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Teaching with mathematical modeling: providing multiple entry points and connections Mary Beth Searcy Appalachian State University Boone, North Carolina USA Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 • • • • • Currículum Nacional Current Implementation: Educación Básica 1° a 6° Básico Matemática Almost a national mathematics curriculum for K-12 Common Core State Standards for Mathematics (CCSSM) http://www.corestandards.org 2012 Implementation: North Carolina one of 13 states in Phase 1 Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 North Carolina (CCSSM) Mathematical Practices Chile (Currículum Nacional) Actitudes Habilidades and Make sense of problems and persevere in solving them Problem Solving Argumentation and Communication Reason abstractly and quantitatively Modeling Construct viable arguments and critique the reasoning of others Representation Demonstrate an organized and methodical working style Model with mathematics Use appropriate tools strategically Be flexible and creative when solving problems Attend to precision Look for and make use of structure Be curious about and interested in learning mathematics Look for and express regularity in repeated reasoning Have a positive attitude about yourself and your abilities. Be hard-working and persevere Express yourself and listen attentively to others Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Teaching with Mathematical Modeling -My journey started with questions Why do I want to use modeling in my classroom? How can I include modeling in the curriculum? How can I make sure that my students make the mathematical connections within modeling? What is teaching with modeling anyway? Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Why Teach with Mathematical Modeling? Reason #1: The Experience of Modeling Students discover something new Generates excitement about answering a question when “THE answer” is not known Students see modeling as a complex process Promotes creativity and communication Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling Example #1: How many barrels of water did Columbus bring on his 1492 journey to the “New World”? Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling: Columbus’ Journey and Water Voyage manifests were lost – we do not know the answer Can be modeled by different levels of students Students Ask questions – what impacts the need for water? Learn more about Columbus’ journey and the world in the late 1400’s Justify their choice of mathematical tools Communicate their solution processes Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Why Teach with Mathematical Modeling? Reason #2: Reinforce Mathematical Concepts Allows more opportunities to use mathematic concepts students have learned. Helps draw connections to other mathematics concepts Allows student to see how mathematical concepts are interpreted in terms of real world situations. Promotes curiosity Helps build foundation for more complex mathematical ideas. Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling Example #2: How much medicine do I have to take to make my sore throat feel better? (tonsillitis) Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling: How much medicine do I need to take? What must I know to answer my question? I will start by thinking about how I will take my medicine each day. Each day •I take a dose when I get up in the morning and •I take a dose when I go to bed. Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Number of Doses Modeling in Grade 1: Tonsillitis and Medicine 2 4 6 Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Number of Doses What happens if …? • you start your first dose when you go to bed on the first day? • you take a dose when you get up in the morning, a dose at lunch, and a dose when you go to bed? 3 6 9 Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Foundational Thinking for Later Grades Next day’s Total = Yesterday’s Total + 2 Also …Total Number of Doses= sum of groups of 2 doses Total Number of Doses = (Number of Days Taken) X 2 2 1 1 1 1 1 1 1 Days taken Total Doses 0 0 1 2 2 4 3 6 Making a Table and Graph leads to a Linear Function Model: y = 2x 2 2 2 2 2 2 2 4 4 8 5 10 6 6 12 7 14 Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile where x = Number of Days & y = Total Number of Doses 16 Number of Doses Counting by Twos leads us to Rates “2 doses per day” 14 12 10 8 6 4 2 0 0 2 4 6 8 Number of Days Taken 10 January 2013 Why Teach with Mathematical Modeling? Reason #3: Introduce New Concepts Explore a familiar situation with mathematics Analyze situation and uncover the need for a new mathematical concept Promotes further research on the situation Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling Example #3: What happens if I am in a classroom with 20 students and one of those students has the flu? Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling: What will happen to the class when one person has the flu? What do I know about the spread of germs? Let us start with a simple idea … Each hour, an infected person will come in contact with two people and thus spread the flu germ to two people. Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Grade 8 Modeling: Catching the flu! Let us explore this situation with an activity. Give each of the twenty students a natural number, starting with 1 up to 20 . Now you have Student No. 1, Student No. 2, Student No. 3, Student No. 4, … , Student No. 20. Using a random number generator from natural numbers 1 to 20, we will select which student comes into class with the flu. Continue to use random number generator to see who comes in contact with “sick” students. Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Grade 8 Modeling: Catching the flu! Number of Hours in Class Student Number -- Those who are Infected 0 19 1 19 11 18 19 11 18 2 5 3 19 11 18 10 5 5 10 9 9 20 20 3 3 8 7 18 3 9 10 12 17 13 18 18 12 11 1 3 10 Continue until all 20 students are “sick” and sitting down at their desks Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Grade 8 Modeling: Catching the flu! Introducing Logistic Function 25 Our modeling activity leads us to a new idea of bounded growth. Number Infected 20 15 10 5 0 0 1 2 3 4 5 6 Number of Hours Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Modeling: Catching the Flu More questions What happens if we change the number of contacts that people have? What happens if we only infect a fraction of the people contacted? Are there infectious diseases that we can model where the entire population does not become infected? What happens if we allow for people to “recover” from their infectious state while others continue to “infect” the population? Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 And so my journey with teaching with modeling continues. And it always leaves me with more questions to ask. Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013 Muchas Gracias Dr. Roberto Araya de Universidad de Chile Ministerio de Educación de Chile Dr. Eric Marland Seminario Internacional sobre Modelamiento Matemático: Santiago, Chile 10 January 2013