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  • Level: GCSE
  • Subject: Maths
  • Word count: 1897

Number Grid Investigation.

Extracts from this document...

Introduction

Mark Johnson 10MR                        

Number Grid Investigation

In this investigation, I am using a 10x10 number grid, with numbers 1 to 100.  I am going to take 2x2 sections of this grid, and calculate the differences between the products of the top-left and bottom-right numbers, and the bottom-right and top-left numbers.  Once I have found a pattern, I will attempt to write a generalisation using algebra.  After I have done this, I investigate larger sections taken from the grid, 3x3, 4x4 and so on.  I will then try to find a general rule for the numbers in any size grid using algebra.

2x2 Sections

12*23=276

13*22=286

   286-276=10

65*76=4940

66*75=4950

   4950-4940=10

29*40=1160

30*39=1170

   1170-1160=10

The difference is ALWAYS 10

Generalisation:

x

x+1

x+10

x+11

x*(x+11) = x²+11x

(x+1)(x+10) = x²+11x+10

   (x²+11x+10) – (x²+11x) = 10

Therefore:

(x+1)(x+10) – x(x+11) = 10

3x3 Sections

13*35=455

15*33=495

   495-455=40

27*49=1323

29*47=1363

   1363-1323=40

65*87=5655

67*85=5695

   5695-5655=40

The difference is ALWAYS 40

Generalisation:

x

x+1

x+2

x+10

x+11

x+12

x+20

x+21

x+22

x(x+22) = x²+22x

(x+2)(x+20) = x²+22x+40

   (x²+22x+40) – (x²+22x) = 40

Therefore:

(x+2)(x+20) - x(x+22) = 40

4x4 Sections

11*44=484

14*41=574

   574-484=90

26*59=1534

29*56=1624

   1624-1534=90

The difference is ALWAYS 90

Generalisation:

X

x+1

x+2

x+3

x+10

x+11

x+12

x+13

x+20

x+21

x+22

x+23

x+30

x+31

x+32

x+33

x*(x+33) = x²+33x

(x+3)(x+30) = x²+33x+90

   (x²+33x+90)

...read more.

Middle

x+31

x*(x+31) = x²+31x

(x+1)*(x+30) = x²+31x+30

   (x²+31x+30) – (x²+31x) = 30

Therefore:

[(x+1)*(x+30)]-[x+(x+31)] = 30

axb rectangles

image05.png

ximage12.png

x+(a-1)

x+10(b-1)image01.png

x+10(b-1)+(a-1)

image13.png

Size of rectangle

Difference

3x2

20

6x3

100

2x4

30

a x b

10ba-10b-10a+10

x*[x+10(b+1)+(a-1)] = x*[x+10b-10+(a-1)] = x²+10bx+10x+ax-x = x²+10bx–11x+ax

[x+(a-1)]*[x+10(b-1)] = [x+(a-1)]*[x+10b-10] = x²+10bx-10x+ax-x+10ba-10b-10a+10

= x²+10bx-11x+ax+10ba-10b-10a+10

(x²+10bx-11x+ax+10ba-10b-10a+10) - (x²+10bx–11x+ax) = 10ba-10b-10a+10

Therefore:

{[x+(a-1)]*[x+10(b-1)]} – {x*[x+10(b+1)+(a-1)]} = 10ba-10b-10a+10

The difference for any rectangle is 10ba-10b-10a+10 when a is equal to the horizontal length and b is equal to the vertical length.  This is also true when using squares. It is true because a square is just a rectangle with equal sides, and the formula is not affected if a and b are equal.

I am now going to continue the investigation with a 10x10 number grid numbered 1–200 with the numbers increasing by 2 instead of 1 each time.

3x2 rectangles

24*48=1152

28*44=1232

   1232-1152 = 80

Difference = 80

2x6 rectangles

52*154=8008

54*152=8208

   8208-8008 = 200

Difference = 200

4x3 rectangles

124*170 = 21080

130*164 = 21320

   21320-21080 = 240

Difference = 240

axb rectangles

...read more.

Conclusion

Therefore, the general rule for a grid where l=6 for any value of i is 6i²ab-6²a-6i²b+6i²

Generalisation:

The co-efficient of each part of the rule is the same as the value of l. This means that the general rule will be li²ab-l²a-li²b+li²

Length of row

Difference

10

10i²ab-10i²a-10i²b+10i²

6

6i²ab-6²a-6i²b+6i²

l

li²ab-li²a-li²b+li²

image08.png

zimage10.png

z+i

(z+i)+i

x

x+i(a-1)

b

x+il(b-1)

x+i(a-1)+il(b-1)

image11.png

x*[x+i(a-1)+il(b-1)] = x*(x+ia-i+ilb-il) = x²+iax-ix+ilbx-ilx

[x+i(a-1)]*[x+il(b-1)] = (x+ia-i)*(x+ilb-il) = x²+ilbx-ilx+iax+li²ab-li²a-ix-li²b+li²

   (x²+ilbx-ilx+iax+li²ab-li²a-ix-li²b+li²) – (x²+iax-ix+ilbx-ilx) = li²ab-li²a-li²b+li²

Therefore:

{[x+i(a-1)]*[x+il(b-1)]} – {x*[x+i(a-1)+il(b-1)]} - li²ab-li²a-li²b+li²

On any number grid with any length of sides and any increment, the difference between the products of the numbers in opposite corners in any rectangle is li²ab-li²a-li²b+li² when a=the horizontal length of the rectangle, b=the vertical length of the rectangle, i=the increment between numbers on the number grid, and l=the length of the rows on the grid.

...read more.

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