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Month: July 2018

A few things you may not know about Pythagoras

Pythagoras could be the first famous mathematician we learn about at school, when we learn the famous formula relating to triangles
h2 = a2 + b2

But how much do we know about the man behind the theorem?

Pythagoras lived 6th century BC.  He was a Greek but settled with his followers on an island off the coast of what is now Italy – But then the ancient Greeks travelled all over the Mediterranean  They lived as a cult in an uneasy relationship with the locals, and were probably driven out before their leader died in 495BC.

The Pythagoreans believed, among other things

  • In re-incarnation – all souls, animal and man, were immortal and moved into a new body when dead
  • Probably for that reason, they were strict vegetarians (though this is in dispute), and nor did they eat beans
  • They were strong believers in numerology – All numbers had meanings,, and quite a few were considered sacred including 3, 7 and 10 – They never gathered in groups greater than 10.
  • That the planets and stars move according to musical equations – and produce  a symphony that we can not hear
  • They were keen athletes; dancing and walks were important features of Pythagorean life.
  • New members had to stay silent for the first five years.
  • Ahead of its time, women were allowed to be full members (Athenian democracy, which cam a little later did not permit women to take part)

But its not clear that they did invent the theorem, which was know to the Chinese and Babylonians Centuries earlier. It is thought that they produced one of the first recognised proofs

 

 

A problem with factors

A problem question I tackled with a student yesterday asked..

What is the smallest number that has 1, 2, 3, 4, 5, 6, 7, 8 and 9 as a factor

I had just taken along a pack of ‘challenging questions’  because he is an able student but needs a challenge. As is happens, this is a question about Lowest Common Multiple, and I had been looking at this a day earlier with another student.

Anyway, yesterday’s student was quick to point out that we can ignore 1 because ‘1 goes into everything’. That was a good start!

He then started listing all the numbers 9 goes into – the 9 times table – and checking off each number against 8, 7, 6 and so on…  When I showed him the quicker way I’m sharing below here, he admitted ‘That would have taken a long time”

The quicker method is to split all the numbers into their prime factors

2 = 2;  3 = 3; 5 = 5; 7 = 7 – thats the prime numbers in the list

4 = 2 x 2;   6 = 2 x 3;   8 = 2 x 2 x 2;  9 = 3 x 3

Now,  consider the answer we are going to get.  As I usually do I’ll give this a letter, but for a change I’ll call it Z!

Z is just a number, even if we don’t know what it is yet. so we will be able to write Z as prime factors, and, those Prime factors are going to be a bit like the prime factors we’ve already found for all the numbers.

We could find Z by combining all the prime factor lists we have, but we will find we don’t need ALL of them.

For example 8 = 2 x 2 x 2  – and we find we don’t also need to add in the 2s from the 4 and the 6! With the 2s in the 8, we have them covered.

Taking that for all the numbers, for each prime number in our list, we only need as many as covers the most in any number.

2 x 2 x 2 – because 8 is the number with the most 2s

3 x 3 – because 2 is the most number of 3s in the prime factors of our numbers

then 5 and 7, because they only appear once in any list

The answer is – 2 x 2 x 2 x 3 x 3 x 5 x 7. You can do this on your calculator and get 2520 – The answer to this question)

[Actually, I try not to use a calculator unless I have to; keeps my mind sharp! Instead I pick numbers from the list that make multiplying in stages easier.
Pick the 2 x 5 = 10.  The 7 x 3 x 3 = 63 – Gives 630. Then double this twice for the other 2 2s  – 1260  then 2520!]

 

 

Tutor Note: On Scales and maps

The last post was longer than I originally planned so I thought I would make this as an extra post to cover a point aimed mainly at fellow tutors and educators.

With some students, getting them to make good use of the paper is challenging for me. They draw in just the bottom corner. Though this experience is more to do with bar charts and similar graphs, its equally the case with maps and scale drawings.  But make mistake, choosing a good scale is NOT a trivial matter, but sometimes it feels it should be easy, which means it can be a hard thing to tutor. Teaching something that people know is hard – quadratic equations maybe – is a challenge  but its a challenge able students are up for.

Trying to get over a point on something where the size of the challenge isn’t immediately obvious is a whole different ball game.

Drawing the Park – The Answer

Yesterday I posed the question of how to draw a map of a playing field.  The first thing to do is decide on the scale. I like to see people use as much of the paper as possible, but there is another consideration too, as I shall show.

With some students, getting them to make good use of the paper is challenging for me. They draw in just the bottom corner. Though this experience is more to do with bar charts and similar graphs, its equally the case with maps and scale drawings.  But make mistake, choosing a good scale is NOT a trivial matter – I’ll make another post on this at the moment, directed at other tutors.

In this case we can use all the paper by using a scale of 1cm = 6m – Note that if we think centimetres for metres, 150 = 6 x 25 and 120 = 6 x 20.   I’d advise against this. In fact I’d advise against any scale that uses factor not based on 5, 2 or 1 (That is 50, 500, 0.5, 20, 200, 0.2, 10, 100 and so on).  These scales make the Maths much easier to understand, both in making the drawing and interpreting it.

I say this from experience without wanting to justify it much further now. Just think of times you have been abroad and the exchange rate is £1 = 60 of the local currency. The mental arithmetic working out how much you are spending becomes tricky.

I’ve waffled a lot today, lets get down to business.   I recommend a scale for this map with  1cm = 10m – 1:1000.

We would draw the full park as rectangle 15cm by 12cm. This only uses part of the paper but we do need space to a title, key and scale – and its better than the 1/4 of the space available I sometimes see.

We draw on the football pitch now. This will be 10cm by 6cm. Where we place it can be ‘trial an error. In fact, if you have scissors to hand (which won’t be likely in an exam!) we can cut out a rectangle with those dimensions and move it round the larger rectangle. We can also cut out a rectangle that is 2cm by 2cm for the play area.   The trees need to be 1cm from the edge and each other and at least 2cm from the football pitch. There is more than one solution but here is mine, with the distances shown in cm in my drawing, as your browser size won’t show the same distances
When you think you’ve finished its worth checking each of you positions and measurements again, to check they comply with the rules

Drawing our own map

In my last post I describe how we could get information about distances from a map using a scale. In this post we will look at how we can use the idea of a scale to draw a map of our own

Think of the following question : Source, my own imagination but I have seen similar questions in Exam papers

My local park is 150m long and 120m wide. We need to plan the park to have one football pitch (100m by 60m), a play area for younger children (20m by 20m) and places to plant 5 trees. Each tree should be at least 10m from the edge of the park and 20m from the football pitch. There should be 10m at least between each item.

We are given a sheet of paper measuring 25cm by 20cm.

See if you can have a go. The answer will be posted tomorrow.

So how do Maps work?

Since I’ve spent a weekend studying maps, I thought a quick post on the maths involved would be in order. Although it might not be immediately obvious, what we are looking at are Ratios – and this can be a pleasant change from mixing paint which is what many questions in Exams about ratio seem to be about!

Now here is a map of one of my favourite places in the world – and comments from anyone who agrees are welcome!  The point is, a map represents the place you want to visit so obviously has to be a lot smaller than the place itself!  (Ok so this reminds me of a favourite Blackadder joke but lets not get off the point)

The map shown here is to the scale 1:25000 if you have it before you. (I can’t make a similar claim from the picture you can see, as that will depend on the size of your browser!)

If you measure on the map that you are 2cm from the car park and pub*, how far do have left to walk?

2 * 25000 = 50000cm.  Which is great but we don’t usually measure walking distances in centimetres. 50000cm = 500m or 0.5km

Activity  : Find a map of your town and measure the distance to a place your often visit.  Don’t assume your map has the scale 1:25000.  The scale for your map will be written somewhere, perhaps even the front of the map.

*Actually the first pub I ever had a pint of beer!

Answer to the Number machine Question

Here is a confession – If anyone read my last blog post they will have seen a mistake with the question I posed at the end – If you are looking now, this has now been corrected!

The question asked to fill the second box in.  The first box says  x 2 , so if we feed in 6, the number in the middle is 12.   So the second box needs to be an operation that gets from 12 to 6.

 

Without the clue now added, you could have at least two different answers.   “Take 6” is what I expected, and with the clue that is now the RIGHT answer.  Without the clue, ‘Divide by 2’ would have worked too.

(So would ‘add -6’ if we are getting pedantic, though really thats the same as subtract 6. Another possible answer would be ‘Raise to the power of 0.721, but I wouldn’t expect people studying Number machines to spot that.  There are probably a lot of other operations that get from 16 to 6 in we go that deep!)

What exactly is a number machine?

This is a question I asked myself when I returned to tutoring Maths a few years ago. Number machines often turn up in questions of Foundation papers. They provide a useful introduction to quite a few things

  • Algebra
  • Functions
  • Computer Programming – Which is what I was doing before I turned to Tutoring.

 

Here is an example Number Machine. The idea is we ‘feed in’ a number on the left, and see what comes out on the right.

A simple starter question would be : If a 4 in entered into this number machine, what would the result be. Number machines work left to right, just like reading. In this case, if the input is 4, we follow the boxes left to right and get  4 x 4 = 12 then + 2 = 14 .

A more advanced question would be – If the output is 8, what is the input.  This asks us to move right to left but also do the operations if reverse. Remember reverse of adding is taking away. The opposite of multiplying is dividing.

8 – reverse of + 2 is -2   so the number between the boxes is 6.  6 Divide by 3 is 2.  So the in number must be 2.

Try putting 2 into the machine and see how this is a reverse of what we did above.

The more complicated questions miss out the instructions. For example.

If we put 6 into this number machine we get 6 out. What is missing in the second box?

Clue – Its an subtraction sum.

I’ll show how to answer that in my next post.

 

All is revealed with regards to dogs – and sheep!

I gave the dog problem from 2 posts again to my Wednesday student yesterday and he solved it in the way it was intended to be solved.

We start by choosing a letter to stand in for the answer we want. Let S be the number of small dogs. Also let L be the number of Large dogs. We have not be asked to find the number of large dogs but this is part of the situation.

So we can say S  = L + 36 – because there are 36 more small dogs than large dogs. Actually L = S – 36 is the same and will lead us to the answer required more quickly.

Also L + S = 49 – the number of dogs. This ‘equation’ I think is intended – but I will return to this point.

We can solve the equations by substituting the first into the second to give  S – 36 + S = 49.

Simplify by adding the S and the S and adding 36 to both sides, we get 2S = 85   and so S = 42.5

“But how can we have half a dog”, asked my Wednesday student, and a very fair question two. This is the more obvious reason why this is a bad question – interestingly bad but bad none the less. If we are going to encourage students to take ‘real life’ problem solving seriously, then the questions we ask should make sense.

But my other reason why I’d want to change this question comes back to the story about the black sheep.  The point of the  story is that as mathematicians we shouldn’t assume anything – or at least we should qualify any answer by stating clearly which further assumptions we have made – I claim the answer to this question is incomplete unless we also say

‘Assuming all dogs are large or small’  – i.e  there are no medium sized dogs!  Without that, we can’t use safely the equation S + L = 49!

 

 

A short trip to Scotland

There is a story about mathematicians that I think I first read when I studying for A-level.

 

An astronomer, a physicist and a mathematician are on a train in Scotland.  The astronomer looks out of the window, sees a black sheep standing in a field. “All the sheep in Scotland are black!”, comments

“Oh no” says the physicist. “Only some Scottish sheep are black.”

This is the cue for the mathematician to get involved. “No,” he tells his friends. “In Scotland, there is at least one sheep, at least one side of which appears to be black, some of the time.”

I remembered this story yesterday when posting the problem about the dogs, which I reckon wasn’t written by the same mathematician. I’ll explain why in my next post, where I will also give the answer!