Advanced Functions and Modeling

Advanced Functions and Modeling : Advanced Functions and Modeling Dr. David Royster, Director Center for Mathematics, Science, and Technology Education UNC Charlotte

Slide2 : Introductory Mathematics Algebra 1 Technical Math 1 Integrated Math 1 Geometry Technical Math 2 Integrated Math 2 Algebra 2 Integrated Math 3 Integrated Math 4 IB Mathematics Level 2 AP Statistics Advanced Functions & Modeling Discrete Mathematics Precalculus AP Calculus Core Courses Advanced Courses North Carolina High School Mathematics Courses, Standard Course of Study (revised 06-Mar-03)

WHAT IS IT? : 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.

WHEN? : 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”.

Courses Meeting new MAR : Courses Meeting new MAR Precalculus Discrete Mathematics AP Statistics AP Calculus Integrated Math 4 IB Mathematics: Level 2 Advanced Functions and Modeling

WHY? : 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.

WHY? : WHY? The study found that students who have completed 4 years of mathematics in high school: averaged 90 points higher on SAT’s 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

WHY? : 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

WHO? : 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%

What does this mean? : 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.

What does this mean? : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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 Let’s 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

NC Standard Course of Study : 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.

NC Standard Course of Study : 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

Frequently Asked Questions : 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?

Frequently Asked Questions : 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?

Frequently Asked Questions : 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?

Frequently Asked Questions : 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?

Frequently Asked Questions : 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

Frequently Asked Questions : 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?

Frequently Asked Questions : 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?

Frequently Asked Questions : 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.

Frequently Asked Questions : Frequently Asked Questions Can students take Discrete Math and then Advanced Functions and Modeling?

Frequently Asked Questions : 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

Exponential Model : 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?

Excel Spreadsheet : Excel Spreadsheet

Other Models : 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

Resources : 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

Web Sites : 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