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A Structured Class Lesson: To find Pick's Theorem

The teacher gives out pin-boards (or squared paper will do) so that each group of children has access to a board or boards, and some elastic bands.

Teacher: Make me a shape with area 4 squares. I'm going to draw mine on the squared board.

Diagram

Now I'll give you about five minutes to find as many more shapes with area 4 as you can … Keep a record, though, of E and I and we'll collect together our results.

Activity!! Time to chat to a few individuals who might need a bit of help/encouragement.

Back at the front … and we fill in a table with two columns already completed:

Diagram

Teacher: Has anybody found a shape with 0 pins inside? or Can anybody fill in the values of E for these I's?

The table is quickly filled in … any gaps being searched for … comparison of results to lose any inconsistencies.

Diagram

A discussion can develop here about what possibilities are available for I and E.

Sometimes the children will automatically look for patterns in the table … but if nothing is forth coming …

Teacher: Anybody notice anything about these numbers?

Students: A's always 4; The E's go up in 2's; I's in one's!; You always get 10 if you add double the I column to the E column …

Teacher writes … E + 2I = 10.

Now over to them … See if you can find rules for A = 5, 6 … etc … and save your results to the end of the lesson …

More activity … more time to talk and help … If people are interested in the number of possible I's and E's at this stage … fine … it is an interesting problem. It's now about 10 minutes before the end of the lesson…

Diagram

See if you can find rules for:

a) triangular pin-boards
b) hexagonal pin-boards etc …

or … to practise linking 3 variables in a different situation Faces, Vertices and Edges or Nodes, Arcs and Regions to generate Euler's Law: V + F = E + 2 or N + R = A + 2.