# Equal Equilateral Triangles

##### Age 14 to 16 Challenge Level:

#### Why do this problem?

The bisected equilateral triangle (30-60-90) is an important shape
for students to become familiar with, and the early questions in
this problem draw out the compound forms built from this
triangle.

The final question, to which the other questions serve as an
introduction, explores surd forms and the circumstances when one
length can, or cannot, be a rational multiple of another.

#### Possible approach

Preceding these questions with 'playtime' using cut out
triangles to form patterns may be a very useful preliminary for
many students. Similarly practice with the interactivity, creating
patterns, means that students can generate arrangement solutions
quickly and spend their maximum effort on comparison and
consideration of possibility and reasoning around
impossibility.

Yellow & Green triangles to print and cut out :

here
#### Key questions

- While looking at the screen shot on the Hints page : are the
green and yellow hexagons really the same size or just close ? How
can we be sure of that? [this draws out the relationship between
the yellow and green triangles, including that they have equal area
- draw attention to the line of symmetry for each shape to help
students visualise this relationship]

- Still looking at the screen shot on the Hints page : can the
larger hexagon be made from green triangles ? If so make it, or if
not explain why this cannot be done.

#### Possible extension

The second of the key questions above is challenging, able students
should be encouraged to reason this thoroughly.

The following problems make a natural follow on from this activity:

The
Square Hole and

Impossible
Triangles?
#### Possible support

There are a number of activities which can provide valuable
auxiliary experiences for students working on this problem :

- Drawing first the equilateral triangle using only a straight
edge and compasses, and then creating the isosceles triangle,
likewise, will give a strong sense for the symmetry of each
triangle and the relationship between them.
- For students who are not able to make any progress with
grasping the last question, 'Can you make an equilateral triangle
from yellow triangles that is the same size as an equilateral
triangle made from green triangles ?' the earlier questions,
although a lead up to this, nevertheless make a very good activity
in themselves. Some children 'play' a long time with cut out
triangles arranged on a table, motivated by the aesthetic appeal of
emerging pattern possibilities, this is excellent grounding for
other mathematical ideas to be built on.