# Opposite Corners of a Square on a Number Grid

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Introduction

Number Grid – Opposite Corners Investigation

Introduction:

This task involves pupils investigating the relationship between the results, when diagonally opposite numbers in various sizes of squares are multiplied, and comparing the products.

Number Grid – Opposite Corners Investigation

Method:

- I will use a number grid from 1-100, like the one below, to collect examples of data from different size squares.
- I will show 4 examples of each size of squares, to confirm each of the differences at the different stages.
- I will look for a pattern and, if I find one, will explain it fully.
- I will make a prediction of what I think will happen next and see if it works.
- I will create an algebraic expression to prove my results.
- I will then test my algebraic expression on the diagonally opposite corners a rectangle, to see if that works as well.
- I will then describe the sequence and relationship of the results.
- I will present my results in a table and write a conclusion.

Prediction:

Middle

10 * 46 = 460

Difference = 160

51 * 95 = 4845

55 * 91 = 5005

Difference = 160

56 * 100 = 5600

60 * 96 = 5760

Difference = 160

So, as I have shown, the difference between the diagonally opposite corner results of a 5*5 square, are always 160.

Data Example 5 – 6*6 Square:

1 * 56 = 56 6 * 51 = 306 Difference = 250 | 5 * 60 = 300 10 * 55 = 550 Difference = 250 |

41 * 96 = 3936 46 * 91 = 4186 Difference = 250 | 45 * 100 = 4500 50 * 95 = 4750 Difference = 250 |

So, as I have shown, the difference between the diagonally opposite corner results of a 6*6 square, are always 250.

Results – Pattern:

Box Size | Examples used | Difference | 1st Difference | 2nd Difference |

2*2 | 12*23 & 13*22 | 10 | ||

19*30 & 20*29 | 10 | Between 2*2 – 3*3= 30 | ||

48*59 & 49*58 | 10 | |||

Conclusion

Conclusion:

From this investigation, I have learned that there is a relationship between the size of the box on the number grid, and difference in the diagonally opposite corners. My prediction was correct, because I said that there would be a pattern and algebraic expression which would work on all box sizes, and there is:

10(n-1)²

I also predicted that there would be a lot to think about and explore, which there is. I also enjoyed discovering the results.

If I was to do the investigation again, I would branch out even more, and test if the expression worked on other quadrilaterals, like rectangles, or even 3D shapes such as cubes.

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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## Here's what a teacher thought of this essay

There is some good investigative work carried out here. However the mathematical content is fairly limited. Generally there should be some evidence of not only the calculation of an nth term but also the multiplication of double brackets. For example (x + 5)(x - 2).

Marked by teacher Cornelia Bruce 18/04/2013