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Number Grid

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Introduction

Number Grid This investigation is based on finding rules for differences. The investigation: Start with a number grid 10 long. Multiply the top left hand number by the bottom right hand number. Multiply the top right hand number by the bottom left hand number. Find the difference. (This means take away the smallest value from the largest value) Investigate In the grid below; Top left hand number = 13 Bottom right hand number = 24 Top right hand number = 14 Bottom left hand number = 23 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 . . . . . . . . . . . . . . . . . . . . So, 13 x 24 = 312 14 x 23 = 322 Difference is largest - smallest = 322 - 312 = 10 Try another 2 x 2 square 56 57 66 67 So, 56 x 67 = 3752 57 x 66 = 3762 Difference is largest - smallest = 3762 - 3752 = 10 (same as above) ...read more.

Middle

3 = 9 x 10 = 90 5 x 5 160 = (5-1=4) 4 = 16 x 10 =160 The size of the square subtract 1, square the answer, multiply it by 10 the pattern continues as w x w = (w-1) x 10 The general rule for the difference using squares of size w on a number grid 10 long is (w-1) x 10 or 10(w-1) Proof for a square w x w Starting position Any number width = w n n+1 n+2 n+3 . . n+(w-1) The amount added n+(1x10) . . . . . . onto n is one less height n+(2x10) . . . . . . than the square (w-1) = w n+(3x10) . . . . . . . . . . . . . . . . . . . . n+10(w-1) . . . . . n+10(w-1) +(w-1) Each row goes up 10, there are one less Need to add a further (w-1) lots of 10 than the height w, so we need to add on to the first square in this row (w-1) lots of 10 (same as the first row) Top left hand number = n Top right hand number n+(w-1) = n+w-1 Bottom left hand number n+10(w-1) ...read more.

Conclusion

. . . . than the square (w-1) = h n+(3xL) . . . . . . . . . . . . . . . . . . . . n+L(h-1) . . . . . n+10(h-1) +(w-1) Each row goes up L, there are one less Need to add a further (w-1) lots of L than the height h, so we need to add on to the first square in this row (h-1) lots of L (same as the first row) Now to find the difference for any size square. Top left hand number = n Bottom right hand number = n+L(h-1)+(w-1) = n+Lh-L+w-1 Top right hand number = n+(w-1) = n+w-1 Bottom left hand number = n+L(h-1) = n+Lh-L Top left hand number x Bottom right hand number n(n+Lh-L+w-1) = n + Lhn-Ln+wn-n Top right hand number x Bottom left hand number (n+w-1) x (n+Lh-L) which gives nxn + nxLh +nx(-L)+ wxn+wxLh+wx(-L)+(-1)xn+(-1)xLh+(-1)x(-L) which gives n + Lhn - Ln +nw +Lwh -Lw - n - Lh + L Difference ( n + Lhn - Ln +nw +Lwh -Lw - n - Lh + L ) - (n + Lhn-Ln+wn-n) Which gives Lwh - Lh - Lw + L = L(wh - h - w + 1) = L(w-1)(h-1) The general rule for rectangles on a L long number grid is L(w-1)(h-1) ...read more.

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