Appendix B:
Elements Of Functional Mathematics
These elements outline aspects of mathematics that are important for all students in their life and work. They emphasize concrete, realistic topics that arise in common situations in news, sports, finance, work, and leisure. These elements can be taught through many different curricula ranging from traditional to reform, from academic to vocational. Students completing any curriculum that includes these elements would be well-prepared to enter a wide variety of technical and academic programs, including a one-year precalculus course.
Numbers and Data
Mental Estimation.Anticipate total costs, distances, times; estimate unknown quantities (e.g., number of high school students in a state or city) using proportional reasoning; order of magnitude estimates; mental checking of calculator and computer results.
Numbers.Examples of whole numbers (integers), fractions (rational numbers), and irrational numbers ( ). Number line; mixed numbers; decimals, percentages, scientific notation. Prime numbers, factors; simple number theory; binary numbers and simple binary arithmetic; units and magnitudes; extreme numbers (e.g., national debt, astronomical distances); number sense; scientific notation.
Calculation.Accurate paper-and-pencil methods for simple arithmetic and percentage calculations; calculator use for complex calculations; spreadsheet methods for problems with a lot of data. Strategies for checking reasonableness and accuracy. Significant digits; interval arithmetic; errors; tolerances. Mixed methods (mental, pencil, calculator).
Coding.Number systems (decimal, binary, octal, hex); ASCII code; check digits. Patterns and criteria for credit card, Social Security, telephone, license plate numbers.
Index Numbers.Examples in the news: stock market averages; consumer price index; unemployment rate; SAT scores. Definitions and deficiencies; uses and abuses.
Information Systems.Collecting and organizing data; geographic information systems (GIS) and management information systems (MIS); visual representation of data.
Measurement and Space
Measurement.Direct and indirect means; estimation; use of appropriate instruments (rulers, tapes, micrometers, pacing, electronic gauges); plumb lines and square corners; calculated measurements; accuracy; tolerances; detecting and correcting misalignments.
Measurement Geometry.Measurement formulas for simple plane figures: triangles, circles, quadrilaterals. Calculation of area, angles, lengths by indirect means. Right triangle trigonometry; applications of Pythagorean theorem.
Dimensions.Linear, square, and cubic growth of length, area, volume. Coordinate notation; dimension as factor in multivariable phenomena.
Geometric Relations.Proof of Pythagorean theorem and of other basic theorems. Construction of line and angle bisectors, finding center of circular arc.
Spatial Geometry.Shapes in space; volumes of cylinders and spheres; calculation of angles in three-dimensions (e.g., meeting of roof trusses). Interpreting construction diagrams; nominal vs. true dimensions (e.g., of 2 x 4s); tolerances and perturbations in constructing three-dimensional objects.
Global Positioning:Map projections, latitude and longitude, global positioning systems (GPS); local, regional, and global coordinate systems.
Growth and Variation
Linear Change.Situations in which the rate of change is constant (e.g., uniform motion); contrast with examples where change is nonlinear (e.g., distance vs. time for falling body). Slope as rate of change; slope-intercept equation, with graphical significance of parameters. Difference between rate of change and value of the dependent variable.
Proportion.Situations modeled by similarity and ratio (e.g., height and shadows, construction cost vs. square footage, drug dose vs. body weight); examples where change is disproportional (e.g., height vs. weight). Calculating missing terms. Mental estimation using proportions.
Exponential Growth.Situations such as population growth, radioactivity, and compound interest, where the rate of change is proportional to size; doubling time and half-life as characteristics of exponential phenomena; symbolic representation (2 n, 10 n); ordinary and log-scaled graphs.
Normal Curve.Situations such as distribution of heights, of repeated measurements, and of manufactured goods in which phenomena distribute in a bell-shaped curve. Examples of situations in which they do not (e.g., income, grades, typographical errors, life spans). Parameters and percentages of normal distribution; z-scores, meaning of 1-, 2-, and 3-s. Area as measure of probability.
Parabolic Patterns.Falling bodies; parabolas; quadratic equations; optimization problems.
Cyclic Functions.Situations such as time of sunrise, sound waves, and biological rhythms that exhibit cyclic behavior. Graphs of sin and cos; relations among graphs; sin2u + cos2u = 1.
Chance and Probability
Elementary Data Analysis.Measures of central tendency (average, median, mode) and of spread (range, standard deviation, midrange); visual displays of data (pie charts, scatter plots, bar graphs, box and whisker charts). Distributions. Quality control charts. Recognizing and dealing with outliers. "Data = Pattern + Noise."
Probability.Chance and randomness; calculating odds in common situations (dice, coin tosses, lotteries); expected value. Binomial probability, random numbers, hot streaks, binomial approximation of normal distribution; computer simulations; estimating area by Monte Carlo methods. Two-way tables; bias paradoxes.
Risk Analysis.Common examples of risks (e.g., accidents, diseases, causes of death, lotteries). Ways of estimating risk. Confounding factors. Communicating and interpreting risk.
Reasoning and Inference
Statistical Inference.Rationale for random samples; double-blind experiments; surveys and polls; confidence intervals. Causality vs. correlation. Multiple factors; interaction effects; hidden factors. Judging validity of statistical claims in media reports. Making decisions based on data (e.g., research methods, medical procedures).
Scientific Inference.Gathering data; detecting patterns; making conjectures; testing conjectures; drawing inferences.
Mathematical Inference.Logical reasoning and deduction; assumptions and conclusions; axiomatic systems; theorems and proofs; proof by direct deduction, by indirect argument, and by "mathematical induction."
Verification.Levels of convincing argument; persuasion and counterexamples; logical deduction; legal reasoning ("beyond reasonable doubt" vs. "preponderance of evidence"; court decisions interpreting various logical options); informal inference (suspicion, experience, likelihood); classical proofs (e.g., isosceles triangle, infinitude of primes).
Variables and Equations
Algebra.Variables, constants, symbols, parameters; equations vs. expressions. Direct and indirect variation; inverse relations; patterns of change; rates of change. Graphical representations; translation between words and graphs. Symbols and functions.
Equations.Linear and quadratic; absolute value; 2 x 2 systems of linear equations; inequalities; related graphs.
Graphs.Interpretation of graphs; sketching graphs based on relations of variables; connection between graphs and function parameters.
Algorithms.Alternative arithmetic algorithms; flowcharts; loops; constructing algorithms; maximum time vs. average time comparisons.
Modeling and Decisions
Financial Mathematics.Percentages, markups, discounts; simple and compound interest; taxes; investment instruments (stocks, mortgages, bonds); loans, annuities, insurance, personal finance.
Planning.Allocating resources; management information systems; preparing budgets; determining fair division; negotiating differences; scheduling processes, decision trees; PERT charts; systems thinking.
Mathematical Modeling.Abstracting mathematical structures from real-world situations; reasoning within mathematical models; reinterpreting results in terms of original situations; testing interpretations for suitability and accuracy; revision of mathematical structure; repetition of modeling cycle.
Scientific Modeling.Role of mathematics in modeling aspects of science such as acceleration, astronomical geometry, electrical current, genetic coding, harmonic motion, heredity, stoichiometry.
Technological Tools.Familiarity with standard calculator and computer tools: scientific and graphing calculators (including solving equations via graphs); spreadsheets (including presentation of data via charts); statistical packages (including graphical displays of data).