Making sense of relationships through modeling
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Making sense of relationships through modeling. Roger Day Glencoe McGraw-Hill School Mathematics Tami S. Martin Illinois State University. Modeling as a tool for reasoning and sense making. Determine relationships among quantities. Make meaningful use of symbols to describe relationships.

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Making sense of relationships through modeling

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Making sense of relationships through modeling

Making sense of relationships through modeling

Roger Day

Glencoe McGraw-Hill School Mathematics

Tami S. Martin

Illinois State University


Modeling as a tool for reasoning and sense making

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.


Paper scraper

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?


Dynamics of modeling

Dynamics of Modeling


Remove the twos

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.


Popping mad

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.


Popping mad1

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?


Water flow

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?


Discussion

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?


Discussion1

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?


Contact info

Contact Info

Roger Day: [email protected]

Tami S. Martin: [email protected]


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