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The equation I used for creating this staircase is

(-1)xx2

I used -1 so my graph will always be oscillating. When you multiply -1 by

and an even number, they will always be positive. And if you multiply -1

by an odd number it will always be negative.

(-1)x This part of the equation will define if the column goes upwardsor downwards.

x2 This part will show the numbers of cubes in that column.

For example if you want to know the number of cubes and the location in

the graph for the height of 45, this would be the steps you have to take

You substitute x in the equation with the number of the height, which is45.

So

(-1)45452

-1 to the power of 45 is -1, which shows that the number and column is

negative. And 45 to the power of two is 2,025. So when we multiply -1 by

2,025, we get -2,025, which shows that the column of the 45 height is

negative and has -2,025 cubes.

Another example that will give us a positive result will be calculating the

80thheight.

So

(-1)80802

-1 to the power of 80 is 1, which shows that the number and the column

will be positive. 80 to the power of 2 is 6,400, so when we multiply 1 by

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6,400 we get 6,400, which shows that the 80thheight will be positive and

it will have 6,400 cubes.

Second Part

Freddie (or Freda) the mathematical flea stands at the bottom of a

staircase. Given that he can only jump up one or two steps at a time

investigate how many different ways he can climb to any given height.

For Freddy to be able to jump to a height of six he has these options.

1.

Jump each step by one.

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2.Give two jumps at a time.

3.Make two jumps of one step and then two of two steps.

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4.Make four jumps of one step and then one of two steps.

5.Make a Jump of two steps, then two jumps of one step and

then two steps.

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6.Make two jumps of two and then two jumps of one.

7.Make one jump of one, then two of two steps and then one

jump of one.

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8.Make a jump of two steps and then four of one step.

9.Make two jumps of one-steps, then one of two steps and

two of one-steps.

These are all the possible ways Freddie can jump up to six steps, if he

could jump one or two steps at a time.

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If we change the rules and Freddie can also jump three steps at a time,

this will be the results.

10. Make two steps of three.

11. Make a jump of three steps, then one of two steps

and one jump of one step.

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12. Make a jump of three steps, then three of one step.

13. Make one jump of one step, then one of two steps and

one jump of three steps.

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14. Make one jump of two steps, then one jump of

three steps and one jump of one step.

15. Make one jump of two steps, then one jump of one step

and one of three steps.

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16. Make three jumps of one step and one of three steps.

Finally another rule we could use would be if Freddie could only jump six

steps!

17. Make a Jump of six steps.

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Investigation by:

Melanie

Moreno