8th through 12th Grade
Mathematics Standards
& 8th Grade Reading/Language Arts Standards
Mathematics
The eighth grade [Mathematics] standards are not listed separately from the high school standards. The standards for grades eight through twelve are organized differently in Mathematics than the standards for grades Kindergarten through Seven. The standards for grades eight through twelve do not mandate that a particular discipline be initiated or completed in a single grade. Completion of these standards is not a prerequisite for successful completion of eighth grade. These standards should be met by the end of twelfth grade.
Grades Eight Through Twelve — Algebra I
Algebra I
Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences. In addition, algebraic skills and concepts are developed and used in a wide variety of problem-solving situations.
1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable:
1.1 Students use properties of numbers to demonstrate whether assertions are true or false.
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents.
3.0 Students solve equations and inequalities involving absolute values.
4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.
5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.
6.0 Students graph a linear equation and compute the x- and y-intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).
7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.
8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.
9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.
11.0 Students apply basic factoring techniques to second- and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.
12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.
14.0 Students solve a quadratic equation by factoring or completing the square.
15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions.
17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.
18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.
19.0 Students know the quadratic formula and are familiar with its proof by completing the square.
20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.
21.0 Students graph quadratic functions and know that their roots are the x-intercepts.
22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.
23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.
24.0 Students use and know simple aspects of a logical argument:
24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.
24.2 Students identify the hypothesis and conclusion in logical deduction.
24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.
25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:
25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.
25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.
25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.
Geometry
The geometry skills and concepts developed in this discipline are useful to all students. Aside from learning these skills and concepts, students will develop their ability to construct formal, logical arguments and proofs in geometric settings and problems.
1.0 Students demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
2.0 Students write geometric proofs, including proofs by contradiction.
3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.
4.0 Students prove basic theorems involving congruence and similarity.
5.0 Students prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
6.0 Students know and are able to use the triangle inequality theorem.
7.0 Students prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
8.0 Students know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
9.0 Students compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and students commit to memory the formulas for prisms, pyramids, and cylinders.
10.0 Students compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
12.0 Students find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
13.0 Students prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
14.0 Students prove the Pythagorean theorem.
15.0 Students use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
16.0 Students perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.
17.0 Students prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
18.0 Students know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them. For example, tan(x) = sin(x)/cos(x), (sin(x)) 2 + (cos(x)) 2 = 1.
19.0 Students use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
20.0 Students know and are able to use angle and side relationships in problems with special right triangles, such as 30° , 60° , and 90° triangles and 45° , 45° , and 90° triangles.
21.0 Students prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
22.0 Students know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
Algebra II
This discipline complements and expands the mathematical content and concepts of algebra I and geometry. Students who master algebra II will gain experience with algebraic solutions of problems in various content areas, including the solution of systems of quadratic equations, logarithmic and exponential functions, the binomial theorem, and the complex number system.
1.0 Students solve equations and inequalities involving absolute value.
2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
3.0 Students are adept at operations on polynomials, including long division.
4.0 Students factor polynomials representing the difference of squares, perfect square trinomials, and the sum and difference of two cubes.
5.0 Students demonstrate knowledge of how real and complex numbers are related both arithmetically and graphically. In particular, they can plot complex numbers as points in the plane.
6.0 Students add, subtract, multiply, and divide complex numbers.
7.0 Students add, subtract, multiply, divide, reduce, and evaluate rational expressions with monomial and polynomial denominators and simplify complicated rational expressions, including those with negative exponents in the denominator.
8.0 Students solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula. Students apply these techniques in solving word problems. They also solve quadratic equations in the complex number system.
9.0 Students demonstrate and explain the effect that changing a coefficient has on the graph of quadratic functions; that is, students can determine how the graph of a parabola changes as a, b, and c vary in the equation y = a(x-b) 2 + c.
10.0 Students graph quadratic functions and determine the maxima, minima, and zeros of the function.
11.0 Students prove simple laws of logarithms.
11.1 Students understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
11.2 Students judge the validity of an argument according to whether the properties of real numbers, exponents, and logarithms have been applied correctly at each step.
12.0 Students know the laws of fractional exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay.
13.0 Students use the definition of logarithms to translate between logarithms in any base.
14.0 Students understand and use the properties of logarithms to simplify logarithmic numeric expressions and to identify their approximate values.
15.0 Students determine whether a specific algebraic statement involving rational expressions, radical expressions, or logarithmic or exponential functions is sometimes true, always true, or never true.
16.0 Students demonstrate and explain how the geometry of the graph of a conic section (e.g., asymptotes, foci, eccentricity) depends on the coefficients of the quadratic equation representing it.
17.0 Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, students can use the method for completing the square to put the equation into standard form and can recognize whether the graph of the equation is a circle, ellipse, parabola, or hyperbola. Students can then graph the equation.
18.0 Students use fundamental counting principles to compute combinations and permutations.
19.0 Students use combinations and permutations to compute probabilities.
20.0 Students know the binomial theorem and use it to expand binomial expressions that are raised to positive integer powers.
21.0 Students apply the method of mathematical induction to prove general statements about the positive integers.
22.0 Students find the general term and the sums of arithmetic series and of both finite and infinite geometric series.
23.0 Students derive the summation formulas for arithmetic series and for both finite and infinite geometric series.
24.0 Students solve problems involving functional concepts, such as composition, defining the inverse function and performing arithmetic operations on functions.
25.0 Students use properties from number systems to justify steps in combining and simplifying functions.
Probability and Statistics
This discipline is an introduction to the study of probability, interpretation of data, and fundamental statistical problem solving. Mastery of this academic content will provide students with a solid foundation in probability and facility in processing statistical information.
1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces.
2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.
3.0 Students demonstrate an understanding of the notion of discrete random variables by using them to solve for the probabilities of outcomes, such as the probability of the occurrence of five heads in 14 coin tosses.
4.0 Students are familiar with the standard distributions (normal, binomial, and exponential) and can use them to solve for events in problems in which the distribution belongs to those families.
5.0 Students determine the mean and the standard deviation of a normally distributed random variable.
6.0 Students know the definitions of the mean, median, and mode of a distribution of data and can compute each in particular situations.
7.0 Students compute the variance and the standard deviation of a distribution of data.
8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
Reading/Language Arts Standards – [Grade 8 ONLY]
READING
1.0 Word Analysis, Fluency, and Systematic Vocabulary Development
Students use their knowledge of word origins and word relationships, as well as historical and literary context clues, to determine the meaning of specialized vocabulary and to understand the precise meaning of grade-level-appropriate words.
Vocabulary and Concept Development
1.1 Analyze idioms, analogies, metaphors, and similes to infer the literal and figurative meanings of phrases.
1.2 Understand the most important points in the history of English language and use common word origins to determine the historical influences on English word meanings.
1.3 Use word meanings within the appropriate context and show ability to verify those meanings by definition, restatement, example, comparison, or contrast.
2.0 Reading Comprehension (Focus on Informational Materials)
Students read and understand grade-level-appropriate material. They describe and connect the essential ideas, arguments, and perspectives of the text by using their knowledge of text structure, organization, and purpose. The selections in Recommended Readings in Literature, Kindergarten Through Grade Eight illustrate the quality and complexity of the materials to be read by students. In addition, students read one million words annually on their own, including a good representation of narrative and expository text (e.g., classic and contemporary literature, magazines, newspapers, online information).
Structural Features of Informational Materials
2.1 Compare and contrast the features and elements of consumer materials to gain meaning from documents (e.g., warranties, contracts, product information, instruction manuals).
2.2 Analyze text that uses proposition and support patterns.
Comprehension and Analysis of Grade-Level-Appropriate Text
2.3 Find similarities and differences between texts in the treatment, scope, or organization of ideas.
2.4 Compare the original text to a summary to determine whether the summary accurately captures the main ideas, includes critical details, and conveys the underlying meaning.
2.5 Understand and explain the use of a complex mechanical device by following technical directions.
2.6 Use information from a variety of consumer, workplace, and public documents to explain a situation or decision and to solve a problem.
Expository Critique
2.7 Evaluate the unity, coherence, logic, internal consistency, and structural patterns of text.
3.0 Literary Response and Analysis
Students read and respond to historically or culturally significant works of literature that reflect and enhance their studies of history and social science. They clarify the ideas and connect them to other literary works. The selections in Recommended Readings in Literature, Kindergarten Through Grade Eight illustrate the quality and complexity of the materials to be read by students.
Structural Features of Literature
3.1 Determine and articulate the relationship between the purposes and characteristics of different forms of poetry (e.g., ballad, lyric, couplet, epic, elegy, ode, sonnet).
Narrative Analysis of Grade-Level-Appropriate Text
3.2 Evaluate the structural elements of the plot (e.g., subplots, parallel episodes, climax), the plot's development, and the way in which conflicts are (or are not) addressed and resolved.
3.3 Compare and contrast motivations and reactions of literary characters from different historical eras confronting similar situations or conflicts.
3.4 Analyze the relevance of the setting (e.g., place, time, customs) to the mood, tone, and meaning of the text.
3.5 Identify and analyze recurring themes (e.g., good versus evil) across traditional and contemporary works.
3.6 Identify significant literary devices (e.g., metaphor, symbolism, dialect, irony) that define a writer's style and use those elements to interpret the work.
Literary Criticism
3.7 Analyze a work of literature, showing how it reflects the heritage, traditions, attitudes, and beliefs of its author. (Biographical approach)
Writing
1.0 Writing Strategies
Students write clear, coherent, and focused essays. The writing exhibits students' awareness of audience and purpose. Essays contain formal introductions, supporting evidence, and conclusions. Students progress through the stages of the writing process as needed.
Organization and Focus
1.1 Create compositions that establish a controlling impression, have a coherent thesis, and end with a clear and well-supported conclusion.
1.2 Establish coherence within and among paragraphs through effective transitions, parallel structures, and similar writing techniques.
1.3 Support theses or conclusions with analogies, paraphrases, quotations, opinions from authorities, comparisons, and similar devices.
Research and Technology
1.4 Plan and conduct multiple-step information searches by using computer networks and modems.
1.5 Achieve an effective balance between researched information and original ideas.
Evaluation and Revision
1.6 Revise writing for word choice; appropriate organization; consistent point of view; and transitions between paragraphs, passages, and ideas.
2.0 Writing Applications (Genres and Their Characteristics)
Students write narrative, expository, persuasive, and descriptive essays of at least 500 to 700 words in each genre. Student writing demonstrates a command of standard American English and the research, organizational, and drafting strategies outlined in Writing Standard 1.0.
Using the writing strategies of grade eight outlined in Writing Standard 1.0, students:
2.1 Write biographies, autobiographies, short stories, or narratives:
a. Relate a clear, coherent incident, event, or situation by using well-chosen details.
b. Reveal the significance of, or the writer's attitude about, the subject.
c. Employ narrative and descriptive strategies (e.g., relevant dialogue, specific action, physical description, background description, comparison or contrast of characters).
2.2 Write responses to literature:
a. Exhibit careful reading and insight in their interpretations.
b. Connect the student's own responses to the writer's techniques and to specific textual references.
c. Draw supported inferences about the effects of a literary work on its audience.
d. Support judgments through references to the text, other works, other authors, or to personal knowledge.
2.3 Write research reports:
a. Define a thesis.
b. Record important ideas, concepts, and direct quotations from significant information sources and paraphrase and summarize all perspectives on the topic, as appropriate.
c. Use a variety of primary and secondary sources and distinguish the nature and value of each.
d. Organize and display information on charts, maps, and graphs.
2.4 Write persuasive compositions:
a. Include a well-defined thesis (i.e., one that makes a clear and knowledgeable judgment).
b. Present detailed evidence, examples, and reasoning to support arguments, differentiating between facts and opinion.
c. Provide details, reasons, and examples, arranging them effectively by anticipating and answering reader concerns and counter-arguments.
2.5 Write documents related to career development, including simple business letters and job applications:
a. Present information purposefully and succinctly and meet the needs of the intended audience.
b. Follow the conventional format for the type of document (e.g., letter of inquiry, memorandum).
2.6 Write technical documents:
a. Identify the sequence of activities needed to design a system, operate a tool, or explain the bylaws of an organization.
b. Include all the factors and variables that need to be considered.
c. Use formatting techniques (e.g., headings, differing fonts) to aid comprehension.
Written and Oral English Language Conventions
The standards for written and oral English language conventions have been placed between those for writing and for listening and speaking because these conventions are essential to both sets of skills.
1.0 Written and Oral English Language Conventions
Students write and speak with a command of standard English conventions appropriate to this grade level.
Sentence Structure
1.1 Use correct and varied sentence types and sentence openings to present a lively and effective personal style.
1.2 Identify and use parallelism, including similar grammatical forms, in all written discourse to present items in a series and items juxtaposed for emphasis.
1.3 Use subordination, coordination, apposition, and other devices to indicate clearly the relationship between ideas.
Grammar
1.4 Edit written manuscripts to ensure that correct grammar is used.
Punctuation and Capitalization
1.5 Use correct punctuation and capitalization.
Spelling
1.6 Use correct spelling conventions.
Listening and Speaking
1.0 Listening and Speaking Strategies
Students deliver focused, coherent presentations that convey ideas clearly and relate to the background and interests of the audience. They evaluate the content of oral communication.
Comprehension
1.1 Analyze oral interpretations of literature, including language choice and delivery, and the effect of the interpretations on the listener.
1.2 Paraphrase a speaker's purpose and point of view and ask relevant questions concerning the speaker's content, delivery, and purpose.
Organization and Delivery of Oral Communication
1.3 Organize information to achieve particular purposes by matching the message, vocabulary, voice modulation, expression, and tone to the audience and purpose.
1.4 Prepare a speech outline based upon a chosen pattern of organization, which generally includes an introduction; transitions, previews, and summaries; a logically developed body; and an effective conclusion.
1.5 Use precise language, action verbs, sensory details, appropriate and colorful modifiers, and the active rather than the passive voice in ways that enliven oral presentations.
1.6 Use appropriate grammar, word choice, enunciation, and pace during formal presentations.
1.7 Use audience feedback (e.g., verbal and nonverbal cues):
a. Reconsider and modify the organizational structure or plan.
b. Rearrange words and sentences to clarify the meaning.
Analysis and Evaluation of Oral and Media Communications
1.8 Evaluate the credibility of a speaker (e.g., hidden agendas, slanted or biased material).
1.9 Interpret and evaluate the various ways in which visual image makers (e.g., graphic artists, illustrators, news photographers) communicate information and affect impressions and opinions.
2.0 Speaking Applications (Genres and Their Characteristics)
Students deliver well-organized formal presentations employing traditional rhetorical strategies (e.g., narration, exposition, persuasion, description). Student speaking demonstrates a command of standard American English and the organizational and delivery strategies outlined in Listening and Speaking Standard 1.0. Using the speaking strategies of grade eight outlined in Listening and Speaking Standard 1.0, students:
2.1 Deliver narrative presentations (e.g., biographical, autobiographical):
a. Relate a clear, coherent incident, event, or situation by using well-chosen details.
b. Reveal the significance of, and the subject's attitude about, the incident, event, or situation.
c. Employ narrative and descriptive strategies (e.g., relevant dialogue, specific action, physical description, background description, comparison or contrast of characters).
2.2 Deliver oral responses to literature:
a. Interpret a reading and provide insight.
b. Connect the students' own responses to the writer's techniques and to specific textual references.
c. Draw supported inferences about the effects of a literary work on its audience.
d. Support judgments through references to the text, other works, other authors, or personal knowledge.
2.3 Deliver research presentations:
a. Define a thesis.
b. Record important ideas, concepts, and direct quotations from significant information sources and paraphrase and summarize all relevant perspectives on the topic, as appropriate.
c. Use a variety of primary and secondary sources and distinguish the nature and value of each.
d. Organize and record information on charts, maps, and graphs.
2.4 Deliver persuasive presentations:
a. Include a well-defined thesis (i.e., one that makes a clear and knowledgeable judgment).
b. Differentiate fact from opinion and support arguments with detailed evidence, examples, and reasoning.
c. Anticipate and answer listener concerns and counter-arguments effectively through the inclusion and arrangement of details, reasons, examples, and other elements.
d. Maintain a reasonable tone.
2.5 Recite poems (of four to six stanzas), sections of speeches, or dramatic soliloquies, using voice modulation, tone, and gestures expressively to enhance the meaning.