The other day I my timehop showed me this lovely little post from last year. It includes “Heron’s Formula” for calculating the area of a triangle, as I read it I remembered thinking it was a little strange that not many people had heard of it before.
Today I was looking through a number of textbooks trying to find a decent set of questions on area, perimeter and volume for my year nines as I wanted to consolidate their learning at the start then move onto surface area. I’m not a fan of textbook misuse- ie “copy the example and try the questions” but I do sometimes use them for exercises as we have a very limited printing budget and some of them have superb exercises. For a fuller picture on.my view of textbooks, read this.
I was looking in one of my favourite textbooks:
And I happened across this:
There it is! Plain as day! Heron’s Formula! In a KS3 textbook!
I was disappointed that its function was described and its name wasn’t and there was no mention of why this worked. It basically reduces the question down from a geometry one to a purely algebraic substitution task and I would question the appropriateness of including it in an exercise on area, but still, I was incredibly exciting to find it there!
Are you a fan of Heron’s Formula? Had you even heard of it? Do you have a favourite textbook? I’d love to hear your views.
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.
Over the weekend I read this post by Jo Morgan (@mathsjem). In it she refers to a website “brilliant.org” which offers some brilliant stuff to be used in maths teaching. I liked the look of it so I set off to investigate and when I accessed the site I was presented with a pop up asking if I wanted to download the app, which I promptly did.
When I accessed the app I was presented with a number of mini quizzes which covered a range of early KS3 material, but on further exploration I discovered a treasure trove of fantastic questions and explanations of topics ranging from very basic up to degree level.
On top of this there are some fantastic puzzles to solve and the app tracks what you have done and how well you have on it. I found it really fun, I enjoyed testing my knowledge on some of tge further mechanics and am looking forward to attempting some of the puzzles.
I think this app is brilliant for anyone who loves maths, whether they be studying it at school, at university, teach it or just enjoy keeping themselves smart.
There is also the chance to subscribe to additional content, I haven’t yet but I am tempted. The additional content seems to be more worked solutions to the quizzes, but I think I’ll see how much I use the app before I upgrade.
I do love the prices too, it’s £tau pounds a month, but if you sign up for a year they reduce it to £pi – a lovely touch.
A great app, if you love maths, get it and if you’re studying maths it will help immensely.
I’d also advise following the app page on Facebook, you get a constant stream of little beauties like this:
Today I was marking my year 9 classes books and came across some work on prime factor decomposition and tests of divisibility. Yesterday I had been arguing with Colin Beveridge (@icecolbeveridge) about the use of formula triangles, (see the comment section on this post) and any other method of anything that was presented in the fashion “do it this way and don’t worry about the why.” I felt a wave of hypocrisy flow over me. I had never explained the tests of divisibility to the class, and furthermore I didn’t even know myself why they worked! I thought I’d explore them and see what I came up with.
If the last digit is divisible by 2, the number is divisible by 2.
This test of divisibility is obvious, the definition of an even number is that it has 2 as a prime factor, and all even numbers have a last number divisible by 2. No further exploration needed. I know why this one works and I’m certain the class do too.
If the sum if the digits is a multiple of 3, then the number is divisible by 3.
This is a fact I’ve known since I was at primary school back in the 1980s, but I can honestly say that I’ve never thought about why it works. I wrote a couple if equations out, realised I should be working in modulo 3 and came up with this:
Which basically boils down to the fact any number can be split into the digits multiplied by a powerful of ten.
A= (x0)10^0 + (x1)10^1 +… + (xn)10^n
As (10^n) mod 3 = 1 for all natural numbers (and 0) then it follows that:
(A)mod 3 = (x0 +x1+…+xn) mod 3
Which implies our test of divisibility. This also implies the test if divisibility for 9 (ie is the sum of digits is divisible by 9 then so is the number) as (10^n)mod9 = 1 for all natural numbers and 0. To prove you just follow the exact working but in mod 9.
A number is divisible by 4 if it’s last two digits are divisible by 4.
This one uses the fact that if two numbers are both divisible by a number then so is their sum.
We know any number bigger than 100 can be expressed 100a + b where a and b are natural numberso and b<100. We also know that as 4 is a factor of 100, 100a is divisible by 4 if a is a natural number. Hence the whole number is divisible by 4 iff the best is divisible by 4, and b is the last two digits.
From this we can deduce the test of divisibility for 8 (ie a number larger than 1000 is divisible by 8 iff the last three digits are divisible by 8) as 1000 is divisible by 8. The proof is the same, but you split the number into 1000a + b where a and b are natural numbers and b is less than 1000.
Testing for 5 and 6
Testing for 5 (the last number is 5 or 0) is another obvious one that needs no further exploration. And the test for 6 is simple, if 2 and 3 are prime factors 6 must be a factor, as all numbers can be expressed as a product of their prime factors and 6 is the product of 2 and 3.
Testing divisibility for 10
Again, this is obvious as we are talking about a base ten system! That leaves just one more test.
To test for divisibility by 7, take the last digit, double it. Take this away from the number you are left with if you subtract the last number and divide by ten. If the answer is 0 or divisible by 7 (ie if the answer mod 7 = 0) then the original number us divisible by 7.
What a ridiculously complicated divisibility test. It looks much nicer algebraically:
(10a + b) mod 7 = 0 (a and b are natural numbers, b < 10)
(a -2b)mod7 = 0
Once I expressed it algebraically I had two simultaneous equations and solved like this:
I enjoyed working through them and now I’m confident I know why these tests work, but I’m fairly sure that the ones that aren’t obvious would be too complicated to explain to year 9. This made me think about their use, and I certainly think it’s fine to use them even without understanding. Which brought me back to the formula triangles debate. This is one that will rage, but as I explained in my post on it, I don’t have a great issue with people who understand the algebra using them. I also don’t have a great issue with people who struggle at maths and aren’t going to pursue it past GCSE using them. My issue is with higher attainers who could and should understand the algebra being taught then with no deeper conceptual understanding. I guess it’s a topic I will need to think more on.
A lot has been said recently on textbooks, the benefits they have and the bad press they get. This has had me thinking a lot about them, and their use in lessons. I rarely use them, certainly not the way they were used in my own schooling, but this doesn’t mean that I don’t think they have their uses.
Why do they have such a bad press?
I think they bad press comes from bad use of textbooks. I remember when I was at school lugging a ridiculously heavy bag around all day every day because there was a huge textbook for each lessons. I remember many lessons which began “Turn to page 6, Stephen (or whoever) can you begin reading.” then after the page was read the class would attempt the exercise. I remember a biology teacher who read the book to us, she’d fire questions off if she thought you weren’t listening. I sat next to Liam, and we’d sussed that you could answer the questions if you had the textbook on the correct page. One time the lesson was on organs, and the question was thrown at Anthony, who sat on the next table along. Liam and I often gave him answers. This time the question was “what’s the largest organ in the body”, I whispered “pipe organ”, which he then shouted out. It was hilarious.
I could go on, but I’m sure you all had your share of uninspiring textbook lessons. I’ve seen them as a professional. I witnessed an A Level lesson where the teacher sat at the front and read the textbook to the class verbatim. It struck me as rather pointless, as they all could have read it themselves. I’ve seen a KS4 teacher, when I was an Nqt, hand out textbooks with the instruction “look at the example on page ten, then attempt the questions”.
All these examples are uninspiring, and not conducive to good learning. But I think it’s unfair to lay the blame on the textbooks themselves.
How can textbooks be used then?
During lessons, there will be a point when you want the students to do some work. Practicing a skill or solving a problem. Using textbook exercises isn’t necessarily worse than a worksheet or questions on the board. In fact, it could be argued it’s better. It’s a greener and cheaper long term alternative to a ton of printed worksheets. The right textbooks have extension work built in, or offer a selection of exercises of differing difficulties. They also usually have plenty of examples, so learners can use them if they’re struggling, then can request help if they still need it.
My favourites of the ones I have are probably these for KS3/4:
And these for KS5:
Within all these books there are some great things, but none of them are what I would call ideal. Each has plenty of flaws. I find that having access to all these, and many more, textbooks allows me to use ideas, examples and exercises from them as and when required. I sometimes think I should write one, it would be great!
So you think the right textbook would be fine?
Not on their own, no. The recent Sutton Trust report showed that a teacher with strong pedagogical subject knowledge is extremely important to the learning of a class. The right textbooks could aid these teachers, not become a band aid to cover for poor teaching or teachers with shallow subject knowledge. I also wouldn’t like to see them used in isolation. There are many other activities that can aid learning. Things that are quick and easy to check pupils have correct without the need to check each bit, resources such as Mathsloops and Tarzia or activities on mini-whiteboards. All these have a place in lessons, and all would be complimentary to the perfect textbook, which would aid, not replace, good teaching. Examples would be additional to the lesson and offer help learners who are still struggling.
Here is the BBC report into the comments by Education Minister Nick Gibb on textbooks.
Here is a nice article from the inside classroom project entitled “Why textbooks matter”.
Exercise books, what are they for?
The name suggests they are there for learners to complete exercises in, but if that’s all they are for then they are pointless. If that is all they are for then we would be better off replacing them with slates! Or the modern day equivalent- mini-whiteboards.
I think they are, or rather should, be tools of learning beyond that. The exercise book a learner finishes should become a tool for revision. They should be able to look back over it and see how to solve problems, see the mistakes they made and see how to correct them.
I’ve often thought a two book system might work. One book to house rules and examples, and one to house work. But again, is practice is the only function of the second, then a rulebook and a whiteboard would be sufficient.
Whether using a one or a two book system I think there can be value in keeping the book that houses exercises, but that value only comes if the work is presented correctly. A well presented book will provide a fantastic tool for revision. Rules, notes and examples will be clear, and exercises will be laid out in a way that enables learners to quickly spot any misconceptions they once had and move past them, ensuring that they don’t make those mistakes again in future.
A recent bit of cover work I set has unearthed this misconception amongst four or five learners who seem to get confused on split variable equations where the x term is negative. This gives me something to go over, and gives those who set out their working a good reference for the future:
Where as this student has no such thing. He is naturally one of the best matheticians in the class, but needs some real training. The majority of his book looks this:
When marking this weekend I was extremely pleased with the presentation from the vast majority, who have crafted brilliant tools that will help them no end, come exam time.
This is how I remember my books looking. Albeit with far messier and much smaller handwriting!
I thought I’d check, but I don’t have any of my old school books, I did find some work from my time at university, I was setting work out well then:
Earlier today I saw a tweet from Luke (@bettermaths) which said that the subject of those evenings #mathschat was “how should we assess year 9, in light of the new 1-9 grading system?”
This got me thinking about year 9. It’s a funny year group in general. Traditionally it falls within key stage 3, but in many schools these days it us counted as key stage 4. This year’s are in a stranger place still, as they will be the first year to go through the new KS4 curriculum and sit the 1-9 exams. And they will do this without having done the new KS3 curriculum, never mind the KS1/2 curricula. This means they run the risk of having gaps in the assumed prior knowledge where said assumed knowledge is on the new curriculum but not the old.
It is important schools address this this year. We need to ensure that we are equipping year 9 with the requirements to access the new curriculum. Edexcel have drafted a transition curriculum for year 9 that is freely available on their website. (I don’t think the other boards have, if you know they have send me the link and I’ll add it!) As all the boards are using the new curriculum with no additional content, this transition scheme should help if you haven’t yet put anything in place.
So how should we assess?
The answer here is pretty obvious to me. We should be assessing against the content they need to know, identifying the gaps in that content and using that to inform our teaching. I believe that that should be the main focus of all our assessment, particularly with this year group who face extra challenges.
I do, however, feel the question is intended to be about tracking, rather than assessment. I think it’s really asking “should we be using levels, a-g grades or 1-9 grades?” This to me is an entirely different question. And I see it as far less important. We are moving into a “life beyond levels” and I see that as an opportunity to take back assessment. To restore it to its former glory as a way to identify gaps, rather than a way to impose a linear model of progress onto learning that doesn’t take a linear form at all. I know at the recent #TLT14 event Tom Sherrington (@headguruteacher) spoke about the removal of levels from reports at his new school and recently wrote this excellent piece in assessment.
So what are you suggesting?
Well obviously we need to have some progress tracking, but does it need to have a numerical value every half term? Should we even be collecting data that often? I believe Kev Bartle (@kevbartle) spoke at #TLT14 about how he’s moved his school to 2 data collections per year, believing that this will mean more accurate data, which I think is a good idea.
So what should we use?
We know, as teachers, what are students need to know to get where they need to be. We know their start points and we know where they need to be at each step. Should we even be quantifying this with numbers or letters? Could we nit be rating them as “On target”, “Above target” “Below target”? In a post level world where progress is king (fingers crossed progress 8 moves us away from a threshold pass!) should we not be assessing against, and reporting on, progress?
#mathschat is a twitter chat which happens Wednesdays at 8pm. Follow the hashtag to join in. And feel free to comment here if you have opinions on this, I’d love to hear them.