Making sense of relationships through modeling

1. Making sense of relationships through modeling Roger Day Glencoe McGraw-Hill School Mathematics Tami S. Martin Illinois State University

2. Modeling as a tool for reasoning and sense making • Determine relationships among quantities. • Make meaningful use of symbols to describe relationships. • Use families of functions to help connect context with the mathematics. • Connect contexts with algebraic representations. • Explore situations that lead to multiple, connected representations, and the interpretation of those representations through connections back to a context.

3. Paper Scraper • Paper Scraper: how-to video; example video • What’s the connection between the number of cards used and the number of levels completed? • Words • Tables • Symbols • Graphs • What other relationships could be explored?

4. Dynamics of Modeling

5. Remove the Twos • Scramble 60 dice in a container and roll them out on the table. Count and remove all the 2s you see. Now repeat the scramble and roll. • Keep track of the number of dice that remain after each roll. Create a table of values (roll number, dice remaining after roll), create a scatter plot of those values, and generate an algebraic representation for your data. • Explain how your representations connect with the context of the problem.

6. Popping Mad! • Have on hand several identical bags—weight and variety—of unpopped microwave popcorn and a microwave oven. • Microwave one bag for exactly 60 seconds. Open the bag and record the number of popped kernels. • Repeat the previous step for subsequent 10-second intervals (70 sec, 80 sec, and so on), each time recording the elapsed time and the number of popped kernels.

7. Popping Mad! • You should now have a set of ordered pairs, relating elapsed time with the number of popped kernels of corn. • Create a scatter plot of (elapsed time in seconds, number of popped kernels). • Examine your scatter plot, What family of functions looks to be an appropriate model for your data? Can you connect the context of the problem with that function family? • What other relationships might be explored using these data and this context?

8. Water Flow • Fill a 2-liter bottle with water. Using a bottle cap with a hole in it, turn the bottle upside down and record the amount of water out of the bottle, in ml, as time passes, in seconds. • Create a scatter plot of (sec, ml) and generate an algebraic representation for your data. • To what family of functions does this relationship belong? Can you justify that based on your knowledge of such functions and the context of the problem?

9. Discussion • What are the key elements of reasoning and sense making that you can drawn upon or emphasize using modeling activities in your classroom? • What actions will you take as a teacher to ensure that the modeling process engenders student thinking and sense-making?

10. Discussion • What challenges to teaching and learning emerge when we engage students in modeling activities such as these? • What are two or three actions you can take to begin (or increase!) implementation of modeling activities in your courses?

11. Contact Info Roger Day: dayr@pontiac.k12.il.us Tami S. Martin: tsmartin@ilstu.edu