Making rectangles, making squares
How many differently shaped rectangles can you build using these equilateral and isosceles triangles? Can you make a square?
Age
11 to 14
Challenge level
Problem
Imagine you have any number of equilateral triangles all of the same size as well as a large number of $30$ $^\circ$ , $30$ $^\circ$ , $120$ $^\circ$ isosceles triangles with the shorter sides the same length as the equilateral triangles.
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Using these triangles how many differently shaped rectangles can you build?
Can you make a square?
Getting Started
The $30$ $^\circ$ , $30$ $^\circ$ , $120$ $^\circ$ isosceles triangles are made by bisecting the equilateral triangles and putting the two pieces together the other way - like this:
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Student Solutions
Lu of St Peter's RC Primary School, sent us the following working:
I use two equilateral triangles and two isosceles triangles to make a basic rectangle.
I have $20$ equilateral triangles and $20$ isosceles triangles, so I can make $10$ basic rectangles.
I can make:
1 row of $10$ basic rectangles long side down ($1$ row of $ 9, 8, 7, 6, 5, 4, 3, 2$ basic rectangles as well).
$2$ rows of $5$ basic rectangles ($2$ rows of $4, 3, 2, 1$ basic rectangles as well).
$1$ row of $10$ basic rectangles ($1$ row of $9, 8, 7, 6, 5, 4, 3, 2$ basic rectangles as well).
$2$ rows of $5$ basic rectangles ($2$ rows of $4, 3, 2, 1$ basic rectangles as well).
$3$ rows of $3$ basic rectangles
Totally I have made $31$ rectangles.
Teachers' Resources
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.
You could print these onto coloured card and laminate them.