New York State MST Standard 3

use models, facts, and relationships to draw conclusions about mathematics and explain their thinking.
use patterns and relationships to analyze mathematical situations.
justify their answers and solution processes.
use logical reasoning to reach simple conclusions.

apply a variety of reasoning strategies.
make and evaluate conjectures and arguments using appropriate language.
make conclusions based on inductive reasoning.
justify conclusions involving simple and compound (i.e., and/or) statements.

construct simple logical arguments.
follow and judge the validity of logical arguments.
use symbolic logic in the construction of valid arguments.
construct proofs based on deductive reasoning.

construct indirect proofs or proofs using mathematical induction.
investigate and compare the axiomatic structures of various geometries.

Key Idea 2
Number and Numeration:

Students use number sense and numeration to develop an understanding of the multiple uses of numbers in the real world, the use of numbers to communicate mathematically, and the use of numbers in the development of mathematical ideas.

Performance Indicators (Benchmarks)

use whole numbers and fractions to identify locations, quantify groups of objects, and measure distances.
use concrete materials to model numbers and number relationships for whole numbers and common fractions, including decimal fractions.
relate counting to grouping and to place-value.
recognize the order of whole numbers and commonly used fractions and decimals.
demonstrate the concept of percent through problems related to actual situations.

understand, represent, and use numbers in a variety of equivalent forms (integer, fraction, decimal, percent, exponential, expanded and scientific notation).
understand and apply ratios, proportions, and percents through a wide variety of hands-on explorations.
develop an understanding of number theory (primes, factors, and multiples).
recognize order relations for decimals, integers, and rational numbers.

understand and use rational and irrational numbers.
recognize the order of the real numbers.
apply the properties of the real numbers to various subsets of numbers.

understand the concept of infinity.
recognize the hierarchy of the complex number system.
model the structure of the complex number system.
recognize when to use and how to apply the field properties.

Key Idea 3
Operations:

Students use mathematical operations and relationships among them to understand mathematics.

Performance Indicators (Benchmarks)

add, subtract, multiply, and divide whole numbers.
develop strategies for selecting the appropriate computational and operational method in problem solving situations.
know single digit addition, subtraction, multiplication, and division facts.
understand the commutative and associative properties.

add, subtract, multiply, and divide fractions, decimals, and integers.
explore and use the operations dealing with roots and powers.
use grouping symbols (parentheses) to clarify the intended order of operations.
apply the associative, commutative, distributive, inverse, and identity properties.
demonstrate an understanding of operational algorithms (procedures for adding, subtracting, etc.).
develop appropriate proficiency with facts and algorithms.
apply concepts of ratio and proportion to solve problems.

use addition, subtraction, multiplication, division, and exponentiation with real numbers and algebraic expressions.
develop an understanding of and use the composition of functions and transformations.
explore and use negative exponents on integers and algebraic expressions.
use field properties to justify mathematical procedures.
use transformations on figures and functions in the coordinate plane.

use appropriate techniques, including graphing utilities, to perform basic operations on matrices.
use rational exponents on real numbers and all operations on complex numbers.
combine functions using the basic operations and the composition of two functions.

Key Idea 4
Modeling/Multiple Representation:

Students use mathematical modeling/multiple representation to provide a means of presenting, interpreting, communicating, and connecting mathematical information and relationships.

Performance Indicators (Benchmarks)

use concrete materials to model spatial relationships.
construct tables, charts, and graphs to display and analyze real-world data.
use multiple representations (simulations, manipulative materials, pictures, and diagrams) as tools to explain the operation of everyday procedures.
use variables such as height, weight, and hand size to predict changes over time.
use physical materials, pictures, and diagrams to explain mathematical ideas and processes and to demonstrate geometric concepts.

visualize, represent, and transform two- and three-dimensional shapes.
use maps and scale drawings to represent real objects or places.
use the coordinate plane to explore geometric ideas.
represent numerical relationships in one- and two-dimensional graphs.
use variables to represent relationships.
use concrete materials and diagrams to describe the operation of real world processes and systems.
develop and explore models that do and do not rely on chance.
investigate both two- and three-dimensional transformations.
use appropriate tools to construct and verify geometric relationships.
develop procedures for basic geometric constructions.

represent problem situations symbolically by using algebraic expressions, sequences, tree diagrams, geometric figures, and graphs.
manipulate symbolic representations to explore concepts at an abstract level.
choose appropriate representations to facilitate the solving of a problem.
use learning technologies to make and verify geometric conjectures .
justify the procedures for basic geometric constructions.
investigate transformations in the coordinate plane.
develop meaning for basic conic sections.
develop and apply the concept of basic loci to compound loci.
use graphing utilities to create and explore geometric and algebraic models.
model real-world problems with systems of equations and inequalities.

model vector quantities both algebraically and geometrically.
represent graphically the sum and difference of two complex numbers.
model and solve problems that involve absolute value, vectors, and matrices.
model quadratic inequalities both algebraically and graphically.
model the composition of transformations.
determine the effects of changing parameters of the graphs of functions.
use polynomial, rational, trigonometric, and exponential functions to model real-world relationships.
use algebraic relationships to analyze the conic sections.
use circular functions to study and model periodic real-world phenomena.
illustrate spatial relationships using perspective, projections, and maps.
represent problem situations using discrete structures such as finite graphs, matrices, sequences, and recurrence relations.
analyze spatial relationships using the Cartesian coordinate system in three dimensions.

Key Idea 5
Measurement:

Students use measurement in both metric and English measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data.

Performance Indicators (Benchmarks)

understand that measurement is approximate, never exact.
select appropriate standard and nonstandard measurement tools in measurement activities.
understand the attributes of area, length, capacity, weight, volume, time, temperature, and angle.
estimate and find measures such as length, perimeter, area, and volume using both nonstandard and standard units.
collect and display data.
use statistical methods such as graphs, tables, and charts to interpret data.

estimate, make, and use measurements in real-world situations.
select appropriate standard and nonstandard measurement units and tools to measure to a desired degree of accuracy.
develop measurement skills and informally derive and apply formulas in direct measurement activities.
use statistical methods and measures of central tendencies to display, describe, and compare data.
explore and produce graphic representations of data using calculators/computers.
develop critical judgment for the reasonableness of measurement.

derive and apply formulas to find measures such as length, area, volume, weight, time, and angle in real-world contexts.
choose the appropriate tools for measurement.
use dimensional analysis techniques.
and compare data.
use trigonometry as a method to measure indirectly.
apply proportions to scale drawings, computer-assisted design blueprints, and direct variation in order to compute indirect measurements.
relate absolute value, distance between two points, and the slope of a line to the coordinate plane.
understand error in measurement and its consequence on subsequent calculations.
use geometric relationships in relevant measurement problems involving geometric concepts.

derive and apply formulas relating angle measure and arc degree measure in a circle.
prove and apply theorems related to lengths of segments in a circle.
define the trigonometric functions in terms of the unit circle.
relate trigonometric relationships to the area of a triangle and to the general solutions of triangles.
apply the normal curve and its properties to familiar contexts.
design a statistical experiment to study a problem and communicate the outcomes, including dispersion.
use statistical methods, including scatter plots and lines of best fit, to make predictions.
apply the conceptual foundation of limits, infinite sequences and series, the area under a curve, rate of change, inverse variation, and the slope of a tangent line to authentic problems in mathematics and other disciplines.
determine optimization points on a graph.
use derivatives to find maximum, minimum, and inflection points of a function.

Key Idea 6
Uncertainty:

Students use ideas of uncertainty to illustrate that mathematics involves more than exactness when dealing with everyday situations.

Performance Indicators (Benchmarks)

make estimates to compare to actual results of both formal and informal measurement.
make estimates to compare to actual results of computations.
recognize situations where only an estimate is required.
develop a wide variety of estimation skills and strategies.
determine the reasonableness of results.
predict experimental probabilities. make predictions using unbiased random samples.
determine probabilities of simple events.

use estimation to check the reasonableness of results obtained by computation, algorithms, or the use of technology.
use estimation to solve problems for which exact answers are inappropriate.
estimate the probability of events.
use simulation techniques to estimate probabilities.
determine probabilities of independent and mutually exclusive events.

judge the reasonableness of results obtained from applications in algebra, geometry, trigonometry, probability, and statistics.
judge the reasonableness of a graph produced by a calculator or computer.
use experimental or theoretical probability to represent and solve problems involving uncertainty.
use the concept of random variable in computing probabilities.
determine probabilities using permutations and combinations.

interpret probabilities in real-world situations.
use a Bernoulli experiment to determine probabilities for experiments with exactly two outcomes.
use curve fitting to predict from data.
apply the concept of random variable to generate and interpret probability distributions.
create and interpret applications of discrete and continuous probability distributions.
make predictions based on interpolations and extrapolations from data.
obtain confidence intervals and test hypotheses using appropriate statistical methods.
approximate the roots of polynomial equations.

Key Idea 7
Patterns/Functions:

Students use patterns and functions to develop mathematical power, appreciate the true beauty of mathematics, and construct generalizations that describe patterns simply and efficiently.

Performance Indicators (Benchmarks)

recognize, describe, extend, and create a wide variety of patterns.
represent and describe mathematical relationships.
explore and express relationships using variables and open sentences.
solve for an unknown using manipulative materials.
use a variety of manipulative materials and technologies to explore patterns.
interpret graphs.
explore and develop relationships among two- and three-dimensional geometric shapes.
discover patterns in nature, art, music, and literature.

recognize, describe, and generalize a wide variety of patterns and functions.
describe and represent patterns and functional relationships using tables, charts and graphs, algebraic expressions, rules, and verbal descriptions.
develop methods to solve basic linear and quadratic equations.
develop an understanding of functions and functional relationships: that a change in one quantity (variable) results in change in another.
verify results of substituting variables.
apply the concept of similarity in relevant situations.
use properties of polygons to classify them.
explore relationships involving points, lines, angles, and planes.
develop and apply the Pythagorean principle in the solution of problems.
explore and develop basic concepts of right triangle trigonometry.
use patterns and functions to represent and solve problems.

use function vocabulary and notation.
represent and analyze functions using verbal descriptions, tables, equations, and graphs.
translate among the verbal descriptions, tables, equations and graphic forms of functions.
analyze the effect of parametric changes on the graphs of functions.
apply linear, exponential, and quadratic functions in the solution of problems.
apply and interpret transformations to functions.
model real-world situations with the appropriate function.
apply axiomatic structure to algebra and geometry.
use computers and graphing calculators to analyze mathematical phenomena.