GCSE: Number Stairs, Grids and Sequences
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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

I am going to investigate by taking a square shape of numbers from a grid, and then I multiply the opposite corners to find the difference of these two results. Firstly I am going to start with a 10x10 grid
4 star(s)I predict if I move the 4x4 square up, I will get the same answer. 12 52x85=4420 55x82=4510 90 My prediction is right. I am going to use algebra to test my results. n n+3 n+30 n+33 (n+3)(n+30)=n�+90+33n n(n+33)=n�+33n Products difference is equal to (n�+90+33n)  (n�+33n) =90 In the same grid I will now work out a 5x5 square. number Left corner x right corner Right corner x left corner Products difference 13 6x50=300 10x46=460 160 14 16x60=960 20x56=1120 160 15 15x59=885 19x55=1045 160 I have noticed that the products difference of 5x5 squares in a 10x10 grid equal to 160.
 Word count: 3671

Number Grid Aim: The aim of this investigation is to formulate an algebraic equation that works out the product of multiplying diagonally opposite corners of a particular shape and finding the difference between the results
3 star(s)Perhaps this means that because it is a 10 x 10 grid, that all the differences would be 10. I would still like to further investigate this theory. The grid below is once again a 2 x 2 box derived from the original 10 x 10 grid. 89 90 99 100 89 x 100 = 8900 90 x 99 = 8910 ? 8910  8900 = 10 This once again confirms what I stated; that the difference between the products of crossmultiplied boxes will always equal 10 in a 10 x 10 grid. I would like to determine if this is definitely correct, so I am going to do it again twice.
 Word count: 4565

To find a relationship between the opposite corners in various shapes and sizes.
81 x 92 = 7452 82 x 91 = 7462 7462  7452 = 10 Again the difference is 10. I am can now definitely see a pattern, so I am going to try and work at a relationship sing algebraic equations. X X+1 X+10 X+11 X = 14 = 14 + 1 = 15 = 14 + 10 = 24 = 14 + 11 = 25 I have found a way to find out the opposite corners in a 2 x 2 square in a 10 by 10 grid.
 Word count: 3284

Number Grids Investigation Coursework
 (top left x bottom right) = 50 x 59  49 x 60 = 2950  2940 = 10 The examples above are consistent with the original example. I shall now use algebra to try and prove that this is the case for all 2 x 2 squares in this grid: Let the top left number equal a, and therefore; a a+1 a+10 a+11 Therefore, if I put this into the calculation I have been using, the difference between the products of opposite corners would be: (top right x bottom left)
 Word count: 6671

Maths coursework. For my extension piece I decided to investigate stairs that ascend along with the numbers, in order to do this the grid was turned upsidedown. I aim to see if there is a pattern within these stairs
+ 13 + 14 + 23 = 62 24 14 15 4 5 6 The stairtotal for this stair shape is 4 + 5 + 6 + 14 + 15 + 23 = 68 Stair number Stair Total 25 194 26 200 67 446 68 452 3 62 4 68 I will then summarize these results in a table: In order to find a formula that I can use to find the stair total when I am given the stair number, I am going to put the stair number as the position and the stair total as the term for
 Word count: 4685

Number Grid Coursework
Data Analysis From the table, it is very easy to see that on all tested locations of the box, the difference of the two products was 10. 5) Generalisation Using this apparently constant number, it can be assumed that for all possible locations of the 2x2 box on the width 10 grid, that the difference is always 10. Therefore, the following equation should be satisfied with any real value of a, where: a is the topleft number in the box; (a + 1)
 Word count: 5993

Number Grids
Without taking the different numbers on the top row into account, this would mean that the difference between the two products would now be the number in the bottom right hand corner. However, because the top left hand square is one less than the top right hand corner, you have to take away the number inside that square away from the higher product. This in theory always gives you a difference of 10. Here is an example: 5 6 15 16 Because you are multiplying 6 x 15 and 5 x 16 you are immediately making a large difference between the two products, as you are multiplying by different numbers.
 Word count: 3515

Maths  number grid
(r+1)(r+10) r (r+11) = r (r+10) +1 (r+10)  r 11r = r +10r +r+10  r 11r =r +11r+10 r 11r =10 This algebra proves my results that for any 2x2 square the answer for the defined difference will always be 10. Furthering my Investigation I am now going to further my investigation by increasing the size of the squares. I am going to be repeating my process used previously but will be looking at randomly selected 3x3 squares, my aim being to see if I can find a trend and a pattern in my results.
 Word count: 6680

Number Stairs
Therefore T = 56. 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 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 Stair number (N)
 Word count: 5404

Number grid
3 x 3 Box n n+1 n+2 n+10 n+11 n+12 n+20 n+21 n+22 4 x 4 Box n n+1 n+2 n+3 n+10 n+11 n+12 n+13 n+20 n+21 n+22 n+23 n+30 n+31 n+32 n+33 I am going to display my results in a table, where I would try finding any patterns within the results I have. 10 x 10 Grid Square Box sizes Difference Pattern 2 x 2 10 =1 x 10 12 = 1 3 x 3 40 = 4 x 10 22 = 4 4 x 4 90 = 9 x 10 32 = 9 Predict 5 x 5
 Word count: 3313

Maths Grids Totals
3 x 3 = 10 x 22 = 10 x 4 = 40). This gives me the formula 10(n1)2. Using this formula, I predict that the difference for an 8 x 8 square will be 10(81)2 = 10 x 72 = 10 x 49 = 490. I will take a random 8 x 8 square from the 10 x 10 grid: 12 13 14 15 16 17 18 19 22 23 24 25 26 27 28 29 32 33 34 35 36 37 38 39 42 43 44 45 46 47 48 49 52 53 54 55 56 57 58
 Word count: 5017

MathsNumber Grid
I will now use this grid to come up with a prediction for the other grids of the same length and width. I will do this by multiplying 63 and 54 together to give me a product of 3402. Afterwards, I will multiply 64 and 53 to give me a product of 3392. Lastly, I will subtract 3392 from 3402 to give me a product difference of 10. I will draw up 3 other grids, to test if my prediction is correct.
 Word count: 4188

GCSE Maths Sequences Coursework
1 2 3 4 5 6 Sequence 1 5 13 25 41 61 1st difference +4 +8 +12 +16 +20 2nd difference +4 +4 +4 +4 Unshaded I can see here that there is not much of a pattern in the 1st difference, but when I calculate the 2nd difference I can see that it goes up in 4's, therefore this is a quadratic sequence and has an Nth term. The second difference is 4 therefore the coefficient of N� must be half of 4 i.e.
 Word count: 3939

number grid
I will then try changing the size of the rectangles and looking for patterns there. I will look at 2 by 4 rectangles, 5 by 3 rectangles and 4 by 5 rectangles. I will also look at changing the size of the number grids to see if this has an affect on the patterns. I will look at a 9 by 9 grid, an 11 by 11 grid and a 5 by 5 grid. I will be looking for patterns in 2 by 2 squares within the different size grids and trying to find an algebraic formula to explain my findings.
 Word count: 7010

Mathematics  Number Stairs
number, for example, 20 T = 6 x 31 + 44 = 230 So: 51 41 42 31 32 33 31 + 32 + 33 + 42 + 41 + 51 = 230 So therefore my prediction works but I must prove it algebraically: n+20 n+10 n+11 n n+1 n+2 n + (n+1) + (n+2) + (n+10) + (n+11) + (n+20) = 6n + 44 T = 6n + 44 has been proved to work. 8 9 10 11 12 1 2 3 T = 6n + 44 4 5 So after the first part of the investigation, I will
 Word count: 4455

Mathematical Coursework: 3step stairs
12 13 14 15 16 17 18 19 10 1 2 3 4 5 6 7 8 9 10 Because I haven't discovered the algebraic equation, this would allow me to find the total stair number in less than minutes. I would have to find the total by using the basic method of adding. The Calculations 1+2+3+11+12+21= 50 5+6+7+15+16+25=74 83+73+63+64+65+74=422 47+48+49+57+58+67=326 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
 Word count: 8908

100 Number Grid
36 x 47 = 1692 37 x 46 = 1702 Product difference = 10 F. 49 x 60 = 2940 50 x 59 = 2950 Product difference = 10 G. 74 x 85 = 6290 75 x 84 = 6300 Product difference = 10 H. 85 x 96 = 8160 86 x 95 = 8170 Product difference = 10 By continuing this investigation, my prediction proves to be correct. As a result, an algebraic formula can be used to work out the product difference.
 Word count: 4202

Staircase Coursework
I will now find a formula for the total, which will make it easier to calculate the sum. I will label a stair algebraically This is an example of stair 1 n stands for the number in the left bottom corner of the step. In order of that if : 1 = n 2 = n + 1 and so on ................ n+20 n+10 n+11 n n+1 n+2 21 11 12 1 2 3 = I am now going to add up the single squares, which gives me the formula n + (n + 1) + (n + 2)
 Word count: 3422

number grid investigation]
The top left number in the grid (letter (a) in the above example) will be represented by the term 'n', which will be referred to in this manner in all proceeding investigations also. This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products. In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables.
 Word count: 7798

number grid
The top left number in the grid (letter (a) in the above example) will be represented by the term 'n', which will be referred to in this manner in all proceeding investigations also. This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products. In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables.
 Word count: 7894

number grid
N x (N + 11). Then multiply top right box with bottom left box e.g. (N +1) (N + 10). Do the same as with the arithmetic product when told to find the difference, by subtracting the N x (N + 11) from the (N +1) (N + 10) using the FOIL method. N (N + 11)  (N + 10) (N + 1) N2 +11N  N2 + 1N +10N + 10 N2 +11N  N2 + 11N + 10 Difference = 10 Cconclusion: the difference for a 2by2 squared box is always 10 Now I will investigate further, by finding the difference of
 Word count: 3063

Algebra Investigation  Grid Square and Cube Relationships
The top left number in the grid (letter (a) in the above example) will be represented by the term 'n', which will be referred to in this manner in all proceeding investigations also. This is only the first section of the investigation. In calculating the formula, enough information will be gained to progress and investigate other factors, variables and measurements that affect the difference between products. In order to see a trend (in the form of an n x n grid) from the sample numeric grids (2 x 2, 3 x 3 etc.) it is necessary to use summary tables.
 Word count: 8966

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
 Word count: 4105

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)
 Word count: 3601

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.
 Word count: 3959