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

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.

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.

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.

A little about Pythagoras

Society

Rules

Mathematical Contributions

GeoGebra

A free application software from geogebra.org

THE THEOREM AND GEOGEBRA

Using the software to visualize the result

Proofs

1. The Pythagorean Theorem proved using triangle similarity. (Attached.)

The Converse

Assume that in a triangle with side lengths a, b and c, we have =

Proof of the Converse

We must have a < c and b < c. Next, construct a right triangle with legs, a and b.

Building on their work with the Pythagorean Theorem to find distances, students use the rectangular coordinate system to verify geometric relationships…

G-SRT

8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

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.

The Equation of a Circle

Euclid and the Theorem

Another proof

Pentagon, Decagon and Hexagon

How to make Pythagorean Triplets QUICKLY

Complex Numbers?

Real (a + bi) = a

Imaginary (a + bi) = b

Modulus of a complex

number

Questions