# Investigate the difference between the products of the numbers in the opposite corners of a rectangle that can be drawn on a 100 square. We were giving as the first rectangle to compare was this

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Introduction

## Maths Coursework

Opposite Corners

April 2005

Investigate the difference between the products of the numbers in the opposite corners of a rectangle that can be drawn on a 100 square.

We were giving as the first rectangle to compare was this

A. 54 55 56

64 65 66

So we have to do

B. 54 × 66 = 3564

64 × 56 = 3584

Then;

C. 3584 – 3564 = 20

The different in this is 20. I am going to investigate if the differences will change if I change the numbers involved.

1a. 1 2 3

11 12 13

b. 1 × 13 = 13

11× 3 = 33

c. 33-13=20

The different in this rectangle is 20, the same as the starter rectangle. From this I am going to try another rectangle the same size as this one and the original.

2a. 84 85 86

94 95 96

b. 84 × 96 = 8064

94 × 86 = 8086

c. 8084 – 8064 = 20

The different in this rectangle is 20 as well. From this it is starting to build up a picture, that all the rectangles this size have the same different of 20. however I will do one more rectangle in this size (2×3)

3a. 81 82 83

91 92 93

b. 81 × 93 = 7533

91 × 83 = 7553

c. 7553 – 7533 = 20

The different in this rectangle is also 20. this indicates that all rectangles of this size will have the difference of 20.

Now I am going to do a rectangle of 2×4 squares. I think that these rectangles different will be 30.

4a. 34 35 36 37

44 45 46 47

b.34 × 47 = 1598

44 × 37 = 1628

c. 1628 – 1598 = 30

This shows that the different in a rectangle the size of 2 × 4 is 30, as I predicted.

Middle

- 36

45 46

55 56

b. 25 × 56 = 1400

55 × 26 = 1430

c. 1430 – 1400 = 30

This rectangle has a different of 30. I m now going to do a rectangle that is the size of 4 × 3. these rectangles should be 60.

19a. 7 8 9

17 18 19

27 28 29

37 38 39

b. 7 × 39 = 273

37 × 9 = 333

c. 333 – 273 = 60

This rectangle, as I prettied has a different of 60, I will do one more rectangle this size before I make it bigger.

20a. 53 54 55

63 64 65

73 74 75

83 84 85

b. 53 × 85 = 4505

83 × 55 = 4565

c.4565 – 4505 = 60

This rectangle has a different of 60. now I am going to make the rectangle bigger by one across (4×4) this should be 90.

21a. 1 2 3 4

11 12 13 14

21 22 23 24

31 32 33 34

b. 1 × 34 = 34

31 × 4 = 124

c. 124- 34 = 90

This rectangle has a different of 60.

The pattern for rectangles with 4 rows high is:

For a rectangle with 4 rows high and 2 columns wide the difference is always 30

For a rectangle with 4 rows high and 3 columns wide the difference is always 60

For a rectangle with 4 rows high and 4 columns wide the difference is always 90

Now I am going to do rectangles with 5 down e.g. 5 × 2. And I will start and 5 × 2.

22a. 27 28

37 38

47 48

57 58

67 68

b. 27 × 68 = 1836

67 × 28 = 1876

c. 1876 – 1836 = 40

This rectangle has different of 40. From this I have predicted that they will go up in 40s for 5 down. E.g. for 5×3 it should be 80 and for 5 × 4 it should be 120 and so on.

Conclusion

With 3 rows down, it should be 20

With 4 rows down, it should be 30

And so on.

So the formula to work these out would be

(10 × Number of Rows) – 10

We can simplify this to

10 × (Number of Rows – 1)

Or 10 × (H – 1)

We must join these two formulas together to get one formula

Which are:

10 × (W – 1)

And

10 × (H – 1)

This gives the formula:

10 × (H – 1) × (W – 1)

Both are multiplied by the same 10.

I will now chose random rectangles and use my formula, then check them against the table I have done.

Size 3 × 3

71 72 73

81 82 83

91 92 93

10 (H-1) × (W-1)

10 (3-1) × ( 3-1)

(10×2) × 2 = 40

Using the formula I have worked out that this rectangle has a different of 40. looking back to the table and to the original way to do this, this size also had a different of 40.

Now using my formula I will try a rectangle the size of 4 × 5

55 56 57 58 59

65 66 67 68 69

75 76 77 78 79

85 86 87 88 89

10 (H-1) × (W-1)

10 (4-1) × (5-1)

(10×3) ×4 = 120

Using the formula the rectangle has a different of 120. and using the table you can see that this is rights for a rectangle this size.

Now using my formula I will try a rectangle that has a size of 10×10

1 /10 10

/10 /10

91 / 10 100

10 (H-1) × (W-1)

10 (10-1) × (10-1)

(10×9) ×9= 810

The different of this rectangle is 810. If you did it the long way, you would find that it would also have a different of 810.

I will now do some graphs to so the relation between the different size rectangles.

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

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