For the 4th part of the investigation, I will be extending the task so that 4 variables will be present in my formula. I will be changing the multiple of this grid and then comparing all of the formulas to see if I can get a general formula which would then include the length of the shape inside the grid, the width of the shape inside the grid, the size of the main grid and the multiple of the main grid.
Squares
For the first part of the investigation, I will be investigating the difference of the products from the opposite numbers in a square block. Firstly, I will start by using a range of sizes so that I can gather results which I can then use to find a formula for the difference of the products from the opposite numbers.
I will start by using a 2 x 2 square box, starting from the top left hand corner of the number grid as this would be an appropriate beginning.
Rectangles
Multiple Grids
Size, Shape and Multiple
Conclusion
In conclusion, I have found many formulas, either to do with the shape of the box inside the grid, the size of the main grid or the multiple of the main grid. While doing this investigation, I noticed a link between all of my formulas and therefore combined all of them together at the end to make one final formula containing 4 variables; the multiple of the grid, the size of the main grid, the length of the box inside the grid and the width of the box inside the grid. I noticed that there is a reason for every number in the formula to be there and investigated why that number was present and what would happen to that number if the circumstances changed. This means that I have done 3 extensions all together; ranging from 1 to 4 variables.
For my beginning task, I got the formula, 10(n-1) 2. This basically means that the “n” value represented my length and width, but as the length and width in a square is the same, I squared the bracket as it would be the same. I then multiplied the squared bracket by 10. For my 1st extension, I got the formula, 10(L-1)(W-1). The “L” and “W” values represented the length and width of the rectangle as they are both different whereas for a square it would be the same. So I just opened out the bracket from the previous formula and added two different values. I then again multiplied the brackets by 10.
For my 2nd extension, I got the formula, 10M2(L-1)(W-1). The “L” and “W” values were again representing the length and width of the shape inside the grid and the “M” value representing the size of the main grid. This meant that in the previous two parts, the reason to why I had multiplied the brackets by 10 was because the size of the main grid was 10. This meant that my extension had helped me to understand why the values I was using were necessary to the investigation.
For my 3rd extension, I got the formula, S x M2(L-1)(W-1). For this extension, I had combined all my findings together to give me one formula to find the difference between the opposite products inside a square/rectangular shaped box, within any sized main grid, with any multiple grid.
Evaluation
Overall, I think my investigation went well because I had extended all my tasks and had up to 4 variables. I completed the given task in as much detail as possible and referred my extensions back to the original given task. If I had more time to complete this assignment, I would have changed the shape of the “box” inside the grid and maybe even involve 3D shapes. I could have also investigated the difference of the products from the opposite numbers inside a rectangular main grid; instead of changing the shape and size of the box inside the grid. Additionally, I could have considered that if the numbers inside the grid included negative numbers, then there may be a change to the general formula.
On the whole, I had no major problems within the course of my assignment, and felt than everything went smoothly. As I had extended my task to 4 variables, I believe that I had done everything I was asked to in the task and maybe more.
