lim joe→∞

Teaching, Mathematics and Teaching Math.

Posts Tagged ‘Maths’

Everything is Mathematical

Posted by forumjoe on September 5, 2012

The Challenge

Ok, I’m swamped at the moment but I wanted to post something involving Latex because I’ve been intending to add it to my blogging skill set.  I had planned to discuss my favourite problem but that will have to wait until later.  For now this is quick and dirty (while still using Latex.  Hooray!)

Today a colleague sent me a link to Everything is Mathematical.  It’s not a resource site, because at the moment there’s only one video.  What it is is a well presented mathematical puzzle/competition site.  If you’re in the UK, you can submit answers to the posed problem and win a prize.  I’m not in the UK, but the site will still be useful.

There’s only been one problem so far.  I’m really impressed by the way Marcus du Sautoy presents the problem in an easy-to-understand way.  There’s no pseudocontext here, there’s no anyqs.  It’s all theory but I still think it’s great to show any class ranging from middle schoolers to upper highschoolers.  The presentation is great, the challenge is easy to understand and adaptable.  I’m REALLY looking forward to what comes next.

The puzzle

The challenge is this:

My paraphrasing is this:

Palindromic numbers read the same forwards and backwards. So, 414 is a palindromic number, so is 1234321

It’s pretty easy to write down all the 2-digit palindromic numbers:

  • 11
  • 22
  • 33
  • 44
  • 55
  • 66
  • 77
  • 88
  • 99

The question is, how many palindromic numbers are there that have 351 digits? 
And, what’s the smallest difference between the two closest ones?

So, I thought I’d have a go at cracking the problem, publishing my solution and exercising my newfound Latex skills.  I’m not completely sure I got it right so feel free to correct me in the comments.  I’m not sure how (or if) I’ll show this to my students yet, because time is tight this time of year.  But I’ll work something out.

The Solution

Anyway, we start with the pattern, and what Du Sautoy gives us.  There’s 9 numbers that are 2 digit palindromes.  For each of those, we can add 10 digits in the middle (0…9) to give us three-digit palindromes, so there are 90.

For four-digit palindromes, we start with our same two-digit palindromes (9 of them).  Now we add two digits into the middle but those two digits must be the same.  This means once again there are 90.

For five-digit palindromes, we start with our three-digit palindromes (90 of them) and we can insert one of ten digits into the middle.  This gives us 900.  The comparison of 6-digit to 5-digit is the same as 4 to 3.  That is, we add two digits but they must be the same, so we’ll end up with 900 again.

The pattern becomes obvious pretty quickly

  • 2 digits – 9 palindromic numbers
  • 3 digits – 90
  • 4 digits – 90
  • 5 digits – 900
  • 6 digits – 900
  • 7 digits – 9000
  • 8 digits – 9000

This pattern is x=9\times10^{\frac{d-2}{2}} if d is even or x=9\times10^{\frac{d-1}{2}} if d is odd.  (You nerds out there, tell me if there’s a better way of writing different-but-related odd/even functions)

So substituting d=351 into x=9\times10^{\frac{d-1}{2}} gives

  • x=9\times10^{\frac{351-1}{2}}
  • x=9\times10^{\frac{350}{2}}
  • x=9\times10^{175}

That’s 90,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,

000,000,000,000,000,000,000,000,000,000,000 numbers.

How far apart are the  closest ones?  Well, if we change any number in the first 175 digits, we have to make one on the symmetrical digit.  The only way we can change just one digit is if it’s the middle one.  In a more simple situation, lets look at some five digit numbers.

  • 55155
  • 55255
  • 55355
  • 55455

These numbers are all 100 apart, and there are no two 5-digit palindromic numbers closer together than that.  Even considering going across a place-value column, the closest I can come up with is (for example) 55955 compared to 56065.  These have a difference of 110, so it’s not helping us get closer to our target.  (Hopefully you would have realised by now, this is not a proof. I don’t have a proof for this).

So assuming the closest difference occurs when we change the middle digit by one, a 351 digit number has a middle digit in the 176th column.  Thus the smallest difference is 10^{175}

The Extension

One thing I love thinking about is how maths works in different numerical bases.  If we’re dealing with base-3, for example we can only use digits 0, 1 and 2.  How many two digit numbers are there?  Two.  11 and 22.  Three Digits?  101, 111, 121, 202, 212, 222 so 6

This means our rule for how many d-digit numbers in a base-3 number system is x=2\times3^{\frac{d-2}{2}} if d is even or x=2\times3^{\frac{d-1}{2}} if d is odd.

In a base-n number system, this generalises to x=(n-1)\times n^{\frac{d-2}{2}} if d is even or x=(n-1)\times n^{\frac{d-1}{2}} if d is odd.

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New Beginnings

Posted by forumjoe on August 21, 2012

Well, I haven’t posted here in 6 months.  I know a number of blogs that don’t post anything for a while and come back with “Sorry I haven’t posted anything for a while” and then go back to dormancy.  Repeat.  I’m determined not to do that.

There are lots of reasons I haven’t posted much (usually because I feel disconnected from the edublogosphere due to teaching under a totally different system) but now I’m using Sam J Shah’s New Blogger Initiation to motivate myself and see if this thing clicks with me.

want to use this blog to help critically reflect on my teaching, to inspire new methods and resources and to share advice on teachings techniques.  It’s only useful if it’s updated regularly, so lets try again.

It starts simply.  Many of the options to make my first post about were based around people starting a new school year.  Being an Australian, we’re in the middle of the toughest part of the year.  Lots of tests coming up, about to have a two-week break, don’t even have time to scratch myself.  So I picked the easy and relavent option to start with:

lim joe→∞

First goal: Explain the title of your blog.

When I was starting my blog (nearly 3 years ago) I asked friends for suggestions.  I got two great ones from @straaken http://brnz.org/hbr/  “Blogorithm” (or “Blogorhythm”) and “Lim Joe Approaches Infinity”.

I liked the first one for the punniness, but it would be more appropriate for a music/maths blog.  It didn’t seem too appropriate for me.  But the second one stuck.  It was Mathsy, it was nerdy and it was appropriate for me.  It signified growth, it showed that I’m getting older, do I get wiser at the same time?  As my age approaches infinity, does my wisdom?  I loved the sort of questions this title signified.  I tweaked it a bit to have a different url that still made sense, and there we have it.

I’m not proud of how much I’ve updated the blog over the years.  I’m not proud of how much I strayed from the core concept of teaching.  But I am proud of quite a few posts here, and most of all I’m proud of the title.  Thanks Straaks!  I’ll buy you a beer next time you’re in Hobart.

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Expectations

Posted by forumjoe on January 10, 2011

To go along with the class discussion on why we study maths, the first lesson needs to discuss expectations. I believe this will help clarify in the students’ minds what they can expect from the course, what they can expect from their teacher, and what the teacher expects from them. If we make this explicit at the beginning of the course, I hope to reduce student misunderstandings about what is required of them.

Here’s the list I’m working on, but I’m sure it’ll grow over the next month.

Expectations:

What can you expect from this course?

  • This course will hopefully teach you what you need to know to pass the course, to survive subsequent years of mathematical study.
  • Does not teach you how to be a good mathematician!

What can you expect from your teacher?

  • I will always attempt to be prompt and reliable.
  • I will always be available to help; before school, after school, at lunch and recess.
  • I will always do my best to help you learn the content and to get the best results you can.

What do I expect from you?

  • I expect you to be polite.  You should show respect to your fellow students and your teacher.
  • I expect you to not waste my time. This sounds selfish, but if you waste my school time, you’re not taking it from me, you’re taking it from other students who need it.
  • I expect you to not disrupt the learning of others.  Everyone has the right to a hassle-free environment.
  • I expect you to use the feedback on tests and homeworks effectively.  This is what I do it for.
  • I expect you to take advantage of my availability.
  • I expect you to do work at home above and beyond what is called for by the homework*.

Homework is a bit of a separate issue, and I’ll be making my expectations of homework submission clear at another time.

 

Does this sound fair to the students?  Would you have liked it if a teacher made this clear when you were at school?  Should there be more expectations in any category?

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Why is Mathematics a core subject in school?

Posted by forumjoe on January 1, 2011

Mathematics contains much that will neither hurt one if one does not know it nor help one if one does know it.

– B. Mencken, 1715

In order to get my students to think a bit more about what they’re studying and why they’re studying it, I’m going to ask them the above question… Why do we study mathematics?

And I’m going to get them to really think about this. First individually, then in groups, but not in the way that can be brushed of with “Because its important”. However, if I’m going to ask my kids this question, I’d better have some idea about what the answer is, so I decided to do some research and discuss the results here.

It is my belief (though I can find no factual evidence of this) that 90% of careers and occupations use no maths higher than Year 8 level. Beyond addition and subtraction (which cash registers and calculators can do anyway) how much maths do you really need to survive in society these days? Why do we force students to study maths beyond this level? Sure, there should be maths classes covering calculus and algebra and trigonometry in highschool for those students who WANT TO BE THERE, and these should be wonderful, happy places where students creatively solve problems and learn about the world, like Lockhart envisioned.

Mathematics is looked at by society at large as a “necessary evil”. Why is it necessary? Do schools make Mathematics a compulsory subject because society sees it as necessary? Or does society see it as necessary simply because “everyone has to do their dues” in the mathematics classroom as a child. Parents expect their children to study maths because they themselves studied maths as a child. But if they’re not going into a career that requires maths, why do they need to study it?

The SIN rule
The Mode of a dataset
The Volume of a right prism

When will people ever need to use these things unless they get a job in a field that requires it? The only reason to understand these concepts in maths is because they’ll need to build on their understanding next year. It’s a repeating cycle. “You need to learn [x] because it will help you learn [y] next year”

Cut The Knot’s manifesto – http://www.cut-the-knot.org/manifesto/need_it.shtml – has a great collection of quotes on this issue from throughout the ages, for both sides of the argument. Here are some interesting ones that I’ll be showing my kids to start discussion on this:

J. B. Mencken, De Chralataneria eruditorium, 1715 (quoted in C. Fadiman, The Mathematical Magpie, p. 256):
Mathematics contains much that will neither hurt one if one does not know it nor help one if one does know it.

Has anything changed since 1715? I think many people would agree with this statement today.

Fran Lebowitz (b. 1951), Social Studies, “Tips for Teens”, 1981.
Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.

Phwoar, no such thing as Algebra!? But what about… oh. How about… hmmm. Surely… no, ok. Maybe you’re right.
Algebra is a great mathematical tool. It’s great for solving mathematical problems and it is a fundamental building block for many subsequent mathematical concepts. But if someone doesn’t learn it, or doesn’t understand it, will it affect their ability to contribute to society?

R. P. Boas, Jr., If This Be Treason…, Amer Math Monthly, 64(1957), 247-249.
When I was teaching mathematics to future naval officers during the war, I was told that the Navy had found that the men who had studied calculus made better line officers than men who had not studied calculus. Nothing is clearer (it was clear even to the Navy) than that a line officer never has the slightest use for calculus.

So now, perhaps, we are approaching the crux of the problem. Why is this so? Why does studying calculus help in an occupation that has no use for calculus? I can think of a whole bunch of possible reasons, but I don’t really know. Hopefully this is where my research will lead next.

There are lots of great quotes in Cut The Knot’s manifesto, and I encourage you to check them out. Many have anecdotal evidence about how studying mathematics helps develop the mind in ways useful for many careers. I shall leave you with one of my favourites. Benjamin Franklin, in 1787:

It seems to me, that if statesmen had a little more arithmetic, or were more accustomed to calculation, wars would be much less frequent.

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Finding Motivation – The Mathematics RPG

Posted by forumjoe on June 14, 2010

{I’m getting a few visitors from Riley Lark’s Virtual Conference on Soft Skills which is great to see. Hi, everyone.
If you’re a teacher and you’ve got a few spare minutes, I’d really appreciate it if you could fill out a survey for me. I’m gathering data about teachers around the world, and I really need more responses from everywhere. There’s only a few questions about how many hours you work, so if you want to help out go to this post of my blog. Cheers!}

Five students, Huh?

I have one Year 8 class that is specifically modified to give attention to those students who need it most.  Five students in the class, with modified content that is supposed to be “within their grasp”.  These are students with identified difficulties in learning mathematics for various reasons.

Although there’s only five students in this class, there’s still a larger range of learning styles, abilities and personalities in this class than in a normal-sized class of 24.  I have to diversify teaching methods and content just as much, if not more, compared to my other classes.  But the biggest problem of all is motivation.

Motivation?

Yeah, motivation.  It’s hard to describe how these students look at the mathematical world, although I’m sure anyone who’s taught a similar class will understand. There’s nothing you can possibly do to make it interesting (not even the WCYDWT suggestions I’ve tried) because a probable response will be a shrugging of the shoulders and a “who cares?”. They’re all awesome kids, and I love them all, but I know I’m not reaching them. Just going through the motions and teaching them what I can. Anything they do pick up is a bonus. It’s a sad state of affairs.

A lack of motivation leads to other problems.  Organisation was an issue, with many students not even bringing anything to class most days.  Absenteeism was high from this class at the beginning of the year.  General refusal to attempt anything asked of them

But over the last four weeks, I have at least got them to look forward to coming to Maths class, which is something that 4 months ago I would have thought impossible.

The Mathematics RPG – Making the entire course into a game

I started awarding XP to students for things that would benefit their learning (re: Behaviour I wanted to see).  These range from “Turning up to class on time” to “submitting homework” and losing points for things that interrupt the flow of conversation.    I started out by giving each student a list of these instructions, with a list of things they can earn and lose points for. Get the sheet here: 8 Modified Maths – The RPG (if you want an editable doc, just ask).

As additional motivation (since points just aren’t enough) I needed a levelling mechanic and a way to specialise their character.  I designed these fairly simplisticly initially, as I just needed something simple I could run with.

The Effect

I had no idea if this was going to work, but anything was worth a try.  Here’s what happened.

The kids didn’t care.  It just seemed like a boring idea, and they thought they’d have to add up their own points or do some maths or something, and it looked too much like Maths.   But I had an excel spreadsheet ready to go to keep track of their score, and had it on the screen during the first lesson I implemented.  Kids started earning some points anyway (just by paying attention) but no one seemed to care.   At the beginning of the next lesson, everyone who was there on time automatically got 10 points, and something surprising happened.  The two who showed up late were UPSET that they didn’t get the points.  Thought they were disadvantaged by not being early, so they worked harder during the lesson to catch up! I’ve never seen these two kids contribute so much in class all year, but all of a sudden they cared about the game.

And from there it’s built and built and built.  Kids have suggested things that should be worth points (and even some things you should lose points for).  If I really want to get attention I can say that participating in this discussion will earn you 20 XP.  They are really excited at the idea of getting to level to choose a specialisation (none of them have it it yet).  They are eager to come to maths to earn more points to get ahead, or to catch up, it’s really exciting to see.  As a side note, they’re getting better results too, but that’s academic.  They’re enjoying coming to class, that’s the important thing.

What would I do differently?

Nothing really.  Starting simple and building on it makes it easy to get to grips with early.  I plan on adding extra mechanics as we go along, when the novelty wears off (it shows no signs of that yet) The only downside is that it’s manageable precisely because there are five students in the class.  I track scores for everyone, I put the leaderboard up on the screen I can see easily who’s got all their gear, etc.  In a class of 24, this could get pretty damn tricky.  I haven’t decided if I’ll attempt to bring a stripped down version to my larger classes or not, but for the moment it’s working.

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Steven Strogatz's Opinionator – NY Times

Posted by forumjoe on May 18, 2010

I’ve just spent an evening reading and re-reading Steven Strogatz series of Opinionator editorials in the new york times: http://opinionator.blogs.nytimes.com/category/steven-strogatz/

I’m absolutely amazed at Strogatz’s ability to explain complex mathematical topics in a way that is understandable by most adults.  The links he provides in his notes sections go to incredible websites that provide hours of enjoyable and informative perusing.  The sort that makes me feel happy about reading about Maths at 10pm at night.

I really wish I could find a way to make my kids read these articles and understand them, but I’ve realised that’s impossible.  By definition, if I show my students an article, they will think it’s boring maths and switch off straight away.  Maybe if I print them out and get the students to paste them to the back of the cereal packet?

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Lockhart's Lament

Posted by forumjoe on September 18, 2009

This is a paper my Head-of-Maths slipped into my pigeon hole the other day, with a sticky note saying “Read this. It should change everything (that you think about Maths).”

http://www.maa.org/devlin/LockhartsLament.pdf

He wasn’t wrong. Written in 2002 by Paul Lockhart, this has become a bible of sorts to several maths teachers, and doing a google search reveals LOTS of people with opinions on it. Anyway, in my effort to spread the knowledge, it’s something every maths teacher (or aspiring maths teacher) should read. It looks daunting, but I suggest that even the casual reader should read the first page, and see if it interests you. If so, print it out, take it home for some casual bed-time reading.

I guess my question for you, dear reader, is what do we do about this? Lockhart suggests lots of problems inherent within the educational culture (and the larger societal culture as well) about the way Mathematics is perceived. He goes on to demonstrate how it should be viewed, but… how do we initiate such large scale change? What’s the first step on the path of revolution?

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Bill The Lizard

Posted by forumjoe on August 27, 2009

My favourite new blog was shown to me by Jonathan a few weeks ago, and it’s the sort of thing I’m looking for. Bill the Lizard is a blog about “programming, math, computer security, learning, science, and technology.” But what interests me the most are the math problems he poses. Many interesting mathematical problems that can be solved by good old-fashioned logical thinking. Often the mathematical solutions require simple mathematics to solve, but can be programmed to analyse the problem further and deeper (The Broken Stick problem is a great example of this.) It’s the sort of thing I’m looking for, but despite the fact that the maths is simple, it’s still a bit beyond some of my grade 7s. What I really want is a website of maths problems like this, but which can be solved using some simpler maths. I like the “Find the fake coin” problem, but you can only go so far with things like that. The key to get kids interested in Mathematics is to make them realise how fun it is, and how beautiful it is, not how useful it is to solve problems. Kids don’t care about problems, they have enough problems in their lives without getting more in the Maths classroom. They need fun, how can we deliver that?

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