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Problem Solving Strand

Students willbuild new mathematical knowledge through problem solving

Use a variety of problem solving strategies to understand new mathematical content

Recognize and understand equivalent representations of a problem situation or a mathematical concept

Students will solve problems that arise in mathematics and in other contexts

Observe and explain patterns to formulate generalizations and conjectures

Use multiple representations to represent and explain problem situations (e.g., verbally, numerically, algebraically, graphically)

Students will apply and adapt a variety of appropriate strategies to solve problems

Choose an effective approach to solve a problem from a variety of strategies (numeric, graphic, algebraic)

Use a variety of strategies to extend solution methods to other problems

Work in collaboration with others to propose, critique, evaluate, and value alternative approaches to problem solving

Students will monitor and reflect on the process of mathematical problem solving

Determine information required to solve a problem, choose methods for obtaining the information, and define parameters for acceptable solutions

Interpret solutions within the given constraints of a problem

Evaluate the relative efficiency of different representations and solution methods of a problem

Reasoning and Proof Strand

Students will recognize reasoning and proof as fundamental aspects of mathematics

Recognize that mathematical ideas can be supported by a variety of strategies

Students will make and investigate mathematical conjectures

Use mathematical strategies to reach a conclusion and provide supportive arguments for a conjecture

Recognize when an approximation is more appropriate than an exact answer

Students will develop and evaluate mathematical arguments and proofs

Develop, verify, and explain an argument, using appropriate mathematical ideas and language

Construct logical arguments that verify claims or counterexamples that refute them

Present correct mathematical arguments in a variety of forms

Evaluate written arguments for validity

Students will select and use various types of reasoning and methods of proof

Support an argument by using a systematic approach to test more than one case

Devise ways to verify results or use counterexamples to refute incorrect statements

Extend specific results to more general cases

Use a Venn diagram to support a logical argument

Apply inductive reasoning in making and supporting mathematical conjectures

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Communicate verbally and in writing a correct, complete, coherent, and clear design (outline) and explanation for the steps used in solving a problem

Use mathematical representations to communicate with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, Venn diagrams, and other diagrams

Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others

Present organized mathematical ideas with the use of appropriate standard notations, including the use of symbols and other representations when sharing an idea in verbal and written form

Explain relationships among different representations of a problem

Communicate logical arguments clearly, showing why a result makes sense and why the reasoning is valid

Support or reject arguments or questions raised by others about the correctness of mathematical work

Students will analyze and evaluate the mathematical thinking and strategies of others

Read and listen for logical understanding of mathematical thinking shared by other students

Reflect on strategies of others in relation to one's own strategy

Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others

Students will use the language of mathematics to express mathematical ideas precisely

Use correct mathematical language in developing mathematical questions that elicit, extend, or challenge other students' conjectures

Represent word problems using standard mathematical notation

Understand and use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale

Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing

Connections Strand

Students will recognize and use connections among mathematical ideas

Understand and make connections among multiple representations of the same mathematical idea

Understand the corresponding procedures for similar problems or mathematical concepts

Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole

Model situations mathematically, using representations to draw conclusions and formulate new situations

Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics

Understand how quantitative models connect to various physical models and representations

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize and apply mathematics to situations in the outside world

Recognize and apply mathematical ideas to problem situations that develop outside of mathematics

Develop an appreciation for the historical development of mathematics

Representation Strand

Students will create and use representations to organize, record, and communicate mathematical ideas

Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts

Recognize, compare, and use an array of representational forms

Use representation as a tool for exploring and understanding mathematical ideas

Students will select, apply, and translate among mathematical representations to solve problems

Select appropriate representations to solve problem situations

Investigate relationships between different representations and their impact on a given problem

Students will use representations to model and interpret physical, social, and mathematical phenomena

Use mathematics to show and understand physical phenomena (e.g., find the height of a building if a ladder of a given length forms a given angle of elevation with the ground)

Use mathematics to show and understand social phenomena (e.g., determine profit from student and adult ticket sales)

Use mathematics to show and understand mathematical phenomena (e.g., compare the graphs of the functions represented by the equations y = x2 and y = - x2 )

Number Sense and Operations Strand

Students will understand numbers, multiple ways of representing numbers, relationships among numbers, and number systems

Identify and apply the properties of real numbers (closure, commutative, associative, distributive, identity, inverse) Note: Students do not need to identify groups and fields, but students should be engaged in the ideas

Students will understand meanings of operations and procedures, and how they relate to one another

Simplify radical terms (no variable in the radicand)

Perform the four arithmetic operations using like and unlike radical terms and express the result in simplest form

Understand and use scientific notation to compute products and quotients of numbers

Solve algebraic problems arising from situations that involve fractions, decimals, percents (decrease/increase and discount), and proportionality/direct variation

Evaluate expressions involving factorial(s), absolute value(s), and exponential expression(s)

Determine the number of possible events, using counting techniques or the Fundamental Principle of Counting

Determine the number of possible arrangements (permutations) of a list of items

Algebra Strand

Students will represent and analyze algebraically a wide variety of problem solving situations

Translate a quantitative verbal phrase into an algebraic expression

Write a verbal expression that matches a given mathematical expression

Distinguish the difference between an algebraic expression and an algebraic equation

Translate verbal sentences into mathematical equations or inequalities

Write algebraic equations or inequalities that represent a situation

Analyze and solve verbal problems whose solution requires solving a linear equation in one variable or linear inequality in one variable

Analyze and solve verbal problems whose solution requires solving systems of linear equations in two variables

Analyze and solve verbal problems that involve quadratic equations

Analyze and solve verbal problems that involve exponential growth and decay

Solve systems of two linear equations in two variables algebraically (See A.G.7)

Solve a system of one linear and one quadratic equation in two variables, where only factoring is required Note: The quadratic equation should represent a parabola and the solution(s) should be integers

Students will perform algebraic procedures accurately

Multiply and divide monomial expressions with a common base, using the properties of exponents Note: Use integral exponents only

Add, subtract, and multiply monomials and polynomials

Divide a polynomial by a monomial or binomial, where the quotient has no remainder

Find values of a variable for which an algebraic fraction is undefined

Simplify fractions with polynomials in the numerator and denominator by factoring both and renaming them to lowest terms

Add or subtract fractional expressions with monomial or like binomial denominators

Multiply and divide algebraic fractions and express the product or quotient in simplest form

Identify and factor the difference of two perfect squares

Factor algebraic expressions completely, including trinomials with a lead coefficient of one (after factoring a GCF)

Determine whether a given value is a solution to a given linear equation in one variable or linear inequality in one variable

Solve all types of linear equations in one variable

Solve literal equations for a given variable

Solve linear inequalities in one variable

Solve equations involving fractional expressions Note: Expressions which result in linear equations in one variable

Solve algebraic proportions in one variable which result in linear or quadratic equations

Understand and apply the multiplication property of zero to solve quadratic equations with integral coefficients and integral roots

Understand the difference and connection between roots of a quadratic equation and factors of a quadratic expression

Students will recognize, use, and represent algebraically patterns, relations, and functions

Use set-builder notation and/or interval notation to illustrate the elements of a set, given the elements in roster form

Find the complement of a subset of a given set, within a given universe

Find the intersection of sets (no more than three sets) and/or union of sets (no more than three sets)

Explain slope as a rate of change between dependent and independent variables

Determine the slope of a line, given the coordinates of two points on the line

Write the equation of a line, given its slope and the coordinates of a point on the line

Write the equation of a line, given the coordinates of two points on the line

Write the equation of a line parallel to the x- or y-axis

Determine the slope of a line, given its equation in any form

Determine if two lines are parallel, given their equations in any form

Determine whether a given point is on a line, given the equation of the line

Determine whether a given point is in the solution set of a system of linear inequalities

Determine the vertex and axis of symmetry of a parabola, given its equation (See A.G.10 )

Find the sine, cosine, and tangent ratios of an angle of a right triangle, given the lengths of the sides

Determine the measure of an angle of a right triangle, given the length of any two sides of the triangle

Find the measure of a side of a right triangle, given an acute angle and the length of another side

Determine the measure of a third side of a right triangle using the Pythagorean theorem, given the lengths of any two sides

Geometry Strand

Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes

Find the area and/or perimeter of figures composed of polygons and circles or sectors of a circle Note: Figures may include triangles, rectangles, squares, parallelograms, rhombuses, trapezoids, circles, semi-circles, quarter-circles, and regular polygons

Use formulas to calculate volume and surface area of rectangular solids and cylinders

Students will apply coordinate geometry to analyze problem solving situations

Determine when a relation is a function, by examining ordered pairs and inspecting graphs of relations

Identify and graph linear, quadratic (parabolic), absolute value, and exponential functions

Investigate and generalize how changing the coefficients of a function affects its graph

Graph linear inequalities

Graph and solve systems of linear equations and inequalities with rational coefficients in two variables (See A.A.10)

Find the roots of a parabolic function graphically Note: Only quadratic equations with integral solutions

Solve systems of linear and quadratic equations graphically Note: Only use systems of linear and quadratic equations that lead to solutions whose coordinates are integers

Determine the vertex and axis of symmetry of a parabola, given its graph (See A.A.41) Note: The vertex will have an ordered pair of integers and the axis of symmetry will have an integral value

Measurement Strand

Students will determine what can be measured and how, using appropriate methods and formulas

Calculate rates using appropriate units (e.g., rate of a space ship versus the rate of a snail)

Solve problems involving conversions within measurement systems, given the relationship between the units

Students will understand that all measurement contains error and be able to determine its significance

Calculate the relative error in measuring square and cubic units, when there is an error in the linear measure

Statistics and Probability Strand

Students will collect, organize, display, and analyze data

Categorize data as qualitative or quantitative

Determine whether the data to be analyzed is univariate or bivariate

Determine when collected data or display of data may be biased

Compare and contrast the appropriateness of different measures of central tendency for a given data set

Construct a histogram, cumulative frequency histogram, and a box-and-whisker plot, given a set of data

Understand how the five statistical summary (minimum, maximum, and the three quartiles) is used to construct a box-and-whisker plot

Create a scatter plot of bivariate data

Construct manually a reasonable line of best fit for a scatter plot and determine the equation of that line

Analyze and interpret a frequency distribution table or histogram, a cumulative frequency distribution table or histogram, or a box-and-whisker plot

Evaluate published reports and graphs that are based on data by considering: experimental design, appropriateness of the data analysis, and the soundness of the conclusions

Find the percentile rank of an item in a data set and identify the point values for first, second, and third quartiles

Identify the relationship between the independent and dependent variables from a scatter plot (positive, negative, or none)

Understand the difference between correlation and causation

Identify variables that might have a correlation but not a causal relationship

Students will make predictions that are based upon data analysis

Identify and describe sources of bias and its effect, drawing conclusions from data

Recognize how linear transformations of one-variable data affect the data's mean, median, mode, and range

Use a reasonable line of best fit to make a prediction involving interpolation or extrapolation

Know the definition of conditional probability and use it to solve for probabilities in finite sample spaces

Determine the number of elements in a sample space and the number of favorable events

Calculate the probability of an event and its complement

Determine empirical probabilities based on specific sample data

Determine, based on calculated probability of a set of events, if: some or all are equally likely to occur; one is more likely to occur than another; whether or not an event is certain to happen or not to happen

Calculate the probability of: a series of independent events; a series of dependent events; two mutually exclusive events; two events that are not mutually exclusive

Information Source: NYSED.GOV CS&IT (standards pdf)