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.