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Archive for November, 2014

Vedic Multiplication

On Friday, timehop told me it was a year since I wrote this post on multiplication methods. I’d forgotten about the post, and tweeted a link to it. A number of people commented and tweeted about it and a nice discussion ensued. Part of the discussion moved to Vedic Multiplication.

I know a few Vedic Multiplication methods, and believe there to be many more. Most of the ones I know link to the most common algorithms but there is this curious one used for numbers close to one hundred.

First, take bother numbers away from 100:

Multiply those numbers and make them your last two digits:

Take one of the differences from 100 away from the other number (it should be the same):

That becomes your first two digits:

It’s an interesting little trick. I don’t see it as something that there is any reason to teach, and I don’t think it promotes understanding at all, but it’s interesting nonetheless. I think it may have a use in lessons, as an interesting introduction to algebraic proof.

How would you prove it?

First, consider the product ab, and apply the same steps:

The last two digits are (100-a)(100-b). The first two are a-(100-b) which equals a+b-100. Or b-(100-a) which also equals a+b-100. To make these the first two digits of a four digit number we need to multiply the expression by 100.

This gives:

(100-a)(100-b)+100(a+b-100)

Which expands to:

10000-100a-100b+ab+100a+100b-10000

Which cancels to:

ab As required.

A nice, accessible, algebraic proof that proves this works. It works for all numbers, not just those close to 100, but if your product (100-a)(100-b) > 99 (ie more than 2 digits) you need to carry the digits.

Categories: Maths, Teaching

Textbooks

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.

I’ve seen a large variety of maths textbooks, I own a fair few. Here are some of them:

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”.

And here is my post on the aforementioned Sutton Trust report (which can be accessed here).

Square Triangles

I often read, and use, Emily Hughes’s (@ilovemathsgames) “Puzzles of the week“. This week’s included this:

I liked the puzzle, it’s fun and it fits well with the sequence of lessons my year 9s have completed recently, but it’s basically a trial and error question. There are 13 square numbers below 180, and it wouldn’t take too long to plug each one it, especially with a computer.

I set myself the challenge of having only one guess, and started to think strategically. Firstly, it’s impossible for the solution to contain an odd number of odd numbers. Ie, there’s 1 even number in the solution or all the numbers are even. Then I thought about making ten with the final digits of the squares. The most obvious seemed to be 0,4,6. Which would mean 100, 64 and 16, which is a solution.

I wondered if there were more, but I’ve considered the problem and I’m pretty sure there isn’t. I am now thinking if something similar can be done for other polygons, can 360 be expressed as the sum of 4 squares? Can 540 be expressed as the sum of 5? Food for thought.

The importance of presentation

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:

Assumed prior knowledge

Today I looked at “the natural logarithm” with my year 13s. They did pretty well on C2 so I assumed there would be some proficiency with log rules. For the majority of the class I was right, but while circulating I found this in one students book:

Wrong, on so many levels! I usually do a recap starter, which is a nice domino activity I created a few years ago on logs to base ten, but for some reason I didn’t with this class. They’re a bright bunch and I figured they’d be fine with log rules as soon as I explained what the natural logarithm was. So instead I chose to look at the graph of Ln(x) as a starter, I certainly won’t be making that mistake again!

Carnival of Mathematics #116

Hello, and welcome to the 116th edition of the Carnival of Mathematics  For those of you who are unaware, a “blog carnival” is a periodic post that travels from blog to blog and has a collection of posts on a certain topic. This is one of two Maths Carnivals, the other being Math(s) Teachers at Play, the current edition can be found here.

Tradition dictates that we begin this round up with a few facts around the number 116. It’s a fine number, as far as even numbers go. It’s a noncototient, which is pretty cool. There are 116 irreducible polynomials of order 6 over a three element field, it is 38 in base thirtysix. 116 years is the length of the 100 years war and 116 is the record number of wins per season in major league baseball, set by the Cubs in 1906 and equalled by the mariners in 2001. It’s prime factorisation is 2^2 x 29, in roman numerals it is CXVI and it can be represented as the sum of two squares! (4^2 + 10^2)

Now, onto the carnival:

In the news

Katie (@stecks) submitted a number of fantastic news articles to this carnival. The first of which is this piece discussing Simon Beck and his geometric snow art. Ed Southall (@solvemymaths) also wrote about geometric art on a similar scale, sharing this about Jim Denevan.

The next article Katie shared was this amazing piece from Alex Bellos (@alexbellos) informing us of the amazingly geeky things Macau have done with their new Magic Square stamps! It almost makes me want to take up stamp collecting!

Maths Applications

Grace, over at “My math-y adventures” tells us how she’s put her undergraduate mathematical studies to good use by creating a model to help her with a real life decision.

And Andrea Hawksley writes this phenomenal piece exploring the maths of dancing.

But where there’s good, there’s bad, and here you can see the sort of fake real life contexts that irk Dave (@reflectivemaths) and myself.

A bit of calculus

John D Cook as submitted a lovely integration trick which I’d not seen before.

Augustus Van Dusen gives us part four of his review of “Inside interesting integrals” by Paul Nahin.

Interesting numbers

3010 tangents have been exploring the infinite with this superb post: “my infinity is bigger than yours”.

Over at the mathematical mystery tour they have shown us how to write pi using primes!

Fawn Nguyen has this great exploration of finding the greatest product.

Geometric thinking

Jo (@mathsjem) has written this excellent piece on circle theorems.

@dragon_dodo has produced this cartoon which illustrates Colin’s (@icecolbeveridge) position on radians.

A bit of fun

The folks over at futility closet have shared this idea about creating an indoor boomerang!

This week saw the UKMT Senior Maths Challenge, and here is my favourite question.

I spent some time at the children’s playground with my daughter, and found this mathematics there.

Manan (@shahlock) over at maths misery turns infuriating into fun talking about fractions and algebra.

Behind the mathematician

Behind the mathematician is a great series from Amir (@workedgechaos) who submitted Martin Noon’s (@letsgetmathing) edition for the carnival! (Mine is available here.)

Podcast

This month saw the annual maths jam conference, which brought with it the exciting prospect of a live special of “Wrong, But useful” my favourite maths podcast!

And that rounds up the 116the edition of the Carnival of Mathematics. I hope you enjoyed reading all these wonderful posts as much as I did. If you have a post for the next carnival submit it here.

Categories: Maths

This week I had the honour of contributing to Amir (@workedgechaos)’s fantastic “Behind the mathematician” series which looks at maths and teaching from the perspectives of different maths teachers. I really enjoy the series and was happy to be asked. Do go and have a read of mine and others.

Behind the Mathematician continues in its quest to publicise the best maths teachers out there. This week the feature contributor is Stephen Cavadino, the man behind cavmaths and one of the most prolific bloggers out there. Whether it’s on the reasons for being a teacher or some ideas on the use of puzzles in the classroom, Stephen’s insight is as valuable as it is comprehensive.

Oh, and anyone who knows the travails of being a parent of a (very) young ‘un and trying to make a real difference in maths teaching deserves a thumbs up from me. Enjoy!

How would you describe your teaching style?

I’m not sure I have a specific style, so perhaps the correct answer would be “varied”. I believe that different classes need to be taught in different ways, and that…

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Categories: Maths

UKMT Maths Challenge Q16

November 7, 2014 1 comment

Above is a photo of question 16 from yesterday’s UKMT Senior Maths Challenge. And what I fantastic question it is!

It shows (incase you can’t make it out on the picture) a rectangle and a circle which have the same centre. The two shorter sides of the rectangle are tangents to the circle, and the rectangle is 6×12. The task is to find the area that is inside both.

After the challenge had been completed one if my Y13s came to ask about it, and it took me a little while to work it out. I played around with some similar triangles, thinking that I would need to calculate the bit of the rectangle outside the circle and subtract them from 72. Then I had the realisation that I could easily split it up into triangles and sectors that I could calculate the area of with great ease.

First I sketched:

Showing the rectangle split into areas and sectors. I then considered the triangles, if I split then into right angled triangles I had a hypotenuse of 6 and a height of 3, so the cosine of the top angle must be 0.5, which means the angle must be pi/3.

I then used the sine of this angle ((rt3)/2) to calculate the opposite side as 3rt3. Then thought “why didn’t I use Pythagoras’s Theorem?!” A quick check using Pythagoras’s Theorem shows us the side is rt27, which simplifies to 3rt3. The area of the triangle then must be 9rt3.

Knowing that the angles at the centre in the two triangles are 2pi/3, that angles round a point add up to 2pi and that the sectors are congruent means we can deduce the angle if each sector is pi/3. We also know the area of a sector is (r^2(theta))/2, which gives the area of each sector as 6pi.

Finally we sum the areas and get 18rt3 + 12pi.

The student had looked at the answers and estimated them using the knowledge that rt3 is around 1.7 and pick is a but more than three. This meant he could eliminate c,d and e, but he didn’t get to the final answer.

The new maths GCSE

The new maths GCSEs have been approved, and schools now face the prospect of selecting which board they are going to go for from next year. A few weeks ago I attended an exam board briefing which included presentations from the three main boards. Today I’ve been working on a presentation to share what I’d learned there with the department and I thought I’d share a few thoughts, and a few of the questions from the sample assessment materials, here.

The compulsory stuff

Obviously, all three boards are constrained by the curriculum set out by the government, if you’re still unfamiliar with the new curriculum (get your act together!) there’s a lovely summary here and you can download it here. They had the option to add extra stuff in, but according to the presentations none of them have. Although stem and leaf diagrams have been removed from the national curriculum but still appear on the Edexcel sample materials.

The exam papers have to run for a total of 4.5 hours, with at least half being calculator assessed. These three have all gone for three 1.5 hour papers, with one non-calculator. I believe the Welsh board have gone for two 2.25 hour papers, one of each. I think the three paper option is sensible, although I would have preferred them to have two non-calculator exams. Edexcel and AQA have assigned 80 marks per paper, where as OCR have assigned 100.

The make up of the tiers will be different, the new 4 will be equal to the old C and the new 7 the old A. That is to say the bottom of each will be the same. Foundation will test 1-5, with 50% coming from 1-3 and 50% coming from 4&5. Likewise, higher will be 4-9 with 50% 4-6 and the rest 7+. The gradings at the top will be slightly different, with the top ten percent getting 9 and the rest being split between 7 and 8. I’m not sure I agree with this. One year a 7 and a 9 maybe a couple of marks different, and one year they may be a ton. It’s a weird hybrid of curve grading and criteria grading and it will potentially confuse folk.

So what are they offering?

As far as the support goes, I think any of the exam boards would be a great call. All offer things that are exceptionally good. AQAs free interactive route maps and all that comes with them are a massive selling point, Edexcel’s free results plus service is phenomenal and OCR are offering a free mock exam service which in the first instance negates the lack of past papers and going forward offers a fresh mock paper so pupils haven’t seen then before. On top of this they all offer an exam paper builder, although OCR charge for this.

So how can we choose?

Given that up until now all three boards seem to be pretty equal, I guess the final choice will cone down to the sample assessment materials. It is here where we see a major difference, particularly in the foundation paper.

AQA, during the briefing, were particularly keen to get across the idea that they’d worked hard to ensure they had the lowest word count. The briefing saw the representatives have some amusing banter over this, with Edexcel suggesting “we’ve always used words, we gone for the best, accessible questions, rather than counting words”! Although I’d question the accedibility of a foundation paper that has questions like this on it:

And this:

I also find that at times the Edexcel foundation papers overcomplicate some of the questions:

And

Then there’s this, which I’m not really sure the point of:

Obviously, the content has to be harder, that’s part of the constraints, but I feel these questions are overly complicated for what they are. There are good questions on there too though. I love this one:

All 6 questions appear on Edexcel foundation SAMs.

The other boards, for me, are much more accessible. OCR have a much more accessible approach to standard form:

But they are still a little overcomplicated at times:

Both from OCR foundation SAMs

AQA have really worked to eliminate unnecessary and confusing wording:

And this question is my favourite on any of the Foundation SAMs:

All 3 taken from AQA Foundation SAMs.

I think I prefer the AQA FN papers at the moment, but I’m still looking at them.

The higher papers are, for me, all superb. I like the fact the content is much tougher, and I feel all three boards have done a great job putting the higher tier papers together. We will need to ensure all members of staff in our departments are secure in their pedagogical subject knowledge of the new curriculum, as there are some tricky topics that haven’t previously been taught until A – Level.

Here are some examples of questions not on the current syllabus:

OCR HIGHER NON-CALCULATOR

AQA HIGHER

OCR HIGHER

EDEXCEL HIGHER

AQA HIGHER

AQA HIGHER

All these questions are, in my opinion, excellent. They are challenging and require a good amount of mathematical thinking. On the whole, I’d be happy sitting my learners any of the higher papers, so I think the decision will probably come down to the foundation papers.

I’m excited by the new curriculum, and I’m now excited by the sample assessments. I can’t wait to get into the teaching of it! I also can’t wait for the exam boards to release the new A Level specs and sample assessments!