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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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. I will do this 2 more times to check that this is not a fluke.

5a. 7     8     9    10

     17   18   19   20

b.   7 × 20 = 140

    17 × 10 = 170

c. 170 – 140 = 30

        

        This rectangle also has a differentce of 30. this means that my prediction was right. My prediction was that a 2×4 rectangle would have a different of 30.

        

        In the next set of rectangles I am going to do a rectangle that is 2×5. I think that the different will be 40. I have gotten this from the previous size of rectangles and I can see a pattern that is in it.

7a.  21   22   23  24  25  

       31   32   33  34  35

b. 21  × 35  =  735

    31  × 25  =  775

c. 735 – 775 = 40

        

In this rectangle the differentce was 40, just as my prediction said it would be. I will do two more of this size before I will change the size completely, and go for a 3 down rectangle.

8a.  85   86   87   88   89

       95   96   97   98   99  

b. 85 × 99  =  8415

    95 × 89  =  8455

c. 8455 – 8415 = 40

Join now!

        This rectangle has a differentce of 40. from all these we can see that my prediction was right.

        I will now do a formula for this:

For rectangles with 2 rows high and 3 columns wide the difference is always 20

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

For rectangles with 2 rows high and 5 columns wide the difference is always 40

So      the number of columns wide – 1 × 10 gives the answer

So   10 × (Width – 1) is ...

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