1 / 17

Tukutuku

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.

tia
Download Presentation

Tukutuku

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

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. T100 = 100 x 101  2 = 5050 How do you find the 100th triangular number? Generalise: Find a formula for the nth triangular number Tn. Tn = 101 100

  5. 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 Snto the tapatoru numbers. Sn= Tn + Tn-1

  6. 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

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

  8. 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

  9. 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

  10. Patiki via Tapawha Look at the fourth Patiki pattern This shows P4 = S4 + S3 = +

  11. Algebra Skills Find a formula for Pn Pn= Sn+ Sn-1 = n2 + (n-1)2 = n2 + n2 - 2n + 1 = 2n2 - 2n + 1

  12. Patiki via Tapawha again Look at P4 and link to tapatoru numbers P4 = 4T2 + number of crosses in the middle

  13. 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

  14. Patiki via Rotation P4 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. = Rotate 45º

  15. Algebra Skills Find a formula for Pn Pn= n2 + (n – 1)2 = 2n2 - 2n + 1 Again!

  16. Patiki via Both Tapatoru and Tapawha Discuss why P4 = S7 – 4Tn-1

  17. 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

More Related