Patterns exist everywhere! I even like to arrange my marshmallows in various patterns although they don't last too long!
In this section, you will find tasks related to solving problems which involve patterns. These range from shape patterns for new mathematicians all the way up to algebraic patterns for advanced mousématicians like me.
But first - a song....
Brie-Anna de Mouse, star of Maths with a Mouse
ACTIVITY TIME
Choo Choo! All aboard! This is where patterns all begins - colours and shapes. Enjoy!
Welcome to the world's most unsusual prison - a prison were doors keep opening and closing and mathematical properties help to decide who gets to leave! With control of the keys, can you work out who is meant to stay and who is free to leave? Good luck solving Transum's Prison Cell Problem!
If you have visited Alice in Fractaland you will have seen many of the great surprises of Pascal's triangle. Not it's your turn to investigate it further in Transum's brilliantly named Pascal's Triangle!
This is a great task and only requires two skills:
ability to count in muliples linked to the times tables
finding the difference between two numbers
Go on, try Times Tables Shifts - it's a great activity from NRich.
Don't play with matches? It is not safe or sensible to play with matches. To keep you safe, these online matches are perfect as they cause absolutely no change of fire, which means that they are great for maths and terrible for starting a campfire for toasting marshmallows! The key to Matchstick Patterns from Transum is looking carefully at how the pattern grows.
Is this task too easy? If it is you now can become an expert finding the nth term for many, many sequences. Let me explain...
Linear sequences increase or decrease by the same amount each time. In the diagram above, it is clear that the first sequence is going up in steps of 2. You can even see the straight line it creates. This is why this is a linear sequence.
The first column gives the first term in the sequence = 1 x2 = 2
The second column gives the second term in the sequence = 2 x2 = 4
The third column gives the third term in the sequence = 3 x2 = 6
and so on...
From this, we can recognise that any number x2 is equal to the number in the sequence.
e.g. the 100th term (number in this sequence) = 100 x 2 = 200
the nth term in this sequence = n x 2 = 2n
In the second diagram, the nth term still relates to 2n as the sequence is increasing by 2 each time, but every term is one more than the two times tables so one more is added to each term (see my pink ducks).
Therefore, th nth term for this sequence
= 2n + 1
Finding the nth term for a linear sequence that is decreasing by a regular amount for each term is similar to find, but a little trickier to understand. I think I can explain…
As the sequence in the first diagram is decreasing by two each time, the nth term will relate to -2 multiplied by end which equals minus
As the sequence in the first diagram is decreasing by two each time, the nth term will not have 2n but will involve -2n as the sequence decreases.
Let's consider the first term.
The first term (in the first sequence above) is 10
Something - 2n = 10
Something - 2 x 1 = 10
Something = 12
12 - 2n = 10
12 - 2 x 1 = 10
What about a different term. Let's try the second term = 8
Something - 2n = 8
something - 2 x 2 = 8
12 - 4= 8
Oh....it's the same!
nth term = 12 - 2n
Wait a minute...the term before the first would have been 12...interesting.
For the second sequence, the term before the first would be 13 and the pattern decreases by 2 each time so that is -2n
nth term = 13 - 2n
That works...I might just be the most clever mouse in the world ever!
