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