Students will
- Discover Greek achievements in mathematics.
- Understand deductive reasoning.
- Demonstrate mathematical proofs of selected theorems.
- Understand Euclid's proof of the Pythagorean Theorem.
- Have students research different proofs of the Pythagorean Theorem and create a poster demonstrating one such proof using print and Web resources. The following Web sites are a good starting point:Have the students create a bulletin board from their posters.
- Have students research a Greek mathematician using print and Web resources. The following Web sites are a good starting point:
- When students have completed their research, ask them to summarize their findings in a one-page report.
- Have students choose a partner. Ask them to share their reports with their partners and answer any questions. Then have each student summarize his or her partner's report for the class, including at least three interesting facts.
- Discuss indirect measurement with the students. On a sunny day, give students yardsticks. Have them measure their heights and shadows and record the results. Then, ask them to measure the shadows of several tall objects, such as trees or flagpoles, and record the results. Return to the classroom and demonstrate how to set up a proportion to find the height of a tall object from the length of its shadow. Then have the students use the shadow lengths of the tall objects they found outside to find the heights of those objects.
- Have students keep a notebook to record the postulates, theorems, and proofs of selected theorems shown in the video. The following are postulates of Thales:
- A circle is bisected by any diameter.
- The base angles of an isosceles triangle are equal.
- Have students help demonstrate the proof of this theorem as seen on the video.
- The angles between two intersecting straight lines are equal.
- Two triangles are congruent if they have two angles and the included side equal.
- An angle in a semicircle is a right angle.
Note: ThisWeb siteis a good source on the life and accomplishments of Thales. - Ask students to demonstrate a proof of the theorem stating that the sum of the interior angles of a triangle is equal to 180?.
- Ask students to demonstrate a proof of Euclid's Proposition 41: If a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle.
- Ask students to demonstrate a proof of the Pythagorean Theorem.
Note: ThisWeb siteshows Euclid's proof of the Pythagorean Theorem.
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Use the following three-point rubric to evaluate students' work during this lesson.
- Three points: Students were highly engaged in class discussions; produced complete reports, including all of the requested information; clearly demonstrated the ability to measure indirectly, and showed a complete understanding of using deductive reasoning in mathematical proofs.
- Two points: Students participated in class discussions; produced an adequate report, including most of the requested information; satisfactorily demonstrated the ability to measure indirectly, and showed a satisfactory understanding of using deductive reasoning in mathematical proofs.
- One point: Students participated minimally in class discussions; created an incomplete report with little or none of the requested information; were not able to measure indirectly, or adequately use deductive reasoning in mathematical proofs.
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axiomDefinition: A statement accepted as true without proof
Context: The Greeks based the study of geometry on definitions and axioms.
deductive reasoning
Definition: The process of reasoning logically from given statements to a conclusion
Context: The Pythagorean Theorem was proved using deductive reasoning.
indirect measurement
Definition: A method of measuring distances by solving a proportion
Context: Thales used indirect measurement to determine the height of a pyramid.
logic
Definition: The formal principles of reasoning
Context: Each statement in a proof is logically justified by a definition, postulate, or an earlier proposition that has already been proven.
mathematical proof
Definition: A demonstration that a mathematical proposition is true based on axioms, definitions, and proven theorems
Context: Euclid's proof of the Pythagorean Theorem made use of the previous proven theorem known as Proposition 41.
theorem
Definition: A proposition that has been or is to be proved on the basis of certain assumptions
Context: In Book 1 of Elements, Euclid's proposition 41 is the theorem "if a parallelogram has the same base with a triangle and is in the same parallels, then the parallelogram is double the triangle."
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National Council of Teachers of Mathematics (NCTM)
The National Council of Teachers of Mathematics provides guidelines for teaching mathematics in grades K-12 to promote mathematical literacy. To view the standards, visit this Web site: http://www.nctm.org/standards/content.aspx?id=16909
This lesson plan addresses the following national standards:
- Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
- Use visualization, spatial reasoning, and geometric modeling to solve problems
Mid-continent Research for Education and Learning (McREL)
McREL's Content Knowledge: A Compendium of Standards and Benchmarks for K-12 Education addresses 14 content areas. To view the standards and benchmarks, visithttp://www.mcrel.org/compendium/browse.asp.
This lesson plan addresses the following national standards:
- Mathematics: Understands and applies basic and advanced properties of the concepts of geometry; Use the Pythagorean theorem and its converse and properties of special right triangles to solve mathematical and real-world problems; Understands the basic concepts of right triangle trigonometry (e.g., basic trigonometric ratios such as sine, cosine, and tangent); Uses trigonometric ratio methods to solve mathematical and real-world problems (e.g., determination of the angle of depression between two markers on a contour map with different elevations); Uses properties of and relationships among figures to solve mathematical and real-world problems (e.g., uses the property that the sum of the angles in a quadrilateral is equal to 360 degrees to square up the frame for a building; uses understanding of arc, chord, tangents, and properties of circles to determine the radius given a circular edge of a circle without the center)
- Science: Physical Science: Understands the structure and properties of matter; Understands the sources and properties of energy
- World History: Understands Greek achievements in mathematics
- Historical Understanding: Understands the impact and achievements of the Hellenistic period (e.g., major lasting achievements of Hellenistic art, mathematics, science, philosophy, and political thought; the impact of Hellenism on Indian art; how architecture in West Asia after the conquests of Alexander reflected Greek and Macedonian influence)
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