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Pythagoras + Geometry Habits of Mind with Geogebra
Lorenzo Rodriguez, CSU Fullerton
Armando Martinez-Cruz, CSU Fullerton
CMC Conference, PS
Saturday, 7 November 2015, 10:30AM - NOONSMOKE TREE F
SESSION POLL CODE: 17003
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Objectives
Learn to concretely illustrate the Pythagorean Theorem with Geogebra.
Use its proofs to promote geometric habits of mind.
Share pre-made files to illustrate these ideas in the classroom.
Use technology appropriately.
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CCSS-Mathematics Grade 8
Students UNDERSTAND the statement of the Pythagorean Theorem and its CONVERSE, and can EXPLAIN why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways.
They APPLY the Pythagorean Theorem to find distances between points on the coordinate plane, to find lengths, and to analyze polygons.
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Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
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A little about Pythagoras
Society
Rules
Mathematical Contributions
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GeoGebra
A free application software from geogebra.org
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THE THEOREM AND GEOGEBRA
Using the software to visualize the result
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Proofs
1. The Pythagorean Theorem proved using triangle similarity. (Attached.)
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The Converse
Assume that in a triangle with side lengths a, b and c, we have =
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Proof of the Converse
We must have a < c and b < c. Next, construct a right triangle with legs, a and b.
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Building on their work with the Pythagorean Theorem to find distances, students use the rectangular coordinate system to verify geometric relationships…
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G-SRT
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
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G-GPE Geometry Expressing Geometric Properties with Equations
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
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The Equation of a Circle
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Euclid and the Theorem
Another proof
Pentagon, Decagon and Hexagon
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How to make Pythagorean Triplets QUICKLY
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Complex Numbers?
Real (a + bi) = a
Imaginary (a + bi) = b
Modulus of a complex
number
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Questions