• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month   # GCSE Maths coursework - Cross Numbers

Extracts from this document...

Introduction

## Cross Numbers

Here is a number square.

It shows all the numbers from 1 to 100.

I can form cross numbers by placing a cross on this grid.

I will use the cross shown below.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

In this investigation, I will try to find a master formula for various functions that I use with a certain cross shape.

To do this, I will follow the procedure given below:

1. I will then pick a number on the grid, (labelled as X).
1. Use a mathematical formula using the 4 numbers around it.                           E.g. (24+33)-(22+13).
1. When an answer is found, I will repeat the formula but pick a different value for X. Repeat this 3 times. If the same answer is achieved, then I will assign an algebraic formula for each number around X.
1. Once I have assigned an algebraic formula for each square, I will replace the numbers in the formula with the algebraic form and justify this mathematically.
1. I will also be changing the horizontal grid value (g) to further my investigation.
1. This will therefore give me a master formula.

## I will use the following symbols

X=center number.

g=number of squares across the horizontal axis.

e.g.

 91 92 93 94 95 96 97 98 99 100 g=10

## Prediction

I predict that whichever grid size I use the number above always = X-g, the number below always = X+g, the number on the left always = (X-1)

Middle

(X+1)

X+g

[(X-1) + (X+1)] + [(X-g) + (X+g)]

= X-1 + X+1 + X-g + X+g

= 4x

If I replace the centre number, I get the following results.

If X=87 then

[(87-1) + (87+1)] + [(87-4) + (87+4)]

= 87-1 + 87+1 + 87-4 + 87+4

= 4x87

= 348

If X=52 then

[(52-1) + (52+1)] + [(52-4) + (52+4)]

= 52-1 + 52+1 + 52-4 + 52+4

= 4x52

= 208

If X=16 then

[(16-1) + (16+1)] + [(16-4) + (16+4)]

= 16-1 + 16+1 + 16-4 + 16+4

= 4x16

= 64

This tells me that whenever I use the above formula, the solution is always 4X ,and if I replace X with any number (apart from the outside edge numbers) I always get 4xX as an answer.

Therefore this is a master formula for this shape, and grid sizes 10x10, 6x10 and 4x10.

 X-g (X-1) X (X+1) X+g

c)

[(X+g) - (X-g)] – [(X+1) - (X-1)]

= [X+g – X+g] – [X+1 – X+1]

= 2g – 2

If I replace X, I get the following results.

If X = 33 then

[(33+4) - (33-4)] – [(33+1) - (33-1)]

= [33+4 – 33+4] – [33+1 – 33+1]

= 2x4–2

=6

If X=83 then

[(83+4) - (83-4)] – [(83+1) - (83-1)]

= [83+4 – 83+4] – [83+1 – 83+1]

= 2x4–2

= 6

If X=49

[(49+4) - (49-4)] – [(49+1) - (49-1)]

= [49+4 – 49+4] – [49+1 – 49+1]

= 2x4–2

= 6

This tells me that whenever I use the above formula, the solution is always 2g – 2, hence I always get 4 as an answer. Therefore this is a master formula for this shape and this grid.

Overall so far, the only formula that works for all three grid sizes (4x10, 6x10 and 10x10) to give the same answer is:

[(X-1) + (X+1)] + [(X-g) + (X+g)]

= X-1 + X+1 + X-g + X+g = 4x (addition formula)

All other formulas work but give a different figure but the same algebraic solution.

For the subtraction formula the solution is g²-1 and for multiplication it is 2g-2.

Dsfsd

Fsd

Fsd

F

Sdf

Sd

F

Sdf

Dsf

Sdf

## Proving that all Algebraic formulas work with all grid sizes

Grid size 4x10:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

If  X=10 and g=4

 X-g (X-1) X (X+1) X+g

The number above x is X-g because 10-4=6

The number below x is X+g because 10+4=14 .

The number to the left is (X-1) because

14-1=13.

The number to the right is (X+1) because                                         14+1=15.

Grid size 8x10

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

If  X=10 and g=4

 X-g (X-1) X (X+1) X+g

The number above x is X-g because 20-8=12 and g=8.

The number below x is X+g: 20+8=28 and g=8

The number to the  left is (X-1) because 20-1=19.

Grid size 6x10

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

If  X=39 and g=6

 X-g (X-1) X (X+1) X+g

The number to the right is (X+1)  because 20+1=21

The number above x is X-g because 39-6=33 and g=6.

The number below x is X+g because 39+6=45 and g=6.

The number to the left is (X-1) because 39-1=38.

The number to the right is (X+1)     because 39+1=40.

 X-g (X-1) X (X+1) X+g

This shows that these

are the master formulas for all values of (g)  with this shape.

To show that this works, If X=8 for g=4 then:

 X-g (X-1) X (X+1) X+g

Conclusion

Grid size 8x10

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80

If X=14

The top left will always be X-(g+1) because 14-(8+1) = 5

The top right will always be X-(g-1) because 14-(8-1)= 7

The bottom left will always equal X+(g-1) because 14+(8-1)=21

The bottom right will equal X+(g+1) because 14+(8+1)=23

Grid size 4x10

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

If X=14

The top left will always be X-(g+1) because 14-(4+1) = 9

The top right will always be X-(g-1) because 14-(4-1)= 11

The bottom left will always equal X+(g-1) because 14+(4-1)=17

The bottom right will equal X+(g+1) because 14+(4+1)=19

This shows that these

 X-(g+1) X-(g-1) X+(g-1) X+(g+1)

are the master formulas for all grid size variations with this shape.

Conclusions:

In this investigation I found several different master formulas and one Universal formula. These are summarised in the table below.

shape

type of formula

master formulas

universal formulas

+

Subtraction

[(X-1) (X+1)] – [(X+g) (X-g)

= g²-1

+ & x

[(X-1) + (X+1)] + [(X-g) + (X+g)] = 4x

+

Multiplication

[(X+g) - (X-g)] – [(X+1) - (X-1)] = 2g-2

x

Subtraction

## [{X+(g+1)}- {X-(g+1)}] – [{X+(g-1)}-{X-(g-1)}]

= 4

x

Multiplication

{X-(g-1)} {X+(g-1)} – {X+(g+1)} {X-(g+1)} = 4g

g= grid size , X= center number of a cross

A master formula is a formula that works for a specific shape on all the three grid sizes ( 10 x 10, 6 x10 and 4 x 10) that I investigated.

A universal formula is a formula that can be used for all shapes and all grid sizes that I investigated.

I would have liked to further my investigation by using a three dimensional grid.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

## Found what you're looking for?

• Start learning 29% faster today
• 150,000+ documents available
• Just £6.99 a month

Not the one? Search for your essay title...
• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month

# Related GCSE Number Stairs, Grids and Sequences essays

1. ## Number Grids Investigation Coursework

for the difference between the products of opposite corners would be: (top right x bottom left) - (top left x bottom right) = (a + 4) (a + 40) - a (a + 44) = a2 + 4a + 40a + 160 - a2 - 44a = a2 + 44a

2. ## GCSE Maths Sequences Coursework

Then in the 3rd stage in the sequence, if you pair off the shaded squares into sets of 3 you are left with 4 sets of 3 squares. This tells us that if you multiply the stage number (N) by 4, you are given the amount of shaded squares in the shape.

1. ## Staircase Coursework

on a 10x10 grid 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56

2. ## Number Grid Investigation.

- (8 X 29) = 72. 33 34 35 36 41 42 43 44 49 50 51 52 57 58 59 60 (33 X 60) - (36 X 57) = 72. The product difference in a 4 X 4 square is clearly 72. All three examples show this.

1. ## Stair shape maths GCSE coursework

Grid: 10 Steps: 3 @1 1+2+3+11+12+21 = 50 Grid: 10 Steps: 3 @4 4+5+6+14+15+24 = 68 Grid: 10 Steps: 3 @7 7+8+9+17+18+27 = 86 The total of all three numbers stairs equals 50, 68, 86 the common difference is 18 you get 18 if you minus two stair shape totals i.e.

2. ## Mathematics Layers Coursework

That is the answer because; This formula works for a four square grid with three layers. This formula works for a four square grid with two layers. I have come to the conclusion that my formula works. Part 3 Investigate the relationship between the number of arrangements and the size

1. ## Number Grid Coursework

To improve the usefulness of my formula, I wondered what would happen to the difference of the two products if I varied the width of the grid on which the 2x2 box was placed. Section 2: 2x2 Box on Width "z" Grid 1)

2. ## algebra coursework

Z+11(X-1) X 2 X 2 square 12 13 22 23 Z = top left number = 12 (in this case) Z+(x-1) = 13 X = 2 - it is the size of the square, therefore Z+ (X-1) = 13 (top right number) 12+ (2-1) = 13 Z + 10(X-1) • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to 