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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?

World Cup Time! – How big is a football Pitch

People may be surprised to know there is no fixed size for a football pitch, just a range

The length of the pitch must be between 90m and 120m, the width must be between 45m and 90m. For some reason a bit of trivia I remember from my childhood is that Doncaster Rovers had the widest pitch – (That quite probably isn’t true now as they moved grounds)

Oh and one other restriction is that a pitch can’t be square – 90m x 90m is not allowed.

Question – In percentage terms how much bigger than the smallest pitch can the largest pitch be?

On these figures, the smallest pitch would be 90m x 45m = 4050m2

The largest pitch could be 120m x 90m = 10800m2

So the largest pitch could be 166% larger than the smallest pitch!

 

In reality that doesn’t happen, the pitch sizes in the Premier League last year ranged from 7140m2 at Bournemouth    to 6400m2 at Stoke City.

Are GCSE’s getting harder – An answer

OK so that title is misleading. I am not going to say if I think for sure GCSEs are getting harder, though I do think the maths exams of the last couple of years are making more demands of Students.

But I will give the answer to the question I posted two days ago.

 

I think what is interesting about this question is that its unlikely the student has seen a question quite like this before.

 

Its hard to know how to prepare for the question. What it is looking for is a ‘feel’ for the situation. I think the new 9-1 exams are designed so that this ‘feel’ is required to get top grades.

So how do we answer it?  Well, we are given the common factorisation of a2 – b2.   We are told for the values of a and b this is a prime number.

But we also are given a multiply sum (a + b)(a – b) with this as the answer.  And what is the only multiply sum that has a given prime number p as the answer?

Its p x 1 !   So either a + b = 1 or a – b = 1.  A + b can’t = 1 because we can’t have two positive whole numbers adding up to 1.

so a – b = 1,  or a = b + 1. In other words they are consecutive numbers, as the question says.

And that’s it – It is only two marks after all. A knowledge of what prime numbers are, but in an unfamiliar context, is what is required for these marks.

And thats what I mean by, an ability, a confidence, of recognising skills and facts learnt when seen in unfamiliar surroundings is what is required to get an 8 or 9 in the new GCSE. It will reward true understanding.

 

 

Volume of a Prism – Exam worked example

Today’s post is a worked solution – of a question from straight off a GCSE paper.

With any question, the first ting to do is check if there are any words that are out of the ordinary.  Here, the word that stands out is ‘Prism’

I first remember using a ‘prism’ in Science lessons, because a ‘triangular’ prism can do beautiful things with a beam of light. But that is a special sort of prism – one with a triangle at each end.  Generally a prism is any solid shape that is the same all the way through.

And that is the clue to working out the volume. Work out the area of the shape of one end, and multiply that by the length.

That leaves us with the hard bit first – but at least we will know we’ve got that ‘out of the way’. How do we work out the area of the end, which is like an L lying on its side.

We need to split this up, but there are 3 ways of doing this. You might be able to see three ways of doing this; the third is a bit harder to spot.

On all of the splits, it will help you to fill in the missing sides. The short one we find by comparing the lengths on the left and right : 7cm – 4cm = 3cm

The longer missing length we can find by comparing the top and bottom. 11cm – 5cm = 6cm

 

 

 

 

 

 

 

 

I am showing the calculation for all three methods here  BUT YOU WOULD ONLY NEED TO CHOOSE ONE!

The first split is into rectangles that are 5cm x 3cm = 15cm and
11cm x 4cm = 44cm2.  Total is 59cm2.

The second alternative is 2 rectangles of 7cm x 5cm  = 35cmand
4cm x 6cm = 24cm2. Total is 59cm2 – And the result should be the same of course!

The third is harder to see but I think quite clever. The area of the shape without the missing part is 7cm x 11cm = 77cm2. The missing part is 6cm x 3cm = 18cm2

. This time we have to take the second area away – its ‘missing’  77-18 = 59cm.  Like I said, this had to be the same answer, but its a good check that it is!

[That’s a good hint with any problem solving. If you want to check your answer, find it in two different ways. If you check an answer by repeating the same steps, there is a chance you’ll make the same mistake, if you made one]


The final step is to calculate the volume by multiply the area of one end by the length. 59
cmx 20cm = 118cm3