developing mathematical modeling skills and other...
TRANSCRIPT
2013 UCLA Curtis Center Mathematics and Teaching Conference
Developing Mathematical Modeling Skills and Other Mathematical Practices Simultaneously
Xuhui Li, Ph.D.
Department of Mathematics and Statistics
California State University – Long Beach
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Common Core State Standards in Math (CCSSM)
Modeling as One of the Standards for Math Practice
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
• Number and Quantity
• Algebra
• Functions
• Modeling
• Geometry
• Statistics and Probability
Modeling as One of the Conceptual Categories in
CCSSM Mathematics Standards for High School
Brainstorm
• What is a mathematical model?
• How is a mathematical model similar to or different from the object or phenomenon it models?
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Math Models as Physical/Pictorial Representations
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Plato’s Cosmological Theory
The universe is composed of five basic elements which are corresponding to the five regular polyhedra (a.k.a. Platonic Solids)
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Johannes Kepler’s Model of the Solar System
(Mysterium Cosmographicum, 1596)
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Regular Polyhedral Structures in the Real World
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Math Models as Graphical Representations
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Math Models as Symbolic/Tabular Representations
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Catenaries and the hyperbolic cosine function
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Math Models of Nature and Architectures
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1. The Pythagorean Theorem a2 + b2 = c2
2. Newton’s 2nd Law of Motion F = ma
3. Newton’s Law of Universal Gravitation 𝐹 = 𝐺𝑚𝑀
𝑟2
4. Euler’s Formula 𝑒𝑖𝑥= 𝑐𝑜𝑠𝑥 + 𝑖𝑠𝑖𝑛𝑥
5. The 2nd Law of Thermodynamics
The Top Ten Laws of Our World (Robert Crease: A Brief Guide to the Great Equations, 2009)
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6. Maxwell’s Equations
7. Einstein’s Mass-Energy Equivalence E = mc2
8. Einstein’s Field Equations
9. Schrödinger’s Equation
10. The Heisenberg Uncertainty Principle
The Top Ten Laws of Our World (Robert Crease: A Brief Guide to the Great Equations, 2009)
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Structures of Basic Types of Models in School Arithmetic and Algebra
• Additive : A + B = C, C – B = A, C – A = B; A ± x = C
• Multiplicative: AB = C, C/B = A, C/A = B; Ax = C
• Proportional: A/B = C/D, B/A = D/C, A/C = B/D, C/A = D/B
• Percent and percent change:
A ÷ B = c%, A ÷ c% = B, B*c% = A;
C = P ± P*r% = P(1 ± r%)
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• Recursive: an – an-1 = c; an / an-1 = c
• Linear: Ax + B = C; y = mx + b; Ax + By + C = 0
• Nonlinear (quadratic, power, radical, polynomial, rational, exponential, logarithmic, trigonometric, etc.)
• Absolute value and piecewise models
• Combinations of the above
• Inequality versions of the above
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Structures of Basic Types of Models in School Arithmetic and Algebra
Brainstorm (Work in groups and make a poster)
• What is mathematical modeling? Provide a one-sentence definition or description
• Use a diagram to demonstrate the major steps in mathematical modeling
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Definitions of Math Model and Modeling
A math model is a math structure that approximates the features of a phenomenon. The process of devising a math model is called math modeling (NCTM, 1991)
A math model is a representation of the behavior of real devices and objects in math terms (Dym, 2004)
A math model is a simplification or idealization of the real world using math terms, symbols, and ideas. Math modeling is the application of math to real-world problems (Albright, 2010)
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The CCSSM on Mathematical Modeling
Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions.
……
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.
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Modeling is Intertwined with Multiple Content Strands
Modeling is best interpreted not as a collection of isolated
topics but rather 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 (★).
Basic Modeling Cycle
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Modeling Involves a Variety of Math Activities
1. Identifying variables in the situation and selecting those that represent essential features
2. Formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables
3. Analyzing and performing operations on these relationships to draw conclusions
4. Interpreting the results of the mathematics in terms of the original situation
5. Validating the conclusions by comparing them with the situation, and then either improving the model or, if it is acceptable
6. Reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.
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Modeling is Interwoven with Other Math Practices
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education.
What is a Mathematical Practice?
The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections.
The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding, procedural fluency, and productive disposition
What is a Mathematical Practice?
The Dual Natures of Math Practice
Productive math activities and
processes
Math expertise, proficiencies,
habits of mind
The Dual Roles of Math Modeling
• An ultimate goal for student learning and math proficiency, accomplished by engaging students in other math practices
• An important vehicle for developing other math expertise and habits of mind
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In building a new stadium, concrete bricks are used to construct stairway seats. The following pictures demonstrate how the first three levels of a stairway are built gradually with squared-shaped bricks.
Model the situation and predict the number of bricks in a stairway with any given levels.
Modeling Activity: The Staircase Problem
Work on the handout in pairs/groups.
During the entire modeling process, pay close attention to:
• What other mathematical practices (as activities or processes) are involved, and when exactly?
• What other mathematical practices (as proficiencies or expertise) could be developed, and where exactly?
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The Handshake Problem
There are 20 attendees in Dr. Li’s session. If every two attendees greet and shake hands once with each other, how many total handshakes are made?
The Staircase Problem:
Differences 2 3 4 5 6
The Handshake Problem:
Differences 2 3 4 5 6
Model (1): Numerical Data Tables
Picture # 1 2 3 4 5 6 … …
# of Bricks 1 3 6 10 15 21 … …
# of students 2 3 4 5 6 7 … … # of
Handshakes 1 3 6 10 15 21 … …
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Model (2): Recursive Relationships (Equations of functions)
The Staircase Problem
S(1) = 1,
S(2) = S(1) + 2 = 3,
S(3) = S(2) + 3 = 6,
… …
S(n) = S(n-1) + n
S(n) + S(n-1) = n2
The Handshake Problem
H(2) = 1,
H(3) = H(2) + 2 = 3,
H(4) = H(3) + 3 = 6,
… …
H(n) = H(n-1) + (n-1)
H(n) + H(n-1) = (n-1)2
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Make sense of these recursive relations from the tables!
Model (3): Finding Explicit Formulas
The Staircase Problem
S(n) = 1 + 2 + 3 + 4 + …... + (n-1) + n = ???
The Handshake Problem
H(n) = 1 + 2 + 3 + …… + (n-1) = ???
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The Legendary Story of Little Gauss
1 + 2 + 3 + 4 + 5 …... + 99 + 100 = 5050
The Pair-up method The Reverse-doubling method
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General Formulas
S(n) = 1 + 2 + 3 + 4 + …... + (n-1) + n = n(n+1)/2
H(n) = 1 + 2 + 3 + 4 + …... + (n-1) = n(n–1)/2
Geometric Proof Proof by Mathematical Induction
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Enter the data tables into a graphing calculator as STAT lists L1, L2, L3, then plot them (STAT PLOT GRAPH), then find regression equations
Models (4) & (5): Plots and Regression Equations
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Equivalent Problem Situations
• Total number of connections in a network of n computers
• Total number of games among n sport teams
• Total number of edges and diagonals of an n-polygon
• Different pairs (2-combination) among n people or objects
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Summary
• What other mathematical practices (as activities or processes) are involved, and when exactly?
• What other mathematical practices (as proficiencies or expertise) could be developed, and where exactly?
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Curtis Math Conference 3/2/2013 Li
The Staircase Problem
In building a stadium, concrete bricks are used to construct stairway seats. The following pictures demonstrate how the first three levels of a stairway are built gradually with squared-shaped bricks.
Model the situation and predict the number of bricks in a stairway with any given levels.
Scaffolding questions:
1. Describe what the stairway looks like when it has 4, 5, 6, 7, and 8 levels.
2. How many bricks are used when the stairway has 1, 2, 3, …., and 8 levels? Record these
numbers in the following table:
Number of levels 1 2 3 4 5 6 7 8
Number of bricks
3. What kinds of patterns and relationships have you observed so far? Describe them clearly in
mathematical (algebraic) terms or equations.
4. How many bricks are used when the stairway has 20 levels? How do you know?
5. Find the direct formula that describes the total number of bricks, B(n), in terms of the number of
levels n in a stairway. (Hint: among others, one strategy is to use finite differences)
6. When a total of 406 bricks are used, how many levels does the stairway have?
During the entire modeling process, pay close attention to:
What other mathematical practices (as activities or processes) are involved, and when exactly?
What other mathematical practices (as proficiencies or expertise) could be developed, and where
exactly?
