Inside Triangles


Considering triangles drawn from corner dot first:






There are no other different ones (where different means could not be placed on top of each other to "match") if you systematically draw all triangles from the side dot and the middle dot. Therefore total of 9.
 
This activity is accessible to all pupils but has the scope to be extended in may directions. In order to make a start, children will need to be familiar with properties of triangles but drawing triangles on the grid will help them to clarify for themselves what they understand by "triangle". If you are not able to use the interactivity, children might find this sheet of dotty grids useful.

You may want to begin this task with the whole class and this way, the notion of "different" will come up quite quickly. How is the group going to define "different"? This is a great discussion point and one where there isn't a right or a wrong answer. You could decide to count triangles which could be picked up and placed exactly on top of another triangle as the same. Or, you could decide that they are different if they are in a different orientation on the grid. The former suggestion makes a more manageable number to count! It might help to suggest working in pairs on this activity so that children are checking they haven't duplicated triangles. The interactivity, if projected or used on an IWB, allows findings to be shared easily.

In order to extend the problem, pupils could be asked to find triangles with three spots inside them or no spots inside ... Differently sized grids could be drawn and compared - for example, you could sort the triangles across differently sized grids eg all right-angled triangles together, or all triangles which are the same shape but different sizes together. At a much higher level, some children could be challenged to explain how they know they have all the triangles each time which could lead into the usefulness of having a system (e.g. drawing all those which have a corner in the corner of the grid first, then all those that have a corner on a "side dot" etc).