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# Circles and Squares

Today I’d like to share with you an exam question that stumped me for a while a week ago.

To find the area of the rectangle we need to find the length of the rectangle and its height.

The Length isn’t too hard, but can you see how to find the height before I post the answer here on Thursday?

# 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  = 1

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