Borders Coursework

Introduction:

The 2D diagrams below show dark cross- shapes that have been surrounded by white squares to make bigger cross- shapes:

The next cross- shape is always made by surrounding the previous cross- shape with small squares.

Part 1

Investigate to see how many squares would be needed to make any 2D cross- shape built up in this way.

Part 2

Extend your investigation to 3 dimensions.

I will be continuing the above 2D sequence until the 9th term, and then forming equations to find out how to calculate the total squares and extra squares used to make any cross shape.  When these 2 formulas for the 2D shapes are found, I will investigate further onto 3D cross- shapes, and find the formula for the total cubes used to form that shape. The 3D drawings will be similar to the 2D drawings but all six faces of the cube will be surrounded rather than just 4 sides as in a square.

+4              +4            +4            +4             +4              +4             +4

1. 2a = 4                        2) 3a + b = 4                         3) a + b + c = 5

        a = 2                             6  + b = 4                             2 + 2 + c = 5

                                                           b = -2                                        c = 1

General equation for Quadratic sequences: an2 + bn + c

So the equation for this sequence is: 2n2 - 2n + 1

To check that the equation is correct I will use values of ‘n’ to make sure.

When n= 4, the total number of squares used are 25.

So: 2 * (42) – (2*4) + 1 = 25

This shows the equation is correct. I will however, check again to be sure of this.

When n = 9, the total squares used are 145.

So: 2* (92) – (2*9) + 1 = 145

This tells that the equation works and is therefore correct.

Part 2: 3D Shapes:

(The 3D drawings are on the triangle dotted paper attached.)

Table to show the amount of squares in total and the amount of extra cubes used for the 3D drawings:

Working out to show the equation (nth term) for the Total Cubes used.

Sequence:

1        7          25          63          129          

+6         +18          +38         +66        

+12              +20            +28            

+8             +8

Cubic sequence :

1        8          27          64          125          

+7         +19          +37         +61        

+12              +18            +24            

+6               +6

General equation for Cubic sequences: an3 + bn2 + cn + d  

(32/3)

36

(108/3)

851/3

(256/3)

1662/3

(500/3)

Difference between sequence and 11/3  n3/ 4/3  n3

-1/3

(-1/3)              

-32/3

(-11/3)

-11

(-33/3)

-221/3

(-67/3)

-372/3

(-113/3)

New sequence:

-1/3-11/3-33/3-67/3          -113/3

-10/3-22/3-34/3-46/3

-12/3-12/3-12/3

General Quadratic equation: bn2 + cn + d

2b= -12/3 (-4)      

b= -6/3 (-2)        

3b + c= -10/3

-18/3 (-6) + c= -10/3

c= 8/3 (2 2/3)

b + c +d= -1/3

-6/3 + 8/3 + d = -1/3

d= -1

Remaining equation for total number of cubes used in any 3D cross- shape: -2n2 + 22/3n – 1

Full equation to work out the total cubes needed to make any 3D cross- shape:

11/3n3 -2n2 + 22/3n – 1

To check the formula works, I will test it using values of ‘n’.

When n= 1, the total cubes used is 1.

So: (1⅓ * 13) – (2 * 12) + (22/3 * 1) – 1

 = 1⅓ - 2 + 22/3  -1 = 1

 This shows that the formula works, but I will check again, in case I have made a mistake.

When n= 5, the total cubes used are 129.

So: (1⅓ * 53) – (2 * 52) + (22/3 * 5) – 1

 = 125⅓ -50  +72/3  -1 = 129

This proves that my formula is correct because the answers match those in my sequence.            

1st term                        2nd term                                         3rd term

4th term                                                                         5th term                                                

Justification of 2D shapes: