Scaffolding Comprehension Lesson

Scaffolding Comprehension Lesson

Short Description

Cut the knot’s webpage for the Pythagorean Theorem (http://www.cut-the-knot.org/pythagoras/) is a great resource for unique proofs of the Pythagorean Theorem. There are 103 different proofs of the Pythagorean Theorem on this page. Some are purely visual; some are geometric; some are trigonometric; some are algebraic. There is a wide variety of complexity for the different proofs. Depending on one’s background certain proofs will be easier or more difficult to understand.

For example, the first proof is fairly complex. Just look at the picture; it’s full of labels and different lines. Then it uses the similarity and congruency of the triangles and rectangles to prove the Pythagorean Theorem. If one comes with a background of geometry and specifically similar triangles and rectangles, that proof might not be so hard to understand.

The age level would probably be 10^th-11^th grade: when the students are learning geometry and what it means to have a mathematical proof of a theorem. I would have the students explore the webpage to look at the wide variety of mathematical proofs available to them. And have them understand that a simple statement such as the Pythagorean Theorem can require a lot of mathematical work to prove mathematically. Or optimistically, some might see the some of the proofs at simple and elegant.

In order to understand many of the proofs, the students must have learned basic algebra and congruency and similarity of triangles. After understanding at least one of the proofs on cut the knot, as a class, we may be able to move on to possibly more difficult proofs on other theorems or equations in geometry.

 

Text Complexity

The complexity is somewhat high because many of the proofs are not given full explanation but rather are left implicit for the reader to deduce for him or herself. Again, it is hard to look back and forth between the picture and the text to construct the meaning of the text for him or herself. The sheer complexity of some of the proofs make this text a difficult read. One definitely must be an active reader to understand what is being said in these proofs.

 

Guiding Questions

                How do we know a²+b² = c²?

And how do mathematicians establish certainty in general?

 

Lesson Plan

Title:  Proofs of the Pythagorean Theorem

Grade Level:  10-11 Geometry

Time Frame: One 75-Minute Class Period

Big Idea: Students will explore various proofs of the Pythagorean Theorem and attempt to understand and share one of the proofs.

 

Objectives/Outcomes/Expectations: [content,     concepts, science process skills, social skills and       applications that students get out of the activity]	Assessment: [how each of the objectives is        measured and recorded]
That students will understand there is more than one way to prove a theorem. That students will understand at least one proof of the Pythagorean Theorem. That students can explain a proof to another.	If they visit the webpage and see all the different proofs. Talk about the different proofs with the students. If they can share the proof with a partner and explain it to the class.

 

Materials Needed per Student:

1. Computer
2. Notebook and pencil

 

Procedures	Academic Adaptation	Behavioral or Social Adaptation	Assistive Technology
Introduction: Today we will be exploring mathematical proofs. The proofs will be about the Pythagorean Theorem.     Launch: 10 mins Have the student write down and/or visualize how they could prove the Pythagorean Theorem. Have them share with a partner. Then bring everyone together as a class and see if anyone came up with novel ways to prove the Pythagorean Theorem.   Main Activity: 30 mins Have the students use a computer to go online and visit the cut-the-knot’s page of proofs of the Pythagorean Theorem. Have the students explore the page with the guiding question of “how do we know Pythagorean Theorem is true?” After they explored for 15 mins, have them focus on one of the 103 proofs to understand it more in depth for the remaining 15 mins.   Sharing: 15 mins Have the students pair up to share their chosen proofs with one another, explaining the proofs to the best of their ability. Presentations: 15 mins Have the students that want to present and explain to the class their chosen proofs.

 

Background Knowledge: Students should be able understand and use ideas of triangle and rectangle similarity and congruency. Students should have a basic understanding of algebra.

 

Bibliography

Hibbing, Anne. Rankin-Erickson, Joan. A Picture is Worth a Thousand Words: Using Visual Images to Improve Comprehension for Middle School Struggling Readers. The Reading Teacher, Vol 56, No 8 (May 2003), pp 758-770. International Reading Association.

Lattimer, Heather. Reading for Learning: Using Discipline-Based Texts to Build Content Knowledge. National Council of Teachers of English. Urbana, Illinois. 2010. pg 62, 101.