### Why do this problem?

This problem offers the opportunity to assess learner's
knowledge, to pose problems, to share ideas and follow different
routes. At one level, it is possible to discuss rational and
irrational numbers as well as utilise Pythagoras' theorem. At
others you might explore symmetry or triangle animals. It links to
a number of other problems on the site including

Triangle Relations and

Equal Equilateral Triangles .

### Possible approach

Just hand out the triangles without any indication of how they
are related or formed. Working in small groups, ask the learners to
"play" with the triangles for a few minutes and write on large
sheets of paper (to share with the rest of the group) what they
consider to be four key mathematical properties of the
triangles.

Invite the class to walk around the room and look at what
other groups have written, then invite them to add anything they
feel is important to their own lists. After this encourage
discussion of the key points and salient features. It is at this
point the relationships between the two triangles can be
established - including their equal areas.

After this you might choose to select a feature that has been
mentioned such as:

"The two
triangles can be put together to form a right-angled
triangle."

to lead into the main problem questions.

Alternatively - why not give the groups more time to come up
with problems of their own first? In my experience one of these is
invariably about rectangles or squares - so you can end up with a
problem that the group has posed themselves, yet meets your
learning objectives (if they require this focus). I have often
allowed groups to choose a problem from the class as a whole to
work on. They can tackle the task as well as discuss how the
problem as posed can be improved.

Different routes offer great opportunities for display and
sharing.

### Key questions

- What properties do the triangles have on their own, or when
joined together?
- Can you write down four things that you think are most
mathematically significant about these shapes?
- Can you pose a question for someone else to answer that
involves these shapes?

### Possible extension

By offering the group opportunities to pose their own problems
it is possible to identify suitable challenges for the most
able.

### Possible support

As with the extension opportunities, this is an ideal problem
for taking those whose background knowledge is less well developed
from a more suitable starting point, for example:

"What equilateral triangles can you make? or

"What rectangles can you make and what are the
smallest/largest number of each triangle that is required in each
case?"

Then of course there are all the problems based on symmetry. (For
example, how many different symmetrical shapes can you make with
just four triangles?)

#### Resources

A grid of equilateral triangles can be downloaded

here
.

A grid of isosceles triangles can be downloaded

here.

I have printed these onto coloured card before laminating and
cutting them out for the class to share.