Number grids. In this investigation I have been attempting to work out a formula that will find the difference between the products of the top left and bottom right of a number grid and the top right and bottom left of a 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 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 In this investigation I have been attempting to work out a formula that will find the difference between the products of the top left and bottom right of a number grid and the top right and bottom left of a 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 A 10x10 number grid If you choose any 2x2 box on a 10x10 number grid then the difference should equal 10...

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I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid

Opposite Corners Opposite Corners * Throughout this piece of work, the horizontal side will be referred to as the length, whilst the vertical side will be referred to as width. Introduction/Aim I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid. I am going to investigate different rectangles, of different areas, lengths, widths and positioning on a grid. After finding a pattern, I will try and prove that it will work for other cases by using algebra and making an algebraic formula. As I go along, I will have to record any ideas I have or patterns I see. Afterwards, I will go on to investigate how this rule may differs on a different sized grid. 2x2 Square, 10x10 Grid 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 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 99 100 This is a 10x10 grid. On it (outlines in red) is a 2x2 square. Firstly, I’m going to see what the difference between the products of the corners is (D): 55x64=3520 54x65= 3510 3520-3510=10, D=10 Now: What

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In this piece course work I am going to investigate opposite corners in grids

Mathematics Course Work Opposite Corners Introduction In this piece course work I am going to investigate opposite corners in grids. I will start by investigating a 7x7 grid. Within this grid I will use 2x2, 3x3, 4x4, 5x5, 6x6 and a 7x7 grid. I will do this to find whether I can find a pattern. I will do this by multiplying the two opposite corners together then subtracting them. I will try to find the patterns and do a formula that will work for all grid sizes and shapes. I will experiment shapes and sizes of all different grids. Prediction I predict that in a 7x7 grid all the opposite corners will be a multiple of 7 and in an 8x8 grid they will be a multiple of 8 and so on. They will only do this if I multiply the two opposite corners then subtract the two from each other. To check my hypothesis I will use 6x6, 7x7, 8x8 and maybe if I have time I will do a 9x9 and 10x10 grid. Also I will be looking at all different shapes and sizes. I hope to find a formula for all grids and all shapes and sizes. 7x7 Grid Here is a grid of numbers in sevens. It is called a seven grid. In this section I will multiply the opposite corners and then subtract them. 2x2 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 In my 7x7 grid I have highlighted three 2x2

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In this coursework, I intend to investigate the differences of the products of the diagonally opposite corners of a rectangle, drawn on a 10x10 grid, with the squares numbered off 1 to 100

Opposite Corners Introduction In this coursework, I intend to investigate the differences of the products of the diagonally opposite corners of a rectangle, drawn on a 10x10 grid, with the squares numbered off 1 to 100. I will aim to investigate the differences for rectangles that are aligned differently, of different lengths and widths. Hypothesis This coursework will aim to establish if there is a pattern from which we could derive a formula from the numbers in the grid. Plan The plan is to use Algebra to analyse the results which will be logged into tables. Subsequently I will test any rules, patterns and theories I find by using predictions and examples. Methodology To start with I will break up my work into sections, my first is: 2x3 Rectangles I will concentrate on one particular aspect 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. I think I can now safely assume that the difference is always constant in relation to the size of the rectangle, therefore I only

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Opposite Corners

Opposite Corners Maths Coursework Investigation The aim of this investigation is to find out the difference between the products of numbers in the opposite corners of any rectangle that can be drawn on a 100 square. I shall start off researching the 2x3 rectangle and then working onto bigger ones. 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 2x3 Rectangle 7 8 9 7 8 9 2 3 1 2 3 88 89 90 98 99 00 x13=13 88x100=8800 7x19=133 1x3=33 98x90=8820 17x9=153 20 20 20 Prediction I predict that when I multiply a 2x3 rectangle the opposite corners will have a difference of 20. 2x2 Rectangle (square) 72 73 82 83 4 5 4 5 33 34 43 44 4x15=60 33x44=1452

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Opposite Corners

Opposite corners 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 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 On a 10*10 square grid, choose any 2*2 square, multiply the corners in that grid and then find the difference between the two corners investigate. 2 13 12 13 16 17 16 17 22 23 * 23 * 22 26 27 * 27 * 26 276 284 332 342 284-276=10 342-332=10 Difference=10 The two answers are the same. I think it would be the same for any 2*2 square. To prove this I will use algebra to show that in any 2*2 square the difference will 10. z- number in the top left corner z z+1 z(z+11)=z²+11z z+10z+11 (z+1)(z+10)=z²+11z+10 (z²+11z+10)-(z²+11z)=10 Difference = 10 This proves that with any 2*2 square the corners multiplied then subtracted always = 10 To further my investigations I am now going to

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Mathematics Coursework: problem solving tasks

My name is.....................; I am currently attending the ................ Amongst the various subjects I am undertaking this year, is GCSE mathematics. My coursework assignment asks that I seek to stipulate a formula that will help me establish exactly the required amount of spacers necessary for different arrangements of tiles. I feel the use of diagrams will be useful to depict different arrangements of tiles. Only from these arrangements, will I then be able to collect related information which will then assist me to compile a set of results. Subsequently; I will then organize my results systematically in table form and hopefully from this table I will become aware of a pattern beginning to form. Once I recognize this pattern, I can then determine a suitable formula as a way to work out the required amount of spacers needed for each tile arrangement without having to draw out all diagrams manually. A formula is a rule written in symbols and letters. Spacers are used when tiling a wall, to make sure the tiles are evenly spaced in straight lines, and that the grout between the tiles is of even thickness. Three types of spacers are used when tiling a wall, these include; * T spacer * + spacers & * L spacers My Prediction 'The L shape spacers required for each tile arrangement will always be 4' Results Table I made a table to show my results in a clear way

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Maths coursework- stair totals. I shall be investigating the total and difference in sets of stairs in different grid sizes. I will be investigating the relationship between stair totals on different grids.

GCSE Mathematics Coursework Introduction In my mathematics GCSE coursework I shall be investigating the total and difference in sets of stairs in different grid sizes. I will be investigating the relationship between stair totals on different grids. This will also include different stair sizes. I will choose to display my findings using a mixture of graphs, tables and grids. I shall also try to work out the formula for each of the number of step sizes. After I have finished investigating the different sized number stairs I will attempt to find out the different sized number grids. At the end of my investigation I will conclude by demonstrating and explaining the relationship between all sizes of stairs. Method I shall be carrying out my stair challenge by finding out the correct information. I shall do this by experimenting with the size of the grid corresponding to the size of the stairs used in the grid. I shall try to work out the formulas by recognising any similarities in the results obtained. Part 1- Three Step grid I have investigated the three step stair (some of my findings can be viewed on the graph paper in the back). During my investigation I believe that I have worked out the formula for the total inside the three step stairs on a 10 x 10 grid. Below is an example of just one of the three sets of stairs that I used to conduct my

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Number Grid Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results

Mathematics Coursework Number Grid Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results. I will start off by working this out on a 2x2 square in a 10 x 10 grid and then will investigate varying the widths and lengths of squares and rectangles. Method: In order to simplify the process, the investigation has been divided into sections according to the size of squares in the grids. 2 x 2 Squares To begin with, I used a 10 x 10 grid, looking at 2 x 2 squares: 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 Then I took 2 x 2 squares from this grid and multiplied the opposing corners to calculate the difference between the two products. 2 1 2 x 12 = 12 2 x 11 = 22 ? 22 - 12 = 10 So the difference between the answers is 10. I then took another 2 x 2 box from the above 10 x 10 grid: 47 48 57 58 47 x 58 = 2726 48 x 57 = 2736 ? 2736- 2726 = 10 The difference is 10 again. Perhaps this

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Opposite Corners. In this coursework, to find a formula from a set of numbers with different square sizes in opposite corners is the aim. The discovery of the formula will help in finding solutions to the tasks ahead as well as patterns involving Opposite

Introduction: The mathematical investigations that are about to be undertaken are all under one puzzle called Opposite Corners. In this coursework, to find a formula from a set of numbers with different square sizes in opposite corners is the aim. The discovery of the formula will help in finding solutions to the tasks ahead as well as patterns involving Opposite Corners. There are a few basic procedures to follow to achieve a basic understanding of the whole puzzle. A box consisting of numbers from 1 to 100, a 10 by 10 grid (arranged in a regular pattern) will aid in initiating an understanding for this piece. Procedure: . Place borders of four lines in order to enclose numbers arranged in a given grid. The enclosed numbers should form a perfect square. 2. Multiply the numbers that are found diagonally opposite and placed in the four corners of the box. 3. From the products obtained after multiplying, find the difference between them. An example is demonstrated on the next page. Below is a 10 by 10 grid. Here the numbers are arranged in 10 columns. 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

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