Home » Posts

Tennis balls!

For this week’s posts I’m dipping into the weird land of Quora. As I may have mentioned before the Maths questions on this Q&A based social media site can be weird and can be trivial, like what is the LCM of 2 and 3. 

That is 6 by the way but I’m not going to dedicate a whole post to that.

Others today ask ‘Why do Negative numbers exist’ which is perhaps deeper than the questioner intended, and ‘What are the 100 ways of asking a woman for her number’, which is perhaps only loosely a Maths question. Though if this can be interpreted as her favourite number (not her phone number which I suspect was the idea) then it could be a more interesting one. What does someone’s favourite number say about them.  Have I posted about my favourite number before? I should do that sometime.

But instead, let’s consider the one above….


This arrangement doesn’t have the correct number of balls, but it gives an idea


Draw a line between the two balls that are opposite. Count how many balls are above that line.

In this case the answer is 5. And the total number of balls is 12.

How did I get from one number to the other (apart from counting)?

There are 5 numbers above and 5 below. – That’s 2 times 5.  Plus the two numbers on the line.

 So that is 2 x 5 + 2 = 12.

I like to generalise these things and look for patterns, and that means , yes, algebra.

So what if the number of balls above the line is a.

The total number of balls is  2a + 2  (Times 2 then add 2)

Now lets think about the problem we are solving. There are 11 balls between the 8th and 20th (Count them,  9th, 10th, 11th…. 19th)
use our formula  – 2 x 11 + 2 = 24.

There are 24 balls in our circle.  We have solved the problem but, better than that, we can answer quickly even if the question setter changed the numbers!

I like that approach; a bit more work and you can answer so many similar questions


Christmassy problem – The answer!

The true love gets 12 Drummers (On the last day, and 12 partridges – each with its own pear tree – 1 on each day!.  To see how many of each of the other gifts look along the marked diagonal!


So its the gift on days 6 and 7 that she gets most of; The geese a-laying and the swans a-swimming; 42 of each..  Lets hope she has a large pond 🙂

A Christmassy Problem!

What does The twelve days ‘True Love’ get most of?

The twelve drummers drumming of course!


But is that really true because she only gets 12 drummers on the last day, on that one day. That’s a total number of 12 drummers.

She gets 11 pipers piping on the last two days, a total of 22 pipers!  So that is the most, yeah, she gets most pipers?

Work back through the twelve days and see what she gets most of.

Answer in the next post, later this week.

A chocolaty problem

Should I be posting more ‘seasonal problems’  Ha  maybe….   I’m a bit of a scrooge  really,  try not tow think of Christmas until December reaches double figures in the date

But here is a problem posted in one of my Facebook groups which I thought I’d share with my diary

Might be the way I think, but I automatically think of Algebra when I see this problem

There are two things that we don’t know,  the weight of the box and the weight of a chocolate.  OK, we have only been asked for the weight of the box, not a chocolate but the weight of each chocolate is there in the problem, and I always like to add extra, especially if its chocolate!

Let b be the weight of the box, c the weight of 1 chocolate

b + 8c = 280g  : Eq1
b + 5c = 199g  : Eq2

Subtract Eq 2 from Eq 1 gived

3c = 81g   – so c = 27g

So from Eq 2  b + 135g  so c = 64g

I always like to use the other equation as a check when solving simultaneous equations

64 + 8 x 27 + 280g  as required by the check

 

 

 

 

 

 

 

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.