Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Tukutuku Adapted from Peter Hughes Tukutuku panels are made from crossed weaving patterns. Here is a sequence of the first four triangular or tapatoru (tapa = side, toru = three) numbers. Another set has been rotated 180 degrees and added as shown below. Build these from tapatoru the pieces. How do you find the 100th triangular number? T100 = 100 x 101 2 = 5050 101 100 Generalise: Find a formula for the nth triangular number Tn. n( n 1) Tn = 2 Tapawha Numbers Let S4 stand for the 4th square or tapawha (tapa = side, wha = four) number. Create S4 from tapatoru pieces. S4 = T4 + T3 Generalise: Link Sn to the tapatoru numbers. Sn = Tn + Tn-1 Algebra Skills Show Sn = Tn +Tn-1 by algebra. Tn +Tn-1 = n(n+1) + n(n-1) 2 2 = n(n+1)+n(n-1) 2 = n(n+1+ n-1) 2 = n2+n+n2-n 2 = 2n2 2 = n2 Patiki Patterns Look at the fourth Patiki (flounder) pattern. Why is it called the fourth one? Write a formula for P4, the 4th Patiki number, in terms of the tapatoru numbers. P4 = T4 + 2T3 +T2 Generalise: Find a formula for Pn Pn = Tn + 2Tn-1 +Tn-2 Algebra Skills Find a formula for Pn Pn = Tn + 2Tn-1 +Tn-2 = n(n+1) + 2 x n(n-1) + (n-2)(n-1) 2 2 2 = n(n+1) + 2n(n-1) + (n-2)(n-1) 2 = n2 + n + 2n2 - 2n + n2 - 3n + 2 2 = 4n2 - 4n + 2 2 = 2n2 - 2n + 1 Patiki via Tapawha Look at the fourth Patiki pattern = This shows P4 = S4 + S3 + Algebra Skills Find a formula for Pn Pn = Sn + Sn-1 = n2 + (n-1)2 = n2 + n2 - 2n + 1 = 2n2 - 2n + 1 Patiki via Tapawha again Look at P4 and link to tapatoru numbers P4 = 4T2 + number of crosses in the middle Algebra Skills Find a formula for Pn Pn = 4Tn-2 + 4n-3 = 4 x (n-2)(n-1) + 4n-3 2 = 2(n-2)(n-1) + 4n-3 = 2n2 - 6n + 4 + 4n - 3 = 2n2 - 2n + 1 Patiki via Rotation P4 is shown below and rotated = Rotate 45º Rotating helps recognise in the fourth pattern there are 4 diagonal lines of 4 white rectangles, and 3 diagonal lines of 3 darker rectangles. So there are 4 x 4 + 3 x 3 = 25 rectangles altogether. Algebra Skills Find a formula for Pn Pn = n2 + (n – 1)2 = 2n2 - 2n + 1 Again! Patiki via Both Tapatoru and Tapawha Discuss why P4 = S7 – 4Tn-1 Algebra Skills Find a formula for Pn Pn = S2n-1 – 4Tn-1 = (2n-1)2 – 4 x (n-1)n 2 = (2n-1)2 - 2(n-1)n = 4n2 - 4n + 1 - 2n2 – 2n = 2n2 - 2n + 1