# Mathematical Modeling in Elementary School by E96nnen

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```									Mathematical Modeling in K-12

December 9, 2011
From Here to There
• Typical first grade place value problem:
– Write two hundred fifty-three as a numeral.
• 253
What does 253 look like?
• Two hundred fifty-three
• Two hundred + fifty + three
• 200 + 50 + 3

SMP 1: make sense of problems
What does 253 look like?
• My father told me that he has been saving
money for Christmas. Right now he has \$253
saved.

2 one hundreds + 1 fifty + 3 ones.

SMP 2: Reason abstractly & quantitativelyc
\$253!

2 hundreds + 1 fifty + 3 ones.
2 hundreds + 2 twenties + 1 ten + 3 ones.

SMP 3: Construct viable arguments and critique the reasoning of others.
What does \$253 look like?
• Using the most number of bills?
• Using the least number of bills?
• How many different ways can we make \$253
using U.S. bills?

SMP 7: Look for and make use of structure.
SMP 8: Look for and express regularity in repeated reasoning.
What does 253 look like?
• The sum of the ages of everyone who lives in my
house is 253 years. What could each person’s age be
if the following people live in my house:
–   My grandmother
–   My grandfather
–   My mother
–   My father
–   My three brothers and sisters.
–   Me

SMP 2: Reason abstractly and quantitatively
What does 253 look like?
• I have \$253 to spend at the mall. Here are
some things I like. What can I buy and still
have more than \$50 left?

Video games         Manga           T-shirts          Jeans        Movie Tickets
\$19.99 each         books          \$9.98 each      \$15.98 each     \$10.00 each
\$6.99 each

SMP 3: Construct viable arguments and critique the reasoning of others.
What else does 253 look like?
New Room, Blue Room
• Marco gets to paint the walls of his room and
put down a new floor in his room. That is,
provided he can determine the cost of this
endeavor and it is not more than \$253! (You
knew this, right?)
• Let’s go to the next slide to determine his
options!

SMP ???
Choices
• Paint                           • Floor tiles
– Marco wants Sherwin              – Marco wants to use large
Williams. The cost                 stone tiles that are 18” x
choices for 1 gallon of            18” in a crème color. His
colors he likes are as             choices are as follows:
follows:                            • Crema at Lowe’s for \$12
• Deep Sea Dive Blue paint           each
Lowe’s \$27                       • Caffe con Leche at Lowe’s
• Loyal Blue paint                   for \$10 each.
Home Depot \$25
• Denim Blue paint
Home Depot \$30.
Parameters
• One can of paint can     • The floor is 12 ft x 12 ft.
cover approximately      • He has all paint brushes
350 sq. feet.              and tools and other
• A diagram of each wall     materials on hand from
of his room in on the      his parents.
next slide.              • The cost of labor is \$0
as he will do this with
the help of his family.
One inch represents 4 ft.
Modeling Cycle

Problem           Formulate   Validate    Report

Compute     Interpret

meaningful problem
based in a context
Standards of Mathematical Practice
• The Standards for Mathematical Practice
describe varieties of expertise that
mathematics educators at all levels should
seek to develop in their students.
Standard 4: Model with Mathematics
• Mathematically proficient students can apply the mathematics they know
to solve problems arising in everyday life, society, and the workplace. In
early grades, this might be as simple as writing an addition equation to
describe a situation. In middle grades, a student might apply proportional
reasoning to plan a school event or analyze a problem in the community.
By high school, a student might use geometry to solve a design problem or
use a function to describe how one quantity of interest depends on
another. Mathematically proficient students who can apply what they
know are comfortable making assumptions and approximations to simplify
a complicated situation, realizing that these may need revision later. They
are able to identify important quantities in a practical situation and map
their relationships using such tools as diagrams, two-way tables, graphs,
flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their
mathematical results in the context of the situation and reflect on
whether the results make sense, possibly improving the model if it has not
served its purpose.
Situation
•   Definition of SITUATION
1. a: the way in which something is placed in relation to its
surroundings
b: site
2. a: position or place of employment: post, job
b: position in life: status
3. position with respect to conditions and circumstances
4. a: relative position or combination of circumstances at a
certain moment
b: a critical, trying, or unusual state of affairs: problem
c: a particular or striking complex of affairs at a stage in the
action of a narrative or drama

http://www.merriam-webster.com/dictionary/situation
What are the “favorite” careers
dreamed of by 4th grade students?
• You gave the assignment, they collected the
data, and on the next slide is what they came
up with:
Future Careers of 4th Grade Students

C ounse llo r                                                 60

Singe r                    22

Acto r                                   45

Spo rts pla ye r                                                      65

Dentist                      25

La wye r                                              55

T e a che r                                      50

Nurse                           30

Docto r                                     48

0   10     20         30   40        50        60        70

Number of Students

SMP ???
1. What was the most popular career chosen by
2. What was the least popular career chosen by
3. Which career was chosen by 55 students?
4. List the careers chosen by the 4th grade
students from most to least popular.
5. Which career was chosen by 65 students?
6. How many students chose a medical career?

SMP ???
Build a function that models a
relationship between two quantities
• The price of gasoline keeps going up.
A          B

1   Price of Gasoline
2

What has the price been over time? The        3
4
Year
1976
Jan
0.605
5     1977        0.627

Bureau of Labor Statistics gives us the       6
7
1978
1979
0.648
0.716
8     1980        1.131

data on the right for the average price of    9
10
1981
1982
1.298
1.358
11     1983        1.230

gasoline in the U.S. from 1976 until 2004.   12
13
1984
1985
1.216
1.148
14     1986        1.194

What can we learn from this data?            15
16
1987
1988
0.862
0.933
17     1989        0.918
18     1990        1.042
19     1991        1.247
20     1992        1.073
21     1993        1.117
22     1994        1.043
23     1995        1.129
24     1996        1.129
25     1997        1.261
26     1998        1.131
27     1999        0.972
28     2000        1.301
29     2001        1.472
30     2002        1.139
31     2003        1.473
32     2004        1.592

SMP ???
A          B

1   Price of Gasoline
2
3     Year        Jan
4     1976        0.605
5     1977        0.627
6     1978        0.648
7     1979        0.716
8     1980        1.131
9     1981        1.298
10     1982        1.358
11     1983        1.230
12     1984        1.216
13     1985        1.148
14     1986        1.194
15     1987        0.862
16     1988        0.933
17     1989        0.918
18     1990        1.042
19     1991        1.247
20     1992        1.073
21     1993        1.117
22     1994        1.043
23     1995        1.129
24     1996        1.129
25     1997        1.261
26     1998        1.131
27     1999        0.972
28     2000        1.301
29     2001        1.472
30     2002        1.139
31     2003        1.473
32     2004        1.592
Average Cost of Gasoline in the U.S. in January

1.600
1.400
1.200
Cost in Dollars

1.000
0.800
0.600
0.400
0.200
0.000
1974   1976   1978   1980   1982   1984   1986   1988     1990    1992    1994    1996    1998    2000    2002    2004
Year

Average Cost of Gasoline in the U.S. in January                     y = 0.0179x - 34.524
2
R = 0.3779
1.600
1.400
1.200
Cost in Dollars

1.000
0.800
0.600
0.400
0.200
0.000
1974   1976   1978   1980   1982   1984   1986   1988    1990    1992    1994    1996    1998    2000    2002    2004
Year
The Corral Problem

•    Rosa needs your help designing a corral for her horses.
Rosa has looked at lots of designs, and has decided
two things:
1.     She wants a corral that is the shape of a rectangle.
2.     She wants the corral to give her horse the largest possible area.
•    Rosa has 16 units of fence to use. The sides of the
corral must be made up of whole units of fence (e.g.,
a side cannot be 2½ units long.)
•    What advice would you give Rosa? Be detailed in

Mathematics Teaching in the Middle School NCTM Volume 5, No. 4 Dec. 1999
SMP ???
Movie Tickets
A movie theater charges \$7.00 per ticket. At that
price, theater owners can expect to sell 1100 tickets.
They also know that for every 10 cent increase in
ticket price, they will sell 20 fewer tickets. However,
for every 10 cent reduction in ticket price, they will
sell an additional 20 tickets. The theater owners
want your class to determine whether they should
raise or lower the price per ticket? They also want
to know what ticket price will maximize their
income?

SMP ???
Clean up!
Your neighborhood wants to host a yard clean-
up on a Saturday next month. They have asked
your class to help plan this event. They want
you to determine how much this will cost for
supplies. They will clean yards, alleys, driveways
and streets. All decisions about how to do this
are yours.

SMP ???
Modeling in the Common Core

• Descriptive Modeling
– Describes or summarizes phenomena in a
compact form.
• E.g., Graphs of observations such as of global
temperature & atmospheric CO2 over time.
• Analytic modeling
– Seeks to explain data on the basis of
deeper theoretical ideas, but with
parameters that are empirically based.
• E.g., exponential growth of bacterial colonies
follows from a constant reproduction rate.
Modeling in the Common Core
• Models devised depend on a number of factors:
– How precise an answer do we want or need?
– What aspects of the situation do we most need to
understand, control, or optimize?
– What resources of time & tools do we have?
– What are the limits of our mathematical, statistical, &
technical skills, & our ability to recognize significant
variables & relationships among them.

Do we have more
than 1 period?
Is this a life
What
Is this a 3rd grade class? do we care technology
What
or death situation?
is
Algebra I? precalculus? most about? available?
the
AP Statistics?
High School Standards Directly
Associated with Modeling
• Number & Quantity
– Quantities
• Reason quantitatively & use units to solve problems.
• Algebra
– Creating Equations
• Create equations that describe numbers or relationships
• Functions
– Building Functions
• Build a function that models a relationship between two quantities
– Linear, Quadratic, & Exponential Models
• Construct & compare linear, quadratic, & exponential models &
solve problems
Summary
• Modeling can be done from Kindergarden
through the Calculus.
• Modeling crosses all domains of mathematics
• Modeling integrates all of the standards of
mathematical practice.
• Modeling takes a situation in the real world,
removes it from the real world and places it in
mathematics, solves it, then replaces it in the real
world to see if the mathematical solution fits.
Mathematical Modeling Test
1. Modeling in mathematics refers to which of
the following (there can be more than one