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Looking at Change Words, Symbols, Graphs, Paper

Looking at Change Words, Symbols, Graphs, Paper. WMC Annual Meeting May 2013 Joseph Georgeson UWM georgeso@uwm.edu. Assumptions: mathematics is the search for patterns- patterns come from problems- therefore, mathematics is problem solving. algebra is the language of mathematics.

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Looking at Change Words, Symbols, Graphs, Paper

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  1. Looking at Change Words, Symbols, Graphs, Paper WMC Annual Meeting May 2013 Joseph Georgeson UWM georgeso@uwm.edu

  2. Assumptions: mathematics is the search for patterns- patterns come from problems- therefore, mathematics is problem solving. algebra is the language of mathematics knowing and describing change is important tables, graphs, equations, words are effective ways to describe change

  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. Graph

  6. 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

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

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

  9. 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

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

  11. Start with a square.

  12. Fold it in half, then unfold.

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

  14. 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.

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

  16. 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.

  17. 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.

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

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

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

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

  22. 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? To answer that we are going to fold cubes from 4” squares and see how that changes when we compare to an 8” square.

  23. What happened? How did the volume change moving from cubes made with 4” squares to those made with 8” squares?

  24. Here is a more elaborate analysis of the question? After measuring the volume of each cube, this table could be filled in.

  25. Here are the results from the last class that did this activity.

  26. what equation that would best fit this set of points

  27. What is “under” the unit that we just folded. Unfolding it 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?

  28. 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.

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

  30. 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 ........

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

  32. this is a Buckyball, 270 units

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

  34. entertaining grandchildren

  35. Sierpinski’s carpet in 3 dimensions-

  36. a model for volume

  37. a wall of cubes!

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

  39. start with any rectangular sheet of paper-

  40. fold to find the midline-

  41. fold the lower right corner up as shown-

  42. fold the upper right corner as shown-

  43. fold over the little triangle-

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