Why do this problem?
offers a good chance to develop concepts surrounding triangles, including their areas. It is an engaging activity which allows pupils to work in twos and to have fruitful disussions.
The interactivity will help to arouse learners' curiosity and explore what happens when different sizes of grids are chosen or other shapes are explored. So, this activity can lend itself to pupils posing their own questions in the form of “I wonder what would happen if...?”. (The further note at the foot of this page offers more support on curiosity in the classroom generally.)
As we all learn in different ways it would be good to allow learners to explore this activity with a number of aids at their disposal, for example:
- Nail boards with elastic bands
- Squared paper
Encourage them to start with the original shape and move one point at a time to see the results. They will need to do this in a systematic way if they are going to find all the possibilities and it is likely that the group will benefit from discussing some strategies for doing this as a whole. This might also require you to model the first few steps in a particular way of being systematic,
for example by moving the "top" corner of the triangle one peg up, then two pegs up etc. You can ask the children what to do next at each stage and talk about what might be "sensible" so that you don't miss any out. They may notice a pattern in the way the possible positions for this point lie so that it might not be necessary to actually test every point on the grid. Depending on the class'
experience, you could use a number of ways of checking whether an angle is a right-angle, for example by using a protractor or simply a corner of paper.
When it comes to looking at the areas of sets of triangles (i.e. triangles which have the same base and height), the pupils will have different ways of calculating these areas and these should be acknowledged and shared. Many children will be able to make the full generalisation about areas of triangles with identical base and height through this investigation.
How are you exploring where the elastic band can go?
How will you make sure you don't miss any triangles out?
How are you finding the areas of the triangles?
What do you notice about the triangles' areas?
In the original Transformations on a Pegboard
problem, one of the challenges was about changing a square into a rectangle so that the area doubles by moving two pegs. You could go on to investigate this further with your class, but rather than find a rectangle, look for any shape which can be made by moving two pegs
which has double the area of the square. How many ways can they do this? Sometimes when you've exhausted a lot of ideas you have to change the rules again so here's an idea - keep the same starting square but have a larger grid, so it looks like this:
Can the group find some new shapes now that by moving two pegs will double the original area? (No matter what the final shape is!) What would happen if you could only move one peg to double the area?
Using nail/peg boards or the interactivity on screen will help all children access this problem, but those with poorly developed motor skills may need help from an adult or fellow pupil.
You may be interested in the following talks given by Professor Susan Engels, which focus on encouraging curiosity and are available on YouTube:
The Rise and Fall of Curiosity
- the extract from 23.50 to 37.15 on adult encouragement and teacher behaviour is particularly worth viewing
The Hungry Mind: The Origins of Curiosity
- the extract from 8.22 to 12.29 on children asking questions is especially useful.