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  • Level: GCSE
  • Subject: Maths
  • Word count: 1834

Number Grids

Extracts from this document...

Introduction

Introduction

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100

We were given this number grid and shown that if a two–by-two square was drawn anywhere on the grid, encompassing four numbers, the difference between the product of the top left number and the bottom right number and the top left number.

For example        (13x22) - (12x23) = 10

Also                (16x25) - (15x26) = 10

We were then told to investigate further.

To give us an idea of where we were headed, our teacher told us we should be aiming at least to find a formula that would allow a person to work out the difference between the products of the corners of any rectangle on such a grid of any length.

The Investigation

The fact that the difference came out to ten no matter where the square was drawn made sense, it could be easily demonstrated algebraically.

Let’s say the top left number was ‘n’. If followed along the grid, the other numbers would come to be: (bottom right) n+11, (top right) n+1 and (bottom left) n+10.

So if worked out, the sum would look like:

image16.png

After this, I tried squares of different sizes around the grid.

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I found that the differences in all the 3x3 squares came out to 40

Eg.

(14x32) - (12x34) = 40

(18x36)

...read more.

Middle

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2x2:

(2x12) - (1x13) = 11

(5x15) – (4x16) = 11

3x3:

(36x56) – (34x58) = 44

(40x60) – (38x62) = 44

4x4:

(81x111) – (78x114) = 99

(86x116) – (83x119) = 99

The pattern came out as:

2x2

3x3

4x4

5x5

 11

44

99

176

So,

11        44        99        176image00.pngimage00.pngimage00.png

      33         55       77                1st Differenceimage00.pngimage00.png

        22        22                2nd Difference

image73.png, determining the coefficient to be used for n2.

n

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pattern

11

44

99

176

11n2

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99

176

275

pattern – 11n2

-33

-55

-77

-99

image18.png

Formula:                                

image19.pngimage01.pngimage01.pngimage01.png

                        -22        -22       -22         Formula for linear part:         

Put together:

image20.png

(Simplifying down to this)

Indeed the only part of the end formula that changed, was the part hypothesized to be directly linked to the length of the grid. With the grid length 10, the formula came out to beimage21.png, with the grid length as 11, it came out to beimage22.png. As this part of the formula appeared to be linked to the grid length, it could now be replaced with an expression representing the grid length so it would work for any grid. Thus the formula now becameimage23.png. If the assumption that one of the image24.pngs were in fact the length, and the other the width continued, the formula then becameimage25.png.

To prove this formula, I looked at the situation algebraically and came up with a list of expressions to represent the real life values.

...read more.

Conclusion

image49.png

Simplifying down to image51.png

This was my first 3D formula.

Now for the second set, the calculation was image31.pngimage45.png-image30.pngimage46.png. When the respective expressions were substituted for these values, it came out to:

image52.png

image53.png

Simplifying down to image54.png

This was my second 3D formula.

I tested the both of these to check that everything was ok with them.

The example I came up for the test was:

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11image11.png

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image14.png

In the example, image55.png, image56.png, image57.png, image58.png, image59.png, image60.png, image61.png and image62.png. Other than that, image63.png, image65.png, image66.png, image67.png, image68.png.image15.png

For the first formula,

image69.png

If done with the formula instead,

image70.png

For the second formula,

image71.png


If done with the formula instead,

image72.png

In both cases, the answer came out the same, both formulas worked. One slight problem was that somewhere in the process I managed to mix up the order I subtracted in so that my first formula always gave a negative answer (as I subtracted a larger value from a smaller one) and my second formula always gave a positive answer (as I subtracted a smaller value from a larger one, like I was supposed to). This isn’t that much of a problem as what the actual number was would always came out the same, the sign in front of it solely depends on the order in which the numbers were subtracted from one another. As the investigation asks for a mere difference, the sign could be ignored altogether.

...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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