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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

    opposite corners

    5 star(s)

    I predict that once again all answers will be the same. 3 X 3 Square 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 (3 x 21) - (1 x 23) = 40 (6 x 24) - (4 x 26) = 40 (10 x 28) - (8 x 30) = 40 I believe the answer will always be 40 for a 3 x 3 square on this grid.

    • Word count: 1638
  3. Marked by a teacher

    I am going to investigate the difference between the products of the numbers in the opposite corners of any rectangle that can be drawn on a 100 square (10x10) grid

    4 star(s)

    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 This is a 10x10 grid. On it (outlines in red) is a 2x2 square. Firstly, I?m going to see what the difference between the products of the corners is (D): 55x64=3520 54x65= 3510 3520-3510=10, D=10 Now: What if the same sized rectangle was placed in a different area of the grid?

    • Word count: 1629
  4. Marked by a teacher

    Maths coursework- stair totals. I shall be investigating the total and difference in sets of stairs in different grid sizes. I will be investigating the relationship between stair totals on different grids.

    3 star(s)

    Below is an example of just one of the three sets of stairs that I used to conduct my investigation: I have worked out the formula for the total inside the three step stairs in a 10 x 10 grid Key: x = corner number t = total inside stairs 1. I worked out the common relationship with the numbers inside the stairs and to be accurate here are two examples: x+x+1+x+2+x+10+x+11+x+20 = 6x+44 2. Noticing that this is the total I realised that it is not so hard to work out the formula for the total.

    • Word count: 1525
  5. Marked by a teacher

    Opposite Corners of a Square on a Number Grid

    3 star(s)

    I think there will be a pattern, and an algebraic expression, which will work for all the box sizes. I think this investigation will be one in which you could branch off into many different sections. I think there will be a lot to explore and a lot to think about. Overall, I think this will be a challenging, but thought-provoking investigation and I think I will enjoy discovering the results. Data Example 1 - 2*2 Square: 12 * 23 = 276 13 * 22 = 286 Difference = 10 19 * 30 = 570 20 * 29 = 580

    • Word count: 1196
  6. Marked by a teacher

    In this coursework, I intend to investigate the differences of the products of the diagonally opposite corners of a rectangle, drawn on a 10x10 grid, with the squares numbered off 1 to 100

    3 star(s)

    What is the same between the two alignments? What About Other Sizes of Rectangles? I will now try rectangles, all in the 2 x X series, with different lengths. I think I can now safely assume that the difference is always constant in relation to the size of the rectangle, therefore I only require one example of each difference. 2 x 4 2 x 5 Therefore, it can be seen that every time I increase the width by one, the difference increases by 10.

    • Word count: 1516
  7. Marked by a teacher

    Opposite Corners

    3 star(s)

    the opposite corners will have a difference of 10. 2x4 Rectangle 1 2 3 4 11 12 13 14 67 68 69 70 77 78 79 80 24 25 26 27 34 45 36 37 24x37=888 1x14=14 67x80=5360 34x27=918 11x4=44 77x70=5390 30 30 30 I have noticed a pattern occurring each time the width increases, the difference increases by 10, by 1. Prediction Using the theory I predict that when I multiply a 2x4 rectangle the opposite corners will have a difference of 30.

    • Word count: 1956
  8. Marked by a teacher

    Opposite Corners.

    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 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

    • Word count: 1921
  9. Marked by a teacher

    My course work in maths is going to consist of opposite corners and/or hidden faces.

    We will now go on to see what a square between 1,2,21,22 will be. Example 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

    • Word count: 1695
  10. Number Grid

    For the 2nd part of the investigation, I will be using 2 variables to extend the task further. I am going to investigate what the difference between the opposite products inside a rectangular shaped box is. I will do this by using the 1st formula and then see if there are any connections or similarities I can make. For the 3rd part of the investigation, I will be using 3 variables to extend the task even further. I will be using the first two formulas to link them with the grid size. I will then find a formula which will relate the shape of the box inside the grid and the size of the main grid.

    • Word count: 1006
  11. Math Grid work

    2 x 2 square 24 25 34 35 46 47 56 57 89 90 99 100 I notice that I get the same number for any same size square so to prove this I will use X and prove why the answer is always 10. I can use a square in which the boxes are the edges of the square on the grid. X X + 1 X + 10 X + 11 If I multiply this out if I were doing what I was doing with the numbers in the grid I would, (X+1)(X+10)...which is = X2+11X+10...and...

    • Word count: 1729
  12. rectangles. I will be trying to develop a formula that will enable me to calculate the sum of all the numbers in a rectangle given

    We would get the value by calculating the sum of all the numbers. A way we could find this value out could be by using a formula. But what formula could we use? n n+1 n+2 n+10 n+11 n+12 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

    • Word count: 1612
  13. Number Stairs

    On my 10x10 grid, I will first use an s-number of 25, which makes the s-total = 194. I will then work systematically, increasing the s-number by 1 every time. However, the s-number is limited in its placement - it cannot be a multiple of 10, or a multiple of 10 -1 - else it would not fit on the grid. I will record results in the following table: s-number (n) | 1 2 3 4 5 s-total (S) | 50 56 62 68 74 I can see from these results I can see that for every increase in n, S increases by 6.

    • Word count: 1467
  14. Maths number grid

    34 x 45 = 1530 44 x 35 = 1540 d = 10 As you can see by my calculations it is not just a 2 x 2 grid in the corner that gives a difference of 10 when the diagonally opposite corners are multiplied. The position of the 2 x 2 grid on a 100 square grid does not change the difference. I am now going to investigate if the difference is the same in a 2 x 3 grid when multiplying diagonally opposite corners.

    • Word count: 1541
  15. Number Grids

    Let's say the top left number was 'n'. If followed along the grid, the other numbers would come to be: (bottom right) n+11, (top right) n+1 and (bottom left) n+10. So if worked out, the sum would look like: After this, I tried squares of different sizes around the 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 38 39 40 41 42 43 44 45 46 47 48 49 50

    • Word count: 1834
  16. Open box. In this investigation, I will be investigating the maximum volume, which can be made from a certain size square piece of card, with different size sections cut from their corners. The types of cubes I will be using are all open topped boxes.

    1cm by 1cm, piece of square card. Length of the section (cm) Height of the section (cm) Depth of the section (cm) Width of the section (cm) Volume of the cube (cm3) 0.1 0.1 0.8 0.8 0.064 0.2 0.2 0.6 0.6 0.072 0.3 0.3 0.4 0.4 0.048 0.4 0.4 0.2 0.2 0.016 0.5 2cm by 2cm, piece of square card Length of the section (cm) Height of the section (cm) Depth of the section (cm) Width of the section (cm)

    • Word count: 1826
  17. Stair shape maths GCSE coursework

    First I will draw a 3 stair shape at the bottom left with the number 1. Then I am going to add up the numbers in the stairs to get the total. Then I am going to move the stairs one step right and find the total of the particular stair shape. Then I am going to repeat this once more moving it one step right and finding its total. Finally I am going to find the relationship between these stair shapes.

    • Word count: 1180
  18. Open box problem

    2.25 27.56250 8 8 2.5 22.50000 8 8 2.75 17.18750 8 8 3 12.00000 8 8 3.25 7.31250 8 8 3.5 3.50000 8 8 3.75 0.93750 8 8 4 0.00000 From the table above it can clearly be seen that the biggest volume gained from an 8 by 8 piece of square sheet is 37.81cm (2dp), this volume lies between the height of 1cm and 1.5cm. Therefore I am going to further investigate the height between these two points to gain the highest possible volume.

    • Word count: 1344
  19. number stairs

    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 Therefore in order for me to see a pattern occurring, I will be moving this 3 step stair, systematically to find out a formula for the total of the three step stair, which works for the entire 3 step stair on a 10 by 10 grid.

    • Word count: 1511
  20. number grid

    For example take 2x2 square and multiply its corners diagonally. 25 26 35 36 25 x 36= 900 26 x 35= 910 910 - 900= 10 77 78 87 88 77 x 88 = 6776 78 x 87= 6786 6786 - 6776= 10 After a few results I observed that any box of 2 by 2 the difference will always be 10 if grid size is 10 by 10. 4 5 6 14 15 16 24 25 26 I decided to try out some square but this time I going to do it bigger like 3 by 3 and 4 by4.

    • Word count: 1212
  21. Maths algerbra

    I done the same to 3/4 - 2/3 9/12 - 8/12 which equalled 1/12 and my second difference. I shall begin to search for an algebraic rule for the first differences Number 1 2 3 4 5 nth n+1th D1 1/6 1/12 1/30 1/42 1/56 To find the nth term I had to take away the n+1th term away from the nth term ; n+1 - n n+2 n+1 I made the denominators the same and got ; (n+1)(n+2) - n(n+2) (n+1)(n+2) (n+1)(n+2) I then multiplied the brackets out by completing the square. For the numerator X n 1 n n � n 1 n 1 X n 2 n n � 2n (n � + 2n + 1)

    • Word count: 1208
  22. This piece of coursework is called 'Opposite Corners' and is about taking squares of numbers from different sized number grids

    Prediction I predict that a 3x3 square from a 5 wide grid, it will have a final difference of 20. Proof: 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 Comparison: N N+2 x N+10 N+12 N (N+12) = N� + 12N (N+2) (N+10) = N� + 10N + 2N + 20 = N� + 12N + 20 Difference: (N� + 12N + 20)

    • Word count: 1877
  23. O-X-O investigation

    Also I will use some abbreviations in this investigation. W=Width L=length T= Number of squares needed for a winning line S= Number of winning lines. Investigation 1 I will start off by investigating square grids with the number of squares that the length and width is will be the number needed for a winning line i.e. L=W=T 3x3 8 winning lines 4x4 10 winning lines 5x5 12 winning lines Size of grid WxL N.o of squares needed for a winning line 'T' N.o of winning lines 'S' 3x3 3 8 4x4 4 10 5x5 5 12 Rule WxW W 2W+2 I got this rule because you can see there

    • Word count: 1184
  24. 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.

    • Word count: 1153
  25. 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)

    • Word count: 1364
  26. 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 cut-out. 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.

    • Word count: 1264

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