Maths Help

Blog dedicated to supporting with any problems, up to and including A level/equivalents (can send in later maths, but can't promise to be able to help!). Just send in an ask or a message and we'll get back to you with an answer ASAP. Will also be posting general interest blog posts here and they'll be a lot of these in the absence of any questions!

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How intuition can make problem-solving easier: Game Theory

How intuition can make problem-solving easier: Game Theory

The content of this post has been motivated by a question which I recently faced in my Optimization course at university. As such, I’m going to add a bit of a warning to this post to say that I’m going to assume a basic level of knowledge of Game Theory (nothing more than A-level, specifically solving 2 x m games). If you’re still interested in reading despite not knowing too much, I thoroughly…

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How can we explain prime factorization to kids?: Prime Climb Product Review

Recently, my partner @ashley_sharman came across the mathematical board game “Prime Climb” and he recommended that we try it out. The game is primarily aimed at kids and sells itself as…

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What does this puzzle tell us about infinity?: Tying up the loose ends

Anyone got a better solution? What does this puzzle tell us about infinity?: Loose ends

This post will be dedicated to tying up the loose ends left in the last post I made about infinity, which can be found here. I recommend you read that before continuing any further, since none of the rest of this post will make sense without it! I’m going to give my solution for how to count the number of steps to reach any given point in the grid from the last post and explain a bit about how an…

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What does this simple puzzle tell us about infinity?: How simple puzzles help us do maths

What does this simple puzzle tell us about infinity?: How simple puzzles help us do maths

Consider the path through a grid below: Seems fairly boring, doesn’t it? However there are a few questions we can ask ourselves about it. Would we ever hit the co-ordinate (17, 15)? (17, 15) here has been chosen fairly arbitrarily, but it’s quite a large co-ordinate to reach simply by continuing the diagram. Can you argue for sure that we will/will not reach this point? Are there any positive…

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Last Biscuit (or Wythoff’s game) is a strategy game where the aim is to beat your opponent by eating the last biscuit from the tin. You start with 2 piles of biscuits, and on your turn you can eith…

Check out my post on the strategy game ‘Last Biscuit’! Also talks a bit about Fibonacci numbers and sequences, as well as the Golden Ratio :)

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Amicable/Perfect numbers

This is the first of hopefully many posts on things which are not on the curriculum but people may find interesting/build their understanding of another topic.

To understand amicable numbers you’ll need a good understanding of factors.

Amicable numbers are two numbers of which the factors (excluding the number itself) of each adds up to the other number in the pair. For example, the lowest of these are 220 and 284, since:

Sum of factors of 220:

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

Sum of factors of 284:

1 + 2 + 4 + 71 + 142 = 220!

Ashley and I found this awesome enough that we have keyrings of them from the brilliant website Mathsgear:

A similar concept to amicable numbers is perfect numbers. A perfect number is a number such that all its factors add up to itself. The smallest known example of this is 6:

Sum of factors of 6:

1 + 2 + 3 = 6

And another small example is 28:

Sum of factors of 28:

1 + 2 + 4 + 7 + 14 = 28

Interestingly, relatively few perfect numbers are known as they get bigger very quickly. Only 4 of these were known to the Greeks, with the largest known one being 8128. The next largest after this is 33550336, which was discovered in 1456!

As well as this, it is not known whether any odd perfect numbers exist. None have been found so far and it is suspected that none exist. However, no one has disproved their existence which is surprising for a problem which is so simple to state.

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Place value

Simply put, place value is the value of where the digit is in the number. So place value is the value of the digits place in the number.

Example: Consider the number 284. This number has 2 hundreds, 8 tens and 4 units or ones. We can write where the columns appear to help us identify its place value.

After hundreds, the next highest placement is thousands then ten thousands, hundred thousands, millions and so on.

Decimal Place Value

Place value can get a little bit trickier when decimal numbers are concerned.

The decimal point is always positioned after the units place, the place value after the decimal point follows a pattern that will help us remember them.

Example: Consider the number 220.284. As we did previously, we can identify that there are 2 hundreds, 2 tens and 0 units. However we now need to deal with the right-hand side of the decimal. This number has 2 tenths, 8 hundredths and 4 thousandths.

Looking at the place value after the decimal point we notice that it goes up in tenths, hundredths, thousandths, etc. which matches the way the place value is described before the decimal point with a couple of subtle differences. These are the suffix –ths and that we start with tens rather than units.

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Whole number definitions

Even: an even number is a whole number which is exactly divisible by 2 with no remainder. An even number will end in 0, 2, 4, 6 or 8.

Example: 12 is even:

·         12 ends in the digit 2.

·         12 ÷ 2 = 6

Odd: an odd number is a whole number which is not exactly divisible by 2 (it will leave a remainder of 1 if divided by two). An odd number will end in 1, 3, 5, 7 or 9.

17 is odd:

·         17 ends in the digit 7.

·         17 ÷ 2 = 8 remainder 1

Prime: a prime number is a number with exactly 2 factors (see definition below). These factors are the number itself and 1.

Example: Is 27 a prime number?

Check for factors of 27.

First check 27 ÷ 2. 2 definitely does not evenly divide 27 because 27 is odd.

However, 27 ÷ 3 = 9, so 3 is a factor of 27.

So, 27 is not prime.

Factor (divisor): a number which divides exactly into another number with no remainder.

Example: The factors of 10 are 1, 2, 5 and 10.

·         10 ÷ 1 = 10

·         10 ÷ 2 = 5

·         10 ÷ 5 = 2

·         10 ÷ 10 = 1

Notice that these factors come in pairs which multiply together to make 10:

·         2 x 5=10

·         1 x 10 = 10

A post related to some more interesting application of factors here.

Multiple: a number which is in the times table of another number.

Example: The multiples of 5 are all the numbers in the 5 times table:

·         1 x 5 = 5

·         2 x 5 = 10

·         3 x 5 = 15

·         7 x 5 = 35

·         15 x 5 = 75

·         100 x 5 = 500

Any number which ends in 0 or 5 will be in the 5 times table and so will be a multiple of 5.

Common factor: a common factor of two numbers is a factor which is shared by both numbers.

Example: Find the common factors of 30 and 48.

The factors of 30 are 1, 2, 3, 5, 10, 15 and 30.

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24 and 48

The common factors of 30 and 48 are 1, 2 and 3, as these appear in both lists of factors.

Common multiple: a common multiple of two numbers is a multiple which is shared by both numbers.

Example: Find some common multiples of 5 and 6.

The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ......

The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ......

The common multiples of 5 and 6, from these lists, are 30 and 60, as they are the only numbers which appear on both lists.

Square: any number which is a whole number multiplied by itself.

Example: Find the first 5 square numbers:

·         1 x 1 = 1

·         2 x 2 = 4

·         3 x 3 = 9

·         4 x 4 = 16

·         5 x 5 = 25

So the first 5 square numbers are 1, 4, 9, 16 and 25.

Cube: any number which is a whole number multiplied by itself twice.

Example: Find the first 5 cube numbers:

·         1 x 1 x 1 = 1

·         2 x 2 x 2 = 8

·         3 x 3 x 3 = 27

·         4 x 4 x 4 = 64

·         5 x 5 x 5 = 125

So the first 5 cube numbers are 1, 8, 27, 64, 125.

Square root: the number which you have to multiply by itself (square) to reach the number you want. This is written as √ e.g: √9 means the square root of 9.

Examples: Find √4, √16 and √64:

·         √4 = 2 because 2 x 2 = 4

·         √16 = 4 because 4 x 4 = 16

·         √64 = 8 because 8 x 8 = 64

Cube root: the number which you have to cube to reach the number you want. This is written as ³√ e.g: ³√8 means the cube root of 8.

Example: Find ³√8, ³√27, ³√125

·         ³√8 = 2 because 2 x 2 x 2 = 8

·         ³√27 = 3 because 3 x 3 x 3 = 27

·         ³√125 = 5 because 5 x 5 x 5 = 125

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Ask us anything (maths related!)

Just started up this blog for maths support, help get us started by sending in any maths questions you’re finding a bit tricky! We should definitely be able to help with anything up to A level (or equivalent) and are more than willing to have a good look at anything harder.

We also intend to post puzzles and explanations of a range of topics soon, follow if you want to keep up with this or send in an ask if there’s any specific topics you want explanations for.

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So the plan for the moment is to go through the OCR GCSE specification, until we get people sending in problems,  and write up posts for each topic. We’ll try and do a post covering the basics of each one, then some interesting stuff around these for a bit of further reading (especially since there’s some nice stuff earlier on in the spec, LCM, HCF, primes, etc.) as well as going over a few examples of exam questions for each one.