1. 1 Math CAMPPP 2011 Math at the Beach!!! Grade 5-8 Sandra Fraser Erik Teather

2. Patterning & Algebra Breakout 2 2

3. Card Sort You will be receiving a card which contains a representation of a pattern. Your task is to find all the people that have a different representation of your pattern. As a group, discuss which representation you find the most clear and which ones you are the most/least comfortable with. 3

4. Learning Goal? What do you think our learning goal was for the card sort? Did we achieve it? 4

5. The Table Problem Think about how you would solve the problem on your own (3 minutes) Pair up and solve the problem in 2 different ways. 5

6. Expectations, Big Ideas, Learning Goals As a table, compare your answers and discuss some of the different ways of solving these problems. On chart paper, determine –The expectation(s) (Grade 8) –The Big Idea(s) –The Learning Goal(s) 6

7. What are you looking for? As a group, determine what a good answer to the table problem would look like. 7

8. Consolidating Question As a group, determine what a good consolidating question would be to determine that your learning goal has been reached. 8

9. Switch! Switch chart papers with another table. How are they similar? different? 9

10. Discussion What did you notice? 10

11. Sample Solutions Examine the following sample solutions and decide if they are a good solution (based on your group’s decision about what a good answer looks like) 11

12. A Different Lens Development of Generalization Strategies Justification Framework –Let’s explore our sample solutions through this different lens. Determine where the solutions fit using these frameworks. 12

13. Generalization Strategies StrategyDescription CountingDrawing a picture or constructing a model to represent the situation to count the desired attributes RecursiveBuilding on the previous term or terms in the sequence to determine subsequent terms (Additive thinking) Whole- object Using a portion as a unit to construct a larger unit by multiplying. There may or may not be an appropriate adjustment for over-or- undercounting. Guess-and- check Guessing a rule without regard to why this rule might work. Usually this involves experimenting with various operations and numbers provided in the problem situation. ContextualConstructing an explicit rule that expresses the co-variation of two sets of data, based on information provided in the situation. An explicit rule can allow for the prediction of any term number in the pattern.

14. Justification Framework Justification LevelDescription Level 0: No JustificationResponses do not address justification Level 1: Appeal to external authority Reference is made to the correctness stated by some other individual or reference material Level 2: Empirical evidenceJustification is provided through the correctness of particular examples Level 3: Generic exampleDeductive justification is expressed for a particular instance. Level 4: Deductive justification Validity is given through a deductive argument that is independent of particular instances

15. Reflection How do you feel about these frameworks? Did these frameworks match your success criteria? Which (frameworks/success criteria) is more beneficial for students? teachers? 15

16. What did we learn? …about the mathematics? …about teaching mathematics? 16

17. Our Learning Goals (Did we meet them?) To explore different representations of patterns and discover that different representations bring about different mathematical concepts. To experience creating learning goals, success criteria and consolidating questions. 17

18. Back to the math Complete handout individually. Compare with your table when everyone has finished. 18