Parents: Boston Public Schools Grade 12

Grade 12 Learning Standards

This information is from the Boston Public Schools Citywide Learning Standards.

ENGLISH LANGUAGE ARTS

Oral Presentation and Discussion
Students will be able to:

  • Use agreed upon rules for informal and formal discussions in small and large groups such as Book Club, Literature Circles, and Buddy Reading.
  • Facilitate discussion groups independent from the teacher; identify and practice techniques to improve group productivity such as discussion guidelines, setting time limits for speakers and deadlines for decision-making.
  • Organize and present ideas in a logical order.
  • Ask for clarification when others' responses are unclear.
  • Actively listen, respond to, and build on ideas generated during group discussions.
  • Use information to inform or change their perspectives.
  • Support their responses with evidence or details; expect and request the same of others.
  • Summarize and evaluate what they have learned from the discussion.
  • Evaluate the productivity of group discussion using group created criteria and make suggestions to address the needs of the group.
  • Deliver informal and formal presentations, giving consideration to audience, purpose and content.
  • Identify elements and organizational structures of effective speeches made for a variety of purposes; work collaboratively to create and use an appropriate rubric or criteria to prepare, improve, and assess presentations.
  • Conduct interviews for research projects and writing.
Language
  • Monitor and evaluate their use of general dictionaries, specialized dictionaries, thesauruses, histories of language, books of quotations, Internet, and other related references.
  • Identify historical changes in conventions for usage and grammar.
  • Evaluate and explain the influences of the English language on world literature and world cultures in oral and written responses.
  • Analyze and explain how the English language has developed and been influenced by other languages in oral and written discussions.
  • Identify and analyze occasions where dialect can be a source of negative or positive stereotypes among social groups.
Reading and Literature
  • Develop fluency, accuracy, and understanding when reading different and more complex texts.
  • Self-monitor, assess, and revise their decisions and work as needed, as well as their development as a reader.
  • Select books for independent reading and study.
  • Use established language for talking about and evaluating books.
  • Use before, during, and after reading strategies to improve comprehension.
  • Use background knowledge to make inferences and predictions and to make personal connections with what they are reading.
  • Set a purpose for reading monitor their progress.
  • Listen critically and ask questions to clarify information.
  • Summarize information to check understanding.
  • Visualize information in text to support comprehension.
  • Identify and analyze the topic and main idea of different texts.
  • Demonstrate their understanding of genres and organizational structure and apply that knowledge to their reading of different texts.
  • Use text features and organizational structure to make meaning of what they are reading.
  • Understand when their comprehension breaks down; know and use self-correcting strategies to make meaning of what they are reading.
  • Analyze and compare texts that express a universal theme, and locate support in the text for the identified theme, orally and in writing.
  • Demonstrate an understanding of intertextuality and different literary theories (i.e., reader response, historical, biographical, structuralism, post-structuralism, gender, psychological, Marxist, new historicist, archetypal) and utilize them when interpreting literary texts.
  • Evaluate and apply their knowledge of how authors use techniques and elements in fiction for rhetorical and aesthetic purposes in oral and written responses.
  • Analyze and evaluate the logic and use of evidence in an author's argument in oral and written responses and group discussions.
  • Analyze, explain, and evaluate how authors use the elements of nonfiction to achieve their purposes in oral and written responses and group discussions.
  • Apply their knowledge of diction and imagery (controlling images, figurative language, understatement, overstatement, irony, paradox) in oral and written responses and in group discussions.
  • Apply their knowledge of how an author uses rhetorical devices in persuasive argument, through individual and group presentations.
  • Analyze and compare style and language across significant cross-cultural and historical literary works.
  • Analyze and evaluate the influence of mythic, traditional, or classical literature on later literature and film.
  • Apply their knowledge of the structure and themes of different dramatic literature in oral and written responses and group discussions.
  • Evaluate the use of different dramatic conventions (monologues, soliloquy, chorus, aside, dramatic irony) in oral and written responses.
Composition
  • Collect ideas for writing from different and more complex texts and sources (dialogue, artifacts, memories, images, etc.).
  • Maintain a process for recording, collecting, referring to, and sharing their ideas for writing, as well as more formal writing products, including drafts.
  • Self-monitor, assess, and revise their decisions and work as needed, as well as their development as a writer.
  • Write for different purposes and for different audiences.
  • Demonstrate their understanding of different genres and organizational structures.
  • Select appropriate genres and organizational structures for drafts.
  • Select appropriate strategies for developing ideas into drafts.
  • Select appropriate strategies for revising the organization and ideas in drafts.
  • Use effective language for talking about pieces of writing (e.g., craft, focus, structure, genre, voice, audience).
  • Use standard English conventions (mechanics, grammar, and spelling) while editing work.
  • Create stories, plays, poems with an explicit or implicit theme, using a variety of literary techniques.
  • Write coherent compositions with a clear focus, an objective presentation of alternate views, rich details, well-developed paragraphs, logical argumentation and a concluding idea.
  • Use effective rhetorical techniques; demonstrate an understanding of purpose, speaker, audience, and form when completing expressive, persuasive, or literary writing assignments.
  • Revise their writing: assess style, word choice, sentence variety, and subtlety of meaning after rethinking how well questions of purpose, audience, and genre have been addressed.
  • Use established conventions of standard English when writing and editing.
  • Analyze a writer's placement of descriptive details about setting, characters, and events in stories.
  • Place their ideas in logical order when writing summaries or reports.
  • Explore a topic of interest: formulate an open-ended question, design and carry out research or inquiry, present results in a well-structured report.
  • Establish criteria and evaluate the quality of their research/inquiry report (e.g., adequacy of question, materials, approach, selection and documentation of sources).
  • Establish and use criteria for assessing the intertexuality or interdisciplinary aspect of literary works from different genres, explaining why the criteria are appropriate before applying them.
  • Evaluate their ability to meet test-taking requirements for different purposes.
Media
  • Analyze and evaluate the aesthetic effects of a media presentation; identify and evaluate the techniques used to create them.
  • Create a media production that synthesizes information from several sources on a related theme or topic.
MATH: ALGEBRA 1A, COURSE 45A

Number Sense and Operations

  • Understand numbers and ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Understand numbers, ways of representing numbers, relationships among numbers.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth roots of positive numbers for any positive integer n.
  • Express and simplify numerical expressions involving real numbers.
  • Understand and demonstrate algebraically and graphically the relationship between operations and their inverses, including exponential.
  • Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers.
  • Express real numbers in fractional and radical form as well as in exponential form using integral and fractional exponents.
  • Use logical reasoning as well as estimation and mental computation to determine the validity of a solution in algebraic, and statistical problems.
Data Analysis, Statistics, and Probability
  • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Understand and apply basic concepts of probability.
  • Formulate questions that can be answered with data and collect, organize and display relevant data to answer them.
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Select, create, and interpret an appropriate graphical representation (e.g., scatterplot, table, stem-and-leaf plot, box-and-whisker plot, circle graph, line graph, and line plot) for a set of data and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.
  • Collect and graph data (using graphing calculators and/or computers when appropriate) and express relationships between variables, both verbally and symbolically.
  • Collect, organize, and analyze data from real problems using graphing calculators and other technology to create tables and graphs.
  • Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot). Use technology when appropriate.
  • Use scatter plots of sets of data points to graph a line of best fit.
  • Use the basic set of operations with the help of Venn diagrams.
  • Solve counting problems using Venn diagrams.
  • Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data.
  • Understand and apply basic concepts of probability.
  • Use tree diagrams, tables, organized lists, basic combinations ("fundamental counting principle"), and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.
  • Carry out probability experiments and discuss the results.
  • Conduct experiments to determine experimental probabilities.
Patterns, Relations, and Algebra
  • Understand patterns, relations, and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive (e.g., Fibonacci Numbers), and linear functional relationships.
  • Use properties of the real number system to judge the validity of equations and inequalities, to prove or disprove statements, and to justify every step in a sequential argument.
  • Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.
  • Demonstrate an understanding of the relationship between various representations of a line.
  • Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "slope y-intercept" formulas.
  • Explain the significance of a positive, negative, zero, or undefined slope.
  • Find linear equations that represent lines parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Solve equations apply to the solution of everyday problems.
Discussion, Presentation, Composition
  • Use agreed upon rules to participate in discussions in large and small groups.
  • Express ideas in an organized way.
  • Explain their mathematical thinking in writing.
  • Maintain a system for collecting, referring to, and sharing their work.
MATH: ALGEBRA, COURSE 451

Number Sense and Operations

  • Understand numbers and ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth roots of positive numbers for any positive integer n; and the inverse relationship between taking the nth root of and the nth power of a positive real number.
  • Recognize when and how to apply the field properties in problems using real numbers.
  • Express and simplify numerical expressions involving real numbers.
  • Pose and solve problems using operations on whole numbers, integers, rational, irrational, and complex numbers.
  • Simplify numerical expressions, including those involving positive integer exponents or the absolute value; apply such simplifications in the solution of problems.
  • Express and simplify numerical expressions involving real numbers.
  • Find the approximate value for solutions to problems involving square roots and cube roots without the use of a calculator. e.g., .
  • Understand and demonstrate algebraically and graphically the relationship between operations and their inverses, including exponential.
  • Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers.
Data Analysis, Statistics, and Probability
  • Formulate questions that can be answered with data and collect, organize and display relevant data to answer them.
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Select, create, and interpret an appropriate graphical representation (e.g., scatterplot, table, stem-and-leaf plot, box-and-whisker plot, circle graph, line graph, and line plot) for a set of data and use appropriate statistics (e.g., mean, median, range, and mode) to communicate information about the data. Use these notions to compare different sets of data.
  • Use logical reasoning as well as estimation and mental computations to determine the validity of a solution in algebraic, geometric, and statistical problems.
  • Collect, organize, analyze and graph data from real problems (using graphing calculators and/or computers when appropriate) and express relationships between variables, verbally, graphically and symbolically.
  • Collect, organize, and analyze data from real problems using graphing calculators and other technology to create tables and graphs.
  • Approximate a line of best fit (trend line) given a set of data (e.g., scatterplot). Use technology when appropriate.
  • Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data.
Patterns, Relations, and Algebra
  • Understand patterns, relations, and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive linear, quadratic, functional relationships.
  • Demonstrate an understanding of relations and functions. Identify the domain, range, dependent, and independent variables of functions.
  • Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.
  • Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "slope y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.
  • Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Solve equations and apply to the solution of problems.
  • Evaluate exponential functions and compare the effects of different growth rates.
  • Describe compounding situations using exponential functions.
  • Evaluate exponential functions and draw their graphs with a graphing calculator. Construct examples of functions on finite sets using diagrams and tables.
  • Describe a sequence recursively and use that description to list its terms with a graphing calculator.
  • Describe a sequence algebraically and use that description to find specific terms.
  • Identify and describe real world examples of step functions.
  • Explain restrictions on the domains of functions.
  • Use graphs to represent functions and to find images of domain elements.
  • Interpret graphs of step function in real world situations.
Discussion, Presentation, Composition
  • Use agreed upon rules to participate in discussions in large and small groups.
  • Express ideas in an organized way.
  • Explain their mathematical thinking in writing.
  • Maintain a system for collecting, referring to, and sharing their work.
MATH: ALGEBRA /GEOMETRY, COURSE 452

Number Sense and Operations

  • Understand numbers and ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Understand numbers, ways of representing numbers, relationships among numbers.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth roots of positive numbers for any positive integer n.
  • Express and simplify numerical expressions involving real numbers.
  • Understand and demonstrate algebraically and graphically the relationship between operations and their inverses, including exponential.
  • Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers.
  • Express real numbers in fractional and radical form as well as in exponential form using integral and fractional exponents.
  • Use logical reasoning as well as estimation and mental computation to determine the validity of a solution in algebraic, and statistical problems.
Patterns, Relations, and Functions
  • Understand patterns, relations, and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Represent and analyze mathematical situations including geometric situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships including geometric relationships.
  • Analyze change in various contexts.
  • Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative, recursive linear, quadratic, functional relationships.
  • Demonstrate an understanding of relations and functions. Identify the domain, range, dependent, and independent variables of functions.
  • Translate between different representations of functions and relations: graphs, equations, point sets, and tabular.
  • Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "slope y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.
  • Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Solve equations and apply to the solution of problems including geometric problems.
  • Evaluate exponential functions and compare the effects of different growth rates.
  • Describe compounding situations using exponential functions.
  • Evaluate exponential functions and draw their graphs with a graphing calculator. Construct examples of functions on finite sets using diagrams and tables.
  • Describe a sequence recursively and use that description to list its terms with a graphing calculator
  • Describe a sequence algebraically and use that description to find specific terms.
  • Identify and describe real world examples of step functions.
  • Explain restrictions on the domains of functions.
  • Use graphs to represent functions and to find images of domain elements.
  • Interpret graphs of step function in real world situations.
Geometry
  • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
  • Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
  • Apply transformations and use symmetry to analyze mathematical situations.
  • Use visualization, spatial reasoning, and geometric modeling to solve problems.
  • Identify figures using properties of sides, angles, and diagonals. Identify the figures' type(s) of symmetry.
  • Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solution of problems.
  • Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses). Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, mid-segments); determine interior angles for regular polygons. Draw and label sets of points such as line segments, rays, and circles. Detect symmetries of geometric figures.
  • Draw congruent and similar figures using a compass, straightedge, protractor, and other tools such as computer software. Make conjectures about methods of constructions. Justify the conjectures by logical arguments.
  • Use compass and straightedge to measure geometric figures to a high degree of accuracy.
  • Draw congruent and similar figures using a compass, straightedge, protractor, or computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments.
  • Recognize and solve problems involving angles formed by transversals of coplanar lines. Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle.
  • Identify congruence and similarity correspondences and properties of the figures to find missing parts of geometric figures, and provide logical justification.
  • Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean Theorem.
  • Use the properties of special triangles (e.g., isosceles, equilateral, 30-60-90, 45-45-90) to solve everyday problems.
  • Using rectangular coordinates, calculate the midpoints of segments, slopes of lines, and distances between two points. Apply the results of these calculations to find the solution to everyday problems.
  • Demonstrate an understanding of the relationship between various representations of a line.
  • Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "point-slope" or "slope y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.
  • Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Explore and use properties of parallel and perpendicular lines, bisectors of angles and segments, triangles and circles through constructions.
  • Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems
  • Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems.
Measurement
  • Understand measurable attributes of objects and the units, systems, and processes of measurement.
  • Apply appropriate techniques, tools, and formulas to determine measurements.
  • Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles.
  • Use diagrams, models, and other manipulatives to determine methods of finding relationships and measurements of the two and three dimensional shapes.
  • Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements.
  • Use appropriate measurement tools along with calculators and computers to solve problems in science, technology, consumer education, and other areas as well.
Data Analysis, Statistics, and Probability
  • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Understand and apply basic concepts of probability.
  • Use the basic set of operations with the help of Venn diagrams.
  • Solve counting problems using Venn diagrams.
  • Solving counting problems using Venn diagrams.
  • Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data.
  • Understand and apply basic concepts of probability.
  • Use tree diagrams, tables, organized lists, basic combinations ("fundamental counting principle"), and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.
  • Carry out probability experiments, discuss the results.
  • Conduct experiments to determine experimental probabilities.
Discussion, Presentation, Composition
  • Use agreed upon rules to participate in discussions in large and small groups.
  • Express ideas in an organized way.
  • Explain their mathematical thinking in writing.
  • Maintain a system for collecting, referring to, and sharing their work.
MATH: GEOMETRY, COURSE 454

Number Sense and Operations

  • Understand numbers and ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Understand numbers, ways of representing numbers, and relationships among numbers.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Identify and use the properties of operations on real numbers, including the associative, commutative, and distributive properties; the existence of the identity and inverse elements for addition and multiplication; the existence of nth roots of positive numbers for any positive integer n.
  • Express and simplify numerical expressions involving real numbers.
  • Use estimation to judge the reasonableness of results of computations and of solutions to problems involving real numbers.
  • Express real numbers in fractional and radical form.
  • Use logical reasoning as well as estimation and mental computation to determine the validity of a solution in algebraic, geometric, and statistical problems.
Patterns, Relations, and Functions
  • Understand patterns, relations, and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Represent and analyze geometric situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative geometric relationships.
  • Analyze change in various contexts.
  • Use properties of the real number system to judge the validity of equations and inequalities, to prove or disprove geometric statements, and to justify every step in a sequential, geometric argument.
  • Translate between different geometric and algebraic representations of functions and relations: graphs, equations, point sets, and tabular.
  • Demonstrate an understanding of the relationship between various representations of a line.
  • Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "slope y-intercept" formulas.
  • Explain the significance of a positive, negative, zero, or undefined slope.
  • Find linear equations that represent lines parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Solve equations apply to the solution of everyday geometric problems.
Data Analysis, Statistics, and Probability
  • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Understand and apply basic concepts of probability.
  • Use the basic set of operations with the help of Venn diagrams.
  • Solve counting problems using Venn diagrams.
  • Solving counting problems using Venn diagrams.
  • Describe and explain how the relative sizes of a sample and the population affect the validity of predictions from a set of data.
  • Understand and apply basic concepts of probability.
  • Use tree diagrams, tables, organized lists, basic combinations ("fundamental counting principle"), and area models to compute probabilities for simple compound events, e.g., multiple coin tosses or rolls of dice.
  • Carry out probability experiments, discuss the results.
  • Conduct experiments to determine experimental probabilities and construct a table to establish theoretical
Geometry
  • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
  • Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
  • Apply transformations and use symmetry to analyze mathematical situations.
  • Use visualization, spatial reasoning, and geometric modeling to solve problems.
  • Identify figures using properties of sides, angles, and diagonals. Identify the figures' type(s) of symmetry.
  • Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solution of problems.
  • Recognize special types of polygons (e.g., isosceles triangles, parallelograms, and rhombuses). Apply properties of sides, diagonals, and angles in special polygons; identify their parts and special segments (e.g., altitudes, midsegments); determine interior angles for regular polygons. Draw and label sets of points such as line segments, rays, and circles. Detect symmetries of geometric figures.
  • Apply congruence and similarity correspondences and properties of the figures to find missing parts of geometric figures and to provide logical justifications.
  • Draw congruent and similar figures using a compass, straightedge, protractor, and other tools such as computer software. Make conjectures about methods of constructions. Justify the conjectures by logical arguments.
  • Use compass and straightedge to measure geometric figures to a high degree of accuracy
  • Draw congruent and similar figures using a compass, straightedge, protractor, or computer software. Make conjectures about methods of construction. Justify the conjectures by logical arguments.
  • Recognize and solve problems involving angles formed by transversals of coplanar lines. Identify and determine the measure of central and inscribed angles and their associated minor and major arcs. Recognize and solve problems associated with radii, chords, and arcs within or on the same circle.
  • Apply properties of angles, parallel lines, arcs, radii, chords, tangents and secants to solve everyday problems.
  • Solve simple triangle problems using the triangle angle sum property and/or the Pythagorean Theorem.
  • Use the properties of special triangles (e.g., isosceles, equilateral, 30-60-90, 45-45-90) to solve everyday problems.
  • Using rectangular coordinates, calculate the midpoints of segments, slopes of lines, and distances between two points. Apply the results of these calculations to find the solution to everyday problems.
  • Demonstrate an understanding of the relationship between various representations of a line. Determine a line's slope and x- and y-intercepts from its graph or from a linear equation that represents the line. Find a linear equation describing a line from a graph or a geometric description of the line, e.g., by using the "point-slope" or "slope y-intercept" formulas. Explain the significance of a positive, negative, zero, or undefined slope.
  • Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Explore and use properties of parallel and perpendicular lines, bisectors of angles and segments, triangles and circles through constructions.
  • Using rectangular coordinates, calculate midpoints of segments, slopes of lines and segments, and distances between two points, and apply the results to the solutions of problems.
  • Find linear equations that represent lines either perpendicular or parallel to a given line and through a point, e.g., by using the "point-slope" form of the equation.
  • Draw the results, and interpret transformations on figures in the coordinate plane, e.g., translations, reflections, rotations, scale factors, and the results of successive transformations. Apply transformations to the solutions of problems.
  • Demonstrate an understanding of the relationship between geometric and algebraic representations of circles.
  • Apply algebraic and geometric principles and practices to solve everyday problems involving circles.
  • Demonstrate the ability to visualize solid objects and to recognize their projections and cross sections.
Measurement
  • Understand measurable attributes of objects and the units, systems, and processes of measurement.
  • Apply appropriate techniques, tools, and formulas to determine measurements.
  • Calculate perimeter, circumference, and area of common geometric figures such as parallelograms, trapezoids, circles, and triangles.
  • Use diagrams, models, and other manipulatives to determine methods of finding relationships and measurements of the two and three-dimensional shapes.
  • Relate changes in the measurement of one attribute of an object to changes in other attributes, e.g., how changing the radius or height of a cylinder affects its surface area or volume Describe the effects of approximate error in measurement and rounding on measurements and on computed values from measurements.
  • Relate geometric and algebraic representations of lines, simple curves, and conic sections.
  • Use appropriate measurement tools along with calculators and computers to solve everyday geometric problems in science, technology, consumer education, and other areas as well.
Discussion, Presentation, Composition
  • Use agreed upon rules to participate in discussions in large and small groups.
  • Express ideas in an organized way.
  • Explain their mathematical thinking in writing.
  • Maintain a system for collecting, referring to, and sharing their work.
MATH: ADVANCED ALGEBRA, COURSE 456

Number Sense and Operations

  • Understand numbers and ways of representing numbers, relationships among numbers, and number systems.
  • Understand meanings of operations and how they relate to one another.
  • Compute fluently and make reasonable estimates.
  • Define complex numbers (e.g., a + bi) and operations on them, in particular, addition, subtraction, multiplication, and division. Relate the system of complex numbers to the systems of real and rational numbers.
  • Simplify numerical expressions with powers and roots, fractional and negative exponents.
Patterns Relations and Algebra
  • Understand patterns, relations, and functions.
  • Represent and analyze mathematical situations and structures using algebraic symbols.
  • Use mathematical models to represent and understand quantitative relationships.
  • Analyze change in various contexts.
  • Describe, complete, extend, analyze, generalize, and create a wide variety of patterns, including iterative and recursive patterns.
  • Identify arithmetic and geometric sequences and finite arithmetic and geometric series. Use the properties of such sequences and series to solve problems, including finding the formula for the general term and the sum, recursively and explicitly.
  • Demonstrate an understanding of the binomial theorem; use it in the solution of problems.
  • Demonstrate an understanding of polynomial and rational functions.
  • Apply polynomial and rational functions to solve "real world" problems.
  • Demonstrate an understanding of the exponential and logarithmic functions.
  • Apply exponential and logarithmic functions to solve "real world" problems.
  • Perform operations on functions, including composition.
  • Find inverses of functions.
  • Given algebraic, numeric and/or graphical representations, recognize functions as polynomial, rational, logarithmic, or exponential.
  • Find solutions to quadratic equations (with real coefficients and real or complex roots) and apply to the solutions of problems.
  • Solve a variety of equations and inequalities using algebraic, graphical, and numerical methods, including the quadratic formula; use graphing calculator technology where appropriate. Include polynomial, exponential, and logarithmic functions; expressions involving the absolute values; and simple rational expressions.
  • Use matrices to solve systems of linear equations; apply to the solution of everyday problems.
  • Use symbolic, numeric, and graphical methods to solve systems of equations and/or inequalities involving algebraic, exponential, and logarithmic expressions; use technology where appropriate.
  • Solve everyday problems that can be modeled using polynomial, rational, exponential, logarithmic, and step functions, absolute values and square roots. Apply appropriate graphical, tabular, or symbolic methods to the solution. Include growth and decay; logistic growth; joint (e.g., I = Prt, y = k(w1 + w2)), and combined (F = G(m1m2)/d2) variation.
  • Identify maximum and minimum values of functions in simple situations; apply to the solution of everyday problems.
  • Describe the translations and scale changes of a given function f(x) resulting from substitutions for the various parameters a, b, c, and d in y = af (b(x + c/b)) + d. In particular, describe the effect of such changes on polynomial, rational, exponential, and logarithmic functions.
Geometry
  • Analyze characteristics and properties of two-dimensional and three-dimensional geometric shapes and develop mathematical arguments about geometrical relationships.
  • Specify locations and describe spatial relationships using coordinate geometry and other representational systems.
  • Apply transformations and use symmetry to analyze mathematical situations.
  • Use visualization, spatial reasoning, and geometric modeling to solve problems.
  • Define the sine, cosine, and tangent of an acute angle. Apply to the solution of problems.
  • Find values of trigonometric functions for acute angles.
  • Solve everyday problems involving right triangles.
  • Derive and apply basic trigonometric identities (e.g., sin2q + cos2q = 1, tan2q + 1 = sec2q) and the laws of sines and cosines.
  • Relate geometric and algebraic representations of lines, simple curves, and conic sections.
  • Change radian measure to degree measure and vice versa.
  • Identify coterminal angles.
  • Find values of trigonometric functions for general angles.
  • Use reference angles to find the values of trigonometric functions.
  • Solve everyday problems using the Law of Sines and the Law of Cosines.
Data Analysis, Statistics, and Probability
  • Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
  • Select and use appropriate statistical methods to analyze data.
  • Develop and evaluate inferences and predictions that are based on data.
  • Understand and apply basic concepts of probability.
  • Select an appropriate graphical representation for a set of data and use appropriate statistics (e.g., quartile or percentile distribution) to communicate information about the data.
  • Find measures of variation for a set of data.
  • Use combinatorics (e.g., "fundamental counting principle," permutations, combinations) to compute probabilities of compound events and solve other problems; use technology as appropriate.
  • Find the probability of two independent events; of two dependent events.
  • Solve everyday problems involving the probability of independent or dependent events.
  • Determine whether a set of data appears to be normally distributed of skewed.
  • Solve everyday problems involving normally distributed data.
Discussion, Presentation, Composition
  • Use agreed upon rules to participate in discussions in large and small groups.
  • Express ideas in an organized way.
  • Explain their mathematical thinking in writing.
  • Maintain a system for collecting, referring to, and sharing their work.
SCIENCE

Earth and Space Science

  • Explain how the transfer of energy through radiation, conduction, and convection contributes to global atmospheric processes, e.g., storms, winds.
  • Explain how the revolution of the earth and the inclination of the axis of the earth cause the earth's seasonal variations (equinoxes and solstices).
  • Explain the dynamics of oceanic currents, including upwelling, density and deep water currents, the local Labrador Current, and the Gulf Stream, and their relationship to global circulation within the marine environment and climate.
  • Describe how glaciers, gravity, wind, temperature changes, waves, and rivers cause weathering and erosion. Give examples of how the effects of these processes can be seen in our local environment.
  • Describe the absolute and relative dating methods used to measure geologic time, e.g., index fossils, radioactive dating, law of superposition, and cross-cutting relationships.
  • Trace the development of a lithospheric plate from its growing margin at a divergent boundary (mid-ocean ridge) to its destructive margin at a convergent boundary (subduction zone). Explain the relationship between convection currents and the motion of the lithospheric plates.
  • Explain how the sun, earth, and solar system formed from a nebula of dust and gas in a spiral arm of the Milky Way Galaxy about 4.6 billion years ago.
Biology
  • Describe the composition and functions of the four major categories of organic molecules (carbohydrates, lipids, proteins, and nucleic acids).
  • Relate cell parts/organelles to their function.
  • Distinguish between plant and animal cells.
  • Explain the role of cell membranes as a highly selective barrier (diffusion, osmosis, and active transport).
  • Describe and compare the processes of mitosis and meiosis, and their role in the cell cycle.
  • Describe the structure and function of DNA, and distinguish among replication, transcription, and translation.
  • Use a Punnett Square to determine the genotype and phenotype of monohybrid crosses.
Chemistry
  • Identify and explain some of the physical properties that are used to classify matter, e.g., density, melting point, ad boiling point.
  • Explain the difference between mixtures and pure substances.
  • Describe the four states of matter (solid, liquid, gas, plasma) in terms of energy, particle motion, and phase transitions.
  • Identify the major components of the nuclear atom (protons, neutrons, and electrons) and explain how they interact.
  • Compare nuclear fission and nuclear fusion and mass defect.
  • Explain the relationship al an element's position on the periodic table to its atomic number and mass.
  • Explain how atoms combine to form compounds through both ionic and covalent bonding.
  • Identify and explain the factors that affect the rate of dissolving, i.e., temperature, concentration, and mixing.
Physics
  • Explain the relationship between mass and inertia.
  • Interpret and apply Newton's first law of motion.
  • Interpret and apply Newton's second law of motion to show how an object's motion will change only when a net force is applied.
  • Interpret and provide examples that illustrate the law of conservation of energy.
  • Differentiate between wave motion (simple harmonic nonlinear motion) and the motion of objects (nonharmonic).
  • Recognize the measurable properties of waves (e.g., velocity, frequency, wavelength) and explain the relationships among them.
  • Distinguish between mechanical and electromagnetic wave.
Scientific Inquiry
  • Use simple tools such as rulers, magnifiers, balances, thermometers, graduated cylinders, etc to observe and measure things carefully.
  • Design and conduct simple science experiments using appropriate controls, variables, equipment and measuring tools. Some questions may be posed by the student and some will be posed by the teacher.
  • Predict, observe, classify and record results clearly in journals or logs.
  • Use technology and mathematics to improve investigations and communications.
  • Communicate scientific procedures and explanations using presentations, charts, simple graphs, discussions and writing.
  • Develop descriptions, explanations, predictions, and models using evidence.
  • Compare results and explanations with scientific knowledge.
Discussion and Presentation
  • Participate in formal and informal discussions in large and small groups, using agreed upon rules to conduct and facilitate them.
  • Organize and present their thoughts in a logical manner.
  • Support their ideas with evidence or details; expect and request the same of others.
  • Actively listen, respond to, and build on ideas generated during discussions.
  • Use the information to inform or change their perspectives.
  • Ask for clarification when others' responses are unclear.
  • Summarize and evaluate what they have learned from the discussion.
  • Evaluate the productivity of discussions using established criteria; make suggestions to improve the discussions.
  • Give oral presentations, using established criteria to prepare, assess, and improve their presentations.
Composition
  • Write frequently in response to readings, other presentations, and observations (e.g., summaries, questions, reactions, connections, predictions, reports).
  • Maintain a system for collecting, referring to, and sharing their thoughts, observations, writings, illustrations, and other work.
  • Write occasional, brief research reports to extend their knowledge beyond classroom presentations; include a clear focus and supporting details.
  • Write, share, assess, and revise frequent responses to MCAS-like, open response (key) questions posed by the teacher.