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Algebra…. The language of maths

This is a rather general post, but since I am about to start on Algebra subjects with one GCSE student, I’ve been giving some thought on how sto start on this subject

A lot of students don’t like algebra. It’s probably the first thing they study in Maths that doesn’t seem to relate to ‘real life’.

But actually it’s very useful to know some algebra if you are going to solve some real life problems.

 

Algebra is the language that maths is written in.

 

Let’s start by considering the most important thing about Algebra – We use letters when we don’t know what numbers are. Each letter ‘represents’ a number.

This can work in different ways.

 

In this first example, you can choose what the numbers are

Choose your own value for a, b and c.

a = 2 b = 3 C= 6

 

Now, with your numbers for a, b and c…   work out

a + b   =  5 b – c  = 3 2 x a =  4
a – b + c  = 7 a x b + c = 12 c ÷ a = 3


These are no right and wrong answers! The answers depend on what numbers you chose. It’s fun, but is it much use?

 

We can make this more useful by turning these bits of algebra into a formula. We do this by making them ‘equal’ something

 

For example

 

 d = a x b + c f = c ÷ a

 

You have already worked these out. They are the last two examples above.

Formulas ARE useful because we can work out things from real life.

d = s x t

What is d if s = 30 and t = 3 ?

[This is how we work out how far we can travel if s is our speed and t is how long our journey is]

In my next post I’ll show how we can ‘simplify expressions that have the same letter repeated.

Rearranging Formulas – a couple of examples

In my post a couple of days ago, I wrote about how a formula can be changed to make another ‘part’ of it the subject. The subject is the element that stands alone and its easier to find a value if the value to be found is the subject

In this post I’ll give a couple of examples.

Most of the world now measures temperature on the Celsius scale, but in a few places, most noticeably perhaps the USA, how hot it is on a weather forecast will be given in degrees Fahrenheit .

There is a simple formula for turning a temperature on one scale into a temperature on the other.

F = 9C/5 + 32  – For example, the boiling point of water is 100 deg C.
F = 9 * 100/5 + 32  = 180 + 32 = 212  – and this is right, 212F is the boiling point of water on the Fahrenheit scale.

But what if we were in the USA, and seeing the weather forecast would like to know the temperature in the more familiar C scale.

We can rearrange the formula so this is  C = . To do this we follow steps familiar to you if you can solved equation. The rule that stays the same is you need to keep the balance – a change you make to one side of the = you make the same change to the other.

F = 9C/5 + 32

(Subtract 32 from both sides)

F – 32 = 9C/5

Multiply both sides by 5/9
C = 5(F-32)/9

You will see that I’ve also switched the sides round, and written the new right side using Brackets.  Its F-32 that needs to be multiplied by the factor 5/9 and we have to remember our BODMAS rules.

Lets try out the new formula.  If 77 degrees F is given as the temperature
C = (77-32) * 5/9  = 45 * 5/9 = 25C – which is a warm day by UK standards.

Now consider v = u + at – which is a formula of motion used at A-Level. Lets re-arrange this to make t the subject.

Take u from both sides (and switch)
at = v – u
t = (v – u)/a 

A few notes on using formulas

Using formulas is part of Maths which can be really useful in real life situations – and when you are using Maths is Science  – and scientists use Maths all the time!

For example  the formula for working out speed* is
s = d/t  where s is the speed, d the distance travelled and t the time taken

So if we know that a car travels 120km in two hours, the speed overall is  120/2 = 60km/h  (Even the unit for speed tells us how to work it out, which is actually how I usually remember it)

Another question we might be asked is – If a car travels at 40km/h for 30 minutes, how far will it travel?

Here we have the speed and the time, and we can plug our numbers into the formula

40 = ? / 0.5  – we can change this to be ? = 20km by multiplying both sides by 0.5 (30 minutes, hence half an hour)

Which is OK, but a bit awkward, especially if we have a lot of similar calculations

So…. the best thing is to re-arrange the formula to the quantity we always want to find is one the left hand side….  and this is where algebra comes in.

s = d/t.  We can rearrange this formula by multiplying both sides by t.  We may not know what t is but it has a value

s [* t] = d/t [* t]  –  so s * t = d

We reverse the formula, so the single letter is on the left.

d = s * t or d = st (because mathematician are sometimes lazy and leave out the multiply sign)

 

 

We call this the ‘subject’ of the formula. In fact, thats how you may be asked to do this in an exam – ‘Make s the subject of the forumla’

I will post with more examples tomorrow

*NOTE: As one of my students mentioned last week, we ‘should be using velocity now rather than speed’.  He had a point, but remember, velocity is the measure where direction is important too..  speed is a useful measure if the direction you are traveling is known or somehow less important.

The Teacher’s response!

 

So..  what happened!     Kate and Jo had to sit the test on Tuesday.  After the test, Kate goes up to the teacher and says angrily ‘You didn’t play by your rules!’

“How do you mean?” asks the teacher, patiently.

So Kate explained to the teacher what she had explained to Jo on Friday.

“That’s very clever,”admitted the teacher. “But tell me, after working all that out, did you expect the test to be today until I told you that it would be?”

“Oh!”  said Jo.  Kate just scowls.

The puzzle of the school test

On a Friday, a teacher say’s to Jo’s class – “Next week you will have a surprise test, but you will not know until you arrive at school that day, that the test is that day.”

Jo is very worried – she hates tests and she hates surprises.  But her friend Kate says – ‘Its OK, we won’t be having a test’”

“But the teacher said…”

“Just think about it,” suggests Kate.  “We can’t have the test on Friday, because if the test is on Friday, we would know it was on Friday when we go home on Thursday and we havn’t had the test yet. And that breaks the rule.”

“So it won’t be Friday,” says Jo. “It can be one of the other days.”

“Once we know it can’t be Fridyam,, it can’t be Thursday for the same reason!”

“I don’t get it.”

“We know it can’t be Friday, so when we leave school on Wednesday, the test must be Thursday.  Breaking the rule!”

“But if we know it can’t be Thursday or Friday,” says Jo, seeing what her friend is saying at last, “then it can’t be Wednesday.”

“Exactlty!  So we can’t have a test unless our teacher breaks the rules!”

So is Kate right? Has the teacher set himself an impossible rule?”

 

 

So.. Does the barber shave himself?

Last week I asked a question about a barber?  Does he shave himself?  He shaves only the men who do no shave themselves, and nobody else.

Let’s look at the possible answers, which would appear to be ‘Yes’ or ‘No’

If the answer is ‘Yes’ then that means he does shave himself. But he only shaves men who don’t shave themselves. That breaks the rules

If the answer is ‘No’  then he is one of the men who doesn’t shave himself, so by the rules, he should shave…  himself!

A Paradox!

Since Russell proposed this paradox a few people have been less than impressed, writing it off as ‘wordplay’, but that seems to me like a rather killjoy approach to an interesting situation.

About the time I learnt about the barber – when I was about 15 – I read about another ‘logical story’…  the story then referred to a ‘Hangman’; but given my total opposition to capital punishment, let’s talk about a surprise test at school. That’s puzzle will be posted tomorrow.

The Problem with barbers

Ah..  I’ve been busy making videos for My Facebook Group and neglecting the blog!!

 

It looks like it was two months ago since I last write here!  Whoops……

I’m coming straight back at you with a problem in Logic!!   (And next week I’ll start something of a ‘series’   Don’t know what yet!

So here is a weekly Conundrum …  and I’ll declare my sources here. This is a ‘puzzle’ set by philosopher Bertrand Russell…  But its relates to ‘Set Theory’  which is part of some Maths courses.  Oh an apology, this puzzle assumes an all male world!

So we have a village..  and everyone in the village has a job,  and one of those jobs is the Barber..  Some of the people in the village shave themselves.  The Job of the Barber is to shave all the men – and only the men – who do not shave themselves

The question is…   Does the barber shave himself?

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.