Advanced Functions and Modeling

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Advanced Functions and Modeling

• Dr. David Royster, DirectorCenter for
  Mathematics, Science, and Technology
  EducationUNC Charlotte

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Advanced Courses
APStatistics
Core Courses
IBMathematicsLevel 2
Integrated Math 1
IntegratedMath 3
IntegratedMath 2
IntegratedMath 4
IntroductoryMathematics
TechnicalMath 1
APCalculus
TechnicalMath 2
Precalculus
Algebra 1
Algebra 2
Geometry
DiscreteMathematics
North Carolina High School MathematicsCourses,
Standard Course of Study(revised 06-Mar-03)
AdvancedFunctions Modeling
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WHAT IS IT?

• Advanced Functions and Modeling is a course to
  study functions and apply mathematical models,
  such as consumer issues, public policy,
  scientific investigation, recreation, etc.

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WHEN?

• In FALL 2006 all entering freshmen into a UNC
  system university must have completed 4
  mathematics courses with at least one beyond
  Algebra 2 read this to mean with Algebra 2 as
  a prerequisite.

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Courses Meeting new MAR

• Precalculus
• Discrete Mathematics
• AP Statistics
• AP Calculus
• Integrated Math 4
• IB Mathematics Level 2
• Advanced Functions and Modeling

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WHY?

• The UNC Office of the President studied freshmen
  entering the UNC system in Fall 1999 in an effort
  to understand retention, student success, and the
  impact of the previous Minimum Admission
  Requirements.

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WHY?

• The study found that students who have completed
  4 years of mathematics in high school
• averaged 90 points higher on SATs
• were at least 12 more likely to be accepted by a
  UNC system school
• were at least 12 more likely to be placed in a
  regular or advanced mathematics class
• were 2-6 more likely to stay after the first
  semester
• were 6-15 more likely to stay into their
  Sophomore year

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WHY?

• averaged 0.3-0.6 higher in first semester GPA
• averaged 0.3-0.5 higher in first year GPA
• were 14-24 more likely to have a first semester
  GPA of C or better
• were 17-28 more likely to have a first year GPA
  of C or better

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WHO?

• This is meant to be a course for those students
  who traditionally did NOT take a 4th mathematics
  course in high school.
• This comprised about 18 of the entering freshmen
  in 1999.
• It could be a much higher percentage of incoming
  freshmen 25-35

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What does this mean?

• This course should be a course which reviews
  algebra in context and allows students to RETAIN
  their mathematical skills as they move into
  college.
• This course should be an engaging course for
  students.

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What does this mean?

• The average student in this course will NOT be a
  future science, engineering or mathematics major.
• The average student in this course will probably
  NOT like mathematics.
• The average student in this course will see no
  use for mathematics.

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NC Standard Course of Study

• Prerequisites

• Describe phenomena as functions
• Translate among representations of relations
• Linear, quadratic, cubic, and exponential
  functions
• Systems of 2 (or more) equations
• Trigonometry (as ratios)
• Logic and deductive reasoning

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NC Standard Course of Study

• Goal 1 The learner will analyze data and apply
  probability concepts to solve problems

1.01 Create and use calculator-generated models
of linear, polynomial, exponential,
trigonometric, power, and logarithmic functions
of bivariate data to solve problems

• Interpret the constants, coefficients, and bases
  in the context of the data.
• Check models for goodness-of-fit use the most
  appropriate model to draw conclusions and make
  predictions.

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NC Standard Course of Study
1.01 Models of linear, polynomial, exponential
functions of bivariate data to solve problems
Linear Models Cell Phone Problem Exponential
Problems Compound Interest Drug
assimilation Polynomial Problems Cone
Volume Traffic Deaths Newspaper Circulation
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NC Standard Course of Study
1.02 Summarize and analyze univariate data to
solve problems.

• Apply and compare methods of data collection.
• Apply statistical principles and methods in
  sample surveys.
• Determine measures of central tendency and
  spread.
• Recognize, define, and use the normal
  distribution curve.
• Interpret graphical displays of univariate data.
• Compare distributions of univariate data.

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NC Standard Course of Study
1.02 Summarize and analyze univariate data to
solve problems.

• Basketball statistics
• Measuring hand span, arm span, etc. Analyze
• Car ratings Car Driver or Consumer Reports -
  data quartiles
• Doing a survey and stats associated different
  types of surveys
• Weather data / climatological data
• Use multiple technologies calculators, Excel,
  spreadsheets

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NC Standard Course of Study
1.03 Use theoretical and experimental probability
to model and solve problems.

• Use addition and multiplication principles.
• Calculate and apply permutations and
  combinations.
• Create and use simulations for probability
  models.
• Find expected values and determine fairness.
• Identify and use discrete random variables to
  solve problems.
• Apply the Binomial Theorem.

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NC Standard Course of Study
1.03 Use theoretical and experimental probability
to model and solve problems.

• Birthday problem
• Collecting cards
• Basketball problem paperwads into trashcan
• Capture release method for figuring out
  population
• Lets Make a Deal Monty Hall problem
• Expected value fairness
• State Fair games, rock-scissors-paper, Lottery
• TI-83 simulations
• Sports series probabilities
• Create a game and analyze it

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NC Standard Course of Study

• Goal 2 The learner will use functions to solve
  problems

2.01 Use logarithmic (common, natural) functions
to model and solve problems justify results

• Solve using tables, graphs, and algebraic
  properties.
• Interpret the constants, coefficients, and bases
  in the context of the problem.

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NC Standard Course of Study
2.01 Use logarithmic (common, natural) functions
to model and solve problems justify results

• Think of logarithms as solving exponential
  problems
• can enrich exponential study
• pH scale, Richter scale, decibel scale some
  applications
• Mathematics of Finance
• Growth and decay
• Half life of substance
• Sound, music, loudness
• Soil pH
• Water pH / water quality
• Re-expression of data

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NC Standard Course of Study
2.02 Use piecewise-defined functions to model and
solve problems justify results

• Solve using tables, graphs, and algebraic
  properties.
• Interpret the constants, coefficients, and bases
  in the context of the problem.

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NC Standard Course of Study
2.02 Use piecewise-defined functions to model and
solve problems justify results

• Tax rate problem / tax brackets (IRS)
• Orange grove problem
• Transportation problem / linear piecewise
• SAT data
• Parachute problem
• Car moving on different surfaces
• Hospital parking / cell phone use
• Birth rate / growth rate

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NC Standard Course of Study
2.03 Use power functions to model and solve
problems justify results

• Solve using tables, graphs, and algebraic
  properties.
• Interpret the constants, coefficients, and bases
  in the context of the problem.

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NC Standard Course of Study
2.03 Use power functions to model and solve
problems justify results
Inverse power models Distance-rate-time Sound /
light intensity (NCTM Illuminations) Projectile
problems air rockets Profit ticket
sales Interpret zeroes, max-min, etc in context
of problem Animation method of undetermined
coefficients Warming cooling problems Splines
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NC Standard Course of Study
2.04 Use trigonometric (sine, cosine) functions
to model and solve problems justify results

• Solve using tables, graphs, and algebraic
  properties.
• Create and identify transformations with respect
  to period, amplitude, and vertical and horizontal
  shifts.
• Develop and use the Law of Sines and Law of
  Cosines.

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NC Standard Course of Study
2.04 Use trigonometric (sine, cosine) functions
to model and solve problems justify results
New concept for these students. Clocks of
different diameters Tuning forks Weight
spring Amplitude / frequency modulation Shuttle
problem latitude, longitude on globe Predator
prey problems Sunrise-sunset Tides Daily max/min
temperature Calculate area of polygonal
region Indirect distance Measuring tall things
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NC Standard Course of Study
2.05 Use recursively-defined functions to model
and solve problems

• Find the sum of a finite sequence.
• Find the sum of an infinite sequence.
• Determine if a given series converges or
  diverges.
• Translate between recursive and explicit
  representations.

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NC Standard Course of Study
2.05 Use recursively-defined functions to model
and solve problems
Medication problem Linear sequences cell phone
charges, movie ticket prices Fibonacci sequence
100 a day for 20 days vs. 0.01 doubled each
day and added. Stones in pyramids Tower of
Hanoi Sphere-packing problem Forest management
problem Discrete model of population
growth Garbage problem / geometric series
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Frequently Asked Questions

• There are seven courses listed that meet the 4th
  course requirement. Are there other courses that
  would meet this requirement, such as
  Trigonometry, Probability and Statistics, and
  Algebra III?
• Suppose a student takes Algebra I in 8th grade,
  and then Geometry, Algebra II, and Pre-Calculus
  in high school and then no math during their
  senior year. They have 4 years of high school
  mathematics, but only 3 units of high school
  credit. Will this count for admission to a UNC
  school?

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Frequently Asked Questions

• Suppose a student takes Algebra I, Algebra II,
  Pre-Calculus, and Discrete Math in grades 9-12,
  but for some reason doesn't take Geometry. Will
  this sequence of courses satisfy the UNC
  requirements?
• Can students receive credits toward high school
  graduation for both Algebra III/Trig AND Advanced
  Functions and Modeling?

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Frequently Asked Questions

• Will the math faculty of non-public high schools
  have the same opportunities for staff development
  offered by the NC Mathematics and Science Network
  (NC MSEN) centers for the new Advanced Functions
  and Modeling course? Will there be a tuition
  charge?
• Our math faculty also needs staff development for
  the other fourth year courses, especially
  Discrete Math. Will this be offered at NC MSEN
  centers as well? If so, when? Will there be a
  fee involved? Will continuing education credit
  be available?

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Frequently Asked Questions

• My math faculty is concerned about the fact that
  a course in Trigonometry has not been included in
  the list of seven acceptable courses. We do not
  feel there is enough trig covered, especially at
  the standard level, in the courses included in
  the acceptable course list. How can we better
  prepare and indicate in course title that we have
  covered a trig curriculum?

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Frequently Asked Questions

• What are the SIMS (Student Information Management
  System) numbers of the seven acceptable 4th math
  courses for UNC Admission?
• Course Title SIMS
• AP Calculus-AB 2076
• AP Calculus-BC 2077
• AP Statistics (indicate AP in SIMS) 2065
• Pre-Calculus 2070
• Discrete Mathematics 2050
• IB Mathematics Level II 2078
• Integrated Mathematics IV 2054
• Advanced Functions and Modeling 2025

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Frequently Asked Questions

• When does our school system need to revise and
  rename a standard-level fourth course as Advanced
  Functions and Modeling?
• Although the new UNC math requirement does not go
  into effect until fall 2006, math textbooks will
  be chosen by high schools this year (2003-04).
  Is there a list of acceptable textbooks for the
  new Advanced Functions and Modeling course?

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Frequently Asked Questions

• In our school system, AP Calculus or AP
  Statistics credit is given only if a student
  takes the AP exam. If not, the student receives
  only honors credit. Honors Calculus and Honors
  Statistics are not listed as acceptable 4th math
  courses. What happens now to students who only
  receive honors credit?

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Frequently Asked Questions

• We presently have two standard-level fourth year
  courses Senior Review Class (for our slowest
  students - those who made a D in Algebra II)
  and Algebra III (for those who did well in
  Algebra II but are not quite ready for Honors
  Pre-Calculus. The Senior Review Class was
  renamed Advanced Functions and Modeling. What do
  we rename our Algebra III?? We want these
  students to meet the admission requirement too.

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Frequently Asked Questions

• Can students take Discrete Math and then Advanced
  Functions and Modeling?

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Frequently Asked Questions

• Who can we call at the Department of Public
  Instruction or the University of North Carolina
  Office of the President if we have questions
  about the new math admissions requirement?
• DPI Bill Scott, Secondary Math Consultant,
  919-807-3842
• UNC-OP Bobby Kanoy, Associate Vice Pres. for
  Academic Affairs, 919-962-1000

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Exponential Model

• A credit card for students advertises an annual
  percentage rate of 1.9, with finance charges
  calculated using simple interest on the monthly
  balance. The minimum payment on the credit card
  is 3 of the outstanding balance or 15,
  whichever is greater unless a balance of less
  than 15 remains, in which case your final
  payment equals this final balance plus the
  interest due on it. Suppose your credit card
  balance is 1,000 on Sept 1, 2003, and that you
  make only the minimum payment on the first of
  each month thereafter. If you make no further
  purchases on the card, when will you pay off the
  account? How much interest will you have paid
  over that time period?

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Excel Spreadsheet
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Other Models

• Amortization
• Temperature Data/Tide Data(Trigonometric
  Functions)
• pH Scale/Richter Scale/decibels(Logarithmic
  Functions)
• Population dynamics/Logistic data(Exponential
  functions)
• Falling bodies(Quadratic functions)
• Difference equations
• Fair Games
• Fair division of resources

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Resources

• COMAP HiMAP, TechMap, et al.
• Society for Industrial and Applied Mathematics
• Connected Curriculum Project
• Shodor Education Foundation
• Math Forum
• NC School of Science Math

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Web Sites

• http//www.ncee.net
• http//www.hsor.org
• http//www.nccee.org
• http//www.shodor.org
• http//www.math.uncc.edu/droyster/AFM