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MATH BENCHMARKS

Math Benchmarks: Grade 10 - Grade 11 - Grade 12
Math Curriculum Alignment - Academics Home Page

Grade 10
 
Number, Number Sense and Operations Standard - Students demonstrate number sense, including an understanding of number systems and operations and how they relate to one another. Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods.
A. Use scientific notation to express large numbers and numbers less than one.
B. Identify subsets of the real number system.
C. Apply properties of operations and the real number system, and justify when they hold for a set of numbers.
D. Connect physical, verbal and symbolic representations of integers, rational numbers and irrational numbers.
E. Compare, order and determine equivalent forms of real numbers.
F. Explain the effects of operations on the magnitude of quantities.
G. Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions.
H. Find the square root of   perfect squares, and approximate the square root of non-perfect squares.
I. Estimate, compute and solve problems involving scientific notation, square roots and numbers with integer exponents.
Measurement Standard - Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies.
A. Solve increasingly complex non-routine measurement problems and check for reasonableness of results.
B. Use formulas to find surface area and volume for specified three-dimensional objects accurate to a specified level of precision.
C. Apply indirect measurement techniques, tools and formulas, as appropriate, to find perimeter, circumference and area of circles, triangles, quadrilaterals and composite shapes, and to find volume of prisms, cylinders, and pyramids.
D. Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems   involving measurements and rates.
E. Estimate and compute various attributes, including length, angle measure, area, surface area and volume, to a specified level of precision.
F. Write and solve real-world, multi-step problems involving money, elapsed time and temperature, and verify reasonableness of solutions.
Geometry and Spatial Sense Standard - Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.
A. Formally define geometric figures.
B. Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence.
C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines.
D. Use coordinate geometry to represent and examine the properties of geometric figures.
E. Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.
F. Represent and model transformations in a coordinate plane and describe the results.  
G. Prove or disprove conjectures and solve problems involving two- and three-dimensional objects represented within a coordinate system.
H. Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.
I. Use right triangle trigonometric relationships to determine lengths and angle measures.
Patterns, Functions and Algebra Standard  - Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities.   Students analyze, model and solve problems using various representations such as tables, graphs and equations.
A. Generalize and explain patterns and sequences in order to find the next term and the nth term.
B. Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations.
C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.
D. Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.
E. Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros.
F. Solve and graph linear equations and inequalities.
G. Solve quadratic equations with real roots by graphing, formula and factoring.  
H. Solve systems of linear equations involving two variables graphically and symbolically.
I. Model and solve problem situations involving direct and inverse variation.
J. Describe and interpret rates of change from graphical and numerical data.
Data Analysis and Probability Standard - Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data.
A. Create, interpret and use graphical displays and statistical measures to describe data; e.g., box-and-whisker plots, histograms, scatterplots, measures of center and variability.
B. Evaluate different graphical representations of the same data to determine which is the most appropriate representation for an identified purpose.
C. Compare the characteristics of the mean, median and mode for a given set of data, and explain which measure of center best represents the data.
D. Find, use and interpret measures of center and spread, such as mean and quartiles, and use those measures to compare and draw conclusions about sets of data.
E. Evaluate the validity of claims and predictions that are based on data by examining the appropriateness of the data collection and analysis.
F. Construct convincing arguments based on analysis of data and interpretation of graphs.
G. Describe sampling methods and analyze the effects of method chosen on how well the resulting sample represents the population.
H. Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes.  
I. Design an experiment to test a theoretical probability, and record and explain results.
J. Compute probabilities of compound events, independent events, and simple dependent events.
K. Make predictions based on theoretical probabilities and experimental results.
Mathematical Processes Standard - Students use mathematical processes and knowledge to solve problems.   Students apply problem-solving and decision-making techniques, and communicate mathematical ideas.
A. Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution.
B. Apply mathematical knowledge and skills routinely in other content areas and practical situations.
C. Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x -intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities.  
D. Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions.
E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.
F. Use precise mathematical language and notations to represent problem situations and mathematical ideas.
G. Write clearly and coherently about mathematical thinking and ideas.
H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner.
 
 
 
Grade 11
Number, Number Sense and Operations Standard - Students demonstrate number sense, including an understanding of number systems and operations and how they relate to one another.   Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods.
A. Demonstrate that vectors and matrices are systems having some of the same properties of the real number system.
B. Develop an understanding of properties of and representations for addition and multiplication of vectors and matrices.
C. Apply factorials and exponents, including fractional exponents, to solve practical problems.
D. Demonstrate fluency in operations with real numbers, vectors and matrices, using mental computation or paper and pencil calculations for simple cases and technology for more complicated cases.
E. Represent and compute with complex numbers.
Measurement Standard - Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies.
A. Explain differences among accuracy, precision and error, and describe how each of those can affect solutions in measurement situations.
B. Apply various measurement scales to describe phenomena and solve problems.
C. Estimate and compute areas and volume in increasingly complex problem situations.
D. Solve problem situations involving derived measurements; e.g., density, acceleration.
Geometry and Spatial Sense Standard - Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.
A. Use trigonometric relationships to verify and determine solutions in problem situations.
B. Represent transformations within a coordinate system using vectors and matrices.
Patterns, Functions and Algebra Standard - Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities.   Students analyze, model and solve problems using various representations such as tables, graphs and equations.
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
B. Use the quadratic formula to solve quadratic equations that have complex roots.
C. Use recursive functions to model and solve problems; e.g., home mortgages, annuities.
D. Apply algebraic methods to represent and generalize problem situations involving vectors and matrices.
Data Analysis and Probability Standard - Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data.
A. Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators.
B. Use descriptive statistics to analyze and summarize data, including measures of center, dispersion, correlation and variability.
C. Design and perform a statistical experiment, simulation or study; collect and interpret data; and use descriptive statistics to communicate and support predictions and conclusions.
D. Connect statistical techniques to applications in workplace and consumer situations.
Mathematical Processes Standard - Students use mathematical processes and knowledge to solve problems.   Students apply problem-solving and decision-making techniques, and communicate mathematical ideas.
A. Construct algorithms for multi-step and non-routine problems.
B. Construct logical verifications or counter-examples to test conjectures and to justify or refute algorithms and solutions to problems.
C. Assess the adequacy and reliability of information available to solve a problem.
D. Select and use various types of reasoning and methods of proof.
E. Evaluate a mathematical argument and use reasoning and logic to judge its validity.
F. Present complete and convincing arguments and justifications, using inductive and deductive reasoning, adapted to be effective for various audiences.
G. Understand the difference between a statement that is verified by mathematical proof, such as a theorem, and one that is verified empirically using examples or data.
H. Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.
I. Communicate mathematical ideas orally and in writing with a clear purpose and appropriate for a specific audience.
J. Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.
 
 
 
Grade 12
 
Number, Number Sense and Operations Standard - Students demonstrate number sense, including an understanding of number systems and operations and how they relate to one another.   Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods.
A. Demonstrate that vectors and matrices are systems having some of the same properties of the real number system.
B. Develop an understanding of properties of and representations for addition and multiplication of vectors and matrices.
C. Apply factorials and exponents, including fractional exponents, to solve practical problems.
D. Demonstrate fluency in operations with real numbers, vectors and matrices, using mental computation or paper and pencil calculations for simple cases and technology for more complicated cases.
E. Represent and compute with complex numbers.
Measurement Standard - Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies.
A. Explain differences among accuracy, precision and error, and describe how each of those can affect solutions in measurement situations.
B. Apply various measurement scales to describe phenomena and solve problems.
C. Estimate and compute areas and volume in increasingly complex problem situations.
D. Solve problem situations involving derived measurements; e.g., density, acceleration.
Geometry and Spatial Sense Standard - Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.
A. Use trigonometric relationships to verify and determine solutions in problem situations.
B. Represent transformations within a coordinate system using vectors and matrices.
Patterns, Functions and Algebra Standard - Students use patterns, relations and functions to model, represent and analyze problem situations that involve variable quantities.   Students analyze, model and solve problems using various representations such as tables, graphs and equations.
A. Analyze functions by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior.
B. Use the quadratic formula to solve quadratic equations that have complex roots.
C. Use recursive functions to model and solve problems; e.g., home mortgages, annuities.
D. Apply algebraic methods to represent and generalize problem situations involving vectors and matrices.
Data Analysis and Probability Standard - Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments that are based on data.
A. Create and analyze tabular and graphical displays of data using appropriate tools, including spreadsheets and graphing calculators.
B. Use descriptive statistics to analyze and summarize data, including measures of center, dispersion, correlation and variability.
C. Design and perform a statistical experiment, simulation or study; collect and interpret data; and use descriptive statistics to communicate and support predictions and conclusions.
D. Connect statistical techniques to applications in workplace and consumer situations.
Mathematical Processes Standard - Students use mathematical processes and knowledge to solve problems.   Students apply problem-solving and decision-making techniques, and communicate mathematical ideas.
A. Construct algorithms for multi-step and non-routine problems.
B. Construct logical verifications or counter-examples to test conjectures and to justify or refute algorithms and solutions to problems.
C. Assess the adequacy and reliability of information available to solve a problem.
D. Select and use various types of reasoning and methods of proof.
E. Evaluate a mathematical argument and use reasoning and logic to judge its validity.
F. Present complete and convincing arguments and justifications, using inductive and deductive reasoning, adapted to be effective for various audiences.
G. Understand the difference between a statement that is verified by mathematical proof, such as a theorem, and one that is verified empirically using examples or data.
H. Use formal mathematical language and notation to represent ideas, to demonstrate relationships within and among representation systems, and to formulate generalizations.
I. Communicate mathematical ideas orally and in writing with a clear purpose and appropriate for a specific audience.
J. Apply mathematical modeling to workplace and consumer situations, including problem formulation, identification of a mathematical model, interpretation of solution within the model, and validation to original problem situation.
          
 
       
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