# Learning Standards for Mathematics

Mathematical Reasoning |
Number and Numeration |

1. Students use mathematical reasoning to analyze.mathematical situations, make conjectures, gather evidence, and construct an argument. Students: • 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 This is evident, for example, when students: prove that an altitude of an isosceles triangle, drawn to the base , is perpendicular to that base. determine whether or not a given logical sentence is a tautology. show that the triangle having vertex coordinates of (0,6), (0,0), and (5,0) is a right triangle. |
2. Students use number sense and numeration todevelop 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. Students: • 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. This is evident, for example, when students: determine from the discriminate of a quadratic equation whether the roots are rational or irrational. give rational approximations of irrational numbers to a specific degree of accuracy. determine for which value of x the expression is undefined. |

**Sample Problems**

Key ideas are identified by numbers (1). Performance indicators are identified by bullets (•). Sample tasks are identified by triangles (). |

**Students will understand mathematics and become
mathematically confident by
communicating and reasoning mathematically, by applying mathematics in
real-world settings, and by solving problems through the integrated study of
number systems, geometry, algebra, data analysis, probability, and trigonometry**

Operations |
Modeling/MultipleRepresentation |

3. Students use mathematical operations andrelationships among them to understand mathematics. Students: • 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. This is evident, for example, when students: determine the coordinates of triangle A(2,5), B(9,8), and C(3,6) after a translation (x,y) (x + 3, y - 1). evaluate the binary operation defined as x * y = x ^{2} + (y + x)^{2} for 3 * 4.identify the field properties used in solving the equation 2(x - 5) + 3 = x + 7. |
4. Students use mathematical modeling/multiplerepresentation to provide a means of presenting , interpreting, communicating, and connecting mathematical information and relationships. Students: • 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. This is evident, for example, when students: determine the locus of points equidistant from two parallel lines. explain why the basic construction of bisecting a line is valid . describe the various conics produced when the equation ax ^{2} + by^{2} = c^{2} is graphed for various values of a, b, and c. |

**Sample Problems**

Used with the permission of New Standards, copyright 1995. |

Measurement |
Uncertainty |

5. Students use measurement in both metric andEnglish measure to provide a major link between the abstractions of mathematics and the real world in order to describe and compare objects and data. Students: • 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. • use statistical methods including measures of central tendency to describe 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. This is evident, for example, when students: change mph to ft/sec. use the tangent ratio to determine the height of a tree. determine the distance between two points in the coordinate plane. |
6. Students use ideas of uncertainty to
illustrate thatmathematics involves more than exactness when dealing with everyday situations. Students: • 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. This is evident, for example, when students: construct a tree diagram or sample space for a compound event. calculate the probability of winning the New York State Lottery. develop simulations for probability problems for which they do not have theoretical solutions. |

**Sample Problems**

Used with the permission of New Standards, copyright 1995. |

Key ideas are identified by numbers (1). Performance indicators are identified by bullets (•). Sample tasks are identified by triangles (). |

**Students will understand mathematics and become
mathematically confident by
communicating and reasoning mathematically, by applying mathematics in
real-world settings, and by solving problems through the integrated study of
number systems, geometry, algebra, data analysis, probability, and trigonometry.**

**Patterns/Functions**

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

This is evident, for example, when students:

determine, in more than one way, whether or not a specific

relation is a function.

explain the relationship between the roots of a quadratic

equation and the intercepts of its corresponding graph.

use transformations to determine the inverse of a function.

**Sample Problem**

Used with the permission of

New Standards, copyright 1995.

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