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

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

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 35203510=10, D=10 Now: What if the same sized rectangle was placed in a different area of the grid?
 Word count: 1629

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.
4 star(s)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

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

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

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

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

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 thoughtprovoking 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

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

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

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

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

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

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

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

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

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

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 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