Unit 13 Section 3 : Patterns and Matchsticks

In the previous section we looked at patterns represented by dots on a grid. In this section we will use matchsticks to represent patterns. We will also look at how some sequences can be described using simple algebra.

Continuing patterns with matches

Look at the pattern below. We want to find the number of matches in the 10th shape.
The number of matches in each shape has been written underneath.

If we look at the number of matches we have added to each shape to get the next, we see that this is the same as the differences in the sequence. The differences are:

We can now see that the differences are increasing by 2 each time. To find the next term after 54, we will add on 16.

The 7th term is 54 + 16 = 70
The 8th term is 70 + 18 = 88
The 9th term is 88 + 20 = 108
The 10th term is 108 + 22 = 130

We can now see that the 10th shape will have 130 matches in it.

Finding a simple rule for a pattern

The diagram below shows the first three shapes of a pattern made from matches.

Work out the answers to the questions on the right, then click the Click on this button below to see the correct answer button to see whether you are correct.

(a) How many matches are there in each of the first three shapes?

(b) How many matches will there be in the 4th shape?

(c) How many matches will there be in the 5th shape?

(d) Can you see a connection between the number of matches and the shape number?

(e) How many matches will there be in shape 10?

(f) How many matches will there be in shape n?

(g) Which shape will have 100 matches in it?

When we find a rule for the nth term of a sequence, it is called a general rule.
We can use it to find the value of any term in the sequence, as long as we know its position.
In the example above, we found the general rule that the nth term is always 4n or 4n.
If we want to know the 36th term we just put 36 in the rule instead of n: the 36th term is 436 = 144.

 

Exercises

Work out the answers to the questions below and fill in the boxes. Click on the Click this button to see if you are correct button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on Click on this button to see the correct answer to see the answer.

You may find it useful to have a pencil and some squared paper to use while you are completing these questions.

Question 1
Here is a pattern formed with matches:

(a) Write in the number of matches in the first 6 shapes.
Separate the numbers with commas, like this:

(b) How many matches will there be in the 7th shape?

(c) How many matches will there be in the 10th shape?

Question 2
Below is another pattern of matchstick shapes.
(a) How many matches are needed for the 6th shape?
matches

(b) Which shape has 97 matches?
Shape number

Question 3
How many matches are needed to make the 8th shape in this pattern?
matches

Question 4
Look at the pattern of rectangles below.
(a) How many matches are needed for the 7th rectangle?
matches

(b) Which rectangle requires 199 matches?
Rectangle number

Question 5
Look at the number of matches in each of the squares in the pattern below.
(a) How many matches would be needed for the 5th shape?

(b) How many matches would be needed for the 6th shape?

(c) What do you have to multiply the shape number by to get the number of matches?

(d) Using algebra, how many matches will there be in the nth shape?

(e) Which shape contains 88 matches?
Shape number

Question 6
Look at the pattern below.
Using algebra, how many matches will there be in the nth shape?


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Produced by A.J.Reynolds January 2001