GCSE: Number Stairs, Grids and Sequences
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GCSE Maths questions
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 Level: GCSE
 Questions: 75

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
4 star(s)10 by 10 grid above), 7 � 18 = 126 8 � 17 = 136 The difference between the products above is 10 Tasks: Investigations to see if any rules or patterns connecting the size of square chosen and the difference can be found. When a rule has been discovered, it will be used to predict the difference for larger squares. A test of the rule will be done by looking at squares like 8 � 8 or 9 � 9 X ?
 Word count: 2865

Opposite Corners
4 star(s)3 4 5 3 5 1 2 3 3 1 13 14 15 * 25 * 23 11 12 13 * 21 * 23 23 24 25 75 115 21 22 23 63 23 11575=40 Difference = 40 6323=40 Opposite corners These answers are the same; just as the answer for the 2*2 squares are the same. I think that any 3*3 square would have a difference of 40. To prove this I will use algebra. z z+1 z+2 z(z+22)=z�+22zz z+10 z+11z+12 (z+2)(z=20)=z�+22z+40 z+20 z+21z+22 (z�+22z+40)(z�+22z)=40 This proves that with any 3*3 square the corners multiplied the subtracted always = 40 Now I am going to further my investigations again.
 Word count: 2183

Mathematics Coursework: problem solving tasks
3 star(s)Step 1 L T + 1 x 1 4 0 0 2 x 2 4 4 1 3 x 3 4 8 4 4 x 4 4 12 9 5 x 5 4 16 16 From the information depicted in the table above it would appear that my prediction stating that the number of L shape spacers needed is always 4, is indeed correct. The obvious reason for this is; because squares and rectangles reliably consist of four corners. So L = 4.
 Word count: 2504

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. To start I used a 5x5 grid:
3 star(s)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 my hypothesis is right and every 2x2 square in a 6x6 grid will have a difference of 6.
 Word count: 2963

In this piece course work I am going to investigate opposite corners in grids
3 star(s)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 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 In my 7x7 grid I have highlighted three 2x2 grids. I will multiply and subtract the opposite corners now.
 Word count: 2254

algebra coursework
Z+10 = 74+10 = 84 ( bottom left number) Z+11= 74+11 = 85 ( bottom right number) Z (Z+11) = Z� + 11Z (Z+1) (Z+10) = Z� +10Z + Z +10 = Z� + 11Z + 10 Z� + 11Z + 10  Z� + 11Z = 10 Ex 3 3 X 3 square on 10 X 10 grid 48 49 50 58 59 60 68 69 70 48 X 70 = 3360 68 X 50 = 3400 3400  3360 = 40 Z Z+2 Z+20 Z+22 Z = top left number = 48 (in this case)
 Word count: 2709

Number Grid Investigation
After doing this, I will further the investigation to changing the numbers within the gird, as I think that this is the only thing that I would not have covered before, this means that the results should be very different from what I have previously tried to find. 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
 Word count: 2556

Number Stairs  Up to 9x9 Grid
88 89 90 78 79 80 68 69 70 On a different number square grid, e.g. 4 by 4 number square grid, the theory would be the same, except that the number above the bottom left hand corner number is going to go up by 4. 13 14 15 16 9 10 11 12 5 6 7 8 1 2 3 4 The total of the numbers inside the stair shape is: * 1st Line: 1+2+3 * 2nd Line: 5+6 * 3rd Line: 9 T=Total T=26 The stair total for this 3step stair is 26. Part 2 I have investigated further and I have found out that the number going diagonal in a 10 by 10 number square grid...
 Word count: 2284

Number Grid
24 31 32 33 34 31 x 4 = 124 34 x 1 = 34 124  34 = 90 4 X 4 BOX ALGEBRA a a+1 a+2 a+3 a+10 a+11 a+12 a+13 a+20 a+21 a+22 a+23 a+30 a+31 a+32 a+33 (a + 30)(a + 3) = a� + 3a + 30a + 90 = a� + 33a + 90 a(a + 33) = a� + 33a (a� + 33a + 90)  (a� + 33a) = 90 5 X 5 BOX 56 57 58 59 60 66 67 68 69 70 76 77 78 79 80 86 87 88
 Word count: 2734

Number grid
= n�+n+10n+10 = n�+11n+10 I will now subtract the two algebraic products to get an overall product. n�+11n+10 n�+11n  +10 This proves that the difference is always 10 I have proved that the above results hold for any 2 by 2 square. This shows that for any 2 by 2 square the difference will always be equal to 10. To extend this investigation I am going to investigate the effect of changing the shape of the box. This variable will look at the effect of changing the length of the rectangle by 1 square each time.
 Word count: 2272

Maths Coursework  Grid Size
'N' will represent the top left number in the square/rectangle. I will calculate the DPD of that formula. If the answer to both the numerical and the algebraic formulas are the same then I will prove my theory by changing the position of the square/rectangle. I will next change the size of the grid and see how this affects the DPD. I will manipulate the numbers again using 'g' to represent grid size. I will then test this on different grid sizes and square/rectangle positions. I will find a pattern between the grid sizes and also have a general formula for grid size and square/rectangle position.
 Word count: 2506

Personal Exercise Program
It is for all these reasons that I will be trying to push myself to my limit. The two areas of specific fitness I will be working on to make my performance levels in football higher are AGILITY and UPPER BODY STRENGTH. Agility is a major component of football. Being able to dodge your opponent and use quick feet to beat them to the ball is vital. Agility is not a weakness of mine, infact I think I am pretty good at it but, it is something I would like to improve.
 Word count: 2263

Number Grid and Stairs
Here are some rules I have found for a 3x3 stair in a 10x10 grid so far. Rules 1 = +6 1 = 60 1 = 6 1 = +60 I am now going to find the algebraic rule for finding the total of the numbers encased in a 3x3 stair anywhere in a 10x10 grid. 6n+44 I will now test my algebraic rule on random places in the grid to see if the rule works in on all the numbers.
 Word count: 2067

Number grid coursework
I shall find the product of the top right and bottom left numbers. Now I shall find difference between these products. Let's now move the square and do same calculations. 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
 Word count: 2284

Investigate Borders  a fencing problem.
Then, I will get those three formulas and get one Universal formula in the end. Diagram of Borders of square: 1x1 Table of results for Borders of square: 1x1 Formula You can always find 'the nth term' using the Formula: 'a' is simply the value of THE FIRST TERM in the sequence. 'd' is simply the value of THE COMMON DIFFERENCE between the terms. To get the nth term you just need to find the values of 'a' and 'd' from the sequence and stick them in the formula. Formula to find the number of squares needed for each border (for square 1x1): Common difference = 4 First term = 4 Formula = Simplification =
 Word count: 2075

Applied Statistics
Thus we have used different approaches to variable selection in order to obtain the final equation. We used Stepwise regression, Backward elimination and Forward selection. In each approach we can see that we obtained the same RSquare value of 0,952 but we also obtained a better equation than with the simple model. We can also see that in each approach the variable Temp2 has been dropped. Omitting Temp2 must have had the least effect on the explanatory power of the model. (Appendix Q1a) We can conclude that the variable Temp2 isn't significant in the model. All of these approaches gave us the same final equation with significant variables (Sig.<0,05).
 Word count: 2295

Investigation into the patterns of mutiplication sqaures
3 X 21 = 63 35 36 37 45 46 47 55 56 57 35 X 57 = 1995 The difference is forty 37 X 55 = 2035 71 72 73 81 82 83 91 92 93 71 X 93 = 6603 The difference is forty. 73 X 91 = 6643 This time the difference is always forty. To prove my results were correct I did another example. 78 79 80 88 89 90 98 99 100 78 X 100 = 7800 My results were correct.
 Word count: 2389

Number Stairs Coursework
from positions 15 (see green), vertically across the grid and calculate the total of the numbers inside it. This can indicate what the relationship between the totals and the position vertically across the grid could be, for example, a key pattern could emerge in the totals. Then I will change the position of the number stair horizontally up the grid from 25  65 and record the total of the numbers inside it. This can indicate what the relationship between the totals and the position horizontally up and down the grid could be. Furthermore, I will find a formula for the total of a 3step stair on every position possible on the number grid.
 Word count: 2237

Number Grid
This is a general hypothesis that I will aim to prove throughout the investigation. Investigate further I will start by developing the task and investigating different sized grids. I will calculate the square grids 2x2, 3x3, 4x4 and 5x5 to begin with to calculate the differences of the products in each grid. I will begin to investigate the numbers in a 2x2 square grid in the 10x10 master grid by using the above rule. 14 15 14x25 = 350 From this calculation there is a difference of 10.
 Word count: 2573

number stairs
54 55 56 57 58 59 60 41 42 43 44 45 46 47 48 49 50 31 32 33 34 35 36 37 38 39 40 21 22 23 24 25 26 27 28 29 30 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 The first 3step stair is made up of 1 + 2 + 3 + 11 + 12 + 21 = 50 If I move the 3step stair 1 unit to the right, the 3step stair would be made up of 2 + 3
 Word count: 2421

Investigating when pairs of diagonal corners are multiplied and subtracted from each other.
40 16 x 49 = 784 19 x 46 = 874 Difference = 90 51 x 84 = 4284 54 x 81 = 4374 Difference = 90 67 x 100 = 6700 70 x 97 = 6790 Difference = 90 In a 4 x 4 box on a 10 x 10 grid the difference is 90. Algebraic Method x x + 3 x + 30 x + 33 x(x + 33) = x� + 33x (x + 3)(x + 30)
 Word count: 2891

Number Stairs
The stairs position is always worked out by the number in the bottom left hand corner. The position= n Investigation for 3 step stairs on a 10 by 10 grid 21 11 12 1 2 3 22 12 13 2 3 4 23 13 14 3 4 5 24 14 15 4 5 6 n Total 1 50 2 56 3 62 4 68 As the stair position goes up by 1 the total goes up by 6 each time. So for the stair in position 5 I predict that its total will be 74. 25 15 16 5 6 7 n+2g n+g n+g+1 n n+1 n+2 The nth term can be worked out by changing
 Word count: 2865

The purpose of this investigation is to look at diagonal differences on different sizes of grids
1x12=12 2x11=22 2212=10 2. 17x28=476 18x27=486 486476=10 3. 24x35=840 25x34=850 850540=10 4. 31x42=1302 32x41=1312 13121302=10 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 99 100 3 by 3 grids 1.
 Word count: 2529

Number Grid Investigation
My first 3x3 grid is (14 x 36)  (16 x 34) = 40 11 12 13 21 22 23 31 32 33 b) My second 3x3 grid is (11 x 33)  (13 x 31) = 40 17 18 19 27 28 29 37 38 39 c) My third 3x3 grid is (17 x 39)  (19 x 37) = 40 23 24 25 33 34 35 43 44 45 d) My fourth 3x3 grid is (23 x 45)  (25 x 43) = 40 2. I predict that in the next 3x3 grid the answer will be 40.
 Word count: 2930

Investigate the difference between the products of the numbers in the opposite corners of a rectangle that can be drawn on a 100 square. We were giving as the first rectangle to compare was this
this indicates that all rectangles of this size will have the difference of 20. Now I am going to do a rectangle of 2�4 squares. I think that these rectangles different will be 30. 4a. 34 35 36 37 44 45 46 47 b.34 � 47 = 1598 44 � 37 = 1628 c. 1628  1598 = 30 This shows that the different in a rectangle the size of 2 � 4 is 30, as I predicted. I will do this 2 more times to check that this is not a fluke. 5a. 7 8 9 10 17 18 19 20 b. 7 � 20 = 140 17 � 10 = 170 c.
 Word count: 2569