"Going Dotty" - Pick's Theorem Investigation: Compound Area, Deriving a General Formula, Sequences

This 1 or 2 lesson investigation is aimed at higher ability classes in KS3 or KS4. The lesson is self-differentiating, as less confident students are given the opportunity to practice conpound area and pattern-spotting, whilst the more confident mathematicians are given the opportunity to derive general formulae for the area of shapes with different properties.

The lesson begins by asking students to find compound area using the method of subtracting smaller areas from the whole.

The investigation progresses to exploring the properties of polygons drawn on dotted paper and deriving rules for the areas of shapes with different numbers of dots on their interior and their perimeter.

Finally, students are asked to generalise a formula for the area of any shape drawn on dotted paper.