GCSE: Number Stairs, Grids and Sequences
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I will take a 2x2 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results
I will take a 3x3 square on a 100 square grid and multiply the two corners together. I will then look at the relationship between the two results, by finding the difference. Test 1 37 38 39 37 x 59= 2183 47 48 49 39 x 57= 2223 57 58 59 2223  2183 = 40 DIFFERENCE 40 Test 2 72 73 74 72 x 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
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Number Grids
49 50 49 x 60 = 2940 59 60 50 x 59 = 2950 The difference between the two answers is 10. 12 13 14 12 x 34 = 408 22 23 24 14 x 32 = 448 32 33 34 448  408 = 40 The difference between the two answers is 40. 78 79 80 78 x 100 = 7800 88 89 90 80 x 98 = 7840 98 99 100 7840  7800 = 40 The difference between the two answers is 40.
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Number Grids  Algebra
This is 1036  1026 = 10. My 2 x 2 square is 51, 52, 61 and 62. The top left times by the bottom right is 51 x 62, this equals 3162. The top right multiplied by the bottom left is 52 x 62 = 3172. To finish I will take the smaller of the two numbers from the larger. This is 3172  3162 = 10. I have found that they all end out that the difference between the two numbers of any 2 x 2 squares on a 1  100 grid is 10. Algebra  2 x 2 X X + 1 X +10 X + 11 (X2 + 10X +1X +10)
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Maths Gridwork
 (x+1)*(x+10) X�+11x  x�+x+10x+10 +11x  x�+11x+10 ?  ? +10 = 10 Therefore my hypothesis was true. 3*3 squares Equation: (TL*BR)(TR*BL) Example 1 43 44 45 53 54 55 63 64 65 (43*65) (45*63) = 40 Example 2 56 57 58 66 67 68 76 77 78 (56*78) (58*76) = 40 Example 3 1 2 3 11 12 13 21 22 23 (1*23) (3*21) = 40 Example 4 8 9 10 18 19 20 28 29 30 (8*30) (10*28) = 40 I predict that the next example will give me a result of 40 Example 5 55 56 57 65 66 67 75 76 77 (55*77) (57*75)
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Maths Number Grids/Sequences
Taking these results into account, I predict for any 2x2 square the result will always be 10. Below is a table of results for 2x2 squares that were randomly chosen from the 10x10 grid. Table 1. 2x2 results. 1st No. multiplication 2nd No. multiplication Difference 5 x 16 = 80 6 x 15 = 90 10 17 x 28 = 476 18 x 27 = 486 10 32 x 43 = 1376 33 x 42 = 1386 10 68 x 79 = 5372 69 x 78 = 5382 10 The results from T.1 show that the difference is a constant 10, so my prediction was correct.
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Number Grids
I can also say that for all 3x3 squares that the difference is 40 and for all 4x4 squares the difference is 90 in 10x10 grids. From this I can therefore say that all squares of the same size have the same difference when the top right and bottom left corner are multiplied and the answer is then subtracted from the product of the top left corner and bottom right corners. An LxL square in a 10x10 grid Since the difference results of any square is the same I will now make the size of the square variable and as it is variable and not a set number it will be known as L for the length of the side.
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Maths Number Stairs
So n+(n+1)+(n+2)+(n+10)+(n+11)+(n+20) = is my formula I will check to see if it works in same way as before. 6 x 31 +44 = 230 230 = 230 Total = 230 I am correct my formula is 4 Step Stair Total = 120 Total = 340 Total = 640 Total = 760 Finding the formula Same process again So n+(n+1)+(n+2)+(n+3)+(n+10) +(n+11)+(n+12)+(n+20)+(n+21)+ (n+30) = To prove it differently this time by rearranging and finding the smallest number. 10n+110=640 10n=640110 10n=530 10n=530/10 n=53 5 Step Stair Instead of writing all the numbers to find my formula I can predict my formula by doing it this way.
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Number grid Investigation
is still 10 59 60 69 70 59 x 70 = 4130 69 x 60 = 4140 4140  4130 = 10 The difference is always 10 After proving and verifying 4 times that 2x2 box in a 10x10 grid difference is 10, I was curious as to what would happen if I changed the size of the box. Therefore I chose to change the box from 2 x 2, (four numbers,) to 3 x 3, (nine numbers,) and used the same process of finding the product of the top left number and bottom right number and vice versa.
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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.
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Maths Coursework
If we take these expressions and put it into the same formula as before we should get 10. (N+10)x(N+1)N(N+11)= 10 (N2+N+10N+10)(N2+11N)= 10 N2+11N+10N211n= 10 Therefore 10 = 10 This shows that this is correct for 2x2 boxes. 3x3 boxes I will now investigate 3x3 boxes that I am once again taking 4 of: 58 59 60 68 69 70 78 79 80 (60x78)(58x80)= 40 21 22 23 52 53 54 62 63 64 42 43 44 52 53 54 62 63 64 26 27 28 36 37 38 46 47 48 (23x41)(21x43)= 40 (44x62)(42x64)= 40 (28x46)(26x48)= 40 Proving algebraically the 3x3 boxes: As the answers are the same as before i.e.
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A box is drawn around 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 num
The equation for this is 12 x 23=276 My next stage is to times the top right hand corner number with the bottom left hand corner number. This is 13 x 22 = 286 286276 = 10 For accuracy reasons I am going to conduct this method to a few more 2x2 boxes to look for common differences. 17 18 27 28 17 x 28 = 476 27 x 18 = 486 486  476 = 10 From this I came to the conclusion that anywhere on the grid where a 2 x 2 square can be drawn the product will always be 10.
 Word count: 892

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.
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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.
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
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In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.
= S2 +11S (S+10)(S+1) = S2 +10S +S + 10 = S2 + 11S + 10 The difference is ten To find this I multiplied the corners of my square and found the difference between the products. Next I investigated 3 x 3 squares within a 10 x 10 grid and found that the difference was 40 Square products difference 1 2 3 11 12 13 21 22 23 1 x 23 = 23 21 x 3 = 63 40 55 56 57 65 66 67 75 76 77 55 x 77 = 4235 75 x 57 = 4275 40 44 45 46 54
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Number Stairs Investigation
Long operation Total (t) 1st Difference (D0) 1 1 + 2 + 3 + 11 + 12 + 21 50 6 2 2 + 3 + 4 + 12 + 13 + 22 56 6 3 3 + 4 + 5 + 13 + 14 + 23 62 6 4 4 + 5 + 6 + 14 + 15 + 24 68 6 5 5 + 6 + 7 + 15 + 16 + 25 74 6 Judging by this, the first part of the overall equation for a 3 step stair is 6n +?
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Number Stairs
Every time it goes up by 6. Stair Number: 71 72 73 74 Stair total: 470 476 482 488 Difference: 6 6 6 Now I know this pattern I can predict that the stair total for stair 75 will be 492 as 488+6. To make sure that is right I have to test prediction. 75+76+77+85+86+95=494 my predict is right We can therefore say that every time you move the stair shape one square to the right you increase the stair total by 6 and every time you move the stair shape one square to the left you decrease the stair total by 6.
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Investigate the size of the cut out square, from any square sheet of card, which makes an open box of the largest volume.
squared This is my first investigation: 30cm by 30cm piece of card. As you can see, the largest volume present in the table is 2000cm cubed, which is what you end up with if have 5cm by 5cm square cutout. At the bottom of the table, you should see that I have checked if the volume can get any higher. That is why I have checked to see 5.1 by 5.1 and 4.9 by 4.9 might have a higher volume than 5cm by 5cm. Clearly, the highest is 5 by 5. I'm going to do a few calculations to see if there are any similarities between this investigation and the next one.
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Opposite corner
corner of this rectangle are: 4x16 = 64 14x6 = 84 The difference between these products is 8464=20 The difference is 20 8 9 10 18 19 20 The product of the number in the opposite corner of this rectangle is: 8x20=160 18x10=180 The differences between these products are: 160180=20 The difference is 20 81 82 83 91 92 93 The product of the number in the opposite corner of this rectangle is: 81x93=7533 91x83=7553 The differences between these products are: 75337553=20 The difference is 20 Now i have found out that a rectangle sized 2 by 3 is on
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10x10 number grid
3 numbers along the top and 2 down. I predict that a similar result will emerge. Once all the data is collected I predict that a pattern in the difference between products and box sizes will also become apparent e.g. a 2x2 square, 3x3 square, 4x4 square will have a noticeable pattern in the product differences. Stephanie Evans  20002303 Method So let start with the number grid below: 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
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How can visual illusions illustrate top down processes in perception? Contrast this with a visual illusion that can be explained through bottom up processes.
The following illusions show examples of how both processes can be used to explain perceptions. An example of a visual illusion that can be explained by top down processing is the Muller Lyer illusion (figure 1). In this illusion the lines in both A and B are the same length however the arrows pointing inwards in A make the line appear longer than when the arrows point outwards as in B. Gregory explained this illusion in 1970 by suggesting that the lines are perceived as being three dimensional rather than twodimensional. This is shown in figure 2 where A is shown as the inside of the room and B shown as the outside.
 Word count: 1688

Investigate the relationships between the numbers in the crosses.
Finding numbers: 36 (a) 36 (b11) (a+9) 45 46 47 (a+11) (b2) 45 46 47 (b) (a+10) 56 (a+20) (b1) 56 (b+9) 36 (c20) 36 (l9) (c11) 45 46 47 (c9) (l) 45 46 47 (l+2) (c10) 56 (c) (l+1) 56 (l+11) 36 (x10) (x1) 45 46 47 (x+2) (x) 56 (x+10) Finding numbers in equations: * If a is known: = (a+9)(a+11)  (a)(a10) = (a +11a+9a+99)  (a +20a) = (a +20a+99)  (a +20a) = 99 * If b is known: = (b)(b2)
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An investigation into the relationship between stairs size and the value.
is always applicable. 25 + 26 + 27 + 35 + 36 + 45 = 194 (6x25) + 44 = 194 26 + 27 + 28 + 36 + 37 + 46 = 200 (6x26) + 44 = 200 These stairs are only one along from each other on the same line. This formula applies to any 3 levelled stairs anywhere on the grid no matter where it is. 45 + 46 + 47 + 55 + 56 + 65 = 314 (6x45)
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Number grid algebraic course work
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 12�23=276 286276= 10 13�22=286 I tested a two by two box two more times to ensure each time I got the answer of 10, also to ensure that the answer was the same in different areas of the grid.
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My investigation will be on 3  step stairs where I will be: Trying to investigate the relationship between the stair total and the position
repeat for 4 and 5 step stairs 7. compare the formulae for 3, 4 and 5 step stair to uncover an overall formula linking them together with the ability to find the total for any stair no matter step stair or stair number 8. change the grid size for the 3  step stairs and see if the outcomes are the same or different 9. find a formula using the grid size and the stair number for the 3  step stair 10.
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Investigating the number of patterns in a certain grid.
y+5 Square 6 extensions Rules for different shapes Change pattern in grid 1 3 5 7 etc. This is the number grid I am going to be investigating: 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
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