1. Van Hiele Levels of understanding shapes in geometry

2. Van Hiele levels Visualization/Recognition Description/Analysis Informed Deduction Formal Logic Rigor/Axiomatic

3. Van Hiele Properties Levels are hierarchical--can’t skip a level Levels are not age dependent Experience with geometry has greatest influence Instruction and language higher than the level the student is on could inhibit their learning teachers need to understand the language and properties of each level

4. REcognition/visualization Lack attention to parts/attributes of shapes Recognize differences between shapes/can compare Learn names of shapes Activities: tangrams find hidden figures examples vs. non-examples manipulate physical models

5. Description/analysis Don’t see relationship between properties Properties understood independent of each other i.e. “a square is not a rectangle” Shapes have properties Activities: geoboards fold, measure, cut, look for symmetry, predict shape change properties and observe, classify

6. Informed deduction Properties are related and logically ordered Follow logical arguments Relationships between figures Activities: express relationships verbally open ended tasks with shapes is converse valid? use deductive language: all, some, none, if-then, what if

7. Formal logic Not typically reached until high school or college Construct deductive arguments Establish interrelationships among theorems Activities: drawings and constructions proofs

8. Axiomatic/rigor College level Highly abstract Compare deductive systems Explore geometries based on postulates Rigorous indirect proof and proof by contrapositive

9. A problem Two brothers discover a quadrilateral shaped island. How can they divide the land fairly between them? Extension: What if they each wanted the same amount of coastline?

10. Patty paper geometry Draw a line segment AB. Find its midpoint by folding only. Make a line a parallel to AB. Make a line b perpendicular to AB.

11. Patty Paper GEometry Angle Bisector given angle extension: draw a point on line, what do you know? Perpendicular Bisector of a line segment given segment extension: draw point on bisector line, what do you know? Perpendicular to line through a point given line and point in space extension: perpendicular through point on line - Parallel to line through a point given line and point in space extension: create parallelogram

12. Patty paper geometry ASA: angle--side--angle SAS: side--angle--side AAA: angle--angle--angle SSS: side--side--side Draw the appropriate pieces of the triangle on your patty paper. For sides, draw two dots at the endpoints so you know those are fixed. For angles draw a dot at the vertex of the angle but draw the sides of the angle at a random length. Cut the paper so you can maniplate each piece for form a triangle. Can you make more than one triangle? What is your conjecture?

13. Proof with alice "Then you should say what you mean," the March Hare went on. "I do," Alice replied; "at least--at least I mean what I say--that's the same thing you know." "Not the same thing a bit!" said the Hatter. "Why, you might just as well say that 'I see what I eat" is the same thing as 'I eat what I see'!" "You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!" "You might just as well say," added the Dormouse, "that 'I breathe when I sleep' is the same thing as 'I sleep when I breathe'!"

14. Contrapositive converse & Inverse

15. other resources for proof If you give a moose a muffin Ad for "A Fish Died" Sherlock Holmes: if-then statements Computer Programming (http://beta.appinventor.mit.edu)http://beta.appinventor.mit.edu Logic Puzzles/LSAT (http://www.logic-puzzles.org/)http://www.logic-puzzles.org/ Rube Goldberg devices (http://www.rubegoldberg.com/)http://www.rubegoldberg.com/

16. Rube Goldberg

17. House project