• 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
  • Level: GCSE
  • Subject: Maths
  • Word count: 2272

Number grid

Extracts from this document...

Introduction

Mathematics Coursework; Number Grid

I have been asked to investigate a mathematical problem, in this piece of work I will explore the problem investigating a number of variables, using algebra to prove that the results hold for any combination of number.

12 x 23 = 276

13 x 23 = 286

The difference between the above two products is equal to 10.

I will repeat this using different numbers to see if there is a recurring pattern.

35 x 46 = 1610

36 x 45 = 1620

The difference between the above two products is equal to 10.

I predict that if I work out the following, the difference will be equal to 10.

38 x 49 =

39 x 48 =

I will now check to see if I am correct.

38 x 49 = 1862

39 x 48 = 1872

I was correct there is a difference of 10.

I am now going to try to prove that this holds for any 2 by 2 square using algebra.

Let n be the top left number and n + 11 be the bottom right number.

n(n+11) = n²+11n

Let n + 1 be the top right number and n + 10 be the bottom left number.

(n+1)(n+10) = n²+n+10n+10

              = n²+11n+10

I will now subtract the two algebraic products to get an overall product.

n²+11n+10

n²+11n       -

        +10

This proves that the difference is always 10



I have proved that the above results hold for any 2 by 2 square.

This shows that for any 2 by 2 square the difference will always be equal to 10.

To extend this investigation I am going to investigate the effect of changing the shape of the box.

...read more.

Middle

This shows that the difference is 60, I will use a 2 by 7 rectangle to see if I am correct.

1 x 17 = 17

7 x 11 = 77

The difference between the above two products is equal to 60, I was correct.

I will now use algebra to try and prove that this holds for any 2 by p rectangle.



Let n be the top left number and n + p + 9 be the bottom right number.

n(n+p)-9 = n²+np+9n

Let n + p - 1 be the top right number and n + 10 be the bottom left number.

(n+10)(n+p-1) = n²+np-9n+10P-10

I will now subtract the two algebraic products to get an overall product.

n²+np-9n+10P-10

n²+np+9n           -

            10p-10

This proves that the difference is always 10P-10


The above algebra tells me that for any 2 by p square the difference is always 10P-10.

To extend the investigation further I am going to investigate the effect of changing the size of the square, I will begin with a 3 by 3 square.

35 x 57 = 1995

37 x 55 = 2035

The difference between the above two products is equal to 40.

I am now going to try to prove that this holds for any 3 by 3 square using algebra.

Let n be the top left number and n + 22 be the bottom right number.

n(n+22) = n²+22n

Let n + 2 be the top right number and n + 20 be the bottom left number.

(n+2)(n+20) = n²+2n+20n+40

              = n²+22n+40

I will now subtract the two algebraic products to get an overall product.

n²+22n+40

n²+22n       -

        +40

...read more.

Conclusion

(n+1)(n+w) = n²+nw+w+1

I will now subtract the two algebraic products to get an overall product.

n²+nw+w+1

n²+nw+1    -

          +w

This proves that the difference is always equal to w

I have proved that the above results hold for any n by w square. W is equal to the size of the overall number grid.

It shows that the rule applies to any combination of letters or numbers, within its limits, there are certain limitations which mean that the solution will not hold for every number, it will not hold for any grid which contains 100 or more numbers, (in other words greater than 10 by 10), this shows that although that the solution does hold for most possibilities certain limitations mean that it cannot fit for absolutely every possibility.

I will now co-ordinate all three of my variables to see if there is any link, when I changed the size of square I discovered that the difference of any m by m square was equal to 10 (p-1)², when I changed 10 (p-1), already I can spot that the only difference between these two differences is that the square difference is squared, and the rectangle difference isn’t.

In conclusion, I have learnt that the by simply exploring and investigating different variables you can discover that there are certain links and connected outcomes, from my investigations I have discovered that the differences that I have found, hold for that particular shape or grid size, I have proved this algebraically, providing the general solution which shows that it holds for any number or letter.

Robert Shaw

Candidate Number: 1075 Centre Number: 41284

...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. Marked by a teacher

    Number Grid Aim: The aim of this investigation is to formulate an algebraic equation ...

    3 star(s)

    So for any 2x2 square taken from any size grid, the difference will be the number of the grid size. I can now work out the difference of any 2x2 square as long as I know the grid size. 5 x 5 grid = Difference of 5 19 x 19

  2. Number Grid Investigation.

    Let's see... 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 Formula 2.

  1. Maths coursework. For my extension piece I decided to investigate stairs that ascend along ...

    + 36 + 37 + 43 + 44 + 51 = 546 ? This shows that my formula must work for all 6-stair shapes on the 8 x 8 grid. To investigate if this formula also works for 6-stair shapes on other grid sizes, I am going to see if

  2. Number Grid Coursework

    p x q box on a width 10 grid, the difference of the two products will always be 10(p - 1)(q - 1), "the length of the box minus 1" multiplied by "the width of the box minus 1", multiplied by 10.

  1. Mathematics - Number Stairs

    + 12 T = 3n + 13 3 T = 6n + 36 T = 6n + 40 T = 6n + 44 T = 6n + 48 T = 6n + 52 4 T = 10n + 90 T = 10n + 100 5 4 Step-Staircase / Grid Width

  2. Maths - number grid

    calculated in Chapter Two I have been able top establish an accurate formula for rectangles in a 10x10 number grid, my result here was: 10(s - 1)(r - 1) Chapter Three So I have now investigated squares and rectangles in a 10x10 grid and have discovered two formulas.

  1. number grid

    and the product of the top right number and the bottom left number in a rectangle. I will do this by finding the formula for changing one side, then I will find the formula for changing the other side and combine the two together.

  2. Mathematical Coursework: 3-step stairs

    19 20 21 22 23 24 25 26 27 10 11 12 13 14 15 16 17 18 1 2 3 4 5 6 7 8 9 73 74 75 76 77 78 79 80 81 64 65 66 67 68 69 70 71 72 55 56 57 58 59

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