An Experience of Modeling
Margaret L. Kidd
Cal State Fullerton
mkidd@fullerton.edu
Session 365 CMC-South
Palm Springs, CA
November 1, 2013
Definition of Mathematical
Modeling
There are many definitions.
There is also a difference in modeling
with mathematics as found in the SMP
and the Mathematical Modeling
Process as found in the high school
curriculum.
Bill McCallum Video
Standards for Mathematical
Practice
The Importance of Mathematical
Practices
What is Modeling Not???
It is not modeling in the sense of, “I do;
now you do.”
It is not modeling in the sense of using
manipulatives to represent mathematical
concepts (these might be called “using
concrete representations” instead.)
It is not modeling in the sense of a
“model” being just a graph, equation, or
function. Modeling is a process. 4
It is not just starting with a real-world
situation and solving a math problem; it
is returning to the real-world situation
and using the mathematics to inform
our understanding of the world. (I.e.
contextualizing and decontextualizing,
see MP.2.)
It is not beginning with the
mathematics and then moving to the
real world; it is starting with the real
world (concrete) and representing it
with mathematics.
Mathematical Modeling
Step 1. Identify a situation.
Read and ask questions about the
problem. Identify issues you wish
to understand so that your
questions are focused on exactly
what you want to know.
Mathematical Modeling
Step 2. Simplify the situation.
Make assumptions and note the
features that you will ignore at first.
List the key features of the
problem. These are your
assumptions that you will use to
build the model.
Mathematical Modeling
Step 3. Build the model and solve the
problem.
Describe in mathematical terms the
relationships among the parts of the
problem, and find an answer to the
problem. Some ways to describe the
features mathematically include:
define variables, write equations make
graphs, gather data, and organize into tables
Mathematical Modeling
Step 4. Evaluate and revise the model.
Check whether the answer makes sense,
and test your model. Return to the original
context. If the results of the mathematical
work make sense, use them until new
information becomes available or
assumptions change. If not, reconsider
the assumptions made in Step 2 and
revise them to be more realistic.
Areas Modeling is Indicated
Modeling is best interpreted not as a collection
of isolated topics but in relation to other
standards. Making mathematical models is a
Standard for Mathematical Practice, and
specific modeling standards appear
throughout the high school standards
indicated by a star symbol (*). The star
symbol sometimes appears on the heading for
a group of standards; in that case, it should
be understood to apply to all standards in that
group.
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Areas Modeling is Indicated
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Areas Modeling is Indicated
 Algebra
 Seeing Structure in Expressions
A-SSE
Write expressions in equivalent forms to solve
problems
3. Choose and produce an equivalent form of an
expression to reveal and explain properties of the
quantity represented by the expression.*
a. Factor a quadratic expression to reveal the zeros of the
function it defines.
b. Complete the square in a quadratic expression to reveal
the maximum or minimum value of the function it defines.
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Teacher and Modeler
Teacher/Problem Writer
Begins with mathematics (content standard).
Focus on mathematics and uses applications to
illustrate, motivate, develop or understand
mathematics.
Results in deeper understanding of the mathematics.
Modeler and Modeling
Begins with life. (Math provides the tools and
pathways.)
Focus on understanding or solving a life problem or
improving a product or situation.
Leads to a solution, decision, recommendation,
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modification, plan.
Getting Ready as a Teacher
Look around you at the real world. Get used
to “noticing” and asking questions.
Be open to your students. Realize you do
not have all of the answers but know where
to find them.
Get used to using the internet to find
answers……and questions to use!
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Modeling in the
Mathematics Classroom
Problems
- Easiest to incorporate in the classroom.
- Should I use the 20% off coupon or the $10 off coupon?
- Pizza which is the better deal?
Lessons
- How should I pack the little boxes into the big box?
- SBAC: Taxicab Problem, Long Jump Problem, MARS
Units
- Extended commitment of time.
- What is the most efficient way to package soda cans?
Curriculum/Course
15
Coupon Choice
Karen had two coupons when
she bought her shoes at the
department store. The clerk said
the $10.00 off coupon is usually
the best. Was the clerk correct?
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What is the Question?
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Pizza Deals
What size pizza is the best deal?
Medium Cheese = $ 9.99
Large Cheese = $11.99
X-Large Cheese = $13.99
Medium = 12” diameter
Large
= 14” diameter
X-Large = 16” diameter
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More than Enough
Problems!
Before you begin writing a lesson plan or
perhaps developing a unit of study,
examine several websites that provide
problems you can use, as is, or adapt for
your students.
Consider, also, the many sources of
authentic modeling problems found in
every-day life.
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Modeling in the
Mathematics Classroom
Problems
- Easiest to incorporate in the classroom.
- Should I use the 20% off coupon or the $10 off coupon?
- Pizza which is the better deal?
Lessons
- How should I pack the little boxes into the big box?
- SBAC: Taxicab Problem, Long Jump Problem, MARS
Units
- Extended commitment of time.
- What is the most efficient way to package soda cans?
Curriculum/Course
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Find a more efficient way to
package soda cans.
What is Efficient?
Efficient =
Space Used
Space Available
Efficient =
Volume of Cans
Volume of Prism
Efficient =
Area of Circles
Area of Polygon
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Standard Six Pack
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Standard Six Pack
The efficiency of the standard six-pack or
twelve-pack is 78.5%. 23
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Triangular Six Pack
What is the efficiency
of the triangular sixpack?
Is a triangular threepack or
ten-pack more or less
efficient than the
triangular six-pack?
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26
27
28
29
Hexagonal Seven Pack
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SODA CANS
BY: FRANKIE AND ERIKA
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RECTANGULAR SIX PACK
The efficiency for a rectangular six pack (or
eight pack) is 78.5%
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Design 1 SEVEN PACK
Area ABCDEF = 74.23 cm 2
Area
GH = 9.03 cm 2
Area
IJ = 9.03 cm 2
Area
IJ7
Area ABCDEF
= 0.85
A
B
G
H
I
J
F
C
E
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D
DESIGN 2 TWELVE PACK
Area ABCDEF = 70.94 cm 2
Area
Area
Area
GH = 5.07 cm 2
GH12 = 60.80 cm 2
GH12
Area ABCDEF
= 0.86
A
B
G
H
F
C
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E
D
COMPARING
The seven pack has an efficiency of 85%.
The twelve pack has an efficiency of 86%.
They are both more efficient than the rectangular six
pack.
A
Area ABCDEF = 74.23 cm 2
B
G
H
I
Area
GH = 9.03 cm 2
Area
IJ = 9.03 cm 2
J
F
C
Area
IJ7
Area ABCDEF
E
= 0.85
D
A
B
Area ABCDEF = 70.94 cm 2
G
H
Area
F
GH = 5.07 cm 2
C
Area
Area
GH12 = 60.80 cm 2
GH12
Area ABCDEF
= 0.86
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E
D
Lesson or Unit Design
Choose a modeling context appropriate for your students and
develop a lesson or plan a unit.
Consider the following while creating your plan:
 Do the problem yourself. Go through the four steps of the
modeling process.
 Scaffolding: What information will you provide? What openended questions will you ask? Brainstorm and list. Consider
your goal and the needs of your students.
The more you scaffold, the less they think and the fewer
opportunities they have to develop “patient problem solving.”





Identify the possible mathematics and standards addressed.
Identify possible challenges for yourself and your students.
Identify tools and materials your students might need.
How will you assess their level of understanding?
What additional support and resources do you need?
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Challenges to
Mathematical
Modeling….
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Systemic inertia
To put in perspective the limited large-scale
implementation of modeling it has proved difficult
to establish any profound innovation in the
mainstream mathematics curriculum.
Compared with, say, a home or a hospital, the
pattern of teaching and learning activities in the
mathematics classrooms we observe today is
remarkably similar to that we, and even our
grandparents, experienced as children.
Modelling in Mathematics Classrooms: reflections on past developments and the future
Hugh Burkhardt, Shell Centre, Nottingham and Michigan State Universities with contributions by Henry Pollak
ZDM 2006 Vol. 38 (2)
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Teacher and Student Role Changes
Responder
Manager
Explainer
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Mathematical Modeling
Mathematical Exercises
Unfamiliar
Familiar
Memorable
Forgettable
Relevant
Irrelevant
Many possible correct answers
One right answer
Lengthy
Brief
Complex
Simple
Discovering processes
Following instructions
Open-ended
Closed (goal chosen by teacher)
Cyclic—constant refining
Linear
Doesn’t appear on a particular page
Appears too often and then not enough
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The modeling
process is
enhanced
by…
4242
The modeling process is enhanced by:
The facilitative skill of the teacher.
The teacher must create a positive
and safe environment where student
ideas and questions are honored
and constructive feedback is given
by the teacher and by other
students. Students do the thinking,
problem solving and analyzing.
1.
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The modeling process is enhanced by:
1. The facilitative skill of the teacher. The teacher must create a positive and safe
environment where student ideas and questions are honored and constructive
feedback is given by the teacher and by other students. Students do the thinking,
problem solving and analyzing.
The content knowledge of the
teacher. The teacher understands
the mathematics relevant to the
context well enough to guide
students through questioning and
reflective listening.
2.
44
The modeling process is enhanced by:
1. The facilitative skill of the teacher. The teacher must create a positive and safe
environment where student ideas and questions are honored and constructive feedback is
given by the teacher and by other students. Students do the thinking, problem solving and
analyzing.
2. The content knowledge of the teacher. The teacher understands the
mathematics relevant to the context well enough to guide students through
questioning and reflective listening.
Teacher and student access to a
variety of representations, and
mathematical tools such as
manipulatives and technological tools
(sketchpad, spreadsheets, internet,
graphing calculators, etc.).
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3.
The modeling process is enhanced by:
1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where
student ideas and questions are honored and constructive feedback is given by the teacher and by other
students. Students do the thinking, problem solving and analyzing.
2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the
context well enough to guide students through questioning and reflective listening.
3. Teacher and student access to a variety of representations, and mathematical
tools such as manipulatives and technological tools (sketchpad, spreadsheets,
internet, graphing calculators, etc.).
Teacher and student understanding of
the modeling process. Teachers and
students who have had prior experience
have better understanding of the
modeling process and the use of
models.
4.
46
The modeling process is enhanced by:
1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where
student ideas and questions are honored and constructive feedback is given by the teacher and by other
students. Students do the thinking, problem solving and analyzing.
2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the
context well enough to guide students through questioning and reflective listening.
3. Teacher and student access to a variety of representations, and mathematical tools such
as manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing
calculators, etc.).
4. Teacher and student understanding of the modeling process. Teachers and
students who have had prior experience have better understanding of the
modeling process and the use of models.
5. Teacher and student understanding of the
context. Background information/experience
may be needed and gained through Internet
searches, print media, video, pictures,
samples, field trips, guest speakers, etc.
47
The modeling process is enhanced by:
1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where
student ideas and questions are honored and constructive feedback is given by the teacher and by other
students. Students do the thinking, problem solving and analyzing.
2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context
well enough to guide students through questioning and reflective listening.
3. Teacher and student access to a variety of representations, and mathematical tools such as
manipulatives and technological tools (sketchpad, spreadsheets, internet, graphing calculators, etc.).
4. Teacher and student understanding of the modeling process. Teachers and students who have had prior
experience have better understanding of the modeling process and the use of models.
5. Teacher and student understanding of the context. Background information/experience
may be needed and gained through Internet searches, print media, video, pictures, samples,
field trips, guest speakers, etc.
6. Richness of the problem to invite open-ended
investigation. Some problems invite a variety of
viable answers and multiple ways to represent
and solve. Some contrived problems may appear
to be real-world but are not realistic or
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cognitively demanding.
The modeling process is enhanced by:
1. The facilitative skill of the teacher. The teacher must create a positive and safe environment where student ideas and questions
are honored and constructive feedback is given by the teacher and by other students. Students do the thinking, problem solving and
analyzing.
2. The content knowledge of the teacher. The teacher understands the mathematics relevant to the context well enough to guide
students through questioning and reflective listening.
3. Teacher and student access to a variety of representations, and mathematical tools such as manipulatives and technological tools
(sketchpad, spreadsheets, internet, graphing calculators, etc.).
4. Teacher and student understanding of the modeling process. Teachers and students who have had prior experience have better
understanding of the modeling process and the use of models.
5. Teacher and student understanding of the context. Background information/experience may be needed and gained through
Internet searches, print media, video, pictures, samples, field trips, guest speakers, etc.
6. Richness of the problem to invite open-ended investigation. Some problems invite a variety of viable
answers and multiple ways to represent and solve. Some contrived problems may appear to be realworld but are not realistic or cognitively demanding.
7. Context of the problem. Selecting realworld problems is important, and real-world
problems that tap into student experience,
(prior and future), and interest are preferred.
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What extra skills do teachers need to
make this a reality? The key elements
here include:
1. Handling discussion in the class in a
non-directive but supportive way, so
that students feel responsible for
deciding on the correctness of their and
others' reasoning and do not to expect
either answers or confirmation from the
teacher;
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giving students time and
confidence to explore each problem
thoroughly, offering help only when
the student has tried, and
exhausted, various approaches
(rather than intervening at the first
signs of difficulty);
2.
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providing strategic guidance and
support without structuring the
problem for the student or giving
detailed suggestions
3.
finding supplementary questions
that build on each student's
progress and lead them to go
further.
4.
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Challenge
Teaching through modeling is a
paradigm shift.
The results are worth it:
Students retain more of what
they learn.
Students are more engaged.
Teachers are much less tired at
the end of the day!
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Let’s Have Some Fun!
With your group, select one of the
four problems.
Solve the problem:
Record your group thinking as you
do each step.
Go through the process as many
times as necessary until you feel
you have a good answer.
Each group will report to everyone.
Elevator Problem
Suppose a building has 5 floors (1 – 5) which are occupied by offices. The ground floor (0) is not used for
business purposes. Each floor has 80 people working on it, and there are 4 elevators available. Each elevator
can hold 10 people at one time. The elevators take 3 seconds to travel between floors and average 22 seconds
on each floor when someone enters or exits. If all of the people arrive at work at about the same time and enter
the elevators on the ground floor, how should the elevators be used to get the people to their offices as quickly
as possible?
Pasture Land
A rancher has a prize bull and some cows on his ranch. He has a large area for pasture that includes a stream
running along one edge. He must divide the pasture into two regions, one region large enough for the cows and
the other, smaller region to hold the bull. The bull’s pasture must be at least 1,000 sq. meters (grazing) and the
cow pasture must be at least 10,000 sq. meters to provide grazing for the cows. The shape of the pasture is
basically a rectangle 120 meters by 150 meters. The river runs all the way along the 120 meter side. Fencing
costs $5/meter and each fence post costs $10.00. Any straight edge of fence requires a post every 20 meters
and any curved length of fence requires a post every 10 meters. Help the rancher minimize his total cost of
fencing.
Faculty
When confronted with a rise of 142 students in a school of 480, and a capacity for 7 new teachers, in what
departments should the new teachers be placed? Placing the new teachers should maintain the ideal studentteacher ratio. The current makeup of the faculty and student enrollment in each department is:
Art 1, 99
Biology 4, 319 Chemistry 3, 294 English 5, 480 French 1, 122 German 1, 51
Spanish 1, 110 Mathematics 6, 613
Music 1, 95 Physics 3, 291 Social Studies 4, 363
A Country and Its Food Supply
The population of a country is initially 2 million people and is increasing at 4% per year. The country’s annual
food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an
additional 0.5 million people per year.
(a) Based on these assumptions, in approximately what year will this country first experience shortages of food?
(b) If the country doubled its initial food supply, would shortages still occur? If so, when? (Assume the other
conditions do not change).
(c) If the country doubled the rate at which its food supply increases, in addition to doubling its initial food
Thank You!
Do not hesitate to contact
me if you have questions
or want more information.
Margaret Kidd
session 365
mkidd@fullerton.edu