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Ritangle Reminiscences: Use of Technology

Each year Integral Maths runs the Ritangle competition, and it is by far my favourite maths competition.  UKMT Maths Challenges cater for those who love time-pressured competitions, letting learners pit their wits, intuition and mathematical flexibility against one another.  Their Team Challenge incorporates collaboration, and is hugely enjoyable, but if anything even more time-pressured and those who excel at rapid mental or written calculation undoubtedly have an edge.  Ritangle, on the other hand, caters to a much broader cohort of budding mathematicians: the fast and the slow, the intuitive and the methodical, the pen-and-paper wizard and the technology geek.

Ritangle encourages the kind of approach I like to take when solving an interesting mathematical problem: I don’t set myself time limits – I often niggle away at a knotty problem for weeks; I don’t work alone – if I get stuck, I seek inspiration or advice, or a new perspective from someone else; I don’t needlessly ignore the benefits of technology or online research.

This post is just a snapshot of a few of my favourite Ritangle questions from previous years, saying what I liked about them and how I solved them.  I’ve deliberately picked out those questions which lend themselves well to the ‘use of technology’, which is another reason I like the competition so much: students who have never done any coding, or even used a spreadsheet, will see the usefulness of these tools for solving certain types of problem, and the emphasis I place on verifying results independently is hopefully one which they carry back into their A-level Maths lessons and beyond.  Instead of just answering a question one way, then asking the teacher if it’s right, find ways to check your result yourself – there are often quick things you can check, but given the time, there’s no substitute for tackling the problem in two entirely different ways and comparing your results.  I will link to GeoGebra files, Google Spreadsheets or Python code I have used.  If you solved one of these problems using a different approach, or if there’s another ‘technology-friendly’ Ritangle question you’d like me to share my method for, leave a comment.  Just remember that discussing any current Ritangle competition questions goes against the spirit of the competition, so if your comment mentions any it won’t get published (at least, not for a few more months!)

Without further ado, the first question, from 2017:

The Ants Go Marching (2017 Q4)

2017_Q4

Why I like it

Ants and Geometric Series: what’s not to love?  One thing I like about it is the visual nature of the problem.  Solvers can quickly get a sense of what is happening, and how an ant following an infinite set of instructions could ever ‘end up’ anywhere, by simply plotting the first few points (either by hand or with GeoGebra).  I also like the fact that solvers can break down the moves in a number of different ways: the north-south moves of each ant form their own geometric progression, independent of the east-west moves, for instance.  But if we can treat the north-south moves as one single GS (just with a negative common ratio), what’s to stop us treating all moves as a single GS (with an imaginary common ratio)?

How to solve it

We can break apart each ant’s moves into North-South and East-West.  By using the infinite sum formula for a geometric series, we can find the eventual position of each ant.  Alternatively, notice that each move can be described by a complex number: the first ant’s moves can be treated as a geometric series with common ratio 0.9i, and (although the numbers aren’t much more elegant to deal with) it feels somewhat satisfying to deal with an entire two-dimensional journey with a single mathematical trick.

How to check it

The most obvious way is brute force: try moving the ants for a while and see where they end up.  I’ve made a deliberately sub-optimal bit of code to represent this: I’m intentionally making it as close to the original problem as possible, not taking advantage of any mathematical insights such as grouping vertical and horizontal motion separately.  This will ensure that if I have made any errors in my assumptions, they won’t pollute this verification process also: Ritangle2017_Q4 Python Snippet.

 

Moving Squares (2017 Q6)

2017_Q6

Why I like it

It’s a superb example of a problem that looks tricky, but just below the surface is simply a quadratic expression wrapped up in some algebraic fractions.  No knowledge beyond the first few weeks of year 12 is required, and students will have the satisfaction of finding a non-obvious solution to an unfamiliar problem without having to lean on any mathematical tools they don’t already have firmly in their grasp.

How to solve it

Write each of the four areas in terms of the variable c.  When you substitute them all into the fraction, the whole thing simplifies, with a bit of work, to the point where all that remains is some careful thinking and completing the square.  No calculus required.

How to check it

A question like this is crying out for some trial-and-error, preferably via a rough and ready GeoGebra manipulative.  GeoGebra handily allows us to not only create movable components and measure them, but also embed calculations, so we can watch as the fraction we want to minimise is evaluated before our eyes, and see if the numbers we found really work.

 

Shifting Squares (2017 Q9)

2017_Q9

Why I like it

It’s a nice question for beginner coders to cut their teeth on.  The numbers are clearly too big to yield their secrets to a lucky guess, or even to a concerted effort with a pocket calculator.  We’ll need variables, conditionals and loops, but this is an ideal example of a question where coding really helps: there’s no fancy method, just do something basic over and over again until we get the result required.

How to solve it

We’ll need to define our variables, then loop through a few thousand possible values for d, and test the various sums to see if they’re all square.  If they are, give the result: Ritangle2017_Q9 Python Snippet

How to check it

Just like coding trivialises the process of finding solutions, the nature of the problem means that verifying the solution is also pretty straightforward: bung the numbers into your calculator and see if it works.

Digitally Challenged (2018 Preliminary Question 1)

2018_P1

Why I like it

There are lots of ways to solve this.  We can make some headway almost immediately simply by trying a few likely numbers.  The more examples we find, the more patterns we are likely to spot, and it’s a question that lends itself well to different members of the team playing to their various strengths: someone can code it up in Python, while someone else uses Excel, and another person plays around with laws of divisibility and combinations.  Each method has its own pros and cons, and the question is a good way to explore them, and to find ways for the different approaches to complement one another.

How to solve it

Brute force, with Python: looping through all 8 digit numbers takes a while.  Checking to see what digits a number is made from is sufficiently complex for a computer (converting from a binary form to decimal, then extracting digits either by modular arithmetic or converting to text strings), that it pays to optimise your code somewhat.  Only looping through multiples of 18 helps.  Restricting your range to values starting at 11111112 and ending with 33333333 is another easy win.  Will anyone spot that we can loop through multiples of 90 without losing any relevant numbers?  Ritangle2018_P1 Python Snippet.

Base 3, with Excel: there are nearly 100 million 8-digit numbers in base 10, but only 6,561 8-digit combinations of three digits.  A nice lazy way to create them is to make use of the BASE function, which can convert a decimal like 6561 into a base-3 number like 22222222.  We can then swap out all the zeroes for 3s, and treat them as decimal numbers.  Test for division using the MOD function, and Bob’s your uncle.  Ritangle2018_P1 Spreadsheet.

Careful counting, by hand: If a number divides by 18, it must also divide by 2.  If the only allowed digits are 1, 2 or 3, it must end in a 2.  Also, since it divides by 18, it divides by 9, and therefore the digits add up to a multiple of 9.  Since we know already it ends in a 2, that means the first 7 digits combined have to add up to either 7, 16 or 25, etc.  But since they can’t be any greater than 3s, the most they can add up to is 21, which means there are only two cases to consider.  The first case is simple: 11111112.  The second takes some more work, but will yield nicely to a systematic approach.  Notice that the order of the digits won’t matter: if the digit sum is right, it will divide by 9.  Try finding all the ways to have the seven digits add to 21 with as many 3s as possible, then gradually reduce them.  The binomial choose or combinations function may come in handy.  For example, four 3s, one 2 and two 1s is a valid combination.  There are 7 choose 4 ways to place the 3s, followed by 3 ways to place the 2, then the 1s must go in the remaining two places.

How to check it

My approach was to use all three methods described above.  When they all yield the same answer, that’s pretty convincing.  It’s also worth noting that, even partial success with, say, Python or Excel, can give fresh insights (“Why do they all end in a 2?” etc), and if you look closely at the results the Python code spits out, you can see the various rearrangements of the digits just as the by-hand method predicts.

 

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A-level Teaching, Uncategorized

Groundhog Day and planning to fail

Some mathematical proofs are essentially magic, aren’t they?  “OK, there’s nothing up my sleeve, right?  Now watch what happens when I multiply by \frac{1-cos(x)}{1-cos(x)}!”  “Stay with me now: we’re going to take logs for reasons that will become clear later, then substitute tan(2t) for x.”

Students say “I understand what you did, but how did you know it would work?  I would never be able to come up with that.”  This is a big problem with the way we teach maths, especially at the more advanced level.  We want students to develop into resilient problem solvers who learn from their mistakes, but most of the time all they see modelled from their teachers is carefully crafted examples, pruned and polished until they are as efficient and sanitized as we can make them, with no hint of the rough-n-ready manner in which they were born.

When’s the last time you stood up in front of a class of students and tried to tackle a problem that you didn’t already know how to solve?  Not just through inadequate subject knowledge or bad planning, but in a deliberate attempt to model failure and how to deal with it for our students.  They already know what success looks like, and half the time we make it look so effortless they get even more demoralised when their own experience doesn’t match up.  Don’t they deserve to watch us flounder, sweat, get stuck and be confused once in a while, too?

Even if we occasionally take a punt at an unfamiliar question in class, I’m guessing for most topics we can see how the problem will play out before we start (or at the very least, its general trajectory), which doesn’t quite fit the bill.  To get well and truly stuck, you either have to be testing the boundary conditions of your subject knowledge comfort zone (Further Maths A-level has a few areas like this for me, still), or tackle something like a UKMT maths challenge problem, or something from the excellent Integral Maths Ritangle competition.

Solving a maths problem to me feels like the film Groundhog Day.  The first time you tackle it, it’s a mess.  You may not get the solution at all, or even close to it.  Perhaps when you come back to it, having reflected on the dead-ends and false trails, you can do better.  Define your variables, because that tripped you up last time.  Clarify your assumptions, and tidy up your notation because those simultaneous equations get messy fast.  A couple more attempts, though, and you can solve it in your sleep.  You even spot shortcuts that would never have occurred to you the first time around.  And when you miraculously pull the perfect substitution out of your hat the next day in class, and they whisper “It worked!  But how did he know to do that??” it’ll look like magic when really it was good old-fashioned hind-sight.  A colleague shared a geometrical problem with me that he said could be done in your head “after you know the answer”.  Which makes perfect sense to mathematicians: solving a problem doesn’t just give you a solution, but an insight into the entire scenario.

When I’m feeling especially reckless, I bring my selected unsolved questions into the classroom.  Narrating my thought processes, and letting the whiteboard fill up in the half-structured half-chaotic manner that back-of-the-envelope problems do, I can show students what it looks like for me to struggle with an unfamiliar problem.  They see the wild goose chases, the uncertainty, the dubious assumptions and second-guessing.  They see me get stuck, and take stock of what I’ve tried and how it helped, if at all.  They see me back-track, and start again from scratch, and get it wrong, and hunt for that dropped minus sign, and look for ways to verify my solution.  And with any luck, in the post-mortem, we’ll spot that elusive, elegant pathway through the maze which is often only visible from the end.

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Uncategorized

The True Scale Multiplication Grid

I came up with a neat idea for a multiplication grid visual the other day, and stuck it up on Twitter where it has been doing the rounds with unprecedented alacrity:

tweet

I’ve loved reading comments and seeing how people are using the grids already, with fellow teachers, students and your own kids (I’m making one on A1 squared paper for my son this weekend – here’s one 3-year-old who will know what multiplication means before he learns his tables, if I can manage it!)  A few of you came up with ideas for variations I could do, including starting the grid from the bottom-left to mimic a Cartesian coordinate grid, and emphasizing square numbers.  I’ve also done one with the prime factorization of numbers on one side of the diagonal, which I quite like.  I’ve put all the images together into a single pdf document to make it easier to access.  It’s on my website at www.thechalkface.net/resources/true_scale_multiplication_grid.pdf:

pdf

Please feel free to mess about with these, share them, modify them, distribute or display them.  I’d love to hear what you get up to, and in particular if you come up with any great ideas for investigations please share them in the comments section below or on twitter: @the_chalkface

My plan with the multiplication table was to give learners a clearer intuition for multiplication.  It is my firm belief that most difficulties students encounter with ‘hard’ topics like proportion, fractions and algebra usually stem from an inadequate grasp of, and familiarity with, multiplication.  There’s no point trying to teach expanding brackets until the distributive property makes sense numerically, for example, and the right sort of visual might help students to see not just the what, but the why.

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Playing with Functions (and the function of play) in A-level Mathematics

I believe we develop our understanding of mathematical concepts in two main ways: making connections, and building familiarity.  New ideas are never wholly new – they build on previous ideas, or take one aspect of something we know and develop it somehow.  But until we spend time messing around with them, our understanding will only remain skin-deep.

Making connections – linking the new idea to prior understanding – will involve well-chosen language, examples and analogies on the part of the teacher.  However incomplete or flawed these are, if chosen carefully so as not to actively hinder future development of ideas, they can form a vital (if temporary) bridge to otherwise inaccessible concepts.  I have found the ADEPT method outlined on www.betterexplained.com to be incredibly helpful in developing and adapting my own explanations of tricky concepts to students.

Building familiarity – playing around with mathematical ideas – should involve both problem solving and open investigation .  Because of the time-consuming nature of play, and the fact that we can’t know in advance what the ‘learning outcomes’ may be, this method of teaching often takes a back seat in the maths classroom.  Fortunately this is not the case everywhere.  http://www.inquirymaths.co.uk/ champions the practice of providing students with a well-chosen prompt and giving them the freedom to explore the ideas that arise.

Bringing together both of these aspects – making connections and building familiarity – the rich tasks provided on www.undergroundmathematics.org are designed to emphasize the links between A-level topics and also encourage more open investigation than is found in the typical textbook or exam question.  The hard part is providing sufficient direction for students to access the problem, but leave enough room for the creative investigative problem solving that maths is really all about.

I recently used a couple of tasks from Underground Maths in a plenary lesson on functions for a year 12 class.  By this point in the year, students have worked with functions and graphs in the context of straight lines, circles, trigonometry, polynomials and calculus, and even statistical distributions or kinematic graphs, and a solid grasp of functions will underpin much of the upcoming year’s content, from parametric curves to circular motion, so I wanted to make sure students weren’t just memorizing the rules for translations and stretches, but developing a sense for functions that goes a bit deeper.

Slippery Slopes:

https://undergroundmathematics.org/calculus-meets-functions/slippery-slopes

This activity encourages students to consider how transformations affect gradient.  It provides sufficient direction to be accessible while keeping the questions broad enough to promote discussion – there’s no one correct approach, and working in a group gave them the freedom to discuss possible methods and gain alternative perspectives.

One student used differentiation to investigate the effect of stretches and translations on cubic functions, and was then able to make the next step, differentiating  to , proving the same results for a much more general case.  We also had discussion about how the same sort of process governs transformations of data (specifically why standard deviation is invariant under translation).

The second part of this activity turned out to be a great way for students to grapple with what changes and what stays the same when functions are transformed.  In particular, the fact that functions were only described in terms of a general function  meant students didn’t get bogged down in specifics, which can make it hard to keep a clear view of the problem, and could focus instead on how the input and output were affected by each transformation.

Can you find…

https://undergroundmathematics.org/trigonometry-triangles-to-functions/can-you-find-trigonometry-edition

Many students of physics already had a good grasp of frequency and amplitude, and were quick to apply their understanding in this area to the mathematical descriptions of horizontal and vertical stretches of trig graphs.  Again, students worked in small groups, but were encouraged to use GeoGebra up at the front to test predictions.  Some students dived straight in, finding valid solutions straight away for the one graph version, while others weren’t sure where to begin.  As they compared their approaches within groups, those who struggled saw methods they could attempt, and those who already had one solution were challenged to find alternatives, and then to generalise further.  Some students ruled out horizontal stretches for the tan graph problem because they wanted to maintain the same vertical asymptote, while others tried anyway, and spotted that the right scale factor would simply move a new asymptote into its place.  Some solutions were considered more elegant than others, but this was often a matter of personal opinion, and it was so nice as a teacher to replace “No, that’s not the right answer” with “Yes, that also works – and can anyone spot any more?  Can we describe them all?”

One student decided that moving a sine wave up by 2 was too boring a solution to the first challenge, so spent a happy 10 minutes looking for ever more obscure ways to solve the same problem.  I love lessons where students ask me questions I can’t answer, and it took me some considerable messing around afterwards for me to find an equation that would create the graph one student had sketched:

functions1

Unless you get the constants just right, the graph unravels like a broken slinky, and if you try to get too fancy you might just end up with this weird scribble:

functions2

This graph prompted a discussion about what it means to be a function, and provided a nice precursor to the whole area of parametric curves.  Whether or not it fit the requirements of the question was no longer the issue – the original activity provided a vital prompt, but when it was no longer needed to direct students’ curiosity, it was simply put to one side.

Keeping with the making connections theme, one student drew a periodic graph that wasn’t quite a trig function, and, with a bit of messing around with modulus function and vertical translations, we were able to reverse-engineer a partial equation:

functions3

An equation for the endless version was beyond us during the lesson, but the floor function came in very handily when I tackled it again after class:

functions4

What’s the difference between an abacus, a calculator and a smartphone?

What’s the difference between 8 \times 8, x^2 and a(x+b)^2+c ?

Some tools do one job, some many, and some have potential beyond what their designers ever imagined.  Functions are to algebra what algebra is to number – they provide a wide angle lens that allows us to ‘see’ the algebraic structures and patterns which would be almost impossible otherwise.  Now every student with a phone can use GeoGebra to investigate functions, I can’t wait to see the impact on understanding.

 

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Why logarithms still make sense

Logarithms are a notoriously tricky topic for A-level students to grasp.  Having just about got their heads around laws of indices, we suddenly expect them to blithely apply a new – seemingly arbitrary – list of rules that turn index rules literally on their heads:

lawsoflogs

Some people argue that logarithms are an outdated topic and have no place in our curriculum.  They were useful back in the day, but smartphones have all but made calculators obsolete, let alone slide rules and log tables.  What need have we for this confusingly different mathematics?

slide rule.png

The key idea underpinning logarithms is this: They are a way of counting multiplicatively.  And there are, in fact, many good reasons for wanting to count in this way, rather than in the traditional additive sense.

I asked my year 12 class how many Christmas trees were sold in the US each year.  One guessed “300 million” and another guessed (I suspect facetiously) “3”.  Given that the true number is around 30 million, which of these students was closer to the truth?  And if you went with the 300 million guy (who overestimated by 270 million) rather than the guy with the ludicrous underestimate (29,999,997 too low) then presumably you are using a different scale to gauge accuracy than a straightforward additive one.  Consciously or not, we usually prefer a logarithmic scale for such comparisons, especially when the values vary considerably in size.  The first guess was 10 times too big, but the second guess was 10 million times too small.

Without thinking too hard, where do you think the number 1,000 belongs on this number line?

numberline

While not many people would have put it right in the middle, most people’s initial reaction is somewhat further along than an additive scale would place it.  The true position is somewhere inside the blob above ‘1’ on an additive scale.  But isn’t there some sense in which a million is ‘halfway’ from a thousand to a billion?  Yes – a multiplicative sense.

Dave gets a 20k pay rise.  Is he much better off?  It depends on what he was on before, doesn’t it?  On the one hand, 20k is £20k, but £520k is only 4% bigger than £500k, while £24k is a 500% increase on £4k.  In fact, whenever we invoke proportion, we are dealing with a logarithmic number sense.

With this new form of number sense, we need new rules for moving along the number line, and that’s where thinking about numbers from the point of view of the exponent starts to really pay off.

If you haven’t yet seen the superb ViHart video on logarithms, I highly recommend it.  It makes sense of this multiplicative number scale beautifully.

numberlinerules

 

When you start to look around you, there is evidence of a multiplicative number sense everywhere.

  • How can going down from 2 cigarettes to 1 be harder than from 20 to 19?
  • In what universe is having a 6th kid less of a big deal than having a 2nd?
  • How could a 10th friend joining a party less noteworthy than the 3rd one?
  • Who drives 10 minutes to save £5 off a £15 item, but not a £150 one?

numbersense.png

Given that logarithms and this logarithmic number sense are such a natural part of how our brains interpret quantities, it makes perfect sense to try and find a rigorous mathematical way to represent and manipulate them.

Once the ideas behind logarithms make more sense, the intuition behind the log rules starts to become clearer:

intuition

The next step is to develop notation.  We need to provide students with a convenient way to ‘read’ the logarithm notation: “What power do I need, with this given base, to result in this number?”

lognotation

And once we’re familiar with this, knowing which bases are most commonly used and why is always helpful:

popularbases.png

And solving real problems early is always helpful, I find.  Compound interest problems should be well within the capacity of A-level students, but as soon as the unknown value is the length of time rather than the interest rate or the amounts, it becomes a problem of trial and error… or logarithms.  It’s worth explicitly demonstrating the distinction between problems involving powers that we know (that are usually solved by rooting) and those involving powers that we don’t know (that need this whole new function to solve):problems

The ‘number of digits’ interpretation of logarithms is a particularly helpful one.  While log10 isn’t exactly the same, if you round it up it gives you exactly that.  A really helpful idea from the excellent new Underground Mathematics website involves working out just how far up an exponential graph will go: https://undergroundmathematics.org/exp-and-log/reach-for-the-stars.  We need to grasp the fact that, as fast as exponential graphs increase, logarithms do the opposite, requiring us to travel literally miles along the horizontal axis just to reach a reasonable height on the y-axis (for the record, 500 miles along gives you about 1 handspan in height on a log10 graph).

graph

I’ve made a PowerPoint with many of these ideas included, an Introducing Logarithms accompanying 8-page booklet, and a follow-on Working With Logarithms 8-page booklet:

powerpoint.png

intro

working.png

As usual, I’d welcome any feedback / ideas / corrections on the ideas & resources included.

Just remember, next time you come second in a pub quiz tie-breaker, to check whether your answer was multiplicatively closest.  Although you should bear in mind that, until our newly enlightened year 12s start administering them, “because of logarithms” is an argument unlikely to carry much weight…

 

 

 

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What are the chances of teaching a great probability lesson?

In my experience a probability lesson runs smoothly about 50% of the time.  Some days those pesky dice will throw sixes all afternoon, or some lucky sod will beat your thousand-to-one odds and win a fiver in the coin-toss challenge.  But if you play your cards right (‘scuse the pun) these lessons can be fun for you and the students as well as a great way to challenge our in-built misconceptions about randomness and chance.

Lucky Streak: Let’s start out with my headline act.  I encourage you to try this out for yourself before you read the spoiler that follows, to convince yourself it works:

Can you tell the difference between a fake list and a real one?

Have you jotted down your fake list yet?  (A string of 0s and 1s in Notepad will do).  Now do the experiment for real (if you’re feeling lazy – or skint – use https://www.random.org/coins/?num=20&cur=60-gbp.1pound).  Did you notice anything odd about the real coin tosses?  The often surprising fact is that runs occur more frequently, and are longer, than we generally assume.  This experiment works better with more coin tosses, but 20 is usually enough to see some discrepancy between an invented list and a genuine one.  According to http://wizardofodds.com/image/ask-the-wizard/streaks.pdf, there is a more than 75% chance of a run of 4, and a 45% chance of a run of 5, though the actual maths involved is a bit beyond high school.  Fortunately he includes a handy graph.  The key idea is simple enough: a run of 5 means four coin tosses must show up the same as the previous one, which occurs with probability one in sixteen.  Makes sense that you’d expect it to happen somewhere in your 20 coin tosses reasonably often.  While the genuine results sometimes don’t have any striking streaks, it’s very rare for invented lists to have any more than 3 in a row before our built-in equalizer tells us we should really be balancing out the proportions.  The other day I tried this with four teams for my plenary activity, and correctly identified the fake from each of the four teams.  Coins, dice and roulette wheels are like Dory from Finding Nemo – getting a tail last time makes it neither more nor less likely to get one this time.  We teachers know this, but are just as likely to fall into the trap of avoiding streaks as our students.

The activity described above is a nice way to get students thinking about what they understand by ‘random’.  When my wife makes a patchwork quilt, a lot of time and effort goes into carefully arranging the various squares of colour to make it look random.  Itunes ‘shuffle’ mode is the same – the algorithm had to be modified because true random song selection would mean you sometimes have to listen to the same song twice in a row, or three from the same artist one after the other.  Once students have got it into their heads that the number 4 has a one in six chance of showing up, they expect it to show up pretty much once in every six throws.  Which, for small numbers of throws, isn’t all that likely.  I use this simple Excel spreadsheet to demonstrate the feebleness of small sample sizes, but also the awesome power of large ones when it comes to making predictions from a probability:

Probability Simulation

This can be modified to reflect whatever probability distribution you choose.  It’s worth pointing out that there are still ‘large’ gaps numerically between the most common and least common outcomes for bigger sample sizes, but they tend to become less significant as a proportion of the whole.

Twenty wrongs makes a right:

Probability Investigation

I tried this recently with more success than I expected.  Students completed the probability experiment for homework (recording the sum of two dice rolls a bunch of times), and I collated the class results in a spreadsheet.  After my trick with the coin tosses they knew they wouldn’t get away with making up results, so their tables were genuine and yet not particularly close to expected values.  For a sample of just 36 dice rolls each this is to be expected, but it was really interesting to show them that even their motley collection of not-too-good-looking results, when combined with each other, averaged out to resemble the theoretical distribution very closely.  Again, the power of large samples coming into its own.

Random Grid

I use the spreadsheet above to emphasize that unlikely events not only happen, but happen with surprising predictability for a large enough sample size.  How can Snickers run entire factories, employ staff and spend millions of pounds all on the off-chance that someone chooses their particular chocolate bar on a whim at the checkout?  Because a tiny percentage of a large enough number is still a decent size, and while it may vary a bit, it’s actually surprisingly unlikely to fall too low.  This all starts to sound like Asimov’s ‘psychohistory’, which takes the intriguing idea of predictability for large numbers to whole new levels.  Nevertheless, it’s one of the most powerful ideas of probability, and I think it’s worth highlighting in between the multiplying of fractions and the drawing of tree diagrams.  Incidentally, one of my self-marking homeworks is on the topic of relative frequency:

Relative Frequency self-marking homework

On a similar theme, I tested a theory with my year 9 class last week on the power of collective guess-work.  Having just shown them how uncannily predictable their combined probability homework results were (however unpredictable each individual result may have been), I told them how the average guess for, say, the weight of a pig or the number of sweets in a jar, has been shown to be better, often, than any one person’s prediction.  They clamoured for proof (in retrospect, possibly just for sweets), so I used a picture I generally reserve for trial and error lessons of a pile of bottles I once counted, and the class average guess for the number of bottles in the pile was within 1.7% of the real total!  I was seriously impressed.

How Many Bottles?

Randomly generate American men: I’ve included links to a few Excel files I use for probability, but it can be really helpful to be able to throw something specific together on the spur of the moment.  So here’s a few Excel formulae you may find useful: =RAND() is the obvious one.  Gives a random number (rectangular distribution) between 0 and 1.  May equal 0 but is always less than 1.  Multiply by it to scale up the range, and add to move the range.  So =5*RAND()+2 gives numbers greater than or equal to 2 but less than 7.  You can even round, but =RANDBETWEEN(1,6) is a simpler way to generate random integers (from 1 to 6 inclusive).  Here’s a clever one for the statistics boffins among you: =NORM.INV(RAND(),0,1) gives a random number normally distributed about the mean of 0 with standard deviation 1.  It uses RAND() as a probability.  Use it to create realistic sample sets like heights or weights.  I have it on good authority that a normal distribution with mean 178cm and standard deviation 8cm describes the heights of American men.  So =NORM.INV(RAND(),178,8) will randomly generate for you an American man.  Well, his height, anyway.  Nifty, no?  For those of you teaching A-level Statistics, the next spreadsheet combines the statistical power of Excel with the actual values provided in the tables in the AQA formula book so you can find not only the correct answers but also the answers your students should be getting.  Includes binomial and normal distributions.

Statistical Distribution Calculator

And if you want a nice interactive way to introduce that beautiful bell curve of the normal distribution, check out my GeoGebra version:

Custom Normal Distribution

False Positive: Conditional probability is set to be a bigger part of the new GCSE, I understand.  The counter-intuitive way that medical testing works makes this a really interesting example of conditional probability.  The question you want to know is “Given that I tested positive, what’s the chance that I’m sick”, and it turns out the biggest factor in determining this probability is the proportion of people who get tested who are actually sick.

False Positive

Random Walks: My Snail Race activity was about as fast-paced and exciting as it sounds, but at least it gave me the chance to test out a nifty method for generating random numbers on the fly.  Two competitors (who are not cooperating) hold up between 1 and 4 fingers simultaneously.  Add the totals together, and use the remainder when divided by 4 to determine the direction to move next (left, up, right or down).  This is a nice two-dimensional way to visualise randomness (and notice how often you double-back on yourself compared to how often you might have done if you were making it up).

“I beat you, you beat him, he beats me.  Wait… what?”: A probability ideas post would not be complete without at least a mention of the ingenious ‘Grime Dice’ developed by MathsGear and available here: http://mathsgear.co.uk/products/non-transitive-grime-dice .  They are non-transitive, so if you memorize the order correctly this is yet another way you can reliably beat your class at an apparently fair game.  Unless you memorize the order backwards, like I did.  That’s a great way to reliably lose.

 

Er, that’s not my card: Of course, for every success there’s bound to be a probability experiment that backfires or just plain doesn’t work.  Last lesson I threw 9 tails in 10 tosses while trying to demonstrate how much more likely 4, 5 or 6 was.  I lost £10 to a student last year when he guessed 10 coin tosses in a row.  More recently I attempted – with much skepticism, it must be said – to sway the random choice of a playing card using auto-suggestion.  A subliminal poster campaign was launched a few days prior to our probability lesson, comprised of the following:

Auto-suggestion

Not only did nobody think of the six of diamonds, but only two people in the whole class even had diamonds as their chosen suit.  My posters must have been sub-subliminal.  Sticking with the null hypothesis on this one for the time being, I reckon.

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Prime version of the 100 grid
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Number sense and the value of primes in the classroom

I loved maths at school.  I hated learning my times tables.  The beauty and underlying structure, the connections and patterns, the tricks and simplifications were all lost on me since the teaching method involved writing down every times table question from 1 to 12, in order, over and over again.  Natural numbers are just that – natural.  They grew inescapably from our desire to accurately describe the world around us, and their fundamental properties are also pleasingly grounded in concrete patterns and structures.  By laying out objects in rectangular groupings we can not only calculate but physically experience how numbers can be separated fairly into a number of groups.  Children should learn their tables, but they should develop a sense of the numbers from their inherent structure, not be required to memorize as-yet meaningless lists of facts.  I want my son to be able to work out 14 times 4 as confidently as 7 times 8, and understand that it’s the same as 3.5 times 16 to boot.  The structure and patterns behind numbers are more mysterious than their simple roots might suggest.  Take a look at these patterns and see if you can figure out what they mean:

Tables Patterns

Each number forms a unique pattern. What determines the complexity and nature of the pattern? Can you deduce the rules?

Some numbers refuse to be broken down nicely into groups.  They resist the urge to be separated beyond the trivial groupings of ‘all together’ or ‘all apart’.  11 sweets won’t share nicely between any group except a group of 11 or a group of 1.  Most normal people might note the anomaly, perhaps try to buy 12 sweets the next time and think no more about it.  Meanwhile mathematicians, in our insatiable desire to look for patterns and rules, try – and, for the most part, gloriously fail – to find an elegant explanation for the distribution of these so-called prime numbers.

Prime Factorization of the 100 grid

Can you see the connection between numbers with the same colour coding?

The study of prime numbers, and number theory in general, is one of the most abstract areas of mathematics.  Until recently, with some clever computer scientists using their very intractability to develop encryption algorithms, there was no real application anywhere in science or technology.  Which makes the topic either pointless or the purest of mathematics depending on your perspective.

As a teacher, I actually hold a position that isn’t quite either of those.  Primes do have specific applications to my field – I use them when making up quadratics to factorise, or fractions problems that will nicely pre-simplify, and they also enable me to do all sorts of mental maths magic on the board without pausing in my explanation (which was what impressed me most at A-level about my teachers).  Here’s a few nifty things you may not have come across yet in your teaching of these topics:

Number of factors from prime decomposition:  By following a straightforward branching method, any number can be separated into its unique product of primes.

primedecomposition

From here, it is worth recognising that every factor is simply a combination, or subset, of those primes.  24 is 2 x 2 x 2 x 3, so its factors can have anything from no 2s up to all 3, and either no 3s or one 3.  We can actually count the factors directly from the prime decomposition.  How many 2s? (0, 1, 2 or 3) and how many 3s? (0 or 1) means 4  options followed by 2 options, giving 8 possible combinations (from no 2s and no 3s: 1, up to three 2s and one 3: 24).  While primes themselves may hide their structure fiendishly well, by following this rule we can find patterns involving the number of factors for different types of number.  Any number which is a power of a single prime will have one greater factors than the power (so 2 to the 6 has 7 factors).  Powers of a number composed of two primes (such as 10) will have a square number of factors (10 has 4, 100 has 9, 1000 has 16, etc).  Some numbers are special to us because of our number system (base 10).  Count up the number of 2s and 5s in the prime factorisation of your number, and the smaller of the two numbers will give you the number of zeroes at the end of your number.  Lastly, if you want an alternative explanation to why square numbers are the only ones with an odd number of factors, use the ‘product of one more than each power’ way of counting factors to see that this will only be odd if every power is even.

To investigate these patterns further, use my Factors Investigation:

Factors Investigation

How does the number of factors link to the prime decomposition?

Recurring decimals: Ever wondered why fractions of powers of 2 (a half, a quarter, etc) look like powers of 5, and vice versa?  As you may suspect, it comes down to the fact that 2 times 5 gives 10, the base of our number system.  It’s quite an elegant way to demonstrate why some fractions give terminating expansions while others are recurring.  To turn an eighth into a fraction whose denominator is a power of 10 would require multiplying top and bottom by 125, and that’s where 0.125 comes from.  Have students use a calculator to identify recurring and terminating decimals for the first few numbers, then look at prime factorisations to see if they can predict which numbers will recur and which terminate:

Recurring Decimals Task

Linking the prime decomposition with whether a decimal expansion terminates or recurs.

The terminating decimals have denominators which contain no primes other than 2 or 5.  Why?  Base 10, of course.  In base 12 we would get terminating decimals from numbers built from 2s and 3s (so a half, a third, a quarter and a sixth would be 0.6, 0.4, 0.3 and 0.2, although a fifth would now be 0.24972497… – that one took me some time!).  Try this with binary and numbers that would be fine in decimal, like 0.2, become recurring (leading to floating point rounding errors, in case you’re curious how computers can sometimes get 0.1+0.2 wrong).

Modular arithmetic: I’m only just beginning to learn about this, and it’s not really that closely related to what I teach, but if you want to be able to say ‘primes are used in, er, computer security’ with a little more confidence, I recommend brushing up on your modular arithmetic and investigating public key encryption.  To whet your appetite, I can convince you that the last digit of the largest known prime has to be a 1 using nothing very complicated:

Prime Ending

How the largest known prime ends, and how we know.

I strongly recommend you carve out some time with or without your classes to investigate prime numbers some more.  Here’s a spreadsheet I’ve created which lists the prime factorisation and factors of numbers, and allows you to quickly and easily examine lists of numbers.  (Thanks to @colinjthomas for writing, implementing and sending me a 5-minute program to factorise the first hundred-thousand numbers all in the time it took my program to work through the first 5 thousand):

Primes Spreadsheet (right-click to download)

Enter any natural number up to 100,000 for its prime factor decomposition and its position relative to nearby primes.

It should be noted that recent Casio calculators have a ‘FACT’ button which allows students to find the prime factorisation of numbers directly (I don’t have this button, so I downloaded an app instead – you never know when you might need to factorise a number in a hurry…)

And lastly, if you haven’t seen datapointed.net’s visualizations of numbers I highly recommend it – a great representation of numbers in terms of their factors:

Visualization of numbers by factor

Primes can only ever be one group, but composite numbers break down in all sorts of cool ways.

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