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Tukutuku Adapted from Peter Hughes

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Tukutuku panels are made from crossed weaving patterns. Here is a sequence of the first four triangular or tapatoru (tapa = side, toru = three) numbers.

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Another set has been rotated 180 degrees and added as shown below. Build these from tapatoru the pieces.

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How do you find the 100th triangular number? 100 101 T 100 = 100 x 101 2 = 5050 Generalise: Find a formula for the nth triangular number T n. T n =

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Tapawha Numbers Let S 4 stand for the 4th square or tapawha (tapa = side, wha = four) number. Create S 4 from tapatoru pieces. S 4 = T 4 + T 3 Generalise: Link S n to the tapatoru numbers. S n = T n + T n-1

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Algebra Skills Show S n = T n +T n-1 by algebra. T n +T n-1 = n(n+1) + n(n-1) 2 2 = n(n+1)+n(n-1) 2 = n(n+1+ n-1) 2 = n 2 +n+n 2 -n 2 = 2n 2 2 = n 2

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Patiki Patterns Look at the fourth Patiki (flounder) pattern. Why is it called the fourth one?

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Write a formula for P 4, the 4th Patiki number, in terms of the tapatoru numbers. P 4 = T 4 + 2T 3 +T 2 Generalise: Find a formula for P n P n = T n + 2T n-1 +T n-2

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Algebra Skills Find a formula for P n P n = T n + 2T n-1 +T n-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 = n 2 + n + 2n 2 - 2n + n 2 - 3n + 2 2 = 4n 2 - 4n + 2 2 = 2n 2 - 2n + 1

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Patiki via Tapawha Look at the fourth Patiki pattern This shows P 4 = S 4 + S 3 =+

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Algebra Skills Find a formula for P n P n = S n + S n-1 = n 2 + (n-1) 2 = n 2 + n 2 - 2n + 1 = 2n 2 - 2n + 1

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Patiki via Tapawha again Look at P 4 and link to tapatoru numbers P 4 = 4T 2 + number of crosses in the middle

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Algebra Skills Find a formula for P n P n = 4T n-2 + 4n-3 = 4 x (n-2)(n-1) + 4n-3 2 = 2(n-2)(n-1) + 4n-3 = 2n 2 - 6n + 4 + 4n - 3 = 2n 2 - 2n + 1

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P 4 is shown below and rotated 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. Patiki via Rotation = Rotate 45º

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Algebra Skills Find a formula for P n P n = n 2 + (n – 1) 2 = 2n 2 - 2n + 1 Again!

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Patiki via Both Tapatoru and Tapawha Discuss why P 4 = S 7 – 4T n-1

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Algebra Skills Find a formula for P n P n = S 2n-1 – 4T n-1 = (2n-1) 2 – 4 x (n-1)n 2 = (2n-1) 2 - 2(n-1)n = 4n 2 - 4n + 1 - 2n 2 – 2n = 2n 2 - 2n + 1

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