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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?
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