the number stair investigation for Maths GCSE answer for 3 number stairs is x times 6 +44 if you make the bottom left hand corner number x e.g. for the step stair 21 11 12 1 2 3 1 would be x so the answer would be 1 times 6+44 = 50 make a table like this to display your results step stair Total difference 1 (1+2+3+11+12+21) 50 - 2 (2+3+4+12+13+22) 56 6 3 (3+4+5+13+14+23) 62 6 4 68 6 5 2 DO THESE 74 6 THEN WRITE THIS- as the difference between the total is always 6 then it must be x times 6 to work out then say the formula must be 6 x + the difference of (6 x- the total of the step stair) = total to work this out i must make another table step-stair (6 x) 6 x equals 6 x -total of step stair difference 1 times 6 6 50-6 44 2 times 6 12 56-12 44 3 times 6 18 62-18 44 4 times 6 24 68-24 44 5 times 6 30 74-30 44 NOW WRITE as the difference is 44 my final formula must be 6 x+44 = total of 3 step stair now do the same thing 4 the 4 step stair and you are nearly their here is my attempt at nicely presented formula. S = Stair size, e.g. a 3 x 3 stair's = 3 G = Grid size, e.g. a 10 x 10 grid = 10 A = Amount of 'n' N = bottom left hand corner of shape in 2 steps: 1) (Sx (S+1) /2) = A 2) AN+ (G+1) x ( (sxsxs-s) /6) = THE ANSWER SOME EXAMPLE WORDINGS TO USE IN YOUR WRITE-UP... + [ (wy-w+x) +3] + [ (wy-w+x) +w] + [ (wy-w+x) +w+1] + [ (wy-w+x) +w+2] + [ (wy-w+x) +2 w] + [ (wy-w+x) +2 w+1] + [ (wy-w+x) +3 w] This can be simplified to: 10 wy-0 w+10 x+10 From these formulae, when x is substituted with 2 and y is substituted with 2 (2 and 2 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 230 is calculated. 12+13+14+15+22+23+24+32+33+42 = 230 Also from these formulae, when x is substituted with 4 and y is substituted with 6 (4 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), an answer of 650 is calculated 54+55+56+57+64+65+66+74+75+84 = 650 This confirms the formula 10 wy-0 w+10 x+10 can be used to calculate the stair total of a 3-step stair on a 10 by 10 grid. Fig. 1 Number Grid From the formula (10 wy-0 w+10 x+10), when x is substituted with 5 and y is substituted with 1 (5 and 1 being the coordinates of the keystone of the step stair positioned lower on the grid above), an answer of 140 is calculated.

5+6+7+8+13+14+15+21+22+29 = 140 From the formula (10 wy-0 w+10 x+10), when x is substituted with 2 and y is substituted with 6 (2 and 6 being the coordinates of the keystone of the step stair positioned higher on the grid above), ... of squares high the stair is w = Width of grid (number of squares) Relationships between different stair heights on a 10 by 10 grid To find a pattern, I kept 'n' constant (n = 1) and I changed the height, 'h'. If you want challenge, use the's-number as the top left of the stair shape... to make life easier, choose the bottom left hand corner... 2. For the second part of the coursework, all I can say at the moment is that the formula is { [s (s-1) ]/2}n + Xg + X. X is the unknown to me: P's = stair size... as in the width and the height... n ='s-number g = grid size... e.g. grid size of 10 has like 10 numbers in each row... X = THE UNKNOWN IMPORTANT: To find X... use the "changing differences" method... aka the "Gregory Newton Formula". I've searched on the internet for this formula and help on it... don't bother... it's like too complicated with all those wacky symbols.

The one my teacher taught is: (If the value of n = 0) +'s x (1st Difference) + [s (s+1) ]/2 x (2nd Difference) + [s (s-1) (s-2) ]/6 x (3rd Difference) Ok I have almost no clue how to do this... but that's about the point I got up to in my coursework. Maybe this might not help you: P Well my best advice is to ask your teacher for help on "changing differences" or the "Gregory Newton Formula."..'s / he should understand where you are up to. Oh yes... and if you want to know why the first part of the final formula is "s (s-1) ]/2" it's because of TRIANGLE numbers... you " ll see if you do all your working out in a systematic and orderly fashion... well I tried to help... I'm not a teacher so maybe you won't understand what I'm talking about.