# JogNog

- using supporting evidence
- vocabulary
- reading comprehension
- text analysis

- biology
- chemistry
- engineering
- physics

- self-reflection
- humility

- applying information
- thinking critically

###### Pros

Quizzes are available on a dizzying array of subjects; tracking student progress is easy.###### Cons

Full access can be costly, and quizzes focus strongly on speed and external praise.###### Bottom Line

Extensive library of quizzes makes this a valuable resource for teachers -- but comes at a price.With their special login, teachers can share quizzes with students and track student progress. Teacher pricing is a little better: teachers can get unlimited access to quizzes for $29.99 for three months or $99.99 for a full year. Teachers can use JogNog's website to create, publish, and share quizzes with their students; these features are not available through the app.

Well-designed progress tracking, rewards, and sharing features make it easy to engage kids, but the focus on external, superficial rewards might be off-putting to some users.

Some quizzes have deeper questions than others, and correct answers disappear before users realize what they've chosen. It's not clear that this drill-and-kill approach will work well for all learners.

Thoughtful in-app FAQs and an extensive YouTube channel offer seemingly unlimited tips for making the most of the resources -- but these tools are aimed more toward teachers than students.

Teachers can use JogNog as a go-to assessment tool for CCSS, MCAS, and SAT review, although that could become personally expensive. Teachers might encourage kids to use the free sections of each quiz to extend and solidify their understanding of in-class concepts. Teachers can explore the JogNog YouTube channel for further ideas on integrating JogNog into the curriculum.

Read More Read LessJogNog is an extensive library of multiple-choice quizzes. Kids complete quizzes to earn Tower Badges and credits toward unlocking other quizzes and upgrading their badges from black to gray to colored. Each quiz offers multiple levels: For a quiz on a poem, for example, levels might be Vocabulary, Comprehension, and Interpretation. Scoring above 70% on one level unlocks the next level. Upon completing a quiz and earning a Tower, kids can select “Praise Me!” to send their results to a parent or teacher and ask for a reward, such as bonus points at school or fewer chores at home.

Quizzes are available for all math and ELA standards for the Massachusetts Curriculum Frameworks for Science and Social Science and for a wide range of high school subjects, including classroom subjects, SAT vocabulary, and driver’s license tests.

Teachers and students can access the first section of each quiz for free; additional quizzes come at a cost either to the teacher (who can purchase them to share with an unlimited number of students) or to the students themselves. Students can purchase additional credits at $0.99 for 10 credits, $4.99 for 60, $9.99 for 130, $19.99 for 270, and $99.99 for 1450 credits; it costs three credits to complete each of the quizzes beyond the intro-level quiz in each subject. While it's possible to get great value from this app for free, purchasing further credits is the best way to fully access its rich resources.

Read More Read LessJogNog features a spectacular amount of content. In addition to taking quizzes aligned to state and national academic standards, kids can access review quizzes for driver's license tests and SAT vocabulary. One thing to consider is that the quizzes focus heavily on speed and rewards. A timer runs while users complete quizzes, and pop-up messages urge users to perform better and faster on the next try. It's also a little frustrating that answers linger; there's little time to see which choice was correct.

In the right context, this could work well as a way for teachers to help students cement their understanding of a wide range of content. Context is key, however; these quizzes are a good supplement to instruction and engagement from a skilled teacher, not a replacement for such expertise.

Read More Read Less## Key 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.2.3a | Compare formal and informal uses of English. |

Vocabulary Acquisition and Use | |

L.2.4 | Determine or clarify the meaning of unknown and multiple-meaning words and phrases based on grade 2 reading and content, choosing flexibly from an array of strategies. |

L.2.4a | Use sentence-level 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 real-life 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 subject-verb and pronoun-antecedent 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 high-frequency 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, position-based 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 multiple-meaning word and phrases based on grade 3 reading and content, choosing flexibly from a range of strategies. |

L.3.4a | Use sentence-level 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 real-life 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 grade-appropriate conversational, general academic, and domain-specific 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 run-ons.* |

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 grade-appropriate 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., small-group discussion). |

L.5: Conventions of Standard English | |

L.5.1 | |

L.5.2 | |

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 multiple-meaning 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, grade-appropriate 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 grade-appropriate general academic and domain-specific 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 | |

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 | |

L.6.2.A | Use punctuation (commas, parentheses, dashes) to set off nonrestrictive/parenthetical elements.* |

L.6.2.B | Spell correctly. |

Vocabulary Acquisition and Use | |

L.6.4 | Determine or clarify the meaning of unknown and multiple-meaning 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, grade-appropriate 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 grade-appropriate general academic and domain-specific 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 | |

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 compound-complex 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 | |

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 multiple-meaning 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, grade-appropriate 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 grade-appropriate general academic and domain-specific 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 | |

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 | |

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 multiple-meaning 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, grade-appropriate 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 grade-appropriate general academic and domain-specific words and phrases; gather vocabulary knowledge when considering a word or phrase important to comprehension or expression. |

## Reading Foundational Skills | |

RF.4: Phonics and Word recognition | |

RF.4.3 | Know and apply grade-level phonics and word analysis skills in decoding words. |

RF.4.3a | Use combined knowledge of all letter-sound correspondences, syllabication patterns, and morphology (e.g., roots and affixes) to read accurately unfamiliar multisyllabic words in context and out of context. |

RF.5: Phonics and Word recognition | |

RF.5.3 | Know and apply grade-level phonics and word analysis skills in decoding words. |

RF.5.3a | Use combined knowledge of all letter-sound correspondences, syllabication patterns, and morphology (e.g., roots and affixes) to read accurately unfamiliar multisyllabic words in context and out of context. |

## Reading Informational | |

RI.3: Key Ideas and Details | |

RI.3.1 | Ask and answer questions to demonstrate understanding of a text, referring explicitly to the text as the basis for the answers. |

RI.3.2 | Determine the main idea of a text; recount the key details and explain how they support the main idea. |

RI.3.3 | Describe the relationship between a series of historical events, scientific ideas or concepts, or steps in technical procedures in a text, using language that pertains to time, sequence, and cause/effect. |

RI.4: Key Ideas and Details | |

RI.4.1 | Refer to details and examples in a text when explaining what the text says explicitly and when drawing inferences from the text. |

RI.4.2 | Determine the main idea of a text and explain how it is supported by key details; summarize the text. |

RI.4.3 | Explain events, procedures, ideas, or concepts in a historical, scientific, or technical text, including what happened and why, based on specific information in the text. |

RI.6: Craft and Structure | |

RI.6.4 | Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings. |

Range of Reading and Level of Text Complexity | |

RI.6.10 | By the end of the year, read and comprehend literary nonfiction in the grades 6–8 text complexity band proficiently, with scaffolding as needed at the high end of the range. |

RI.7: Craft and Structure | |

RI.7.4 | Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze the impact of a specific word choice on meaning and tone. |

Range of Reading and Level of Text Complexity | |

RI.7.10 | By the end of the year, read and comprehend literary nonfiction in the grades 6–8 text complexity band proficiently, with scaffolding as needed at the high end of the range. |

RI.8: Craft and Structure | |

RI.8.4 | Determine the meaning of words and phrases as they are used in a text, including figurative, connotative, and technical meanings; analyze the impact of specific word choices on meaning and tone, including analogies or allusions to other texts. |

Range of Reading and Level of Text Complexity | |

RI.8.10 | By the end of the year, read and comprehend literary nonfiction at the high end of the grades 6–8 text complexity band independently and proficiently. |

RI.9-10: Range of Reading and Level of Text Complexity | |

RI.9-10.10 | By the end of grade 9, read and comprehend literary nonfiction in the grades 9–10 text complexity band proficiently, with scaffolding as needed at the high end of the range. |

RI.11-12: Range of Reading and Level of Text Complexity | |

RI.11-12.10 | By the end of grade 11, read and comprehend literary nonfiction in the grades 11–CCR text complexity band proficiently, with scaffolding as needed at the high end of the range. |

## Reading Literature | |

RL.3: Key Ideas and Details | |

RL.3.1 | Ask and answer questions to demonstrate understanding of a text, referring explicitly to the text as the basis for the answers. |

RL.3.2 | Recount stories, including fables, folktales, and myths from diverse cultures; determine the central message, lesson, or moral and explain how it is conveyed through key details in the text. |

RL.3.3 | Describe characters in a story (e.g., their traits, motivations, or feelings) and explain how their actions contribute to the sequence of events. |

RL.4: Craft and Structure | |

RL.4.4 | Determine the meaning of words and phrases as they are used in a text, including those that allude to significant characters found in mythology (e.g., Herculean). |

RL.4.5 | Explain major differences between poems, drama, and prose, and refer to the structural elements of poems (e.g., verse, rhythm, meter) and drama (e.g., casts of characters, settings, descriptions, dialogue, stage directions) when writing or speaking about a text. |

RL.4.6 | Compare and contrast the point of view from which different stories are narrated, including the difference between first- and third-person narrations. |

Integration of Knowledge and Ideas | |

RL.4.7 | Make connections between the text of a story or drama and a visual or oral presentation of the text, identifying where each version reflects specific descriptions and directions in the text. |

RL.4.8 | (Not applicable to literature) |

RL.4.9 | Compare and contrast the treatment of similar themes and topics (e.g., opposition of good and evil) and patterns of events (e.g., the quest) in stories, myths, and traditional literature from different cultures. |

RL.6: Craft and Structure | |

RL.6.4 | Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of a specific word choice on meaning and tone. |

RL.7: Craft and Structure | |

RL.7.4 | Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of rhymes and other repetitions of sounds (e.g., alliteration) on a specific verse or stanza of a poem or section of a story or drama. |

RL.7.5 | Analyze how a drama’s or poem’s form or structure (e.g., soliloquy, sonnet) contributes to its meaning. |

RL.8: Craft and Structure | |

RL.8.4 | Determine the meaning of words and phrases as they are used in a text, including figurative and connotative meanings; analyze the impact of specific word choices on meaning and tone, including analogies or allusions to other texts. |

## Speaking & Listening | |

SL.6: Presentation of Knowledge and Ideas | |

SL.6.4 | Present claims and findings, sequencing ideas logically and using pertinent descriptions, facts, and details to accentuate main ideas or themes; use appropriate eye contact, adequate volume, and clear pronunciation. |

## Writing | |

W.6: Production and Distribution of Writing | |

W.6.4 | Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. (Grade-specific expectations for writing types are defined in standards 1–3 above.) |

W.7: Production and Distribution of Writing | |

W.7.4 | Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. (Grade-specific expectations for writing types are defined in standards 1–3 above.) |

W.8: Production and Distribution of Writing | |

W.8.4 | Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. (Grade-specific expectations for writing types are defined in standards 1–3 above.) |

## Expressions And Equations | |

6.EE: Apply And Extend Previous Understandings Of Arithmetic To Algebraic Expressions. | |

6.EE.1 | Write and evaluate numerical expressions involving whole-number 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 real-world 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 One-Variable 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 real-world 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 real-world 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 real-world 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 real-world 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 Real-Life And Mathematical Problems Using Numerical And Algebraic Expressions And Equations. | |

7.EE.3 | Solve multi-step real-life 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 real-world 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 real-world 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 distance-time graph to a distance-time 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 non-vertical 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. |

8.F.5 | Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. |

## Geometry | |

2.G: Reason With Shapes And Their Attributes. | |

2.G.1 | Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. |

2.G.2 | Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. |

2.G.3 | Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape. |

3.G: Reason With Shapes And Their Attributes. | |

3.G.1 | Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories. |

3.G.2 | Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape. |

4.G: Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles. | |

4.G.1 | Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures. |

4.G.2 | Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. |

4.G.3 | Recognize a line of symmetry for a two-dimensional figure as a line across the figure such that the figure can be folded along the line into matching parts. Identify line-symmetric figures and draw lines of symmetry. |

5.G: Classify Two-Dimensional Figures Into Categories Based On Their Properties. | |

5.G.3 | Understand that attributes belonging to a category of two- dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. |

5.G.4 | Classify two-dimensional figures in a hierarchy based on properties. |

Graph Points On The Coordinate Plane To Solve Real-World And Mathematical Problems. | |

5.G.1 | Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., x-axis and x-coordinate, y-axis and y-coordinate). |

5.G.2 | Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. |

6.G: Solve Real-World And Mathematical Problems Involving Area, Surface Area, And Volume. | |

6.G.1 | Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. |

6.G.2 | Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. |

6.G.3 | Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. |

6.G.4 | Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. |

7.G: Draw, Construct, And Describe Geometrical Figures And Describe The Relationships Between Them. | |

7.G.1 | Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. |

7.G.2 | Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. |

7.G.3 | Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. |

Solve Real-Life And Mathematical Problems Involving Angle Measure, Area, Surface Area, And Volume. | |

7.G.4 | Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. |

7.G.5 | Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. |

7.G.6 | Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. |

8.G: Solve Real-World And Mathematical Problems Involving Volume Of Cylinders, Cones, And Spheres. | |

8.G.9 | Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. |

Understand And Apply The Pythagorean Theorem. | |

8.G.6 | Explain a proof of the Pythagorean Theorem and its converse. |

8.G.7 | Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. |

8.G.8 | Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. |

Understand Congruence And Similarity Using Physical Models, Trans- Parencies, Or Geometry Software. | |

8.G.1 | Verify experimentally the properties of rotations, reflections, and translations: |

8.G.1.a | Lines are taken to lines, and line segments to line segments of the same length. |

8.G.1.b | Angles are taken to angles of the same measure. |

8.G.1.c | Parallel lines are taken to parallel lines. |

8.G.2 | Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. |

8.G.3 | Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. |

8.G.4 | Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two- dimensional figures, describe a sequence that exhibits the similarity between them. |

8.G.5 | Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. |

## Measurement And Data | |

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.5.a | A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area, and can be used to measure area. |

3.MD.5.b | A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units. |

3.MD.6 | Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units). |

3.MD.7 | Relate area to the operations of multiplication and addition. |

3.MD.7.a | Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. |

3.MD.7.b | Multiply side lengths to find areas of rectangles with whole- number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. |

3.MD.7.c | Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. |

3.MD.7.d | Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems. |

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 two-step “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 one-step 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 “one-degree angle,” and can be used to measure angles. |

4.MD.5.b | An angle that turns through n one-degree angles is said to have an angle measure of n degrees. |

4.MD.6 | Measure angles in whole-number degrees using a protractor. Sketch angles of specified measure. |

4.MD.7 | Recognize angle measure as additive. When an angle is decomposed into non-overlapping 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 different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, 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 whole-number 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 whole-number 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. |

5.MD.5.c | Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world 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. |

## Number And Operations In Base Ten | |

2.NBT: Understand Place Value. | |

2.NBT.1 | Understand that the three digits of a three-digit 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.1.a | 100 can be thought of as a bundle of ten tens — called a “hundred.” |

2.NBT.1.b | The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones). |

2.NBT.2 | Count within 1000; skip-count by 5s, 10s, and 100s. |

2.NBT.3 | Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. |

2.NBT.4 | Compare two three-digit 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 two-digit 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. |

2.NBT.8 | Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. |

2.NBT.9 | Explain why addition and subtraction strategies work, using place value and the properties of operations.3 |

3.NBT: Use Place Value Understanding And Properties Of Operations To Perform Multi-Digit 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 one-digit 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 Multi-Digit Whole Numbers And With Decimals To Hundredths. | |

5.NBT.5 | Fluently multiply multi-digit whole numbers using the standard algorithm. |

5.NBT.6 | Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit 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.1 | Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. |

5.NBT.2 | Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. |

5.NBT.3 | Read, write, and compare decimals to thousandths. |

5.NBT.3.a | Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1000). |

5.NBT.3.b | Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. |

5.NBT.4 | Use place value understanding to round decimals to any place. |

4.NBT: Generalize Place Value Understanding For Multi-Digit Whole Numbers. | |

4.NBT.1 | Recognize that in a multi-digit 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 multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit 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 multi-digit whole numbers to any place. |

Use Place Value Understanding And Properties Of Operations To Perform Multi-Digit Arithmetic. | |

4.NBT.4 | Fluently add and subtract multi-digit whole numbers using the standard algorithm. |

4.NBT.5 | Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit 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 whole-number quotients and remainders with up to four-digit dividends and one-digit 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 50-pound 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.a | Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) |

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.5 | Interpret multiplication as scaling (resizing), by: |

5.NF.5.a | Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication. |

5.NF.5.b | Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1. |

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 non-zero 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 non-zero 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 1/3-cup 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 | |

2.OA: Add And Subtract Within 20. | |

2.OA.2 | Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. |

Represent And Solve Problems Involving Addition And Subtraction. | |

2.OA.1 | Use addition and subtraction within 100 to solve one- and two-step 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.3 | Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. |

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 one-digit 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.2 | Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. |

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 two-step 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 unknown-factor 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 one-digit 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.1 | Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. |

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 whole-number 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. |

## 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 real-world 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 Real-World 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. |

## Statistics And Probability | |

6.SP: Develop Understanding Of Statistical Variability. | |

6.SP.1 | Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. |

6.SP.2 | Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. |

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 | Summarize numerical data sets in relation to their context, such as by: |

6.SP.5.a | Reporting the number of observations. |

6.SP.5.b | Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. |

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.3 | Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. |

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 seventh-grade science book are generally longer than the words in a chapter of a fourth-grade 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 long-run 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.7 | Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. |

7.SP.7.a | Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. |

7.SP.7.b | Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? |

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.1 | Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. |

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 two-way table. Construct and interpret a two-way 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 3/4-cup 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 real-world 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 real-world 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 real-world 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 real-world 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 Multi-Digit Numbers And Find Common Factors And Multiples. | |

6.NS.2 | Fluently divide multi-digit numbers using the standard algorithm. |

6.NS.3 | Fluently add, subtract, multiply, and divide multi-digit 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 real-world 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 real-world 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 real-world contexts. |

7.NS.2.b | Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero 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 real-world 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. |