Yesterday I was teaching perimeter of compound shapes with my year 9s. After they had been solving problems based on shapes made up solely from squares and rectangles one if the pupils asked “Sir, will the perimeter always be even?” I thought this was a great point for discussion so I opened it up for the class.
They decided that for the type of shape we were looking at the perimeter would indeed always be even, as you had to cover every distance twice, meaning 2 was always a factor of the perimeter. I was impressed with their reasoning, but a little disappointed that one had to prompt non-integer side lengths. When I did they quickly dismissed this with “of course we knew that sir, but be were only considering whole numbers!”
I then asked them to consider is this applied to all shapes. They quickly concluded that triangles could have odd perimeters with integer side lengths, and circles, then they extrapolated to any shape with an odd number if sides. They concluded quickly that regular polygons with an even number of sides couldn’t have an odd perimeter if it had integer side lengths. Irregular polygons provided a more difficult challenge, and I left them to ponder it over the weekend!
I seem to be reading a lot lately about “Mocksteds”, these mock inspections that seem to be all the rage, and other seems to me that they can cause more damage than good. I have a number of friends who have been subjected to these over the last few months, all felt heavily pressured, all felt they had had their already massive workloads increased and some felt undermined and have lost confidence in themselves. I can’t imagine any of these are positive outcomes for the schools.
So you think it’s all nonsense?
No, I can see there may be a need to review and evaluate a school, it’s the manner in which it is often done that I take issue with. Part if the philosophy in my school is that the “why” and the “how” are as important as the “what”. The reasoning behind the review should be shared, and when designing the how staff welfare should be taken into consideration. During my career so far I’ve been through Ofsted inspections, internal “mock inspections”, external “mock inspections”, internal “reviews” and external “reviews”. On the face of it, the latter four seem very similar but in reality there can be a major difference.
We had an external review in the winter term, it lasted a day, reasoning was shared beforehand, and it seemed like the senior leadership had gone out of their way to ensure that staff remained relaxed about the process, while still putting their best foot forward. The process went well and I didn’t notice any colleagues crying in the cupboards.
On the other side, I have a friend who teaches at a different school who has recently been through a “mocksted”, she was put under an immense amount of undue pressure and was left questioning her own practice.
I have friends at another school who have been through a series of events that would test the strongest of people. They are expecting Ofsted, so are in the midst of Ofsted fever, they are being hit with no notice “Marking” scrutinies, they had an external company come in to do a mock inspection to ensure they are ready, and had an internal review conducted by SLT from within their Academy chain to make sure they were ready for that. If ofsted come next week that’s three two day review processes in a term. One staff member received 5 observations in 6 days. As if teachers weren’t under enough pressure.
The worst though, is this harrowing account, where school staff were told that Ofsted were here, but actually it was an external review team conducting a mocksted. A fact not shared until after the inspectors had left.
What are the outcomes
If a review is conducted in the right manner it will find areas to develop for the school, it will build teachers confidence in themselves and their leadership and will ultimately improve outcomes for the learners in our care. If a review is conducted wrongly it can lead to increased stress, reduced confidence, reduced staff well being, increased workload and can lead to learners getting a raw deal.
Of course we need to know how are schools are performing, we need to know where staff strengths and weaknesses are in order to tailor our development plans, but it doesn’t need to be a painful process.
It’s world book day today, and despite dressing up as Rudy Baylor from “The Rainmaker” (you know, suit, shirt, tie etc) my pupils didn’t believe I was dressed up! I have been working on a pupil facing site, and wanted to include a page on suggested further reading, so far I have:
You all have textbooks to help with your course, and revision guides. If you want any further books/websites to specifically help with your course, try the revision page.
This page is a selection of books that are about maths, but not specifically related to the course. The maths in them should be accessible to A level maths students and they will help deepen your knowledge of maths.
For those of you considering further study they may be particularly helpful in shaping the direction you go in, and may provide excellent fodder for UCAS statements and university interview discussions.
These books are available from all good bookshops, we are looking at getting some into the library, and Mr Cavadino has a few of them which I’m sure he’ll lend you if you ask nicely.
Fermat’s Last Theorem – Simon Singh
This is Mr Cavadino’s favourite book. It is based around an enigma known as Fermat’s Last Theorem. Fermat was an amateur mathematician, but a brilliant one. He did Maths for the live of it and he came up with, proved and solved many great mathematical theorems and puzzles. When he died he left a number of theorems unproven. Slowly as the years progressed mathematicians proved them all, except one. His last theorem. One he posed with a note “I have a truly marvellous proof that this margin is too narrow to contain”. In the book, Simon Singh looks at the evolution of maths, and how this amazing theorem drove so many people to make so many amazing discoveries. If you only read one maths book in your life, make it this one.
The Simpsons and Their Mathematical Secrets – Simon Singh
Have you ever watched an episode of The Simpsons, or its sister show Futurama, and noticed a maths reference? Well so did the author. It turns out it’s not just a coincidence, but that the writing team are all mathematicians! For years they’ve been sneaking mathematics into the world’s most popular cartoon. The book looks at the maths they’ve included, why they’ve included it and how it relates to the episodes it’s in. If you like maths, and the Simpsons, then this book is for you.
The Code Book – Simon Singh
Another classic from the pen of Simon Singh. This one looks at the evolution of cryptography and cryptanalysis over the millennia and includes some fascinating accounts of where codes and encryption have been used throughout history. If you feel you may be interested at pursuing this as a career, or just have a passing interest in it, then make sure you read the book!
From Here to Infinity – Ian Stewart
Ian Stewart is a Professor of Mathematics and has written many fantastic maths books. This one is a particularly good one for learners who are interested in picking Maths at university. The book tracks the evolution of mathematics and gives a great introduction to many of the mathematical topics that will be covered on the course. The maths in this one does get quite heavy, and there may be a coupe of points where you can’t follow it. This shouldn’t matter as you should all be able to follow the majority of it and if you do read it and want to discuss any of it then mention it to your teacher. The variation of topics included in the books gives a good start point to future mathematicians who are unsure which areas of maths they would like to study.
Music of the Primes – Marcus Du Sautoy
Marcus Du Sautoy is a another Professor of Mathematics and he has also written many great maths books. This one is based around unsolved problems in Maths. He says the reason he wrote it was that when Fermat’s Last Theorem was getting a lot of press the non-maths world seemed to be of the feeling that when it was solved that was it, we had “done” maths. This is obviously not the case, as maths is infinite, and he uses this book to explore some of the big unsolved problems of the subject.
I am also going to include more books by Marcus Du Sautoy, some Rob Eastaway ones and Tony Crilly’s “How big is infinity”, then add books as I read them, but I’d love feedback on it and I’d love suggestions on what other books to include.
It may not surprise you to discover that Simon Singh (@SLSingh) is one of my favourite authors. I have previously reviewed “The Simpsons and Their Mathematical Secrets“, and “Fermat’s Last Theorem“, the latter of which is still my favourite ever book.
“The Code Book” came out quite a while ago, but I’ve only just read it, and as with Simon’s other books I was hooked pretty much straight away. The narrative Simon weaves throughout the ages is amazing. Seamlessly switching between the hardcore maths of the subject and the historic events that drove the discoveries. What did Mary Queen of Scots use codes for? What about Julius Caeser? How brilliant was Alan Turing?
I was lost I’m a world of espionage, war strategies and amazement at how cryptographers (code makers) and cryptanalysists (code breakers) managed to keep in out doing each other, whether the driver was military power or purely academic.
The book covers some heavy maths, but it is broken down into terms that anyone with a high school education should be able to follow. At times I felt it was broken down a bit too much, but I realise I gave quite a strong mathematical background, and the subject of codes may appeal to people who don’t.
This book is a great read, a must read for anyone with the slightest interest in codes, which is probably a growing number in tge wake of the imitation game! I think if I’d read it as a teenager I may have ended up in cryptanalysis! There is also a version of the book aimed at young adults.
Hello, and welcome to the 83rd Edition of the monthly blog carnival “Math(s) Teachers at Play”. For those of you unaware, a blog carnival is a periodic post that travels from blog to blog. They take the form of a compilation post and contain links to current and recent posts on a similar topic. This is one of two English language blog carnivals around mathematics. The other is “The carnival of mathematics“, the current edition of which can be found here.
It is traditional to start with some number facts around the edition number, 83 is pretty cool, as it happens. Its prime, which sets it apart from all those lesser compound numbers. Not only that, its a safe prime, a Chen prime and even a Sophie Germain prime, you can’t get much cool than that can you? Well yes, yes you can, because 83 is also an Eisenstein prime!!!! Those of you who work in base 36 will know it for its famous appearance in Shakespeare’s Hamlet: “83, or not 83, that is the question…..”
Firstly, to whet you appetite, here is a little puzzle, courtesy of Chris Smith (@aap03102):
So, what delights do we have for you within the carnival?
Firstly, we had a few submissions that were based around having fun learning maths.
Firstly, Mike Lawler (@mikeandallie) submitted this on a 242 sided Zonohedron: This project plus the follow up project, were projects out of Zome Geometry that we did in the open space in our new house (i.e. we don’t have much furniture yet!) Really fun project for kids. Lots to learn about geometry, symmetry, and especially perseverance! Really shows how amazing the Zometool sets are as learning aides, too.
Pedagogy and Reflections
There’s a few post around the pedagogy of teaching mathematics, including reflections on what’s been tried in the classroom.
Cody Meirick submitted this on “Maths Investigations”: The developers of this series argue that math should not be viewed as a history lesson, teaching formulas and concepts that mathematicians “invented” centuries ago. Instead, math time should be an active and even creative process, allowing students to learn through experimentation and exploration.
Benjamin Leis asks “can we get there from here?”: I’ve been blogging about my experiences running a math club for the first time. This one was a planning exercise to figure out how to make a particular problem accessible to the kids.
Rodi Steinig has submitted this nice little post around tessellation. In the 5th of our 6th Math Circle session about Escher and Symmetry, middle-school students make some discoveries about assumptions, and also discuss the pros and cons of inventing your own math.
The superb Ed Southall (@solvemymaths) has produced more posts in his excellent complements series, aimed at helping to further subject knowledge within the profession. The latest instalment is on Highest Common Factors (or Greatest Common Divisors, to those of you across the pond) and Lowest Common Multiples.
Kris Boulton (@krisboulton) explores the question: “If we cannot see the learning win a lesson hat are exit tickets for?”
Also this month many people have shared great resources, here are some brilliant posts on that.
Jo Morgan (@mathsjem) has produced another of her maths gems series. The series looks at great ideas and resources Jo has discovered recently, and it is another of my favourite series. This is issue 25.
Teachers at play
Well that rounds up edition 83, I hope you have enjoyed it. If you want to submit a post to the next carnival you can do so here. If you’d like to host contact Denise (@letsplaymath). And make sure you catch next moths carnival which will be hosted by John Golden (@mathhombre) over at mathhombre.
Today I was teaching one of my classes about drawing graphs. It was going well, we’d moved from linear to quadratic and discussed the shape. After a few minutes of plotting quadratic graphs from equations one of the students asked, “Sir, these parabolas are all symmetrical, right?” we discussed it briefly and he decided that yes they were. Then he carried on working. I was circulating the class and I noticed he was flying through the work so I asked him to explain why he was going so much faster. He said, “I’ve found a short cut sir, cos they’re symmetrical you once you see when they start repeating you can easily just work out the points from the ones you know.” I was impressed by his mathematical thinking, although I wasn’t as impressed with his explanation, although he did manage to refine it and explained it much better to one of his class mates who had overheard our discussion and wanted further explanation.
The student had seemed surprised when I approved of his method. I think he thought I’d chastise him for not working all the points had. But a massive part of mathematics is pattern spotting and he’d spotted a pattern within the shape of a quadratic graph and used it to streamline his working and get to the correct answer. By spotting this he was working at a level that is streets ahead of the actual work set and I think this needs to be encouraged. It’s made me think about things I teach, and if I’m building in enough opportunities to think like this. I know I do build them in, and encourage the thinking through questioning, but I’d never considered asking about this so I figured I’d try to think where else I could find ideas like this.
This is the third in a series of posts which have been written for a site aimed at our A level students. The first two (fractions and indices) have led to people giving me great feedback on things I could improve and have enabled me to ensure the site, when launched, will be in great shape to help our learners. The third installment is on rounding, a topic I’ve seen really bright students get wrong and one I’ve known senior teachers to teach incorrectly. (It’s a pandemic!) Please get in touch if I’ve omitted anything, made an error, or just lack clarity anywhere.
It seems ridiculous to even contemplate a help page for A level students which is based on rounding, it’s something that really should have been learned before now. Yet every year we see students who struggle to round correctly and lose daft marks through rounding errors. We hope this page will help you minimise the rounding errors and stop losing silly marks.
Don’t round until the end
One way people can go wrong with rounding is to round early. This can lead to an answer being far enough away from the correct answer to lose marks, even though the maths is correct all the way through. Rather than round, use the “ans” button on your calculator, or write the whole display down each time. This may still technically be rounded, but the full display should have sufficient accuracy to ensure your final answer is correct. When you are rounding at the end of a question, it is always best to write the full display on one line, then the rounded answer on the next.
Before we go into rounding, I feel I should mention exact answers. This is a related area where people lose marks. The exact answer isn’t “all the digits of your calculator display”, this is still rounded. If a question asks for an exact answer it will want it in surd form, or in terms if a known irrational constant such as pi.
These are easy, this is the number of digits after the decimal point. 3.547 has 3 decimal places (3dp), 7.4147 has 4dp
To round to a specific number of decimal places you need to pay attention to the digit that falls one place after. If you are asked to round 4.735 to 2dp it’s the same as saying “to the nearest hundredth”. You look at the 3rd decimal place, the midpoint between 4.73 and 4.74 is 4.735, so if it’s that or higher we round up, if it’s lower we round down.
You only look at the next digit. If you were asked to round 4.4444444445 to 1 dp you would round it to 4.4 as 4.4444444445 is lower than 4.45 YOU DO NOT ROUND FROM THE RIGHT. That would lead to every 4 becoming a 5 and the answer being wrong. I know it’s obvious, but I’ve known A grade A level candidates mess up here!
For some reason these are like kryptonite to some students. Even some students who are generally fine with rounding. Significant figures are about the numbers which hold most significance. In the number 78315 the 7 is the most significant number, as it holds the highest place value. This is the number we use as our first significant figure. We then round the same as for decimal places, use the next digit to see if we are closest to the number we have, or if we need to round up. So 78315 would round to 80000 to 1sf 78000 to 2sf and 78300 to 3sf
Likewise for decimal numbers. In 0.00537 the 5 holds the highest place value, so is most significant. This would round to 0.005 to 1sf and 0.0054 to 2sf. An issue some have is when a 0 appears after a non zero digit. Ie in 0.05071, I this case we still start from the most significant, so it’s 0.05 to 1sf and 0.051 to 2sf etc.
A problem some have is when it’s something like 4.745 and you’re rounding to significant figures. People get confused and round to decimal places, when actually the first significant figure falls before the decimal point. So in this case we’d get 4.7 (2sf).
What should I round to?
Usually exams will specify what to round to, in which case you should round to that. If it’s not specified then as long as you have used exact values then an exact answer should be fine, if you have used a rounded value then round to whatever that was rounded to. For example, if you are working with gravity in M1 you take g to equal 9.8 which is the constant rounded to two significant figures, so round your answer to that.