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Step Stairs

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Introduction

Number squares

In this investigation I will be looking at ‘square step shapes’. These are shapes such as the one pictured below:

image00.png

These shapes are put onto a grid such

...read more.

Middle

5

6

7

8

9

10

When placed on one of these grids the numbers inside of each box of the step shape are added to give you a total. The number in the bottom left-hand corner of the step shape is the number for that step shape.

        E.g.image01.png

This is shape placed on a 10x10 grid. It is shape number one, the first shape, as the number in the bottom left hand corner is 1. All the numbers in the shape added together equals 81. So the total for stair shape 1 on a 10x10 grid is 81.

I am going to begin my investigation by trying to find the rule that allows you, if you are given a shape number, to work out the total for a shape on a 10x10 grid. I will begin by putting the shape numbers, numbers inside that shape and shape totals onto a grid. I already know number one from my example and I will use the method I used in my example to work out the subsequent numbers and totals. However, for times’ sake I will not draw out each shape.

10x10 grid:

Shape number

Numbers in shape

Shape total

1

1,2,3,11,12,21

50

2

2,3,4,12,13,22

56

3

3,4,5,13,14,23

62

4

4,5,6,14,15,24

68

5

5,6,7,15,16,25

74

6

6,7,8,16,17,26

80

7

7,8,9,17,18,27

86

8

8,9,10,18,19,28

92

11

11,12,13,21,22,31

110

12

12,13,14,22,23,32

116

...read more.

Conclusion

8 x 8 grid:

An 8 x 8 grid will use this pattern:

image05.png

Giving this formula:

6n+36

I know this formula is correct so I wont test it.

7 x 7 grid:

A 7 x 7 grid will use this pattern:

image06.png

Giving this formula:

6n+32

6 x 6 grid:

A 6 x 6 grid will use this pattern:

Giving this formula:

6n+28

5 x 5 grid:

A 5 x 5 grid will use this pattern:

image07.png

Giving this formula:

6n+24

I am now going to try and find out a formula that, given the shape number, will allow you to work out a shapes total on any size grid.

Shape size

Grid Size

Formula

6

10

6N +44

6

9

6N +40

6

8

6N +36

6

7

6N +32

6

6

6N +28

6

5

6N +24

From this I have noticed that the number is always 6N for a 6-box stair shape and that the ‘plus’ number is always the grid size times 4 plus 4. So if this were written into a rule it would be:

6N +4G+4

(G =the grid size)

        I have also noticed that the 4 in the four G relates to the following:

In each case G repres-

ents 10.

...read more.

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