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GCSE: Number Stairs, Grids and Sequences

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  • Marked by Teachers essays 18
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  1. GCSE Maths questions

    • Develop your confidence and skills in GCSE Maths using our free interactive questions with teacher feedback to guide you at every stage.
    • Level: GCSE
    • Questions: 75
  2. Marked by a teacher

    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.

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    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. 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

    • Word count: 941
  3. Marked by a teacher

    Opposite Corners Maths investigation.

    3. 85 x 77 = 6545 75 x 87 = 6525 20 Yes, it appears my prediction is correct. All 2x3 rectangles on the grid have a difference of 20. However what if the rectangle is aligned differently on the grid, so the shorted sides are at the top and bottom? Will the difference for that still be 20? 1. 46 x 65 = 2990 45 x 66 = 2970 20 Nope, it is still 20. I will now try rectangles with the same height of two, but different lengths.

    • Word count: 907
  4. Numberical method

    Decimal research For function f(x) = , there is a change sign between the interval of -2 and -3 means there is a root. In this method i am going to use Excel spreadsheet to do decimal search by take increments in of the size 0.1 between the interval -2 and -3 and work out the function value for each. To work out x I'll use the formula r+1= r+0.1 given function of f(x) ==> f(x)+1= x -3 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2 f(x)

    • Word count: 771
  5. maths grid coursework

    n+g n+(w-1)+g 3x3 sqaure on a 10 size grid n n+(w-2) n+(w-1) n+g n+(w-2)+g n+(w-1)+g n+2g n+(w-2) +2g n+(w-1)+ 2g I can simpifly this into a forumula G(n-1)� and i realise i have 2 varibles that can be changed the number of sqaures grid size RESULTS Size of squares Difference between answers 2x2 10 3x3 40 4x4 90 5x5 160 From these results i can conclude that the formula for this pattern (when n is the number in and G equals grid size) G(n-1)� i will now prove this by finding the square of 9 by 9 on a 12 sqaure grid example 9=n 12=G algerbraic equation Rectangles 10 squared grid 2x3 8 9 10 18 19

    • Word count: 987
  6. Number Stairs

    shape is 26 +27 + 28 + 36 + 37 + 46 = 200 The stair-total for this stair shape is 27 + 28+ 29 + 37 + 38 + 47= 206 The stair-total for this stair shape is 3 + 4 + 5 + 13 + 14 + 23 = 62 The stair-total for this stair shape is 4 + 5 + 6 + 14 + 15 + 23 = 68 This table summarizes these results : Stair number 24 25 26 25 3 4 Stair Total 188 194 200 206 62 68 In order to find a formula

    • Word count: 587
  7. Mathematics Borders

    Now I will find the formula for the square numbers of the sigma notation. Sigma Notation 12 12+22 12+22+32 12+22+32+42 12+22+32+42+52 12+22+32+42+52+62 Value 1 5 14 30 55 91 1 5 14 30 55 91 4 9 16 25 36 2nd difference --> 5 7 9 11 3rd difference --> 2 2 2 The 3rd difference is denoted by the term 6a. Therefore in order to find a, I will replace the 3rd difference value 6a = 2 a = 2 = 1 6 3 By the given formula, Un = an3 + bn2 +cn + d, I will now be able to arrive to the generalized formula.

    • Word count: 861
  8. maths stairs

    How I got the formula is explained in part 1. (Diagrams are above) I am unable to do a '1 by 1', '2 by 2' as there is not enough room to get 1 result. n+n+1+n+2+n+3+n+4+n+7=6n+16 I have worked out that if the corner square is 1 in a 3 by 3 grid the total will be. 1+2+3+4+5+8=22 I worked this out by adding all of the numbers inside the stair and finding the total. I came up with the formula of 16+6n. As there is still the same number of 'n' in the diagram, the only change is that of the numbers.

    • Word count: 731
  9. Opposite Corners

    I will test any rules, patterns and theories I find by using predictions and examples. I will record any ideas and thoughts I have as I proceed. Plan: Firstly use a 2x2 box on a 10x10 grid in 5 positions Move on to 3x3 in 3 positions. Then 4x4 in 3 positions. Then predict what the difference will be for a 5x5 box. Test the prediction. Prove using algebra why the difference is always the same. Find formula for a square on a 10x10 grid. Prove formula works Investigate rectangles(with same method). Change grid size.

    • Word count: 786
  10. 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 = 286 286-276 = 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
  11. 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

    • Word count: 611
  12. 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 +?

    • Word count: 442
  13. 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)

    • Word count: 885
  14. Investigating how the numbers worked on a number grid.

    (n + 30) = n2 + 33n + 90 The difference between these two equations is 90. In order to test the rule I tried a few more 4 x 4 grids. 35 36 37 38 45 46 47 48 55 56 57 58 65 66 67 68 5 6 7 8 15 16 17 18 25 26 27 28 35 36 37 38 5 by 5 n n+4 n+40 n+44 53 54 55 56 57 63 64 65 66 67 73 74 75 76 77 83 84 85 86 87 93 94 95 96 97 5 6 7 8 9 15 16 17 18 19 25 26 27 28 29 35 36 37 38 39 45

    • Word count: 882
  15. Number Grid

    second box 45*56=2520 55*46= 2530 2530-2520= 10 The product of the top right hand number and the bottom left hand number in each square of numbers minus the product of the top left hand number and the bottom right hand number regardless of the position of the square is always equal to ten (10). From this I tried to get a formula for the nth term as follows The first number (2) is n and the second number 3 is n+1, The number below n (12)

    • Word count: 880
  16. Corner to corner

    71 72 73 81 82 83 91 92 93 71x93=6603 73x91=6643 6643-6603=40 Here is a table for the difference for a ten by ten grid: Length of Square Difference 2 10 3 40 4 90 5 160 6 250 7 360 8 490 9 640 10 810 The difference is calculated by the (length of the square - 1)� x10, that is how I was able to calculate the other differences of the different length of square. (L-1)� x 10= difference [algebraic form] L = length of the square The limitations with this formula are that the shape has to

    • Word count: 622
  17. You are given a 9x9 number grid with a 2x2 square on it. Investigate when pairs of diagonal corners are multiplied and subtracted from each other.

    This is: By changing the size of the grid I found that the width of the grid is equal to the difference while still sing a 2x2 square. I tested this using a 5x5 and 7x7 number grid as follows: Dif = (a+1)(a+9)-a(a+6) Dif = (a+1)(a+9)-a(a+8) = 1(+1x5+1+5-1(-1x6 = 1(+1x7+1+7-1(-1x8 = 5 = 7 As soon as the size of the square is changed the rule does not work, although the difference is still a multiple of the grid width.

    • Word count: 686
  18. Number grid

    (2x2) = (2 rows x 2 columns) 4 numbers (3x3) = (3 rows x 3 columns) 9 numbers (4x4) = (4 rows x 4 columns) 16 numbers I will begin my investigation by randomly selecting two (2x2) number grids. I will then multiply the top left by the bottom right of the (2x2) number grids and then multiply the top right by the bottom left, after that I will subtract the two totals to find the difference. if there is any pattern then I will randomly select a third number grid and try to predict the result. I will do the same thing again but instead of using a (2x2)

    • Word count: 681
  19. For 3-step stairs, investigate the relationship between the stair total and the position of the stair shape on the grid. Investigate further the relationship between the stair totals and other step stairs on other number grids.

    PART 2 Investigate further the relationship between the stair totals and other step stairs on other number grids. PART 1 The total of the squares inside the stair is all the squares added together. The stair number is the number in the bottom left hand corner of the stair. We can call this number n. In order to see a pattern between the totals of the shapes, we can arrange information in a table. n 6n +44 1 6 50 2 12 54 3 18 64 4 24 68 25 150 194 As you can see from the above table we can come to a formula of: T = 6n + 44 T is the stair total, n the stair number and 44 is the remaining number.

    • Word count: 937
  20. The Magic of Vedic Mathematics.

    Suppose you want to find the square of a number which is one less than the number whose square is known, you can use the following method: Square of 79 will be given as, (80)^2 - (80 + 79) = 6400 - 159 = 6241. Finding the square of a number near 50. Now, if you want to find the square of 51, the formula will be, (5)^2+1/ (1)^2 = 25 + 1/ 01 = 2601. This is what you do: The LHS of the answer is given as (5)^2 + 1 and the RHS is given as the square of the difference of the number from 50.

    • Word count: 834
  21. The Open Box Problem.

    (All measurements will be measured in centimetres) Question One This is an example of what I will do. 10x10 cut out size 2cm 20x20 cut out size 3cm 30x30 cut out of 4 cm Question Two Some examples on how cut out will look like in question two 10x20 cut out size 3 10x30 cut out size 4 20x30 cut out size 5 Conclusion All the formulas I found were based on the results and graphs shown.

    • Word count: 568
  22. As Australia is one of the most popular tourist destinations in the world it is necessary to observe the tourist precincts Australia has to offer.

    There is lots of signage around the square and also a small information booth at the front corner of the square for information of where anything is situated and a general overview of what's happening in the square and around the city. One of the major attractions within the square is the art gallery. It is a great place for tourists to visit and explore the art inside Melbourne. The Ian Potter NGV has a lot of art pieces where locals and tourists can visit. Admission is free and they also have tours that show traditional Aboriginal and Australian art.

    • Word count: 823
  23. Defences At Kennilworth Castle.

    You can tell that soldiers stayed in the gatehouse because of the latrines on the second floor. There was also a fireplace on the second floor. This must have been where the soldiers boiled the oil or water to pour down the murder holes in the ceiling. THE CASTLE WALLS The castle had curtain walls. One wall surrounded the keep. The other was a much thicker, taller wall surrounding the entire castle. This had all sorts of arrow slits in them.

    • Word count: 631
  24. Maths number stairs coursework.

    15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 For this coursework I have been asked to investigate; *The relationship between the stair total and the position of the stair shape on the grid. *Investigate further the relationship between the stair totals and the other step stairs on the number grids.

    • Word count: 403
  25. Opposite Corners.

    work out as all that you need to do is look at the n figures and see how the other numbers relate to them. N 2x3 squares (3 = width) 1x13= 13 11x3= 33 18x10= 180 8x20= 160 66x58= 3828 56x68= 3808 The difference is always 20. Nn n+1 n+2 n+3 Ln+10 n+11 n+12 n+13 Hn(n+13) Difference is 20 (n+3) (n+10) 3x3 Squares 1x23= 23 21x3= 63 43x65= 2795 63x45= 2835 68x90= 6120 88x70= 6160 The difference is always 40.

    • Word count: 732
  26. Number stairs

    The overlapped numbers will be added up to create a total figure this is the number i am trying to work towards. The formula created will have to be formed to calculate the total from using the bottom left hand number on the stair shape.

    • Word count: 218

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