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  • Level: GCSE
  • Subject: Maths
  • Word count: 4459

Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted.

Extracts from this document...

Introduction

Jonathan Ransom

Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted

Introduction

The purpose of this investigation is to explore the answer when the products of opposite corners on number grids are subtracted and to discover a formula, which will give the answer in all cases. I hope to learn some aspects of mathematics that I previously did not know.

The product is when two numbers are multiplied together.

There is one main rule: the product of the top left number and the bottom right number must be subtracted from the product of the top right and bottom left numbers. It cannot be done the other way around.

I plan to achieve my aim by attempting a series of different methods and comparing the answers to see if they are the same. The first method I will do is to draw diagrams, using excel. These will give me a clearer picture of what is involved in this investigation. I will write all the calculations of how I worked out the answer beside each diagram.

There are many different types of number grid, but I will start with square grids. I will also assume that the starting number is one and that the numbers in the grid are consecutive. The numbers in the grid will go horizontally from left to right.

image00.png

Examples

2 x 2 Grid          

1image12.pngimage01.png

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3 x 3 Grid

1image23.png

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4 x 4 Grid

1image34.png

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No patterns are visible yet so I have done two more diagrams.

5 x 5 Grid

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6 x 6 Grid

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Using the same method:

(6 x 31) – (1 x 36)

186 – 36 = 150

Answer   = 150

I do not need to draw any more diagrams because it is clear what is being done to get the answer and I now have enough data to make a table of results.

...read more.

Middle

Sequences

Now I have tried grids with number sequences. First where the numbers increase by 2.

2 x 2 Gridimage03.png

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 I have checked this by doing another 2 x 2 grid but with different numbers.

2 x 2 Gridimage04.png

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I have now done a 3 x 3 grid just to check this rule.

3 x 3 Gridimage05.png

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Now I have decided to try number grids where the numbers increase by 3, to see what happens.

2 x 2 Gridimage06.png

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I have realised that when the increase in the numbers is 1 the answer is 1 times as much, (1²).

When the increase in numbers is 2 the answer is 4 times as much, (2²)

When the increase in numbers is 3 the answer is 9 times as much, (3²)

So the increase squared is how many times more the answer is, than the answer in a grid where the numbers increase by one.

So presumably the answer for a grid where the numbers increase by 4 would be 16 (4²) times as much.

So if I check it:

2 x 2 Gridimage07.png

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 I have made a table with this data. And from it you can predict the answer for any square grid and for different increase in the numbers.

Grid Size

Answer when

Answer when

Answer when

Answer when

increase in

increase in

increase in

increase in

numbers is 1

numbers is 2

numbers is 3

numbers is 4

2 x 2

2

8

18

32

3 x 3

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48

108

192

4 x 4

36

144

324

576

5 x 5

80

320

720

1280

Therefore the formula for number grids where:

The increase in numbers is 2 = 4(N³ - 2N² + N)         

The increase in numbers is 3 =        9(N³ - 2N² + N)          

The increase in numbers is 4 =        16(N³ - 2N² + N)        

The increase in numbers is 5 =         25(N³ - 2N² + N)        

So the overall formula is N³ - 2N² + N multiplied by the increase in the numbers squared.

A = Increase² (N³ - 2N² + N)

Rectangles

Now I will investigate rectangular number grids. I have chose to keep the width (W) the same but vary the depth (D) of the grid.

2 x 3 Grid

image08.png

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2 x 4 Grid

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2 x 5 Grid

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...read more.

Conclusion

All the formulae I have, contain either of two main expressions: N³ – 2N² + N or

D (W²) + W – W² - WD.

This is the case for all the formulae except the ones for squared and cubed numbers.

In this investigation I have learnt how to use difference tables and how they help to work out formulae. I have learnt a lot about working with formulae and simplifying them.

There are many more extensions that can be investigated. Squared and Cubed decimals and squared sequences, for example. But there is simply not enough time to investigate all types. This is an endless investigation.

The main question was to investigate square grids with consecutive numbers. I believe I have gone a step further and investigated a wide range of extensions and given an equation for each, that works.

I drew all the diagrams that I did so that it was easier to work out the answer.

I have stuck to my plan all the way through except, in my plan I did not mention anything about extensions.  

The part of the project I found most difficult was simplifying the equations for squared and cubed number grids. I could not simplify the formula for cubed numbers so I left it as it was.

All the calculations I have done took a long time but were worth doing in order to discover and prove a formula. The graph that I did on page 6 did not help whatsoever in my investigation, but until I did it I did not know that.

All the formulae I have, work and therefore I can use them to predict larger sized grids that are impossible to draw.

...read more.

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