# To investigate the patterns generated from using rules in a square grid

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Introduction

Coursework Task – Number Grids

Aim - To investigate the patterns generated from using rules in a square grid.

Keywords (Internet Search) – Algebraic expressions, proof, nth terms, sequences

Textbook Reference– Key Maths Higher Chapters 4 and 8.

## Investigate | Write up |

Investigate results from a 2 by 2 square on your chosen grid | I have decided to investigate a 2 by 2 square because……………………………….. After a few results I observed that a ……………….. |

Investigate results from a 3 by 3, 4 by 4 square on your chosen grid | I decided to try out……………………………….. I put my results into table and observed that……… I therefore expect…………………………………. |

Make a prediction and test | I predict that for a 5 by 5 square I will always …….. I tested this by using……………………………….. |

Algebraic Proof | I will now attempt to prove my results algebraically. I conjecture that………………………………….. I will use “n” to represent ………………..,this means that…………………………………………. Using the rule I get the expressions..….and…..… By subtracting I get………………………………. This proves that……………………………………. |

Link results together and extend | By putting results in a table I have noticed that………………………………. This suggests that………………………………. I will now try a larger grid size. Having checked a further series of results I conclude /conjecture/expect……………………………. |

Initial Conclusion | Summarise with bullet points. I found that ………………………………………. I will now try to extend and prove the results further |

Algebraic Proof 1 | My results suggest that for an “n by n” square on an “m by m” grid ……………………………………. To prove this I will let “n” be……………………… Therefore using the diagram below the corners are…. This gives………………….which proves…………. |

Algebraic Proof 2 | If I were to consider rectangles then…………….. I could prove this by…………………………… |

Extension | I decided to use cuboids in three dimensions by……. I therefore found……………………………………. |

Final Conclusion | In this project I found that………………………. I have successfully……………………………….. If I were to extend this project further I would…….. |

Appendices | Bibliography Internet print outs Rough Work |

Middle

I will aim to investigate the differences for rectangles that are aligned differently, of different lengths, widths and when they are squares.

I plan to use algebra to prove any rules I discover which I will hopefully discover by analysing my results that I will log in tables. I will test any rules, patterns and theories I find by using predictions and examples.

I will record any ideas and thoughts I have as I proceed.

Method:

To start with I will break up my work into sections, my first is:

2x3 Rectangles

I will concentrate on one particular aspect of at a time.

What is the difference between the products of the corners?

Is the difference the same for a rectangle drawn anywhere in the grid?

I now have several questions:

Is the difference different when the rectangle is aligned so that its shortest sides are at the top and bottom?

What is the same between the two alignments?

What About Other Sizes of Rectangles?

I will now try rectangles, all in the 2 x X series, with different lengths.

Conclusion

So far so good

And now I will use the above expression to find the difference (which should be 10)

L-1 (G (H-1))

2-1 (10 (2-1))

1 x (10 x 1)

1x 10 = 10

It seems fine so far.

I will now check it with a 3 x 4 rectangle on a 10x10 grid. I Know the difference should be 60 from previous workings.

L-1 (G (H-1))

4-1 x (10 (3-1))

3 x (10 (2))

3 x 20 = 60

The expression is true !

L-1 (G (H-1)) is the universal rule with a the brackets in different positions:

L-1 (G (H-1)) = L-1 x (Grid Width) x (H-1) = Difference

Evaluation

I believe I have completed my aims and succeeded in attaining the goals I set myself. I have found the rule to work out the difference for any rectangle of any size on any grid. I have done this through progressive investigation and through the use of algebraic methods.

I have taken this course work as far as I can in the time that I have been allotted. The only other area that is left to be explored would be drawing a cuboid in a cube. Though I imagine this would be a lot more complicated, I am left wondering the possibility of the existence of a rule that would cover any, or even an infinite number of dimensions.

I am happy with the work that I have done and can not think of anything I could have carried out better or improved upon.

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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