1. Unfolding Mathematics with Unit Origami OETC integratED February 27, 2014 Joseph Georgeson UWM georgeso@uwm.edu

2. Assumptions: mathematics is the search for patterns- patterns come from problems- therefore, mathematics is problem solving. Problems don’t just come from a textbook. Knowing and describing change is important. Math should be fun.

3. Example-The Ripple Effect How do the number of connections change as the number of people grows?

4. Two people. One connection. Three people. Three connections. Four people. ? connections.

5. peopleconnections 21 33 46 510 615 721

6. Graph

7. Verbal explanation: In a group of 10 people, each of the 10 would be connected to 9 others. But, those connections are all counted twice. Therefore the number of connections for 10 people is

9. This cube was made from 6 squares of paper that were 8 inches on each side.

10. The volume or size of each cube changes as the size of the square that was folded changes. Here are some other cubes, using the same unit, but starting with square paper of other sizes.

11. First, we are going to build a cube. This process is called multidimensional transformation because we transform square paper into a three dimensional cube. Another more common name is UNIT ORIGAMI

12. two very useful books-highly recommended. Unit Origami, Tomoko Fuse Unfolding Mathematics with Unit Origami, Key Curriculum Press

13. Start with a square.

14. Fold it in half, then unfold.

15. Fold the two vertical edges to the middle to construct these lines which divide the paper into fourths. Then unfold as shown here.

16. Fold the lower right and upper left corners to the line as shown. Stay behind the vertical line a little. You will see why later.

17. Now, double fold the two corners. Again, stay behind the line.

18. Refold the two sides back to the midline. Now you see why you needed to stay behind the line a little. If you didn’t, things bunch up along the folds.

19. Fold the upper right and lower left up and down as shown. Your accuracy in folding is shown by how close the two edges in the middle come together. Close is good-not close could be problematic.

20. The two corners you just folded, tuck them under the double fold. It should look like this.

21. Turn the unit over so you don’t see the double folds.

22. Lastly, fold the two vertices of the parallelogram up to form this square. You should see the double folds on top.

23. This is one UNIT. We need 5 more UNITS to construct a cube.

24. Change- The volume of the cube will change when different size squares are folded. The cubes you just made were made from 5.5” squares. What if the square was twice as long? How about half as long? What about any size square?

25. Volume: How could we answer the question? How does volume change as the size of paper used to make the cube changes? We are going to measure volume using non-standard units spoonfuls of popcorn

27. What happened? Do you notice a pattern? Compare your results with those of other groups. Are they the same, close, very different?

28. original square number of spoonfuls 1 2 3 4 5 5.5 6 7 8 9 10 Gathering Data

29. Reporting Data Graph-GeoGebra Table-Geogebra Generalizing-GeoGebra

30. Another Method Uncovering the Mathematics

31. What is “under” the unit that we just folded? Unfolding one unit reveals these lines. The center square is the face of the cube. If the square is 8” by 8”, what is the area of the square in the middle?

32. This design might be helpful as well as suggest a use of fold lines as an activity for students learning to partition a square and apply understanding of fractions in this area model.

33. What functional relation would relate the edge of the square (x) and the resulting length, surface area, and volume of the resulting cube?

34. length of original square resulting length of cube resulting area of one face of cube resulting volume of cube 20.7070.50.354 41.41422.828 62.1214.59.546 82.828822.627 103.53512.544.194

35. other uses for this unit: model of volume, surface area, and length Sierpinski’s Carpet in 3 dimensions model the Painted Cube problem construct stellated icosahedron with 30 units, stellated octahedron with 12 units or........

36. here is a stellated icosahedron- 30 units are required

37. this is a Buckyball, 270 units

38. a science fair project- determining how many structures the unit can make

39. entertaining grandchildren

40. Sierpinski’s carpet in 3 dimensions-

41. a model for volume

42. a wall of cubes!

43. Have you ever wanted an equilateral triangle? or How about a regular hexagon? or A tetrahedron? or What about a truncated tetrahedron?

44. start with any rectangular sheet of paper-

45. fold to find the midline-

46. fold the lower right corner up as shown-

47. fold the upper right corner as shown-

48. fold over the little triangle-

50. sources that would be helpful: handout: this keynote is available in pdf form at http://piman1.wikispaces.comhttp://piman1.wikispaces.com Unit Origami, Tomoko Fuse Unfolding Mathematics using Unit Origami, Key Curriculum Press geogebra.org Fold In Origami, Unfold Math, http://www.nctm.org/publications/article.aspx?i d=28158