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. A valuable source of ideas comes from Nrich, see these activities and Exploring Mathematics on dotty grids.
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 search for further activities returned this Desmos resource, Exploring Triangle Area with Geoboards.