Category Archives: proof

Palindromes

No 2 in a series of posts based on Thinking Mathematically (1985) by Mason, Burton, Stacey

This is the sort of exercise I can envisage taking a number of different paths depending on what my students do with it which is exciting. The book walks through a generalisation by looking for the lowest 4-digit palindromic number, 1001 and then noting that subsequent palindromic numbers can be found by adding 110. Since 1001 and 110 are multiples of 11, then all numbers in this series are multiples of 11. However, this series misses out other palindromic numbers, e.g. 7557 so we need to refine it further.

I am intrigued to see if this is indeed a path my students would follow or if we would discover something else in these numbers. Depending on the class, I might start by asking “how many 4 digit palindromic numbers are there?”  Before getting into the general, I would see this as an opportunity for purposeful practice of long division if that was something that my students require.  Some students might need a fair amount of direction to reach a proof, but I would aim to make sure that all students left this lesson with an appreciation of that proof even if I had to lead them through it.

 

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Geometric proof

A quick post to share a useful site that I learned about today.  Thanks to  who shared this during this evening’s #mathscpdchat on Twitter.

I’ve always struggled teaching students how to explain their reasoning when solving angle problems to a sufficient level to gain full marks on a GCSE question.  Here is a nice site that scaffolds this for students by presenting a series of mathematically concise statements that students need to chain together into a coherent and complete reasoning.

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I would get students to write these in their books so they get a good sense of what is required in a full and coherent proof.

There are lots of good examples on this site.

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It was obviously built a while ago, and sadly it doesn’t cover circle theorems, but I have e-mailed the author, so I’ll post any updates here!

 

Proving Pythagoras

Although I’m sure I’ve taught Pythagoras lots of times, I have never really looked at the proofs before either for my own subject knowledge or with students. This may be because I was always happy when students had the understanding of how to apply the theorem and were able to find the missing side and so I left it at that.

Looking at proofs is a good way to deepen understanding of a topic, but generally shouldn’t be attempted the first time the topic is introduced, one of the points made in this comprehensive review of literature on how students approach proof in mathematics written by Danny Brown.

There are something like 140 different proofs of Pythagoras, cut-the-knot.org lists 118 geometric proofs here.

I decided to work through three:

Proof 1

On squared paper, students draw two adjoining squares of side length a and b as follows:

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Next they draw diagonal lines.  The first thing that needs proving is that these two lines are perpendicular which can be done by finding the gradient of each of them.

Screen Shot 2016-06-07 at 21.30.34.pngWe are now starting to get closer to a square of side c. A bit of cutting and rearranging and hopefully they establish that the area that they started with, a²+b² can be re-arranged to form c².

Here is a lovely Geogebra showing how these squares could tessellate for form Pythagorean Tiles.

Proof 2

This one is worth drawing although the scissors won’t help much here.  This is a Geogebra drawing of it (click on it to adjust the lengths):

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A few of my students went down a blind alley with this one assuming that a is double b.  That is why it is useful to have the dynamic drawing to show that this is not the case. The crux to this one is seeing that the red square in the middle has side length (a-b) and then multiplying out (a-b)² to get the area of that square.

Proof 3

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The third one I chose is fairly simple if you can remember the formula for the area of a trapezium! And really, once you’ve played this video over 10 times, nobody will ever forget that!

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Fun with Fibonacci

This is an old one but fun, and a good way to use algebra to show why a trick works.  It’s a similar to showing how Magic Squares work.  It’s not a formal proof as such, but I think it’s a good way to introduce the topic.

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Once students have grasped the basic concept of a Fibonacci Series (something which, in my experience they often see at Primary School even if they can’t remember what it is called), then you are ready to start the trick.

Fibonacci series don’t have to start with a 1 and a 1 as in the diagram above.  You start by asking students which two numbers they want to start with.

Then they get ready to be wowed with your powers of mental arithmetic. Tell them that you will be able to add up the first 10 digits of this sequence in your head faster than they can on calculators. Get one student up to the board to write down the numbers one by one.  TOP TIP here: make sure you have the numbers 1-10 written in a vertical column and that the chosen student writes down each term in the sequence against the numbers.  You should end up with something like this on your board:

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As soon as term 7 goes up on the board, you start calculating.  You should be able to find the sum of the first 10 terms before they even get to term 10 and this is why:

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I quite like doing the calculation on a miniwhiteboard, then writing the answer face down on a students’ desk and then walking to the other side of the room.  Once they have finally totalled the column of numbers on their calculator, you ask a student to have a look under the whiteboard.

And like all good magicians, you DO then go on to reveal the secrets of your trick!

Hippocrates First Theorem

Another one from the fabulous Don Steward:

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You could of course just go straight for the algebraic proof but it does require a level of confidence with surds.  So you might want to scaffold this task. Maybe start by putting some numbers in for the radius of the smaller semi-circle, maybe 2.  You could then do it again with 4 and ask students if they are convinced by that.  (Here are some examples to warn against the dangers of extrapolating what appears to be a pattern).  If you do take the numbers approach it’s good calculator practice.  Can you type the whole expression for the area of the curved shape into the calculator to get an exact answer?

And here is a little GeoGebra drawing to go with it.

 

Stumped by a Foundation GCSE Maths question

I have just returned from an inspiring morning at #mathsmeetglyn organised by @mathsjem watching Don Steward give a fantastic, brain-stretching whistle-stop tour of some of the great problems on his site.

The theme was “Generalising” – he started off by saying that his current mission is to get some generalising into every single lesson because without it, well, you’re not doing real maths.

I filled 10 pages of notes, some of which I’ll come back to, but wanted to share how he took this Edexcel Exemplar Foundation GCSE question and “generalised” it.

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Now, first up, I don’t mind admitting that after working at it for about 3-4 minutes I was completely stumped.  Don had added “You can’t cut the tiles” to the question which was essential in my view, but didn’t actually help me. I was completely fixated on the fact that all tiles have to be in the same orientation.  They don’t.

I think it’s a pretty poor question, probably mainly because I couldn’t do it. But there is a serious point around what questions like this are really testing. If there is a simple “trick” you need to get, is that fair?

A debate to be had there, I’m sure, but more interesting was what Don did with it next.

What other areas can you fit 40 x 30cm “carpet tiles” into and how many do you need? Start with:

120cm x 60cm
120cm x 70cm
120cm x 80cm
120cm x 90cm

Do you need to go any further?  Can you write a general statement from this? i.e. can you prove that all multiples of 10 will work if the width is 120cm?  What other widths does this work for and why? And then, of course, what happens if you try different sized carpet tiles.

It feels a bit like one of those Maths GCSE coursework questions that were set in the days before I was a teacher.  But I really like the idea of taking what is a pretty bad question and turning into some interesting maths.

Are you sure you can prove that?

A nice couple of demonstrations of what makes a proof, i.e. just because you’ve got lots of examples doesn’t mean you’ve proven it.

Mathematicians of the 18th century proved that numbers 31; 331; 3331; 33331; 333331; 3333331; 33333331 are all primes.  This was no mean feat without a calculator.  It was a big tempation to think that all numbers of such kind are primes.  But, the next number is not a prime:

333333331 = 17 x 19607843 

Another classic example is the question of how many areas you get when you cut a circle with chords formed by joining points on the circumference.

CircleCuttingCircumference_1000

You might think you’ve spotted a pattern of doubling each time (or 2^n).  And indeed the next one is 16.  But the one after that is 31.

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The formula is not quite so straightforward and involves combinations:

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In its expanded form it looks even more crazy!

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Wolfram has more details on this problem here.

And some nice discussion of the problem here.