Home » Posts

Understanding sequences

Growing up , Loving maths, I found out a few things for myself, and I’ll post about one of those next time.

But this is a simple rule with numbers my Dad showed me.  Try adding up the odd numbers one by one, and see what you get

1  = 1seqQ

1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7 = 16
1 + 3 + 5 + 7 + 9 = 25

 

You will recognise these numbers – all the squares!  And this is the secret behind a type of questions I’ve seen on exam papers recently

 

 

 

 

There are a lot of different sequence questions, and its best not to leap in until you have really recognised. But if nothing appears to you, a next step is to look at how the numbers are increasing. Actually write the numbers in the gaps. If you worked them out in your head you might not see the pattern.

Here the gaps are :  7   11  15  19

And you will see straight away they are increasing steadily – By 4 each time.  – And when you see this – that the second level of differences are constant – you will know you have a ‘quadratic’ sequence.

That is to say, the expression in terms of n will be of the form

an2 + bn + c   – where b and c could be 0 but a is not (If a was 0, the first differences would be constant)

If you play with these sequences for long enough, you will find something very interesting but for this posts I’ll let you in on the secret

To find what a should be, take half of the ‘constant’ you find by following the differences. In this case that was 4, so a = 2.

So our sequence rule starts 2n2

We are now going to write our sequence again, and underneath, write double the square numbers

4    11   22  37   56
2     8    18  32    50

Takes away the bottom from from the top, you get

2  3  4  5  6

or n + 1 for each position

This means our full answer is

2n2 + n + 1

 

What can we say about a difficult question

A contact of mine on Facebook kindly provided me with a set of harder problem style questions recently for one of my more capable students…  These are questions where you can’t just apply the maths you know – You have to think a bit.

I’m not going to use them all for blog posts, but this was an interesting one I think.

First thought – How can we tell that? Your calculator isn’t going to help; Numbers that big are not going to give you the last digit.

So if we can’t tell directly, what can we tell? Actually, we need to start playing, and playing with numbers is something I love to do.

Let’s start by looking at what powers of 4 are like – That’s going to help us

You don’t have to go very far before seeing that every second number – every power of 4 to an odd number – end in a 4.

Continue that pattern on and we can see 4 to power 333 is going to end in 4.

 

 

 

 

What about powers of 3  –  and here I did need my calculator – 3  9  27  81  243   729    2187  6561

Every fourth number in this list 4th, 8th – and indeed the one before the 3 would be 1, 3^ 0) ends in a 1.

All the numbers divisible by 4 in fact.  So we can say for sure that 3 to the power 444 end in a 1.

And a number ending in a 4 plus a number ending in a 1 will end in a 5.

WE have shown what we were asked to show, and we didn’t have to work out the whole number

 

 

 

Where the tutor went wrong

Ahh  I thought I’d posted the answer to my last post the day after, and now I see I didn’t. Whoops

In my last post I described a question where I couldn’t see the answer.

When I took the question away and studied it, I realised I hadn’t read the question in full and it wasn’t my small doubts on Circle Theorems that were the issue.

However good you get at Maths there will always be some thinks you spot quicker than others – but in all cases, reading the quesion is so important!

The line I have underlined tells me that < CBA is x degrees – I was trying to think of another Circle Theorem that told me that! The alternate angle theorem tells us that <BAC is x degrees.  Once we know the opposite angles are the same, so are the sides, so AC = BC

When the tutor doesn’t listen to himself!

Here is a thing I always say to my students…  make sure you read the question ..

And here is a confession..  though I can do algebra and number skills in my head, if I ever have to stop and think, its on the Geometry and Angles questions.

So yesterday was with one of my brightest students, and we’d finished the main activity for the day so I brought out my reserve exam questions…  and the first one was about the Alternative Angle Theorem.

I know this theorem well and I know I know it… but I don’t

always believe I know I know it!  And yesterday I looked at the question and my student was looking at me expectantly…   Later in the evening I realised the main problem was I hadn’t read the question!  Doh, as Homer Simpson would say

 

 

Let’s have a look at the question – and how much easier it was once I’d read ALL the information!

I could immediately see that this was an Alternative Angle question – that is a picture of exactly when it applies.  But ‘Prove AC = BC’ – I couldn’t see how to use the theorem to prove that. There didn’t seem to be enough information in the DIAGRAM!! Its where my moment of confidence crisis kicked in and I’m sharing this with you to show it happens to tutors too (especially when watched by eager students). The way out if this happens to you is to sit back, stay calm – possibly come back to this question later – and read ALL the information given

 

So who can spot what I didn’t read!!  The answer to this – and the whole question – will be given tomorrow

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