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.

Texts for understanding proofs of the Pythagorean Theorem

1.       Pickover, Clifford. The Math Book. Sterling. 2012.

2.       The text describes the history, gives a definition, and some examples of the Pythagorean Theorem.

3.       The text is pretty simple if you have a background in mathematics. There are some nice historical comments. The hardest part to understand is probably the part where Fermat asks for the hypotenuse and the sum of a and b be perfect squares. That is a bit tricky to understand, especially because the answers are such large numbers. The text’s word choice do not include longer words, and there isn’t excessive sentence length.

4.       If someone wants to know the history of the Pythagorean Theorem, this is a great place to start.

5.       If you were mathematician Pierre de Fermat, what question would you pose to other mathematicians?

1.       Sultan, Alan and Artzt, Alice. The Mathematics That Every Secondary School Math Teacher Needs To Know.  New York and London: Routledge. 2011.

2.       Although the very beginning and very end are missing, this text contains all the meat of the proof. It is a beautiful proof, simple and elegant, but if math isn’t your strong suit I could see this being difficult to follow.

3.       The text is filled with math specific terminology. And if you don’t know math vocabulary very well, you won’t understand the text. The sentences aren’t excessively long and are usually to the point. This makes the text pretty dense though. For example, it’s hard to make sense of the definition and formula for the trapezoid. And other parts of the text. This is because the text requires you to look back and forth at the picture, the written text, and the equation. This requires work and if you don’t know what you’re doing it is tough stuff. Although, the text does make it possible to do the work step by step.

4.       If you want learn a cool proof of the Pythagorean Theorem, here one is.

5.       Why is the picture drawn as such for this proof?

 

1.       Berlinghoff, William. Gouvea Fernando. Math through the Ages: A Gentle History for Teachers and Others. Farmington, Maine: Oxton House Publishers. 2002.

2.       This text include various proofs of the Pythagorean Theorem, its history, and its use.

3.       The text isn’t overly complex. In fact, it’s quite simple, you just need to know a little math, such as squaring a length gives you the area of a square composed of the length you squared. On the page I copied, there are 2 proofs without words, so considering the text to be just the written words will miss out on a lot of the information on the page. The proofs without words require a knowledge of geometry, specifically how to find area of different parts of the pictures and then to string it all together so that it is understandable. That’s the main thing with these math texts: you have to look back and forth between the text and the pictures to glean an understanding.

4.       If you want to know purely visual proofs of the Pythagorean Theorem, this is a good place to do it.

5.       Can you think of a different proof of the Pythagorean Theorem other than what’s given?

 

1.       Angie Head. Pythagorean Theorem. 1997. http://jwilson.coe.uga.edu/EMT668/emt668.student.folders/HeadAngela/essay1/Pythagorean.html

2.       This is a really nice short essay on the history of Pythagorean Theorem and various proofs of it.

3.       The complexity isn’t very high, especially for the history part which reads well. The texts of the proofs themselves are difficult because one has to repeatedly look back and forth between the picture and the text, so that they may have a visual representation of the proof as it goes on. There aren’t words of excessive complexity and the sentences are kept short.

4.       This would be a good place to go for a brief history and list of some of the simpler proofs the Pythagorean Theorem.

5.       Are there proofs of PT that you don’t understand but would like to? See if anyone close to you understands the proof you would like to understand.

 

1.       Cut the Knot. http://www.cut-the-knot.org/pythagoras/

2.       This webpage has 103 unique proofs of the PT.

3.       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.

4.       This read would be good for you if you wanted to collect the various proofs of the PT.

5.       Are any of the 103 proofs incorrect?

 

1.       Math is Fun. Pythagorean Theorem Algebra Proof. 2013. http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html

2.       This website is simple and gives a nice, easy proof of the Pythagorean Theorem.

3.       This website is pretty simple to understand and requires little mathematical knowledge to understand the proof because they explain it all thoroughly. The words were not obscure and the sentences were short. Everything is well organized, making it easy to understand overall.

4.       If you just want one simple proof of the Pythagorean Theorem, this is the place to go.

5.       Could this proof be made easier to understand?

Overall, these texts can be used in combination to yield the best communication possible. Using the easy to understand ways of writing along with well-described pictures with necessary arrows of reference would give the reader the best understanding.

Process Analysis

So, I knew I had to do something visual, and one of the first things that came to mind was the Pythagorean Theorem. I had just done a recording of the proof in the Explain Everything app in the Change and Change Strategies class on an iPad. So, I thought I could just use that to be my visualization. The problem was the file wasn't compatible with non-Apple products. So I had to use a different path to get this done.

I searched online for images of the Pythagorean Proof online. There were some examples of the proof on Google Images, but none of them, after coloring them in Microsoft Paint, were able to be uploaded successfully, without blur, to Blogger, so I had to use something else.

I decided to draw my own proof of the Pythagorean Theorem. I used Microsoft Paint to do so. It turned out quite well. :)

When it came to describing the proof with words to accompany the visualization, I made sure to specifically define words that people might have a problem with. After the day in class in which we critically analyzed Maram's visualization, I realized that I too had to more specifically define and walk people through the visualization. People might not understand that A^2 in the visual proof stands for the area of the square next to side A, which in the equation is just A multiplied by itself. So I had to draw that out further. People might not know what the legs and the hypotenuse refer to, so I had to define those. I even described the proof itself in words in case the visualization was insufficient.

Generally, I explained everything specifically and walked through all the knowledge step by step.

Visualization of the Pythagorean Theorem

 

Many remember the Pythagorean Theorem that they learned in middle school. It goes A^2 + B^2 = C^2. It is a theorem about right triangles, triangles with one angle equal to 90 degrees. There are 3 sides to a right triangle: the 2 legs, adjacent to the right angle, and 1 hypotenuse, the side opposite the right angle and also the longest side of a right triangle. In the equation above, A and B are the lengths of the legs of the right triangle, and C is the length of the hypotenuse. So, what A^2 + B^2 = C^2 represents is A multiplied by itself plus B multiplied by itself is equivalent to C multiplied by itself. This can be represented visually as the area of squares because the area of a square is the length of one side length multiplied by itself.

 

In the figure below, there are two equally sized big squares, each containing 4 congruent (the same) yellow triangles. In the figure on the left, there is a red square in the middle. The side length of the square is C, the hypotenuse, and so the area of the square is C^2 or the length of the hypotenuse multiplied by itself.

 

In the figure on the right, there is a blue square and a green square. The blue square's area is B^2 because B is its side length and also the length of one of the longer leg of the right yellow triangle. The green square's area is A^2 because A is its side length and is also the length of the shorter leg of the right yellow triangles.

 

To say that A^2 + B^2 = C^2, after realizing that both big squares are equal in area, we can remove the 4 right triangles from both sides and to find that we are left with equivalent areas in both figures. On one side is the red square, C^2, and on the other side are the blue and the green squares, A^2 and B^2. The area of this side is then A^2 + B^2. And because the areas of both sides are equivalent C^2 = A^2 + B^2.

 



Where I am and where I want to go

So, I used to believe in biological evolution. It was the only thing that made sense with the evidence. From this belief, in college, I came up with this idea of where we were headed in the future. If you look at the story of evolution, it says we started out as molecules, which grouped together in cells, which grouped together in multi-cellular organisms, which (in some species: ants, bees, and termites) grouped together in hives. So you can see this repeated grouping through evolutionary history. Humans are something in between individuals and groups. The next evolutionary step for humans seemed obvious to me. Humans had to move to the next highest level of evolution groups or tribes or hives, whichever term you want to use.

There are examples of this sort of thing happening in science fiction. There is the Borg in Star Trek. There is the tree of life in Avatar. There is the Zerg in StarCraft. And I'm sure there are plenty of other examples.

Some might say we are already there in some aspects. We are connected to each other through the internet. We live our days working in companies everyone working together in the giant buildings like honeycomb or termite hives. And then we have a family unit in our house. We divide ourselves into groups with group labels based on politics, sports, and entertainment.

The sort of group I envisioned in my college days were people connected together through the internet and connected to the internet via brain-computer interfaces, effectively being brain-to-brain interfaces through the computer. Since biologically, the brain is the seat of consciousness, people would be able to communicate thoughts directly to one another, bypassing all the middle men of our body, no need for eyes, ears, nose, tongue, skin. All these senses can be reproduced electrically by stimulating the brain in the correct way. We would probably spend our time simulating various entertainment environments sharing or co-experiencing the fun.

Another implication of brain-to-brain interfaces would be a huge reduction in selfishness and greatly increased empathy. The idea of one individual being separate from another would go by the way-side and we would have a communal sense of self. The group-self composed of the merging of minds into one super-being.

So, here I am right now. And the previous paragraphs are about where I want to go. It is obviously a ways off, and it may never happen especially if I become a math teacher. But it was my dream, and I still like the idea connecting and bringing people together.

This has implications for education. If people ever develop brain-computer interfaces, we may be able to download the content and ways of thinking into the students' brain. If such a thing ever happens, we would be able to greatly shorten the length of education needed for students, and they can move on to more important things like enjoying life.

Introduction

This blog will be about my history, thoughts, and interests. To give some background on who I am I'll tell you a summarized version of my history.

When I was very young, I came to believe that the ultimate trait a person could have was intelligence. This is because, I thought, if you are intelligent, you can learn to become whatever else you want to be. So I saw intelligence as a sort of meta-trait. During school, I developed an ego and was ego-driven to perform well in school.

I really tried hard in school to get A's, partly for scholarships, partly for my ego because I wanted to be the smartest kid there was.

I liked math and science because it was not ambiguous. There was a right answer to the problems. I was good at it, and I enjoyed learning and being the best at it.

I went on to college with a free ride, due my working hard in high school. I wanted to work with stem cells because in high school, I had gotten brain damage from either concussions from football or a nasty flu I caught my sophomore year. I wanted to fix my brain and recover the clarity of thought I once had.

As time went on, my thinking ability recovered, and my interests moved towards other topics. I became interested in brain-computer interfaces. I thought it would be really cool if we could have direct control over things in the real world by simply thinking. I also wanted the ability to directly conjure up experiences by directly stimulating the brain and share these experiences with other people with brain-to-brain interfaces.

I went on to graduate school with the intent of working with brain-computer interfaces. I didn't get very far because I had to leave due to health problems.

I went back to grad school, but it still didn't work out. Part of me would have enjoyed being a professor and sharing my thoughts with the students in the lecture hall. So I thought that the next best thing would be a teacher at the high school level. I would still get to teach material to the students.

And that brings me to now. I am in a teacher certification program for math, writing a blog as part of a class assignment.

I hope the following posts peak your interests.