### providing multiple entry points and connections

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
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

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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


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?


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
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
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
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
“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?
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