number grid
Algebraic Investigation 1: Square Boxes on a 10x10 Grid In this first investigation, the difference in products of the alternate corners of a square, equal-sided box on a 10x10 gridsquare will be investigated. It is believed that the products and their differences should demonstrate a constant pattern no matter what dimensions are used; as long as they remain equal. In order to prove this, both a numeric and algebraic method will be used in order to calculate this difference. The numeric method will help establish a baseline set of numbers for testing, and to help in the establishment of a set of algebraic formulae for use on an n x n gridsquare. In the example gridsquare below, the following method is used in order to calculate the difference between the products of opposite corners. (a) (b) (c) (d) Stage A: Top left number x Bottom right number = (a) multiplied by (d) Stage B: Bottom left number x Top right number = (c) multiplied by (b) Stage B - Stage A: (c)(b) - (a)(d) = The difference The overall, 10 x 10 grid that is used for the first investigation will be a standard, cardinal gridsquare, which progresses in increments of 1. The formulae calculated will mainly be applicable to this grid, as other formats of gridsquares will require others formulae to provide valid results. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26
Number Grid
NUMBER GRID Look at this number grid: 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 0 x 10 grid * A box is drawn round four numbers. * Find the product of the top left number and the bottom right number in this box. * Do the same with the top right and bottom left numbers. * Calculate the difference between these products. Investigate further. The purpose of this investigation is to prove or disprove that there is a correlation between the products of corner numbers of any size box within any size grid. I shall calculate the diagonal difference (d) for a box. There are two ways to calculate the difference between the products: V W Y Z i. W x Y - V x Z ii. V x Z - W x Y I shall begin with the 10 x 10 grid, as shown above, and a 2 x 2 box. 2 x 2 box #1 I have substituted numbers for the letters in my two formulae. 2 1 2 i. d = 2 x 11 - 1 x 12 = 22 - 12 = 10 ii. d =1 x 12 - 2 x 11 = 12 - 22 = -10 d = +/-10 I will use formula i, because formula ii creates negative numbers, which could make my calculations more complex than
Number Grid
Number Grid I have been given the following task: I will now carry out this investigation in four different parts. The 1st part includes 1 variable; which is the one given to me on the task sheet. I am going to investigate what the difference between the opposite products inside a square shaped box is. I will calculate the differences using the grid for 4 different sized square boxes and then put my results into a table. After doing this for all 4, I will look for a pattern with all my data and try to come up with a general formula which will give me my nth term. After getting my formula, I will predict an nth term using the formula and also calculate the differences using the grid and see whether my formula is correct. For the 2nd part of the investigation, I will be using 2 variables to extend the task further. I am going to investigate what the difference between the opposite products inside a rectangular shaped box is. I will do this by using the 1st formula and then see if there are any connections or similarities I can make. For the 3rd part of the investigation, I will be using 3 variables to extend the task even further. I will be using the first two formulas to link them with the grid size. I will then find a formula which will relate the shape of the box inside the grid and the size of the main grid. For the 4th part of the investigation, I will be
Number grid
Samantha Whittaker Number grid 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 Pick out 2x2 squares Multiply the diagonals Find the difference 34 35 44 45 INVESTIGATE 27 28 37 38 82 83 92 93 68 69 78 79 9 0 9 20 First number Second number Third number Fourth number stx4th 2ndx3rd Difference 34 35 44 45 530 540 0 27 28 37 38 026 036 0 82 83 92 93 7626 7636 0 68 69 78 79 5372 5382 0 9 0 9 20 80 90 0 I have put my results into a table so that they are easier to analyse and compare. What I have found is; when you take a 2x2 grid from a 10x10 grid, times the diagonals, the difference between the products of the diagonals is always 10. 36 37 46 47 If this rule is correct, then by using this grid: I predict that the difference between the Product of 36 and 47 compared with that of 37 and 46 will equal 10. 36x47=1692 37x46=1702 Difference=10 My prediction was correct as the difference between 36x47 (1692), and 37x46 (1702) was 10. To prove that this theory will work for any 2x2
number grid
NUMBER GRID COURSE WORK 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 To investigate the patterns generated from using rules in a square grid The aim of the investigation is to find differences of small n x n squares in 10 x 10 square and then to see if there is any rule or pattern which connects the size of square chosen and the difference. In order to find the difference of n x n square first step is to take nxn square and then multiply the corners diagonally. For example take 2x2 square and multiply its corners diagonally. 25 26 35 36 25 x 36= 900 26 x 35= 910 910 - 900= 10 77 78 87 88 77 x 88 = 6776 78 x 87= 6786 6786 - 6776= 10 After a few results I observed that any box of 2 by 2 the difference will always be 10 if grid size is 10 by 10. 4 5 6 4 5 6 24 25 26 I decided to try out some square but this time I going to do it bigger like 3 by 3 and 4 by4. 4 x 26= 104 6 x 24=144 144 - 104=40 31 32 33 34 41 42 43 44 51 52 53 54
Number Grid.
Mathematics Coursework Number Grid 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 Using the following rule: Find the product of the top left number and the bottom right number in the square. Do the same thing with the bottom left and top right numbers in the square. Calculate the difference between these numbers. I N V E S T I G A T E!!!! I am going to work out a formula to work out the difference between the top right and bottom left numbers, and top left and bottom right numbers. I will work out the difference to many different sized number squares with in the grid. I will change the shape of the number pattern and I will also change the size of the number grid itself. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00
Number grid.
Maths coursework number grid In this project, I am going to investigate a number grid. Using a set of instructions, that, have been given to me. The instructions are to find the product of various numbers. Before I start I will like to explain exactly why I have colour coded my work. If you look at my project you will find that there are certain numbers in colour the reason for this is, that it makes it easier to understand what is being multiplied and what is being subtracted. The numbers that were initially given to me were 12, 13, 22, 23 presented in a number grid marked by a two by two box. I was told to find the product of top left, (12) and the bottom right number (23). We then had to do the same to the top right (13) and bottom left (22). Once I we had worked out both products I had to calculate the difference. I am now going to give two examples to show you what I had to do. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 54 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 5 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 Example One 2 3 22 23 12 13 DIFFERENCE x 23 x 22
Number Grid
Number Grid Coursework Danny Stone Mr. Williams' Class 1LA Number Grid Coursework 25th October 2004 I am investigating the patterns which form when I have different size grids, and in these grids there will be different sized windows. Example: 5 x 5 Grid, with 3 x 3 window 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 I will be finding the difference between two products. The products will always be: (top right x bottom left) - (top left) x (bottom right) In this example, it can be represented as: (9 x 17) - (7 x 19) Using a range of different grid and window sizes, I will hope to find an equation to fit any window size, for example, a 5 x 5 grid. I will then hope to find the equation for any grid size. Throughout I will use algebra as further evidence of what I have found. So, my hope is to find a formula to find the difference between any two products in any window, and on any grid size. 5 x 5 Grid 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 Window Size Sum Difference 2 x 2 (2 x 6) - (1 x 7) 5 3 x 3 (5 x 13) - (3 x 15) 20 4 x 4 (4 x 16) - (1 x 19) 45 5 x 5 (5 x 21) - (1 x 25) 80 L X L* 5(L-1)2 45 *L = Length of sides of window This is the formula to find the difference between any two products (following the 'corner to corner' rule) on a 5 x 5 grid. I will now use algebra
Number Grid
Higher Tier Task - Number Grid 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 Use the following rule: find the product of the top left number and the bottom right number in the square. Do the same thing with the bottom left and the top right numbers in the square. Calculate the difference between these numbers. INVESTIGATE! I will begin with 2x2 windows on a 10x10 grid. 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 00 34 x 25 = 850 49 x 58 = 2842 24 x 35 = 840- 48 x 59 = 2832- 10 10 88 x 97 = 8536 63 x 72 = 4536 87 x 98 = 8526- 73 x 62 = 4526- 10 10 3 x 12 = 36 6 x 15 = 90 2 x 13 = 26- 5 x 16 = 80- 10 10 For all of these windows you can
Number Grid
Number Grid Introduction I am going to be looking at a 10x10 square and drawing a square around a 2x2 square around 4 numbers and multiplying the top left number by the bottom right number and multiplying the top right number by the bottom left. I will find out the difference and do 2 more and see if there is a pattern. I will then investigate this further. I will first be looking at a 10x10 square. 2x2 Example 1 Top Left x Bottom Right-12 x 23= 276 Top Right x Bottom Left-13 x 22= 286 286-276=10 The difference is 10. I will now look at 2 more examples and see if this continues Example 2 Top Left x Bottom Right-35 x 46= 1610 Top Right x Bottom Left- 36 x 45= 1620 1620-1610=10 Example 3 Top Left x Bottom Right- 81 x 92=7452 Top Right x Bottom Left- 82 x 91=7462 7462-7452= 10 The difference of all 3 examples was 10 so it must be the same for any 2 x 2 in that particular number grid. General Rule For 2x2 If we look at the 2x2 algebraically then it would look like this:- Top Left x Bottom Right- (X)(X+11)=X2+11X Top Right x Bottom Left- (X+1)(X+10)=X2+11X+10 Prove X=2 Top Left x Bottom Right-(2)2+11(2)=4+22=26 Top Right x Bottom Left-(2)2+11(2)+10=4+22+10=36 36-26=10 QED This proves the difference is always 10 3x3 I will now be looking at a 3x3 square and seeing if it follows a rule as well. Example 1 Top Left x Bottom Right- 36 x 58= 2088 Top
