GCSE: Number Stairs, Grids and Sequences
Meet our team of inspirational teachers

Investigate the products of 2x2 number squares within a large 10x10 number grid.
57 58 67 68 * I will now find the products of the top left & bottom right numbers, and the top right & bottom left numbers. 57 x 68 = 3876 58 x 67 = 3886 * I will then write down what I notice I have noticed that the difference is 10 & that they are both even numbers I will now do this again for 2 more sets of 2x2 number squares. 12 13 22 23 12 x 23 = 276 I have noticed that the difference 13 x 22 = 286 between these two values is 10, and they are both even numbers.
 Word count: 3556

Random Sample.
Number Of Shape Calculations Total 1 1+2+3+11+12+21 50 2 2+3+4+12+13+22 56 3 3+4+5+13+14+23 62 4 4+5+6+14+15+24 68 5 5+6+7+15+16+25 74 Comment: There is clearly a pattern here; the totals are increasing by 6, we call this a linear sequence. From this information I predict that the next total will be 80 and the 8th will be 92. 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53
 Word count: 3157

Investigate The Answer When The Products Of Opposite Corners on Number Grids Are Subtracted.
 (1 x 36) 186  36 = 150 Answer = 150 I do not need to draw any more diagrams because it is clear what is being done to get the answer and I now have enough data to make a table of results. This is another method where I can look for any obvious patterns in the data. Table of Results Grid size Number of Product of top right Product of top left Answer (Products Squares number and bottom number and bottom Subtracted)
 Word count: 4459

Number Grid Investigation.
67 68 77 78 I did (67 X 78)  (68 X 77) = 10 This shows that the product difference is 10. Below are two more examples to prove it will work anywhere on the grid. 42 43 52 53 I did (42 X 53)  (43 X 52) = 10 TL X BR  TR X BL 12 13 22 23 I did (12 X 23)  (13 X 22) = 10 TL X BR  TR X BL I am now going to prove algebraically that in a 10 X 10 grid, with a 2 X 2 square the difference will always be 10.
 Word count: 6037

Algebra  Date Patterns.
Why did the operation keep resulting in the number seven, no matter what data I processed through it? My earlier recognition of a potential relationship was now strengthened; it seemed to me that the pattern would produce a value that was equal to the number of columns involved in the grid as a whole. In order to prove this I would have to create an algebraic formula for this particular event. First I assembled an algebraic representation of the box: n n+1 n+7 n+8 n = root number Using this I could concoct a generic formula that could be used with every box in a calender to find the difference.
 Word count: 3094

What the 'L'  L shape investigation.
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 I calculated up to seven three by three LShapes and found their LSum (as this is the last LShape that will fit across a nine by nine grid).
 Word count: 3629

To find any relationships and patterns in the total of all the numbers in 'Number Stairs', that might occur if this stair was in a different place on the grid.
+ (x + 2) + (x + 3) + (x + 10) + (x + 11) + (x + 20) = 6x + 1 + 2 + 3 + 10 + 11 + 20 = 6x + 44 25+26+27+35+36+45= 194 The stair total for this threestep stair is 194 This formula should work with every number stair that can fit onto a 10 by 10 grid. To support my judgement I will repeat the formula another two times on the same threestair grid, but in a different position to the previous three stairs.
 Word count: 3089

Number Grid Coursework.
= 455 difference = 40 48 x 39 = 1872 82 x 64 = 5248 38 x 49 = 1862 difference = 10 62 x 84 = 5208 difference = 40 96 x 87 = 8352 88 x 70 = 6160 86 x 97 = 8342 difference = 10 68 x 90 = 6120 difference = 40 4 x 4 5 x 5 41 x 14 = 574 46 x 10 = 460 11 x 44 = 484 difference = 90 6 x 50 = 300 difference = 160 67 x 40 = 2680 62 x 26 = 1612 37
 Word count: 3793

Number Stairs
To generate a sequence, I will need a starting value and a rule to find the next term. My starting value will be the Stair Total of the first 3step stair, and the rule to find the next term will be found by working through the Stair Total difference of first 3step stair and the second 3step stair, by subtracting the two totals. 12 11 12 1 2 3 1+2+3+11+12+12=50 The Stair Total of this 3Sstair is 50 22 12 13 2 3 4 2+3+4+12+13+22=56 The Stair Total of this 3Sstair is 23 13 14 3 4 5 3+4+5+13+14+23=62 The
 Word count: 4332

Number Stairs
In doing so, I will use different size grids and use different size stairs. Again, investigating relationships and discovering formulae for each problem I encounter. I am going to start with a 10 by 10 grid with a 3step stair. 91 92 93 94 95 96 97 98 99 100 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50
 Word count: 3711

Investigate the difference between the products of the numbers in the opposite corners of any rectangles that can be drawn on a 100 square.
I will also do one test at a random site on the grid to make sure that results are as close to perfect as possible. I predict that the difference will increase as the shape moves across or down the grid. 1 2 11 12 1 x 12 = 12 2 x 11 = 22 22  12 = 10 5 6 15 16 5 x 16 = 80 5 x 15 = 90 90  80 = 10 9 10 19 20 9 x 20 = 180 10 x 19 = 190 190  180 = 10 From this
 Word count: 5216

Boxes made in the shape of a cube are easy to stack to make displays in supermarkets. Investigate
2x5=10 As you can see my formula was correct! 2d Triangles I shall try another shape: I will find a formula for this pattern by, first of all, drawing a difference table: Number of Layers (N) Number Of Boxes (B) 1st difference 2nd Difference 1 1 2 2 3 1 3 3 6 1 4 4 10 1 5 5 15 As this table goes into a second difference this means it is a quadratic equation, implying the highest power of N is N2.
 Word count: 5257

My task is to investigate a 2x2 box on a 100 square
94= 6768 82 83 84 74 x 92= 6808 92 93 94 6808  6768 = 40 DIFFERENCE = 40 Test 3 1 2 3 1 x 23 = 23 11 12 13 3 x 21 = 63 21 22 23 63  23 = 40 DIFFERENCE = 40 Prediction I predict that in a 3 x 3 square the difference will always be 40 Proof 26 27 28 26 x 48 = 1248 36 37 38 28 x 46 = 1288 46 47 48 1288  1248 = 40 DIFFERENCE = 40 Algebra I will assign a letter to the first number in the 3x3square, n.
 Word count: 4357

Number Grid.
What I will do is pick out random numbers in a square this is a two by two pattern. 12 13 22 23 I am going to multiply the top right and bottom left numbers together and the top left and bottom right numbers together. 12 x 23 = 276 13 x 22 = 286 I am going to then work out the difference between these two numbers. 286  276 10 By looking at this I can see I have come to the difference of 10.
 Word count: 3243

For my investigation I will be finding out patterns and differences in a number grid.
Once I have found out the differences I will then write up a statement and the predictions for each of my 6x6 and 7x7 squares. After I have done some data calculation to prove my differences are correct I will then move on to working out differences for rectangular shapes for instance 2x3, 2x4, 2x5, 3x4, 3x5, 4x5 after doing this I will then do another set of statements and predictions to work out 2x6, 2x7, 3x6, 3x7, 4x6, 4x7, and 5x6, 5x7 this would be the second part complete of my investigation.
 Word count: 9480

Number stairs.
+44 Stair 3: 21 + 22 + 23 + 31 + 32 + 41 = 170 Y+Y+1+Y+2+Y+10+Y+10+1+Y+20= 6Y+44=170 6Y=17044 6Y=126�21=6 Y=21 6(21) +44 Stair 4: 25 + 26 + 27 + 35 + 36 + 45 = 194 L+L+1+L+2+L+10+L+10+1+L+20= 6L+44=194 6L=19444=150 6L=150�25=6 L=25 6(25) +44 Conclusion: Based on all of the 4 3 step stairs that I have investigated, I can now say that I have found a formula which works with all of the other 3 step stairs. The formula which I have found is 6A+44 which just simply means 6AX4X10+4.
 Word count: 4730

For other 3step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. To start the investigation a 10x10numbered grid square is used as illustrated below in table 1:
 10 x  11 X x + 1 x + 2 By adding the values from the equation (20  10  9 + 1 + 2) = [36] Thus we can use the algebra equation 6x  36 = [the total value of squares in a 10x10 square] Using the above logic and method we can use it in other grids, such as an 11x11 and a 12x12 numbered grid square. Using the same 3step stair approach we can use the theory for the 11x11 and 12x12numbered square to find a pattern.
 Word count: 6878

Number grid
The 10 that is underlined in the above equation, proves that the latter part of the equation will always be 10 more than the first half of the equation. So, by using a 2x2 grid and multiplying the diagonals, the difference will always be 10. I am now going to investigate whether there is any relationship between the products of the diagonals of a 3x3 grid. 42 43 44 52 53 54 62 63 64 78 79 80 88 89 90 98 99 100 6 7 8 16 17 18 26 27 28 18 19 20 28 29 30 38
 Word count: 6538

I am going to investigate taking a square of numbers from a grid, multiplying the opposite corners and finding the difference of these two results
So I can see like in the 5x5 grid there is a pattern. If I am right every 2x2 square in a 6x6 grid should have a difference of 6. To check if I am right I will take one more square out of the grid. 16 17 22 23 16 x 23 = 368 17 x 22 = 374 374  368 = 6 This shows that I am right and every 2x2 square in a 6x6 grid will have a difference of 6.
 Word count: 5357

Number Grid Product Differences Investigation
 (73x82) = 10. This shows that the product difference is the same anywhere in the grid. Then we were told that we had to find a general formula to work out the product difference in 10x10 grid but with a 3x3 or 5x5 matrices inside. I firstly found out the product difference in a 3x3, 4x4, 5x5 and a 6x6 grid. Below are the differences of the matrices. Here is another 2 more examples to prove that the formula above works 74 75 84 85 I did (74x85)  (75x84) = 10. This shows that the product difference is the same anywhere in the grid. 59 60 69 70 I did (59x70)  (60x69) = 10.
 Word count: 3124

Maths Grid Investigation
grids of all sizes not yet done Page 26: Justifying My Results, formula tested on grids of all sizes not yet done Page 27: Further Investigation, formula found for any rectangle grid inside any square grid. Page 28: Justifying My Results, formula tested on grids of all sizes not yet done Page 29: Justifying My Results, formula tested on grids of all sizes not yet done Page 30: Conclusion Statement I have been told to consider the following table of numbers: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
 Word count: 5120

GCSE Maths coursework  Cross Numbers
and the number on the right always = (X+1). Therefore I predict this is a master formula for every number I pick. Xg (X1) X (X+1) X+g Algebraic Investigation I am going to investigate 3 different formulas on a shape, for 3 grid sizes using the assigned cross shape. a) Xg (X1) X (X+1) X+g [(X1) (X+1)]  [(X+g) (Xg) = (X�1)  (X�g�) = X�1x�+g� = g�1 b) Xg (X1) X (X+1) X+g [(X1) + (X+1)] + [(Xg) + (X+g)] = X1 + X+1 + Xg + X+g = 4x Xg (X1)
 Word count: 3674