Table of Contents
- Content Home
Welcome
Module Overview- Pre-Assessment
- Unit 1: Teaching and Learning the Standards for Mathematical Practice
- Unit 2: Overarching Habits of Mind: MP1 and MP6
- Unit 3: Reasoning and Explaining (MP2 and MP3)
- 3.0 Unpacking MP2 and MP3
- 3.1 Beginning to Reason: Definitions and Conjectures
- 3.2 Explaining and Justifying
- Taxonomy of Questions in Mathematical Discourse
- 3.3 Identifying Flaws in Reasoning
- 3.4 Making Arguments More Viable
- 3.5 Summary
- Unit 4: Modeling and Using Tools (MP4 and MP5)
- Unit 5: Seeing Structure and Generalizing (MP7 and MP8)
- Unit 6: Summary and Next Steps
- Post-Assessment
Certificate of Completion- Glossary
Resources- Acknowledgements
Module Evaluation
Unit 3: Reasoning and Explaining (MP2 and MP3)
3.4.5 Student-Generated Math Challenge
Led by a student with special needs, the class developed their own challenge: What happens when adding odd and even numbers?
Max, Spencer, and Haley worked on this challenge during their own time.
Max:
Along the way we discovered that theories have been tested; some were successful and some were not. A successful one was even + even = even and odd + odd = even, but even + odd = odd.
even plus even: 4 + 4 = 8 (2, 4, 6, 8) even
odd plus odd: 3 + 9 = 12 (2, 4, 6, 8, 10, 12) even
even plus odd: 4 + 9 = 13 (2, 4, 6, 8, 10, 12, 14) odd
Spencer:
The theory we had was that if you plus a number plus itself, no matter what, you will get an even number.
1785 + 1785 = 3570 … even!
1853 + 1853 = 2706 … even!
The other theory we had was that if you add an even plus an odd number, you will get an odd number no matter what.
19 + 24 = 43
18 + 17 = 35
Haley:
odd + odd = even
1 + 1 = 2 3 + 3 = 6 5 + 5 = 10 7 + 7 = 14 9 + 9 = 18
1 + 3 = 4 3 + 5 = 8 5 + 7 = 12 7 + 9 = 16
1 + 5 = 6 3 + 7 = 10 5 + 9 = 14
1 + 7 = 8 3 + 9 = 12
1 + 9 = 10
Conjecture: For odd and odd will always equal to even.
even + even = even
0 + 0 = 0 2 + 2 = 4 4 + 4 = 8 6 + 6 = 12 8 + 8 = 16
0 + 2 = 2 2 + 4 = 6 4 + 6 = 10 6 + 8 = 14
0 + 4 = 4 2 + 6 = 8 4 + 8= 12
0 + 6 = 6 2 + 8 = 10
0 + 8 = 8
Conjecture: For even and even will always equal to even.
odd + even = odd
1 + 0 = 1 3 + 0 = 3 5 + 0 = 5 7 + 0 = 7 9 + 0 = 9
1 + 2 = 3 3 + 2 = 5 5 + 2 = 7 7 + 2 = 9 9 + 2 = 11
1 + 4 = 5 3 + 4 = 7 5 + 4 = 9 7 + 4 = 11 9 + 4 = 13
1 + 6 = 7 3 + 6 = 9 5 + 6 = 11 7 + 6 = 13 9 + 6 = 15
1 + 8 = 9 3 + 8 = 11 5 + 8 = 13 7 + 8 = 15 9 + 8 = 17
Conjecture: For odd and even, it will always be odd.
This student-generated challenge fits the descriptors of Ball’s reasoning community, stating how public knowledge (e.g., the definition of odd and even) becomes the foundation for new knowledge under construction (e.g., developing rules for adding odd and even numbers).
View this final odd/even video of three English learner students describing viable arguments.
Children must be shown how to cultivate a climate of debate, questioning and multiple interpretations. They must think about how to disagree with each other in ways that allow the other person to hear what is being said."
Calkins, 2001
Mathematics: Kindergarten through Grade Twelve (K–12) Standards for Mathematical Practice. Brought to you by the California Department of Education, the Regents of the University of California, and the California Mathematics Project.
For questions or feedback, please e-mail commoncoreteam@cde.ca.gov.
