• 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
  • 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.

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

See related essaysSee related essays

Related GCSE Number Stairs, Grids and Sequences essays

  1. Step-stair Investigation.

    21X+35g+35 = (21*11) + (35*8) + 35 = 546 11+12+13+14+15+16+19+20+21+22+23+27+28+29+30+35+36+37+43+44+51= 546 31 32 33 34 35 36 25 26 27 28 29 30 19 20 21 22 23 24 13 14 15 16 17 18 7 8 9 10

  2. What the 'L' - L shape investigation.

    the L-Shape to different positions and investigate the relationship between the L-Sum, the L-Numbers and the grid size. Using the difference method as used in Part 1 on 4 by 4 grid I have found: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

  1. Number Grid Investigation.

    Let's see... (75 X 87) - (77 X 85) = 20. My formula is correct. I could use this formula to work out any size of square such as: 18 X 2 = 170 97 X 2 = 970 and so on...

  2. Mathematics - Number Stairs

    + (n+2) + (n+3) + (n+4) + (n+10) + (n+11) + (n+12) + (n+13) + (n+20) + (n+21) + (n+22) + (n+30) + (n+31) + (n+40) = 15n + 220 8 9 10 11 12 1 T = n T = n T = n T = n T = n 2

  1. Investigate the difference between the products of the numbers in the opposite corners of ...

    any square on a 10 x 10 grid it doesn't matter where it is positioned, the difference remains the same. I predict that this is the same for any rectangle on a grid. I will now try out several other shapes to test this theory.

  2. Maths - number grid

    46x67 3102 - 3082 Difference = 20 As can be seen, whether the rectangle is horizontal or vertical the outcome is the same. Because the outcome is the same I am going to concentrate on horizontal rectangles only. I am now going to use algebra to prove my defined difference of 20 is correct when using 3x2 rectangles.

  1. number grid

    Therefore: a(a + 33) = a� + 33a (a + 3)(a + 30) = a� + 33a + 90 (a� + 33a + 90) - (a� + 33a) = 90 According to this, the difference between the product of the top left number and the bottom right number, and the

  2. Mathematical Coursework: 3-step stairs

    Nevertheless using the annotated notes on the formula my formula would look like this now: > 6N =6n x 1= 6 > 6+b=54 Now I would need to find the value of b in order to use my formula in future calculations.

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