Growing Patterns

What kind of pattern is it?

It is a growing pattern.

How many buttons do you think will be in the next row?

5.

How do you get five?

Because it will be one more than the row before it which has four buttons.

How many buttons do you think will be in the 10th row?

45.

How did you get that?
10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

Is there another faster way of calculating it? Maybe doing up a table will help. Or if you can also try rearranging the shape of the triangle.

Row number Number of buttons
1 1
2 3
3 6
4 10
5 15

What I learnt at this point:

It was easy to recognise and describe the vertical rule, both by looking at the numbers in a table, or by working off the visual representation.You just take the previous number, and add on one number more than the number you added on to the last number to get the previous answer.

1+2=3, 3+3=6, 6+4=10, 10+5=15

But to recognise the horizontal pattern or see how the numbers in the second column related to the numbers in the first column was much more difficult.

Eventually I found the the rule was: take the row number, square it, add it to the row number, and divide the whole answer by two e.g. 5 x 5 = 25; 25+ 5 = 30; 30/2 = 15

As I reflect on the reasons why it is important to recognise the horizontal pattern and not just the vertical I realise itt is because the point of identifying patterns is to use it to make predictions; and being able to identify a horizontal pattern was necessary to efficiently predict the output when given the input, or the backtrack by figuring out the input from being given the output.