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Problem Solving Strand

Students will build new mathematical knowledge through problem solving

Use a variety of strategies to understand new mathematical content and to develop more efficient methods

Construct appropriate extensions to problem situations

Understand and demonstrate how written symbols represent mathematical ideas

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

Observe patterns and formulate generalizations

Make conjectures from generalizations

Represent problem situations verbally, numerically, algebraically, and graphically

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

Understand that there is no one right way to solve mathematical problems but that different methods have advantages and disadvantages

Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem

Work backwards from a solution

Use proportionality to model problems

Work in collaboration with others to solve problems

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

Interpret solutions within the given constraints of a problem

Set expectations and limits for possible solutions

Determine information required to solve the problem

Choose methods for obtaining required information

Justify solution methods through logical argument

Evaluate the efficiency of different representations 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

Evaluate conjectures by distinguishing relevant from irrelevant information to reach a conclusion or make appropriate estimates

Students will develop and evaluate mathematical arguments and proofs

Provide supportive arguments for conjectures

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

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

Apply inductive reasoning in making and supporting mathematical conjectures

Communication Strand

Students will organize and consolidate their mathematical thinking through communication

Provide a correct, complete, coherent, and clear rationale for thought process used in problem solving

Provide an organized argument which explains rationale for strategy selection

Organize and accurately label work

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

Share organized mathematical ideas through the manipulation of objects, numerical tables, drawings, pictures, charts, graphs, tables, diagrams, models and symbols in written and verbal form

Answer clarifying questions from others

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

Analyze mathematical solutions shared by others

Compare strategies used and solutions found by others in relation to their own work

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

Increase their use of mathematical vocabulary and language when communicating with others

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

Recognize connections between subsets of mathematical ideas

Connect and apply a variety of strategies to solve problems

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

Students will recognize and apply mathematics in contexts outside of mathematics

Recognize and provide examples of the presence of mathematics in their daily lives

Apply mathematical ideas to problem situations that develop outside of mathematics

Investigate the presence of mathematics in careers and areas of interest

Recognize and apply mathematics to other disciplines, areas of interest, and societal issues

Representation Strand

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

Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations

Explain, describe, and defend mathematical ideas using representations

Recognize, compare, and use an array of representational forms

Explain how different representations express the same relationship

Use standard and non-standard representations with accuracy and detail

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

Use representations to explore problem situations

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

Use representation as a tool for exploring and understanding mathematical ideas

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

Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects)

Use mathematics to show and understand social phenomena (e.g., determine profit from sale of yearbooks)

Use mathematics to show and understand mathematical phenomena (e.g., use tables, graphs, and equations to show a pattern underlying a function)

Number Sense and Operations Strand

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

Develop and apply the laws of exponents for multiplication and division

Evaluate expressions with integral exponents

Read, write, and identify percents less than 1% and greater than 100%

Apply percents to: Tax; Percent increase/decrease; Simple interest; Sale price; Commission; Interest rates; Gratuities

Students will compute accurately and make reasonable estimates

Estimate a percent of quantity, given an application

Justify the reasonableness of answers using estimation

Algebra Strand

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

Translate verbal sentences into algebraic inequalities

Write verbal expressions that match given mathematical expressions

Describe a situation involving relationships that matches a given graph

Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship

Use physical models to perform operations with polynomials

Students will perform algebraic procedures accurately

Multiply and divide monomials

Add and subtract polynomials (integer coefficients)

Multiply a binomial by a monomial or a binomial (integer coefficients)

Divide a polynomial by a monomial (integer coefficients) Note: The degree of the denominator is less than or equal to the degree of the numerator for all variables.

Factor algebraic expressions using the GCF

Factor a trinomial in the form ax2 + bx + c; a=1 and c having no more than three sets of factors

Apply algebra to determine the measure of angles formed by or contained in parallel lines cut by a transversal and by intersecting lines

Solve multi-step inequalities and graph the solution set on a number line

Solve linear inequalities by combining like terms, using the distributive property, or moving variables to one side of the inequality (include multiplication or division of inequalities by a negative number)

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

Understand that numerical information can be represented in multiple ways: arithmetically, algebraically, and graphically

Find a set of ordered pairs to satisfy a given linear numerical pattern (expressed algebraically); then plot the ordered pairs and draw the line

Define and use correct terminology when referring to function (domain and range)

Determine if a relation is a function

Interpret multiple representations using equation, table of values, and graph

Geometry Strand

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

Construct the following, using a straight edge and compass: Segment congruent to a segment; Angle congruent to an angle; Perpendicular bisector; Angle bisector

Students will identify and justify geometric relationships, formally and informally

Identify pairs of vertical angles as congruent

Identify pairs of supplementary and complementary angles

Calculate the missing angle in a supplementary or complementary pair

Determine angle pair relationships when given two parallel lines cut by a transversal

Calculate the missing angle measurements when given two parallel lines cut by a transversal

Calculate the missing angle measurements when given two intersecting lines and an angle

Students will apply transformations and symmetry to analyze problem solving situations

Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations)

Draw the image of a figure under rotations of 90 and 180 degrees

Draw the image of a figure under a reflection over a given line

Draw the image of a figure under a translation

Draw the image of a figure under a dilation

Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation

Students will apply coordinate geometry to analyze problem solving situations

Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change

Determine the y-intercept of a line from a graph and be able to explain the y-intercept

Graph a line using a table of values

Determine the equation of a line given the slope and the y-intercept

Graph a line from an equation in slope-intercept form ( y = mx + b )

Solve systems of equations graphically (only linear, integral solutions, y = mx + b format, no vertical/horizontal lines)

Graph the solution set of an inequality on a number line

Distinguish between linear and nonlinear equations ax2 + bx + c; a=1 (only graphically)

Recognize the characteristics of quadratics in tables, graphs, equations, and situations

Measurement Strand

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

Solve equations/proportions to convert to equivalent measurements within metric and customary measurement systems Note: Also allow Fahrenheit to Celsius and vice versa

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