Using Repeating Patterns to Think Functionally

1. Using Repeating Patterns toThink Functionally National Council of Teachers of Mathematics Annual Meeting April 22-24 2010 San Diego CA Beth Schefelker & Melissa Hedges Mathematics Teaching Specialists Milwaukee Public Schools Milwaukee WI The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding by the National Science Foundation.

2. Session Goals: • To connect repeating and growing patterns to their underlying functional relationships • To examine recursive and explicit functional relationships in patterns

3. Concept Map for Patterns Recursive Strategies Using repeated patterns to think functionally Repeating Patterns Patterns Functional Relationships Explicit or Functional Strategies Assessment Resource Banks English, Mathematics and Science arb.nzcer.orgnz

4. Mathematical Knowledgefor Teaching (MKT) • What do we want Kindergarten students to know about AB patterns? • What do we want 2nd, 3rd, and 4th graders to know about AB patterns? • What do we want 8th graders to understand?

5. Extending the work of patterns Pattern exploration leads naturally to the development of rules to describe patterns and strategies to work a given pattern.

6. Investigating an AB Pattern • Use the tiles to build a pattern of yellow, red. • Analyze the pattern • Identify the unit. • What comes next? • Share out your insights about the structure of the pattern.

7. Thinking beyond recursive thinking… Research shows that students tend to use the recursive strategy in their pattern work…however its usefulness in the type of pattern work students will encounter in their school years is limited.

8. Examining a pattern • What is the unit? • What comes next? • What are the next three shapes in this pattern?

9. Thinking recursively… • How would children answer these questions using recursive thinking? • What would the 10th shape look like? • What would the 21st shape be? • Draw the 35th shape? • Next = Now + Change

10. Beyond recursive thinking What should we be doing with patterns between Kindergarten and Grade 8?

11. Moving Toward Functional Strategies • Return to the yellow, red tile pattern you built earlier. • Create a number sequence associated with the repeating pattern. 1 2 3 4 5

12. Surfacing relationships 1 2 3 4 5 • Discuss relationships that emerge. • What do you notice about the yellow tiles? red tiles?

13. Intentionally surfacing thinking… • What color is the 17th tile? • Is the 20th tile red? Why? • What color is the 100th tile? • What number is the 100th red tile? • What number is the 100th yellow tile? Explain the differences among these questions and how you solved them

14. Thinking functionally Functional strategies and rules are very powerful because they allow the user to work out any number in a pattern without knowing its predecessor. • What can you say about the position of any red tile? • What can you say about the position of any yellow tile?

15. Investigating Red, Orange, Yellow • Build an ABC pattern using red, orange, and yellow tiles. • Create a number sequence associated with the repeating pattern. • Use the relationships of the ordinal position and the constant increment to analyze the pattern.

16. Consider the following questions as you work. • What color is the 25th tile? • What position on the number line is the 13th yellow tile? • What position on the number line is the 19th red?

17. ROYGBIV • Think about a repeating pattern with the unit shown above… • Where would you find the 30th yellow? • Where would you find the 45th indigo ?

18. How have your thoughts changed? • What do we want Kindergarten students to know about AB patterns? • What do we want 2nd, 3rd, and 4th graders to know about AB patterns? • What do we want 8th graders to understand?

19. Where to find this session… • Milwaukee Mathematics Partnership • mmp.uwm.edu • schfeba@milwaukee.k12.wi.us • hoganme@milwaukee.k12.wi.us Thank you for coming!