Area – Scissors style

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

What did we learn??

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!!

Area

• What is it?• Rectangle:

• Convenient Formula: Area =

Parallelograms

Area =

Triangles

• Area =

• Complement

A Different Proof

• Decompose

Trapezoids

• Area = b1

b2

b1

b2

Trapezoid Area Contest

• Which team can come up with the most to find the area of a trapezoid?

• Catalog them on your poster paper.

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

Tangrams

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?

Bolya-Gerwein Theorem

• The answer is yes!• Proved independently by Bolyai and Gerwein

in the 1830’s

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

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

QP

Q

R

Q

P

R

What does this look like?

• Three properties of an equivalence relationship

• P~P Reflexivity• P~Q then Q~P Symmetry• P~Q and Q~R then P~R Transitivity• Can you give me other examples?

How do we prove the Bolyai Gerwein Theorem?

Steps

1. Every Triangle is equi to a Rectangle2. Parallelograms with a common base and the

same height are Equi3. Two rectangles with the same area are Equi4. Every polygon can be dissected into triangles5. Every polygon is Equi to a rectangle6. Two polygons with the same area are Equi

Step 1

?

Step 2

?

Step 3

?

Step 4

?

Step 5

?

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.

?

Step 1: Triangles to Rectangles

• Step 1 is for freeMidpoints are all we shall needPlease twist and shout now

Step 2: Parallelograms with a common base and same height

• Symmetry is neatTessellate the plane with copiesParallel translates

There’s a special case somewhere around here….

Step 3: Any two rectangles with the same area

• My head is hurtingParallelogram aspirinMake the sides bases

J HC

A D

B E

FG

I

Another Way… A

B

A

B

Step 4: Dissect a polygon into triangles

• Induct: Base Case: n =3, a triangle, we are doneInduction Step:

The base case is threeN is the number of sidesFind a diagonal

Step 5: Polygons are equi to rectangles

Use Step 4 freelyMake rectangles with the same

baseSquish them together

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 GerweinRectangles of the same areaWe are almost done

Extensions

• We can ask, what are the minimum number of cuts necessary?

• What kind of motions are allowed?– Parallel translation– Central SymmetriesIn general, you need both, the Hadwiger-Glur result

classified what polygons you need just parallel translations for

Hadwiger-Glur

l +

c

-

a

b

-

Jl = a-b-c

Two polygons can are equi through parallel translations alone if they have the same Jl for every line l. Furthermore, the only shapes that are equi to a square by using parallel translations alone are centrally symmetric polygons.

Classical Dissections

Aha! Solutions, Martin EricksonWikipedia, Henry Dudeney

Third Dimension

• Hilbert’s Third Problem• Max Dehn showed that the

regular tetrahedron and the Cube of the same volume were not Equi in 1902.

• Still an open problem in Non-Euclidean geometries

References• Dissections: plane & fancy by Greg N. Frederickson• Aha! Solutions by Martin Erickson• Equivalent and Equidecomposable Figures by V.G. Boltyanskii• http://mathworld.wolfram.com/Dissection.html• http://www.cut-the-knot.org/Curriculum/Geometry/CarpetWithHole.shtml• The “Two Basics” mathematics teaching approachand open ended problem solving in China by Zhang, Dianzhou1and Dai,

Zaiping