Diagonals

Purpose

The purpose of this activity is to engage students in recognising non-linear patterns, in this case a quadratic relationship. Students might use tables, graphs and equations to represent the relationship between sides of a polygon and the number of diagonals for the polygon.

Description of Mathematics

The background knowledge presumed for this task is outlined in the diagram below:

This activity should be used in a ‘free exploration’ way with an expectation that students will justify the solutions that they find.

Activity

Mathematically speaking, a diagonal of a polygon is a line connecting any two opposite vertices (corners).  For example, a pentagon has five diagonals.

A picture containing object, clockDescription generated with high confidence

Note that diagonal AC is the same as diagonal CA.

If a pentagon has five diagonals, how many diagonals do polygons with other numbers of sides have?

Is there a pattern for the number of diagonals that allow you to predict the number of diagonals for a hectogon (100 sides)?

The procedural approach (show more)

  • The student uses patterns in the numbers within a table to find a general rule.

The conceptual approach (show more)

  • The student uses the spatial situation to generalise the number of diagonals emanating from each vertex and extend that idea to generalise the total number of vertices.

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