MATHEMATICS - DOE Elementary Math...MATHEMATICS Content Standards ELEMENTARY 1 Acknowledgements The American Samoa Department of Education (ASDOE) appreciates the effort of teachers,

American SamoaDepartment of Education

MATHEMATICSContent Standards

ELEMENTARY111

Acknowledgements

The American Samoa Department of Education (ASDOE) appreciates the effort of teachers, principals, leadership, community members and content specialists who gave their time to review and revise the standards for Samoan Studies, English language arts (ELA), math, science and social studies. We have benefitted from the hard work of many national experts who developed the common core state standards for English language arts and math (www.corestandards.org) and acknowledge the close reading we did to revise the standards for these content areas. We have also greatly profited from national expert, Dr. Margo Gottlieb, and her guidance as we created grade level English language proficiency (ELP) standards for ELA, math, science and social studies from the previously adopted TESOL ELP standards (www.tesol.org/s_tesol/seccss.asp?CID=95&DID=1565). We are especially grateful to the late Kalolo Iosefo who championed standards for Samoan Language Proficiency during the revision process. Collaboratively, we have worked towards a common goal—fewer topics and more in-depth learning across the five subject areas for the students of American Samoa with the language support they so rightly deserve. The American Samoa Department of Education Content & Performance Standards were developed in partnership with Pacific Resources for Education and Learning (PREL) and supported by the American Recovery and Reinvestment Act (ARRA).

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http://www.corestandards.org/

http://www.tesol.org/s_tesol/seccss.asp?CID=95&DID=1565

Table of Contents

Acknowledgements......................................................................................................................... 2

English Language Proficiency Introduction..................................................................................... 4

Mathematics Introduction................................................................................................................ 10

Kindergarten ................................................................................................................................... 18

Grade 1........................................................................................................................................... 23

Grade 2........................................................................................................................................... 29

Grade 3........................................................................................................................................... 35

Grade 4........................................................................................................................................... 41

Grade 5........................................................................................................................................... 48

Grade 6........................................................................................................................................... 55

Grade 7........................................................................................................................................... 62

Grade 8........................................................................................................................................... 69

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ENGLISH LANGUAGE PROFICIENCY STANDARDS�

AMERICAN SAMOA DEPARTMENT OF EDUCATION (ASDOE) STANDARDS FOR CONTENT AND LANGUAGE

The American Samoa Department of Education’s (ASDOE) newly revised content standards describe expectations for student learning in specific subject areas, K–12. Content standards for English language arts, Samoan studies, math, social studies, and science, and aligned standards for English language proficiency (ELP) for all but Samoan studies, provide a framework for professional reflection and planning that enable content-rich curricula and literacy for all. They offer teachers, schools, and the Department a guide to plan and assess for learning, as well as set goals within a school improvement process. The revised standards continue to build upon the expectations established in the original standards and incorporate current research and suggestions for improvement from many experienced practitioners, especially those who participated in the revision process. They offer a sharper focus than the original standards on knowledge (what learners need to understand); practices (what learners will be capable of doing); performance (how learners demonstrate understanding); and language (what social and academic language learners will be able to use). A focus on core content, intellectual thinking, and literacy give the revised standards a singular goal of preparing students for college, career, and citizenship. Standards for Samoan studies, English language arts, math, social studies and science The organizing structure for the content standards is hierarchical and includes strands, standard statements, benchmarks, and sample performance indicators. Strands are overarching organizers for standards. They are typically one or two words in length. Standards statements broadly describe the knowledge and skills we expect students to have at the end of grade 12. Benchmarks are learning targets that describe what students should know and be able to do (related to each standard) at the end of a grade level or at the end of a course. They describe advancement toward meeting standards in a coherent progression from kindergarten through grade 12. Sample performance indicators provide further detail of key aspects of a learning target and are examples of what students should be able to know and do for each grade related to the benchmark. The coding system Each benchmark is coded for easy identification. The first number represents the grade level, the second is the standard, and the third is a benchmark associated with the standard. For example, 7.2.1 means grade 7, standard 2, benchmark 1. Standards for English language proficiency Who Are English Language Learners?

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The term “English Language Learners” (ELLs) refers to students who are developing their proficiency in understanding and using the English language. ELLs come from diverse backgrounds. Some were born in foreign countries with varied experiences in learning English. Others were born in English-speaking countries, such as the U.S. and U.S. territories, to families where languages other than English are spoken at home and do not learn English until they go to school. In American Samoa, a majority of students are ELLs because of their developing proficiency in English, the language or medium of instruction in all content areas but Samoan studies. What Is English Language Proficiency? English language proficiency (ELP) is the extent to which students understand information and perform tasks in English. There are three factors that affect the language proficiency of ELLs: sociocultural, linguistic, and cognitive. Sociocultural factors such as attitude, motivation, investment, home cultures, family expectations, and literacy and proficiency in first languages can influence the process of teaching and learning a languageincluding. Linguistic factors describe the language features an ELL needs to know in order to succeed in school; example language features include phonology (system of sounds in a language), grammar, vocabulary, discourse (genres and organization of texts), and formal and informal registers (formal and informal language). An understanding of language features in an ELL’s first language can help support learning of these features in English, since knowledge and skills acquired in a first language can be transferred to a second language. Finally, cognitive factors refer to learning processes that increase an ELL’s knowledge of English. Examples of cognitive factors include learning strategies that promote higher-order thinking skills (e.g., analyzing, evaluating, synthesizing), metacognitive skills (thinking about thinking), and metalinguistic skills (thinking about language use). What Is Social and Academic Language? Language holds an important role at schools because it serves as a medium of learning: we use language for speaking, listening, reading, and writing during class. The language of school is both social and academic. Social language refers to the different types of language used to interact with people in everyday situations. It includes communication skills between speakers of various linguistic and cultural backgrounds (intercultural), and communication skills for classroom routines (instructional). Here are some examples of social language: language used to organize for recess, language used to get to know new classmates on the playground, and language used to ask for permission to use the restroom. Academic language represents the language used by students to learn content, to communicate their understanding, and to participate in classroom activities. There are three levels of academic language: word level, which includes general academic vocabulary not related to specific content area (e.g., compare and contrast), specialized academic vocabulary related to a content area (e.g., water cycle and reproduction in science), and technical vocabulary related to specific content topic (e.g., centromere for mitosis in biology class); sentence level, which includes language patterns and grammatical structures for each content area (e.g., signal words for different text structures in language arts); and extended text level, which includes organization of oral and written academic language, in general and content-specific (e.g., bibliography for writing across content areas; different types of graphs in math). The language of school has multiple components, including different registers (informal and formal language), discourse structures (vocabulary, grammar, and organization of oral and written language), and functions (e.g., to describe something, to provide an explanation, to make a prediction).

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For ELLs, ELP at school means not only the development of the four language domains of speaking, listening, reading, and writing in English in each content area, but also learning how to use English to communicate inside and outside of classroom, and using academic English for the purpose of content learning. What Are the American Samoa Department of Education ELP Standards? The ASDOE ELP Standards are grade-level language expectations set for ELLs in American Samoa for listening, speaking, reading, and writing. They draw upon the framework and standards developed by Teachers of English to Speakers of Other Languages (TESOL) in 2006. There are five ASDOE ELP Standards for each grade (kindergarten through grade 12) in each of the four core content areas. The standards address both the social language needed for social, intercultural, and classroom communication, and the academic language needed for ELLs to learn in the core content areas of language arts, mathematics, science, and social studies: ELP Standard 1: English language learners communicate for SOCIAL, INTERCULTURAL, and INSTRUCTIONAL PURPOSES within the school setting. ELP Standard 2: English language learners communicate information, ideas, and concepts necessary for academic success in the area of LANGUAGE ARTS. ELP Standard 3: English language learners communicate information, ideas, and concepts necessary for academic success in the area of MATHEMATICS. ELP Standard 4: English language learners communicate information, ideas, and concepts necessary for academic success in the area of SCIENCE. ELP Standard 5: English language learners communicate information, ideas, and concepts necessary for academic success in the area of SOCIAL STUDIES. Each ELP Standard is represented in the form of a matrix with the following components: sample content topics, language domains, stages of language proficiency, and sample performance indicators. Educators can use the ELP Standards as tools to support ELLs from different language proficiency levels to develop social and academic language skills. For example, if a teacher is responsible for teaching American Samoa history, s/he can draw on examples from ELP Standard 1 for American Samoa history to support ELLs in developing social language skills during class, and examples from ELP Standard 5 for American Samoa history to support ELLs in developing academic language skills related to the history of American Samoa. Five Stages of English Language Proficiency Each ELP Standard is divided into sample content topics for one of the 4 language domains—listening, speaking, reading, and writing. The topics originate from an analysis by TESOL of the national and state academic content standards in ESL (English as a Second Language), language arts, mathematics, science, and social studies. In addition, select ASDOE teachers reviewed the topics to ensure they are representative of the revised ASDOE content standards (2011) and topics commonly taught in the content areas. The ELP Standards address the social and academic language needs for ELLs across five stages of English language proficiency, as defined by TESOL (2006). The ASDOE made a decision to focus on three of the five stages (5, 3, and 1) in this initial

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phase. Using ELP standards will be new to many teachers. Most teachers will more easily be able to distinguish between stages 5, 3, and 1 initially, than stages 3 and 4. As the majority of teachers gain more expertise with ELP Standards, stages 4 and 2 will be incorporated. The stages are ordered by the highest proficiency first, stage 5. If the language evidence demonstrated by the student does not meet the descriptors of stage 5, then the teacher reviews the descriptors of the next stage, stage 3, and repeats this until the descriptors of a stage align with the language evidence demonstrated by the student. The three stages currently highlighted in ASDOE ELP Standards are described below: Stage 5: ELLs at this stage can use English fluently with minimal instructional support and in different contexts. They can work with modified grade-level materials, and can produce a variety of texts with different linguistic features incorporated (e.g., idiomatic expressions, colloquialisms) and minimal errors. However, they are not necessarily fully proficient across all language domains and standards. Stage 4: ELLs at this stage have acquired language skills adequate for most day-to-day communication needs. Occasional errors occur with sentence structures and word choices. They may have difficulty understanding and using some idioms, figures of speech, and words with multiple meanings. They can communicate in English in new or unfamiliar settings, but may have occasional difficulty with complex structures and abstract academic concepts. They can read with significant fluency and occasional comprehension questions, and can identify specific information from texts, but may have difficulty understanding decontextualized concepts and complex sentence structures. Stage 3: ELLs at this stage can understand some specialized academic vocabulary and more complex speech with occasional need for repetition. Their vocabulary knowledge includes words and phrases for many everyday situations. They can use English in different situations, with simple sentences that are understandable, but marked by grammatical errors. They may have difficulty expressing all their thoughts due to limited vocabulary and control over language structure. Their reading proficiency depends on their experiences with themes, concepts, or genres. They can make meaning from texts connected to their prior knowledge, and can write more complex and different types of texts, with significant amount of nonconventional features. Stage 2: ELLs at this stage can understand phrases and short sentences. They can use memorized phrases, groups of words, general academic vocabulary, and formulaic expressions to communicate in simple, everyday situations, with basic errors. Their reading and writing proficiency depends on their first language literacy skills and other sociocultural factors. They can read simple words and phrases, and find specific and predictable information in simple everyday print. There are often significant errors in their writing. Stage 1: ELLs initially have little to no understanding of English. They can respond nonverbally to simple commands, statements, questions, and body language. As their oral comprehension increases, they can begin understanding one-step questions, directions, and commands with visual support, and use single words, short phrases, and memorized chunks of language. They initially

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make meaning from texts using non-print features (e.g., illustrations, graphs, maps, tables), and can gradually make more, but often incomplete, meaning from printed words. They can produce simple texts that may include nonconventional features, such as invented spelling, grammatical inaccuracies, pictorial representations, and linguistic features transferred from first language (e.g., sentence structure). Sample Performance Indicators Sample performance indicators (SPIs) are examples of observable and measurable language behaviors that show what ELLs at stages 1, 3, and 5 of language proficiency can achieve for each sample content topic and its corresponding language domain. The SPIs generally consist of three elements: language function, content, and support or strategy. For example:

Standard 1: English language learners communicate for social, intercultural, and instructional purposes within the school setting

Domain Topic Stage 5 Stage 3 Stage 1

R

eadi

ng

Information gathering at school or at home Research Multiculturalism

Summarize key ideas about topics of interest from multiple diverse sources.

Identify main ideas and supporting details from illustrated information about topics of interest in small groups.

Locate key words on topics of personal interest using illustrated text (e.g., from Samoa artists, Internet, or newspapers) with a partner.

The above strand of SPIs demonstrates the language needs and use for ELLs in stages 5, 3, and 1 to be able to gather information (sample content topic) during reading (language domain) in the general classroom (ELP Standard 1). The three elements for each SPI have been highlighted with corresponding font features. Teachers can refer to SPIs to provide targeted and differentiated instruction and support for ELLs. For example, when practicing information gathering through reading about main idea and supporting details, a teacher can refer to the above SPI strand and assign ELLs at stage 1 with a partner (e.g., non-ELL) to practice identifying key words related to a topic from illustrated texts. For ELLs at stage 3, the teacher can provide illustrated texts with these students and allow them to work in small groups (e.g., mixed-ability group) to identify main ideas and supporting details related to the topic. Finally, for ELLs at stage 5, the teacher can provide them with different sources on the same topic (including illustrated texts), dictionaries to look up meaning, and ask them to summarize the key ideas about the topic. The same instructional activity, with or without additional scaffolding support, can be implemented during small-group instruction for ELLs from the same stage to reteach or review the focused skill. What Is the Relationship Between English Language Proficiency and Academic Achievement? Consider the following project commonly found in schools: Students conduct research to identify historical events of a country and deliver oral presentation on findings. ELLs must be able to understand and apply different components of academic English in order to complete this multistep assignment. Because of the multiple levels and huge spans of

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features related to general and specific content, academic language often becomes the most difficult area for ELLs to learn. The English language proficiency (ELP) standards provide a way to highlight not only the social language needed to participate in school, but most importantly, the academic language to learn content. Gottlieb, Katz, and Ernst-Slavit (2009) state the following six principles to show the relationships between academic language proficiency, language and literacy development, and academic achievement:

1. “Language proficiency is an outgrowth of cumulative experiences both inside and outside of school.” ELLs experience language learning constantly, including before and after school. Classroom activities that connect to the experiences and knowledge ELLs bring into the classroom help support and extend their language development.

2. “Language proficiency can reflect complex thinking when linguistic complexity is reduced and support is present.” ELLs from any language proficiency level are capable of higher-order thinking, as long as the language used is adjusted to their proficiency levels, and if ELLs are provided with scaffolding support.

3. “Both social and academic language proficiencies are necessary for school success.” Teachers need to incorporate both social and academic language learning opportunities for their students.

4. “Academic language proficiency works in tandem with academic achievement.” Students need to learn academic language in order to understand and show their understanding of the content. Content also drives the academic language students need to learn and succeed.

5. “Academic language proficiency is developed through sustained content-based language instruction.” In order for ELLs to reach full proficiency for academic success, they need to receive ongoing language support. One way to provide ongoing language support is through core content classes.

6. “Native language proficiency—in particular, academic language proficiency in the native language—facilitates the acquisition of academic language proficiency in English.” Foundational knowledge and skills from first language can be transferred to a new language if first language is strongly developed.

How Can We Help Ensure the Success of Our English Language Learners? The integration of language expressed in the ELP standards, with content outlined in the academic content standards, ultimately facilitates ELLs’ achievement. By pairing the two sets of standards, teachers and students alike gain an understanding of the grade-level language needed to access the skills and knowledge of the core content areas. The SPIs from the ELP standards show how ELLs at different language proficiency stages can understand and use language to learn specific content skills and knowledge. To ensure the success of our ELLS, teachers can utilize the ELP standards to develop differentiated instruction by scaffolding the academic language skills needed to learn grade-level content skills and knowledge, and to create learning and assessment activities for different stages of language proficiency. ��

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MATHEMATICS: INTRODUCTION

Mathematics is about patterns, relationships, and structures; it is about logical analysis,

deduction, abstraction, and calculation within these patterns, relationships, and structures.

When patterns are found, often in widely different areas of science and technology, the

mathematics of these patterns can be used to explain, predict, and in some cases

manipulate natural happenings and situations. Mathematics has a pervasive influence on

our everyday lives and contributes to the wealth of the country.

The fundamental goals of learning mathematics are multifold:

To help students become numerate. Fundamental to numeracy is the ability to estimate

and reckon (guess) with numbers, no matter how small or large, which is the habit of

comparing numbers with meaningful reference points from previous experience. Students

receive frequent practice in estimation and reckoning.

To assist students in navigating their lives in this ever-changing complex, modern world. This of course involves dealing with such things as taxes, loans, credit cards,

purchases, budgeting, and shopping. Our younger generation needs to be able to handle

money wisely, which requires a good understanding of parts, proportion, and percentages.

To enable students to understand information around us. Being able to read through

the scientific information and make sense of it requires knowing big and small numbers,

statistics, probability, and percentage.

To prepare students for further studies in mathematics and science. Many students

may choose to go to college for higher education in math or science. An in-depth knowledge

of various types of functions such as algebraic, trigonometric, logarithmic, exponential, and

inverse is critical, along with the basic understanding of limits, differentiation, and integration.

Their applications are all essential in achieving higher education goals.

To teach logical reasoning through inductive and deductive reasoning. Geometry is

useful for developing logical reasoning. Theorems and related problems in geometry are

taught by the method of two-column proof writing, which involves deductive and inductive

reasoning.

Mathematics education provides students the opportunities to explore and gain knowledge

about “how” something works, often called procedural understanding. Conceptual

knowledge is important for the development of procedural fluency, while fluent procedural

knowledge supports the development of further understanding and learning. Development of

higher order thinking skills, reasonableness of the answers, logical reasoning, problem-

solving skills, responsible decision-making, and interpretation of data are all integral parts of

teaching and learning mathematics. Therefore, mathematics education contributes to

students’ becoming productive citizens in a diverse society.

By grade 8, students’ mathematical sensitivity should be sharpened. Students need to start

perceiving logical subtleties (refinement or detail) and appreciate the need for sound

mathematical arguments before making conclusions. Mathematical reasoning and

conceptual understanding are not separate from content; they are intrinsic (essential) to the

mathematical discipline that students master at more advanced levels.

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To allow flexibility in teaching the material, the standards for grade 9 through 12 are not

mandated as end-of-grade disciplines; rather, the standards are included as related to each

course: Pre-Algebra, Algebra I, Geometry, Algebra II, Pre-Calculus, and Calculus.

The ASDOE believes that, through participating in the lucid K–12 standards-based

mathematics curriculum, students will be challenged and actively engaged in their own

learning processes, and they will gain proficiency in the following:

x Gathering, interpreting, and analyzing the data they come across

x Exploring the interdependency of various math disciplines and the role played by

mathematics in the core areas

x Connecting mathematics with the community and using appropriate problem-solving

skills to address and solve the problems faced by its members

Organization of Standards for Mathematics

Framed by four strands, the ASDOE mathematics program comprises eight standards.

Strand: Number Sense, Properties, and Operations

Standard 1: Students investigate numbers, ways of representing numbers, relationships

among numbers, and number systems.

Understand and express numbers (concepts, representations, magnitude, orders and comparison, and place value) and relationships among numbers (place value, estimation, multiples and factors, equivalency, conversions, ratio and rate).

Standard 2: Students perform mathematical operations and determine how they relate to

one another.

Perform operations of whole numbers, fractions, mixed numbers, decimals, and proportions.

Strand: Patterns, Algebra, and Functions

Standard 3: Students investigate patterns, functions, and quantitative relationships and

apply that knowledge to modeling, analyzing, or solving mathematical and real-world

problems.

Identify and describe objects in space using direction, orientation, and perspective; recognize and use relative shapes and sizes of figures and objects; understand and express relationships among points, lines, and angles.

Identify, create, extend, and represent patterns (e.g., numerical patterns or objects) and quantitative relationships using words, mathematical formulas, pictures, tables, graphs, and rules.

Use variables and symbolic expressions, and algebraic equations and inequalities to represent linear, non-linear, and simple exponential relationships.

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Standard 4: Students use symbolic forms, operations, and properties to represent, model,

solve, and analyze mathematical situations and structures.

Identify, classify, compare, and analyze properties and relationships of geometric shapes and space.

Use tables, graphs, and rules to determine functional relationships that are linear, non-linear, or simple exponential relationships, and to represent them as algebraic equations.

Strand: Measurement, Geometry, and Spatial Sense

Standard 5: Students explore properties, relationships, and transformations of shapes and

space using coordinate geometry and other representational systems.

Identify, describe and explain relationships among shapes, such as congruency, similarity, symmetry, motion geometry, and transformations of figures.

Standard 6: Students explore concepts and practical applications of units, geometric

constructions, and measurements of plane and geometric shapes and space.

Identify estimates and measures to a particular unit, and utilize measurement attributes of geometric shapes (e.g., perimeter, area), units, and tools for measurement.

Strand: Data Analysis, Statistics, and Probability

Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect,

organize, analyze, display, and make inferences based on data.

Collect, organize, and read data in charts, graphs, and tables, and investigate frequency and distribution of data using range and central tendency (e.g., mode, mean, median).

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life

situations.

Identify and use the concept of probability and the likelihood of the possible outcomes of an event, and use theoretical or experimental probability to predict the outcome of a situation in real life.

ASDOE Math Standards Proposed Implementation Plan for CCSS A decision was made that 100% of the Common Core State Standards (CCSS) will now be

introduced in this document. As a result, the remaining Common Core mathematics

standards have now been included for the first time. These are coded with the letter “E” (for

Extra) throughout the document. These additions complete almost 100% adoption of the

Common Core Standards for mathematics.

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In addition, the evidence concerning college and career readiness – and therefore the

transition from high school to post-secondary education for college and careers – shows

clearly that the knowledge, skills, and practices important for readiness include a great deal

of mathematics prior to the boundary defined by (+) symbols in these standards. The

‘boundary’ concepts are considered optional for teachers, depending on availability of time

and/or students’ readiness.

Nevertheless, the fact that two years are given for the implementation of the revised

standards, Table 1 outlines, for administrators and teachers in particular, a recommended

pathway towards implementing the remaining Common Core Standards so that a 100%

adoption of the Common Core Standards will take effect after two years:

Table 1. ASDOE Math Standards Proposed Implementation Plan (2012–2013)

Grade Level/

HS Course

AS Extra Standards

Extended Common Core Math Standards

Extended American Samoa Math Standards

(Benchmarks

Code) (Standards Code)

Year 1 Year 2

K K.2.3

K.3.1

K.5.4

K.6.2

K.6.3

K.7.1

K.8.1

NONE ¥ ¥

1 1.1.4

1.3.1

1.6.1

1.7.1

1.8.1

NONE ¥ ¥

2 2.1.4

2.2.1

2.3.1

2.8.1

NONE ¥ ¥

3 3.4.1

3.5.1

3.7.1

3.8.1

3.MD.5(b)

3.OA.8

(E) 3.4.1

(E) 3.6.1(b)

4 4.1.2

4.4.1

4.6.5

4.7.1

4.8.1

4.NF.1

4.NF.2

4.NF.5

4.NF.6

4.NF.7

(E) 4.1.3

(E) 4.1.3

(E) 4.2.4

(E) 4.1.2

(E) 4.1.3

5 5.4.1

5.5.1

5.8.1

5.MD.5(c) (E) 5.6.3(d)

6* 6.5.1

6.5.2

6.5.3

6.6.1

6.NS.6(a)

6.NS.6 (b)

6.NS.6(c)

6.NS.7

6.NS.6(a)

6.NS.6 (b)

6.NS.6(c)

6.NS.7

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Grade Level/

HS Course

AS Extra Standards

Extended Common Core Math Standards

Extended American Samoa Math Standards

(Benchmarks

Code) (Standards Code)

Year 1 Year 2

6.8.5

6.NS.7(a)

6.NS.7(b)

6.NS.7(c)

6.NS.7(d)

6.NS.8

6.EE.9

6.EE.9

6.NS.7(a)

6.NS.7(b)

6.NS.7(c)

6.NS.7(d)

6.NS.8

7* 7.1.3

7.3.2

7.5.1

7.RP.2(c)

7.NS.1(d)

7.G.2

7.G.3

7.SP.2

7.SP.6

7.SP.7(a)

7.SP.7(b)

7.SP.8(c)

7.RP.2(c)

7.NS.1(d)

7.G.2

7.SP.2

7.SP.2

7.G.3

7.SP.7(a)

7.SP.7(b)

7.SP.8(c)

8 8.2.2

8.5.1

8.6.2

8.6.3

8.7.1

8.7.2

8.8.1

8.G.1

8.SP.1

8.SP.2

8.SP.3

(E) 8.5.4

(E) 8.7.3

(E) 8.7.3

(E) 8.7.3

*In these courses, the corresponding Extended American Samoa Math Standards have not

been added from the Extended Common Core Math Standards. Teachers must therefore

refer to the official Common Core State mathematics standards document when

implementing the extended American Samoa math standards for Year 1 and Year 2.

Standards for Mathematical Practices

The ASDOE’s standards for mathematics include grade-level content and English language

proficiency (ELP) standards as well as standards for mathematical practices. In alignment

with the Common Core State Standards, the math standards specify skills and

understandings that students should develop at each grade, whereas the standards for

mathematical practice identify “processes and proficiencies” that students should develop

throughout their mathematical education at all levels, K–12. ELP standards support

mathematical literacy needed for mathematics achievement.

The standards for mathematical practices describe the mathematical “habits of mind” and

varieties of expertise that all students should develop, with the help of their teachers at all

levels. These practices rest on important “processes and proficiencies” with longstanding

importance in mathematics education. All mathematics teachers should attend to the need

to connect mathematical practices to mathematical content in their instruction.

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All ASDOE students are expected to practice the following processes and proficiencies:

1. Develop appropriate strategies by making sense of the problems and persevere in

solving them

2. Develop abstract, logical, and quantitative mathematical reasoning

3. Model mathematics through applying the mathematics students know

4. Communicate fluently in order to learn and express their mathematical

understanding with precision

5. Select and use tools (e.g., technology) appropriately and strategically for learning

and solving problems

6. Construct appropriate mathematical images, whether physical, visual, or mental, to

assist in processing information, making connections, and solving problems

7. Look closely for and make use of any noticeable mathematical pattern or structure, a

process that may involve connecting mathematical ideas to each other or to real-

world phenomena

8. Develop estimation skills as useful, efficient strategies for solving problems and

dealing with situations in daily life

Six of the above stated processes and proficiencies are as defined in the Common Core

State Standards. Introducing “construct appropriate mathematical images” (which includes

the ability to do mental mathematics) and “develop estimation skills” not only brings needed

attention to these types of skills and proficiencies, but also emphasizes their relevancy to

American Samoa students who must cope with a wide variety of mathematically-related

situations in everyday life. The following paragraphs describe how visualization and

estimation are understood as practices.

Images, Visualization, and Mental Mathematics Students identify images held, accessed, made, modified, and worked with as they engage

in a mathematical task. An “image” is often defined as any physical, visual, or mental

representation or idea that the students may have for a particular concept or topic. These

images are often specific, mathematically limiting, and context-dependent, and students’

work with images in large part determines for them what is mathematically meaningful (or

not meaningful) while they are working on a task.

When students visualize, they are thinking constructively and creatively, using pictures and

mental images to understand mathematical concepts and making connections among them.

When a student demonstrates the use of mental constructs about concepts, he or she is

able to use that particular image or visualization without doing the activity itself. The student

is no longer tied to a particular action, but is now able to carry out the activity through

internalization of the constructed images, often called a “mental plan.”

Estimation as a Skill Estimation is a strategy for determining approximate values or quantities. It is also said to be

a process whereby one approximates, through rough calculations, the worth, size, or

amount of an object or quantity that is present in a given situation. The approximation, or

estimate, is a value that is deemed close enough to the exact value or measurement to

answer the question being posed. Students need to know how, when, and what strategy to

use when estimating.

Estimation has been seen as an important, even necessary, daily living skill that all students

should have, whether they are working with paper and pencil, doing mental mathematics, or

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using manipulatives or technological tools. When coupled with mental mathematics, the two

processes require the manipulation of numbers, shapes, or patterns in one’s head. Both are

key skills in reasoning and problem solving. Estimation is used to make mathematical

judgments and develop useful, efficient strategies for dealing with situations in daily life.

Estimation is therefore seen as a central mathematical process applicable across a wide

range of mathematical content areas, and it should be viewed as a necessary component of

the school mathematics curriculum.

While more than one practice standard can be identified with one particular benchmark, it is

up to each teacher to choose which one(s) to focus on, or which one(s) can be used to help

his or her students best.

16

8. Develop estimation skills as useful, efficient strategies for solving problems and

dealing with situations in daily life

Six of the above stated processes and proficiencies are as defined in the Common Core

State Standards. Introducing “construct appropriate mathematical images” (which includes

the ability to do mental mathematics) and “develop estimation skills” not only brings needed

attention to these types of skills and proficiencies, but also emphasizes their relevancy to

American Samoa students who must cope with a wide variety of mathematically-related

situations in everyday life. The following paragraphs describe how visualization and

estimation are understood as practices.

Images, Visualization, and Mental Mathematics Students identify images held, accessed, made, modified, and worked with as they engage

in a mathematical task. An “image” is often defined as any physical, visual, or mental

representation or idea that the students may have for a particular concept or topic. These

images are often specific, mathematically limiting, and context-dependent, and students’

work with images in large part determines for them what is mathematically meaningful (or

not meaningful) while they are working on a task.

When students visualize, they are thinking constructively and creatively, using pictures and

mental images to understand mathematical concepts and making connections among them.

When a student demonstrates the use of mental constructs about concepts, he or she is

able to use that particular image or visualization without doing the activity itself. The student

is no longer tied to a particular action, but is now able to carry out the activity through

internalization of the constructed images, often called a “mental plan.”

Estimation as a Skill Estimation is a strategy for determining approximate values or quantities. It is also said to be

a process whereby one approximates, through rough calculations, the worth, size, or

amount of an object or quantity that is present in a given situation. The approximation, or

estimate, is a value that is deemed close enough to the exact value or measurement to

answer the question being posed. Students need to know how, when, and what strategy to

use when estimating.

Estimation has been seen as an important, even necessary, daily living skill that all students

should have, whether they are working with paper and pencil, doing mental mathematics, or

using manipulatives or technological tools. When coupled with mental mathematics, the two

processes require the manipulation of numbers, shapes, or patterns in one’s head. Both are

key skills in reasoning and problem solving. Estimation is used to make mathematical

judgments and develop useful, efficient strategies for dealing with situations in daily life.

Estimation is therefore seen as a central mathematical process applicable across a wide

range of mathematical content areas, and it should be viewed as a necessary component of

the school mathematics curriculum.

While more than one practice standard can be identified with one particular benchmark, it is

up to each teacher to choose which one(s) to focus on, or which one(s) can be used to help

his or her students best.

171717

MATH GRADE K

NUMBER SENSE, PROPERTIES, AND OPERATIONS

Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. K.1.1 Identify number names and the count sequence, and write numbers with the appropriate numerals

Sample Performance Indicators: x Identify and match numbers (1–100) to picture cards, objects, etc. x Arrange number cards in a correct sequence x Write numbers from 0 to 20, and represent a number of objects with a written numeral 0–20 (with

0 representing a count of no objects) x Count to 100 by ones and by tens, with and without visual supports x Count forward beginning from a given number within a known sequence (not only beginning at 1)

K.1.2 Identify the relationship between numbers and quantities, and connect counting to cardinality Sample Performance Indicators: x Match numbers with objects:

o Say the number names in the standard order when counting objects, pairing each object with one and only one number name, and each number name with one and only one object

o Identify the last number name as the number of objects counted, recognizing that the number of objects is the same regardless of their arrangement or the order in which they are counted

o Explain that each successive number name refers to a quantity that is one larger x Count to answer “how many?” questions about as many as 20 things arranged in a line, a

rectangular array, or a circle, or as many as 10 things in a scattered configuration

K.1.3 Compare groups of objects and numbers between 1 and 10 to show less, more, same, or equal Sample Performance Indicators: x Identify whether the number of objects in one group (up to ten objects) is greater than, less than,

or equal to the number of objects in another group (e.g., by using matching and counting strategies, and by using manipulatives)

x Compare two numbers between 1 and 10 presented as written numerals

Standard 2: Students perform mathematical operations and determine how they relate to one another. K.2.1 Calculate addition and subtraction facts with sums and differences up to 20, and solve simple word

problems Sample Performance Indicators: x Identify addition as putting together and adding to and subtraction as taking apart and taking from:

o Demonstrate or represent addition and subtraction with objects o Act out situations, with verbal explanations and expressions, or show equations by using

fingers, sounds (e.g., claps), mental images, or drawings (which need not show details, but

Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly, logically, and quantitatively. 3. Model with mathematics. 4. Communicate with precision. 5. Use appropriate tools strategically. 6. Construct appropriate mathematical images. 7. Look for and make use of mathematical pattern or structure. 8. Develop estimation skills.

FINAL July 2012 181818

MATH GRADE K

should show the mathematics in the problem) x Add and subtract within 5 x Represent addition and subtraction within 10, using objects or drawings x Solve addition and subtraction word problems

K.2.2 Work with numbers to gain foundations for place value and identify addition facts using composition and decomposition of numbers Sample Performance Indicators: x Decompose numbers less than or equal to 10 into pairs in more than one way (e.g., by using

objects or drawings) and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1)

x Find the number that makes 10 when added to the given number (e.g., using objects or drawings), record the answer with a drawing or equation, and speak the number out loud

x Compose and decompose numbers from 11 to 19 into ten ones and some further ones (e.g., by using objects or drawings); record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); and demonstrate understanding that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones

PATTERNS, ALGEBRA, AND FUNCTIONS

Standard 3: Students investigate patterns, functions, and quantitative relationships and apply that knowledge to modeling, analyzing, or solving mathematical and real-world problems. K.3.1 Recognize different or repeated patterns

Sample Performance Indicators: x Identify a pattern involving sounds, shapes, or numbers, and also recognize any differences x Demonstrate repeating patterns by identifying, reproducing, and extending a given pattern using

manipulatives, sounds, and/or actions

MEASUREMENT, GEOMETRY, AND SPATIAL SENSE

Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. K.5.1 Identify and describe different geometric shapes or objects and relative positions in the environment

Sample Performance Indicators: x Describe geometric objects using names of shapes: squares, circles, triangles, rectangles, cubes,

cones, cylinders, and spheres x Describe the relative positions of objects in the environment using terms such as above, below,

far, near, left/right, beside, in front of, behind, and next to (i.e., use appropriate language to describe objects by proximity, position, and direction)

x Correctly name shapes regardless of their orientations or overall size x Identify shapes as 2-D (i.e., lying in a plane, flat) x Identify shapes as 3-D (i.e., solid)

K.5.2 Analyze and compare shapes Sample Performance Indicators: x Analyze and compare 2-D and 3-D shapes, in different sizes and orientations, using informal

language to describe their similarities, differences, parts (e.g., number of sides and vertices/corners) and other attributes (e.g., having sides of equal length)

x Construct and combine simple shapes to form larger shapes (e.g., model shapes in the world by building shapes from components such as sticks and clay balls, and draw shapes)


MATH GRADE K

x Compose simple shapes to form larger shapes (e.g., explain or draw how two triangles can be joined with full sides touching to make a rectangle)

K.5.3 Recognize the difference between a “whole” and a “part of a whole” using geometry Sample Performance Indicators: x Draw a diagram (e.g., a rectangle or a circle) to show a “whole” and a “half” (or “one-half”) x Model different parts of a “whole” by folding a strip of paper

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space. K.6.1 Describe and compare measurable attributes of objects

Sample Performance Indicators: x Describe measurable attributes of objects, or a single object such as length or weight x Compare two objects with a measurable attribute in common to see which object has more of or

less of the attribute (e.g., directly compare the heights of two children, describe one child as taller or shorter, and describe differences)

x Measure attributes of objects using nonstandard unit (e.g., measure length and weight to the nearest nonstandard unit), and discuss findings

K.6.2 Tell time to the hour using analog clocks, and recognize value of money Sample Performance Indicators: x Tell time to the hour using a 12-hour analog clock x Draw pictures to show time to the hour (e.g., 7 o’clock) based on models x Identify the values of coins and bills (e.g., using play money, a penny, or any money value up to

$10)

DATA ANALYSIS, STATISTICS, AND PROBABILITY

Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. K.7.1 Recognize and explain simple tables, charts, and graphs

Sample Performance Indicators: x Explain data sets presented in picture graphs or pictographs x Explain data sets presented in tables, line graphs, pie charts, and flow charts that are familiar (e.g.,

posted on a bulletin board, presented in a grade-level text, or created for a story or class activity)

K.7.2 Classify objects and count the number of objects in each category Sample Performance Indicators: x Classify objects into two main categories x Count the number of objects in each category x Sort the categories by count, but limit to less than or equal to 10

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. K.8.1 Introduce the terms likely, never, always, and less likely

Sample Performance Indicators: x Talk about probability using the terms likely, never, always, and less likely, as used in everyday

situations x Discuss and give simple everyday probability examples


MATH GRADE K

An aligned instructional assessment system begins with standards. Standards relate grade or grade-level cluster expectations for students, teachers, administrators, and other stakeholders. Educators of language learners must be cognizant of the correspondence between language and content standards and the influence of a student’s language proficiency on academic achievement. Corresponding Representation of Mathematics in English Language Proficiency Standards Standard 1: English language learners communicate for social, intercultural, and instructional purposes within the school setting.

Language Domain

Topic ELP Stage 5: Advanced

ELP Stage 3: Intermediate

ELP Stage 1: Beginning

Listening

Rules; Games

Respond to commands from teachers while engaged in math games to count, working with a partner

Follow multistep directions given orally during math games to count, working with a partner

Locate objects from math counting games (e.g., manipulatives for counting; play coins) from pictures or realia, as directed orally

Speaking

Social behavior

Adapt expressions appropriate to familiar audiences (e.g., talking with elders, peers, siblings) during math lessons

Use basic forms of polite expressions in conversations with adults during math lessons

Restate basic forms of polite expressions (e.g., “please”; “thank you”) during math lessons

Reading

Classroom Construct meaning from illustrated math text books

Sort labels of math objects and explain grouping (e.g., “These are cards with numbers on them.”; or, “These are things to help us count.”)

Match math objects (e.g., flash cards, manipulatives, number line) in the classroom with their proper names and use, working with a partner

Writing

Family Copy and illustrate notes about family members and total numbers of people using a combination of words and phrases with invented spellings

Describe family members and total numbers of people, using a combination of drawings, letters, scribble writings, or words with invented spellings

Describe family members (e.g., parents, siblings, grandparents) and show total numbers of people, using drawings


MATH GRADE K

FINAL July 2012

Standard 3: English language learners communicate information, ideas, and concepts necessary for academic success in the area of mathematics.

Language Domain




Listening

Patterns

Follow oral descriptions to extend patterns using manipulatives

Identify what comes next based on oral descriptions of patterns with visual support

Repeat pattern sounds with physical movement (e.g., clap, snap, stomp), working as a whole group

Speaking

Numbers; Operations

Discuss grade-level math stories using comparative language (e.g., “The girl has more toys than the boy.”)

Use comparative phrases (e.g., less than, more than, same as, equal to) with a partner to describe the relationship between two groups of objects

Use number words to count objects (e.g., crayons; cubes), working with a partner

Reading

Measurement; Geometry

Identify words for shapes, order, or position from pictures, flash cards, and illustrated word walls

Sort labeled pictures (e.g., circle, square, rectangle, etc.) based on observable and measurable attributes (e.g., sides, corners)

Match pictures of real-life items with shapes, working in a small group

Writing

Time (analog)

Create illustrated stories about a time-related sequence of events (e.g. a typical school day) using drawings, words, phrases, and invented spellings

Draw and label special times of day (e.g., waking up, going to school, spending time with family) using writings that include letters, scribble writings, and words with invented spellings

Draw pictures to show time to the hour (e.g., 7 o’clock), based on models

222222

MATH GRADE 1


Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. 1.1.1 Count to 120 by ones and tens, starting at any number less than 120, and read and write numbers

Sample Performance Indicators: x Count to 120 by ones and tens, starting at any number less than 120 x Skip-count by 2s and 10s, starting at any number less than 120, with and without visual support x Count backward by ones, starting at any number less than 120, with and without visual support x Identify and write the missing numbers on a number chart from 0–120 x Read and write numerals and represent a number of objects with a written numeral

1.1.2 Develop understanding of place value for “ones” and “tens” Sample Performance Indicators: x Identify how the two digits of a 2-digit number represent amounts of tens and ones with the

following as special cases: o 10 can be thought of as a bundle of ten ones—called a “ten” o The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven,

eight, or nine ones o The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven,

eight, or nine tens (and 0 ones) x Organize to show amounts of tens and ones using manipulatives x Determine the place values of whole numbers up to 120

1.1.3 Compare and order numbers to 120 using less, more, same, or equal Sample Performance Indicators: x Compare two 2-digit numbers based on meanings of the tens and ones digits, recording the

results of comparisons with the symbols >, =, and < x Show two numbers on a number line to compare more or less x Mentally calculate “10 more” or “10 less” than a two-digit number, without having to count, and say

the number and explain the reasoning used

1.1.4 Recognize numerical relationships on a 100’s chart Sample Performance Indicators: x Identify, label, predict, and show any 2-digit number using a 100’s number chart x Recognize odd and even numbers from 1 to 100 x Pair up a group of objects (up to 10) to show an odd or even number of members (e.g., by pairing

objects or counting them by 2s)



MATH GRADE 1

Standard 2: Students perform mathematical operations and determine how they relate to one another. 1.2.1 Interpret and apply concepts of addition, subtraction, and comparison

Sample Performance Indicators: x Discuss and explain the difference between addition (e.g., in all, put together, total, etc.), and

subtraction (e.g., take away, difference, etc.) x Identify and apply subtraction as an unknown-addend in solving a problem (e.g., subtract 10 – 8

by finding the number that makes 10 when added to 8) x Solve addition and subtraction word problems involving situations of adding to, taking from,

putting together, taking apart, and comparing, with unknowns in all positions, and demonstrate by using objects, drawings, and equations with a symbol for the unknown number in the problem

x Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20 and demonstrate by using objects, drawings, and equations with a symbol for the unknown number to represent the problem

1.2.2 Represent and apply properties of operations to solve problems involving addition and subtraction Sample Performance Indicators: x Relate counting to addition and subtraction (e.g., counting forward by 2 is equivalent to adding 2

each time) x Add and subtract within 20, utilizing strategies such as: counting on; making ten (e.g., 8 + 6 = 8 +

2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13)

x Solve problems using properties of operations: the commutative property of addition (e.g., if 8 + 3 = 11 is known, then 3 + 8 = 11 is also known), and the associative property of addition (e.g., to add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12)

1.2.3 Add and subtract using place value understanding and properties of operations Sample Performance Indicators: x Add within 100, including adding a 2-digit number and a 1-digit number, and adding a 2-digit

number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method; and explain the reasoning used

x Show that in adding two-digit numbers, the tens and ones are added separately, and that sometimes it is necessary to compose a ten

x Subtract multiples of 10 in the range 10–90 from multiples of 10 in the range 10–90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method; and explain the reasoning used

x Discuss how to add and subtract two 2-digit numbers to 100 without regrouping

1.2.4 Work with addition and subtraction equations Sample Performance Indicators: x Discuss and illustrate the meaning of the equal sign (e.g., describe equality as a “balance” and

inequality as an “imbalance,” concretely and pictorially using numbers from 0 to 20) x Determine if equations involving addition and subtraction are true or false (e.g., which of the

following equations are true and which are false? 6 = 6; 7 = 8 – 1; 5 + 2 = 2 + 5; 4 + 1 = 5 + 2) x Determine the unknown whole number in an addition or subtraction equation relating three whole

numbers (e.g., determine the unknown number that makes the equation true in each of the equations: 8 + ? = 11; 5 = ? – 3; 6 + 6 = ?)


MATH GRADE 1


Standard 3: Students investigate patterns, functions, and quantitative relationships and apply that knowledge to modeling, analyzing, or solving mathematical and real-world problems. 1.3.1 Create own numerical and geometric patterns

Sample Performance Indicators: x Create and describe a geometrical pattern using objects and shapes x Create and describe own numerical patterns x Demonstrate repeating patterns by identifying, reproducing, and extending a given pattern using

manipulatives and numbers


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. 1.5.1 Distinguish shapes through their attributes using appropriate language

Sample Performance Indicators: x Describe defining attributes of shapes (e.g., triangles are closed and three-sided) x Describe nondefining attributes of shapes (e.g., color, orientation, overall size) x Build or draw shapes that possess certain defining attributes x Collect and display specific objects or geometric shapes with similar attributes (e.g., being

congruent or similar, or having a line of symmetry)

1.5.2 Combine several geometric shapes to make a larger shape Sample Performance Indicators: x Compose 2-D shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles)

to create a composite shape, and compose new shapes from the composite shape x Compose 3-D shapes (cubes, right rectangular prisms, right circular cones, and right circular

cylinders) to create a composite shape, and compose new shapes from the composite shape

1.5.3 Identify different fractions using geometry Sample Performance Indicators: x Fold and model different fractions using a strip of paper x Partition circles and rectangles into two and four equal shares; describe the shares using the

words halves, fourths, and quarters; use the phrases (one or a) half of, a fourth of, and a quarter of; and describe the whole as two of, or four of, the shares

x Explain for the above examples how decomposing into more “equal shares” creates smaller shares

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space. 1.6.1 Measure lengths indirectly and by iterating length units

Sample Performance Indicators: x Identify specific measurable attributes of an object x Order three objects by length, and compare lengths of two objects directly and indirectly (i.e., by

using a third object) x Express the length of an object as a whole number of length units, by laying multiple copies of a

shorter object (the length unit) end-to-end, and understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps (Note: limit to contexts where the object being measured is spanned by a whole number of length units with no


MATH GRADE 1

gaps or overlaps) x Measure length and weight of an object using nonstandard units (e.g., hand-span, common

cultural measuring objects, etc.) x Measure length and weight of an object using standard units, and report findings

1.6.2 Tell and write time to the hour and half-hour, and recognize simple compositions of money Sample Performance Indicators: x Tell and write time in hours and half-hours, using analog and digital clocks x Draw and label special times of day (e.g., waking up, going to school, spending time with family) x Identify compositions of coins and/or bills using appropriate amounts of $ and ¢ (e.g., 50¢ equals 5

dimes or 2 quarters)


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. 1.7.1 Identify, represent, and interpret statistical data as presented in simple tables, charts, or graphs

Sample Performance Indicators: x Identify simple tables, charts, and graphs x Explain data sets presented in tables, line graphs, picture graphs, pie charts, and flow charts that

are familiar (e.g., posted on a bulletin board, presented in a grade-level text, or created for a story or class activity)

x Organize, represent, and interpret data with up to three categories x Identify the total number of data points, telling how many are in each category, and how many are

in one category, more or less than in another

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. 1.8.1 Interpret the terms likely, never, always, and less likely

Sample Performance Indicators: x Explain possible meanings of the terms likely, never, always, and less likely in everyday situations x Discuss and give simple everyday probability examples


MATH GRADE 1

An aligned instructional assessment system begins with standards. Standards relate grade or grade-level cluster expectations for students, teachers, administrators, and other stakeholders. Educators of language learners must be cognizant of the correspondence between language and content standards and the influence of a student’s language proficiency on academic achievement. Corresponding Representation of Mathematics in English Language Proficiency Standards Standard 1: English language learners communicate for social, intercultural, and instructional purposes within the school setting. Language Domain




Listening

Rules; Games

Follow multistep oral instructions in various contexts of math lessons (e.g., small groups or with a partner)

Follow a series of oral commands for counting activities, supported by gestures or realia and working with a partner

Copy responses to one-step oral instructions for counting activities, supported by gestures or realia, working as a whole group

Speaking

Social behavior

Retell through demonstration about a personal experience in needing to measure the length of different objects at home, and the tools used

Describe a personal experience in needing to measure the length of an object at home, and the tool used for measurement, using drawings as support

Answer “yes or no” questions about an experience using tools needed to measure length of objects

Reading

Classroom Connect ideas about digital and analog clocks in illustrations to new situations (e.g., ways to tell time)

Answer questions related to illustrations of digital and analog clocks, working in small groups

Match icons of different parts of a digital or analog clock, working with a partner

Writing

Family Produce illustrated writings to describe ways to measure weights of different objects, using traditional methods and standard units

Describe methods to measure weight of different objects, using traditional methods and standard units, and using drawings and graphic organizers

Label illustrations of measuring the weight of different objects, using traditional methods and standard units, and working with a partner


MATH GRADE 1

FINAL July 2012

Standard 3: English language learners communicate information, ideas, and concepts necessary for academic success in the area of mathematics. Language Domain




Listening

Time (digital and analog)

Respond to oral directions/questions, and show time in hours and half-hours on the clock (e.g., “Put the hour hand on number 6 and the minute hand on number 6. What time is it?”; or, “Write number 6 for the hour and 30 for the minutes. What time is it?”)

Respond to oral directions (“Show me six thirty.”) to show time in hours and half-hours on digital and analog clock, working in small groups

Respond to one-step oral directions, and point to time in hours and half-hours (e.g., “Put the hour hand on number 6.”; or, “Write number 6 for the hour.”), working with a partner

Speaking

Whole numbers

Name numbers that correspond to numbers of objects shown on picture cards or shown by realia

State missing numbers on number charts, working in small groups

Mimic and repeat counting aloud by ones, twos, or tens, modeled by teacher or peers, and with visual support

Reading

Basic operations

Interpret key information, and solve word problems for addition and subtraction

Match key words and phrases from word problems associated with addition and subtraction, with illustrations

Identify words and symbols for addition and subtraction (e.g., +; in all; put together; total for addition), working with a partner

Writing

Two- and three-dimensional shapes

Compare real-life objects based on their shapes and attributes, using key vocabulary, phrases, and illustrations

Describe real-life objects and their shapes, using key vocabulary, phrases, and illustrations, and working with a partner

Draw and label real-life objects with key vocabulary to describe their shapes and attributes (defining and nondefining), using Samoan and English and working with a partner

282828

MATH GRADE 2


Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. 2.1.1 Count to 1,000, starting at any number, and also read and write numbers

Sample Performance Indicators: x Count within 1,000, starting at any number x Skip-count by 2s, 5s, 10s, and 100s x Read and write numbers to 1,000 using base-10 numerals, number names, and expanded

form x Organize and model the numbers by utilizing base-10 blocks or connecting cubes

2.1.2 Identify, represent, and compare 3-digit numbers using place value understanding Sample Performance Indicators: x Determine the place values of whole numbers to 1,000 using amounts of hundreds, tens, and

ones (e.g., 803 equals 8 hundreds, 0 tens, and 3 ones) x Recognize as special cases that:

o 100 can be thought of as a bundle of ten tens—called a “hundred” (e.g., 803 equals 8 hundreds, 0 tens, and 3 ones)

o 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)

x Compare two 3-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons

x Construct a number line to show two numbers that compare more or less using <, >, or =

2.1.3 Determine the fractional part of a whole number, an object, or a set of objects Sample Performance Indicators: x Display and model objects to show the fractional part of a whole x Demonstrate the link between objects and numbers (e.g., using a set of objects, a drawing,

or paper-folding)

Standard 2: Students perform mathematical operations and determine how they relate to one another. 2.2.1 Add and subtract using place value understanding and properties of operations

Sample Performance Indicators: x Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a

given number 100–900, and speak and record the answer x Add and subtract within 20 using mental strategies, and demonstrate knowing from memory

all sums of two one-digit numbers x Add and subtract within 100 using strategies based on place value, properties of operations,

and/or the relationship between addition and subtraction



MATH GRADE 2

x Add up to four 2-digit numbers using strategies based on place value and properties of operations

x Add and subtract within 1,000, using base-10 blocks; concrete models or drawings and strategies based on place value (e.g., regrouping); properties of operations and/or the relationship between addition and subtraction; and relate the strategy to a written method (noting that in adding or subtracting 3-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)

x Add and subtract two 3-digit numbers to 1,000 without regrouping x Draw or use objects to explain why addition and subtraction strategies work using place

value and the properties of operations x Complete or construct a word problem using appropriate language and the terms for addition

and subtraction

2.2.2 Work with equal groups of objects to gain foundations for multiplication Sample Performance Indicators: x Distinguish odd and even numbers

o Organize and display a number chart to show odd and even numbers o 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) x Write an equation to express an even number as a sum of two equal addends x Find and report the total number of objects using addition of rectangular arrays, arranged

with up to 5 rows and up to 5 columns x Discuss and write an equation to express the total as a sum of equal addends


Standard 3: Students investigate patterns, functions, and quantitative relationships and apply that knowledge to modeling, analyzing, or solving mathematical and real-world problems. 2.3.1 Describe and extend patterns to determine the next term in a linear or geometric pattern

Sample Performance Indicators: x Complete and extend a given linear or geometric pattern x Determine the rule for a repeating pattern x Create own repeated pattern

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and analyze mathematical situations and structures. 2.4.1 Represent and solve problems involving addition and subtraction

Sample Performance Indicators: x Solve one- and two-step word problems involving situations of adding to, taking from, putting

together, taking apart, and comparing, with unknowns (or missing terms) in all positions (e.g., by using drawings and equations with a symbol for the unknown number to represent the problem); use only addition and subtraction within 100

x Solve word problems involving addition and subtraction within 100 and with lengths that are given in the same units (e.g., by using ruler drawings), and solve equations with a symbol for the unknown number to represent the problem

x Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram


MATH GRADE 2


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. 2.5.1 Sort shapes and objects by specific attributes and partition shapes into smaller shapes

Sample Performance Indicators: x Classify geometric properties of 2-D or 3-D shapes, and identify triangles, quadrilaterals

(squares and rectangles), circles, pentagons, hexagons, cubes, cones, cylinders, and pyramids

x Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces (e.g., triangles, quadrilaterals, pentagons, hexagons, and cubes)

x Sort shapes and objects by sizes, directly or visually, but not compared by measuring x Partition a rectangle into rows and columns of same-size squares, and count to find the total

number x Partition circles and rectangles into two, three, or four equal shares, and explore

relationships between these different parts (e.g., How many quarters in one-half?) x Describe the shares using the words halves, thirds, (a) half of, a third of, etc.; describe the

whole as two halves, three thirds, four fourths; and recognize that equal shares of identical wholes need not have the same shape

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space. 2.6.1 Measure and estimate lengths in standard units

Sample Performance Indicators: x Measure the length of an object by selecting and using appropriate tools such as rulers,

yardsticks, meter sticks, and measuring tapes x Measure to determine how much longer one object is than another, expressing the length

difference in terms of a standard length unit x Compare the measurements of two objects using the standard units, and describe how the

two measurements relate to the size of the unit chosen x Estimate lengths using units of inches, feet, centimeters, and meters

2.6.2 Work with time and money, and estimate a total value of coins and/or bills Sample Performance Indicators: x Tell and write time from analog and digital clocks to the exact minutes, and use the

abbreviated terms a.m. and p.m. x Create illustrated stories about a time-related sequence of events (e.g., a typical school day),

using drawings or words x Estimate the amount of coins and bills (e.g., counting game money) x Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using

$ and ¢ symbols appropriately (e.g., if you have 2 dimes and 3 pennies, how many cents do you have?)


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. 2.7.1 Create, organize, represent, and interpret data


MATH GRADE 2

Sample Performance Indicators: x Generate a measurement data by measuring lengths of several objects to the nearest whole

unit, or by making repeated measurements of the same object x Show the measurements by making a line plot, where the horizontal scale is marked off in

whole-number units x Create a table, chart, or graph using a given data set x Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up

to four categories x Solve simple “put-together,” “take-apart,” and “compare” problems, using information

presented in a bar graph

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. 2.8.1 Apply the terms likely, never, always, and less likely to a simple experiment using concrete

materials Sample Performance Indicators: x Apply the terms likely, never, always, and less likely correctly in simple experiments using

concrete materials (e.g., using cards, dice, coins) x Discuss and/or present simple probability experiments


MATH GRADE 2





Listening

Directions; Instructions

Follow multistep oral instructions in various contexts of math lessons (e.g., working in small groups or with partners)

Follow a series of oral commands for money counting activity, supported by gestures or realia, and working with a partner

Copy response to one-step oral instructions for money counting activity, supported by gestures or realia, and working as a whole group

Speaking

Feelings; Emotions; Needs;

Discuss personal opinions about the possible outcomes to orally-stated hypothetical situations of simple experiments (e.g., “What would the outcomes be if seven more blue cards were added to the pile?”)

Share personal opinions about the possible outcomes of simple experiments (e.g., “I think it will be more likely”), working in small groups

Answer “choice” questions to show personal opinions about the possible outcomes of simple experiments (e.g., “Will it be more likely or less likely for us to pick a red card?”)

Reading

Messages; Information

Infer ideas about money based on representations from local market advertisements

Answer questions related to money representations from local market advertisements (e.g., “How much is a bag of rice?”)

Match representations of money (e.g., price) from local market advertisements, working with a partner

Writing

Social, cultural, and school traditions

Produce illustrated writings to describe the shapes and attributes of objects from local traditions, using key vocabulary and sentences

Describe the shapes and attributes of objects from local traditions using some key vocabulary, phrases, and illustrations

Draw and label objects from local traditions that represent 2-D and 3-D shapes (e.g., shape of a fale)


MATH GRADE 2

FINAL July 2012


Language Domain




Listening

Time (digital and analog)

Show time in exact minutes on a clock, and tell time in response to oral directions and questions

Respond to oral directions (“Show me six twenty.”) to show time in exact minutes on a digital and analog clock, working in small groups

Show time to the exact minutes based on one-step oral directions (e.g., “Put the minute hand to number 4.”; or, “Write number 20 for the minutes.”), working with a partner

Speaking

Money; Whole numbers; Estimation

Explain the rationale behind hypothetical purchases (e.g., “What would you buy if you had $50, and why?”) using visually supported materials (e.g., a local market advertisement)

Use key vocabulary to estimate types of products to purchase based on criteria (e.g., “What can you buy for about $20?”) and visually supported materials (e.g., a local market advertisement), working with a partner

State vocabulary related to estimation to describe pictures of coins and bills (e.g., a lot of nickels; about 25¢), working with a partner

Reading

Basic operations

Apply strategies to add and subtract math problems

Match strategies to add and subtract with numerical examples, working with a partner

Identify strategies to add and subtract using manipulatives (e.g., base-10 blocks, cubes), working in small groups

Writing

Two- and three- dimensional shapes

Write an illustrated passage to describe partitioned shapes, using key vocabulary and extended sentences

Describe equal shares of partitioned shapes using key vocabulary, illustrations, and sentence frames

Draw and label partitioned shapes using key vocabulary (e.g., halves, one-third)

343434

MATH GRADE 3

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Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. 3.1.1 Round whole numbers to the nearest 10 or 100 using place value understanding

Sample Performance Indicators: x Identify place value of numbers (digits) using a place-value chart x Determine the place values of whole numbers to 10,000 using amounts of thousands, hundreds, tens,

and ones (e.g., 38,006 equals 38 thousands, 0 hundreds, 0 tens, and 6 ones) x Explain and apply rules of rounding numbers x Apply understanding of place value in order to round whole numbers to the nearest 10 or 100

3.1.2 Develop understanding of fractions as numbers, their representations on the number line, the notion of equivalent fractions, and comparing fractions by size Sample Performance Indicators: x Identify a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts

(e.g., 1/2 is one part of two parts; 1/4 is one part of four parts; 1/10 is one part of ten parts) x 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, and 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

x Identify a fraction a/b as the quantity formed by the parts of size 1/b x Represent a fraction a/b on a number line diagram by marking off a length 1/b from 0, and recognize

that the resulting interval has size a/b, and that its endpoint locates the number a/b on the number linex Explain equivalence of fractions in special cases, and compare fractions by reasoning about their

size: o Describe two fractions as “equivalent” (equal) if they are of the same size or the same point on a

number line o Recognize and generate simple equivalent fractions (e.g., 1/2 = 2/4; 4/6 = 2/3), and explain why

the fractions are equivalent (e.g., by using a visual fraction model) o Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers

(e.g., 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)

o Compare two fractions with the same numerator or the same denominator by reasoning about their size

o Recognize that the comparison above is valid only when the two fractions refer to the same whole, and record the result of the comparison with either the symbol >, =, or <, and justify the conclusion (e.g., by using a visual fraction model)

Standard 2: Students perform mathematical operations and determine how they relate to one another. 3.2.1 Represent and solve problems involving multiplication and division using grouping



MATH GRADE 3

Sample Performance Indicators: x Interpret or describe a context in which a total number of objects can be expressed as products of

whole numbers (e.g., interpret 8 × 3 as the total number of objects in 8 groups of 3 objects each) x Interpret whole-number quotients of whole numbers (e.g., interpret 84 ÷ 3 as the number of objects in

each share when 84 objects are partitioned equally into 3 shares) x Describe a context in which a total number of objects, a number of shares, or a number of groups can

be expressed mathematically (e.g., 5 x 7; 56 ÷ 8) x Draw pictures of “groupings” to illustrate examples of dividing a 2–digit number by a 1-digit number

with a remainder (e.g., 15/4 = 3R3) x Represent and solve problems involving multiplication and division within 100 and 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)

x Determine the unknown whole number in a multiplication or division equation relating three whole numbers (e.g., determine the unknown number that makes the equation true in each of the equations 8 × ? = 48; 5 = _ ÷ 3; 6 × 6 = ?)

3.2.2 Perform multidigit arithmetic in base 10 using place value understanding and properties of operations Sample Performance Indicators: x Add and subtract within 1,000, using strategies and algorithms based on place value x Add and subtract within 1,000, using strategies and algorithms based on properties of operations x Add and subtract within 1,000, using strategies and algorithms based on the relationship between

addition and subtraction x Multiply one-digit whole numbers by multiples of 10 in the range 10–90, using strategies based on

place value and properties of operations x Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80; 5 × 60)

3.2.3 Explain the relationship between multiplication and division, and apply properties of multiplication Sample Performance Indicators: x Explain the relationship between multiplication and division: o Apply commutative property of multiplication (e.g., if 6 × 4 = 24 is known, then 4 × 6 = 24 is also

known) o Apply associative property of multiplication (e.g., if 3 × 5 × 2 can be found by first 3 × 5 = 15 then

15 × 2 = 30, or, by first 5 × 2 = 10 then 3 × 10 = 30) o Apply distributive property of multiplication [e.g., 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] x Relate division to multiplication as an unknown-factor problem (e.g., find 32 ÷ 8 by finding the number

that makes 32 when multiplied by 8) x Multiply and divide within 100, using strategies such as the relationship between multiplication and

division or properties of operations (e.g., show that if 8 × 5 = 40, then 40 ÷ 5 = 8) x Recite or recall all products of two one-digit numbers x Multiply a 2- to 4-digit number by a 1-digit number


Standard 3: Students investigate patterns, functions and quantitative relationships and apply that knowledge to modeling, analyzing or solving mathematical and real-word problems. 3.3.1 Identify and explain arithmetic patterns using properties of operations

Sample Performance Indicators: x Identify and explain arithmetic patterns using properties of operations (e.g., observe that 4 times a

number is always even, and explain why 4 times a number can be decomposed into two equal


MATH GRADE 3

addends) x Describe and extend patterns to determine the next term in a linear or geometric pattern

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and analyze mathematical situations and structures. 3.4.1 Describe and represent relationships of quantities using mathematical expressions, equations, and

inequalities, and solve two-step word problems using the four operations Sample Performance Indicators: x Represent relationships of quantities using equations or inequalities, with letters standing for the

unknown quantities x (E) Solve two-step word problems using the four operations x (E) Assess the reasonableness of the answers by mentally computing answers and comparing with

the solutions x (E) Assess and explain the reasonableness of the answers by estimating answers to whole numbers,

including rounding


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. 3.5.1 Draw and identify lines and angles

Sample Performance Indicators: x Draw and label points, lines, and line segments x Identify angles using objects in the classroom (e.g., clock, scissors, chairs, desks) x Draw angles and describe their differences

3.5.2 Identify and classify relationships among geometric shapes or objects with shared attributes Sample Performance Indicators: x Describe how 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)

x Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space. 3.6.1 Recognize and describe concept of area as an attribute of plane figures and use it for measurement

Sample Performance Indicators: x Describe a unit square as a square with side length 1 unit to have one square unit of area x (E) Define what an area of n square units is (i.e., a plane figure which can be covered without gaps or

overlaps by n unit squares) x Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised

units) x Express the area of each part as a unit fraction of the whole (e.g., partition a shape into 4 parts with

equal areas, and describe the area of each part as 1/4 of the area of the shape)

3.6.2 Calculate the perimeter of polygons, area of geometric shapes (unit squares), and volume of geometric shapes (unit cubes) by relating area to the operations of multiplication and addition Sample Performance Indicators: x Find the perimeter, given the side lengths; or, find an unknown side length and exhibit rectangles with


MATH GRADE 3

the same perimeter and different areas, or with the same area and different perimeters x Solve real-world and mathematical problems involving perimeters of polygons x 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 x Use tiling to show a concrete example that the area of a rectangle with whole-number side lengths a

and b + c is the sum of a × b and a × c x Recognize area as additive, and find areas of rectilinear figures by decomposing them into non-

overlapping rectangles and adding the areas of the non-overlapping parts, and apply this technique to solve real-world problems (e.g., tiles needed for the classroom floor or a countertop)

3.6.3 Solve problems involving measurement and estimation of liquid volumes and masses of objects Sample Performance Indicators: x Measure and estimate liquid volumes and masses of objects using standard units of grams (g),

kilograms (kg), and liters (l) x 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 to represent the problem, such as a beaker with a measurement scale)

3.6.4 Solve problems involving measurement and estimation of intervals of time Sample Performance Indicators: x Tell and write time to the nearest minute, in analog or digital x Measure time intervals in minutes x Solve word problems involving addition and subtraction of time intervals in minutes (e.g., by

representing the problem on a number line diagram)


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. 3.7.1 Generate, collect, organize, represent, and interpret a simple data set with several categories

Sample Performance Indicators: x Create a survey (e.g., in a group or with a partner), and collect data x Present data using graphs (e.g., bar, pictograph), charts, and/or tables x Generate data by measuring lengths using rulers marked with halves and fourths of an inch x Display measured data by making a line plot, where the horizontal scale is marked off in appropriate

units (whole numbers, halves, or quarters) x Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories x Solve one- and two-step “how many more” and “how many less” problems, using information

presented in scaled bar graphs (e.g., draw a bar graph in which each square in the bar graph might represent five items)

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. 3.8.1 Determine possible outcomes of simple experiments using charts and tables

Sample Performance Indicators: x Determine possible outcomes of a simple experiment x Collect data through a simple experiment (e.g., tossing a coin), and organize data using tables, charts,

and graphs or tree diagrams


MATH GRADE 3


Language Domain




Listening

Directions; Instructions

Follow multistep oral instructions in various contexts of math lessons (e.g., working in small groups or with partners)

Follow a series of oral commands for data collection activity, supported by gestures or realia

Copy responses to one-step oral directions for data collection activity, supported by gestures or realia, and working with a partner

Speaking

Feelings; Emotions; Needs

Discuss key concepts about fractions that still need clarification, using extended sentences

Share in small groups key concepts about fractions that still need clarification, using phrases

Identify key concepts about fractions that still need clarification, working with a partner, and using Samoan and English

Reading

Messages; Information

Read survey questions created by peers, and answer independently

Read and answer survey questions that are created by peers and are visually supported with illustrations and examples, working in small groups

Identify key words from survey questions created by peers (e.g., “favorite food” in “What is your favorite food?”), and give an answer, working with a partner

Writing

Social, cultural, and school traditions

Create illustrated writing to describe traditional ways to tell time and to measure intervals of time, using key vocabulary and extended sentences

Describe traditional ways to tell time and to measure intervals of time, using some key vocabulary and simple sentences

List key words to show traditional ways to tell time and to measure intervals in time


MATH GRADE 3

FINAL July 2012


Language Domain




Listening

Time Respond to more complex oral questions (e.g., “John eats breakfast from six fifteen to six twenty-five. How many minutes did it take for him to eat breakfast?”), and show time interval on clock

Respond to oral questions (e.g., “What is ten minutes after six fifteen?”) and show time intervals on clock, working in small groups

Shade or show time intervals on clock based on oral descriptions (e.g., “from six fifteen to six twenty-five”)

Speaking

Whole numbers; Estimation

Explain the rationale behind the rounding of particular whole numbers to the nearest 10 or 100 (e.g., rationale behind 68 becoming 70 if rounding to the nearest 10)

Use sequential signal words, key vocabulary (round up, round down), and simple sentences to describe the process of rounding whole numbers to the nearest 10 or 100

State key vocabulary related to estimation and whole numbers (e.g., rounding, round up, round down) that correspond to number examples shown on cards, working with a partner

Reading

Basic operations

Apply strategies to multiply and divide to solve math problems

Match strategies to multiply and divide with examples from illustrated texts

Identify key words for strategies to multiply and divide from illustrated texts, working with a partner

Writing

Two- and three- dimensional shapes

Write an illustrated paragraph to describe attributes of different types of quadrilaterals, using key vocabulary and extended sentences

Write to describe attributes of different types of quadrilaterals, using some key vocabulary and sentence frames

Label examples of quadrilaterals (e.g., rhombus, rectangle, square), working with a partner

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404040

MATH GRADE 4

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Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. 4.1.1 Recognize, categorize, compare, and round decimals using place value understanding

Sample Performance Indicators: x Recognize and categorize place values of whole numbers (up to 12 digits); that is, show that in a

multidigit 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

x Read, write, and compare decimals to thousandths using base-ten numerals, number names, and expanded forms (e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × [1/10] + 9 × [1/100] + 2 × [1/1,000])

x Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of the comparison. In addition, o Compare two decimals to hundredths by reasoning about their size o Explain or show why a comparison is valid only when the two compared decimals refer to the

same whole o Justify the conclusion using a visual fraction model

x Round decimals using place value understanding

4.1.2 Multiply and divide decimals by a power of 10 Sample Performance Indicators: x Explain patterns in the number of zeros of the product when multiplying a number by powers of 10 x Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a

power of 10, and use whole-number exponents to denote powers of 10 x (E) Rewrite and identify decimals using decimal notation for fractions with appropriate

denominators 10 or 100 (e.g., rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram)

(E) 4.1.3 Recognize and generate equivalent fractions, and compare two fractions with different numerators and different denominators Sample Performance Indicators: x Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) x Explain (e.g., using visual fraction models) how the number and size of the parts differ even

though two fractions themselves are the same size x Recognize and generate more equivalent fractions using the principle above x 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) x Compare two decimals to hundredths by reasoning about their size x Explain or show why a comparison is valid only when the two compared fractions refer to the



MATH GRADE 4

same whole x Record the result of such comparisons with symbols >, =, or <, and use a visual fraction model to

justify the conclusions

Standard 2: Students perform mathematical operations and determine how they relate to one another. 4.2.1 Become familiar with factors and multiples of whole numbers, and solve problems using the four

operations with whole numbers Sample Performance Indicators: x Recognize that a whole number is a multiple of each of its factors x Find all factor pairs for a whole number in the range 1–100 x Determine whether or not a given whole number in the range 1–100 is a multiple of a given one-

digit number x Determine whether or not a given whole number in the range 1–100 is prime or composite x 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), and represent verbal statements of multiplicative comparisons as multiplication equations

x 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)

x Solve multistep word problems posed with whole numbers and having whole-number answers, using the four operations or other strategies, including problems in which remainders must be interpreted; represent these problems using equations with a letter standing for the unknown quantity; and assess the reasonableness of answers using mental computation and estimation strategies, including rounding

4.2.2 Perform multidigit arithmetic using place value understanding and properties of operations Sample Performance Indicators: x Add and subtract multidigit whole numbers using the standard algorithm x Multiply a whole number of up to four digits by a 1-digit whole number, and multiply two 2-digit

numbers, using strategies based on place value and the properties of operations x Find whole-number quotients and remainders with up to 4-digit dividends and 1-digit divisors,

using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division

x Illustrate and explain any of the above calculations using equations, rectangular arrays, and/or area models

4.2.3 Multiply a fraction by a whole number, using visual fraction models and equations Sample Performance Indicators: x Explain a fraction a/b as a multiple of 1/b [e.g., 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)] x Explain a multiple of a/b as a multiple of 1/b, which in general is n × (a/b) = (n × a)/b, and use this

understanding to multiply a fraction by a whole number [e.g., use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5]

x Solve word problems involving multiplication of a fraction by a whole number, using visual fraction models and equations to represent the problem (e.g., 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 the answer lie?)

4.2.4 Add and subtract fractions using like denominators Sample Performance Indicators:


MATH GRADE 4

x Describe a fraction a/b with a > 1 as a sum of fractions 1/b x Explain addition and subtraction of fractions as joining and separating parts referring to the same

whole x (E) Express a fraction with the denominator 10 as an equivalent fraction with the denominator

100, and use this technique to add two fractions with respective denominators 10 and 100 (e.g., express 1/10 as 10/100, and 1/10 + 7/100 = 17/100)

x Add and subtract fractions with like denominators x Decompose a fraction into a sum of fractions with the same denominator in more than one way,

record each decomposition by an equation, and justify the decomposition (e.g., 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)

x 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)

x 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)


Standard 3: Students investigate patterns, functions and quantitative relationships and apply that knowledge to modeling, analyzing or solving mathematical and real-word problems. 4.3.1 Generate and analyze numerical and geometrical patterns

Sample Performance Indicators: x Generate a number or shape a pattern that follows a given rule x Identify and explain apparent features of the pattern that were not explicit in the rule itself (e.g.,

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, and explain informally why the numbers will continue to alternate in this way)

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and analyze mathematical situations and structures. 4.4.1 Explain how a change in one quantity (or variable) can produce a change in another quantity (or

variable) Sample Performance Indicators: x Represent measurement quantities using diagrams, such as number line diagrams that feature a

measurement scale x Apply rules using functions, and create relevant function tables


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. 4.5.1 Discuss the concepts of angles and measure of angles, draw and identify lines and angles, and

classify shapes by properties of their lines and angles Sample Performance Indicators: x Describe angle as formed wherever two rays share a common endpoint x Describe angle as measured with reference to a circle with its center at the common endpoint of

the rays, and recognize that an angle that turns through 1/360 of a circle is called a one-degree angle, and can be used to measure angles; an angle that turns through n one-degree angles is said to have an angle measurement of n degrees


MATH GRADE 4

x Describe and classify angles (right, acute, and obtuse) according to their measurements x Draw points, lines, line segments, rays, angles, and perpendicular and parallel lines, and identify

these in 2-D figures

4.5.2 Classify shapes by properties of their lines and angles and identify and draw lines of symmetries Sample Performance Indicators: x Classify 2-D figures based on the presence or absence of parallel or perpendicular lines, or the

presence or absence of angles of a specified size x Recognize right triangles as a category, and identify right triangles x Form questions about figures (e.g., “Which figures have parallel or perpendicular lines?”; “Which

figures have angles of a specified size?”) x Recognize the property of symmetry in relation to points, lines, rays, angles, parallel lines and

planes, perpendicular lines and planes, and identify relationships on the road map x Recognize a line of symmetry for a 2-D figure as a line across the figure such that the figure can

be folded along the line into matching parts x Identify line-symmetric figures, and draw lines of symmetry

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space. 4.6.1 Measure or sketch angles in whole-number degrees using a protractor

Sample Performance Indicators: x Measure and draw a specified angle using a protractor x Sketch a specified angle without the use of a protractor x Recognize angle measure as additive (i.e., 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) x 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)

4.6.2 Recognize area as an attribute of plane figures, and understand concepts of area measurement Sample Performance Indicators: x Describe a unit square as a square with side length 1 unit to have one square unit of area x Recognize that 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 x Apply the area and perimeter formulas for rectangles in real-world and mathematical problems

(e.g., 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)

4.6.3 Solve problems involving measurement and conversion of measurements within a single system of measurement, and from a larger unit to a smaller unit Sample Performance Indicators: x Determine relative sizes of measurement units within one system of units (including km, m, cm; kg,

g; lb, oz.; l, ml; hr, min, sec) x Express measurements in a larger unit in terms of a smaller unit, using a single system of

measurement x Record measurement equivalents in a two-column table (e.g., know that 1 ft is 12 times as long as

1 in, and express the length of a 4 ft snake as 48 in) x Generate a conversion table for feet and inches listing the number pairs (e.g., [1, 12], [2, 24], or [3,

36]) x Solve word problems involving distances, liquid volumes, masses of objects, and money, using the

four operations, including problems involving simple fractions or decimals, and problems that


MATH GRADE 4

require expressing measurements given in a larger unit in terms of a smaller unit x Represent measurement quantities using diagrams such as number line diagrams that feature a

measurement scale

4.6.4 Measure and estimate intervals of time, and relate them to appropriate events Sample Performance Indicators: x Tell and write time x Measure time intervals, and relate them to appropriate events x Use the four operations to solve word problems involving intervals of time in various units (e.g.,

seconds, hours, days, weeks, months, years)


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. 4.7.1 Analyze outcomes of simple experiments using charts and tables, and make a line plot to display a

data set of measurements in fractions of a unit Sample Performance Indicators: x Collect data through a simple experiment (e.g., survey); organize data using pictographs, tables,

charts, and bar graphs; and analyze, compare, interpret, and discuss outcomes x Make a line plot to display a data set of measurements, in fractions of a unit (1/2, 1/4, 1/8), and

use the line plot to find and interpret specific information about the data x Solve problems involving addition and subtraction of fractions by using information presented in

line plots (e.g., the difference in lengths between the longest and shortest specimens in an insect collection)

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. 4.8.1 Represent in an organized way all possible outcomes for a simple probability situation

Sample Performance Indicators: x List all possible outcomes of a simple experiment x Determine the outcomes of a simple event in a clear and organized manner x Collect data through a simple experiment (e.g., survey); organize data using tables, charts, and

graphs or tree diagrams; and predict and draw conclusions about the data


MATH GRADE 4


Language Domain




Listening

Body language; Gestures

Demonstrate cross-cultural communication with nonverbal behavior in different learning contexts in the math classroom

Demonstrate examples of nonverbal behavior common to different cultures in response to oral explanations of math problems

Act out nonverbal behavior common to own cultures in response to oral explanations of math problems by peers, working in small groups

Speaking

Tone; Volume; Dress; Intonation; Register

Describe and explain place value of multidigit whole numbers with adjusted speech patterns appropriate to the audience and situation

Describe place value of multidigit whole numbers using some appropriate speech patterns according to the audience and situation

Repeat words and phrases used to describe place value of multidigit whole numbers according to the audience or situation

Reading

Strategies Identify key information and questions in decontextualized math word problems, with use of mnemonic devices

Identify key information and questions with use of mnemonic devices (e.g., QTIPS), and solve illustrated word problems, working in small groups

Use illustrations to identify key information and questions in word problems, working in small groups

Writing

Interests; Opinions; Preferences; Wishes

Write and evaluate survey questions, using key vocabulary and extended sentences

Write and illustrate survey questions, using some key vocabulary and simple sentences

List topics for survey questions, working with a partner in Samoan and English


MATH GRADE 4

FINAL July 2012


Language Domain




Listening

Patterns; Relations; Functions

Identify rules and what comes next in shape or number patterns, based on oral descriptions

Draw shape or number patterns and identify rules, based on oral descriptions

Point to graphical representations that show shape or number patterns described in oral statements, working with a partner

Speaking

Basic operations (addition, subtraction, multiplication, division)

Explain ways to apply basic operations to solve problems in everyday situations

Use sequential signal words to outline steps to solve problems using basic operations, working with a partner

Restate descriptions of basic operations with the use of manipulatives (e.g., “Here are three groups of four.”), working in Samoan and English

Reading

Two-dimensional figures; Polygons; Angles

Make inferences on geometric relationships among figures based on a grade-level text

Distinguish figures from visually supported descriptions

Sort figures by characteristics and properties from labeled visuals or objects (e.g., three sides; four angles)

Writing

Data analysis Summarize and apply information to new situations based on results from simple experiments in graphs, tables, or charts

Organize, display, and describe data from simple experiments in graphs, tables, or charts, working in small groups

Label variables or data sets from simple experiments (e.g., survey on favorite hobby) on graphs, tables, or charts, working with a partner

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474747

MATH GRADE 5


Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. 5.1.1 Recognize and categorize the place value system, and use it to compare and round decimals

Sample Performance Indicators: x Recognize and show that, in a multidigit 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 x Read, write, and compare decimals to thousandths using base-ten numerals, number names, and

expanded forms (e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × [1/10] + 9 × [1/100] + 2 × [1/1,000])

x Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons

x Round decimals to any place using place value understanding

5.1.2 Explore properties and patterns in using powers of 10 Sample Performance Indicators: x Explain patterns in the number of zeros of the product when multiplying a number by powers of 10 x Explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a

power of 10 x Express powers of 10 using whole-number exponents

Standard 2: Students perform mathematical operations and determine how they relate to one another. 5.2.1 Perform operations with multidigit whole numbers and with decimals (to hundredths)

Sample Performance Indicators: x Multiply multidigit whole numbers using the standard algorithm x Find whole-number quotients of whole numbers with up to 4-digit dividends and 2-digit divisors,

using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division

x Illustrate and explain the above calculation using arrays or area models x 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

x Divide decimals (up to hundredths) by a whole number using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction

5.2.2 Apply and extend previous understandings of multiplication to multiply fractions and division to divide fractions or whole numbers by a fraction Sample Performance Indicators:

Mathematical Practices 1. Make sense of problems and persevere in solving them. 2. Reason abstractly, logically, and quantitatively. 3. Model with mathematics. 4. Communicate with precision. 5. Use appropriate tools strategically. 6. Construct appropriate mathematical images. 7. Look for and make use of patterns or structure. 8. Develop estimation skills.


MATH GRADE 5

x Interpret the product (a/b) × q as a part of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b using a fraction model; and in general, (a/b) × (c/d) = ac/bd (e.g., [2/3] × 4 = 8/3 and [2/3] × [4/5] = 8/15)

x Create a story context for the products (a/b) × q and a × q ÷ b x Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the

appropriate unit fraction side lengths; 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

x Interpret multiplication as scaling (resizing) and fraction as division of the numerator by the denominator (a/b=a/b)

x Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, by using visual fraction models or equations to represent the problem (e.g., 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)

x Compare 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

x Explain why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (i.e., recognize multiplication by whole numbers greater than 1 as a familiar case)

x Explain why multiplying a given number by a fraction less than 1 results in a product smaller than the given number, and relate the principle of fraction equivalence a/b = (n × a)/(n × b) to the effect of multiplying a/b by 1

x Solve real-world problems involving multiplication of fractions and mixed numbers (e.g., using visual fraction models or equations to represent the problem)

5.2.3 Use equivalent fractions as a strategy to add and subtract fractions Sample Performance Indicators: x 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; in general, a/b + c/d = (ad + bc)/bd (e.g., 2/3 + 5/4 = 8/12 + 15/12 = 23/12)

x Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators. Apply these strategies were appropriate (e.g., by using visual fraction models or equations to represent the problem.)

x Estimate mentally any addition or subtraction involving fractions, and assess the reasonableness of the answers (e.g., recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2)

5.2.4 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions Sample Performance Indicators: x Interpret division of a unit fraction by a nonzero whole number, and compute such quotients (e.g.,

create a story context for [1/3] ÷ 4; use a visual fraction model to show the quotient; and use the relationship between multiplication and division to explain that [1/3] ÷ 4 = 1/12 because [1/12] × 4 = 1/3)

x Interpret division of a whole number by a unit fraction, and compute such quotients (e.g., create a story context for 4 ÷ [1/5], and use a visual fraction model to show the quotient, and use the relationship between multiplication and division to explain that 4 ÷ [1/5] = 20 because 20 × [1/5] = 4)

x Solve real-world problems involving division of unit fractions by nonzero whole numbers and division of whole numbers by unit fractions, using visual fraction models and equations to represent the problem (e.g., how much chocolate will each person get if 3 people share 1/2 lb of


MATH GRADE 5

chocolate equally? How many 1/3 cup servings are in 2 cups of raisins?)


Standard 3: Students investigate patterns, functions and quantitative relationships and apply that knowledge to modeling, analyzing or solving mathematical and real-word problems. 5.3.1 Generate and analyze a pair of patterns, and graph them on a coordinate plane

Sample Performance Indicators: x Generate two numerical patterns using two given rules x Identify apparent relationships between corresponding terms x Form ordered pairs consisting of corresponding terms from the two patterns, graph the ordered

pairs on a coordinate plane, and explain (e.g., 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; observe that the terms in one sequence are twice the corresponding terms in the other sequence; and explain informally why this is so)

5.3.2

Solve real-world mathematical problems by graphing points on the coordinate plane Sample Performance Indicators: x Define a coordinate system using a pair of perpendicular lines (axes) that intersect at the origin x Use the convention that the names of the two axes and the coordinates correspond to determine

the x- and y-coordinates and their respective distance from the origin x Write and determine a number pair for a point, and locate the point for a given number pair, and

use a grid map with given locations and directions (i.e., students identify ordered pairs) x 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 x State how far the first number is from the origin in the direction of one axis, and how far the

second number is from 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)

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and analyze mathematical situations and structures. 5.4.1 Use variables to explain how one quantity changes as a function of another

Sample Performance Indicators: x Represent measurement quantities using diagrams such as number line diagrams that feature a

measurement scale x Apply rules using functions and create relevant function tables

5.4.2 Write and interpret numerical expressions Sample Performance Indicators: x Evaluate expressions involving parentheses, brackets, or braces in numerical expression x Write simple expressions that record calculations with numbers, and interpret numerical

expressions without evaluating them (e.g., express the calculation “add 8 and 7, then multiply by 2” as 2 × [8 + 7], and recognize that 3 × [18932 + 921] is three times as large as 18932 + 921, without having to calculate the indicated sum or product)


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. 5.5.1 Draw and identify lines and angles, and classify shapes by properties of their lines and angles

Sample Performance Indicators:


MATH GRADE 5

x Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines, and identify these in 2-D figures

x Classify 2-D 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

5.5.2 Identify and draw lines of symmetries, and classify 2-D figures into categories based on their properties Sample Performance Indicators: x Recognize and identify line-symmetric figures, and draw lines of symmetry for a 2-D figure as a

line across that figure such that the figure can be folded along the line into matching parts x Define properties of spheres, cubes, cones, pyramids, cylinders, and polygons x Create models by using cut out shapes already learned, and explain using the names of shapes x Create different models using origami x Classify 2-D figures in a hierarchy based on properties, and discuss

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space. 5.6.1 Convert like measurement units within a given measurement system

Sample Performance Indicators: x Convert among different-sized standard measurement units within a given measurement system

(e.g., convert 5 cm to 0.05 m) x Solve multistep, real-world problems, involving measurements and unit conversions

5.6.2 Recognize area and perimeter as attributes of plane figures and volume as an attribute of solid figures, and explain concepts of area, perimeter, and volume measurements Sample Performance Indicators: x Apply the area and perimeter formulas for rectangles in real-world and mathematical problems

(e.g., find the width of a rectangular room given the length and area of the flooring) x Apply the area and perimeter formulas for rectangles in real-world and mathematical problems

(e.g., 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)

x Explain a unit cube as one-cube unit of volume of side length 1 unit x Measure volumes by counting unit cubes using cubic cm, cubic m, and cubic ft and improvised

units

5.6.3 Relate volume to the operations of multiplication and addition, and solve real-world and mathematical problems involving volume Sample Performance Indicators: x 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

x Represent threefold whole-number products as volumes (e.g., to represent the associative property of multiplication)

x Apply the formulas V = l × w × h and V = 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

x (E) 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, and apply this technique to solve real-world problems


MATH GRADE 5


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. 5.7.1 Collect, organize, read, and display a data set of measurements in charts, graphs, and tables

Sample Performance Indicators: x Represent, compare, and interpret a set of data using tables, charts, and graphs (e.g., line, circle,

bar graph) to represent, compare, and interpret a set of data x Make a line plot to display a data set of measurements in fractions of a unit (1/2,1/4, 1/8), and use

the line plot to find and interpret specific information about the data (e.g., 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 was redistributed equally)

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. 5.8.1 Conduct simple experiments where probabilities of all outcomes are equal

Sample Performance Indicators: x List all possible outcomes, and explain reasoning x Determine outcomes of a simple event in a clear and organized manner x Predict possible outcomes for similar experiments


MATH GRADE 5





Listening

Body language; Gestures

Demonstrate cross-cultural communication with nonverbal behavior in different learning contexts in the math classroom

Demonstrate examples of nonverbal behavior common to different cultures in response to oral explanations on strategies used to perform operations with decimals

Act out nonverbal behavior common to own cultures in response to oral explanations by peers on strategies used to perform operations with decimals

Speaking

Tone; Volume; Dress; Intonation; Register

Discuss outcomes of simple experiments and predict possible outcomes for similar experiments with adjusted speech patterns appropriate to the audience and situation

Discuss outcomes of simple experiments using some appropriate speech patterns according to the audience and situation

Repeat words and phrases used to discuss outcomes of simple experiments according to the audience or situation

Reading

Strategies Identify key information and questions in decontextualized math word problems that involve fractions

Refer to key vocabulary in illustrated word problems involving fractions to make meaning, working with a partner

Use illustrations to identify key information and questions in word problems that involve fractions, working in small groups

Writing

Interests; Opinions; Preferences; Wishes

Evaluate pros and cons of using different strategies to solve multiplication or division problems with decimals

Describe the rationale behind choosing particular strategies to solve multiplication or division problems with decimals

List and illustrate key vocabulary to show the rationale behind choosing particular strategies to solve multiplication or division problems with decimals


MATH GRADE 5

FINAL July 2012


Language Domain




Listening

Patterns; Relations; Functions

Identify the relationship between corresponding terms, based on oral description

Draw representations that show the relationship between corresponding terms, based on oral description

Point to graphical representations that show the relationship between corresponding terms described in oral statements, working with a partner

Speaking

Basic operations (addition, subtraction, division, multiplication)

Explain ways to apply basic operations to solve problems in everyday situations

Use sequential signal words to outline steps to solve problems using basic operations, working with a partner

Restate descriptions of basic operations with use of manipulatives (e.g., “Here are three groups of four.”) in Samoan and English, working with a partner

Reading

Two-dimensional figures; Polygons; Angles

Make inferences about geometric relationships among figures based on grade-level text

Distinguish figures from visually supported descriptions

Sort figures by characteristics and properties from labeled visuals or objects (e.g., three sides, four angles)

Writing

Data analysis Summarize and apply to new situations information based on results from simple experiments in graphs, tables, or charts

Organize, display, and describe data from simple experiments in graphs, tables, or charts, working in small groups

Label variables or data sets from simple experiments (e.g., survey on favorite hobby) on graphs, tables, or charts, working with a partner

545454

MATH GRADE 6


Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems.

6.1.1 Determine the ratio between two quantities, the unit rate a/b associated with a ratio a:b with b � 0, and use rate language (with appropriate units) in the context of a ratio relationship Sample Performance Indicators: x Write ratios in their simplest form

x Write ratios and rates, and find unit rates (e.g., “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.”)

x Compare and order ratios (e.g., “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.”)

6.1.2 Apply and extend previous understandings of numbers to the system of rational numbers Sample Performance Indicators:

x Show examples where positive and negative numbers are used together to describe quantities having opposite directions or values

x Identify a rational number as a point on the number line, and then extend number line diagrams and coordinate axes to represent points on the line and in the plane with negative number coordinates

Standard 2: Students perform mathematical operations and determine how they relate to one another.

6.2.1 Compute fluently with multidigit numbers, and find common factors and multiples Sample Performance Indicators: x Divide multidigit numbers fluently using the standard algorithm

x Add, subtract, multiply, and divide multidigit decimals fluently using the standard algorithm for each operation

x 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

x Express (using the distributive property) 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 [e.g., express 36 + 8 as 4 (9 + 2)]

6.2.2 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions Sample Performance Indicators:



MATH GRADE 6

x Interpret and compute quotients of fractions (e.g., by using visual fraction models)

x Solve word problems involving division of fractions by fractions, using equations to represent the problems (e.g., create a story context for [2/3] ÷ [3/4] and use a visual fraction model to show the quotient)

x Discuss the relation between multiplication and division of the general form (a/b) ÷ (c/d) = ad/bc (e.g., explain that [2/3] ÷ [3/4] = 8/9 because 3/4 of 8/9 is 2/3 such as, “How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?”; and “How many 3/4 cup servings are in 2/3 of a cup of yogurt?”; and “How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mile?”)

6.2.3 Use ratio and rate reasoning to solve real-world and mathematical problems Sample Performance Indicators: x Make tables of equivalent ratios relating quantities with whole-number measurements, find

missing values in the tables, compare ratios using tables, and plot the pairs of values on the coordinate plane

x Solve unit rate problems including those involving unit pricing and constant speed (e.g., if it takes 7 hours to mow 4 lawns, then, at that rate, “How many lawns can be mowed in 35 hours?”; and “At what rate are lawns being mowed?”)

x Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity), and solve problems involving finding the whole, given a part and the percent

x Convert measurement units using ratio reasoning, and manipulate and transform units appropriately when multiplying or dividing quantities


Standard 3: Students investigate patterns, functions and quantitative relationships and apply that knowledge to modeling, analyzing or solving mathematical and real-word problems.

6.3.1 Write, read, and evaluate expressions in which letters stand for numbers, and analyze tables, graphs, and rules to determine functional relationships Sample Performance Indicators:

x Write and evaluate numerical expressions involving whole-number exponents

x Write expressions that record operations with numbers and with letters standing for numbers (e.g., express the calculation “Subtract y from 5” as 5 – y)

x Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient), and view one or more parts of an expression as a single entity (e.g., describe the expression 2 [8 + 7] as a product of two factors and view [8 + 7] as both a single entity and a sum of two terms)

x Evaluate expressions at specific values of their variables, including expressions that arise from formulas used in real-world problems

x Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order or order of operations (e.g., use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2 unit)

x (+) Analyze tables, graphs, and rules to determine functional relationships, and explain process

x Interpret data to write an equation for a function, and solve the equation to find missing values

x Represent linear functions using ordered pairs and graphs

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and analyze mathematical situations and structures. 6.4.1 Generate equivalent expressions using properties of operations, use variables to represent numbers,


MATH GRADE 6

and write expressions when solving a real-world or mathematical problem Sample Performance Indicators: x Show or explain how a variable can represent an unknown number, or, depending on the purpose

at hand, any number in a specified set

x Apply the properties of operations to generate equivalent expressions (e.g., apply the distributive property to the expression 3 [2 + x] to produce the equivalent expression [6 + 3x])

x Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them)

6.4.2 Investigate reasoning behind solving an equation or inequality, and explain how inequality relates to familiar constraints and conditions in a real-world or mathematical problem Sample Performance Indicators:

x Explore 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?”

x Determine (using the substitution method) whether a given number in a specified set makes an equation or inequality true

x Solve linear equations and inequalities in one variable with whole-number coefficients and constants with rational number solutions

x Write and solve equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers

x Solve simple algebraic equations expressed in real-world and mathematical problems

x 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; and represent solutions of such inequalities on number line diagrams


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems.

6.5.1 Describe and compare regular or nonregular polygons, and find the angle measures of regular polygons Sample Performance Indicators: x Describe and compare the sides and angles of regular or irregular polygons

x Calculate the sum of interior angles in polygons (e.g., 180° in a triangle, and 360° in a quadrilateral)

6.5.2 Define properties of points, lines, parallel lines, and perpendicular lines, and introduce the basic concept of the Pythagorean theorem Sample Performance Indicators: x Identify and classify properties of points, lines, segments, parallel lines, and perpendicular lines

x Evaluate sums of square roots

x Describe what a right triangle is

x (+) Illustrate relationship of the side-lengths of right triangles using sum of square areas x (+) Infer the basic concept of the Pythagorean theorem

6.5.3 Identify and analyze properties of, and relationships among, acute, right, and obtuse triangles Sample Performance Indicators: x Construct and compare triangles in different orientations, including scalene, isosceles, equilateral,

right, obtuse, and acute triangles

x Classify triangles by side lengths and angle measures


Sofai Tuatoo

Text

MATH GRADE 6

x Describe properties of and relationships among acute, right, and obtuse triangles

x Solve problems involving angles and side measures of triangles

Standard 6: Students explore concepts and practical applications of units, geometrical constructions, and measurements of plane and geometric shapes and space.

6.6.1 Describe measurable attributes of objects and the units, systems, and processes of measurement Sample Performance Indicators:

x Demonstrate understanding of both metric and customary systems of measurement

x Convert units between simple metric and customary measurements

x Select appropriate methods for estimating measurements

6.6.2 Use the coordinate plane to determine transformation and side-length of a polygon Sample Performance Indicators:

x Draw polygons in the coordinate plane given the coordinates for the vertices

x Find the length of a side joining the points with the same first or second coordinates in a coordinate plane

x Apply coordinate techniques in the context of solving real-world and mathematical problems

x Perform a combination of translation(s), rotation(s), and/or reflection(s) on a single 2-D shape, with and without technology, and draw and describe the image

x Perform a combination of successive transformations of 2-D shapes to create a design, and identify and describe the transformations

x Identify and plot points in the first quadrant of a Cartesian plane using whole-number ordered pairs

x Perform and describe single transformations of a 2-D shape in the first quadrant of a Cartesian plane

6.6.3 Use representations of real-world and mathematical problems to solve area, surface area, and volume problems Sample Performance Indicators: x Develop and apply the area of right triangles, other triangles, special quadrilaterals, and polygons

by composing into rectangles or decomposing into triangles and other shapes

x Apply techniques of composing and decomposing in the context of solving real-world and mathematical problems

x 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

x Represent 3-D figures using nets made up of rectangles and triangles, and find the surface area of these figures

x Apply the described techniques in the context of solving real-world and mathematical problems

x (+) Find the perimeter, area, surface area, and volume of a geometric shape using the appropriate formulas (e.g., develop and apply a formula for determining the perimeter of polygons, the area of rectangles, and the volume of right rectangular prisms)


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data.

6.7.1 Develop understanding of statistical variability Sample Performance Indicators: x (+) Identify a statistical question as one that anticipates variability in the data related to the

question and accounts for it in the answers (e.g., “How old am I?” is not a statistical question, but


MATH GRADE 6

“How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages)

x Explain or show how a set of data collected to answer a statistical question has a distribution that can be described by its center, spread, and overall shape

x Recognize that for a numerical data set a measure of center summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number

6.7.2 Summarize numerical data sets in relation to their context Sample Performance Indicators:

x Display numerical data in plots on a number line, including dot plots, histograms, and box plots

x Describe or determine quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describe any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered

x Relate the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations.

6.8.1 Use theoretical or experimental probability to explain the outcome of a situation in real life Sample Performance Indicators: x List, determine, and predict possible outcomes of a simple event or experiment

x Explain the odds of a desired outcome using everyday examples and identify whether the outcome is certain, impossible, likely, or unlikely


MATH GRADE 6


Language Domain




Listening

School behavior and activities

Follow multistep oral instructions for different learning contexts in the math classroom

Follow multistep oral directions for individual and groups tasks in the math classroom, supported by gestures and visuals

Follow one-step oral instructions for math tasks (e.g., “Explain your calculations to a partner, and check your answers using a calculator.”), supported by visuals or gestures

Speaking

School life; Social interaction

Express humor and sarcasm in everyday conversations (e.g., “Sure I plan to study all weekend for the test on the Pythagorean theorem.”)

Engage in everyday conversations with peers in small groups (e.g., “Please tell me tonight’s math assignment.”)

Respond to and offer introductions, compliments, and farewells in everyday conversations with peers in the math classroom

Reading

Information gathering at school or at home; Multiculturalism

Compare and contrast between finding areas, surface area, and volume using mathematical formulas and traditional methods from multiple diverse sources

Identify main ideas and examples about finding area, surface area, and volume using mathematical formulas and traditional methods from illustrated information, working in small groups

Locate key words about finding area, surface area, and volume using mathematical formulas and traditional methods from illustrated and modified texts, working with a partner

Writing

Negotiating solutions to problems; Interpersonal or cultural misunderstand-ings

Write key vocabulary and extended sentences (e.g., “I think the greatest common factor [GCF] you found is incorrect. Recheck the factors for both numbers.”) to provide descriptive feedback on peer’s work to find GCF

Write key vocabulary and simple sentences to provide descriptive feedback (e.g., “I think this is incorrect. I got a different number.”) on peer’s work to find GCF, using support such as a reference sheet and sentence frames

List key vocabulary (e.g., factor) to provide feedback on peer’s work to find GCF, using support such as a reference sheet


MATH GRADE 6

FINAL July 2012


Language Domain




Listening

Statistics; Metric units; Measures of central tendency

Evaluate and use appropriate graphical representations based on oral descriptions of real-life situations to solve statistical problems with metric units and measures of center

Draw visual or graphic representations of metric unit and measure of center based on oral descriptions, working with a partner

Select visual or graphical representations of metric units and measure of center based on oral descriptions

Speaking

Area; Volume; Two- and three- dimensional shapes

Explain differences in finding area, surface area, and volume of 2-D and 3-D figures using specialized or technical vocabulary

Describe dimensions of 2-D and 3-D figures or diagrams using some specialized vocabulary (e.g., “You multiply length, height, and width.”)

Name 2-D and 3-D figures from real objects or diagrams

Reading

Data sets; Plots; Data interpretation

Interpret data sets presented in dot plots, histograms, or box plots from modified grade-level text

Make meaning of data sets presented in dot plots, histograms, or box plots, working with a partner

Identify different components (e.g., of a dot plot, histogram, or box plot)

Writing

Probability; Ration

Explain and justify the evaluation of outcomes of everyday situations, based on a variety of scenarios

Evaluate the outcome of everyday situations from illustrated scenarios, using key vocabulary and simple sentences

List key vocabulary (certain, impossible, likely, or unlikely) to evaluate outcomes of everyday situations from illustrated scenarios, working with a partner

616161

MATH GRADE 7


Standard 1: Students investigate numbers, ways of representing numbers, relationships among numbers, and number systems. 7.1.1

Determine the unit rates associated with ratios of fractional quantities measured in like or different units Sample Performance Indicators: x Identify, write, and compare ratios and rates (e.g., 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)x Compute unit rates associated with ratios of lengths, areas, and other quantities measured in like

or different units

7.1.2 Recognize and represent proportional relationships between quantities Sample Performance Indicators: x 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)

x Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships

x 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

7.1.3 Apply and extend previous understandings of rational numbers to the coordinate system Sample Performance Indicators: x Identify and represent points (including rational numbers) on the real number line and in the plane

with negative number coordinates x Order rational numbers using positions on the real number line, and find their absolute values x Solve real-world and mathematical problems by graphing points in all four quadrants of the

coordinate plane, and include the use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate

Standard 2: Students perform mathematical operations and determine how they relate to one another. 7.2.1 Apply and extend previous understandings of addition and subtraction to add and subtract rational

numbers Sample Performance Indicators: x Describe situations in which opposite quantities combine to make 0 (e.g., a hydrogen atom has 0

charge because its two constituents are oppositely charged) x Show 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 x Show that a number and its opposite have a sum of 0 (i.e., additive inverses)



MATH GRADE 7

x Interpret sums of rational numbers by describing real-world contexts x Subtract rational numbers by adding the additive inverse, p – q = p + (–q) x 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 x Represent addition and subtraction on a horizontal or vertical number line diagram

7.2.2 Apply and extend previous understandings of multiplication and division, and of fractions, to multiply and divide rational numbers Sample Performance Indicators: x Show how the multiplication rules for fractions work with rational numbers, and demonstrate the

distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying numbers

x Interpret and describe products of rational numbers in real-world contexts x Show how integers can be divided, provided that the divisor is not zero, and how every quotient of

integers (with a nonzero divisor) is a rational number such that if p and q are integers, then –(p/q) = (–p)/q = p/(–q)

x Interpret quotients of rational numbers by describing real-world contexts x Apply properties of operations as strategies to multiply and divide rational numbers x Convert a rational number to a decimal using long division, and demonstrate knowing that the

decimal form of a rational number terminates in 0s or eventually repeats x Solve real-world and mathematical problems involving the four operations with rational numbers

7.2.3 Use proportional relationships to solve multistep ratio and percent problems Sample Performance Indicators: x Discuss the concepts of simple interest, tax, markups and markdowns, gratuities and

commissions, fees, percent increase and decrease, and percent error x Solve multistep ratio and percent problems using proportional relationships


Standard 3: Students investigate patterns, functions and quantitative relationships and apply that knowledge to modeling, analyzing or solving mathematical and real-word problems. 7.3.1

Solve multistep real-life and mathematical problems posed with positive and negative rational numbers in any form Sample Performance Indicators: x Apply properties of operations to calculate numbers in any form x Convert between forms (whole numbers, fractions, and decimals) as appropriate x Assess the reasonableness of answers using mental computation and estimation strategies (e.g.,

“If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10of 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”), and use the estimate as a check on the exact computation

7.3.2 Model linear relationships as functions between quantities, and analyze and represent them as algebraic equations Sample Performance Indicators: x Construct a function to model a linear relationship between two quantities x Define and interpret unit rate in terms of the situation it models and in terms of its graph or a table

of values x Determine and interpret the rate of change and initial value of the function from a description of a


MATH GRADE 7

relationship, or from two (x, y) values, including reading these from a table or from a graph x Analyze tables, graphs, and rules to determine functional relationships including linear, nonlinear,

and simple exponential relationships x Represent functional relationships as algebraic equations x Generate and graph ordered pairs using function tables x Write linear equations from real-world situations

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and analyze mathematical situations and structures. 7.4.1 Use properties of operations to generate equivalent expressions and represent and analyze

quantitative relationships between variables Sample Performance Indicators: x Apply properties of operations as strategies to add, subtract, factor, and expand linear

expressions with rational coefficients x Rewrite an expression in different forms in a problem context and explain how the quantities in it

are related (e.g., a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05”) x 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 x Represent two quantities in a real-world problem using variables that change in relationship to one

another x Analyze the relationship between the dependent and independent variables using graphs and

tables, and relate these to the equation (e.g., 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)

7.4.2 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 Sample Performance Indicators: x 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, and solve equations of these forms fluently x Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations

used in each approach (e.g., the perimeter of a rectangle is 54 cm, and its length is 6 cm, so what is its width?)

x 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

x Graph the solution set of an inequality, and interpret it in the context of the problem (e.g., as a salesperson, you are paid $50 per week, plus $3 per sale, and this week you want your pay to be at least $100, so write an inequality for the number of sales you need to make, and describe the solutions)


Standard 5: Students explore properties, relationships, and transformations of shapes and space using coordinate geometry and other representational systems. 7.5.1 Use the Pythagorean theorem to find the measure of a side of a right triangle

Sample Performance Indicators: x Describe and represent the Pythagorean theorem using a right triangle x Calculate an unknown length of a right triangle

Standard 6: Students explore concepts and practical applications of units, geometrical constructions,


MATH GRADE 7

and measurements of plane and geometric shapes and space. 7.6.1 Solve real-world and mathematical problems involving angle measure, area, surface area, volume,

and scale drawing Sample Performance Indicators: x Describe features of different polygons x Find the sum of angle measure in polygons x Describe an informal derivation of the relationship between the radius, circumference, and area of

a circle x Solve real-world and mathematical problems using the formulas for the area and circumference of

a circle x Develop and apply formulas to find area, surface area, and volume of 2-D and 3-D objects

composed of triangles, quadrilaterals, polygons, cubes, and right prisms, and discuss x Write and solve simple equations for an unknown angle of a figure in a multistep problem using

facts about supplementary, complementary, vertical, and adjacent angles, and present steps x Compute actual lengths and areas from a scale drawing, and reproduce a scale drawing at a

different scale x Draw, construct, and describe area and circumference of a circle, including use of scale drawings

for geometric figures x Solve problems involving scale drawings of geometric figures


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect, organize, analyze, display, and make inferences based on data. 7.7.1 Use random sampling to draw inferences about a population

Sample Performance Indicators: x Explain how statistics can be used to gain information about a population by examining a sample

of the population x Explain how generalizations about a population from a sample are valid only if the sample is

representative of that population x Explain random sampling and its role in drawing inferences about a population

7.7.2 Assess the degree of visual overlap of two numerical data distributions with similar variabilities, and measure the difference between the centers Sample Performance Indicators: x Informally assess the degree of visual overlap of two numerical data distributions with similar

variabilities, and measure the difference between the centers by using a multiple of a measure of variability (e.g., assess the visual overlap when the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, or is about twice the variability [mean absolute deviation] on either team, and recognize on a dot plot that the separation between the two distributions of heights is noticeable)

x Find the mode, median, and mean of a data set x Draw informal comparative inferences about two populations using measures of center and

measures of variability for numerical data from random samples

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life situations. 7.8.1 Explain the concept of chance using examples with numbers between 0 and 1 that express the

likelihood of the events occurring


MATH GRADE 7

Sample Performance Indicators: x Explain the concept of chance, and give examples (e.g., a probability near 0 indicates an unlikely

event; a probability around 1/2 indicates an event that is neither unlikely nor likely; a probability near 1 indicates a likely event; and larger numbers indicate the greater likelihood of an event)

x Determine the probability of a simple event and identify whether the outcome is certain, impossible, likely, or unlikely

x Distinguish the difference in the probability of sampling, with and without replacement x Explain the odds of a desired outcome using everyday examples

7.8.2 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation Sample Performance Indicators: x Explain 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 x Identify and list the outcomes in a sample space which composes an event described in everyday

language (e.g., rolling double sixes with dice) x Represent the sample spaces for compound events using methods such as organized lists, tables,

and tree diagrams


MATH GRADE 7


Language Domain




Listening

School behavior and activities

Follow multistep oral instructions for different learning contexts in the math classroom

Follow multistep oral directions for individual and groups tasks in the math classroom, supported by gestures and visuals

Follow one-step oral instructions for math class tasks (e.g., “Share your calculations with a partner.”), supported by visuals or gestures

Speaking

School life; Social interaction

Express humor and sarcasm in everyday conversations (e.g., “Sure I plan to study all weekend for the test on transformations.”)

Engage in everyday conversations with peers in small groups (e.g., “Please explain to me the directions of this activity.”)

Respond to and offer introductions, compliments, and farewells in everyday conversations with peers in the math classroom

Reading

Information gathering at school or at home; Multiculturalism

Compare and contrast between metric and traditional systems of measurement from multiple diverse sources

Identify main ideas and examples about metric and traditional systems of measurement from illustrated information, working in small groups

Locate key words about metric and traditional systems of measurement from illustrated and modified text, working with a partner

Writing

Negotiating solutions to problems; Interpersonal or cultural misunderstandings

Write key vocabulary and extended sentences (e.g., “I think the median you calculated is incorrect. Recheck your division”) to provide descriptive feedback on a peer’s work to find measures of center

Write key vocabulary and basic sentences to provide descriptive feedback (e.g., “I think this is incorrect because I got a different number”) on a peer’s work to find measures of center, using support such as a reference sheet and sentence frames

List key vocabulary (e.g., median) to provide feedback on a peer’s work to find measures of center, using support such as a reference sheet


MATH GRADE 7

FINAL July 2012


Language Domain




Listening

Statistics; Metric units; Measures of central tendency

Evaluate and use appropriate graphical representations based on oral descriptions of real-life situations to solve statistical problems with metric units and measures of center and variability

Draw visual or graphic representations of metric units, and measures of center and variability, based on oral descriptions from everyday situations, working with a partner

Select visual or graphical representations of metric units and measures of center and variability, based on oral descriptions from everyday situations, and working with a partner

Speaking

Area; Volume; Two- and three-dimensional shapes

Explain differences in the figures that compose 2-D and 3-D objects, and in finding area, surface area, and volume

Describe figures that compose 2-D or 3-D objects and ways to find area, surface area, and volume, using some specialized vocabulary (e.g., “This figure is made of three triangles. To find the area, you multiply ”)

Name figures that compose 2-D or 3-D objects, and formulas used to find area, surface area, and volume

Reading

Data sets; Plots; Data interpretation

Interpret data presented in texts and plots from modified grade-level text

Make meaning of data sets presented in texts and plots, working with a partner

Identify different components of data sets and plots, working with a partner

Writing

Probability; Ration

Explain and justify the desired outcome of an everyday example with the concept of chance, using specialized vocabulary

Describe the desired outcome of an everyday example with the concept of chance, using some specialized vocabulary and sentence frames

Draw and label the odds of a desired outcome related to an everyday example using the concept of chance

�

686868

MATH GRADE 8�


Standard 1: Students investigate numbers, ways of representing numbers, relationships among

numbers, and number systems.

8.1.1 Classify, express, compare, and perform computations with rational numbers

Sample Performance Indicators: x Define and perform computations, expressing rational numbers as a ratio of two integers with a

nonzero denominator

x Order or compare rational numbers (fractions, decimals, percents, or integers) using various

models and representations (e.g., number line)

x Explain and discuss the difference between rational and irrational numbers using examples

x Express a rational number in decimal expansion

x Convert into a rational number a decimal expansion which terminates or eventually repeats

8.1.2 Identify, estimate, and compare sizes of irrational numbers


x Define an irrational number, and show examples

x Compare the size of irrational numbers using rational approximation, and locate them

approximately on a number line diagram

x Estimate the value of expressions using rational approximation of irrational numbers (e.g., 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

Standard 2: Students perform mathematical operations and determine how they relate to one

another.

8.2.1 Apply and extend previous understandings of arithmetic operations to rational or irrational numbers in

solving real-world and mathematical problems

Sample Performance Indicators: x Show how rational numbers are added, subtracted, multiplied, and divided, in writing and orally

x Show how irrational numbers are added, subtracted, multiplied, and divided, and outline steps

x Solve real-world and mathematical problems involving the four operations with rational numbers

8.2.2 Work with radicals and integer exponents fluently

Sample Performance Indicators: x Apply the properties of integer exponents to generate equivalent numerical expressions (e.g.,

) 32 u 3�5 3�3 1/33 1/27x Use square root and cube root symbols to represent solutions to equations of the form x 2 p and

px 3 , where p is a positive rational number (e.g., evaluate square roots of small perfect squares

and cube roots of small perfect cubes, and show why ¥2 is irrational)

Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly, logically, and quantitatively.

3. Model with mathematics.

4. Communicate with precision.

5. Use appropriate tools strategically.

6. Construct appropriate mathematical images.

7. Look for and make use of patterns or structure.

8. Develop estimation skills.


MATH GRADE 8�

x 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 one is more than the other (e.g.,

estimate the population of the United States as 8103u and the population of the world as 9107u ,

and determine that the world population is more than 20 times larger)

x Perform operations with numbers expressed in scientific notation, including problems where both

decimal and scientific notations are used, and explain

x Work with 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)


Standard 3: Students investigate patterns, functions and quantitative relationships and apply that

knowledge to modeling, analyzing or solving mathematical and real-word problems.

8.3.1 Define, evaluate, and compare linear functions

Sample Performance Indicators: x Define function as a rule that assigns to each input exactly one output, and explain how the graph

of a function is the set of ordered pairs consisting of an input and the corresponding output

x Compare properties of two functions each represented in a different way—algebraically,

graphically, numerically in tables, or by verbal descriptions (e.g., 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)

x Interpret the equation y = mx + b as defining a linear function whose graph is a straight line, and

give examples of functions that are not linear (e.g., the function A = s2 defines the area of a square

as a function of its side length and is not linear because its graph contains the points [1,1], [2,4],

and [3,9], which are not on a straight line)

8.3.2 Model relationships between quantities using functions

Sample Performance Indicators: x Construct a function to model a linear relationship between two quantities, and explain

x 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

x 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, and present information

x Describe qualitatively the functional relationship between two quantities by analyzing various

features of a graph (e.g., where the function is increasing or decreasing, linear or nonlinear)

x Sketch a graph that exhibits the qualitative features of a function that has been described verbally

8.3.3 Recognize the connections between proportional relationships, lines, and linear equations


x Graph proportional relationships, interpreting the unit rate as the slope of the graph, and explain

the graph

x Compare two different proportional relationships represented in different ways (e.g., compare a

distance-time graph to a distance-time equation to determine which of two moving objects has

greater speed)

x Explain why the slope m is the same between any two distinct points on a nonvertical line in the

coordinate plane using similar triangles

x 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

Standard 4: Students use symbolic forms, operations, and properties to represent, model, solve, and

analyze mathematical situations and structures.


MATH GRADE 8�

8.4.1 Analyze and solve linear equations and inequalities


x Solve linear equations in one or more variables with whole numbers

x Give examples of linear equations in one variable with one solution, with infinitely many solutions,

or with no solutions

x Solve linear equations with rational number coefficients, including equations whose solutions

require expanding expressions using the distributive property and collecting like terms

x Solve linear inequalities in one or more variables with whole numbers

x Solve multistep equations and inequalities posed with positive and negative coefficients

8.4.2 Analyze and solve pairs of simultaneous linear equations


x Explain how the solutions to a system of two linear equations in two variables correspond to points

of intersection of their graphs

x Solve systems of two linear equations in two variables algebraically, estimate solutions by graphing

the equations, and discuss

x Solve simple cases by inspection (e.g., 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x +

2y cannot simultaneously be 5 and 6)

x Solve real-world and mathematical problems leading to two linear equations in two variables (e.g.,

given coordinates for two pairs of points, determine whether the line through the first pair of points

intersects the line through the second pair)


Standard 5: Students explore properties, relationships, and transformations of shapes and space

using coordinate geometry and other representational systems.

8.5.1 Evaluate and find angles in polygons using properties of a triangle

Sample Performance Indicators: x Find the sum of angle measures in polygons

x Calculate the angles of acute, right, and obtuse triangles

8.5.2 Use and apply the Pythagorean theorem, and its converse, to solve problems


x Explain the proof of the Pythagorean theorem and its converse

x Determine unknown side lengths in right triangles for real-world and mathematical problems in 2-D

or 3-D

x Apply the Pythagorean theorem to find the distance between two coordinate points

8.5.3 Identify parallel and perpendicular lines, and calculate the angles formed by a transversal

Sample Performance Indicators: x Identify and apply the relationships of vertical, adjacent, complementary, and supplementary angles

x Identify congruent angles formed by a transversal, and explain using a diagram

x Calculate angle measures of parallel lines cut by a transversal, and outline the steps

x Identify and apply the relationships of angles to the angles formed by perpendicular lines, and to

angles formed by parallel lines and a transversal

8.5.4 Apply congruence and similarity using physical models


x (E) Verify experimentally the properties of rotations, reflections, and translation, and discuss


MATH GRADE 8�

findings

o Lines are taken to lines, and line segments to line segments of the same length

o Angles are taken to angles of the same measure

o Parallel lines are taken to parallel lines

x Use 2-D transformation to demonstrate congruency between two geometric figures

x Use 2-D transformation (including dilations) to demonstrate similarity between two geometric

figures

x Describe the effect of dilations, rotations, reflections, and translations on 2-D figures, using

coordinates

x 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 about the angle-angle

criterion for similarity of triangles; and explain orally

Standard 6: Students explore concepts and practical applications of units, geometrical constructions,

and measurements of plane and geometric shapes and space.

8.6.1 Solve real-world and mathematical problems involving volumes of geometric containers

Sample Performance Indicators: x Develop and apply formulas to determine volumes of cones, right cylinders, and spheres

x Outline the sequence of steps and talk about the techniques used to develop volume formulas

x Apply appropriate techniques to solve real-world and mathematical problems involving volumes of

geometric containers

8.6.2 Draw geometric shapes with given conditions


x Construct triangles from three measures of angles or sides, and then determine when the

conditions give rise to a unique triangle, more than one triangle, or no triangle

x Describe the 2-D figures that result from slicing 3-D figures (e.g., plane sections of right rectangular

prisms or pyramids)

x Draw and interpret top, front, and side views of 3-D objects composed of right rectangular prisms

8.6.3 Draw, construct, and describe area and circumference of a circle, including use of scale drawing for

geometric figures


x Describe an informal derivation of the relationship between the radius, circumference, and area of a

circle

x Solve problems using the formulas for the area and circumference of a circle, and explain steps

x 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


Standard 7: Students demonstrate statistical thinking and apply statistical methods to collect,

organize, analyze, display, and make inferences based on data.

8.7.1 Describe random processes that underlie statistical experiments


x Explain statistics as a process for making inferences about population parameters based on a

random sample from that population

x Distinguish between correlation and causation using linear model graphs

8.7.2 Estimate the measures of central tendency for a data set


MATH GRADE 8�


x Apply prior understanding of mean, median, mode, and range of a group of objects or numbers to

estimate the measures of central tendency for a data set

x Use measures of center and measures of variability for numerical data from random samples to

draw informal comparative inferences about two populations

8.7.3 Identify or describe the patterns of association in a bivariate categorical data (having two variables),

represented in a two-way table

Sample Performance Indicators: x Construct and interpret a two-way table, summarizing data on two categorical variables collected

from the same subjects

x Use relative frequencies calculated for rows or columns to describe possible association(s) between

the two variables (e.g., 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, and assess if

there is evidence that those who have a curfew also tend to have chores)

x (E) Construct and interpret scatter plots for bivariate measurement data to investigate patterns of

association between two quantities

x (E) Describe patterns such as clustering, outliers, positive or negative association, linear

association, and nonlinear association

x (E) Use the equation of a linear model to solve problems in the context of bivariate measurement

data, interpreting the slope and intercept (e.g., 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)

Standard 8: Students use concepts of chance and probability to explain outcomes of real-life

situations.

8.8.1 Define, determine, and demonstrate probability of events using simple experiments


x Define probability, outcomes, events, and sample space, and use categories or characteristics of

the outcomes as unions, intersections, or complements of other events in the descriptions

x Represent and count outcomes using a tree diagram

x Determine the probability of a simple event, and identify whether the outcome is certain, impossible,

likely, or unlikely

x Demonstrate probability of events using simple experiments, and participate in a gallery

presentation

x Find the probability of a sampling, with or without replacement, and explain reasoning

x Calculate the odds of a desired outcome in a simple experiment


MATH GRADE 8�

An aligned instructional assessment system begins with standards. Standards relate grade or

grade-level cluster expectations for students, teachers, administrators, and other stakeholders.

Educators of language learners must be cognizant of the correspondence between language

and content standards and the influence of a student’s language proficiency on academic

achievement.

Corresponding Representation of Mathematics in English Language Proficiency Standards

Standard 1: English language learners communicate for social, intercultural, and instructional

purposes within the school setting.

Language

Domain

Topic ELP Stage 5:

Advanced

ELP Stage 3:

Intermediate

ELP Stage 1:

Beginning

Listening

School behavior

and activities

Follow multistep oral

instructions for

different learning

contexts in the math

classroom

Follow multistep oral

directions for individual

and groups tasks in the

math classroom,

supported by gestures

and visuals

Follow one-step oral

instructions for math

classroom tasks

(e.g., “Show your

graph to a partner.”),

supported by visuals

or gestures

Speaking

School life;

Social interaction

Express humor and

sarcasm in everyday

conversations (e.g.,

“Sure I am going to

study all weekend for

the probability test.”)

Engage in everyday

conversations with

peers in small groups

(e.g., “Would you like to

go first?”)

Respond to and

offer introductions,

compliments, and

farewells in

everyday

conversations with

peers in the math

classroom

Reading

Information

gathering at school

or at home;

Multiculturalism

Compare and

contrast between

statistics and

traditional ways to

collect and interpret

data from multiple

diverse sources

Identify main ideas and

examples about

statistics and traditional

ways to collect and

interpret data from

illustrated information,

working in small groups

Locate key words

about statistics and

traditional ways to

collect and interpret

data from illustrated

and modified text,

working with a

partner

Writing

Negotiating

solutions to

problems;

Interpersonal or

cultural

misunderstandings

Write key vocabulary

and extended

sentences (e.g., “I

think the area you

calculated is

incorrect. Recheck

the formula that you

used.”) to provide

descriptive feedback

on peer’s work to

calculate

measurement

Write key vocabulary

and simple sentences

to provide descriptive

feedback (e.g., “I think

you used the wrong

formula.”) on peer’s

work to calculate

measurement, using

support such as a

reference sheet and

sentence frames

List key vocabulary

(e.g., area formula)

to provide feedback

on peer’s work to

calculate

measurements,

using support such

as a reference sheet


MATH GRADE 8�

FINAL July 2012

Standard 3: English language learners communicate information, ideas, and concepts

necessary for academic success in the area of mathematics.

Language

Domain

Topic ELP Stage 5:

Advanced

ELP Stage 3:

Intermediate

ELP Stage 1:

Beginning

Listening

Statistics;

Metric units;

Measures of

central

tendency

Evaluate and use

appropriate

graphical

representations

based on oral

descriptions to show

metric units and

measures of center

and variability for

data from random

samples

Draw visual or graphic

representations of

metric unit and measure

of center and variability,

based on oral

descriptions of data

from random samples,

and working with a

partner

Select visual or graphical

representations of metric

units and measure of

center and variability,

based on oral

descriptions of data from

random samples, and

working with a partner

Speaking

Area;

Volume;

Two- and

three

dimensional

shapes

Compare and

contrast ways to

calculate

measurement in

different real-life

situations that

involve 2-D and 3-D

figures, using key

vocabulary

Describe ways to

calculate measurement

in real-life situations that

involve 2-D and 3-D

figures, using some key

vocabulary

Name formulas to

calculate measurement

involving 2-D and 3-D

figures in real-life

situations

Reading

Data sets;

Plots;

Data

interpretation

Interpret linear

model graphs and

data presented in

the grade-level text

Make meaning of data

sets presented in text

and linear model

graphs, working with a

partner

Identify different

components of data sets

and linear model graphs,


Writing

Probability;

Ration

Explain probability,

outcomes, events,

and sample space

using specialized

vocabulary and an

everyday example

Define probability,

outcomes, events, and

sample space using

some specialized

vocabulary, sentence

frames, and an

illustration of an

everyday example,


Draw and label

definitions of probability,

outcomes, events, and

sample space with the

use of everyday

examples, working with a

partner

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Documents

MATHEMATICS - DOE Elementary Math...MATHEMATICS Content Standards ELEMENTARY 1 Acknowledgements The American Samoa Department of Education (ASDOE) appreciates the effort of teachers,