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Tukutuku Adapted from Peter Hughes. Tukutuku panels are made from crossed weaving patterns. Here is a sequence of the first four triangular or tapatoru.

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Presentation on theme: "Tukutuku Adapted from Peter Hughes. Tukutuku panels are made from crossed weaving patterns. Here is a sequence of the first four triangular or tapatoru."— Presentation transcript:

1 Tukutuku Adapted from Peter Hughes

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

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

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

5 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

6 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

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

8 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

9 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) = n(n+1) + 2n(n-1) + (n-2)(n-1) 2 = n 2 + n + 2n 2 - 2n + n 2 - 3n = 4n 2 - 4n = 2n 2 - 2n + 1

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

11 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

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

13 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 n - 3 = 2n 2 - 2n + 1

14 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 x 3 = 25 rectangles altogether. Patiki via Rotation = Rotate 45º

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

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

17 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 n 2 – 2n = 2n 2 - 2n + 1


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