Calculations
512 – 470 = 42
92 – 50 = 42
512 – 92 = 420
470 – 50 = 420
42 x 10 = 420
I have now established that as you move from left to right with the 3-step stairs, the difference is 42. This is a common number. The common number for moving the 3-step stair from up to down is 420. This shows that to work out the difference of up to down is multiplied by 10 using the difference of 42 from right to left.
Key
- 3-step stair 9
- 3-step stair 10
- 3-step stair 11
Calculations
3-step stair 9 – 27+ 17+18+7+8+9 = 86
3-step stair 10 – 97+87+88+77+78+79 = 506
3-step stair 11 – 92+82+83+72+73+74 = 476
3-step stair 12 – 22+12+2+13+3+4 = 56
506 – 476 = 30
86 – 56 = 30
506 – 86 = 420
476 – 56 = 420
I have worked out that if you move the 3-step stair one block inwards (like shown above) and calculate the totals for each of them and then take away one from the other, you are left with a difference of thirty, like shown in the calculation section. But more importantly, when taking away the top 3-step stairs by the bottom stairs, the final result is 420, like before. I am now able to see that finding the difference from left to right is not relevant at this point.
Formula calculations: trial and error
I will begin using the corner number and the number of squares in the 3-step stair to work out the stair total and I will use 3-step stair 8 as an example to calculate the formula.
6 = number of squares in the 3-step stair
n = corner number
t = total
3-step stair 8
Stair total = 50
6 x n – 10 = -4
6 x n + 10 = 16
6 x n + 20 = 26
6 x n + 30 = 36
6 x n + 40 = 46 – need four more to achieve the total
6 x n + 44 = 50
6n + 44 + t (the formula)
As you may see from above, I have moved onto trial and error to work out the formula. I understood that in a 2-step stair there are six squares. I predicted that the corner number had some part in the formula and so selected it to use in my formula seen as it was a low number and I was trying to reach a number that was larger than itself (50) therefore I began by multiplying. This formula would work for every 3-step square.
Part two: 4-step stair
The aim of this section is to investigate further the relationship between the stair totals and other step stairs on other number grids.
Here is a 4-step stair number grid:
Key
- 4-step stair 13
- 4-step stair 14
- 4-step stair 15
I have forwarded from part one to extended from part one and have moved onto a 4-step stair rather than a 3-step stair like before.
Calculations
4-step stair 13 – 37+27+17+7+28 +18 +8 +19+9+10 = 180
4-step stair 14 – 97+87+77+67+88+78+68+79+69+70 = 780
4-step stair 15 – 91+81+71+61+82+72+62+73+63+64 = 720
4-step stair 16 – 31+21+11+1+22+12+2+13+3+4 = 120
780 – 720 = 60
180 – 120 = 60
780 – 180 = 600
720 – 120 = 600
60 x 10 = 600
Once again, like doing the first calculation, if you multiply the smallest answer by 10, you end up with the largest number as an answer. This may vary well be a coincidence or it may contribute towards the final formula for the 4-step stair. I am presuming that the number of squares will have something to do with the part two formula so I will try and include it somewhere in the equation. Likewise with the Corner Square as the only difference is an increase in the number of squares. I will also be using 4-step stair 16 as an example.
Calculations: trial and error
4-step stair 13
Stair total = 120
11 = number of squares in the 4-step stair
n = corner number
t = total
11 x n + 10 = 21
Number of squares x Corner Square.
11 x n + 88 = 99
Number from first formula, doubled
(n + 11) x 10 = 120
This formula works for 4-step stair 13 but does it work for the others?
4-step stair 14
Stair total = 180
(n + 11) x 10 = t
(7 + 11) x 10 = 180
This is proof that my formula for part two of the 4-step stair sector is correct. The formula is (n + 11) x 10 = 10. I have discovered another formula for this section, too, 10n + 110 = t and I have tested it three times:
4-step stair 13 – (stair total = 180) (10 x 7) + 110 = 180
4-step stair 14- (stair total = 780) (10 x 67) + 110 = 780
4-step stair 15- (stair total = 720) (10 x 61) + 110 = 720
So the two formulas are 10n + 110 = t and (n + 11) x 10
Part two: 5-step stair
Key
- 5-step stair 17
- 5-step stair 18
- 5-step stair 19
Calculations
5-step stair 17 -46+36+26+16+6+37+27+17+7+28+18+8+19+9+10 = 310
5-step stair 18- 96+86+76+66+56+87+77+67+57+78+68+58+69+59+60=1060
5-step stair 19 – 91+81+71+61+51+82+72+62+52+73+63+53+64+54+55 = 985
5-step stair 20 – 41+31+21+11+1+32+22+12+2+23+13+3+14+4+5 = 235
1060-985= 75
310-235= 75
1060-310= 750
985-235= 750
75 x 10 = 750
As you can see if I multiply the lowest total by ten, I am left with the largest total for an answer. I have also come to notice that the amount of squares in each step stair goes up in triangle numbers:
Part Two- Different number grids
For this particular section, I will be focusing on different number grids and how to find the formula for each times table.
I will begin this section with the 2 times table, like so:
Key
- 3-step stair 21
- 3-step stair 22
- 3-step stair 23
Calculations
3-step stair 21 – 16+18+20+36+38+56= 184
3-step stair 22 – 152+154+156+172+174+192= 1000
3-step stair 23 – 166+168+170+186+188+206= 1084
3-step stair 24 – 2+4+6+22+24+42= 100
1084-1000= 84
184-100=84
1084-184=900
1000-100= 900
Taking away 3-step stair 23 and 3-step stair 22 totals from each other results in the same answer as taking away 3-step stair 21 and 3-step stair 24 from each other. This also shows that the difference from left to right is 84 and the difference from top to bottom is 190. However, whereas before on all other calculations, I am able to multiply the lowest total by ten to reveal the largest total, I can not do this on the two times table number grid. This may be because you are only able to do it on the original number grid… I will find out when doing the three times table.
Trial and error
There are still 6 squares so the corner number, assuming that it will still be this, is multiplied by the corner number (6n):
6 = number of squares in the 3-step stair
n = corner number
t = total
+44 was the original number to be added in the original formula so seen as the number grid has just doubled then I will start of by doubling this number to +88:
6n + 88 = t
As an example, I will use 3-step stair 24:
3-step stair 24
Stair total = 100
(6 x 2) +88 = 100
This proves that my formula works for the two times table but now I must test to see whether it works on the three times table. From working out a formula for the three times table I will be able to work out a formula overall for all times tables.
Key
- 3-step stair 25
- 3-step stair 26
- 3-step stair 27
Calculations
3-step stair 25 – 63+33+36+3+6+9= 150
3-step stair 26 – 84+54+57+24+27+30= 276
3-step stair 27 – 294+264+267+234+237+240= 1536
3-step stair 28 – 273+243+246+213+216+219= 1410
1536-1410= 126
276-150= 126
1536-276= 1260
276-150= 126
126 x 10 = 1260
As you may see, I am able to multiply the smallest total by ten to reveal the largest number. I cannot find any reasonable explanation to this and therefore it is an anomalous result. This may not affect my formula though:
6 = number of squares in the 3-step stair
n = corner number
t = total
As expected, I will be using this formula 6n + 132 = t on 3-step stair 25:
3-step stair 25
Stair total = 150
(6 x 3) + 132 = 150
6 = number of squares in the 3-step stair
n = corner number
t = total
a = times table
This proves that my formula of 6n + 132 works. I am now able to say that if ‘a’ represents the time’s table you are trying to work out then this is my overall formula for the times table number grids:
6n + 44a = t
C:\My Documents\maths cwk\Kelly Duggan\NDO\10Y1.doc Tuesday, 24 June 2003
