MATHEMATICS COURSEWORK- NUMBER STAIRS
My investigation is based on number stairs. An example of a number stair is below:
This is a 3 step stair because both the length and the width of the stair are 3 steps
For the first part of my investigation this 3-step stairs (above is) is going to be placed on different number grids such as a 10 by 10 or 9 by 9 Number Grid etc
Secondly for my first part of my investigation I also need to find a formula for each grid. These formulas must be able to work out the stair total for the 3-step on a number of different size grids such as 10 by 10 or 9 by 9 etc
A stair total is all the values added together in the 3-step stairs
After I have found a formula for a couple of number grids I am going to work out a general formula for any grid size possible.
Below is a stair drawn on a 10 by 10 Number Grid:
On the 10 by 10 Number Grid (on the previous page) a 3-step stair is highlighted
The total of the numbers inside the stair shape is:
91+81+82+71+72+73=470
The stair total for this 3-step stair is 470
PART 1
For other 3-step stairs, investigation the relationship between the stair total and the position of the stair shape on the gird
To start my investigation I am going to start by using a 10 by 10 Number grid below:
I have highlighted a 3-step stair above on my 10 by 10 Number Grid
The total of the numbers inside the stair shape is:
91+81+82+71+72+73=470
The stair total is all the six numbers in 3-step stair added together so for this 3-step stair the stair total is 470
On the right is a portion of the 10 x 10 grid squares and
there is 6 boxes which are representing the numbers
91, 81, 82,71,72,73 and these are the 3-step stair
From this diagram of the 3-step stair I can start to establish if there is a pattern
I need to find a pattern so that I can find an algebra formula to represent this pattern and use the formula for the 10 by 10 Number Grid
By looking at the 3 step stair diagram we know that there are 6 boxes and I will assume in a 3 step stair the bottom left box is equal to X, therefore in my 3-step stair x=71
Now I am going to use the values in algebra equations and below and this is how it is going to look like:
- The 1st square = 91 then the formula is x + 20 = 91
-The 2nd square = 81 then the formula is x + 10 = 81
-The 3rd square = 82 then the formula is x + 11 = 82
-The 4th square =71 it is simply just x = 71
-The 5th square = 72 then the formula is x + 1 = 72
-The 6th square = 73 then the formula is x + 2 = 73
The above algebra formulas are shown below in a portion of the 3-step stair:
Now I am going to add the value inside the 3-step stair above so: Total= X +(x+20) + (x+10) + (x+11) + (x+1) + (x+2) = 44
As a result we can find the total value of 3-step stairs in a 10 by 10 Number Grid by using the algebra equation: T=6x - 44
T= The total value of the 3-step stairs
X= The bottom left hand corner stair number so in this case X=71
6x is the number of the squares in the 3-step stair
44 is the sum of all the algebra formula in the 3-step stair and they are: Total= X +(x+20) + (x+10) + (x+11) + (x+1) + (x+2)
In conclusion the algebra formula to find the total inside the 3-step stairs for a 10 by 10 Number Grid is:
T= 6x + 44
I am going to test my formula for this portion of a 3-step stair:
T=6x + 44
T= (6 x 35) + 44
T= 210 + 44
T=254
The total for all the stair values added together without a formula is=
55 + 45 + 46 + 35 + 36+ 37= 254
My formula to find the total values in a 3-step stair on a 10 by 10 grid is correct. This is because all the stair values in a 3-step stairs added together on a 10 by 10 Number grid gives 254 and I also get 254 when I use my formula. This means I have proved for my formula to be correct
Now I am going to use the same logic and method for other grids such as 11 x 11 and 12 x 12 and 13 x 13 etc. I will use the same 3 step-stair approach and I can then use the algebra formula for the 11 x 11 and 12 x 12 Number Grids to find a pattern
Below is an 11 by 11 Number Grid that I am going to investigate on. I am going to use the 11 by 11 Number Grid to find an algebra formula to work out the 3-step stair total on a 11 by 11 Number Grid
00
01
02
03
04
05
06
07
08
09
10
89
90
91
92
93
94
95
96
97
98
99
78
79
80
81
82
83
84
85
86
87
88
67
68
69
70
71
72
73
74
75
76
77
56
57
58
59
60
61
62
...
This is a preview of the whole essay
02
03
04
05
06
07
08
09
10
89
90
91
92
93
94
95
96
97
98
99
78
79
80
81
82
83
84
85
86
87
88
67
68
69
70
71
72
73
74
75
76
77
56
57
58
59
60
61
62
63
64
65
66
45
46
47
48
49
50
51
52
53
54
55
34
35
36
37
38
39
40
41
42
43
44
23
24
25
26
27
28
29
30
31
32
33
2
3
4
5
6
7
8
9
20
21
22
2
3
4
5
6
7
8
9
0
1
I have highlighted a 3-step stair on the 11 by 11 Number grid
The total of the numbers inside the highlighted 3-step stair is:
00+ 89 + 90 + 78 + 79 + 80 = 516
The stair total is all the six numbers in 3-step stair added together so for this 3-step stair the stair total is 516
On the right is a portion of the 11 x 11 grid squares and
there is 6 boxes which are representing the numbers
00, 89, 90, 78, 79, 80 and these are the 3-step stair
From this diagram of the 3-step stair I can start to establish if there is a pattern
I need to find a pattern so that I can find an algebra formula to represent this pattern and use the formula for the 11 by 11 Number Grid
By looking at the 3 step stair diagram we know that there are 6 boxes and I will assume in a 3 step stair the bottom left box is equal to X, therefore in my 3-step stair x=78
Now I am going to use the values in algebra equations and below and this is how it is going to look like:
-The 1st square = 100 then the formula is x + 22 = 100
-The 2nd square = 89 then the formula is x + 11 = 89
-The 3rd square = 90 then the formula is x + 12 = 90
-The 4th square = 78 it is simply just x = 78
-The 5th square = 79 then the formula is x + 1 = 79
-The 6th square = 80 then the formula is x + 2 = 80
The above algebra formulas are shown below in a portion of the 3-step stair:
Now I am going to add the value inside the 3-step stair above so: Total= X + (x+22) + (x+11) + (x+12) + (x+1) + (x+2) = 48
In conclusion the algebra formula to find the total inside the 3-step stairs for an 11 by 11 Number Grid is:
T=6x - 48
I am going to test my formula for this portion of a 3-step stair:
T=6x + 48
T= (6 x 38) + 48
T= 228 + 48
T=276
The total for all the stair values added together without a formula is=
60 + 49 + 50 + 38 + 39 + 40= 276
My formula to find the total values in a 3-step stair on an 11 by 11 grid is correct. This is because all the stair values in a 3-step stairs added together on a 11 by 11 Number grid gives 276 and I also get 276 when I use my formula. This means I have proved for my formula to be correct
Now I am going to use the same logic and method for one other grid and that is a 12 by 12 Number Grid. I will use the same 3 step-stair approach and I can then use the algebra formula for the 10 x 10 and 11 x 11 and 12 x 12 to find a pattern to create a general formula for any grid size
Below is a 12 by 12 Number Grid that I am going to investigate on. I am going to use the 12 by 12 Number Grid to find an algebra formula to work out the 3-step stair total on a 12 by 12 Number Grid
09
10
11
12
13
14
15
16
17
18
19
20
97
98
99
00
01
02
03
04
05
06
07
08
85
86
87
88
89
90
91
92
93
94
95
96
73
74
75
76
77
78
79
80
81
82
83
84
61
62
63
64
65
66
67
68
69
70
71
72
49
50
51
52
53
54
55
56
57
58
59
60
37
38
39
40
41
42
43
44
45
46
47
48
25
26
27
28
29
30
31
32
33
34
35
36
3
4
5
6
7
8
9
20
21
22
23
24
2
3
4
5
6
7
8
9
0
1
2
I have highlighted a 3-step stair on the 12 by 12 Number grid
The total of the numbers inside the highlighted 3-step stair is:
09 + 97 + 98 + 85 + 86 + 87 = 553
The stair total is all the six numbers in 3-step stair added together so for this 3-step stair the stair total is 553
On the right is a portion of the 12 x 12 grid squares and
there is 6 boxes which are representing the numbers
09, 97, 98, 85, 86, 8 and these are the 3-step stair
From this diagram of the 3-step stair I can start to establish if there is a pattern
I need to find a pattern so that I can find an algebra formula to represent this pattern and use the formula for the 12 by 12 Number Grid
By looking at the 3 step stair diagram we know that there are 6 boxes and I will assume in a 3 step stair the bottom left box is equal to X, therefore in my 3-step stair x=85
Now I am going to use the values in algebra equations and below and this is how it is going to look like:
- The 1st square = 109 then the formula is x + 24 = 109
-The 2nd square = 97 then the formula is x + 12 = 97
-The 3rd square = 98 then the formula is x + 13 = 98
-The 4th square = 85 it is simply just x = 85
-The 5th square = 86 then the formula is x + 1 = 86
-The 6th square = 87 then the formula is x + 2 = 87
The above algebra formulas are shown below in a portion of the 3-step stair:
Now I am going to add the value inside the 3-step stair above so: Total= X + (x+24) + (x+13) + (x+11) + (x+1) + (x+2) = 52
In conclusion the algebra formula to find the total inside the 3-step stairs for a 12 by 12 Number Grid is:
T=6x + 52
I am going to test my formula for this portion of a 3-step stair:
T=6x+52
T= (6 x 29) + 52
T= 174+52
T=226
The total for all the stair values added together without a formula is=
53 + 41 + 42+ 29+ 30 + 31= 226
My formula to find the total values in a 3-step stair on a 12 by 12 grid is correct. This is because all the stair values in a 3-step stairs added together on a 12 by 12 Number grid gives 226 and I also get 226 when I use my formula. This means I have proved for my formula to be correct
If I closely observe the results from the 3-step stairs from the 3 number grids, such 10x10, 11x11 and 12x12 I can see there is a constant number that is consistent, which is [4]
Below is a table that I have drawn by using the constant number [4]
GRID SIZE
ALGEBRA FORMULA
3 x 3
6 x + 16
4 x 4
6 x + 20
5 x 5
6 x + 24
6 x 6
6 x + 28
7 x 7
6 x + 32
8 x 8
6 x + 36
9 x 9
6 x + 40
0 x 10
6 x + 44
1 x 11
6 x + 48
2 x 12
6 x + 52
3 x 13
6 x + 56
4 x 14
6 x + 60
From the table above and using my results from the investigations Number Grids 10 x 10, 11 x 11 and 12 x 12 I can see a pattern emerging
The value in the algebra formula increases by 4 every time the size of the grid box increases by 1. An example of this is the 10 by 10 Number Grid, where the 3-step formula is 6x-36 where x is the number in the bottom left hand square, then I increased the grid size by 1 so 11 x 11 and using the same 3-step stair approach the formula is 6x-40, etc
This proves that my theory of the constant number of [4] is consistent every time the grid size increases by [1]
For any 3-step numbered grid box as shown in my table above or in the testing examples I am going to do below, and [x] as the value from the bottom left hand square and the algebra theory used to calculate the formula the total value of the squares can be found. This theory has been put to the test using a 10x10, 11x11 and 12x12 gird squares. The results are definite and consistent, proving the theory to be accurate and reliable.
Below I have tested out each formula for each grid size to make sure that the formulas are right and to prove that the formulas are correct:
From me testing all 12 algebra formulas, it clearly shows that the algebra formula works and the theory is correct, accurate and reliable.
In the second part of my testing formulas I added the numbers and this was because I wanted to make sure that the algebra formula gives the correct results and in all cases it did, which again proves all the algebra equation to be correct.
Till this point I have found many algebra formulas for my 3-step stair on a number of grids such as:
0 by 10 Number Grid and the algebra formula is 6 x + 44
1 by 11 Number Grid and the algebra formula is 6 x + 48
2 by 12 Number Grid and the algebra formula is 6 x + 52 etc.
Now I need to use these formulas and continued to work out a general formula for any size grid
In my general formula I need to include the value of the grids size e.g. if it is a 10 x 10 [10] or 11 x 11 [11] represented as algebra to give us the total of the numbers added together
To start my investigation for the general formula I need to establish the highest common factors, by using our values above, which are 36, 40 & 44 and show them as shown below:
I know that the above values increase by the constant number [4] and also that in any 3-step grid square if the grid size increases by 1 then the constant number is added to the value (The Highest Common Factor)
The above values increase by the constant number [4] and also that in any 3-step number grid, if the grid size increases by 1 then the constant number is added to the value (The Highest Common Factor)
In the above diagram the grid size is increasing by 1, such as from 10 to 11 and then to 12, using the highest common factor for 3-step grids (4) the calculations are 9, 10 & 11 for example:
1 x 4 = 44 12 x 4 = 48 13 x 4 = 52
I can now use 4 as the constant number and [n] as the grid size in my General Formula. Now the algebra formula is starting to emerge and is starting to look like this: 6x + 4n
My next step is to test my general formula, I am going to use 15 by 15 Number grid as it the next in my table after 14 by 14 Number Grid
On the left is a portion of the 15 x 15 number grid and there is 6 boxes which are representing the numbers 1, 2, 3, 16, 17 and 31 these are the 3-step stair
I am going to test my formula for this portion of a 3-step stair:
T=6x+4n
T= (6 x 1) + (4x15)
T= 6+60
T=66
The total for all the stair values added together without a formula is=
+2+3+16+17+31= 70
[The stair total for this 3-step stair is 70]
My general formula to find the total values in a 3-step stair on a 15 by 15 grid is incorrect. This is because all the stair values in a 3-step stairs added together on a 15 by 15 Number grid gives 70 but when I use my formula I get 66. This means I need to increase n, and I can start this by 4 for example (n+4) so my general formula now looks likes: 6x + 4n + 4 and now I am going to test it to see if it works
On the left is a portion of the 15 x 15 number grid and there is 6 boxes which are representing the numbers 1, 2, 3, 16, 17 and 31 these are the 3-step stair
I am going to test my formula for this portion of a 3-step stair:
T=6x + 4n + 4
T= (6 x 1) + (4 x 15) + 4
T= 66+6
T=70
The total for all the stair values added together without a formula is=
+2+3+16+17+31= 70
[The stair total for this 3-step stair is 70]
My general formula to find the total values in a 3-step stair on a 15 by 15 grid is correct. This is because all the stair values in a 3-step stairs added together on a 10 by 10 Number grid gives 70 and I also get 70 then I use my formula. This means I have proved for my general formula to be correct
To prove my algebra formula is correct I am going to test it on two other 3-step grids. I have chosen grid sizes 23 and 33 Number Grids.
From the processes I have gone through, I have devised an algebraic formula which when I tested, in each case gave me positive results for every 3-step stair in any grid size.
I have also linked the general formula any 3-step stair grid size to the stairs. Below is an example of the 3 step stair with the grid size (n) included:
The total of the 3-step stair:
T= X + (x + 2n) + (x + n) + (x + n + 1) + (x + 1) + (x + 2)
Now I will simply that to get:
T=6x + 4n + 4
THE GENERAL FORMULA FOR ANY 3-STEP STAIR GRID SIZE IS:
T=6x + 4n + 4
PART 2
Investigate further the relationship between the stair totals and other step stairs on other number grids.
Using my algebra equations and my proven theory for 3-step, I am going to use the same approach and theory as I used in the 3-step exercise, I can apply the same to the 4-step stair to find the general formula
FOUR-STEP STAIRS:
To start my investigation I am going to start by using a 10 by 10 Number
grid below:
Below is a portion of a 4 step- stair in algebraic terms:
The total of the terms added in the second portion of the 4-step stair is:
T= X + (x+3n) + (x+2n) + (x+n+1) + (x+n) + (x+n+1) + (x+n+2) + (x+1) + (x+2) + (x+3)
I can simplify this down to:
T= 10x + 10n + 10
T= The total value of the 4-step stairs
X= The bottom left hand box which is the corner stair number
N= The grid size
I am going to test this general formula on a 23 by 23 and a 33 by 33 Number Grid below:
From the processes I have gone through before, I have devised an algebraic for any 4-step stair on any grid size.
THE GENERAL FORMULA FOR ANY 4-STEP STAIR GRID SIZE IS:
T= 10x + 10n + 10
FIVE- STEP STAIRS:
To start my investigation I am going to start by using a 10 by 10 Number grid below:
Below is a portion of a 5 step- stair in algebraic terms:
The total of the terms added in the second portion of the 5-step stair is:
T= X + (x+4n) + (x+3n) + (x+3n+1) + (x+2n) + (x+2n+1) + (x+2n+2) + (x+n) + (x+n+1) + (x+n+2) + (x+n+3) + (x+1) + (x+2) + (x+3) + (x+4)
I can simplify this down to:
T= 15x + 20 (n+1)
T= The total value of the 5-step stairs
X= The bottom left hand corner stair number so in this case X=41
X= is the number of the squares in the 5-step stair
To prove my general formula is correct I am going to test it on two other 5-step grids. I have chosen grid sizes 23 and 33 Number Grids.
From the processes I have gone through, I have devised an algebraic formula which when I tested, in each case gave me positive results for every 5-step stair in any grid size.
THE GENERAL FORMULA FOR ANY 3-STEP STAIR GRID SIZE IS:
T=15x + 20 (n+1)
SIX- STEP STAIRS:
To start my investigation I am going to start by using a 10 by 10 Number grid below:
Below is a portion of a 6 step- stair in algebraic terms:
The total of the terms added in the second portion of the 5-step stair is:
T= X + (x+6n) + (x+4n+8) + (x+4n+7) + (x+3n+6) + (x+3n+5) + (x+3n+4) + (x+2n+4) + (x+2n+3) + (x+2n+2) + (x+2n+1) + (x+n+2) + (x+n+1) + (x+n) + (x+9) + (x+8) + (x+1) + (x+2) + (x+3) + (x+4) + (x+5)
I can simplify this down to:
T= 21x + 35n + 1
T= The total value of the 6-step stairs
X= The bottom left hand corner stair number
X= is the number of the squares in the 6-step stair
To prove my general formula is correct I am going to test it on two other 6-step grids. I have chosen grid sizes 6 and 8 Number Grids.
From the processes I have gone through, I have devised an algebraic formula which when I tested, in each case gave me positive results for every 6-step stair in any grid size.
THE GENERAL FORMULA FOR ANY 6-STEP STAIR GRID SIZE IS:
T=21x+ 35 (n+1)
CONCLUSION
From the investigations and tests that I conducted in this investigation I can conclude that by using the general formula to calculate the total of the grid numbers:
The step stairs, regardless to whether it is a 3-stepped, 4-stepped, or 5-stepped etc or where its position is in a numbered grid, such as 10x10, or 23x23 etc the general formula will always give the total of the grid numbers in that step stair.
The investigation and tasks carried out in Part 1 and Part 2 show clearly illustrates the common relationship when using the general formula
THE COMPLETE AND ENTIRE FORMULA
The aim of The Complete And Entire Formula is to calculate the total value of the numbers in a step stair.
From my investigations and the results of my table I can state the hypothesis as:
Each time the step stair increases so does the amount of squares within the step, which is called the formula of triangular numbers.
Also as the size of the grid increase by x number of squares the value of the squares total will also change again known as the triangular numbers.
Below is a representation of triangular numbers:
P = 1
3n
P = 2
6n
P = 3
0n
P = 4
5n
P = 5
21n
From the triangular numbers diagram the 1st set of numbers are referred to as the "a" numbers (1,4,10,20,35,56)
2nd set as the "c" numbers (1st difference - 3,6,10,15,21,28)
3rd set as the "d" numbers (2nd difference - 3,4,5,6,7)
Last set "e" numbers (common difference all the 1's).
Then P is referred to as the step index.
Consequently if P = 5 then there will be 21 squares (1st difference) in the step stair, making it a 6-step stair as shown above on the next page:
`
I have found another pattern. If I look at the '1st difference' column, I noticed that all the numbers are triangular numbers. This could be help to find the second part of the formula '1/2 n² + 1/2 n'
The "Complete & Entire Formula" (which I was not able to find) would have demonstrated that no matter what the size of the step stair (such as the number of squares), the position of the step stair (for example 1,11,12,21,22,23 or 81,82,83,91,92,93) and the grid size (e.g. 10x10, 23x23) the result will always be the total of the numbers in the step stair.
This relationship is known as the "triangular numbers"+
Maryam Ahmad Mathematics GCSE Coursework-Number Stairs
