1. Communities of Exemplary Practice Patterns, Formulas, and Problem-Solving Summer 2012 Workshop

2. Mathematicians and 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 InputRule Output Number of ? Number of Tables Seats Input: # of tablesOutput: # of seats 13 24 3 4 5 6

6. What patterns do you see? What formulas can you create? Input: # of tables nOutput: # of seats S 13 24 35 46 57 68 + 2 +1+1

7. Which pattern/formula do we desire?

8. The +2 Pattern appears…where? Numerically, as shown in the table, going across Input: # of tables nOutput: # of seats S 13 24 35 46 57 68 + 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 2Pool 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 n th pool. Find the a formula for the number of white tiles in the n th pool. Pool 1 Pool 2Pool 3

12. Many solutions that are based on the geometry…