[b]1) Overview
[/b]Pick’s Theorem gives the area of a simple lattice polygon (A) in terms of the number of interior lattice points (i) and the number of boundary lattice points (b): [math]A = i + \frac{b}{2} -1[/math]. The result is simple yet powerful, and links to students’ prior knowledge of finding polygonal areas by dissection and/ or compensation. Less able students can appreciate elegance of mathematics by solely recognizing the theorem while more able students can try to understand the idea of proof.
Reference: [url]http://www.cut-the-knot.org/ctk/Pick.shtml[/url]
[b]2) Learning objectives[/b]
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[*]recognize Pick’s Theorem
[*]learn to observe number relationship and think inductively
[*]appreciate the elegance of mathematics
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[b]3) Description of the underlying pedagogical/ teaching approach or rationale[/b]
The applet provides students with autonomy to draw any lattice polygon (maximum number of vertices is 12). Also, the applet automatically counts the interior lattice points and boundary lattice points and highlights them with different colours. This facilitates students to explore the number relationships instead of spending much time on counting.
[b]4) Teacher’s note[/b]
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[*]Lattice points in a plane are points with integer coordinates. A simple lattice polygon means all the vertices of the polygon are lattice points and the sides do not intersect each others.
[*]Teachers may ask the students to draw any simple lattice polygons they like and count the number of interior lattice points (i) and the number of boundary lattice points (b), as shown in the spreadsheet in the lower part of the applet. The applet also gives half of b and measures the area of the polygon. Teachers may ask students to report their own set of numbers and record them on blackboard. Students can then observe the many i, b/2 and A’s to see if there is any relationship. It should not be difficult to observe that [math]A = i + \frac{b}{2} – 1[/math]. Then each student can draw another polygon to verify the result (Pick’s Theorem).
[*]Teachers may let students appreciate the power & elegance of the formula by drawing a lattice triangle with three slanted sides. In the old days, to find the area of this type of triangle, students may need to use a rectangle to surround the triangle, evaluate the area of the rectangle and three compensatory triangles, and finally find the area concerned by subtraction. Now with Pick’s formula, students can find the area with ease. It may be surprising and exciting for students to see that the traditional method and Pick’s formula gives exactly the same answer.
[*]For more able students, teachers may ask them to explain why the formula always works. Teachers can lead students to investigate, in order, rectangles with horizontal/ vertical sides, then right-angled triangles with horizontal/ vertical legs, then triangles with three slanted sides, then any quadrilaterals, and finally any polygons. Primary students need not write the proof elaborately but they can benefit from thinking mathematically.
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