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
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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