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?
• What about coins…..?
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
$19.99 each
Manga
books
$6.99 each
T-shirts
$9.98 each
Jeans
$15.98 each
Movie Tickets
$10.00 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
– Marco wants Sherwin
Williams. The cost
choices for 1 gallon of
colors he likes are as
follows:
• Deep Sea Dive Blue paint
Lowe’s $27
• Loyal Blue paint
Home Depot $25
• Denim Blue paint
Home Depot $30.
• Floor tiles
– Marco wants to use large
stone tiles that are 18” x
18” in a crème color. His
choices are as follows:
• Crema at Lowe’s for $12
each
• Caffe con Leche at Lowe’s
for $10 each.
Parameters
• One can of paint can
cover approximately
350 sq. feet.
• A diagram of each wall
of his room in on the
next slide.
• The floor is 12 ft x 12 ft.
• He has all paint brushes
and tools and other
materials on hand from
his parents.
• 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
meaningful problem
based in a context
students care about.
Formulate
Validate
Compute
Interpret
Report
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
60
C ounsellor
22
Singer
45
Actor
65
Sports player
25
Dentist
55
Lawyer
50
Teacher
30
Nurse
48
Doctor
0
10
20
30
40
Number of Students
SMP ???
50
60
70
Questions we could ask…
1. What was the most popular career chosen by
4th grade students?
2. What was the least popular career chosen by
4th grade students?
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.
What has the price been over time? The
Bureau of Labor Statistics gives us the
data on the right for the average price of
gasoline in the U.S. from 1976 until 2004.
What can we learn from this data?
SMP ???
A
1
B
Price of Gasoline
2
3
Year
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
Jan
A
1
B
Price of Gasoline
2
3
Year
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
Jan
Average Cost of Gasoline in the U.S. in January
1.600
C o s t in D o lla r s
1.400
1.200
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
R
2
= 0.3779
1.600
C o s t in D o l l a r s
1.400
1.200
1.000
0.800
0.600
0.400
0.200
0.000
1974
1976
1978
1980
1982
1984
1986
1988
1990
Year
1992
1994
1996
1998
2000
2002
2004
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.
2.
•
•
She wants a corral that is the shape of a rectangle.
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
your explanation.
SMP ???
Mathematics Teaching in the Middle School NCTM Volume 5, No. 4 Dec. 1999
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 cleanup 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.
Is this a life
or death situation?
Do we have more
than 1 period?
What technology
Is this a 3rd grade class?
What do we care
Algebra I? precalculus?
the most about?is available?
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
correct answer.):
Answer: It depends on the question being asked!