I have always found the above task, suggested in the National Strategies exemplification examples (Geometry & Measures page 10 of pdf) an excellent one when looking at types of triangles and their properties. Students are required to find the number of unique triangles on a 3×3 grid. The task can be extended by asking students how many unique triangles there are on a 4×4 grid (29). A good discussion can be had on being systematic in approach. Owen on TES has provided an excellent (free) resource on quadrilaterals on a 3×3 grid.
Two excellent resources are the Nrich Geoboard (and note environments not requiring Flash) or this one from the Math Learning Centre.
The following slideshow illustrates these two resources and ends with what I think is a great exploration:
- With GeoBoard create triangles on a GeoBoard of any size from 2×2 to 5×5.
- Geoboard paper (square dot paper). Create geometric figures on square dot paper
- ..or on Isometric dot paper
- Areas with Geoboard
- Triangles on the Geoboard find areas
A valuable source of ideas comes from Nrich, see these activities. (Nrich also have a Virtual Geoboard in their generic interactive resources collection (scroll down the page). Also from Nrich, we have Exploring Mathematics on dotty grids.
A search for further activities returned this Desmos resource, Exploring Area with Geoboards; Triangles.