Tricky Triangles

Activity One

This pattern is taken from a section of tapa cloth. The design is based on tessellations of triangles.
Students will need to count the triangles systematically for questions 1–4 and record their counting carefully. A knowledge of symmetry is very useful in solving these problems. For example, students can use the following method to count the number of small brown triangles:

Students can use the total number of small brown triangles to help find the number of small white
triangles. In the central section of the tapa cloth design, there is one more row of small white triangles
than there is of brown triangles, so the total number is: (1 + 2 + 3 + 4 + 5) + (5 + 5) = 25.

In question 3, students may realise that each row contains two more triangles than the previous row: 1 + 3 + 5 + 7 + 9 = 25.
In question 4, students will need to think how many triangles of different sizes there are in the pattern.

Question 5 asks students to generate the next section of the pattern. Students may extend the pattern
down or sideways. Two possible answers are given in the Answers section.
Students can find the next pattern using translation (shifting) and half-turn rotation.

Answers to Activities

Activity One
1. 20
2. 25
3. 25
4. 65
5. a. Answers will vary. They include:
the next section sideways:

i.

or the next section downwards:
ii.

b. Answers will vary, but the two examples
above will give i. 35 or ii. 15 new triangles.
Activity Two
Answers will vary.