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Introduction

Clare Heirene 10I

Mathematics

Coursework:

Stacking By Clare Heirene

Risca

Community

Comprehensive

School

##### Contents

Introduction                                                                        Page 3

Plan                                                                                        Page 3

2-d designs                                                                        Page 4

3-d designs                                                                        Page 10

Investigation into Square-Based Pyramids                                Page 24

Conclusion                                                                        Page 34

##### Introduction

The question:

Boxes made in the shape of a cube are easy to stack tomake displays in supermarkets.

Investigate!

## Plan

I will carry out this investigation by following these points:

1.        Simplify the question by using 2-d shapes.

2.        Draw 2-d designs.

3.        Draw 3-d designs.

4.        Evaluate my work.

#### Detailed Plan

To investigate each shape I will follow a pattern:

1. I will state which shape I am investigating.
2. Draw the shape or a bird’s eye view to show the shape.
3. Draw a difference table to show whether it is a linear or quadratic formula.
4. Make a guess as to what the formula might be.
5. Check it. If it is right prove it by using the next shape in the series. If it is not right work out the next part of the formula.

2-d Squares and Rectangles

To simplify this investigation I will start off with an easy shape and work my way onto more complex shapes.    To find a formula for this pattern I will draw a difference table.

Number of Layers (N)        Number of boxes (B)        1st Difference (1st)

1        2
2
2        4
2
3        6
2
4        8

This shows that each time you add a layer you add two more boxes. I will try and find a formula for this by multiplying the difference by the length of the side.

 N x 1st  = 2 4 6 8 B= 2 4 6 8

As you can see the numbers are the same, meaning the formula for this pattern is:

B=2N

I shall prove his formula to be right by trying the next shape in the sequence.

2x5=10 As you can see my formula was correct!

2-d Triangles

I shall try another shape:     I will find a formula for this pattern by, first of all, drawing a difference table:

Middle

3

4

5

Number Of Boxes

1

10

35

84

165

Pattern

1x1

1x1+3x3

1x1+3x3+5x5

1x1+3x3+5x5+7x7

1x1+3x3+5x5+7x7+9x9

As there is a third difference this means that N will be cubed. To find out the relationship between the number in the third difference and the number that I multiply N3 by I will first test 2N3 then 3N3.

Number Of Layers            2N3              1st Difference                    2nd Difference                     3rd Difference

1                2

14

2                16                                        24

38                                                12

3                54                                        36

74                                                12

4                128                                        48

122

5                250

When N3 is multiplied by two the third difference has a constant of

12. The relationship between 2 and 12 is either multiplying by six or adding twelve. I will see which is right by drawing a difference table for 3N3 .

Number Of Layers            2N3              1st Difference                    2nd Difference                     3rd Difference

1                3

21

2                24                                        36

57                                                18

3                81                                        54

111                                                18

4                192                                        72

183

5                375

As you can see the number in the third difference column is 18. This is  3 x 6. This shows me that the relationship between the number that is multiplied by N3 and by the number in the third difference column is:

The number in the 3rd difference column

6

Using this formula I can now work out the formula for my first 3-d pyramid.

8/6 = 1 1/3

This shows me that the first part of the formula should be:

1 1/3 N3

To find the next part of the formula I will use the formula above and use it to look at the difference between the answers from 1 1/3 N3 multiplied by the layers.

No. Of Layers                    No. Of Boxes                            1 1/3 N3                                    1st Difference

1                        1                        1 1/3                                1/3

2                        10                        10 2/3                                2/3

3                        35                        36                                3/3

4                        84                        85 1/3                                4/3

5                        165                        166 2/3                                 5/3

To find the number of boxes using the formula 11/3 N3 you have to add a fraction and the numerator of the fraction is equal to the number of layers. This means that the overall formula for this stacking design is:

B= 1 1/3 N3 N

3

To prove this formula I will try it on the next shape in the series:

Conclusion

((1/6 (2x1½)2)-((1½-½) x 2)x6)N

6

((1/6 (3)2)-(1 x 2)x6)N

6

(((1/6 9)-2)x6)N

6

((1½-2)x6)N

6

(3/6x6)N

6

This is my prediction  -3N

6

The answer is   -N that is the same as-3N

2                                 6

This has proven my formula correct, this means that this is the last part of my formula.  I can now add this onto the rest of my formula and this will give me the formula to workout the total number of boxes in any 3D square based pyramid.  My complete formula is:

B= ((1/3(2A2)) x 2) N3 + ((-1/2(2A)2) + 2A) N2 +

((1/6 (2A)2)-((A-½) x 2)x6)N

6

I do not have to prove this formula correct because I have been proving it as I went along.  I have already proven that the individual parts of the formula are correct and this means that when they are put together they will create the correct formula.

I can now use this formula to workout any formula for any three-dimensional square-based pyramid and that will give me the formula for the number of boxes depending on the number of layers.

Conclusion

I have completed the task, which was to investigate cubes and how well they stack. I have come up with these formulas:

Shape:

Formula:

2-d Rectangles

B=2L

2-d Triangles

B= ½ L2 + ½ L

2-d Pyramids (A=1)

B=L2

2-d Pyramids (A=1/2)

B= ½ L2 + ½ L

2-d Pyramids (A= 2)

## B= 2L2

3-d Cubes

B = L3

3-d Cuboids

B=2L2

3-d Pyramids (A=1)

B= 1 1/3 N3 N

3

3-d Pyramids (A =1 ½)

B= 3x63 – 1 ½ x 62 – 6

2

3-d Pyramids (A= 2)

B= 5 1/3 N3 -4 N2 N

3

3-d Pyramids (A= ½)

B = 1/3 N3 + ½ N2 + N

6

Any Pattern with a repeating 2nd difference

½ 2nd difference x N2 + The remaining difference

Any square based Pyramid

where A= the distance of the step and N = The number of layers.

B= ((1/3(2A2)) x 2) N3 + ((-1/2(2A)2) + 2A) N2 +

((1/6 (2A)2)-((A-½) x 2)x6)N

6

If I were to extend this investigation I would try looking at other shapes, as well as cubes e.g. Pyramids, Prisms etc.

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