Tri's
Many schools that I have worked in, which is quite a lot, have used nail boards with elastic bands to do some work on shape. Sometimes they have some plastic ones which do the same job and are less painful when you lean on them. Well this challenge is to use an idea that started with using nailboards like these but has changed into using 9 dots arranged in a square to be like a small 3 by 3
nail board.
If you use three lines, (like you may have had three elastic bands for a nail board) see what triangles you can make just using these nine dots. The lines MUST go from dot to dot, (like an elastic band has to go from nail to nail).
The recording of these can get very confused so I suggest that you draw each new triangle on a new set of 9 dots. There are some to print off here . 


One thing which makes this challenge a little different is that if you produce the same shaped triangle but place it in a different position on the 9 dots then it can be counted as being different. 
Notice also that there are some triangles that you can make that do not have their vertices (corners) on the dots but they still obey the rules:
1) Use three straight lines only.
2) Each line must go from a dot to a dot.
3) You can only use these 9 dots arranged as shown.


Well now, what about the smallest one?
Have you used any kind of system to get all the ones that are the same shape but put in different places?
How many of each triangle have you found?
Finally, the usual question for you to ask, "I wonder what would happen if ...?"
Don't forget to send in solutions and ideas and things that you have found out.
Why do this problem?
This problem is ideal for helping pupils get a broader idea about triangles. It also gives the pupils a chance to use their natural curiosity to pose questions for them to solve themselves. Asking the question "I wonder what would happen if I...? is a good way to start, and perseverance will possibly take them to a
new understanding of triangles.
Possible approach
If you are working with young children then I would use nail boards for this challenge, but provide the youngsters with prepared dotted paper with 3 by 3s easily seen (you could print off
this sheet of grids). It is best to use those boards that have the nails as far apart as
possible (the old inch ones are good and it is important to use three bands, one for each line that constructs the triangle). They may need help initially in transferring their ideas from the boards to the paper.
With older children you may be able to dispense with the boards almost straight away and get them using the paper only. This allows the children to work quite quickly.
You could use the
virtual geoboard to share ideas on the interactive whiteboard.
Key questions
Tell me about this triangle you've made.
Is this triangle SMALLER, THE SAME SIZE or BIGGER than this one?
Possible extension
The area of these triangles can be considered by some pupils, but difficulties will probably arise when it comes to the smaller triangles that are "hung" between the dots with no vertex at a dot. Fairly obvious extensions of using 4 by 4 dots and using four lines to produce squares (probably best to start with a 4 by 4 for squares) will captivate some children. Last but not least those
children who like doing things neatly and take care in presentation can produce delightful work that can be displayed in school and attract attention to the work of investigations in school.
For more extension work
Consider a cube with $9$ holes in each face, feed string through a hole and out through another one on another face. What triangles or tetrahedrons can you form inside the cube?
Possible support
Many youngsters will be encouraged when their teacher joins in the activity with them.