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Area – Scissors style

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Puzzle! 3 RULES: 1. You can not talk, point, nudge, indicate 2. You can’t take pieces from others, you can only give them! 3. You can’t request pieces from others

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What did we learn??

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Why Teach Area? One of the most intuitive ideas in math Nice interplay of algebra and geometry Good scaffold to higher level topics in calculus Human Nature to find Area!!

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Getting to Know Each Other Get into groups of at most 4 Take a minute to introduce yourself, say your name and your favorite geometric shape!

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Area What is it? Rectangle: Convenient Formula: Area =

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Parallelograms Area =

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Triangles Area = Complement

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A Different Proof Decompose

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Trapezoids Area =

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Trapezoid Area Contest Which team can come up with the most to find the area of a trapezoid? Catalog them on your poster paper.

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Finding Area In general to find the area of something, break it into smaller pieces OR Add shapes we know the area of to make shapes we know the area of Decomposition vs Complementing

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Tangrams

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Converse If polygon P can be decomposed into pieces that are rearranged to make Q, then P and Q have the same area. Is the opposite true? If P and Q are polygons of equal area, can it be decomposed into pieces that can be put together to make Q?

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Bolya-Gerwein Theorem The answer is yes! Proved independently by Bolyai and Gerwein in the 1830’s

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Equi Two polygons are “equi” if you can cut one into pieces and rearrange those pieces to get the second polygon If P and Q are polygons that are equi, we say P~Q

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Properties P~P If P~Q then Q~P If P~Q and Q~R then P~R.

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Q P Q R

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Q P R

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What does this look like? Three properties of an equivalence relationship P~P Reflexivity P~Q then Q~PSymmetry P~Q and Q~R then P~RTransitivity Can you give me other examples?

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How do we prove the Bolyai Gerwein Theorem?

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Steps 1.Every Triangle is equi to a Rectangle 2.Parallelograms with a common base and the same height are Equi 3.Two rectangles with the same area are Equi 4.Every polygon can be dissected into triangles 5.Every polygon is Equi to a rectangle 6.Two polygons with the same area are Equi

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Step 1 ?

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Step 2 ?

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Step 3 ?

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Step 4 ?

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Step 5 ?

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Step 6 Finish it Your turn to work. Haiku and Graphical hints on the board Record any ideas you have that seem significant on the large poster paper. Keep track of the proof as a team. ?

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Step 1: Triangles to Rectangles Step 1 is for free Midpoints are all we shall need Please twist and shout now

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Step 2: Parallelograms with a common base and same height Symmetry is neat Tessellate the plane with copies Parallel translates There’s a special case somewhere around here….

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Step 3: Any two rectangles with the same area My head is hurting Parallelogram aspirin Make the sides bases

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Another Way… A B A B

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Step 4: Dissect a polygon into triangles Induct: Base Case: n =3, a triangle, we are done Induction Step: The base case is three N is the number of sides Find a diagonal

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Step 5: Polygons are equi to rectangles Use Step 4 freely Make rectangles with the same base Squish them together

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Step 6: Finish it! So any two polygons with the same area can be made equi to some rectangles. Since these rectangles have the same area, they are equi and we are done!!!! Bolyai and Gerwein Rectangles of the same area We are almost done

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Extensions We can ask, what are the minimum number of cuts necessary? What kind of motions are allowed? – Parallel translation – Central Symmetries In general, you need both, the Hadwiger-Glur result classified what polygons you need just parallel translations for

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Hadwiger-Glur l + c - a b - J l = a-b-c Two polygons can are equi through parallel translations alone if they have the same J l for every line l. Furthermore, the only shapes that are equi to a square by using parallel translations alone are centrally symmetric polygons.

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Classical Dissections Aha! Solutions, Martin Erickson Wikipedia, Henry Dudeney

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Third Dimension Hilbert’s Third Problem Max Dehn showed that the regular tetrahedron and the Cube of the same volume were not Equi in Still an open problem in Non-Euclidean geometries

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References Dissections: plane & fancy by Greg N. Frederickson Aha! Solutions by Martin Erickson Equivalent and Equidecomposable Figures by V.G. Boltyanskii The “Two Basics” mathematics teaching approachand open ended problem solving in China by Zhang, Dianzhou1and Dai, Zaiping

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