1. Spatial reasoning plays a key foundational role in the development of ALL thinking! This is especially true for science & mathematics.

2. Paper folding is a spatial reasoning task:
It will make you a more organized , logical thinker if you focus on folding in your head before you touch the paper.

3. Paper folding is a spatial reasoning task:
It will make you a more flexible thinker which affects your success with problem solving.

4. Paper folding is a spatial reasoning task:
It will develop your skill, understanding and recall with multiplication facts.

5. Paper folding is a spatial reasoning task:
It will develop your multiplicative reasoning which links to fractions, decimals, place value, division...

6. Paper folding is only a spatial reasoning task if you challenge yourself to think it through, see in your head

7. Here’s a challenge to try....

8. Think and gesture, but do not act: Can you see in your head a strategy for folding a piece of paper into 3 equal parts?

9. Use gestures, no words, no speaking to show a partner or the whole class.

10. Have several students demonstrate to the class
Have several students demonstrate to the class. You follow their gestures to see if the folding works.

11. Can anyone translate a student’s gestures into diagrams?
Encourage students to help you model through a series of diagrams how to fold into 3 equal parts or pieces.

12. Depending on how you position the paper to start you will see:

13. Some strategies I have seen involve folding front to back like a fan:

14. Rolling in a cylinder and eyeballing to get equal parts.

15. Eyeballing and folding in to about the one third mark, then folding over.
The page overlaps itself.

16. Plan, position the paper once you decide to start, COMMIT to the fold
If the folds are not smooth and crisp, throw out and start over.
After 3 papers, stop. Reflect and compare.
Once the class have discussed and agreed on some ways to fold in 3 encourage the thinking above.

17. What do you need to do to improve?

18. Your goal: Become proficient
and accurate at folding a paper into 3 equal parts.
Practice 3 tries a day for 3 to 5 days.
(if the tries are serious)
Those who need to practice, those who are confident practice with different size papers.

19. If you are accurate, you still practice..
Good players do not get to skip practice
(and neither did Wayne Gretsky!)
Use different sizes paper to challenge yourself if you are confident but still practice 3 times a day for a week.

20. If you are confident and accurate with thirds then each day the class is practicing, you use 3 different sizes of paper to make your thirds.

21. You might also set a timer to document your time each day.
Not a race, just a comfortable pace. Are you consistent, not are you getting faster.

22. After 5 days of practice link the model to
division

23. One (1) piece of paper divided (÷) into 3 equal parts, each part is one third of the whole paper.

24. After 5 -7 minutes of daily practice for a week, move on.
Think but do not act:
Can you see in your head a strategy for folding a piece of paper into 5 equal parts?

25. Gesture, no words, to show us or a partner.
Draw a series of diagrams and we will see if we can follow....

26. There will be a “fan folding way and an eyeball estimate and then fold way. Both require you to fiddle and try and fiddle and try.

27. Plan, position the paper but COMMIT to the fold once you decide to start.
If the folds are not smooth and crisp, throw out and start over.
After 3 tries stop for today.

28. Every day for 7 days, the class will practice accurately and precisely folding a paper into 5 equal parts.

29. You have learned to model the equation:

30. If you have mastered fifths before the 7 days is up, then each day the class is practicing, you use 3 different sizes of paper to make your fifths.
You do not get to skip practice
(and neither did Wayne Gretsky!)

31. Once you are good at folding in 3 and folding in 5, practice the following sequence.
Day 1: Fold 3 one way and 4 the other. Diagram the results.
Save your best.

34. When you fold 3 in one direction and 4 in the other, you end up with 12 parts.
I know 12 because I counted by 3s. Then I counted by 4s, still 12. 4 3

35. Day 2: fold 3 one way and 5 the other. Diagram the results
Day 2: fold 3 one way and 5 the other. Diagram the results. See 15 parts?
Save your best.

36. When you fold 3 in one direction and 5 in the other, you end up with 15 parts.
Can you count by 3s? How about by 5s. 5 3

37. Day 3: fold 3 one way and 3 the other. Diagram the results. See 9 parts?
Save your best.

38. When you fold 3 in one direction and 3 in the other, you end up with 9 parts.
I knew because 3 threes is 9.
Save your best. 3 3

39. Day 4: Fold 3 one way and 6 the other. Practice to get accurate.
Save your best. Draw a diagram.

40. When you fold 3 in one direction and 6 in the other, you end up with 18 parts.
Did you count by 3s to see?
Did your diagram your best. 6 3

41. Day 5: Compare all the folds to this point. They all have 3 one way.
Do you see anything else?

42. Day 6: fold 4 one way and 5 the other.

43. Day 6: Diagram the results. See 20 parts?
Save your best.

44. Day 7: fold 4 one way and 4 the other.

45. Day 7: Diagram the results. See 16 parts?
Save your best.

46. Day 8: Fold 4 one way and 6 the other.

47. Day 8: Diagram the results. See 24 parts?
Save your best.

48. Day 9: fold 5 one way and 5 the other..

49. Day 9: Diagram the results.
See 25 parts?
Save your best.

50. Day 10: fold 5 one way and 6 the other.

51. Day 10:
Diagram the results. See 30 parts?
Save your best.

52. Email Mrs. Lorway and tell her you are ready for the challenge!!!

53. Nora Newcombe is one of the researchers who has begun to write about the findings that show a positive link between spatial skills and achievement in math and science.

54. Spatial thinking and the development of "number sense" are linked.
Early spatial intelligence predicts a child's performance in mathematics.
(Newcombe et al 2015; Verdine et al 2014).

55. Children who are better at visualizing spatial relationships develop stronger arithmetic abilities in primary school .
Zhang et al, 2014

56. Mix and Cheng, 2012, gave students tasks that practiced spatial reasoning skills.
The tasks had nothing to do with numbers.

57. After several weeks, they checked the students’ arithmetic skills and found there was a very clear improvement.
Especially on missing number questions like
5 + ? = 8

58. Paper-folding activities in mathematics address some of the goals for teaching and learning mathematics in the 21st century as identified by English (2002):
mathematical modeling
visualizing
algebraic/relational thinking
problem-solving

59.  There is evidence that early spatial ability predicts a young child's reading skills.
Franceschini et al 2012

60. Ready for a fold?

61. Carefully observe the folded sheet. How many fold do you see?

62. What do you note or notice?
How could you describe the differences between them?

63. Fold 1.) This fold runs vertically, (up and down).
This is a “mountain” fold.
It looks like you fold to the back

64. Fold 2.) This fold runs diagonally from top left to bottom right.
This is a “valley” fold. (It looks like you fold forward)

65. Fold 2.) Diagonal actually is used to describe a line segment that connects interior angles of a polygon. In this case the “diagonal” fold line follows the diagonal of the square.
We like to name a line that sits on a slant as diagonal but it might be more correct to call it oblique.... Something to think about

66. Now imagine folding the paper as shown
Now imagine folding the paper as shown. If you did, what shape would the paper turn into. So draw what the paper will look like
if you complete both folds.

67. Getting started can be tough
Getting started can be tough. Try gesturing in the air to pretend you are folding, or start by drawing the paper with its fold lines

68. Explain and compare with a partner
Explain and compare with a partner Make changes if you think you need to

69. These are some examples students have drawn
These are some examples students have drawn. Can you explain what each person is “thinking”.

70. Go ahead and fold to see if your visualization was accurate.....

71. When I look at the picture, I suspect that folding the top right vertex down to meet the bottom left vertex is the way to start..... Do you see why?

72. Ready for a fold?

73. Can you explain or diagram what shape is made when the paper is folded using the crease pattern shown?

74. Compare with a partner......

75. As a class, describe the shape that has been created.

76. Explain the sequence of folds. Where did you start. Then what
Explain the sequence of folds. Where did you start? Then what.... Be specific...

77. Ready for a fold?

78. Can you explain or diagram what shape is made when the paper is folded using the crease pattern shown?

79. Compare with a partner......

80. As a class, describe the shape that has been created.

81. Explain the sequence of folds. Where did you start. Then what
Explain the sequence of folds. Where did you start? Then what.... Be specific...

82. Ready for a fold?

83. Can you explain or diagram what shape is made when the paper is folded using the crease pattern shown?

84. Compare with a partner......

85. As a class, describe the shape that has been created.

86. Explain the sequence of folds. Where did you start. Then what
Explain the sequence of folds. Where did you start? Then what.... Be specific...

87. Ready for a fold?

88. How many folds do you need to make to have this paper......
Look like this?
Can you do the folding in your head?

89. Can you think it through, without folding paper?

90. I am thinking: fold the white half backward, then fold the orange quarter up and to the back.
Two folds.

91. Ready for a fold?

92. How many folds do you need to make to have this paper......
End up like this?
I can do it in 2 folds.

93. What if you start with the same paper but want to end up like this?

94. It takes me 3 folds, what about you?

95. Can you explain what fraction of the whole each of the coloured sections is?
How would you show a proof?