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  • Level: GCSE
  • Subject: Maths
  • Word count: 2284

Number grid coursework

Extracts from this document...

Introduction

Number Grid Coursework.

For this coursework I shall investigate and explain thoroughly the
patterns, rules and formulae found in a number grid when placing a
square at any point in the grid, multiplying the top left and bottom
right corners and the top right and bottom left corners and finding
the difference. In the beginning of my investigation I’d like to write the main purpose of it. I want to find one common formula that would help me to find difference on any x by x square on 10 by 10 number grid. Let’s start the investigation!

I have a 2 by 2 square in a 10 by 10 number grid.

I am going to investigate it and find the difference.

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First I shall find the product of the top left number and the bottom right number in the square.

image00.png

I shall find the product of the top right and bottom left numbers.

image01.png

Now I shall find difference between these products.

image26.png

Let’s now move the square and do same calculations.

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...read more.

Middle

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1)*  image23.pngimage23.png

2)    image24.pngimage24.png

       3)     image25.pngimage25.png

*        The calculations that are being done every time (i.e. multiplying the top left and bottom
right corners and the top right and bottom left corners and finding
the difference) will only be shown by numbers (i.e. 1), 2), 3)) in the further content of the coursework.

I predict that my next 3 by 3 square will result in the answer 40.

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  1. image27.pngimage27.png
  2. image28.pngimage28.png
  3. image29.pngimage29.png

My prediction was correct. There was a difference of 40. This may mean that it doesn't matter which 3 by 3 square you select within a 10 by 10 number grid the difference will still be 40.

I will now prove my conclusion using algebra.

Any top left number will be image07.pngimage07.png, any top right number will be image30.pngimage30.png, any bottom left number will be image32.pngimage32.png, any bottom right number will be image33.pngimage33.png in any 3 by 3 square on a 10 by 10 number grid.

  1. image34.pngimage34.png
  2. image35.pngimage35.png
  3. image36.pngimage36.png

This proves my conclusion and means that all 3 by 3 squares taken from a 10 by 10 number grid result in the answer 40.

I will now investigate 4 by 4 squares in a 10 by 10 number grid.

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  1. image38.pngimage38.png
  2. image39.pngimage39.png
  3. image40.pngimage40.png

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...read more.

Conclusion

  1. image03.pngimage03.png
  2. image04.pngimage04.png
  3. image05.pngimage05.png

All 5 by 5 squares on a 10 by 10 number grid result in the answer 160.

Now, using algebra I will prove that an x by x square will result in image06.pngimage06.png

Any top left number will be image07.pngimage07.png, any top right number will be image09.pngimage09.png, any bottom left number will be image10.pngimage10.png, any bottom right number will be image11.pngimage11.png in any x by x square on a 10 by 10 number grid.

  1. image12.pngimage12.png
  2. image13.pngimage13.png
  3. image15.pngimage15.png

I can see that image16.pngimage16.png repeats in all parts of equation. To simple equation I will renameimage16.pngimage16.png. It will now be called T.

So image17.pngimage17.png

image18.png

Now I will put image19.pngimage19.png instead of T.

So it is:

image21.png

This means that an x by x square would result inimage21.pngimage21.png. This means my
prediction was correct. This is the result for any x by x grid taken from
a 10 by 10 number grid.

In this investigation I found a formula that can be used to find difference in any x by x square on a 10 by 10 number grid. This was the main purpose of my investigation, so I can say that the investigation passed successfully.

If I were to extend this investigation further, I would experiment with different sized master grids

...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.

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