This lesson has students investigate area using three area axioms. Students are lead through the investigation by key questions. Additional guidance may be necessary for students to comprehend the ideas explained in the overview.
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Topic: Axioms of Area
The Student Will investigate and apply the foundational ideas of area.
Prior Knowledge:.axiomatic method
Motivational Problem: Draw a strange, asymmetric diagram (perhaps a map of the United States). Ask "How would we go about measuring the area of this region?"
The primary idea to elicit from the motivational problem is that we would probably look to break down the unusual figure into a number of small, familiar figures whose areas are easy to find. Regions like squares, triangle, rectangles, etc. But if we didn't know the formulas for the areas of these figures, what then? We must postulate the area of one, and hope the others will follow. Which area seems most foundational?
The three postulates of area are:
The area of a square is side squared.
If two figures are congruent, then they have the same area (Area Congruence Postulate)
The area of a region is the sum of the areas of its non-overlapping parts (Area Addition Postulate)
Assume these postulates, you can find the area of a rectangle by extending each side by the other lenght, thereby creating a square--the square can be decomposed into two squares and two congruent rectangles, and by the AAP and some algebra, you can prove that A=bh. Key Questions: What is area? Why is the area of a square side squared? Why is the area addition addition postulate true? What postulate is AAP reminiscent of?
Looking Ahead: From these posulates, we will derive all the area formulas.