Pythagorean Play, Perception, Puzzles, and...

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Pythagorean Play, Perception, Puzzles, and Proofs

AMTNJ Convention October 26, 2019

Mark Schlawin

• Retired mathematics teacher at Princeton Charter School

• Volunteer mathematics and science tutor at Isles Youth Institute, Trenton NJ

• Contracted provider of program analysis and workshops for mathematics

teachers


Mark Schlawin [email protected]

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Introduction 3 Assessment strategies 4 A simple but hard problem 5 An extension of simple but hard problem 6 A paper-folding proof 8 All the squares 14 Resources 19

Books Internet Geometric building sets



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Introduction Participants will learn to help students discover and apply the Pythagorean theorem by solving challenging puzzles and problems using geometric pin boards, paper folding, and dot paper. The methods shown here have been used successfully with a wide range of student ages and abilities, ranging from 5th grade through accelerated algebra and geometry. Hands-on activities reinforce area, distance, and perimeter concepts, improve geometric perception and reasoning, and lead to discovery and application of the Pythagorean theorem. This material works well as small-group activities, often leading to well-focused (and sometimes positively competitive) group work.



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Assessment Strategies The following can be used to assess and improve the students' grasp of these activities, in ascending order of sophistication: • Describe the objectives of the activity in mathematical terms • Perform the construction. • Show another student how to do it, evaluate by other student’s success. • Write a set of instructions that can be understood by a fellow student to

carry out the construction or activity. • Write an informal proof with the construction or a diagram of the

construction to explain why it shows what it does. The written tasks are especially good preparation for the GEPA—no, the ASK; no, the PARCC, no the NJ SLA. Yes, a first version of some of this goes way back!



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A Pythagorean based line length puzzle Fold a piece of paper in half the long way, and open back up. Bring the bottom right corner to the center-line, continuing the crease exactly to the left hand corner. Make a dot where the bottom right corner touches the center line, then open the paper again. If the bottom of the paper is two units wide, how far from the bottom is the dot?

Fold right corner to Middle crease and run resulting crease exactly to left corner.



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[Since the former bottom edge with a length of 2 units became the hypotenuse of a right triangle with a horizontal leg 1 unit in length, the other leg, and thus the height of the dot, is √3.] Folding some more irrational line lengths

First, fold a sheet of paper lengthwise twice ¾ (a double “hot dog” fold).

1) If the rectangle’s width is 4 units, how can a line √7 units in length be formed? √5? 2) With four vertical divisions, how many different heights can be obtained by this method of folding, and what are they? Hints: The arrows have not been placed randomly! The Pythagorean theorem is key!



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If we have four equal vertical divisions, we can pair hypotenuses of 4, 3, and 2 with bases as follows 4,3 4& −3& = 7, so height is √7 4,2 4& −2& = 12, so height is 2√3 4,3 4& −1& = 15, so height is √15 4,3 3& −2& = 5, so height is √5 4,3 3& −1& = 8, so height is 2√2 4,3 2& −1& = 3, so height is √3 Obviously, additional folds could produce other lengths.



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A paper-folding proof. Don’t tell your students what they are doing until the end! Find a partner. Each of you needs a sheet of square paper. Origami paper looks sharper, but patty paper is cheap and shows creases as nice white lines. Pictures of the steps will follow on the next page. Left-hand partner only: Fold #1: Fold a corner-to-corner diagonal on a square piece of paper. (Turn it so the diagonal goes from the lower left to the upper right so we all look the same.) Mark a dot on the diagonal, maybe about two-thirds of the way up. Call this Dot 1 Fold #2: Fold the lower left-hand corner onto your dot Mark new dots where this new fold strikes the left edge (Dot 2) and bottom edge (Dot 3) of your paper. Make a new fold (#3) through Dots 1 and 2. This fold will be parallel to the top and bottom of your paper. Make a new fold (#4) through Dots 1 and 3. This will be parallel to the sides of your paper. Fold corner-to-corner diagonals (Folds #5 and #6) in each of the rectangles.



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You will have created a large square, a small square and two identical rectangles divided into four identical triangles. Turn your paper and admire your work.



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Voila!

And now for the right-hand partner--



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Use the length of the big square on the first sheet to measure and make dots on all four sides of the second sheet. Make sure the longer segments and shorter segment are in the same order on each side.

Then make folds connecting the dots on adjacent sides. You will have created four triangles congruent to each other and congruent to the triangles that divided the rectangles in the first square. (Why are they congruent? Why is the central figure a square?)



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So now you have a square surrounded by four triangles. Compare your two sheets. What have you proven and how?

The area of the outer squares is the same. The area of the four triangles on each sheet is the same. So, the area of the two smaller squares on the first sheet must equal the area of the square on the second sheet. In other words, the area of the squares constructed on the two legs of a right triangle equals the area of the square constructed on the hypotenuse. The Pythagorean Theorem!

Slay an Ox!

Note that we never had to use any algebra. Pythagoras might have used similar geometric arguments, since there was no algebra then.



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This proof can also be done by drawing on a square and by placing rubber bands on a Geoboard (or virtual Geoboard?). Extensions of this might include having one student dictate instructions (perhaps without looking!) while another carries out the activity. Also, have the students justify all the congruencies.



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All the Squares—A Group Challenge On your Geoboard, connect the pegs to create all the unique size squares with an integral area that is less than or equal to 64. I mean, that the area is an integer, nothing fancy like calculus. In today’s presentation we will draw squares on dot paper rather than play with rubber bands. Use a pencil. You don’t need to make every possible square of each size, just show one instance of each size of square. Read the directions carefully, then begin. First group to get them all wins. (Meaningless prizes, extra-credit, bragging rights, etc.)



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Okay, you have a good start

But you are obeying a rule that doesn’t exist! There are many more! Back to work!



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What about these, for example?

What was the rule you obeyed that doesn’t exist? There are total of 29 unique squares with integer area! How many can you find? Get your inspiration by drawing some, but then see if you can find a systematic algebraic way of finding them.



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Here they are!

a b c2 count 1 0 1 1 1 1 2 2 2 0 4 3 1 2 5 4 2 2 8 5 3 0 9 6 1 3 10 7 2 3 13 8 4 0 16 9 1 4 17 10 3 3 18 11 2 4 20 12 5 0 25 13 1 5 26 14 2 5 29 15 4 4 32 16 3 5 34 17 6 0 36 18 1 6 37 19 2 6 40 20 4 5 41 21 3 6 45 22 7 0 49 23 1 7 50 24 4 6 52 25 2 7 53 26 3 7 58 27 5 6 61 28 8 0 64 29



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Thanks for coming and participating. This presentation should be up on the AMTNJ conference website, and feel free to email me any questions.



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Books Paper Folding for the Mathematics Class Donovan A. Johnson NCTM Publications

Geometric concepts illustrated or proven by paper folding. Geometric Exercises in Paper Folding T. Sundara Row Dover Publications

Out of print classic, but available from used booksellers. Mathematical Origami David Mitchell Tarquin Publications

Great introduction to modular origami geometric constructions, Origami for the Connoisseur Kunihiko Kasahara & Toshie Takahama Japan Publications

Advanced origami, including many geometric constructions both modular and single-sheet. Includes Haga Theorems.

Dot Paper Geometry Charles Lund Cuisenaire Company of America

Dot paper basics. Pythagoras's Bow Tie Ron Ritchhart Cuisenaire Company of America

Pre-algebra activies with Geoboards Patty Paper Geometry Michael Serra Key Curriculum Press

The famous geometry man himself The Geometry of Wholemovement Bradford Hansen-Smith Wholemovement Publications

Idiosyncratic folding of geometric figures from paper plates, including philosophy concerning mathematical primacy of circle.

Paper Polygons Burnett, Irons, and Turton Origo Publications How to Enrich Geometry Using String Designs NCTM Victoria Pohl



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Paper Folding Resources on the Internet

There are literally millions (23 million and counting) of origami websites, many devoted to mathematical topics. This is just a sample.

http://en.wikipedia.org/wiki/Huzita's_axioms A paper folding Euclid; develops whole theory of paper folding mathematics axiomatically. https://mathigon.org/origami https://en.wikipedia.org/wiki/Mathematics_of_paper_folding http://www.origami.gr.jp/index0.html Home page of Japan Origami Society; many resources

https://www.pinterest.com/pin/511791945149377641/?nic=1a Kirigami basics

http://www.lwcd.com/paper-folding/geometry.html Paper folding geometry https://www.instructables.com/id/Teaching-Math-Through-Paper-Folding/

Geoboard activities https://mathgeekmama.com/geoboard-activity-cards/

Physical Building Sets

Zome Geofix Fractiles Straws and Connectors Zoobs

Pythagorean Play, Perception, Puzzles, and...

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