New York State Standards for Mathematics: Grade 8

NY.3. Mathematics, Science, and Technology: Students will understand the concepts of and become proficient with the skills of mathematics, communicate and reason mathematically; become problem solvers by using appropriate tools and strategies, through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.

3.1. Problem Solving Strand: Students will build new mathematical knowledge through problem solving.

8.PS.1. Use a variety of strategies to understand new mathematical content and to develop more efficient methods.

8.PS.2. Construct appropriate extensions to problem situations.

8.PS.3. Understand and demonstrate how written symbols represent mathematical ideas.

3.2. Problem Solving Strand: Students will solve problems that arise in mathematics and in other contexts.

8.PS.4. Observe patterns and formulate generalizations.

8.PS.5. Make conjectures from generalizations.

8.PS.6. Represent problem situations verbally, numerically, algebraically, and graphically.

3.3. Problem Solving Strand: Students will apply and adapt a variety of appropriate strategies to solve problems.

8.PS.7. Understand that there is no one right way to solve mathematical problems but that different methods have advantages and disadvantages.

8.PS.8. Understand how to break a complex problem into simpler parts or use a similar problem type to solve a problem.

8.PS.9. Work backwards from a solution.

8.PS.10. Use proportionality to model problems.

8.PS.11. Work in collaboration with others to solve problems.

3.4. Problem Solving Strand: Students will monitor and reflect on the process of mathematical problem solving.

8.PS.12. Interpret solutions within the given constraints of a problem.

8.PS.13. Set expectations and limits for possible solutions.

8.PS.14. Determine information required to solve the problem.

8.PS.15. Choose methods for obtaining required information.

8.PS.16. Justify solution methods through logical argument.

8.PS.17. Evaluate the efficiency of different representations of a problem.

3.5. Reasoning and Proof Strand: Students will recognize reasoning and proof as fundamental aspects of mathematics.

8.RP.1. Recognize that mathematical ideas can be supported by a variety of strategies.

3.6. Reasoning and Proof Strand: Students will make and investigate mathematical conjectures.

8.RP.2. Use mathematical strategies to reach a conclusion.

8.RP.3. Evaluate conjectures by distinguishing relevant from irrelevant information to reach a conclusion or make appropriate estimates.

3.7. Reasoning and Proof Strand: Students will develop and evaluate mathematical arguments and proofs.

8.RP.4. Provide supportive arguments for conjectures.

8.RP.5. Develop, verify, and explain an argument, using appropriate mathematical ideas and language.

3.8. Reasoning and Proof Strand: Students will select and use various types of reasoning and methods of proof.

8.RP.6. Support an argument by using a systematic approach to test more than one case.

8.RP.7. Devise ways to verify results or use counterexamples to refute incorrect statements.

8.RP.8. Apply inductive reasoning in making and supporting mathematical conjectures.

3.9. Communication Strand: Students will organize and consolidate their mathematical thinking through communication.

8.CM.1. Provide a correct, complete, coherent, and clear rationale for thought process used in problem solving.

8.CM.2. Provide an organized argument which explains rationale for strategy selection.

8.CM.3. Organize and accurately label work.

3.10. Communication Strand: Students will communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

8.CM.4. 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.

8.CM.5. Answer clarifying questions from others.

3.11. Communication Strand: Students will analyze and evaluate the mathematical thinking and strategies of others.

8.CM.6. Analyze mathematical solutions shared by others.

8.CM.7. Compare strategies used and solutions found by others in relation to their own work.

8.CM.8. Formulate mathematical questions that elicit, extend, or challenge strategies, solutions, and/or conjectures of others.

3.12. Communication Strand: Students will use the language of mathematics to express mathematical ideas precisely.

8.CM.9. Increase their use of mathematical vocabulary and language when communicating with others.

8.CM.10. Use appropriate language, representations, and terminology when describing objects, relationships, mathematical solutions, and rationale.

8.CM.11. Draw conclusions about mathematical ideas through decoding, comprehension, and interpretation of mathematical visuals, symbols, and technical writing.

3.13. Connections Strand: Students will recognize and use connections among mathematical ideas.

8.CN.1. Understand and make connections among multiple representations of the same mathematical idea.

8.CN.2. Recognize connections between subsets of mathematical ideas.

8.CN.3. Connect and apply a variety of strategies to solve problems.

3.14. Connections Strand: Students will understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

8.CN.4. Model situations mathematically, using representations to draw conclusions and formulate new situations.

8.CN.5. Understand how concepts, procedures, and mathematical results in one area of mathematics can be used to solve problems in other areas of mathematics.

3.15. Connections Strand: Students will recognize and apply mathematics in contexts outside of mathematics.

8.CN.6. Recognize and provide examples of the presence of mathematics in their daily lives.

8.CN.7. Apply mathematical ideas to problem situations that develop outside of mathematics.

8.CN.8. Investigate the presence of mathematics in careers and areas of interest.

8.CN.9. Recognize and apply mathematics to other disciplines, areas of interest, and societal issues.

3.16. Representation Strand: Students will create and use representations to organize, record, and communicate mathematical ideas.

8.R.1. Use physical objects, drawings, charts, tables, graphs, symbols, equations, or objects created using technology as representations.

8.R.2. Explain, describe, and defend mathematical ideas using representations.

8.R.3. Recognize, compare, and use an array of representational forms.

8.R.4. Explain how different representations express the same relationship.

8.R.5. Use standard and non-standard representations with accuracy and detail.

3.17. Representation Strand: Students will select, apply, and translate among mathematical representations to solve problems.

8.R.6. Use representations to explore problem situations.

8.R.7. Investigate relationships between different representations and their impact on a given problem.

8.R.8. Use representation as a tool for exploring and understanding mathematical ideas.

3.18. Representation Strand: Students will use representations to model and interpret physical, social, and mathematical phenomena.

8.R.9. Use mathematics to show and understand physical phenomena (e.g., make and interpret scale drawings of figures or scale models of objects).

8.R.10. Use mathematics to show and understand social phenomena (e.g., determine profit from sale of yearbooks).

8.R.11. Use mathematics to show and understand mathematical phenomena (e.g., use tables, graphs, and equations to show a pattern underlying a function).

3.19. Number Sense and Operations Strand: Students will understand meanings of operations and procedures, and how they relate to one another.

8.N.1. Operations: Develop and apply the laws of exponents for multiplication and division.

8.N.2. Operations: Evaluate expressions with integral exponents.

8.N.3. Operations: Read, write, and identify percents less than 1 percent and greater than 100 percent.

8.N.4. Operations: Apply percents to: Tax; Percent increase/decrease; Simple interest; Sale price; Commission; Interest rates; Gratuities.

3.20. Number Sense and Operations Strand: Students will compute accurately and make reasonable estimates.

8.N.5. Estimation: Estimate a percent of quantity, given an application.

8.N.6. Estimation: Justify the reasonableness of answers using estimation.

3.21. Algebra Strand: Students will represent and analyze algebraically a wide variety of problem solving situations.

8.A.1. Variables and Expressions: Translate verbal sentences into algebraic inequalities.

8.A.2. Variables and Expressions: Write verbal expressions that match given mathematical expressions.

8.A.3. Variables and Expressions: Describe a situation involving relationships that matches a given graph.

8.A.4. Variables and Expressions: Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship.

8.A.5. Variables and Expressions: Use physical models to perform operations with polynomials.

3.22. Algebra Strand: Students will perform algebraic procedures accurately.

8.A.6. Variables and Expressions: Multiply and divide monomials.

8.A.7. Variables and Expressions: Add and subtract polynomials (integer coefficients).

8.A.8. Variables and Expressions: Multiply a binomial by a monomial or a binomial (integer coefficients).

8.A.9. Variables and Expressions: 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.

8.A.10. Variables and Expressions: Factor algebraic expressions using the GCF.

8.A.11. Variables and Expressions: Factor a trinomial in the form ax to the power of 2 + bx + c; a=1 and c having no more than three sets of factors.

8.A.12. Equations and Inequalities: Apply algebra to determine the measure of angles formed by or contained in parallel lines cut by a transversal and by intersecting lines.

8.A.13. Equations and Inequalities: Solve multi-step inequalities and graph the solution set on a number line.

8.A.14. Equations and Inequalities: 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).

3.23. Algebra Strand: Students will recognize, use, and represent algebraically patterns, relations, and functions.

8.A.15. Patterns, Relations, and Functions: Understand that numerical information can be represented in multiple ways: arithmetically, algebraically, and graphically.

8.A.16. Patterns, Relations, and Functions: Find a set of ordered pairs to satisfy a given linear numerical pattern (expressed algebraically); then plot the ordered pairs and draw the line.

8.A.17. Patterns, Relations, and Functions: Define and use correct terminology when referring to function (domain and range).

8.A.18. Patterns, Relations, and Functions: Determine if a relation is a function.

8.A.19. Patterns, Relations, and Functions: Interpret multiple representations using equation, table of values, and graph.

3.24. Geometry Strand: Students will use visualization and spatial reasoning to analyze characteristics and properties of geometric shapes.

8.G.0. Constructions: Construct the following, using a straight edge and compass: Segment congruent to a segment; Angle congruent to an angle; Perpendicular bisector; Angle bisector.

3.25. Algebra Strand: Students will identify and justify geometric relationships, formally and informally.

8.G.1. Geometric Relationships: Identify pairs of vertical angles as congruent.

8.G.2. Geometric Relationships: Identify pairs of supplementary and complementary angles.

8.G.3. Geometric Relationships: Calculate the missing angle in a supplementary or complementary pair.

8.G.4. Geometric Relationships: Determine angle pair relationships when given two parallel lines cut by a transversal.

8.G.5. Geometric Relationships: Calculate the missing angle measurements when given two parallel lines cut by a transversal.

8.G.6. Geometric Relationships: Calculate the missing angle measurements when given two intersecting lines and an angle.

3.26. Algebra Strand: Students will apply transformations and symmetry to analyze problem solving situations.

8.G.7. Transformational Geometry: Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations).

8.G.8. Transformational Geometry: Draw the image of a figure under rotations of 90 and 180 degrees.

8.G.9. Transformational Geometry: Draw the image of a figure under a reflection over a given line.

8.G.10. Transformational Geometry: Draw the image of a figure under a translation.

8.G.11. Transformational Geometry: Draw the image of a figure under a dilation.

8.G.12. Transformational Geometry: Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation.

3.27. Algebra Strand: Students will apply coordinate geometry to analyze problem solving situations.

8.G.13. Coordinate Geometry: Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change.

8.G.14. Coordinate Geometry: Determine the y-intercept of a line from a graph and be able to explain the y-intercept.

8.G.15. Coordinate Geometry: Graph a line using a table of values.

8.G.16. Coordinate Geometry: Determine the equation of a line given the slope and the y-intercept.

8.G.17. Coordinate Geometry: Graph a line from an equation in slope-intercept form (y=mx+b).

8.G.18. Coordinate Geometry: Solve systems of equations graphically (only linear, integral solutions, y=mx+b format, no vertical/horizontal lines).

8.G.19. Coordinate Geometry: Graph the solution set of an inequality on a number line.

8.G.20. Coordinate Geometry: Distinguish between linear and nonlinear equations ax to the power of 2 + bx + c; a=1 (only graphically).

8.G.21. Coordinate Geometry: Recognize the characteristics of quadratics in tables, graphs, equations, and situations.

3.28. Measurement Strand: Students will determine what can be measured and how, using appropriate methods and formulas.

8.M.1. Units of Measurement: Solve equations/proportions to convert to equivalent measurements within metric and customary measurement systems. Note: Also allow Fahrenheit to Celsius and vice versa.