students can be challenged a lot further than I ever thought.
I decided to use the UKMT's Team Maths Challenge resources from last year's competition to run an internal team maths competition with a class of high-ability year 8 students. The Team Maths Challenge is designed to be done by a team of two year 8s and two year 9s, and it takes place in March each year. Running it with just year 8s meant the students would have to work harder as they wouldn't have the older students to help them. Also it is only October, so they have got 5 months less experience than they should have when doing this competition, so that makes it even more challenging still.
I decided to only do the group challenge round and the crossnumber round, as I only had two hours. They started with the group challenge. They responded very enthusiastically, and there was much dialogue between the groups of four. The jottings they were doing were vast and full of impressive maths. When I went over to see what individuals were doing, I had a few comments saying it was hard, but most were too absorbed to even talk to me. One student, about two-thirds of the way through, announced that he had finally got the answer. The other teacher who was with me asked him, "What, you've spent all that time just answering one question?" (as there are ten questions altogether). The student gave the best reply I could ask for. He said, "Yes, but it was worth it".
They got all the way through the forty minutes without giving up, and when I announced there was only one minute to go, the room reverberated with pencil scratchings and the discussions became noticeably higher-pitched. I collected up the answer sheets. It was then that I noticed.
You see, there is a twist to this story. I mentioned above that the challenge would be difficult for the year 8s, for various reasons. But what I hadn't taken into account was that I had accidentally photocopied the wrong materials for them. Yes, I had in fact given them the Senior Team Maths Challenge. Yes, the one designed for year 12 and 13 A-Level students. Yes, the one that even students with targets of A and A* in A-Level maths find difficult. I gave them that.
So these intelligent little 12 year-olds sat for forty minutes tackling problems designed for those who have learnt a lot more mathematical techniques than they. Take a look at the problems I gave them here. They were able to access the questions, give them a good go, and even answer some of them correctly! Yes, many of the groups got one or two answers correct! And what is wonderful is that they weren't put off by the fact that there were several questions that they had no idea how to attempt. They just took the questions they felt they could make a start on, and ran with them. Like the boy who spent 25 minutes on one question - which was worth it. He never even checked with me afterwards to see if he had got it right. The satisfaction came from reaching a conclusion.
We may feel that we challenge our students. But this suggests that students can be challenged more than we give them credit for. What's great about the questions I gave them, was that they don't necessarily rely on knowledge of mathematical techniques (although I think some involved Pythagoras or Trig) so they can be accessed by all ages.
May I suggest you try this with your year 8s and see if you get a similar reaction? It would be interesting to see if they respond similarly or if they give up. The group that I did this with have been taught by a very skilled teacher whose strength is in getting students to be resilient and independent, so this may be why this worked.
Oh, and if you've never entered your students into the UKMT Team Maths Challenge, make sure you do this year!
Emma x x x
I always thought I was good at giving students challenging learning activities. But on Friday I learnt something surprising. Students can be challenged a lot further than I ever thought.
I decided to use the UKMT's Team Maths Challenge resources from last year's competition to run an internal team maths competition with a class of high-ability year 8 students. The Team Maths Challenge is designed to be done by a team of two year 8s and two year 9s, and it takes place in March each year. Running it with just year 8s meant the students would have to work harder as they wouldn't have the older students to help them. Also it is only October, so they have got 5 months less experience than they should have when doing this competition, so that makes it even more challenging still.
I decided to only do the group challenge round and the crossnumber round, as I only had two hours. They started with the group challenge. They responded very enthusiastically, and there was much dialogue between the groups of four. The jottings they were doing were vast and full of impressive maths. When I went over to see what individuals were doing, I had a few comments saying it was hard, but most were too absorbed to even talk to me. One student, about two-thirds of the way through, announced that he had finally got the answer. The other teacher who was with me asked him, "What, you've spent all that time just answering one question?" (as there are ten questions altogether). The student gave the best reply I could ask for. He said, "Yes, but it was worth it".
They got all the way through the forty minutes without giving up, and when I announced there was only one minute to go, the room reverberated with pencil scratchings and the discussions became noticeably higher-pitched. I collected up the answer sheets. It was then that I noticed.
You see, there is a twist to this story. I mentioned above that the challenge would be difficult for the year 8s, for various reasons. But what I hadn't taken into account was that I had accidentally photocopied the wrong materials for them. Yes, I had in fact given them the Senior Team Maths Challenge. Yes, the one designed for year 12 and 13 A-Level students. Yes, the one that even students with targets of A and A* in A-Level maths find difficult. I gave them that.
So these intelligent little 12 year-olds sat for forty minutes tackling problems designed for those who have learnt a lot more mathematical techniques than they. Take a look at the problems I gave them here. They were able to access the questions, give them a good go, and even answer some of them correctly! Yes, many of the groups got one or two answers correct! And what is wonderful is that they weren't put off by the fact that there were several questions that they had no idea how to attempt. They just took the questions they felt they could make a start on, and ran with them. Like the boy who spent 25 minutes on one question - which was worth it. He never even checked with me afterwards to see if he had got it right. The satisfaction came from reaching a conclusion.
We may feel that we challenge our students. But this suggests that students can be challenged more than we give them credit for. What's great about the questions I gave them, was that they don't necessarily rely on knowledge of mathematical techniques (although I think some involved Pythagoras or Trig) so they can be accessed by all ages.
May I suggest you try this with your year 8s and see if you get a similar reaction? It would be interesting to see if they respond similarly or if they give up. The group that I did this with have been taught by a very skilled teacher whose strength is in getting students to be resilient and independent, so this may be why this worked.
Oh, and if you've never entered your students into the UKMT Team Maths Challenge, make sure you do this year!
Emma x x x
Friday, 4 October 2013
7 Habits to Get Your Year 7s into
(Substitute "year sevens" with "seventh graders" if you're American, "S1s" if you're Scottish, or "first years" if you're posh/old/a wizard).
Year 7s are so cute, aren't they? So eager to learn, so willing to please. Some are actually shorter than me, which is nice. By Christmas they'll have had their growth spurts and be taller than me. That's why I like this term. (FYI if you want a mental picture of me, I'm 150cm tall, 50kg, and look a bit like Garth from Wayne's World but slightly more feminine).
The thing with year 7s is that they are lovely little blank slates. They're a fresh batch of play-doh just waiting to be moulded. At my academy, we keep our classes throughout their school career. So it's important that you get your year 7s into good habits early on, to make your life easier later.
Here are the good habits I'd like to get my year 7s into:
1) Leaving the answer as a fraction
To some students, an answer of 3/5 doesn't look finished - because you haven't actually carried out the division. They would much rather put 0.6 because it looks like a proper answer. We need to stamp this out! Fractions are infinitely superior to decimals. The use of fractions should be encouraged from day one. Don't you just hate A level students who convert all their fractions to decimals? Think ahead, teachers!
2) Lining up the equals signs
Some teachers are very anal about this and I admit I'm not really one of them. But it does make algebra look a lot more beautiful when there is neat line of =s down the page.
3) Drawing margins
Why do maths exercise books not have margins pre-printed like all the other exercise books?! It drives me mad having to remind students to draw margins every day. It's amazing how some of them still forget - even my top set year 11s! We need to try some Pavlovian conditioning to get them to automatically reach for a ruler and pencil as soon as they open their books.
4) Drawing diagrams
To solve any geometrical problem, the first step should be to draw a diagram. This is something good mathematicians do automatically. I don't know which is the cause and which is the effect, but it's worth getting our students into this habit.
5) Resilience
Perhaps the most important characteristic of a mathematician is resilience. Try something. If it doesn't work, try something else. Don't tippex out your first attempt. Don't sit there with a blank page because you're scared of writing something that's wrong. If we can instil this attitude into our youngest students, they will grow up to be good mathematicians, whatever their attainment level.
6) Using a calculator properly
Calculators are great. I can honestly say I haven't done bus-stop division with pen and paper for a good 10 years. Because I own a calculator. My computer has a calculator. My phone has a calculator. Even my tape measure has a built-in calculator. Don't diss calculators. However, some students become instantly stupid as soon as they pick one up. They don't question whatever answer it spits out. Students need to be taught to estimate the answer first to check if it's roughly right. Also, calculators are really sophisticated these days, and for example you can type in the entire quadratic formula in one go without pressing equals in between or using complicated nested brackets. Teach students how to do this. Teach them about the magical s<=>d key. Explain how the fraction key works. Get them to make frequent use of the "ans" key. And most importantly, get them to buy their own and bring it in every lesson. But make sure it's a Casio! (Sorry Sharp, but you make my life so difficult. Please stop making calculators.)
7) Taking pride in their exercise book
That's "jotter" if you're Scottish. Or "notebook" if you're American. (Or "parchment" if you're a wizard).
When an exercise book gets filled up, students are supposed to keep it. What do most students do? Throw it away. How awful is this? The problem is that many students see their exercise book as the place where they do work, not the place where they write down things to help them understand. Also some books are just horribly messy! I find that if your work is neat, you take more pride in your book, and hence you put more effort into your work. I know presentation is not about learning and hence presentation-focused comments is considered ineffective marking, but I think good presentation does lead to better learning.
Those are my personal picks. Are there any you would like to add?
Emma x x x
Year 7s are so cute, aren't they? So eager to learn, so willing to please. Some are actually shorter than me, which is nice. By Christmas they'll have had their growth spurts and be taller than me. That's why I like this term. (FYI if you want a mental picture of me, I'm 150cm tall, 50kg, and look a bit like Garth from Wayne's World but slightly more feminine).
The thing with year 7s is that they are lovely little blank slates. They're a fresh batch of play-doh just waiting to be moulded. At my academy, we keep our classes throughout their school career. So it's important that you get your year 7s into good habits early on, to make your life easier later.
Here are the good habits I'd like to get my year 7s into:
1) Leaving the answer as a fraction
To some students, an answer of 3/5 doesn't look finished - because you haven't actually carried out the division. They would much rather put 0.6 because it looks like a proper answer. We need to stamp this out! Fractions are infinitely superior to decimals. The use of fractions should be encouraged from day one. Don't you just hate A level students who convert all their fractions to decimals? Think ahead, teachers!
2) Lining up the equals signs
Some teachers are very anal about this and I admit I'm not really one of them. But it does make algebra look a lot more beautiful when there is neat line of =s down the page.
3) Drawing margins
Why do maths exercise books not have margins pre-printed like all the other exercise books?! It drives me mad having to remind students to draw margins every day. It's amazing how some of them still forget - even my top set year 11s! We need to try some Pavlovian conditioning to get them to automatically reach for a ruler and pencil as soon as they open their books.
4) Drawing diagrams
To solve any geometrical problem, the first step should be to draw a diagram. This is something good mathematicians do automatically. I don't know which is the cause and which is the effect, but it's worth getting our students into this habit.
5) Resilience
Perhaps the most important characteristic of a mathematician is resilience. Try something. If it doesn't work, try something else. Don't tippex out your first attempt. Don't sit there with a blank page because you're scared of writing something that's wrong. If we can instil this attitude into our youngest students, they will grow up to be good mathematicians, whatever their attainment level.
6) Using a calculator properly
Calculators are great. I can honestly say I haven't done bus-stop division with pen and paper for a good 10 years. Because I own a calculator. My computer has a calculator. My phone has a calculator. Even my tape measure has a built-in calculator. Don't diss calculators. However, some students become instantly stupid as soon as they pick one up. They don't question whatever answer it spits out. Students need to be taught to estimate the answer first to check if it's roughly right. Also, calculators are really sophisticated these days, and for example you can type in the entire quadratic formula in one go without pressing equals in between or using complicated nested brackets. Teach students how to do this. Teach them about the magical s<=>d key. Explain how the fraction key works. Get them to make frequent use of the "ans" key. And most importantly, get them to buy their own and bring it in every lesson. But make sure it's a Casio! (Sorry Sharp, but you make my life so difficult. Please stop making calculators.)
7) Taking pride in their exercise book
That's "jotter" if you're Scottish. Or "notebook" if you're American. (Or "parchment" if you're a wizard).
When an exercise book gets filled up, students are supposed to keep it. What do most students do? Throw it away. How awful is this? The problem is that many students see their exercise book as the place where they do work, not the place where they write down things to help them understand. Also some books are just horribly messy! I find that if your work is neat, you take more pride in your book, and hence you put more effort into your work. I know presentation is not about learning and hence presentation-focused comments is considered ineffective marking, but I think good presentation does lead to better learning.
Those are my personal picks. Are there any you would like to add?
Emma x x x
Friday, 28 June 2013
Introduction to Mechanics: Lesson Plan and a Speed Riddle
First, the riddle:
The first answer I got from the class was 90mph. Is that what you think the answer is? If you do, you're wrong. Sorry.
(30+90)/2 = 60, but this is not the average. You would spend longer driving at 30mph than at 90mph, and this average does not take that into account.
Hint Number One:
What is the definition of "average speed"? Total distance covered divided by total time taken.
At this point the calculators started to come out. Pens and paper had yet to make an appearance. I then started to get some bizarre answers like 2mph. This is an example of why a calculator without pen and paper is a dangerous thing.
Hint Number Two:
To average 60mph, how long should the entire journey take?
Speed = distance/time. So 60 = 2/t. So t = 2/60 hours. This would be 2 minutes. Have you worked out the answer yet?
Hint Number Three:
How long have they been travelling so far?
Speed = distance/time. So 30 = 1/t. So t = 1/30 hours which is... 2 minutes.
So it is impossible.
Surprised? I was. It still don't really see how it can be impossible. Surely if you go fast enough you can catch up? It seems wrong somehow.
Now, onto the rest of the lesson.
I borrowed from the science prep room a mechanical weighing scale. The kind you have in your bathroom. Side note: what is it about science prep room technicians that makes them so formidable? I have never returned a borrowed item so promptly!
Anyway, I put the scales on the floor and stood on them. I asked the class to look at the number displayed. I then pointed out I was wearing heavy clothes and block heels and a big watch and I'd just eaten lunch. Then I asked the class what would happen to the number if I put my hands on the back of the chair in front whilst standing on them. They correctly told me the scales would say I've lost weight. Then I asked if there was a way for me to make the scales think I've gained weight. That's a more interesting question. I'll leave you to think about that.
Then I asked them to consider a person weighing themself whilst in a lift. What would happen to your weight when the lift is going up? The class was split almost exactly in half on this. Some thought you would gain weight because the lift is pushing the scales into your feet. Some said you would lose weight because the lift is pushing your feet off the scales. They were also divided over what happens when the lift is going down.
Mechanics is a very sciency bit of maths, and when there's a debate in science there's only one way to settle it: an experiment! So I simply declared, "To the lift!" The students were surprised and I think a little bit excited. We all went to the lift and took it in turns to go in groups of four up to the third floor then down to the ground floor, then back.
We returned to the classroom and discussed our findings and tried to explain them.I won't tell you the result of our experiment, but if you ever get the opportunity to try it out, please do!
Next I held up a tennis ball and a basket ball (borrowed from the PE department, who are a lot less scary than the lab techs) and told them we were going to do another experiment. I asked them which ball they thought would hit the floor first if we dropped them at the same time from the same height.
Again, the class was divided. We discussed mass, surface area, rigidness, air resistance... It was a good discussion. And then we gleefully left the classroom to do our experiment. Half of us went to the top floor (the third floor, or the fourth floor if you're American), and the rest went down to the bottom. Our school is kind of open so that from the top floor you can see all the way to the bottom if you lean over the balcony on the inside. This made it ideal. (The building won an award recently for its awesome architecture).
There were a few students in the corridor working on the computers or printing stuff so we drew a bit of an audience. And when the two balls hit the ground (at the same time? Well, that would be telling...) we definitely drew some attention to ourselves! It was loud.
So we went back to the classroom and discussed our findings. We talked about how the experiment was kind of rubbish because there were too many variables. So I told them we were going to watch a better experiment where these variables were controlled. That's when I showed them this clip from Brainiac. Sorry about the Arabic subtitles.
That concluded the lesson. I think it was a great way to introduce the mechanics module and give them a bit of a taster. Hopefully it has made them excited to start their A level maths in September!
Emma x x x
You are driving along a 2-mile long bridge. You drive the first mile at an average speed of 30mph. You want your average speed for the whole bridge to be 60mph. What speed do you need to drive at for the second mile?This is how I began my mechanics taster session with next year's AS students. Have you worked it out yet?
The first answer I got from the class was 90mph. Is that what you think the answer is? If you do, you're wrong. Sorry.
(30+90)/2 = 60, but this is not the average. You would spend longer driving at 30mph than at 90mph, and this average does not take that into account.
Hint Number One:
What is the definition of "average speed"? Total distance covered divided by total time taken.
At this point the calculators started to come out. Pens and paper had yet to make an appearance. I then started to get some bizarre answers like 2mph. This is an example of why a calculator without pen and paper is a dangerous thing.
Hint Number Two:
To average 60mph, how long should the entire journey take?
Speed = distance/time. So 60 = 2/t. So t = 2/60 hours. This would be 2 minutes. Have you worked out the answer yet?
Hint Number Three:
How long have they been travelling so far?
Speed = distance/time. So 30 = 1/t. So t = 1/30 hours which is... 2 minutes.
So it is impossible.
Surprised? I was. It still don't really see how it can be impossible. Surely if you go fast enough you can catch up? It seems wrong somehow.
Now, onto the rest of the lesson.
I borrowed from the science prep room a mechanical weighing scale. The kind you have in your bathroom. Side note: what is it about science prep room technicians that makes them so formidable? I have never returned a borrowed item so promptly!
Anyway, I put the scales on the floor and stood on them. I asked the class to look at the number displayed. I then pointed out I was wearing heavy clothes and block heels and a big watch and I'd just eaten lunch. Then I asked the class what would happen to the number if I put my hands on the back of the chair in front whilst standing on them. They correctly told me the scales would say I've lost weight. Then I asked if there was a way for me to make the scales think I've gained weight. That's a more interesting question. I'll leave you to think about that.
Then I asked them to consider a person weighing themself whilst in a lift. What would happen to your weight when the lift is going up? The class was split almost exactly in half on this. Some thought you would gain weight because the lift is pushing the scales into your feet. Some said you would lose weight because the lift is pushing your feet off the scales. They were also divided over what happens when the lift is going down.
Mechanics is a very sciency bit of maths, and when there's a debate in science there's only one way to settle it: an experiment! So I simply declared, "To the lift!" The students were surprised and I think a little bit excited. We all went to the lift and took it in turns to go in groups of four up to the third floor then down to the ground floor, then back.
We returned to the classroom and discussed our findings and tried to explain them.I won't tell you the result of our experiment, but if you ever get the opportunity to try it out, please do!
Next I held up a tennis ball and a basket ball (borrowed from the PE department, who are a lot less scary than the lab techs) and told them we were going to do another experiment. I asked them which ball they thought would hit the floor first if we dropped them at the same time from the same height.
Again, the class was divided. We discussed mass, surface area, rigidness, air resistance... It was a good discussion. And then we gleefully left the classroom to do our experiment. Half of us went to the top floor (the third floor, or the fourth floor if you're American), and the rest went down to the bottom. Our school is kind of open so that from the top floor you can see all the way to the bottom if you lean over the balcony on the inside. This made it ideal. (The building won an award recently for its awesome architecture).
There were a few students in the corridor working on the computers or printing stuff so we drew a bit of an audience. And when the two balls hit the ground (at the same time? Well, that would be telling...) we definitely drew some attention to ourselves! It was loud.
So we went back to the classroom and discussed our findings. We talked about how the experiment was kind of rubbish because there were too many variables. So I told them we were going to watch a better experiment where these variables were controlled. That's when I showed them this clip from Brainiac. Sorry about the Arabic subtitles.
That concluded the lesson. I think it was a great way to introduce the mechanics module and give them a bit of a taster. Hopefully it has made them excited to start their A level maths in September!
Emma x x x
Thursday, 20 June 2013
A New Way to Teach Dividing Fractions
How do you divide a fraction by another fraction?
For example, how would you do something like this:
My guess is you would flip the second fraction upside down and then multiply like so:
If you are a maths teacher, is this how you teach students?
Do you think your students understand why this method works? And, be honest, do you understand why it works?
Well today in the maths office at my academy one of my colleagues showed us a new method he'd thought of.
It works like this:
I think this is a little bit more intuitive.
My colleague got the idea from one of his year sevens who had answered this question without showing any working out:
The answer is quite obviously three. How many quarters are there in three quarters? Three, duh. But I am quite certain most of my A* students would perform the technique of flipping and timesing without even thinking. My colleague was impressed that this student had used some common sense. He wondered whether the same idea could be applied to fractions with different denominators. It is a little bit less obvious that 21/28 divided by 20/28 is 21/20, but it's not entirely unbelievable. Whereas the "trick" of flipping and timesing can look a little bit like magic to some students.
I haven't tried teaching this method so I can't comment on its effectiveness yet. But as a mathematician it appeals to me. It's quite neat. And in case you were wondering, yes this works with algebraic fractions too.
If you're going to be teaching fractions soon, why not try this out? If you do, please let me know how it goes.
Do you think this is a good method?
Emma x x x
For example, how would you do something like this:
My guess is you would flip the second fraction upside down and then multiply like so:
If you are a maths teacher, is this how you teach students?
Do you think your students understand why this method works? And, be honest, do you understand why it works?
Well today in the maths office at my academy one of my colleagues showed us a new method he'd thought of.
It works like this:
I think this is a little bit more intuitive.
My colleague got the idea from one of his year sevens who had answered this question without showing any working out:
The answer is quite obviously three. How many quarters are there in three quarters? Three, duh. But I am quite certain most of my A* students would perform the technique of flipping and timesing without even thinking. My colleague was impressed that this student had used some common sense. He wondered whether the same idea could be applied to fractions with different denominators. It is a little bit less obvious that 21/28 divided by 20/28 is 21/20, but it's not entirely unbelievable. Whereas the "trick" of flipping and timesing can look a little bit like magic to some students.
I haven't tried teaching this method so I can't comment on its effectiveness yet. But as a mathematician it appeals to me. It's quite neat. And in case you were wondering, yes this works with algebraic fractions too.
If you're going to be teaching fractions soon, why not try this out? If you do, please let me know how it goes.
Do you think this is a good method?
Emma x x x
Thursday, 14 March 2013
Official Outstanding Lesson: Permutations and Combinations
In this post I am going to describe to you the lesson that I taught this week which was graded as Outstanding by an OfSTED inspector.
Obviously I can't guarantee that if you do this lesson you will get a 1, because a lot of it is in the delivery, and in how your students respond. There was a sort of magic spark in my lesson, which I think was there because of the good relationship I have with the group.
My lesson was structured as a series of problems relating to permutations and combinations. I didn't tell my students what perms and combs were until right near the end of the lesson. Not one of my students could have told you the learning objectives, until the very end. But who cares about learning objectives when you're busy doing PROPER MATHS? FYI, the only resources you need for this lesson are a deck of cards and, optionally, some mini whiteboards and pens.
1) The Dramatic Intro
Here is a rough transcript of what I said at the start of the lesson:
"All of you in here are extremely lucky. I am going to show you something no one has ever seen before. *dramatic pause*. This is a once in a lifetime experience, literally. In the whole history of the human race, no one has ever seen this before, and there's a very good chance that no one ever will. I'm not just doing this because we have a special visitor today (nervous glance at inspector) but because I love you all very much [I tend to babble when I get nervous]. Are you ready? *Another dramatic pause*"
Then I took out a deck of cards and shuffled them. Then I fanned them out and showed them all of the cards. Then I just looked at my students, paused, then said, "Amazing, yeah!" In my most excited voice.
There was kind of a confused awkward silence, followed by a chorus of "no"s. I feigned disbelief. Then I explained to them that that particular order of cards has never been seen before by anybody, living or dead, and probably won't be seen again for an extremely long time.
I then explained the maths. I said, let's assume that playing cards have been around for about 500 years, because that's roughly what Wikipedia suggests. Let's assume the population of the world has been a constant 6 billion for the past 500 years. And let's assume that everybody shuffles one deck of 52 cards every second. How many orders of cards would that make? I let them work it out. I gave them mini whiteboards to do their working out on, because pupils are much more likely to start attacking a problem if they're not worried about making mistakes. With MWBs they can erase all evidence of stupidity later. They worked out the answer (I leave that to you as an exercise!) and then I got them to calculate the number of possible orderings (permutations) of 52 cards. Both tables of pupils worked out very quickly that it would involve factorials. Again I will leave the answer to you as an exercise.
[*edited to add* I've just remembered a terrible joke I made. This is pretty cringey, but I'll include it anyway. A student asked the sensible question of what to do about leap years. I said to ignore leap years, and pretend that everyone gets the day off from shuffling cards once every four years. Then I said, "oh no, but that contradicts the song!" So they're all like, "What song?" and I said, "You know, Everyday I'm Shuffling!". Pretty good considering I thought of it on the spot! ]
We had a discussion about the hugeness of the result. The students were pretty amazed, I think. The inspector was too!
2) The iPhone Problem
I moved swiftly on to my next problem. Here is roughly how I introduced the problem:
"You're sitting with your friend, and their iPhone is on the table. They leave it there and get up to go to the toilet. You want access to their phone, because you want to frape them, or whatever [is frape an appropriate word to use during an observed lesson?] but they have a passcode on it. It's four digits, and you happen to know each digit is different, so there are no repeated numbers. The question is, do you have time to try every possible passcode before your friend comes back from the toilet?
The thing I like about this question is that students have to make certain assumptions in order to answer it. Like how long the friend will spend going to the toilet (actually that ends up being kind of irrelevant once you've worked out the answer), and how long it takes you to type in the four numbers. My two tables ended up with different lengths of time that it would take to try all the possibilities, because they made different assumptions.
You can have a little discussion then about phone security and whether a four-digit passcode is enough. That will take care of your PMSSC. I didn't really go into it that much though.
3) The Baby Chris Problem
"Chris' mum gave Chris five wooden alphabet blocks to play with. She gave him the letters A, H, M, S and T. She left the room to make a cup of tea, and when she came back, baby Chris had spelt out the word MATHS. So Chris' mum is going around saying he's a genius. Do you think her claim is valid?"
The students calculated the number of possibilities and decide whether the probability is low enough to claim he's a genius. The class decided it was just a coincidence.
So then I gave them part two:
"This time Chris' mum gives him all 26 alphabet blocks to play with. She leaves the room, and when she comes back, he has taken those same five blocks and spelt out MATHS. Now do you think her claim is valid?"
This is trickier to calculate, and the first thing many of my students wrote down was 26!. I had to question them a bit to steer them in the right direction. The probability is about 1 in 8 million, so we concluded that he probably is a genius.
At that point my inspector left, but I'll tell you the rest of the lesson anyway.
4) An exam style question
Each of the digits 3, 4, 5, 6 and 7 are written on a separate piece of card. Four of the cards are picked at random and then laid down to make a four digit number.
a) How many even numbers are possible?
b) How many odd numbers smaller than 5000 are possible?
I then formalised the idea of permutations slightly by showing them the notation. I didn't give them the formula though, because I find the formula is quite unhelpful. I think it's easier to just think about the question and work it out. Someone said, "Oh! That's what that's for! I always wondered that!" Meaning the key on the calculator.
I only spent a couple of minutes on that, and moved swiftly onto the next problem.
5) The lottery
My students have all memorised by now that the odds of winning the lottery are 1 in 14 million. I suppose I must go on about it a lot! But none of them has ever thought to work out where that number comes from. I think they were quite excited to actually be able to work it out. The key thing I wanted them to notice is that it is a different type of question because in the lottery, the order of the numbers doesn't matter, so it's not the same as finding permutations.
They worked out that they needed to divide by the number of arrangements of 6 numbers. Then it took them a little to work out what that was. The most common guess was 36.
I took their workings out and demonstrated that what they'd done was divide 49P6 by 6!. I said that finding the number of combinations will always be like that. Then I said, "If only there were a key on your calculator which could work it out for you!" At which point they all realised that what they'd just done was work out nCr. They'd used nCr in binomial expansions in core maths but hadn't realised (or forgotten) the connection with combinatorics. They were very relieved that there was a shortcut! But I think it is so important to show them where the formula comes from. I proved the formula using the nPr formula that we'd already derived.
6) The Cricket Team Problem
This was just a question I took from the textbook. A cricket team consists of 6 batsmen, 4 bowlers, and 1 wicket-keeper.
So there you go, my outstanding lesson on perms and combs. I really enjoyed teaching it, and I think my students enjoyed it too.
If you decide to use this lesson, please let me know in the comments below how it went for you.
Outstanding Emma x x x
Obviously I can't guarantee that if you do this lesson you will get a 1, because a lot of it is in the delivery, and in how your students respond. There was a sort of magic spark in my lesson, which I think was there because of the good relationship I have with the group.
My lesson was structured as a series of problems relating to permutations and combinations. I didn't tell my students what perms and combs were until right near the end of the lesson. Not one of my students could have told you the learning objectives, until the very end. But who cares about learning objectives when you're busy doing PROPER MATHS? FYI, the only resources you need for this lesson are a deck of cards and, optionally, some mini whiteboards and pens.
1) The Dramatic Intro
Here is a rough transcript of what I said at the start of the lesson:
"All of you in here are extremely lucky. I am going to show you something no one has ever seen before. *dramatic pause*. This is a once in a lifetime experience, literally. In the whole history of the human race, no one has ever seen this before, and there's a very good chance that no one ever will. I'm not just doing this because we have a special visitor today (nervous glance at inspector) but because I love you all very much [I tend to babble when I get nervous]. Are you ready? *Another dramatic pause*"
Then I took out a deck of cards and shuffled them. Then I fanned them out and showed them all of the cards. Then I just looked at my students, paused, then said, "Amazing, yeah!" In my most excited voice.
There was kind of a confused awkward silence, followed by a chorus of "no"s. I feigned disbelief. Then I explained to them that that particular order of cards has never been seen before by anybody, living or dead, and probably won't be seen again for an extremely long time.
I then explained the maths. I said, let's assume that playing cards have been around for about 500 years, because that's roughly what Wikipedia suggests. Let's assume the population of the world has been a constant 6 billion for the past 500 years. And let's assume that everybody shuffles one deck of 52 cards every second. How many orders of cards would that make? I let them work it out. I gave them mini whiteboards to do their working out on, because pupils are much more likely to start attacking a problem if they're not worried about making mistakes. With MWBs they can erase all evidence of stupidity later. They worked out the answer (I leave that to you as an exercise!) and then I got them to calculate the number of possible orderings (permutations) of 52 cards. Both tables of pupils worked out very quickly that it would involve factorials. Again I will leave the answer to you as an exercise.
[*edited to add* I've just remembered a terrible joke I made. This is pretty cringey, but I'll include it anyway. A student asked the sensible question of what to do about leap years. I said to ignore leap years, and pretend that everyone gets the day off from shuffling cards once every four years. Then I said, "oh no, but that contradicts the song!" So they're all like, "What song?" and I said, "You know, Everyday I'm Shuffling!". Pretty good considering I thought of it on the spot! ]
We had a discussion about the hugeness of the result. The students were pretty amazed, I think. The inspector was too!
2) The iPhone Problem
I moved swiftly on to my next problem. Here is roughly how I introduced the problem:
"You're sitting with your friend, and their iPhone is on the table. They leave it there and get up to go to the toilet. You want access to their phone, because you want to frape them, or whatever [is frape an appropriate word to use during an observed lesson?] but they have a passcode on it. It's four digits, and you happen to know each digit is different, so there are no repeated numbers. The question is, do you have time to try every possible passcode before your friend comes back from the toilet?
The thing I like about this question is that students have to make certain assumptions in order to answer it. Like how long the friend will spend going to the toilet (actually that ends up being kind of irrelevant once you've worked out the answer), and how long it takes you to type in the four numbers. My two tables ended up with different lengths of time that it would take to try all the possibilities, because they made different assumptions.
You can have a little discussion then about phone security and whether a four-digit passcode is enough. That will take care of your PMSSC. I didn't really go into it that much though.
3) The Baby Chris Problem
"Chris' mum gave Chris five wooden alphabet blocks to play with. She gave him the letters A, H, M, S and T. She left the room to make a cup of tea, and when she came back, baby Chris had spelt out the word MATHS. So Chris' mum is going around saying he's a genius. Do you think her claim is valid?"
The students calculated the number of possibilities and decide whether the probability is low enough to claim he's a genius. The class decided it was just a coincidence.
So then I gave them part two:
"This time Chris' mum gives him all 26 alphabet blocks to play with. She leaves the room, and when she comes back, he has taken those same five blocks and spelt out MATHS. Now do you think her claim is valid?"
This is trickier to calculate, and the first thing many of my students wrote down was 26!. I had to question them a bit to steer them in the right direction. The probability is about 1 in 8 million, so we concluded that he probably is a genius.
At that point my inspector left, but I'll tell you the rest of the lesson anyway.
4) An exam style question
Each of the digits 3, 4, 5, 6 and 7 are written on a separate piece of card. Four of the cards are picked at random and then laid down to make a four digit number.
a) How many even numbers are possible?
b) How many odd numbers smaller than 5000 are possible?
I had to help them quite a bit with this question, but it's a nice one. FYI, the answers are 48 and 30 respectively.
I then formalised the idea of permutations slightly by showing them the notation. I didn't give them the formula though, because I find the formula is quite unhelpful. I think it's easier to just think about the question and work it out. Someone said, "Oh! That's what that's for! I always wondered that!" Meaning the key on the calculator.
I only spent a couple of minutes on that, and moved swiftly onto the next problem.
5) The lottery
My students have all memorised by now that the odds of winning the lottery are 1 in 14 million. I suppose I must go on about it a lot! But none of them has ever thought to work out where that number comes from. I think they were quite excited to actually be able to work it out. The key thing I wanted them to notice is that it is a different type of question because in the lottery, the order of the numbers doesn't matter, so it's not the same as finding permutations.
They worked out that they needed to divide by the number of arrangements of 6 numbers. Then it took them a little to work out what that was. The most common guess was 36.
I took their workings out and demonstrated that what they'd done was divide 49P6 by 6!. I said that finding the number of combinations will always be like that. Then I said, "If only there were a key on your calculator which could work it out for you!" At which point they all realised that what they'd just done was work out nCr. They'd used nCr in binomial expansions in core maths but hadn't realised (or forgotten) the connection with combinatorics. They were very relieved that there was a shortcut! But I think it is so important to show them where the formula comes from. I proved the formula using the nPr formula that we'd already derived.
6) The Cricket Team Problem
This was just a question I took from the textbook. A cricket team consists of 6 batsmen, 4 bowlers, and 1 wicket-keeper.
These need to be chosen from a
group of 18 cricketers comprising 9 batsmen, 7 bowlers and 2 wicket-keepers.
How many different ways are there
of choosing the team?
This question is actually really easy using nCr, but it was useful because they could see how to use nCr rather than working it out fully themselves. This was a good question to end on, because they found it quick and easy, which felt like sort of a reward for having derived the formulae earlier.
Then I set the homework. I didn't really feel like I needed a plenary, because we'd been plenarying all the way through.
So there you go, my outstanding lesson on perms and combs. I really enjoyed teaching it, and I think my students enjoyed it too.
If you decide to use this lesson, please let me know in the comments below how it went for you.
Outstanding Emma x x x
Thursday, 23 August 2012
Playing to Lose
(I was contemplating calling this post "Every Loser Wins", but I hate that song).
As all of you should know by now, I'm a winner. I win things. So when my friend Stacy challenged me to a game of Noughts and Crosses today, I was prepared to win. Until she said this:
"The winner is the person who loses. The aim is to NOT get three in a row".
As you can imagine, I was flummoxed. Winning is easy, losing is difficult. She told me to go first, which, now that I think about it, was a sneaky way of increasing my chance of winning - and hence losing. I obviously avoided the middle square. I figured the corner squares were also too good to use. So I opted for a side square. That decision was pretty easy. The rest was not so trivial. Stacy lost the game, and hence won. Apparently I am such a winner that even when I'm trying to lose, I win.
Please please please grab your partner/your child/your flatmate/the guy next to you on the bus and challenge them to lose a game of Noughts and Crosses. It's the only way you can really think about the strategy involved.
Have fun!
Emma x x x
As all of you should know by now, I'm a winner. I win things. So when my friend Stacy challenged me to a game of Noughts and Crosses today, I was prepared to win. Until she said this:
"The winner is the person who loses. The aim is to NOT get three in a row".
As you can imagine, I was flummoxed. Winning is easy, losing is difficult. She told me to go first, which, now that I think about it, was a sneaky way of increasing my chance of winning - and hence losing. I obviously avoided the middle square. I figured the corner squares were also too good to use. So I opted for a side square. That decision was pretty easy. The rest was not so trivial. Stacy lost the game, and hence won. Apparently I am such a winner that even when I'm trying to lose, I win.
Please please please grab your partner/your child/your flatmate/the guy next to you on the bus and challenge them to lose a game of Noughts and Crosses. It's the only way you can really think about the strategy involved.
Have fun!
Emma x x x
Wednesday, 15 August 2012
Why We Actually Won The Olympics
Before this year, I'd never watched so much as one minute of an
Olympic event. Not even China's opening ceremony last time (I have a
phobia of fireworks). I am an incredibly competitive person, and the
idea of watching a massive competition into which I'm not even entered,
really holds no interest to me. Who cares what happens? The bottom line
is, I'm not coming home with a shiny medal around my neck.
But then I went to spend a week in the house of a family who were interested in the Olympics, and (this is important) actually had television (we stopped buying our TV licence several years ago as our favourite shows only air in Japan anyway). I didn't have much else to do that week, no longer being a fifteen minute walk from a city centre (and hence, shops) so I found myself watching the Olympics A LOT. And I found myself enjoying it. I finally got it: sure, I wasn't going to win anything, but my country would, and in a way, me and my country are one and the same. I suddenly understood what Nick Hornby was on about in Fever Pitch (OK, I never actually read it, because, I enjoyed About A Boy and all, but seriously, a book entirely about football?) and I felt that I was Team GB. It was a great feeling.
And then we didn't win.
Those of you who know me well know that I don't particularly enjoy not winning. Please note this only applies to competitions I deem worthy: me always failing to win at Laser Quest, Mario Kart, and anything that involves Geography does not bother me in the slightest. Because they're stupid. But the Olympics was something I wanted to win, something I thought I was really good at (by me, I mean Team GB of course). I wasn't going to let stupid countries like China and the USA beat me. It's not fair: they already have crazy cheap designer clothes and fifty flavours of Pop Tarts, why do they have to have this too?!?
Naturally I did what I always do when I am told I have not won. I set about moving the goalposts, scouring the rule book, and examining the data to try and find a way that means I actually won. But whichever way you count the medals, we were not higher than third place.
There was only one thing I could do: "To the math cave!!"
I'm sure that anyone who is even slightly mathematically minded automatically saw a problem with the scoring system. I don't mean the fact that it's all done on the number of Golds and not on a 5-3-1 system, because we still wouldn't win then. I'm sure that when most people looked at the scoreboard, their first thought was "Well of course those two countries are winning, they're absolutely massive". China and the USA are well known as having huge populations, along with India, Brazil and Indonesia (interesting point: this is probably the one geographical fact that I actually know without having to look it up. Maths teachers from my Academy will know why). Is it really fair that we are being compared to countries with 5 or even 21 times more people than us? (I had to look that up). If we randomly divided the USA into five pieces, and selected the best athletes from each piece, would they still be the best? And if we cut China into 21 pieces? I don't think so.
So, here comes the maths: I have created a table of the competing countries, and their number of golds per 10 million people. This is like doing population density (urgh, there is way too much geography in this post) but instead it's gold density. Except out of the number of people, not the area of land. Hmm. OK.
As you can see, we're actually 3 times better than the USA, and loads better than China. So, we actually should have w... Wait a minute. What?!?! Looking further down the list I can see some pretty disturbing numbers. I'd better give you the whole table. This is it ordered by golds:
And this is it ordered by gold density:
So Grenada (a country I'm not even sure I've heard
of) has stormed the league table with a whopping 95, making our 4.7 look
pretty pathetic. We don't even make the top ten! Random places like
Lithuania and Slovenia (who?) have done well, and places that I'm sure
only exist for people to take holidays in have managed to come second
and third!
I was really hoping that maths would prove we won the Olympics. I suppose this is why I should work out all the data BEFORE writing a blog post. It's not my fault: how was I supposed to know that our little Island actually has a lot of people on it? I felt so sure we were one of the smallest countries. I mean, we only have two types of Skittles over here! Big countries have at least five (including the sour ones, mmmm)! I am outraged.
And before anyone mentions it: the failure of this post has absolutely nothing to do with my lack of geographical knowledge.
If you're worried the kids will google and find this blog and get the answer from here, fear not. As if any teenager is going to read through this long, dull piece! I'm surprised you've got this far, to be frank.
That last bout of googling I did to find out the main exports and national bird of Grenada revealed something interesting to me: Grenada is a Commonwealth country. Do you know what that means? Yep, it's owned by Queen Elizabeth. Who? Yep, the Queen of England. So technically, technically, we won the Olympics. I knew it!!!
Emma x x x
But then I went to spend a week in the house of a family who were interested in the Olympics, and (this is important) actually had television (we stopped buying our TV licence several years ago as our favourite shows only air in Japan anyway). I didn't have much else to do that week, no longer being a fifteen minute walk from a city centre (and hence, shops) so I found myself watching the Olympics A LOT. And I found myself enjoying it. I finally got it: sure, I wasn't going to win anything, but my country would, and in a way, me and my country are one and the same. I suddenly understood what Nick Hornby was on about in Fever Pitch (OK, I never actually read it, because, I enjoyed About A Boy and all, but seriously, a book entirely about football?) and I felt that I was Team GB. It was a great feeling.
And then we didn't win.
Those of you who know me well know that I don't particularly enjoy not winning. Please note this only applies to competitions I deem worthy: me always failing to win at Laser Quest, Mario Kart, and anything that involves Geography does not bother me in the slightest. Because they're stupid. But the Olympics was something I wanted to win, something I thought I was really good at (by me, I mean Team GB of course). I wasn't going to let stupid countries like China and the USA beat me. It's not fair: they already have crazy cheap designer clothes and fifty flavours of Pop Tarts, why do they have to have this too?!?
Naturally I did what I always do when I am told I have not won. I set about moving the goalposts, scouring the rule book, and examining the data to try and find a way that means I actually won. But whichever way you count the medals, we were not higher than third place.
| Rank by Gold | Country | Gold | Silver | Bronze | Total |
|---|---|---|---|---|---|
| 1 |  United States of America | 46 | 29 | 29 | 104 |
| 2 |  People's Republic of China | 38 | 27 | 23 | 88 |
| 3 |  Great Britain | 29 | 17 | 19 | 65 |
There was only one thing I could do: "To the math cave!!"
I'm sure that anyone who is even slightly mathematically minded automatically saw a problem with the scoring system. I don't mean the fact that it's all done on the number of Golds and not on a 5-3-1 system, because we still wouldn't win then. I'm sure that when most people looked at the scoreboard, their first thought was "Well of course those two countries are winning, they're absolutely massive". China and the USA are well known as having huge populations, along with India, Brazil and Indonesia (interesting point: this is probably the one geographical fact that I actually know without having to look it up. Maths teachers from my Academy will know why). Is it really fair that we are being compared to countries with 5 or even 21 times more people than us? (I had to look that up). If we randomly divided the USA into five pieces, and selected the best athletes from each piece, would they still be the best? And if we cut China into 21 pieces? I don't think so.
So, here comes the maths: I have created a table of the competing countries, and their number of golds per 10 million people. This is like doing population density (urgh, there is way too much geography in this post) but instead it's gold density. Except out of the number of people, not the area of land. Hmm. OK.
| Country | Golds | Population (10 millions to 3sf) | G/10mill (2 dp) |
| USA | 46 | 31.5 | 1.46 |
| China | 38 | 135 | 0.28 |
| Great Britain | 29 | 6.23 | 4.65 |
As you can see, we're actually 3 times better than the USA, and loads better than China. So, we actually should have w... Wait a minute. What?!?! Looking further down the list I can see some pretty disturbing numbers. I'd better give you the whole table. This is it ordered by golds:
| Country | Golds | Population (10 millions to 3sf) | G/10mill (2 dp) |
| USA | 46 | 31.5 | 1.46 |
| China | 38 | 135 | 0.28 |
| Great Britain | 29 | 6.23 | 4.65 |
| Russia | 24 | 14.3 | 1.68 |
| North Korea | 13 | 2.46 | 5.28 |
| France | 11 | 6.54 | 1.68 |
| Germany | 11 | 8.19 | 1.34 |
| Hungary | 8 | 0.996 | 8.03 |
| Italy | 8 | 6.08 | 1.32 |
| Kazakhstan | 7 | 1.68 | 4.17 |
| Australia | 7 | 2.27 | 3.08 |
| Japan | 7 | 12.8 | 0.55 |
| New Zealand | 6 | 0.443 | 13.54 |
| Netherlands | 6 | 1.68 | 3.57 |
| Ukraine | 6 | 4.56 | 1.32 |
| Cuba | 5 | 1.12 | 4.46 |
| Jamaica | 4 | 0.271 | 14.76 |
| Czech Republic | 4 | 1.05 | 3.81 |
| South Korea | 4 | 5 | 0.80 |
| Iran | 4 | 7.51 | 0.53 |
| Croatia | 3 | 0.429 | 6.99 |
| Spain | 3 | 4.62 | 0.65 |
| South Africa | 3 | 5.06 | 0.59 |
| Ethiopia | 3 | 8.43 | 0.36 |
| Brazil | 3 | 19.2 | 0.16 |
| Lithuania | 2 | 0.319 | 6.27 |
| Norway | 2 | 0.503 | 3.98 |
| Denmark | 2 | 0.558 | 3.58 |
| Switzerland | 2 | 0.795 | 2.52 |
| Azerbaijan | 2 | 0.924 | 2.16 |
| Belarus | 2 | 0.946 | 2.11 |
| Romania | 2 | 1.9 | 1.05 |
| Poland | 2 | 3.85 | 0.52 |
| Kenya | 2 | 4.27 | 0.47 |
| Turkey | 2 | 7.47 | 0.27 |
| Grenada | 1 | 0.0105 | 95.24 |
| Bahamas | 1 | 0.035 | 28.57 |
| Trinidad and Tobego | 1 | 0.132 | 7.58 |
| Slovenia | 1 | 0.206 | 4.85 |
| Latvia | 1 | 0.207 | 4.83 |
| Georgia | 1 | 0.45 | 2.22 |
| Ireland | 1 | 0.459 | 2.18 |
| Serbia | 1 | 0.712 | 1.40 |
| Dominican Republic | 1 | 0.945 | 1.06 |
| Sweden | 1 | 0.951 | 1.05 |
| Tunisia | 1 | 1.07 | 0.93 |
| Venezuela | 1 | 2.72 | 0.37 |
| Uzbekistan | 1 | 2.91 | 0.34 |
| Uganda | 1 | 3.29 | 0.30 |
| Canada | 1 | 3.49 | 0.29 |
| Algeria | 1 | 3.71 | 0.27 |
| Argentina | 1 | 4.01 | 0.25 |
| Colombia | 1 | 4.67 | 0.21 |
| Mexico | 1 | 11.2 | 0.09 |
And this is it ordered by gold density:
| Country | Golds | Population (10 millions to 3sf) | G/10mill (2 dp) |
| Grenada | 1 | 0.0105 | 95.24 |
| Bahamas | 1 | 0.035 | 28.57 |
| Jamaica | 4 | 0.271 | 14.76 |
| New Zealand | 6 | 0.443 | 13.54 |
| Hungary | 8 | 0.996 | 8.03 |
| Trinidad and Tobego | 1 | 0.132 | 7.58 |
| Croatia | 3 | 0.429 | 6.99 |
| Lithuania | 2 | 0.319 | 6.27 |
| North Korea | 13 | 2.46 | 5.28 |
| Slovenia | 1 | 0.206 | 4.85 |
| Latvia | 1 | 0.207 | 4.83 |
| Great Britain | 29 | 6.23 | 4.65 |
| Cuba | 5 | 1.12 | 4.46 |
| Kazakhstan | 7 | 1.68 | 4.17 |
| Norway | 2 | 0.503 | 3.98 |
| Czech Republic | 4 | 1.05 | 3.81 |
| Denmark | 2 | 0.558 | 3.58 |
| Netherlands | 6 | 1.68 | 3.57 |
| Australia | 7 | 2.27 | 3.08 |
| Switzerland | 2 | 0.795 | 2.52 |
| Georgia | 1 | 0.45 | 2.22 |
| Ireland | 1 | 0.459 | 2.18 |
| Azerbaijan | 2 | 0.924 | 2.16 |
| Belarus | 2 | 0.946 | 2.11 |
| France | 11 | 6.54 | 1.68 |
| Russia | 24 | 14.3 | 1.68 |
| USA | 46 | 31.5 | 1.46 |
| Serbia | 1 | 0.712 | 1.40 |
| Germany | 11 | 8.19 | 1.34 |
| Italy | 8 | 6.08 | 1.32 |
| Ukraine | 6 | 4.56 | 1.32 |
| Dominican Republic | 1 | 0.945 | 1.06 |
| Romania | 2 | 1.9 | 1.05 |
| Sweden | 1 | 0.951 | 1.05 |
| Tunisia | 1 | 1.07 | 0.93 |
| South Korea | 4 | 5 | 0.80 |
| Spain | 3 | 4.62 | 0.65 |
| South Africa | 3 | 5.06 | 0.59 |
| Japan | 7 | 12.8 | 0.55 |
| Iran | 4 | 7.51 | 0.53 |
| Poland | 2 | 3.85 | 0.52 |
| Kenya | 2 | 4.27 | 0.47 |
| Venezuela | 1 | 2.72 | 0.37 |
| Ethiopia | 3 | 8.43 | 0.36 |
| Uzbekistan | 1 | 2.91 | 0.34 |
| Uganda | 1 | 3.29 | 0.30 |
| Canada | 1 | 3.49 | 0.29 |
| China | 38 | 135 | 0.28 |
| Algeria | 1 | 3.71 | 0.27 |
| Turkey | 2 | 7.47 | 0.27 |
| Argentina | 1 | 4.01 | 0.25 |
| Colombia | 1 | 4.67 | 0.21 |
| Brazil | 3 | 19.2 | 0.16 |
| Mexico | 1 | 11.2 | 0.09 |
I was really hoping that maths would prove we won the Olympics. I suppose this is why I should work out all the data BEFORE writing a blog post. It's not my fault: how was I supposed to know that our little Island actually has a lot of people on it? I felt so sure we were one of the smallest countries. I mean, we only have two types of Skittles over here! Big countries have at least five (including the sour ones, mmmm)! I am outraged.
And before anyone mentions it: the failure of this post has absolutely nothing to do with my lack of geographical knowledge.
The Point of This Post
This post has been a cluster-fudge of poor formatting and lazy researching, so you probably want the payoff now. Well here it is: whatever country you're in, think about doing the following activity at the start of the new term. Announce to the class: "I don't know what you've heard on TV, but Grenada actually won the Olympics". They'll be like, what? Show some pictures of the beautiful country of Grenada, and pictures of, uh... *googles* nutmeg and mace and the uh... Grenada dove. Then, with or without giving them any extra information or instructions, but giving them access to the internet, get them to find out why Grenada actually won. I'm sure that intelligent pupils will be able to work it out, and you can give them lots of hints if need be. Extension: get them to prove that another country actually won the Olympics (e.g. by looking at the number of golds per GDP or something).If you're worried the kids will google and find this blog and get the answer from here, fear not. As if any teenager is going to read through this long, dull piece! I'm surprised you've got this far, to be frank.
My Final Conclusion
That last bout of googling I did to find out the main exports and national bird of Grenada revealed something interesting to me: Grenada is a Commonwealth country. Do you know what that means? Yep, it's owned by Queen Elizabeth. Who? Yep, the Queen of England. So technically, technically, we won the Olympics. I knew it!!!
Emma x x x
Saturday, 16 June 2012
A Grouping Activity
In today's blog post, I'm going to be sharing with you one of the first classroom activities I was ever taught. As a would-be teacher I mean, not as a student.
It was the second day of my PGCE, the first day that we were separated into subject groups, although a lot of us maths lot managed to find each other on the first day anyway. I suppose geeks, like insects, give off pheromones that signal to others of the same species.
We were each given a number. For the sake of simplicity, let's say we were given the numbers 1 to 36. As memory serves, there were actually over forty of us. But let's not let accuracy get in the way of a good story.
On each of the six tables there was a piece of paper saying one of the following:
Prime numbers
Square numbers
Triangle numbers
Numbers greater than 20
Even Numbers
Numbers with more than four factors.
We had to join a group corresponding to our number. As I recall my number was 3. I could have joined either the prime group or the triangle group. Some numbers could have belonged to three different groups. The number 5, for example, had no choice in the matter. The catch was this: every group had to have exactly six members.
I could tell you the solution to this problem, but I'll leave that for you to work out. Feel free to email me (nqtpi@gmail.com) if you get stuck. The one hint I will give you is that a girl named Lydia worked out pretty quickly that because both of our numbers were triangular but not square, we would have to be in the triangle group. And that is the story of how how me and Lydia became friends* and how Team Hopper (previously known as Team Triangle Numbers) was formed.
*OK not really, but I repeat: let's not let accuracy get in the way of a good story.
I think this activity would be a cool way of getting a class of pupils into groups. It's pretty challenging, so maybe best saved for A level classes? Obviously it will need adapting, as if you have 36 students in your A level maths class, you are clearly working at the wrong school.
Emma x x x
PS My blogiversary is coming up, so expect some interesting posts in the near future.
It was the second day of my PGCE, the first day that we were separated into subject groups, although a lot of us maths lot managed to find each other on the first day anyway. I suppose geeks, like insects, give off pheromones that signal to others of the same species.
We were each given a number. For the sake of simplicity, let's say we were given the numbers 1 to 36. As memory serves, there were actually over forty of us. But let's not let accuracy get in the way of a good story.
On each of the six tables there was a piece of paper saying one of the following:
Prime numbers
Square numbers
Triangle numbers
Numbers greater than 20
Even Numbers
Numbers with more than four factors.
We had to join a group corresponding to our number. As I recall my number was 3. I could have joined either the prime group or the triangle group. Some numbers could have belonged to three different groups. The number 5, for example, had no choice in the matter. The catch was this: every group had to have exactly six members.
I could tell you the solution to this problem, but I'll leave that for you to work out. Feel free to email me (nqtpi@gmail.com) if you get stuck. The one hint I will give you is that a girl named Lydia worked out pretty quickly that because both of our numbers were triangular but not square, we would have to be in the triangle group. And that is the story of how how me and Lydia became friends* and how Team Hopper (previously known as Team Triangle Numbers) was formed.
*OK not really, but I repeat: let's not let accuracy get in the way of a good story.
I think this activity would be a cool way of getting a class of pupils into groups. It's pretty challenging, so maybe best saved for A level classes? Obviously it will need adapting, as if you have 36 students in your A level maths class, you are clearly working at the wrong school.
Emma x x x
PS My blogiversary is coming up, so expect some interesting posts in the near future.
Thursday, 22 March 2012
A Counting Game
Every so often I'll post about an activity I did or a technique I used that was good. I find I often forget these good things when I should be reusing them. Hopefully by recording them on the blog I'll remember them.
So, what I'm sharing with you this week is an activity I did with my year 7 class. I told them they were going to have a little competition (which was met with a few "YES!"s), and that it would be a competition against me. This excited them because they know I'm both extremely competitive and extremely clever. They would love the opportunity to take me down a peg or two (as would most people, probably).
The competition was this: I will play against one person at a time. We will start from zero, and we will take it inturns to add either 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10. The winner is the person who reaches 100. It's very similar to 21 dares, or those games where you have to avoid being the one who takes the last matchstick.
Now obviously, there is a trick that means you always win. Work this out for yourself, I don't want to spoil your fun! (Hint: whoever gets up to 89 has won, because whatever the other person says, you will be able to reach 100 on your next go. By extension, whoever gets to 78 has won, as they can get to 89, and so on). I took on about 7 students in total and beat all of them. As I did it, pupils started to notice things. This is what the pupils discovered, in order:
1) If Miss says 89, then she's won
2) Whoever goes first always wins (false!)
3) You need to stop her getting to 89
4) Get to 78, because then she can't go to 89!
I then extended them to realise that there were certain numbers that you should always try to aim for to make sure you win, and they spotted the pattern of these numbers. On the next go, a student beat me. And as much as it pains me to admit it, he beat me fair and square (I got my numbers mixed up). You can imagine how happy that made the class: "We beat Miss, and she got bare A*s!"
The class was then desperate to show off this newfound skill by challenging the head of maths to a competition. This hasn't been arranged yet but I think it's a really nice idea. I hope they beat him!
You can obviously extend the activity by considering a different target instead of 100, and different numbers that you're allowed to add.
I'm going to do the same activity with my other classes to see how it turns out. If you try out this activity, please let me know how it goes!
Emma x x x
So, what I'm sharing with you this week is an activity I did with my year 7 class. I told them they were going to have a little competition (which was met with a few "YES!"s), and that it would be a competition against me. This excited them because they know I'm both extremely competitive and extremely clever. They would love the opportunity to take me down a peg or two (as would most people, probably).
The competition was this: I will play against one person at a time. We will start from zero, and we will take it inturns to add either 1, 2, 3, 4, 5, 6, 7, 8, 9 or 10. The winner is the person who reaches 100. It's very similar to 21 dares, or those games where you have to avoid being the one who takes the last matchstick.
Now obviously, there is a trick that means you always win. Work this out for yourself, I don't want to spoil your fun! (Hint: whoever gets up to 89 has won, because whatever the other person says, you will be able to reach 100 on your next go. By extension, whoever gets to 78 has won, as they can get to 89, and so on). I took on about 7 students in total and beat all of them. As I did it, pupils started to notice things. This is what the pupils discovered, in order:
1) If Miss says 89, then she's won
2) Whoever goes first always wins (false!)
3) You need to stop her getting to 89
4) Get to 78, because then she can't go to 89!
I then extended them to realise that there were certain numbers that you should always try to aim for to make sure you win, and they spotted the pattern of these numbers. On the next go, a student beat me. And as much as it pains me to admit it, he beat me fair and square (I got my numbers mixed up). You can imagine how happy that made the class: "We beat Miss, and she got bare A*s!"
The class was then desperate to show off this newfound skill by challenging the head of maths to a competition. This hasn't been arranged yet but I think it's a really nice idea. I hope they beat him!
You can obviously extend the activity by considering a different target instead of 100, and different numbers that you're allowed to add.
I'm going to do the same activity with my other classes to see how it turns out. If you try out this activity, please let me know how it goes!
Emma x x x
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