1. Paper Folding The NYS Next Generation Mathematics Learning Standards
Paper Folding The NYS Next Generation Mathematics Learning Standards
Teri Calabrese-Gray, CVES 
Jessica Sheridan, WFL BOCES
Allison Peet, TST BOCES 

2. Objectives Investigate the area of various polygons by folding paper
Identify the Standards for Mathematical Practice and explain how the task aligns to each one
Determine the NYS Next Generation Mathematics Learning Standards in the Folding Paper task and justify your reasoning

3. Paper Folding 
Jo Boaler "A special thank you to Mark Driscoll for an engaging task that blends Number Sense and Geometry through paper folding. Great activity to kick off the school year with this task in order to build classroom collaboration."

4. Jo Boaler Video
Doing Math: It's about Thinking Deeply and Slowly - It's not about Speed!     .

5. Paper Folding Math Topics: Area, Fractions, Geometry Concepts:
Number Sense, Shape & Space
Mathematical Practices:
MP1, MP2, MP3, MP4, MP5, MP6, MP7, MP8
Grades:
Low Floor High Ceiling, 4, 5, 6, 7, 8, 9, 10

6. Materials
Square pieces of paper for each student
Markers (optional)

7. Task Instructions
For each part of the task, start with a square sheet of paper and make folds to construct a new shape. Then, explain how you know the shape you constructed has the specified area.

8. Convincing Mathematical Explanations

9. Convincing Mathematical Explanations
Convincing mathematical explanations stand up to any challenge and can convince others of a mathematical result.
Two conditions must be satisfied: 
Convincing explanations use facts, not opinions, to support claims.
Convincing explanations are complete and don’t leave any gaps or holes.

10. Convincing Mathematical Explanations
Convincing explanations use facts, not opinion. The figure below is a square. 
What is the measure of ∠ABC? 
Turn and talk to an elbow partner at your table to provide a convincing mathematical explanation.

11. Turn and talk to an elbow partner at your table to provide a convincing mathematical explanation.

12. Convincing Mathematical Explanations
Opinion: ∠ABC is 90 degrees because it looks like an L.
Fact: ∠ABC is 90 degrees because it is one of the 4 angles in a square and all 4 angles of a square are 90 degrees.

13. Convincing Mathematical Explanations
Convincing explanations are complete and don’t leave any gaps or holes. The figure below is a square. What is the name of the figure in bold? B A C D
Turn and talk to an elbow partner at your table to provide a convincing mathematical explanation.

14. Turn and talk to your other elbow partner at your table to provide a convincing mathematical explanation.
Turn and talk to an elbow partner at your table to provide a convincing mathematical explanation.

15. Convincing Mathematical Explanations
Explanation with gaps: This figure is an isosceles right triangle. I know this because two of the triangle’s sides are sides of the square, so they must be the exact same length.

16. Convincing Mathematical Explanations
Complete explanation: This figure is an isosceles right triangle. I know this because two of the triangle’s sides are sides of the square, so they must be the exact same length and ∠DAB is a right angle because it is one of the 4 angles of the square, and all angles in a square are 90 degrees.

17. Warm-Up Investigating Area by Folding Paper

18. Warm-Up for Investigating Area by Folding Paper
If you fold a square piece of paper along one of its diagonals:
What do you get?
Write response independently on recording sheet

19. Warm-Up for Investigating Area by Folding Paper
If you fold a square piece of paper along one of its diagonals:
What do you get?
What kind of triangle is this? 
  Write response independently on sheet of paper

20. Warm-Up for Investigating Area by Folding Paper
If you fold a square piece of paper along one of its diagonals:
What do you get?
What kind of triangle is this? 
What is the relationship between the area of this triangle and the area of the original square? 
Write response independently on sheet of paper

21. Warm-Up for Investigating Area by Folding Paper
If you fold a square piece of paper along one of its diagonals:
What do you get?
What kind of triangle is this?
What is the relationship between the area of this triangle and the area of the original square?
Explain how you decided on the relationship.
Write response independently on sheet of paper

22. Numbered Heads Together
Number off at your table.
Stand up and "put your heads together" showing answers, discussing and teaching each other.
Once your group comes to consensus, please sit down.
We will call a number and that person will report for your group.

23. Warm-Up for Investigating Area by Folding Paper
2. Where can you fold a square in order to construct a rectangle with exactly ½ the area of the square? How do you know your rectangle has exactly ½ the area of the square?
Practice Numbered Heads Together Engagement Strategy

24. Task Instructions
Throughout the Paper Folding task, keep a running list of academic vocabulary used at your table.

25. Task Instructions
1. Construct a square with exactly ¼ the area of the original square. Convince yourself that it is a square and has ¼ of the area.
Convince Yourself
Use same number from Number Heads Together Engagement Strategy 
1 is the person doing the convincing 
2 is a friend
3 is a skeptic
4 is the person doing the convincing
5 is a friend
6 is a skeptic
Keep repeating as you go around the table

26. How do you know?
Explain how you know it has ¼ the area.

27. How do you know?
Sentence Starter: To create a square with ¼ the area of the original square piece of paper, I folded: __________________________________________________________________

28. How do you know?
Frame: To create a _____________with______ the area of the original square piece of paper, I folded:____________________ ________________________________

29. Task Instructions
2. Construct a triangle with exactly ¼ the area of the original square. Convince a friend that it has ¼ of the area.
Convince a Friend
Use same number from Number Heads Together Engagement Strategy 
1 is the person doing the convincing 
2 is a friend
3 is a skeptic
4 is the person doing the convincing
5 is a friend
6 is a skeptic
Keep repeating as you go around the table

30. How do you know?
Explain how you know it has ¼ the area.

31. How do you know?
Sentence Starter: To create a triangle with ¼ the area of the original square piece of paper, I folded: ___________________________________________________________________________________________________

32. How do you know?
Frame: The way I folded the paper to create a ______________ with ¼ the area and a triangle with ¼ the area of the original piece of paper was different because: __________________________________________________________________

33. Comparing Task 1 to Task 2 - How do you know?
Explain the difference between folding a square to make a square ¼ the area of the original square (Task 1) compared to folding a square to make a triangle ¼ the area of the original square (Task 2).
Randomly select someone to come up and explain to all. 

34. How do you know?
Sentence Starter: The way I folded the paper to create a square with ¼ the area and a triangle with ¼ the area of the original square piece of paper was different because: __________________________________________________________

35. Task Instructions
3. Construct another triangle, also with ¼ the area, that is not congruent to the first one you constructed. Convince a skeptic that it has ¼ of the area.
Convince a Skeptic
Use same number from Number Heads Together Engagement Strategy 
1 is the person doing the convincing 
2 is a friend
3 is a skeptic
4 is the person doing the convincing
5 is a friend
6 is a skeptic
Keep repeating as you go around the table

36. How do you know?
Explain how you know it has ¼ the area.

37. How do you know?
Sentence Starter: To create a triangle with ¼ the area of the original square piece of paper, I folded: ___________________________________________________________________________________________________

38. Comparing Task 2 to Task 3
Explain the difference between the triangle you folded in Task 2 with the triangle you folded in Task 3.  How do you know the triangles are not congruent? 

39. Comparing Task 2 to Task 3 - How do you know
Sentence Starter: I know that the two triangles I have created in Task 2 and Task 3 are not congruent because: _________________________________ __________________________________________________________________

40. Comparing Task 2 to Task 3 - How do you know?
Frame: I know that the two triangles I have created in Task 2 and Task 3 are not ___________________because they cannot be transformed or manipulated in a way that allows me to place one exactly over the other.

41. Task Instructions
4. Construct a square with exactly ½ the area of the original square. Convince yourself that it is a square and has ½ of the area.
Work independently at your table (convince yourself) then others at your table (convince a friend ).  Finally convince your table.

42. How do you know?
Explain how you know it has ½ the area.

43. Task Instructions
5. Construct another square, also with ½ the area, that is oriented differently from the one you constructed in 4. Convince someone not at your table that it has ½ of the area.
Work independently at your table.  When done, circulate the room and share your solution with others. Ask volunteers to come forward and share their solutions and how the orientation is different.  

44. How do you know?
Explain how you know it has ½ the area.

45. Academic Language
Learners should have opportunities to see, hear, and write key mathematical ideas during this activity. There are some specific terms that learners need to understand in order to engage in this task, and there are some additional terms and phrases that may surface as the learners engage with the task. 

46. Academic Language
    As the task is introduced, solved by the learners, and discussed, ensure that learners have opportunities to experience (i.e., through discussion, pictures, and the use of gestures) and to build understanding for key words and phrases. 

47. Word Chart for Investigating Area by Folding Paper 
(Click on the link above or copy and paste the link to access the collaborative word chart.) 
Popcorn or Whip Around
Call on participants to share their word and then ask them to enter the word they shared in the Word Chart (Google doc)

48. Word Chart for Investigating Area by Folding Paper
Words and Phrases
Academic Language Meaning 
Everyday Language Meaning 
Other Forms of the Word or Phrase
Related Words or Phrases
Examples of word use w/students
Construct
To create new elements with previously measured or constructed elements.
To build; to assemble. To put something together.
Constructing Constructed Constructs
Build Make Create Form To put together
Words shared may include:
Congruent 
Area 
Triangle; square 
Construct a square; construct a triangle; square or triangle that you constructed 
Side lengths are equal
Oriented differently
Base 
Height 
Reflection 
Fold 
Specified

49. Debriefing the Task Graffiti Wall followed by a Gallery Walk
Participants will use the SMP handout at their tables to serve as a reference.

50. Standards for Mathematical Practice
Make sense of problems and persevere in solving them
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of others
Model with mathematics
Use appropriate tools strategically
Attend to precision
Look for and make use of structure
Look for and express regularity in repeated reasoning
Graffiti Wall followed by a Gallery Walk
Post eight large pieces of chart paper for each Standard for Mathematical Practice around the room. Draw a circle in the middle of each piece of chart paper with MP1, MP2, etc. written inside the circle.  Hand out the MP reference sheet to all learners as well.  Ask learners to take a marker and write specific examples from the task that align to each of the mathematical practices.  Allow them to weigh in on all 8 MPs and then ask them to circulate around the room (Gallery Walk) and review all the responses.  

51. Connecting to the NYS Next Generation Mathematics Learning Standards
Show participants where the NYS Next Generation Mathematics Learning Standards are located on EngageNY.  The mathematics standards have been added to the shared folder as well for easier access for the next activity.

52. IdeaBoardz
What Next Generation Mathematics Learning Standards were addressed in the Paper Folding Activity? Record your thoughts online at
Assign each table a grade level 4, 5, 6, 7, 8, 9, 10 and repeat for additional tables.  Once your table thinks they have completed the list of standards for their grade level, find the table working on the same grade level and compare the standards they listed.  Once you come to consensus go to IdeaBoardz and list your grade level standards.  

53. Extension Activity
Present this task if time allows, if not go to slide 55.

54. Think Think Think What would be the total area if you constructed an:
 8th square?
9th square?
Nth square?
Think Think Think 

55. Notice Think Wonder 1. What do you notice?​ 2. What do you think?​
3. What do you wonder?
Notice Think Wonder

56. 3-2-1 Reflection

57. "Mathematics is not about numbers, equations, computations, or algorithms: it is about UNDERSTANDING!" William Paul Thurston THE END!