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• Level: GCSE
• Subject: Maths
• Word count: 1581

Number grid algebraic course work

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Introduction

Number grid algebraic course work

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Firstly I was given this number grid, the intrusions were…

• A box is drawn round four numbers.
• Find the product of the top left number and the bottom left in this box.
• Do the same with the top right and bottom left numbers.
• Calculate the difference between these products.
• Investigate further.

The first thing I did was follow these instructions. Then I changed the box size and looked for patterns.

A two by two box…

 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

12×23=276                 286-276= 10

13×22=286

I tested a two by two box two more times to ensure each time I got the answer of 10, also to ensure that the answer was the same in different areas of the grid. I will do this for each different size boxes.

 78×89=694279×88= 6952 9×20=180               10×19= 190 6952-6942 =10 190-180=10

Middle

46

52

53

54

55

56

 12×56=67216×52=832 1×45=455×41=205 96×60=5760100×56=5600 832-672=160 205-45=160 5760-5600=160

I have noticed that all these numbers are divisible by ten. I’m going to divide by 10 to see what answers I get…

 box Difference Divide by 10 answer 2×2 10 10÷10= 1 3×3 40 40÷10= 4 4×4 90 90÷10= 9 5×5 160 160÷10= 16

I have notice that when I divide by 10 I get all square numbers as my answers. These answers are all less than the box size chosen. Eg 2×2    2-1=1     1²

 box Difference Divide by 10 answer Square no 2×2 10 10÷10= 1 1² 3×3 40 40÷10= 4 2² 4×4 90 90÷10= 9 3² 5×5 160 160÷10= 16 4²

So from this I can make an equation to test any box…

10 (n-1)(n-1)=10 (n-1)²

Now I will test rectangle boxes to further my investigation. I predicted that this formula would not work on the rectangles. I will have to alter the formula later.

A three by two box…

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12×24=288               308-288=20

22×14=308

I tested the three by two box twice more in different places to see if it would be the same answer…

 5×17=857×15=105 67×79=529369×77=5313 105-85=20 5313-5293=20

Now I will test the box as a two by three and see if this will alter my answer…

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Conclusion

128

4293-4165=128

I can use the same formula on this grid except I have to change it slightly… this was my formula for the 10 grid

10(c-1)(r-1)

as this is a 8 grid and all my difference are divisible by 8 I will change my formula to

8(c-1)(r-1)

I can check my answers using this formula

8(5-1)(5-1)=128                  8(2-1)(2-1)=8

I now know that I can use my formula to find my difference of rectangles in this grid. Eg.

A two by three grid…

 11 12 13 19 20 21

11x21=231           247-231=16

13x19=247

8(2-1)(3-1)=16

The answers are the same so I think it is more efficient to use the formula to find the answers…

8(2-1)(4-1)=24        8(4-1)(2-1)=24

8(2-1)(5-1)=32        8(5-1)(2-1)=32

I will now further my investigation using algebra…

If I know that it’s a two by grid…

2

 X

2

TL=X  TR=X+1

BL=X+10   BR=X+11

TL X BR         TRXBL

(X)(X+11)        (X+1)(X+10)

X²+11X         X²+10+X+10X   X²+10+11X

This helps to p

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