IXL  Math and English
 reading comprehension
 text analysis
 using supporting evidence
 vocabulary
 algebra
 geometry
 graphing
 ratio
 perseverance
 applying information
 memorization
 partwhole relationships
Pros
Excellent coverage of math and ELA curriculum, with detailed reporting available for multiple users.Cons
Digital skillanddrill may not give kids the depth of understanding that other activities could.Bottom Line
Straightforward worksheets help kids practice and build fluency in a huge range of math and ELA skills.Free
You can create a student roster by registering on the developer's website. This allows you to generate progress reports.
Graphite Expert Review
Learning Scores
The app is all about practicing math and language arts skills, which can be a little tedious. However, the awards are motivating, and the digital experience is certainly more engaging than traditional paperandpencil worksheets.
Kids can reach a challenge mode after completing basic questions, and feedback for correcting errors is thorough and helpful.
Content is easy to navigate, and inputting answers is straightforward: Enter manually or choose from four options. A link to tech support is accessible directly from the app.
IXL  Math and English is a handy tool that makes it easy for kids to practice gradeappropriate math and language arts skills. On the home screen, kids select math or language arts and their grade level. Then, kids choose a topic and begin answering the questions. Number of problems completed, score, and time appear at the top of the screen. IXL  Math and English offers praise for correct answers and stepbystep explanations for incorrect answers. Kids earn prizes and awards for reaching milestones and mastering skills. After registering on the developer's website, users can access progress reports to evaluate kids' strengths and problem areas. This is particularly useful for differentiating assignments in the classroom. A subscription gives access to both the website and the app.
Read More Read LessWith a preK through precalculus math curriculum and activities appropriate for second through 10thgrade language arts, IXL  Math and English topics align with Common Core State Standards and with individual states' requirements, depending on the home state chosen at registration. Each grade level is packed with virtual worksheets arranged by topic. If kids get an answer wrong, the feedback is excellent and gives a stepbystep explanation of the correct answer. This is a powerful tool and can greatly improve kids' comprehension of difficult concepts. Kids have to choose the correct answer from four options, or they have to enter the answer manually. For math, kids can use the touchscreen as scratch paper to work through a problem. For language arts, preK through firstgrade questions can be read aloud inapp. The variety of question types engages and challenges kids, and rewards help motivate them to keep practicing. Kids earn medals when they master skills, and parents and teachers can generate reports on the developer's website by creating a roster of users.
Read More Read LessThis tool is best for practicing and building fluency in math and for accessing understanding of grammar and language skills in English. But since it gives kids such great feedback for incorrect answers, there's a teaching component that you could use as well. For example, you might assign some of the worksheets as preassessments before starting a unit. Discuss any questions that were particularly challenging, and keep a record of the preassessment scores. Ask kids to share anything they learned from their mistakes. As you work through the unit, use the worksheets to reinforce concepts. As kids make progress and earn high scores, challenge them to improve their speed. When the unit is wrapping up, assign the same worksheet topics that you used for the preassessment and compare kids' scores.
Read More Read LessKey Standards Supported
Language  
L.2: Conventions of Standard English  
L.2.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.2.1a  Use collective nouns (e.g., group). 
L.2.1b  Form and use frequently occurring irregular plural nouns (e.g., feet, children, teeth, mice, fish). 
L.2.1c  Use reflexive pronouns (e.g., myself, ourselves). 
L.2.1d  Form and use the past tense of frequently occurring irregular verbs (e.g., sat, hid, told). 
L.2.1e  Use adjectives and adverbs, and choose between them depending on what is to be modified. 
L.2.1f  Produce, expand, and rearrange complete simple and compound sentences (e.g., The boy watched the movie; The little boy watched the movie; The action movie was watched by the little boy). 
L.2.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.2.2a  Capitalize holidays, product names, and geographic names. 
L.2.2b  Use commas in greetings and closings of letters. 
L.2.2c  Use an apostrophe to form contractions and frequently occurring possessives. 
L.2.2d  Generalize learned spelling patterns when writing words (e.g., cage → badge; boy → boil). 
L.2.2e  Consult reference materials, including beginning dictionaries, as needed to check and correct spellings. 
Knowledge of Language  
L.2.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.2.3a  Compare formal and informal uses of English. 
Vocabulary Acquisition and Use  
L.2.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grade 2 reading and content, choosing flexibly from an array of strategies. 
L.2.4a  Use sentencelevel context as a clue to the meaning of a word or phrase. 
L.2.4b  Determine the meaning of the new word formed when a known prefix is added to a known word (e.g., happy/unhappy, tell/retell). 
L.2.4c  Use a known root word as a clue to the meaning of an unknown word with the same root (e.g., addition, additional). 
L.2.4d  Use knowledge of the meaning of individual words to predict the meaning of compound words (e.g., birdhouse, lighthouse, housefly; bookshelf, notebook, bookmark). 
L.2.4e  Use glossaries and beginning dictionaries, both print and digital, to determine or clarify the meaning of words and phrases. 
L.2.5  Demonstrate understanding of figurative language, word relationships and nuances in word meanings. 
L.2.5b  Distinguish shades of meaning among closely related verbs (e.g., toss, throw, hurl) and closely related adjectives (e.g., thin, slender, skinny, scrawny). 
L.2.6  Use words and phrases acquired through conversations, reading and being read to, and responding to texts, including using adjectives and adverbs to describe (e.g., When other kids are happy that makes me happy). 
L2.5a  Identify reallife connections between words and their use (e.g., describe foods that are spicy or juicy). 
L.3: Conventions of Standard English  
L.3.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.3.1a  Explain the function of nouns, pronouns, verbs, adjectives, and adverbs in general and their functions in particular sentences. 
L.3.1b  Form and use regular and irregular plural nouns. 
L.3.1c  Use abstract nouns (e.g., childhood). 
L.3.1d  Form and use regular and irregular verbs. 
L.3.1e  Form and use the simple (e.g., I walked; I walk; I will walk) verb tenses. 
L.3.1f  Ensure subjectverb and pronounantecedent agreement.* 
L.3.1g  Form and use comparative and superlative adjectives and adverbs, and choose between them depending on what is to be modified. 
L.3.1h  Use coordinating and subordinating conjunctions. 
L.3.1i  Produce simple, compound, and complex sentences. 
L.3.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.3.2a  Capitalize appropriate words in titles. 
L.3.2b  Use commas in addresses. 
L.3.2d  Form and use possessives. 
L.3.2e  Use conventional spelling for highfrequency and other studied words and for adding suffixes to base words (e.g., sitting, smiled, cries, happiness). 
L.3.2f  Use spelling patterns and generalizations (e.g., word families, positionbased spellings, syllable patterns, ending rules, meaningful word parts) in writing words. 
L.3.2g  Consult reference materials, including beginning dictionaries, as needed to check and correct spellings. 
L3.2c  Use commas and quotation marks in dialogue. 
Knowledge of Language  
L.3.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.3.3a  Choose words and phrases for effect.* 
L.3.3b  Recognize and observe differences between the conventions of spoken and written standard English. 
Vocabulary Acquisition and Use  
L.3.4  Determine or clarify the meaning of unknown and multiplemeaning word and phrases based on grade 3 reading and content, choosing flexibly from a range of strategies. 
L.3.4a  Use sentencelevel context as a clue to the meaning of a word or phrase. 
L.3.4b  Determine the meaning of the new word formed when a known affix is added to a known word (e.g., agreeable/disagreeable, comfortable/uncomfortable, care/careless, heat/preheat). 
L.3.4c  Use a known root word as a clue to the meaning of an unknown word with the same root (e.g., company, companion). 
L.3.4d  Use glossaries or beginning dictionaries, both print and digital, to determine or clarify the precise meaning of key words and phrases. 
L.3.5  Demonstrate understanding of figurative language, word relationships and nuances in word meanings. 
L.3.5a  Distinguish the literal and nonliteral meanings of words and phrases in context (e.g., take steps). 
L.3.5b  Identify reallife connections between words and their use (e.g., describe people who are friendly or helpful). 
L.3.5c  Distinguish shades of meaning among related words that describe states of mind or degrees of certainty (e.g., knew, believed, suspected, heard, wondered). 
L.3.6  Acquire and use accurately gradeappropriate conversational, general academic, and domainspecific words and phrases, including those that signal spatial and temporal relationships (e.g., After dinner that night we went looking for them). 
L.4: Conventions of Standard English  
L.4.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.4.1a  Use relative pronouns (who, whose, whom, which, that) and relative adverbs (where, when, why). 
L.4.1b  Form and use the progressive (e.g., I was walking; I am walking; I will be walking) verb tenses. 
L.4.1c  Use modal auxiliaries (e.g., can, may, must) to convey various conditions. 
L.4.1d  Order adjectives within sentences according to conventional patterns (e.g., a small red bag rather than a red small bag). 
L.4.1e  Form and use prepositional phrases. 
L.4.1f  Produce complete sentences, recognizing and correcting inappropriate fragments and runons.* 
L.4.1g  Correctly use frequently confused words (e.g., to, too, two; there, their).* 
L.4.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.4.2a  Use correct capitalization. 
L.4.2b  Use commas and quotation marks to mark direct speech and quotations from a text. 
L.4.2c  Use a comma before a coordinating conjunction in a compound sentence. 
L.4.2d  Spell gradeappropriate words correctly, consulting references as needed. 
Knowledge of Language  
L.4.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.4.3a  Choose words and phrases to convey ideas precisely.* 
L.4.3b  Choose punctuation for effect.* 
L.4.3c  Differentiate between contexts that call for formal English (e.g., presenting ideas) and situations where informal discourse is appropriate (e.g., smallgroup discussion). 
Vocabulary Acquisition and Use  
L.4.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grade 4 reading and content, choosing flexibly from a range of strategies. 
L.4.4a  Use context (e.g., definitions, examples, or restatements in text) as a clue to the meaning of a word or phrase. 
L.4.4b  Use common, gradeappropriate Greek and Latin affixes and roots as clues to the meaning of a word (e.g., telegraph, photograph, autograph). 
L.4.4c  Consult reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation and determine or clarify the precise meaning of key words and phrases. 
L.4.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.4.5a  Explain the meaning of simple similes and metaphors (e.g., as pretty as a picture) in context. 
L.4.5b  Recognize and explain the meaning of common idioms, adages, and proverbs. 
L.4.5c  Demonstrate understanding of words by relating them to their opposites (antonyms) and to words with similar but not identical meanings (synonyms). 
L.4.6  Acquire and use accurately gradeappropriate general academic and domainspecific words and phrases, including those that signal precise actions, emotions, or states of being (e.g., quizzed, whined, stammered) and that are basic to a particular topic (e.g., wildlife, conservation, and endangered when discussing animal preservation). 
L.5: Conventions of Standard English  
L.5.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.5.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
Knowledge of Language  
L.5.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.5.3a  Expand, combine, and reduce sentences for meaning, reader/listener interest, and style. 
L.5.3b  Compare and contrast the varieties of English (e.g., dialects, registers) used in stories, dramas, or poems. 
Vocabulary acquisition and Use  
L.5.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grade 5 reading and content, choosing flexibly from a range of strategies. 
L.5.4a  Use context (e.g., cause/effect relationships and comparisons in text) as a clue to the meaning of a word or phrase. 
L.5.4b  Use common, gradeappropriate Greek and Latin affixes and roots as clues to the meaning of a word (e.g., photograph, photosynthesis). 
L.5.4c  Consult reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation and determine or clarify the precise meaning of key words and phrases. 
L.5.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.5.5a  Interpret figurative language, including similes and metaphors, in context. 
L.5.5b  Recognize and explain the meaning of common idioms, adages, and proverbs. 
L.5.5c  Use the relationship between particular words (e.g., synonyms, antonyms, homographs) to better understand each of the words. 
L.5.6  Acquire and use accurately gradeappropriate general academic and domainspecific words and phrases, including those that signal contrast, addition, and other logical relationships (e.g., however, although, nevertheless, similarly, moreover, in addition). 
L.6: Conventions of Standard English  
L.6.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.6.1.A  Ensure that pronouns are in the proper case (subjective, objective, possessive). 
L.6.1.B  Use intensive pronouns (e.g., myself, ourselves). 
L.6.1.C  Recognize and correct inappropriate shifts in pronoun number and person. 
L.6.1.D  Recognize and correct vague pronouns (i.e., ones with unclear or ambiguous antecedents). 
L.6.1.E  Recognize variations from standard English in their own and others' writing and speaking, and identify and use strategies to improve expression in conventional language. 
L.6.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.6.2.A  Use punctuation (commas, parentheses, dashes) to set off nonrestrictive/parenthetical elements.* 
L.6.2.B  Spell correctly. 
Knowledge of Language  
L.6.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.6.3.A  Vary sentence patterns for meaning, reader/listener interest, and style. 
L.6.3.B  Maintain consistency in style and tone. 
Vocabulary Acquisition and Use  
L.6.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grade 6 reading and content, choosing flexibly from a range of strategies. 
L.6.4.A  Use context (e.g., the overall meaning of a sentence or paragraph; a word's position or function in a sentence) as a clue to the meaning of a word or phrase. 
L.6.4.B  Use common, gradeappropriate Greek or Latin affixes and roots as clues to the meaning of a word (e.g., audience, auditory, audible). 
L.6.4.C  Consult reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation of a word or determine or clarify its precise meaning or its part of speech. 
L.6.4.D  Verify the preliminary determination of the meaning of a word or phrase (e.g., by checking the inferred meaning in context or in a dictionary). 
L.6.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.6.5.A  Interpret figures of speech (e.g., personification) in context. 
L.6.5.B  Use the relationship between particular words (e.g., cause/effect, part/whole, item/category) to better understand each of the words. 
L.6.5.C  Distinguish among the connotations (associations) of words with similar denotations (definitions) (e.g., stingy, scrimping, economical, unwasteful, thrifty). 
L.6.6  Acquire and use accurately gradeappropriate general academic and domainspecific words and phrases; gather vocabulary knowledge when considering a word or phrase important to comprehension or expression. 
L.7: Conventions of Standard English  
L.7.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.7.1a  Explain the function of phrases and clauses in general and their function in specific sentences. 
L.7.1b  Choose among simple, compound, complex, and compoundcomplex sentences to signal differing relationships among ideas. 
L.7.1c  Place phrases and clauses within a sentence, recognizing and correcting misplaced and dangling modifiers.* 
L.7.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.7.2a  Use a comma to separate coordinate adjectives (e.g., It was a fascinating, enjoyable movie but not He wore an old[,] green shirt). 
L.7.2b  Spell correctly. 
Knowledge of Language  
L.7.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.7.3a  Choose language that expresses ideas precisely and concisely, recognizing and eliminating wordiness and redundancy.* 
Vocabulary Acquisition and Use  
L.7.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grade 7 reading and content, choosing flexibly from a range of strategies. 
L.7.4a  Use context (e.g., the overall meaning of a sentence or paragraph; a word’s position or function in a sentence) as a clue to the meaning of a word or phrase. 
L.7.4b  Use common, gradeappropriate Greek or Latin affixes and roots as clues to the meaning of a word (e.g., belligerent, bellicose, rebel). 
L.7.4c  Consult general and specialized reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation of a word or determine or clarify its precise meaning or its part of speech. 
L.7.4d  Verify the preliminary determination of the meaning of a word or phrase (e.g., by checking the inferred meaning in context or in a dictionary). 
L.7.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.7.5a  Interpret figures of speech (e.g., literary, biblical, and mythological allusions) in context. 
L.7.5b  Use the relationship between particular words (e.g., synonym/antonym, analogy) to better understand each of the words. 
L.7.5c  Distinguish among the connotations (associations) of words with similar denotations (definitions) (e.g., refined, respectful, polite, diplomatic, condescending). 
L.7.6  Acquire and use accurately gradeappropriate general academic and domainspecific words and phrases; gather vocabulary knowledge when considering a word or phrase important to comprehension or expression. 
L.8: Conventions of Standard English  
L.8.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.8.1a  Explain the function of verbals (gerunds, participles, infinitives) in general and their function in particular sentences. 
L.8.1b  Form and use verbs in the active and passive voice. 
L.8.1c  Form and use verbs in the indicative, imperative, interrogative, conditional, and subjunctive mood. 
L.8.1d  Recognize and correct inappropriate shifts in verb voice and mood.* 
L.8.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.8.2a  Use punctuation (comma, ellipsis, dash) to indicate a pause or break. 
L.8.2b  Use an ellipsis to indicate an omission. 
L.8.2c  Spell correctly. 
Knowledge of Language  
L.8.3  Use knowledge of language and its conventions when writing, speaking, reading, or listening. 
L.8.3a  Use verbs in the active and passive voice and in the conditional and subjunctive mood to achieve particular effects (e.g., emphasizing the actor or the action; expressing uncertainty or describing a state contrary to fact). 
Vocabulary Acquisition and Use  
L.8.4  Determine or clarify the meaning of unknown and multiplemeaning words or phrases based on grade 8 reading and content, choosing flexibly from a range of strategies. 
L.8.4a  Use context (e.g., the overall meaning of a sentence or paragraph; a word’s position or function in a sentence) as a clue to the meaning of a word or phrase. 
L.8.4b  Use common, gradeappropriate Greek or Latin affixes and roots as clues to the meaning of a word (e.g., precede, recede, secede). 
L.8.4c  Consult general and specialized reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation of a word or determine or clarify its precise meaning or its part of speech. 
L.8.4d  Verify the preliminary determination of the meaning of a word or phrase (e.g., by checking the inferred meaning in context or in a dictionary). 
L.8.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.8.5a  Interpret figures of speech (e.g. verbal irony, puns) in context. 
L.8.5b  Use the relationship between particular words to better understand each of the words. 
L.8.5c  Distinguish among the connotations (associations) of words with similar denotations (definitions) (e.g., bullheaded, willful, firm, persistent, resolute). 
L.8.6  Acquire and use accurately gradeappropriate general academic and domainspecific words and phrases; gather vocabulary knowledge when considering a word or phrase important to comprehension or expression. 
L.910: Conventions of Standard English  
L.910.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.910.1a  Use parallel structure.* 
L.910.1b  Use various types of phrases (noun, verb, adjectival, adverbial, participial, prepositional, absolute) and clauses (independent, dependent; noun, relative, adverbial) to convey specific meanings and add variety and interest to writing or presentations. 
L.910.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.910.2a  Use a semicolon (and perhaps a conjunctive adverb) to link two or more closely related independent clauses. 
L.910.2b  Use a colon to introduce a list or quotation. 
L.910.2c  Spell correctly. 
Knowledge of Language  
L.910.3  Apply knowledge of language to understand how language functions in different contexts, to make effective choices for meaning or style, and to comprehend more fully when reading or listening. 
L.910.3a  Write and edit work so that it conforms to the guidelines in a style manual (e.g., MLA Handbook, Turabian’s Manual for Writers) appropriate for the discipline and writing type. 
Vocabulary Acquisition and Use  
L.910.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grades 9–10 reading and content, choosing flexibly from a range of strategies. 
L.910.4a  Use context (e.g., the overall meaning of a sentence, paragraph, or text; a word’s position or function in a sentence) as a clue to the meaning of a word or phrase. 
L.910.4b  Identify and correctly use patterns of word changes that indicate different meanings or parts of speech (e.g., analyze, analysis, analytical; advocate, advocacy). 
L.910.4c  Consult general and specialized reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation of a word or determine or clarify its precise meaning, its part of speech, or its etymology. 
L.910.4d  Verify the preliminary determination of the meaning of a word or phrase (e.g., by checking the inferred meaning in context or in a dictionary). 
L.910.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.910.5a  Interpret figures of speech (e.g., euphemism, oxymoron) in context and analyze their role in the text. 
L.910.5b  Analyze nuances in the meaning of words with similar denotations. 
L.910.6  Acquire and use accurately general academic and domainspecific words and phrases, sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when considering a word or phrase important to comprehension or expression. 
L.1112: Conventions of Standard English  
L.1112.1  Demonstrate command of the conventions of standard English grammar and usage when writing or speaking. 
L.1112.1a  Apply the understanding that usage is a matter of convention, can change over time, and is sometimes contested. 
L.1112.1b  Resolve issues of complex or contested usage, consulting references (e.g., MerriamWebster’s Dictionary of English Usage, Garner’s Modern American Usage) as needed. 
L.1112.2  Demonstrate command of the conventions of standard English capitalization, punctuation, and spelling when writing. 
L.1112.2.b  Spell correctly. 
L.1112.2a  Observe hyphenation conventions. 
Knowledge of Language  
L.1112.3  Apply knowledge of language to understand how language functions in different contexts, to make effective choices for meaning or style, and to comprehend more fully when reading or listening. 
L.1112.3a  Vary syntax for effect, consulting references (e.g., Tufte’s Artful Sentences) for guidance as needed; apply an understanding of syntax to the study of complex texts when reading. 
Vocabulary Acquisition and Use  
L.1112.4  Determine or clarify the meaning of unknown and multiplemeaning words and phrases based on grades 11–12 reading and content, choosing flexibly from a range of strategies. 
L.1112.4a  Use context (e.g., the overall meaning of a sentence, paragraph, or text; a word’s position or function in a sentence) as a clue to the meaning of a word or phrase. 
L.1112.4b  Identify and correctly use patterns of word changes that indicate different meanings or parts of speech (e.g., conceive, conception, conceivable). 
L.1112.4c  Consult general and specialized reference materials (e.g., dictionaries, glossaries, thesauruses), both print and digital, to find the pronunciation of a word or determine or clarify its precise meaning, its part of speech, its etymology, or its standard usage. 
L.1112.4d  Verify the preliminary determination of the meaning of a word or phrase (e.g., by checking the inferred meaning in context or in a dictionary). 
L.1112.5  Demonstrate understanding of figurative language, word relationships, and nuances in word meanings. 
L.1112.5a  Interpret figures of speech (e.g., hyperbole, paradox) in context and analyze their role in the text. 
L.1112.5b  Analyze nuances in the meaning of words with similar denotations. 
L.1112.6  Acquire and use accurately general academic and domainspecific words and phrases, sufficient for reading, writing, speaking, and listening at the college and career readiness level; demonstrate independence in gathering vocabulary knowledge when considering a word or phrase important to comprehension or expression. 
Arithmetic With Polynomials And Rational Expressions HSA.APR  
Perform Arithmetic Operations On Polynomials: Rewrite Rational Expressions  
HSA.APR.6  Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. 
HSA.APR.7  (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. 
HSS.CP.1  Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). 
HSS.CP.2  Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 
HSS.CP.5  Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. 
Use The Rules Of Probability To Compute Probabilities Of Compound Events In A Uniform Probability Model  
HSS.CP.6  Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. 
HSS.CP.7  Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model. 
HSS.CP.8  (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(BA) = P(B)P(AB), and interpret the answer in terms of the model. 
HSS.CP.9  (+) Use permutations and combinations to compute probabilities of compound events and solve problems. 
Counting And Cardinality  
K.CC: Compare Numbers.  
K.CC.6  Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.1 
K.CC.7  Compare two numbers between 1 and 10 presented as written numerals. 
Count To Tell The Number Of Objects.  
K.CC.4  Understand the relationship between numbers and quantities; connect counting to cardinality. 
K.CC.4.a  When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. 
K.CC.4.b  Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. 
K.CC.4.c  Understand that each successive number name refers to a quantity that is one larger. 
K.CC.5  Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects. 
Know Number Names And The Count Sequence.  
K.CC.1  Count to 100 by ones and by tens. 
K.CC.2  Count forward beginning from a given number within the known sequence (instead of having to begin at 1). 
K.CC.3  Write numbers from 0 to 20. Represent a number of objects with a written numeral 020 (with 0 representing a count of no objects). 
Creating Equations  
HSA.CED: Create Equations That Describe Numbers Or Relationships  
HSA.CED.1  Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 
HSA.CED.2  Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 
Expressions And Equations  
6.EE: Apply And Extend Previous Understandings Of Arithmetic To Algebraic Expressions.  
6.EE.1  Write and evaluate numerical expressions involving wholenumber exponents. 
6.EE.2  Write, read, and evaluate expressions in which letters stand for numbers. 
6.EE.2.a  Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. 
6.EE.2.b  Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. 
6.EE.2.c  Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in realworld problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area of a cube with sides of length s = 1/2. 
6.EE.3  Apply the properties of operations to generate equivalent expressions. 
6.EE.4  Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. 
Reason About And Solve OneVariable Equations And Inequalities.  
6.EE.5  Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. 
6.EE.6  Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. 
6.EE.7  Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. 
6.EE.8  Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. 
Represent And Analyze Quantitative Relationships Between Dependent And Independent Variables.  
6.EE.9  Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. 
7.EE: Solve RealLife And Mathematical Problems Using Numerical And Algebraic Expressions And Equations.  
7.EE.3  Solve multistep reallife and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. 
7.EE.4  Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. 
7.EE.4.a  Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? 
7.EE.4.b  Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. 
Use Properties Of Operations To Generate Equivalent Expressions.  
7.EE.1  Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 
7.EE.2  Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” 
8.EE: Analyze And Solve Linear Equations And Pairs Of Simultaneous Linear Equations.  
8.EE.7  Solve linear equations in one variable. 
8.EE.7.a  Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). 
8.EE.7.b  Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. 
8.EE.8  Analyze and solve pairs of simultaneous linear equations. 
8.EE.8.a  Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. 
8.EE.8.b  Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6. 
8.EE.8.c  Solve realworld and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair. 
Understand The Connections Between Proportional Relationships, Lines, And Linear Equations.  
8.EE.5  Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distancetime graph to a distancetime equation to determine which of two moving objects has greater speed. 
8.EE.6  Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. 
Work With Radicals And Integer Exponents.  
8.EE.1  Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27. 
8.EE.2  Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. 
8.EE.3  Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger. 
8.EE.4  Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. 
Functions  
8.F: Define, Evaluate, And Compare Functions.  
8.F.1  Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 
8.F.2  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change. 
8.F.3  Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line. 
Use Functions To Model Relationships Between Quantities.  
8.F.4  Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 
Linear, Quadratic, And Exponential Models  
HSF.LE: Construct And Compare Linear, Quadratic, And Exponential Models And Solve Problems  
HSF.LE.2  Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). 
HSF.LE.3  Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 
HSF.LE.4  For exponential models, express as a logarithm the solution to abct =dwherea,c,anddarenumbersandthebasebis2,10,ore; evaluate the logarithm using technology. 
Interpret Expressions For Functions In Terms Of The Situation They Model  
HSF.LE.5  Interpret the parameters in a linear or exponential function in terms of a context. 
Measurement And Data  
1.MD: Measure Lengths Indirectly And By Iterating Length Units.  
1.MD.1  Order three objects by length; compare the lengths of two objects indirectly by using a third object. 
1.MD.2  Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of samesize length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. 
Represent And Interpret Data.  
1.MD.4  Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. 
Tell And Write Time.  
1.MD.3  Tell and write time in hours and halfhours using analog and digital clocks. 
2.MD: Measure And Estimate Lengths In Standard Units.  
2.MD.3  Estimate lengths using units of inches, feet, centimeters, and meters. 
2.MD.4  Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit. 
Relate Addition And Subtraction To Length.  
2.MD.5  Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. 
2.MD.6  Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent wholenumber sums and differences within 100 on a number line diagram. 
Represent And Interpret Data.  
2.MD.9  Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in wholenumber units. 
Work With Time And Money.  
2.MD.7  Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. 
2.MD.8  Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have? 
3.MD: Geometric Measurement: Recognize Perimeter As An Attribute Of Plane Figures And Distinguish Between Linear And Area Measures.  
3.MD.8  Solve real world and mathematical problems involving perimeters of polygons, including finding the perimeter given the side lengths, finding an unknown side length, and exhibiting rectangles with the same perimeter and different areas or with the same area and different perimeters. 
Geometric Measurement: Understand Concepts Of Area And Relate Area To Multiplication And To Addition.  
3.MD.5  Recognize area as an attribute of plane figures and understand concepts of area measurement. 
3.MD.7  Relate area to the operations of multiplication and addition. 
Represent And Interpret Data.  
3.MD.3  Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one and twostep “how many more” and “how many less” problems using information presented in scaled bar graphs. For example, draw a bar graph in which each square in the bar graph might represent 5 pets. 
3.MD.4  Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters. 
Solve Problems Involving Measurement And Estimation Of Intervals Of Time, Liquid Volumes, And Masses Of Objects.  
3.MD.1  Tell and write time to the nearest minute and measure time intervals in minutes. Solve word problems involving addition and subtraction of time intervals in minutes, e.g., by representing the problem on a number line diagram. 
3.MD.2  Measure and estimate liquid volumes and masses of objects using standard units of grams (g), kilograms (kg), and liters (l).6 Add, subtract, multiply, or divide to solve onestep word problems involving masses or volumes that are given in the same units, e.g., by using drawings (such as a beaker with a measurement scale) to represent the problem.7 
4.MD: Geometric Measurement: Understand Concepts Of Angle And Measure Angles.  
4.MD.5  Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement: 
4.MD.5.a  An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “onedegree angle,” and can be used to measure angles. 
4.MD.5.b  An angle that turns through n onedegree angles is said to have an angle measure of n degrees. 
4.MD.6  Measure angles in wholenumber degrees using a protractor. Sketch angles of specified measure. 
4.MD.7  Recognize angle measure as additive. When an angle is decomposed into nonoverlapping parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve addition and subtraction problems to find unknown angles on a diagram in real world and mathematical problems, e.g., by using an equation with a symbol for the unknown angle measure. 
Represent And Interpret Data.  
4.MD.4  Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection. 
Solve Problems Involving Measurement And Conversion Of Measurements From A Larger Unit To A Smaller Unit.  
4.MD.1  Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ... 
4.MD.2  Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 
4.MD.3  Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. 
5.MD: Convert Like Measurement Units Within A Given Measurement System.  
5.MD.1  Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real world problems. 
Geometric Measurement: Understand Concepts Of Volume And Relate Volume To Multiplication And To Addition.  
5.MD.3  Recognize volume as an attribute of solid figures and understand concepts of volume measurement. 
5.MD.3.a  A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of volume, and can be used to measure volume. 
5.MD.3.b  A solid figure which can be packed without gaps or overlaps using n unit cubes is said to have a volume of n cubic units. 
5.MD.4  Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. 
5.MD.5  Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. 
5.MD.5.a  Find the volume of a right rectangular prism with wholenumber side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold wholenumber products as volumes, e.g., to represent the associative property of multiplication. 
5.MD.5.b  Apply the formulas V=l×w×handV=b×h for rectangular prisms to find volumes of right rectangular prisms with whole number edge lengths in the context of solving real world and mathematical problems. 
Represent And Interpret Data.  
5.MD.2  Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. 
K.MD  
Classify Objects And Count The Number Of Objects In Each Category.: Describe And Compare Measurable Attributes.  
K.MD.1  Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object. 
K.MD.2  Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of” the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter. 
Number And Operations In Base Ten  
1.NBT: Extend The Counting Sequence.  
1.NBT.1  Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. 
Understand Place Value.  
1.NBT.2  Understand that the two digits of a twodigit number represent amounts of tens and ones. Understand the following as special cases: 
1.NBT.3  Compare two twodigit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. 
Use Place Value Understanding And Properties Of Operations To Add And Subtract.  
1.NBT.4  Add within 100, including adding a twodigit number and a onedigit number, and adding a twodigit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding twodigit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 
1.NBT.6  Subtract multiples of 10 in the range 1090 from multiples of 10 in the range 1090 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 
2.NBT: Understand Place Value.  
2.NBT.1  Understand that the three digits of a threedigit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases: 
2.NBT.2  Count within 1000; skipcount by 5s, 10s, and 100s. 
2.NBT.3  Read and write numbers to 1000 using baseten numerals, number names, and expanded form. 
2.NBT.4  Compare two threedigit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons. 
Use Place Value Understanding And Properties Of Operations To Add And Subtract.  
2.NBT.5  Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 
2.NBT.6  Add up to four twodigit numbers using strategies based on place value and properties of operations. 
2.NBT.7  Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 
3.NBT: Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic.4  
3.NBT.1  Use place value understanding to round whole numbers to the nearest 10 or 100. 
3.NBT.2  Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction. 
3.NBT.3  Multiply onedigit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations. 
5.NBT: Perform Operations With MultiDigit Whole Numbers And With Decimals To Hundredths.  
5.NBT.5  Fluently multiply multidigit whole numbers using the standard algorithm. 
5.NBT.6  Find wholenumber quotients of whole numbers with up to fourdigit dividends and twodigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 
5.NBT.7  Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 
Understand The Place Value System.  
5.NBT.3  Read, write, and compare decimals to thousandths. 
5.NBT.4  Use place value understanding to round decimals to any place. 
K.NBT Work With Numbers 11–19 To Gain Foundations For Place Value.  
4.NBT: Generalize Place Value Understanding For MultiDigit Whole Numbers.  
4.NBT.1  Recognize that in a multidigit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division. 
4.NBT.2  Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 
4.NBT.3  Use place value understanding to round multidigit whole numbers to any place. 
Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic.  
4.NBT.4  Fluently add and subtract multidigit whole numbers using the standard algorithm. 
4.NBT.5  Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 
4.NBT.6  Find wholenumber quotients and remainders with up to fourdigit dividends and onedigit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 
Number And Operations—Fractions  
5.NF: Apply And Extend Previous Understandings Of Multiplication And Division To Multiply And Divide Fractions.  
5.NF.3  Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie? 
5.NF.4  Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 
5.NF.4.b  Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 
5.NF.6  Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 
5.NF.7  Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1 
5.NF.7.a  Interpret division of a unit fraction by a nonzero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. 
5.NF.7.b  Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4. 
5.NF.7.c  Solve real world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3cup servings are in 2 cups of raisins? 
Use Equivalent Fractions As A Strategy To Add And Subtract Fractions.  
5.NF.1  Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 
5.NF.2  Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions 
4.NF: Build Fractions From Unit Fractions By Applying And Extending Previous Understandings Of Operations On Whole Numbers.  
4.NF.3  Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. 
4.NF.3.a  Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. 
4.NF.3.b  Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. 
4.NF.3.c  Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. 
4.NF.3.d  Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 
4.NF.4  Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 
4.NF.4.a  Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4). 
4.NF.4.b  Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.) 
4.NF.4.c  Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie? 
Extend Understanding Of Fraction Equivalence And Ordering.  
4.NF.1  Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. 
4.NF.2  Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 
Understand Decimal Notation For Fractions, And Compare Decimal Fractions.  
4.NF.5  Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100.4 For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. 
4.NF.6  Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. 
4.NF.7  Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. 
3.NF: Develop Understanding Of Fractions As Numbers.  
3.NF.1  Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. 
3.NF.2  Understand a fraction as a number on the number line; represent fractions on a number line diagram. 
3.NF.2.a  Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. 
3.NF.2.b  Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. 
3.NF.3  Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 
3.NF.3.a  Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. 
3.NF.3.b  Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. 
3.NF.3.c  Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. 
3.NF.3.d  Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 
Operations And Algebraic Thinking  
1.OA: Add And Subtract Within 20.  
1.OA.5  Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 
1.OA.6  Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). 
Represent And Solve Problems Involving Addition And Subtraction.  
1.OA.1  Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.2 
1.OA.2  Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 
Understand And Apply Properties Of Operations And The Relationship Between Addition And Subtraction.  
1.OA.3  Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) 
1.OA.4  Understand subtraction as an unknownaddend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. 
Work With Addition And Subtraction Equations.  
1.OA.7  Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 
1.OA.8  Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = � – 3, 6 + 6 = �. 
2.OA  
Add And Subtract Within 20.: Represent And Solve Problems Involving Addition And Subtraction.  
2.OA.1  Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 
Work With Equal Groups Of Objects To Gain Foundations For Multiplication.  
2.OA.4  Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 
3.OA: Multiply And Divide Within 100.  
3.OA.7  Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. 
Represent And Solve Problems Involving Multiplication And Division.  
3.OA.1  Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 
3.OA.3  Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 
3.OA.4  Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?. 
Solve Problems Involving The Four Operations, And Identify And Explain Patterns In Arithmetic.  
3.OA.8  Solve twostep word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3 
3.OA.9  Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. 
Understand Properties Of Multiplication And The Relationship Between Multiplication And Division.  
3.OA.5  Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 
3.OA.6  Understand division as an unknownfactor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. 
4.OA: Gain Familiarity With Factors And Multiples.  
4.OA.4  Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1–100 is prime or composite. 
Generate And Analyze Patterns.  
4.OA.5  Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. 
Use The Four Operations With Whole Numbers To Solve Problems.  
4.OA.2  Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1 
4.OA.3  Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 
5.OA: Analyze Patterns And Relationships.  
5.OA.3  Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. 
Write And Interpret Numerical Expressions.  
5.OA.1  Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 
5.OA.2  Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. 
K.OA: Understand Addition As Putting Together And Adding To, And Under Stand Subtraction As Taking Apart And Taking From.  
K.OA.1  Represent addition and subtraction with objects, fingers, mental images, drawings2, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. 
K.OA.2  Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. 
K.OA.3  Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). 
K.OA.5  Fluently add and subtract within 5. 
Quantities  
HSN.Q: Reason Quantitatively And Use Units To Solve Problems.  
HSN.Q .1  Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 
HSN.Q .2  Define appropriate quantities for the purpose of descriptive modeling. 
HSN.Q .3  Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 
Ratios And Proportional Relationships  
6.RP: Understand Ratio Concepts And Use Ratio Reasoning To Solve Problems.  
6.RP.1  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 
6.RP.2  Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”1 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 
6.RP.3.a  Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. 
6.RP.3.b  Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed? 
6.RP.3.c  Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent. 
6.RP.3.d  Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. 
7.RP: Analyze Proportional Relationships And Use Them To Solve RealWorld And Mathematical Problems.  
7.RP.1  Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. 
7.RP.2  Recognize and represent proportional relationships between quantities. 
7.RP.2.a  Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. 
7.RP.2.b  Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. 
7.RP.2.c  Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. 
7.RP.2.d  Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 
Reasoning With Equations And Inequalities  
HSA.REI: Represent And Solve Equations And Inequalities Graphically  
HSA.REI.10  Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 
HSA.REI.12  Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. 
Solve Equations And Inequalities In One Variable  
HSA.REI.3  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 
HSA.REI.4  Solve quadratic equations in one variable. 
HSA.REI.4.a  Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. 
HSA.REI.4.b  Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. 
Solve Systems Of Equations  
HSA.REI.6  Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 
HSA.REI.7  Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3. 
HSA.REI.8  (+) Represent a system of linear equations as a single matrix equation in a vector variable. 
Understand Solving Equations As A Process Of Reasoning And Explain The Reasoning  
HSA.REI.2  Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 
Seeing Structure In Expressions  
HSA.SSE: Interpret The Structure Of Expressions  
HSA.SSE.1  Interpret expressions that represent a quantity in terms of its context. 
HSA.SSE.1.a  Interpret parts of an expression, such as terms, factors, and coefficients. 
HSA.SSE.1.b  Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. 
HSA.SSE.2  Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). 
Write Expressions In Equivalent Forms To Solve Problems  
HSA.SSE.3  Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★ 
HSA.SSE.3.a  Factor a quadratic expression to reveal the zeros of the function it defines. 
HSA.SSE.3.b  Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 
HSA.SSE.3.c  Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 
HSA.SSE.4  Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★ 
Similarity, Right Triangles, And Trigonometry  
HSG.SRT: Apply Trigonometry To General Triangles  
HSG.SRT.11  (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and nonright triangles (e.g., surveying problems, resultant forces). 
HSG.SRT.9  (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. 
HSG.SRT.7  Explain and use the relationship between the sine and cosine of complementary angles. 
HSG.SRT.8  Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ 
Statistics And Probability: 6.SP  
6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. 
Summarize And Describe Distributions.  
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 
6.SP.5.c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 
6.SP.5.d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 
7.SP: Draw Informal Comparative Inferences About Two Populations.  
7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventhgrade science book are generally longer than the words in a chapter of a fourthgrade science book. 
Investigate Chance Processes And Develop, Use, And Evaluate Probability Models.  
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 
7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 
7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 
7.SP.8.a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 
7.SP.8.b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. 
7.SP.8.c  Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? 
Use Random Sampling To Draw Inferences About A Population.  
7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. 
8.SP: Investigate Patterns Of Association In Bivariate Data.  
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 
8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 
8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. 
8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? 
The Number System  
6.NS: Apply And Extend Previous Understandings Of Multiplication And Division To Divide Fractions By Fractions.  
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? 
Apply And Extend Previous Understandings Of Numbers To The System Of Rational Numbers.  
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation. 
6.NS.6  Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 
6.NS.6.a  Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite. 
6.NS.6.b  Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 
6.NS.6.c  Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. 
6.NS.7  Understand ordering and absolute value of rational numbers. 
6.NS.7.a  Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. 
6.NS.7.b  Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is warmer than –7 oC. 
6.NS.7.c  Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of –30 dollars, write –30 = 30 to describe the size of the debt in dollars. 
6.NS.7.d  Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars. 
6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 
Compute Fluently With MultiDigit Numbers And Find Common Factors And Multiples.  
6.NS.2  Fluently divide multidigit numbers using the standard algorithm. 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. 
6.NS.4  Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). 
7.NS: Apply And Extend Previous Understandings Of Operations With Fractions To Add, Subtract, Multiply, And Divide Rational Numbers.  
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 
7.NS.1.a  Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. 
7.NS.1.b  Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts. 
7.NS.1.c  Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in realworld contexts. 
7.NS.1.d  Apply properties of operations as strategies to add and subtract rational numbers. 
7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 
7.NS.2.a  Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing realworld contexts. 
7.NS.2.b  Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real world contexts. 
7.NS.2.c  Apply properties of operations as strategies to multiply and divide rational numbers. 
7.NS.2.d  Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 
7.NS.3  Solve realworld and mathematical problems involving the four operations with rational numbers. 
8.NS: Know That There Are Numbers That Are Not Rational, And Approximate Them By Rational Numbers.  
8.NS.1  Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 
8.NS.2  Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 
The Real Number System  
HSN.RN: Extend The Properties Of Exponents To Rational Exponents.  
HSN.RN.2  Rewrite expressions involving radicals and rational exponents using the properties of exponents. 
Use Properties Of Rational And Irrational Numbers.  
HSN.RN.3  Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. 
Using Probability To Make Decisions: HSS.MD  
HSS.MD.4  (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? 
Calculate Expected Values And Use Them To Solve Problems  
HSS.MD.1  (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. 
HSS.MD.2  (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. 
Use Probability To Evaluate Outcomes Of Decisions: Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.  
HSS.MD.5  (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. 
HSS.MD.5.a  Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast food restaurant. 
HSS.MD.6  (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 
Vector And Matrix Quantities: Perform Operations On Matrices And Use Matrices In Applications.  
HSN.VM.7  (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 
HSN.VM.8  (+) Add, subtract, and multiply matrices of appropriate dimensions. 
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