GCSE: Number Stairs, Grids and Sequences
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Opposite Corners
2 x 3 Rectangles 1 2 3 11 12 13 To keep things simple I have started with rectangles with a width of 2 squares. I kept the width to two squares and increased the length by one square. (see results table above). I discovered that the width increases by 10 every time the length increases by 1. The difference can be worked out for all rectangles with a width of 2 squares by using several formulas: 1.
 Word count: 516

Number Grid.
3 by 3 boxes 4 by 4 boxes. 5 by 5 boxes Results: Term (n) 1 2 3 4 5 Differences in opposite corners 0 10 40 90 160 Term (n)...... 1 2 3 4 5 Differences in Opposite corners...... 0 10 40 90 160 10 30 50 70 20 20 20 We know that the nth term will have to be squared.
 Word count: 471

For this coursework  stair shape  I am going to investigate the relationship between the stair total and the position of the stair shape on the grid. To do this I am going to create tables, charts, graphs, algebra equations and try to find the n'th ter
I also plan to show my working outs, and my method of my data capture. Once I have done this I will predicate the n'th term then test to see if my formula works. Data This data that I have collected was from a 10, by 10 grids. The data is presented in a table below. Number inside stair Shape Stair Total 1+2+3+11+12+21 21 11 12 1 2 3 50 6 2+3+4+12+13+22 56 22 12 13 2 3 4 6 3+4+5+13+14+23 62 23 13 14 3 4 5 6 4+5+6+14+15+24 68 24 14 15 4 5 6 *Number in Red is the different that the Stair total is going up in.
 Word count: 672

Borders Maths Coursework
With this shape there are now 12 borders. This shape has 16 borders I can now see a pattern in the number of square, which is that there are 4 more each time.
 Word count: 279

Maths Grid Investigation
What I think will happen I think that the diagonal difference will always be 32 with a 3x3 grid. I think that the diagonal difference for a 2x2 grid will be 8 each time. I am also going to investigate the diagonal difference of a grid sized 4x4; I think the difference will be 72 each time. I came to these conclusions by doing a series of preliminary investigations. Preliminary Investigations For a 3x3 grid: 1 2 3 9 10 11 17 18 19 1 x 19 = 19 3 x 17 = 51 51  19 = 32 46
 Word count: 915

A garden centre sells square slabs which can be used to surround ponds
To work out the perimeter of a square the formula is 4n. Using this 4x1=4. I need 8 slabs so that it goes all the way around the pond. I need four more. That makes 8. I will try using the formula 4n+4 to see how many slabs are needed. It works for a square when the sides are 2x2 and 3x3. 4x2 = 8. 8+4 = 12 4x3 = 12. 12+4 = 16 The amount of slabs are correct.
 Word count: 528

Routes On Polyhedra
at G. My method for the cuboid in the diagram above is to start at A (*) and then find all the routes that began AB, then all the routes that began AD and finally AE.
 Word count: 283

Squares and columns Investigation
= 16 11 12 13 14 15 2 (2+3+7+8) = 20 16 17 18 19 20 3 (3+4+8+9) = 24 21 22 23 24 25 4 (4+5+9+10) = 28 5 (5+6+10+11) = 32 6 (6+7+11+12) = 36 7 (7+8+12+13) = 40 This is the step size they are all going up in fours The reason I put number 5 in as a different colour is because it doesn't really exist because its on the end row, also numbers 10, 15, 20 and 25 don't exist, they would only exist if I added another number to the end of the row, it also shows what it would be anyway so it keeps the pattern going.
 Word count: 528

Number Grid.
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
 Word count: 358

Diagonal Differences
This is equal to the number of rows. Once I noticed this I realised that the 3X3 grids diagonal difference was equal to the number of rows X4. Also the number of rows in the 4X4 square are timed by 9 And in a 5X5 square the rows are timed by 16.
 Word count: 570

Investigation 1Square shaped pieces of card
Height is also the same as to the size of the cutout corners of an open box. I am going to begin by investigating a square with a side length of 12 cm. Using this side length, the maximum whole number I can cut off each corner is 5cm, as otherwise I would not have any box left. I am going to begin by looking into whole numbers being cut out of the box corners.
 Word count: 499

Opposite Corners coursework
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 Highlight any 2x2 square within this grid.
 Word count: 304

The Open Box Problem
I should also be able to come up with a formula to ..... INVESTIGATING SQUARE SHEETS OF CARD 1. For any sized square sheet of card, investigate the size of the cut out square which makes an open box of the largest volume. USING A 100MM SQUARE SHEET OF CARD I will begin this investigation by working out the size of the square cut which makes an open box of the largest volume, using a 100mm square sheet of card .
 Word count: 268

Maximum Box Investigation
First of all, I'm going to try a perfect cube, which i predict to be the best shape. With no lid, the cube has 5 sides, so the 576cm� must be divided by 5. This gives us 115.2cm� for each of the cube's faces. The square root of 115.2cm� is 10.73cm, and this gives us the length of each of the edges in the cube.
 Word count: 477