tukutuku adapted from peter hughes. tukutuku panels are made from crossed weaving patterns. here is...
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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 (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?
100
101
T100 = 100 x 101 2
= 5050
Generalise: Find a formula for the nth triangular number Tn.
Tn = 2
)1( nn
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 SkillsShow 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 PatternsLook 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 TapawhaLook 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 againLook 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
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.
Patiki via Rotation
=Rotate 45º
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