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Communities of Exemplary Practice Patterns, Formulas, and Problem-Solving

This workshop focused on the importance of understanding patterns in mathematics, particularly how they relate to geometry and problem-solving. Participants explored various representations of problems, emphasizing the Iowa Core Standards' principles, such as making sense of problems and using structure. Through engaging activities, students learned to investigate patterns in numbers, shapes, and data, all while developing mathematical habits of mind. Participants also created formulas and recursive representations for real-world problems, emphasizing the interconnections between geometry and algebra.

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Communities of Exemplary Practice Patterns, Formulas, and Problem-Solving

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  1. Communities of ExemplaryPracticePatterns, Formulas, and Problem-Solving Summer 2012 Workshop

  2. Mathematiciansand scientists want to understand how and why. From the Iowa Core, Standards for Mathematical Practice: • Make sense of problems and persevere in solving them.

  3. Mathematicians look to use patterns to understand • Look for and make use of structure. Students who look for patterns in their environment expect things to make sense and develop a habit of finding relationships and making predictions. Students should investigate patterns in number, shape, data, change, and chance. They should be given opportunities to learn how to represent those patterns numerically, geometrically and/or algebraically.

  4. How Many Seats? We have a long skinny room and triangle tables that we need to arrange in a row with their edges touching, as shown. Each side can hold one “seat,” shown with a circle. Can patterns help us find an easy to answer the question: How many seats can fit around a row of triangle tables?

  5. Student Worksheet Triangle Rule Machine Input Rule Output Number of ? Number of Tables Seats

  6. What patterns do you see?What formulas can you create? + 1 + 2 Recursive Representation Exact, closed-form solution

  7. Which pattern/formula do we desire?

  8. The +2 Pattern appears…where? • Numerically, as shown in the table, going across + 2

  9. The +2 Pattern appears…where? • Geometrically

  10. Connections between Geometry and Algebra • Encourage students to relate different representations of the problem • Consider the classic pool problem: Pool 1 Pool 2 Pool 3

  11. The Pool Problem • Find the number of gray tiles in Pool 5. • Use a table to represent the number of gray tiles in Pools 1,2,3, and 5. • Find a formula for the number of gray tiles in the nth pool. • Find the a formula for the number of white tiles in the nth pool. Pool 1 Pool 2 Pool 3

  12. Many solutions that are based on the geometry… • Determine how to “see” the shapes geometrically so that the formula would be

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