• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month
Page
1. 1
1
2. 2
2
3. 3
3
4. 4
4
5. 5
5
6. 6
6
7. 7
7
8. 8
8
9. 9
9
10. 10
10
11. 11
11
12. 12
12
13. 13
13
14. 14
14
15. 15
15
16. 16
16
• Level: GCSE
• Subject: Maths
• Word count: 3623

Maths Coursework

Extracts from this document...

Introduction

Introduction:

The task we have been set for this piece of coursework is to investigate numbers presented in a number grid. A box is placed around 4 of the number and the question asked was to find the product of the top left number and the bottom right number. This solution was then taken away from the top right number and the bottom left number like this:

 12 13 22 23

(13 x 22)-(12 x23) = 10

Aim:

My aim in this coursework is to experiment using 3 different variables in boxes.

These will be: size, shape and size of grid.

I will investigate boxes that are the same size, bigger, smaller, in different proportions and those that are different in shape.

What I mean by this is that I will take boxes of the following proportions:

2x2, 3x3, 4x4, 5x5, 2x3, 2x4, 2x5, 3x4, 3x5, 4x5.

I will take 4 boxes of each of type of the above that are squares. I will take these results by using the random function on my calculator.  The figure given will be any number in the box.

Investigation:

2x2 boxes:

My first boxes that I will be investigating are the 2x2 boxes of which I am taking 4.

Theses are the boxes I have taken:

1:

 35 36 45 46

(36x45)-(35x46) = 10
2:

 57 58 67 68

(58x67)-(57x68) = 10
3:

 88 89 98 99

(89x98)

Middle

4

5

6

7

14

15

16

17

24

25

26

27

(7x24)-(4x27) = 60

 33 34 35 36 43 44 45 46 53 54 55 56

(36x53)-(33x56) = 60

Lorenzo Brusini

Proving algebraically the 3x4 boxes:

The answers for 3x4 boxes all answer to 60. I shall try to use the algebraic box once again where n is the smallest number in the box:

 N N+1 N+2 N+3 N+10 N+11 N+12 N+13 N+20 N+21 N+22 N+23

I shall try to use the formula:

(N+3)x(N+20)-N(N+23) = 60

(N2+20N+3N+60)-(N2+23N) = 60

N2+23N+60-N2-23N = 60

Therefore 60 = 60

3x5 boxes:

 54 55 56 57 58 64 65 66 67 68 74 75 76 77 78

(58x74)-(54x78) = 80

 13 14 15 16 17 23 24 25 26 27 33 34 35 36 37

(17x33)-(13x37) = 80

 2 3 4 5 6 12 13 14 15 16 22 23 24 25 26

(6x22)-(2x26) = 80

 36 37 38 39 40 46 47 48 49 50 56 67 68 69 60

(40x56)-(36x60) = 80

Proving algebraically the 3x5 boxes:

The answers for 3x5 boxes all answer to 80. I shall try to use the algebraic box once again where n is the smallest number in the box:

 N N+1 N+2 N+3 N+4 N+10 N+11 N+12 N+13 N+14 N+20 N+21 N+22 N+23 N+24

Lorenzo Brusini

I shall try to use the formula:

(N+4)x(N+20)-N(N+24) = 80

(N2+20N+4N+80)-(N2+24N) = 80

N2+24N+80-N2-24N = 80

Therefore 80 = 80

4x5 boxes:

 43 44 45 46 47 53 54 55 56 57 63 64 65 66 67 73 74 75 76 77

(47x73)-(43x77) = 120

 22 23 24 25 26 33 34 35 36 37 43 44 45 46 47 53 54 55 56 57

(26x53)-(22x57) = 120

 66 67 68 69 70 76 77 78 79 80 86 87 88 89 90 96 97 98 99 100

(70x96)-(66x100) = 120

 35 36 37 38 39 45 46 47 48 49 55 56 57 58 59 65 66 67 68 69

(39x65)-(35x69) = 120

Proving algebraically the 4x5 boxes:

The answers for 4x5 boxes all answer to 120. I shall try to use the algebraic box once again where n is the smallest number in the box:

 N N+1 N+2 N+3 N+4 N+10

Conclusion

2+10N+9-N2-10N = 9

Therefore 9 = 9

 N N+3 N+27 N+30

(N+3)x(N+27)-N(N+30) = 81

(N2+3N+27N+81)-(N2+30N) = 81

N2+30N+81-N2-30N = 81

Therefore 81 = 81

 N N+3 N+18 N+21

(N+3)x(N+18)-N(N+21) = 54

(N2+3N+18N+54)-(N2+21N) = 54

N2+21N+54-N2-21N = 54

Therefore 54 = 54

 N N+4 N+27 N+31

(N+4)x(N+27)-N(N+31) = 108

(N2+4N+27N+108)-(N2+31N) = 108

N2+31N+108-N2-31N = 108

Therefore 108 = 108

AxB grid:

 N N+A-1 N+G(B-1) N+G(B-1)+(A-1)

Where: A = The horizontal side of the box,

B = The vertical side of the box,

N = The smallest number in the box,

G = The size of the grid.

I shall put this box into the formula I used for all the other boxes:

(N+A-1)(N+G(B-1))-N(N+1G(B-1)+A-1)

=(N+A-1)(N+GB-G)-N(N+GB-G+A-1)

=N2+GBN-GN+AN+GAB-GA-N-GB+G-N2-GBN+GN-AN+N

= GAB-GA-GB+G

This can be factorised to make:

G(A-1)(B-1)

Lorenzo Brusini

I shall show that this formula works by using it on every rectangular box:

 Size of box Formula Answer 2x2 6(1x1) 6 2x2 7(1x1) 7 2x2 8(1x1) 8 2x2 9(1x1) 9 4x4 6(3x3) 54 4x4 7(3x3) 63 4x4 8(3x3) 72 4x4 9(3x3) 81 3x4 6(2x3) 36 3x4 7(2x3) 42 3x4 8(2x3) 48 3x4 9(2x3) 54 4x5 6(3x4) 72 4x5 7(3x4) 84 4x5 8(3x4) 4x5 9(3x4)

Looking at the results of the formula it shows that it works and can be used for any square or rectangle in any size grid.

Lorenzo Brusini

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 Grid Aim: The aim of this investigation is to formulate an algebraic equation ...

3 star(s)

determine if the difference is always 20 if arranged the other way. 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

2. GCSE Maths Sequences Coursework

I am now going to put the information I found from each of the shapes into tabular form so that I can understand it better and perhaps spot a pattern emerging. If I find a pattern I will use it to determine an Nth term and see if my predictions are right.

1. How many squares in a chessboard n x n

This is 8 x 8 square plus previous sequence (49 +36 + 25 + 16 + 9 + 4 + 1). Since n = 8, as stated earlier, it's mathematical form is 82 + 49 + 36 + 25 + 16 + 9 + 4 + 1 or 64 +

2. Number Grids Investigation Coursework

x 4 squares, to see if this is the same in all 4 x 4 squares in this grid. 7 8 9 10 17 18 19 20 27 28 29 30 37 38 39 40 (top right x bottom left)

1. Number Grid Maths Coursework.

20 7,10,17,20 (4x2) 30 32,35,52,55 (4x3) 60 48,50,88,90 (3x5) 80 71,75,91,95 (5x3) 80 Results From these results of the numerical examples we can again notice that every rectangle, like the squares, of the same size has the same difference. This can be seen by the results of the 5x3, with a difference of 80, and 2x3 rectangles, with a difference of 20 each time.

2. algebra coursework

44Z = 160 Table Square size Result Difference 2 X 2 10 3 X 3 40 30 4 X 4 90 50 5 X 5 160 70 The Square formula for a 10X10 grid X times X square X Z Z+(X-1)

1. Number Grids

t can be used for any number, and the difference will always come out correct no matter what number you replace it with. The formula will also tell us that the square number theory is correct, as it fits all the differences between 2 x 2 and 5 x 5 (in the table).

2. Maths Grids Totals

6 x 7 34 35 36 37 38 39 44 45 46 47 48 49 54 55 56 57 58 59 64 65 66 67 68 69 74 75 76 77 78 79 84 85 86 87 88 89 94 95 96 97 98 99 39 x 94 = 3666 34 x 99 = 3366 3666 - 3366 = 300.

• Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to