• 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
• Level: GCSE
• Subject: Maths
• Word count: 3344

# &quot;Multiply the figures in opposite corners of the square and find the difference between the two products. Try this for more 2 by 2 squares what do you notice?&quot;

Extracts from this document...

Introduction

Gurprit Singh Khela

Opposite Corners-coursework

“Multiply the figures in opposite corners of the square and find the difference between the two products.

Try this for more 2 by 2 squares what do you notice?”

Investigate!

In this investigation I will research various squares and rectangles within selected grids such as 10 by 10, 11 by 11 and so forth. I will find patterns between the differences and the squares and rectangles within the grids. By the end of this investigation my aim is to achieve a formula that will connect and link the shape within the grid whether it is a square, rectangle etc. to the size of the grid itself. If I have time I may possibly go into 3 Dimensions and investigate the addition of the next dimension and see how this will affect the 2D work so far and how it can be linked.

To start off with I will take a 2 by 2 square out of the top left hand corner of my grid which for now is 10 by 10 size.

 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

2 x 11 =22

1 x 12 =12

I have multiplied the opposite corners and now will subtract the smaller number from the larger one as explained in the question above.

22 – 12 = 10

By subtracting the 2 totals I have found the difference which is 10.I wonder if this difference of 10 will remain constant if I change the position of the 2 by 2 square on the grid. To test this I will move the square along a place and see what the difference is.

 2 3 12 13

3 x 12 = 36

Middle

33) = x2 + 33x

(x + 3)(x + 30) = x2 + 33x + 90

(x2 + 33x + 90) – (x2 + 33x) = 90

The difference is 90 and I have proven this with the use of algebra and numbers therefore my prediction was wrong and now I shall collate the differences so far and see what I can conclude on.

Here are my results so far:

 Square Size Difference First Difference Second Difference 2 x 2 10 3 x 3 40 +30 4 x 4 90 +50 +20 5 x 5 ???? ???? ?????

This is interesting as the first difference isn’t constant but the second difference is 20.Looking at the sequence I predict that the 20 will remain the same for the second difference and the main difference for the 5 by 5 square will be 160.This is because I will add the 20 to the already increasing first difference of 50 which will be 70.Then I will add the 70 to the 4 by 4 main difference of 90 which will make 160.Therefore a 5 by 5 square in my opinion should have a difference of 160.Now I will test my new prediction.

 1 2 3 4 5 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45

1 x 45 = 45

5 x 41 = 205

205 – 45 = 160

 x x + 1 x + 2 x + 3 x + 4 x + 10 x + 11 x + 12 x + 13 x + 14 x + 20 x + 21 x + 22 x + 23 x + 24 x + 30 x + 31 x + 32 x + 33 x + 34 x + 40 x + 41 x + 42 x + 43 x + 44

x(x + 44) = x2 + 44x

(x + 4)(x + 40) = x2 + 44x + 160

(x2 + 44x + 160) – (x2 + 44x) = 160

My theory is correct and the difference has been proven to be 160.I have now found a pattern and have enough results to come up with a formula that will give me the results for any size square on a 10 by 10 size grid.

Conclusion

n= one side of the square

I have expanded my formula because now it will give me the difference from any size square within any size grid. To test my formula I will choose a 3 by 3 square on a 12 by 12 firstly work it out with the formula then the full way and see if the answers match.

With the formula:

x(n – 1) 2

12(3 – 1)2

The difference in theory is 48

I will now test the difference to see if it remains the same:

 1 2 3 13 14 15 25 26 27

3 x 25 = 75

1 x 27 = 27

75 – 27 = 48

The difference is 48

This proves that my formula will work with any grid and any size square within it at any place on the grid. I have now reached the limit of how squares within a 2 dimensional grid can be investigated. To add another variable into my work I am going to investigate rectangles. This will add a further variable because as with squares both the length and width are represented by 1 number whereas rectangles can have 2 completely different numbers. Therefore I am expecting to work out a much different formula from the one above with the inclusion of another variable. This formula should provide me with the following:

• Difference to any size square or rectangle on a size grid in any location on the grid.

I will work out the difference to a 2 by 3 rectangle, all the work from this point will remain on a 10 by 10 grid until stated otherwise.

 1 2 3 11 12 13

1 x 13 = 13

3 x 11 = 33

Difference is 20

 x x + 1 x + 2 x +10 x + 11 x + 12

(x + 2)(x + 10) = x2 + 12x + 20

x(x + 12) = x2 + 12x

Difference is 20

2 x 4:

 1 2 3 4 11 12 13 14

4 x 11 = 44

1 x 14 = 14

Difference is 30

 x x + 1 x + 2 x + 3 x +10 x + 11 x + 12 x + 13

Page

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. ## Opposite Corners. In this coursework, to find a formula from a set of numbers ...

4 star(s)

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

2. ## I am going to investigate the difference between the products of the numbers in ...

4 star(s)

So the formula for 2x?n? was 1 x 10(L-1), And the formula for 3x?n? was 2 x 10(L-1), And the formula for 4x?n? was 3 x 10(L-1). One possible step would be to replace the 1, 2 and 3 with (W-1).

1. ## I am going to investigate by taking a square shape of numbers from a ...

4 star(s)

� * =10(6-1) � * =10x25 * =250 My prediction is right. number Left corner x right corner Right corner x left corner Products difference 19 11x88=968 18x81=1458 490 20 1x78=78 8x71=568 490 I predict that the result for an 8x8 square in a 10x10 grid will be 490 by using this formula: * 10(b-1)

2. ## I am going to investigate taking a square of numbers from a grid, multiplying ...

3 star(s)

I did this by looking at patterns. Now I am going to prove it is right by using algebra and get an expression at the end of it, which could be used to work out the difference of any 2x2 square, in any grid. I am going to call the first corner in the 2x2 square 'x'.

1. ## Number Grid Investigation.

This is correct. (12 X 42) - (14 X 40) = 56. Both formulas are correct. You just have to change the first number accordingly to match the width of the grid. I can now abbreviate my formulas. Abbreviated formulas 'z' => width of grid.

2. ## For other 3-step stairs, investigate the relationship between the stair total and the position ...

X+4 X+5 Using the algebra equation and adding the values we get -385 If we closely examine the results from the 6-step stairs from the 3 numbered grid boxes, i.e. 10x10, 11x11 and 12x12 we can see there is a constant number that is consistent, which is [35] We can

1. ## Investigation of diagonal difference.

I predict that for a 2x2 cutout on a 5 x 5 grid, the diagonal difference will be 5. 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 .

2. ## Number Grid Investigation.

My findings show also by using algebra in the equations, I can construct an algebraic formula. By using `N` in the top left of the algebraic table, the equation, when fully worked out will always give the correct constant answer in any square selection size in any size grid.

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