Overview

Arizona Adult Education Standards Initiative

The Arizona Adult Education Standards Initiative (Standards Initiative) represents a proactive effort by Arizona’s adult education community to ensure consistency in program content and student outcomes for adult learners throughout the state. The Initiative is sponsored by the Arizona Department of Education – Division of Adult Education and developed by an outstanding cadre of the state’s adult educators.

The fundamental goal of this multi-year project is to ensure high levels of achievement for all adult learners in Arizona. As such, there are several critical reasons why it is so important to the future of adult education in Arizona as well as in the nation.

Value to the Adult Learner

The Standards Initiative provides consistent content and performance standards for implementation in all programs funded by the Arizona Department of Education.

Value to Programs and Instructional Practices

The Standards Initiative improves articulation and allows adult educators to assess

student performance and measure program effectiveness with greater accuracy.

In addition, exemplars of curriculum alignment developed by adult educators

during the spring and summer of 2000 also provide outstanding examples of

curricula in each of the content areas based on the standards.

Value to the State of Arizona

The Standards Initiative establishes a strong foundation for effective delivery of services to all adult learners. Moreover, the Initiative offers benchmarks for learning and program performance and sets forth high expectations for quality and accountability.

Value to the Profession of Adult Education

The Standards Initiative raises the bar on instructional performance and accountability which, in turn, increases the credibility of adult education within the field of teaching and learning. In addition, the Arizona Adult Education Standards complement similar efforts on the national level (i.e., Equipped for the Future published by the National Institute for Literacy) by providing the framework for adult learners to maximize their potential in the community, family, and workplace.

HOW THE ADULT EDUCATION STANDARDS WERE DEVELOPED

The process used to develop the adult education content and performance standards was designed by the Arizona Department of Education (Division of Adult Education) with the assistance of two consulting firms: Leadership Learning Systems, Inc. (based in Arizona and Illinois) and StandardsWork (Washington, D.C.).

In order to create a clear focus and ensure leadership of the Initiative from professionals the field, an open invitation was extended to adult educators statewide requesting participation in the Standards Initiative. The initial team convened in January 1998, to inaugurate the Initiative. As a result of their thoughtful dialogue and discussion, the following critical statements were created to direct the work of the Standards Initiative.

Beliefs

We believe adult learners are

· multi-faceted, unique individuals

· capable of learning

· motivated by diverse life experiences

· exploring ways to improve their lives through relevant educational

opportunities.

We believe adult education is a learner-centered, interactive process which

· values and supports the individual in defining and achieving personal

goals

· develops and improves basic and life skills in the community, family,

and workplace.

Vision

Adult education standards are the cornerstone for quality teaching, quality learning, and quality lives.

Mission

The Arizona Adult Education Standards Initiative provides the framework for

Adult learners to maximize their potential in the community, family, and

workplace. The project provides consistency and continuity of educational services throughout the state as well as an easily understood model which

communicates the contributions of adult education.

The approach used to create the Arizona Adult Education content and performance standards combined both process and substance. The process was highly participatory and encompassed active involvement and input of more than 200 adult educators across the state during the period of February 1998 – June 2000. The substance focused on the articulation and continuous improvement of rigorous and realistic standards for adult learning in specific subject areas including reading, writing, mathematics, science, social studies, ESOL, and citizenship test preparation.

THE STANDARDS INITIATIVE TIMELINE

A Steering Committee of adult educators provided overall guidance and direction throughout this period. Facilitation of the process was provided by Gail A. Digate of Leadership Learning Systems, Inc. and consultation in developing content and performance standards was provided by Susan Pimentel of StandardsWork.

A brief description of each phase of the Arizona Adult Education Standards Initiative appears below:

Phase I: January – December 1998

Teams of adult educators met to draft content standards in reading, writing, mathematics, ESOL, and Citizenship Test Preparation. These teams consulted a variety of resources, including the Arizona K-12 Academic Standards. A description of the relationship of the adult education content standards and the K-12 academic standards is provided on page 7. Several external, expert reviewers provided feedback and comments for continuous improvement to the original drafts.

Phase II: January – June 1999

During the second year of the Initiative, expanded teams of adult educators met to:

· conduct focus groups with adult educators and adult learners to solicit comments and suggestions on the drafts of the content standards. Focus group sessions were held in

Flagstaff, Phoenix, Tucson and Yuma.

· A second external review was conducted by Susan Pimentel of StandardsWork in August 1998.

· Following adoption of the content standards by the Steering Committee, initial work began on the development of performance standards in reading, writing, mathematics, ESOL and Citizenship Test preparation in September.

· Two additional teams were established to plan future implementation efforts: Professional Development and Marketing/Communications.

Phase III: July 1999 – June 2000

During the third year of the Initiative, the focus of work included:

· Initial release of the content standards in reading, writing, mathematics, ESOL,

and Citizenship Test preparation at the 1999 Arizona Adult Education Conference

(September 29 – October 2)

· Regional focus groups to solicit input on performance standards. These sessions

were held in Flagstaff, Holbrook, Phoenix, Tucson, and Yuma.

· Revision of both content and performance standards to reflect the federal

requirements of an additional level in ABE, the division of Adult Secondary Education (ASE – formerly GED preparation) into two levels, and adding two additional ESOL levels.

· Creation of content standards in science and social studies

· Regional focus groups to solicit and gather input on drafts of content standards

in science and social studies

· Establishment of a work team to develop recommendations regarding appropriate

assessment strategies in alignment with the content standards and federal requirements to document educational gain (Note: Recommendations will be

submitted to the Arizona Department of Education – Division of Adult Education)

in the autumn, 2000).

· Creation and training of a cadre of adult educators to support implementation of

the Arizona Adult Education Standards Initiative (i.e., Standards Specialists)

· Creation of curriculum alignment exemplars in reading, writing, mathematics,

and ESOL by teams of adult educators from programs across the state

(i.e., Curriculum Aligners)

· Implementation of a four-day summer institute which brought together more

than 100 adult educators (i.e., Standards Specialists and Curriculum Aligners)

to complete development of curriculum alignment exemplars and begin

articulation of strategies and action plans designed to support implementation of the Standards Initiative in adult education programs throughout the state.

It was during this institute that the State Director of Adult Education remarked that what began as a curriculum frameworks “project” indeed had become a major “initiative” destined to transform adult education in the state of Arizona and ensure “extraordinary” education to every adult learner.

Phase IV: July 2000 – September 2001

The following activities are anticipated for implementation in the next phase of the Standards Initiative:

· Pilot project to implement and “test” assessment strategies

· Consultation and support to adult education programs by Standards Specialists to

implement content and performance standards in reading, writing, mathematics,

ESOL, and citizenship test preparation

· Development of performance standards in science and social studies (including

input and feedback from the field via the Arizona Department of Education

(Division of Adult Education) website

· Focus groups with representatives of community colleges regarding implications

of the Arizona Adult Education Standards Initiative for adult learners’ matriculation to community college programs

· Evaluation of the Standards Initiative (1998 – 2001)

Phase V: July 2001 – June 2004

· Complete implementation of content and performance standards in

reading, writing, mathematics, science, social studies, ESOL, and citizenship

test preparation

(Note: Implementation of science and social studies content and performance

standards is required of adult education providers by July 1, 2002.)

· Complete implementation of assessment strategies

(Note: Implementation of assessment strategies is required of adult education providers by July 1, 2002.)

· Periodic review and revision of content and performance standards as needed

(e.g. commitment to continuous improvement)

THE IMPORTANCE OF SCIENCE, SOCIAL STUDIES AND TECHNOLOGY

The first edition of the Arizona Adult Education Standards was released in September, 1999, and contained content standards in Reading, Writing, Mathematics, ESOL and Citizenship Test Preparation. This (second) edition provides updated content and performance standards in these disciplines along with content standards in Science and Social Studies.

The purpose of including content standards in Science and Social Studies is to ensure that students who so choose would have access to instruction in these disciplines. As adult literacy education in Arizona is not compulsory, adult learners choose to take the courses that enable them to reach educational goals that further their ability to function in the family, the community and the workplace. Making available to adult learners a solid foundation in the physical, natural and social sciences enables them to invest in their own personal and professional development.

As technological advances propel rapid changes in how people live and work, all adult learners will need to develop and refine skills that keep them competitive and productive in the workplace. Now and into the future, access to, and basic computer and Internet skills, will enable adult learners to function successfully in the family, the community and the workplace. During FY2000, the ADE provided resources and training to make all ADE-funded programs Internet-connected: for administration, instruction and professional development. During the next three years, the ADE will continue to provide resources and training to enable adult education and family literacy teachers to become computer literate and Internet savvy. By the end of FY2004, it is expected that Arizona’s Adult Education Standards will be revised to reflect a much greater expertise with technology on the part of adult educators, and a much higher expectation of adult learners

with regard to basic computer and Internet skills.

RELATIONSHIP OF THE ARIZONA ADULT EDUCATION

CONTENT STANDARDS TO ARIZONA K-12 ACADEMIC STANDARDS

The initial charge from the State Director of Adult Education in January 1998 to develop content standards in adult education carried with it the need to craft world-class standards (not minimal competencies) and to customize these standards for adult learners. That said, content standards contained herein reflect sensible criteria for usefulness, intelligibility, rigor and measurability. In addition, content standards focus on academics, contain the right mix of skills and content, and represent a reasonable pattern of cumulative learning that is manageable (given the constraints of time).

A critical element in the process of developing content standards in adult education involved benchmarking the drafts of content standards to world-class levels and then reviewing them for relevancy, intelligibility and measurability.

As Arizona’s academic standards for students in grades K-12 are considered to be among the best in the nation, adult educators used this document as a valuable resource in both crafting and reviewing the adult education standards. Comparing what students in K-12 are capable of accomplishing with expectations for adult learners helped to aim higher when judging the potential of adult learners.

In summary, the focus in consulting the Arizona K-12 Academic Standards was to align the documents (i.e., content standards in Adult Education and K-12) in terms of rigor and comprehensiveness. However, no attempt was made to gain a direct one-to-one correspondence between the two documents as the two systems of education clearly serve different populations with specific needs, and facing diverse challenges and opportunities.

HOW TO READ CONTENT AND PERFORMANCE STANDARDS

If you are confused about the language of standards, you are not alone. This section provides definitions for standards-related terms and an analogy (using a non-academic example) to illustrate several important concepts. The analogy appears in italics .

Goal

A goal is the end result of a learning experience. A goal often is not measurable in an immediate sense. It reflects a state of being rather than a state of action. A goal expresses a purpose for instruction but does not designate the specific abilities that the learner must possess.

To improve running skills

Content Standard

A content standard supports the goal. It defines what a learner must know and be able to do. A content standard (also referred to as an exit standard) is brief, crisp, and written to the point. It uses jargon-free English so instructors and adult learners can understand it easily.

The learner is able to run one mile.

Indicators and Sub-Indicators

Indicators and sub-indicators contain all the knowledge and skills a learner needs to master the more broadly stated content standard. In essence, indicators and sub-indicators detail the content standard. Educators may refer to indicators and sub-indicators as “further domain specifications” or “benchmarks” that describe the skills, habits, and understandings that the learner must master.

Indicator: The learner understands the physiology of the body and knows how

to run safely.

Sub-indicators: • Understands physiology of muscles, bones, and

Cardiovascular system

• Understands how to warm up and cool down safely

• Understands how to pace self and breathe

correctly while running

• Uses correct foot position when running

(i.e., heel-toe-heel running)

• Observes the rules of the road (e.g., face traffic, observe

signs, run on sidewalk or shoulder of the road)

Sample Activities

Sample activities are designed to illustrate the indicators and sub-indicators. They are not required; rather, sample activities are provided to offer instructors some useful ideas, suggestions, and possible ways to bring the standards and indicators to life. In addition, sample activities reflect several core competencies (including communication skills, interpersonal skills, and critical thinking skills) which can be demonstrated within several contexts or settings (including the community, family, and workplace). Sample activities are included in this document as resources for instruction. Sample activities in science and social studies have been cross-referenced to content standards in reading, writing, and mathematics.

Core Competencies

Core competencies, the application of knowledge and skills in communication, interpersonal relations, and critical thinking, are designed as a fundamental element in sample activities.

Communication and interpersonal skills reflect the learner’s ability to engage in an interactive process while clearly expressing ideas that lead to mutual understanding. The following skill areas are demonstrated in these activities: speaking, listening, reading, and writing. A learner who communicates effectively is able to respond to an audience, demonstrate a clear sense of purpose, organize information, and deliver information using appropriate language and nonverbal behaviors.

Interpersonal skills encompass the ability to interact appropriately with individuals or groups in a variety of settings. Effective interpersonal interactions require the use of critical thinking skills such as analysis, synthesis, evaluation, and application in addition to the effective demonstration of communication skills (e.g., speaking, listening, reading, and writing).

The outcome of an activity is influenced by the environment or circumstances in which the activity occurs and the skills applied (e.g., communication, interpersonal, and/or critical thinking).

A sample activity may involve the learner in the process of entering a charity

run in support of cancer research.

Performance Standard

A performance standard indicates how competent or adept a learner’s demonstration must be to show attainment of the content standard. In other words, a performance standard defines “how good is good enough” to meet the content standard. Performance standards specify the quality of learner performance – acceptable, excellent, or something less. The level of performance is determined by the extent to which students demonstrate command over the concepts of skills outlined in the content standards. Such command must include both quality and quantity.

Performance standards:

• Specify particular concepts and skills that the learner must know and be able

to do as defined by the content standards (often in greater detail with some

additional explanation of the type, quality, range and depth of the performance

expectations)

• Define several different levels of achievement that outline the extent to which

the learner demonstrates command over the concepts and skills within the content

standards. The Arizona Adult Education Standards Initiative has adopted four

levels of proficiency:

Beginning (a ways to go before passing)

Approaching (getting closer)

Met (passing)

Exceeds (excellent performance, beyond passing)

• Establish the difficulty of material with which the learner must work (e.g.,

vocabulary lists, spelling lists, reading lists or reading difficulty levels).

A learner at one proficiency level is able to display most of the knowledge, skills, and processes at that particular level (e.g., met level) and lower proficiency levels (e.g., approaching and beginning levels). Once assessment strategies have been adopted, the proficiency levels and their descriptors are intended to inform and guide interpretation of

the scores. In short, each proficiency level descriptor is a statement of the knowledge, skills, and abilities expected to be held by the average learner who is associated with that level.

In an attempt to ensure consistency across the various disciplines, the following terms were adopted by the Performance Standards Work Team:

• Occasionally, seldom Able to demonstrate skills and command of the

concepts up to 49% of the time

• Sometimes Able to demonstrate skills and command of the

concepts up to 50 – 74% of the time

• Often; most of the time Able to demonstrate skills and command of the

concepts up to 75 – 89% of the time

• Consistently Able to demonstrate skills and command of the

concepts up to 90 – 100% of the time

Returning to the sports analogy, consider time trials for Olympic runners as a vehicle to motivate and measure performance. For example, Olympic runners are not simply told they have to run fast in order to qualify for the 100-yard dash. Rather, they know exactly what times they need to beat. Without performance standards, a deliberate stroll could constitute running a mile.

The learner is able to run one mile in seven minutes.

Curriculum

Curriculum is best characterized as descriptions of what should take place in the classroom and describes in greater detail the topics, themes, units, and questions contained in the content standards. Curriculum serves as a guide for instructors; addressing teaching techniques, recommending activities, scope and sequence, and modes of presentation considered most effective.

In addition, curriculum indicates those textbooks, materials, activities, and equipment that best help the learner achieve the content standards. Unlike content standards, curriculum can vary from region to region or program to program as well as from teacher to teacher,

provided that the focus remains on delivering the “big” ideas and concepts that the content standards require the learner to understand and apply. Content standards are the framework for curriculum.

Curriculum within the sports analogy example include units on

physiology, questions and topics to cover, suggested reading material,

and training sessions needed in order to ensure the learner is able to

run one mile safely and efficiently.

Assessment

Assessment defines the nature of evidence required to demonstrate that the content standard has been met (e.g., essay, solution to a mathematical problem, answers to questions in reference to a reading passage).

In the charge to the Assessment Strategies Work Team (January, 2000), Karen M. Liersch, State Director of Adult Education specified the following requirements for assessment in adult education in Arizona:

· It will insure reliability and validity

· It will provide for pre-, interim, and post-testing

· It will be aligned to and test the Arizona Adult Education Content

Standards in Reading, Writing, Mathematics, and ESOL

· It will be criterion – or standards - referenced

· It will inform instruction

· It will serve as an accountability measure

· It will be adaptable to a variety of instructional environments

· The Assessment will accommodate learners with special needs

Assessments for the sports analogy might require the learner to run one mile,

demonstrating ability to use proper form and observe safety rules of running

(this would be an example of performance-based assessment).

Another approach might ask the learner to complete a written test,

Demonstrating understanding of physiology of running (this would be an

example of a criterion-referenced test, including multiple choice and

short answer questions).

Again, the performance standard specifies the learner’s degree

of proficiency on those demonstrations or assessments, defining

what it means to run the mile in one of three ways or levels:

expert, competent, or less than competent fashion.

Mathematics

Standard: The adult learner develops and applies math

strategies to a variety of situations.

Pre-Literacy (Beginning ABE Literacy).............................2

ABE I (Beginning Basic Education)...............................….4

ABE II (Low Intermediate Basic Education).....................7

ABE III (High Intermediate Basic Education)...................9

ASE I/GED (Low Adult Secondary Education)...........…12

ASE II (High Adult Secondary Education)…….........….14

Standard: The adult learner develops and applies math strategies to a variety

of situations.

Pre-Literacy

Indicator A: Develops and applies number sense to solve a variety of real-life problems

and to determine if the results are reasonable

1. Recognizes relationships between real-life representations, number names, and symbolic representation of numbers

a. Writes and reads whole numbers between 0 and 100 as numerals

2. Relates counting, grouping, and place value concepts to whole numbers

a. Places in correct sequence whole numbers between 0 and 100

3. Performs the operations of addition and subtraction of one-digit numbers

a. Adds and subtracts whole numbers between 0 and 9 correctly

4. Uses coins and currency

a. Recognizes symbols for currency (e.g., $, ¢)

b. Identifies coins and currency using pennies, nickels, dimes, quarters, half-dollars, and bills

Indicator B: Applies data collection, data analysis, and probability to interpret, predict,

and/or solve real-life problems

1. Constructs and reads tables, charts and graphs

a. Collects and records data from a simple survey of at least 5 respondents

b. Organizes data according to choice from a simple survey of at least 5 respondents

c. Identifies choice receiving largest and smallest number of responses from a simple

survey of at least 5 respondents

d. Constructs a display of data indicating responses from a simple survey of at least 5

respondents

Indicator C: Applies algebraic concepts and methods to explore, analyze or solve real-life

problems

1. Creates, describes, and extends a variety of patterns and formulates generalizations to make predictions

a. Replicates a pattern using manipulatives or objects (tangrams)

2. Represents and describes mathematical ordering and grouping relationships

a. Determines the next number in a sequence of numbers up to a hundred

Indicator D: Uses geometric properties, relationships, and methods to identify, analyze

and solve real-life problems

1. Identifies basic geometric shapes

a. Names simple polygons (e.g., triangle, square, rectangle)

b. Names simple solid geometric forms using own vocabulary

Indicator E: Applies knowledge of standard measurements to real-life situations

1. Selects the appropriate measurement with U.S. customary units for an object or event

a. Selects the appropriate device to measure the given attribute of an object or event (e.g., ruler, thermometer, measuring cup, scale, stop watch)

ABE I

Indicator A: Develops and applies number sense to solve a variety of real-life problems

and to determine if the results are reasonable

Demonstrates an understanding of number meanings and relationships

a. Places numbers between 0 and 1000 on a number line

b. Describes fractions (halves, thirds, fourths) as parts of a whole

c. Distinguishes between odd and even numbers

2. Recognizes relationships between real-life representations, number names, and symbolic

representation of numbers

a. Expresses and reads whole numbers between 0 and 1000 as numerals

b. Reads and writes whole numbers between 0 and 1000 as number words

c. Matches a fraction to a pictorial representation of halves, thirds, and fourths

d. Matches a number word to a pictorial representation of halves, thirds, and fourths

Represents and uses numbers in equivalent forms

a. Writes whole numbers between 0 and 1000 in expanded notation (e.g., 89 = 80 + 9)

b. Makes a model to represent a fractional representation of halves, thirds and fourths

Uses coins and currency

a. Expresses equal relationships of coins and currency using pennies, nickels, dimes,

quarters, half-dollars, and bills up to $5.00

Demonstrates the meaning of operations and the relationships between them

a. Explains that addition joins groups

b. Explains that subtraction decreases, takes away, compares, or finds the difference

c. Uses addition to check subtraction problems and vice versa

6. Performs the operations of addition, subtraction, multiplication, and division on whole

numbers

a. Adds, subtracts up to 500, multiplies, and divides single digit whole numbers correctly

b. Selects appropriate operation in addition or subtraction to solve one-step word problems involving whole numbers up to 500.

c. Selects appropriate operation in multiplication and division to solve one-step word problems with single digit numbers

7. Selects and uses appropriate techniques to facilitate computation while solving problems and determining the reasonableness of results

a. Rounds whole numbers to tens and hundreds

b. Uses estimation to check the reasonableness of results in solving one-step word problems in addition and subtraction of whole numbers up to 500

c. Uses estimation to check the reasonableness of results in solving one-step word problems in multiplication and division of single-digit numbers

Indicator B: Applies data collection, data analysis, and probability to interpret, predict,

and/or solve real-life problems

1. Constructs, reads, analyzes, and interprets tables, charts, and graphs

a. Makes and labels a graph (horizontal bar, vertical bar, circle graph, pictograph) from data

Predicts and measures the likelihood of events

a. Collects and records data from a simple one-step probability experiment

b. Organizes (e.g., sorts, sequences, tallies data from a simple one-step probability

experiment)

c. Predicts the possible outcomes from a simple one-step probability experiment

d. Predicts the most likely or least likely outcome in a simple one-step probability

experiment

e. Compares the outcome of the experiment to the prediction

Indicator C: Applies algebraic concepts and methods to explore, analyze or solve real-life

problems

1. Creates, describes, and extends a variety of patterns and formulates generalizations to

make predictions

a. Communicates orally the description of the pattern in a series of objects

b. Communicates orally a description of the pattern in a sequence of numbers

c. Extends a pattern using manipulatives or objects

d. Extends a pattern occurring in a sequence of numbers

2. Represents and describes mathematical ordering and grouping relationships

a. Identifies the pattern in skip counting (e.g., 2, 4, 6 – add 2 to each number)

b. Determines the next number in a skip counting pattern (e.g., 2, 4, 6 _____?)

Indicator D: Uses geometric properties, relationships, and methods to identify, analyze

and solve real-life problems

Identifies and describes basic geometric shapes

a. Identifies the characteristics of simple polygons (e.g., triangle, square, rectangle)

b. Identifies the characteristics of simple solid geometric figures (e.g., cube and rectangular container)

Indicator E: Applies knowledge of standard measurements to real-life situations

Demonstrates that a single object or event can be measured in different ways (e.g., length,

mass/weight, time, capacity, temperature, area, volume)

a. Determines what attributes of an object or event are measurable

b. Identifies the appropriate type of measurement for each attribute of an object or event

Identifies the appropriate measurement with U.S. customary units

for an object or event including:

a. Length - inches, feet and yards

b. Capacity - cups, gallons

c. Weight - ounces, pounds, tons

d. Area - square unit

e. Volume - cubic unit

f. Time - second, minute, hour, day, month, year, decade, century

g. Temperature - degrees on Fahrenheit scale, degrees on Celsius scale

Compares units of measurement to determine equal relationships using U.S. customary units (e.g., 2 cups = 1 pint, 3 cups > 1 pint)
Makes estimation of measurement

a. Using U.S. customary units, estimates a measurement of a given object or event and compares the estimation to actual measurement

b. Evaluates the reasonableness of the estimation

Applies measurement

a. Solves real-life problems involving measurements using U.S. customary units

ABE II

Indicator A: Develops and applies number sense to solve a variety of real-life problems

and to determine if the results are reasonable

1. Demonstrates an understanding of number meanings and relationships

a. Places numbers between 0 and 10,000 on a number line

b. Describes mixed numbers as parts of a whole

2. Recognizes relationships between real-life representations, number names, and symbolic representation of numbers.

a. Expresses and reads whole numbers between 0 and 10,000 as numerals

b. Reads and writes whole numbers between 0 and 10,000 as number words

c. Matches a mixed number to a pictorial representation

d. Matches a number word to a pictorial representation of mixed numbers

3. Represents and uses numbers in equivalent forms

a. Writes whole numbers between 0 and 10,000 in expanded notation (e.g., 89 = 80 + 9)

b. Makes a model to represent a fractional representation of mixed numbers

4. Uses coins and currency

Expresses equal relationships of coins and currency using pennies, nickels, dimes,

quarters, half-dollars, and bills up to $100.00

5. Demonstrates the meaning of operations and the relationships between them

Explains that multiplication is repeated addition of equal numbers and/or groups
Explains that division is repeated subtraction or placing items into groups of equal size
Uses multiplication to check division problems and vice versa

6. Performs the operations of addition, subtraction, multiplication, and division on whole

numbers

a. Adds, subtracts, multiplies, and divides whole numbers between 0 and 1,000 correctly

b. Selects appropriate operation to solve one-step word problems involving whole numbers

between 0 and 1,000

7. Selects and uses appropriate techniques to facilitate computation while solving problems

and determining the reasonableness of results

a. Rounds whole numbers to thousands

b. Uses estimation to check the reasonableness of results in solving one-step word problems

using whole numbers between 0 and 1,000

Indicator B: Applies data collection, data analysis, and probability to interpret, predict,

and/or solve real-life problems

1. Constructs, reads, analyzes, and interprets tables, charts, and graphs

a. Interprets and analyzes data from pictographs and bar graphs where each symbol represents one unit

b. Interprets and analyzes data on a pictograph where each symbol represents multiple units

2. Predicts and measures the likelihood of events

a.. Describes events that have 100% probability or 0% probability

b. Identifies outcomes that are more likely, less likely, or equally likely to occur

c. Describes the concept of sample

Indicator C: Applies algebraic concepts and methods to explore, analyze or solve real-life

problems

1. Creates, describes, and extends a variety of patterns and formulates generalizations to make predictions

a. Communicates in written form the description of the pattern in a series of objects

b. Communicates in written form a description of the pattern in a sequence of numbers

c. Extends simple geometric and number pattern

d. Creates simple geometric and number patterns

2. Represents and describes mathematical ordering and grouping relationships

a. Sorts and classifies objects according to observable attributes

b. Finds the missing element in a number sentence involving addition, subtraction, multiplication, and division

Uses words such as all, none, some, and many to make reasonable statements

d. Describes a rule for a simple pattern

Indicator D: Uses geometric properties, relationships, and methods to identify, analyze

and solve real-life problems

1. Identifies and describes basic geometric shapes

a. Compares and contrasts the characteristics of simple polygons (e.g., triangle, square,

rectangle)

b.Compares and contrasts the characteristics of simple solid geometric figures (e.g., cube

and rectangular container)

c. Identifies characteristics of lines which intersect, are parallel, or are perpendicular

Indicator E: Applies knowledge of standard measurements to real-life situations

1. Demonstrates that a single object or event can be measured in different ways (e.g., length,

mass/weight, time, capacity, temperature, area, volume)

Identifies the appropriate type of measurement for each attribute of an object or event and justifies answer

2. Demonstrates the appropriate measurement with U.S. customary and metric units

for an object or event including:

a. Length - inches, feet and yards, millimeters, centimeters, meters, kilometers

b. Capacity - cups, gallons, milliliters, liters

c. Weight - ounces, pounds, tons, grams, kilograms

d. Area - square unit

e. Volume - cubic unit

f. Time - second, minute, hour, day, month, year, decade, century

g. Temperature - degrees on Fahrenheit scale, degrees on Celsius scale

3. Compares units of measurement to determine more or less relationships using U.S. customary and metric units (e.g., 2 cups = 1 pint, 3 cups > 1 pint)

4. Makes estimation of measurement

a. Using U.S. customary or metric units, estimates a measurement of a given object or event and compares the estimation to actual measurement and justifies the answer

b. Evaluates the reasonableness of the estimation and justifies the answer

5. Applies measurement

a. Solves real-life problems involving measurements using U.S. customary and metric units

ABE III

Indicator A: Develops and applies number sense to solve a variety of real-life problems

and to determine if the results are reasonable

1. Develops concepts, number sense, and number relationships relating to whole numbers,

fractions, decimals, and percents

a. Describes a fraction of any quantity as the relationship between the given numerator

part(s) related to the entire number of part(s) in the whole denominator

b. Describes a decimal as the fractional representation of the quantity expressed as a

whole number and/or tenths, hundredths, thousandths, etc.

c. Describes percents as a fraction or as parts out of 100

d. Reads and writes fractions, decimals, and percents as numerals and number words

e. Expresses and reads whole numbers between 1000 and 1,000,000,000 as numerals

f. Reads and writes whole numbers between 1000 and 1,000,000,000 as number words

g. Writes whole numbers between 1000 and 1,000,000,000 in expanded notation

h. Places in correct sequence whole numbers between 1000 and 1,000,000,000

i. Places in correct sequence fractions, decimals, and percents in same groups or mixed

groups

j. Expresses a quantity in equivalent fraction, decimal, and percent form

2. Performs the operations of addition, subtraction, multiplication, and division using whole

numbers, fractions, decimals, and percents

a. Selects and uses correctly the operations of addition, subtraction, multiplication, and division in story problems involving whole numbers

b. Selects and uses correctly the operations of addition, subtraction, multiplication, and division in story problems involving fractions and decimals

c. Identifies the whole, part, and percent in problems involving percent

d. Solves word problems involving averaging of rational whole numbers, fractions, or

decimals

e. Solves word problems involving the order of operations

3. Applies number theory concepts to represent numbers in various ways

a. States the prime factors for a given whole number

b. Names the square root of a number with a perfect square

c. States the multiples of a given number

d. Defines prime and composite numbers

e. Sorts numbers by their properties

4. Selects and uses appropriate techniques and information to facilitate computation while solving problems and determining the reasonableness of results

a. Rounds decimals to tenths, hundredths, and thousandths place

b. Rounds fractions to nearest whole and/or half

c. Uses estimation to check the reasonableness of results using whole numbers, fractions, decimals, and percents in solving problems

d. Distinguishes between relevant and irrelevant information

e. Recognizes the degree of precision needed

Indicator B: Applies data collection, data analysis, and probability to interpret, predict,

and/or solve real-life problems

1. Constructs, reads, analyzes and interprets graphs, tables, and charts

a. Interprets and analyzes data from circle and line graphs

b. Formulates questions from graphs, tables, and charts

c. Solves word problems using graphs, tables, and charts

2. Determines probabilities through experiments and/or simulations and compares the results

with prediction

a. Predicts possible outcomes in an experiment in which the possible number of outcomes changes (e. g., two-step probability)

b. Compares the outcome of the experiment to the predictions

Indicator C: Applies algebraic concepts and methods to explore, analyze or solve real-life

problems

1. Translates and differentiates the language of algebra

a. Describes and uses a variable and a constant in a real life situation

b. Defines a term, expression, equation and inequality

c. Simplifies an expression by combining like terms (e.g., 3x + 2 + 2x + 3 = 5x + 5)

d. Translates a written phrase into an expression

e. Correctly uses mathematical symbols <, >, =, ≠

2. Identifies order of operations

a. Uses the correct order of operations in solving algebraic expressions

3. Represents and describes how changing the value of one variable in a relationship results in a change in another ("When I am 9, 3 times my age = 27. When I am 10, 3 times my age = 30. In the equation 3x = y, when x = 9, y = 27")

Indicator D: Uses geometric properties, relationships, and methods to identify, analyze and

solve real-life problems

1. Identifies, describes and measures basic geometric shapes and angles using definitions and appropriate measuring devices (e.g., protractor, ruler, compass)

a. Draws, measures, and classifies angles as right, acute, obtuse, straight, or reflex

b. Identifies the properties of geometric figures using definitions of similarity, congruent, and symmetry

c. Identifies and describes properties of alternate interior, corresponding, complementary, and supplementary angles

d. Classifies triangles by their angles and sides as equilateral, isosceles, scalene, acute, obtuse and right

e. Labels and identifies the characteristics of a circle, cylinder, parallelogram, pentagon, hexagon, octagon, decagon, rhombus, and trapezoid (e.g., radius, diameter, base, height)

Indicator E: Applies knowledge of standard measurements to real-life situations

1. Estimates and uses U.S. customary and metric measurement to describe and make

comparison

a. Converts measurement units to equivalent units within a given system

b. Compares estimated measurements between U.S. customary and metric systems and

Fahrenheit and Celsius systems

2. Estimates, uses, and describes measures of distance, perimeter, area, volume, capacity, weight, mass, and angles

a. Differentiates between perimeter, area, and volume of polygons and solids using concrete and illustrative modes

b. Differentiates between weight and mass

c. Differentiates between capacity and volume

d. Records estimates and measurements for:

Distance in scale drawings

Circumference

Degrees of angles

3. Uses formulas and procedures to solve problems involving measurement

a. Uses given formulas to find:

Area and perimeter of simple polygons

Surface area of rectangular containers

Volume of rectangular containers

ASE I/GED

Indicator A: Develops and applies number sense to solve a variety of real-life problems

and to determine if the results are reasonable

1. Develops concepts, number sense, and number relationships relating to integers and rational numbers (e.g., whole numbers, decimals, fractions)

a. Estimates the square root of any whole number to the nearest whole number

b. Places integers in correct sequence

c. Adds, subtracts, multiplies, and divides positive and negative numbers

2. Demonstrates the relationships between the operations of addition, subtraction, multiplication, and division as they relate to integers

a. Explains the effect of addition, subtraction, multiplication, and division on positive and negative numbers

3. Selects and uses appropriate techniques while solving problems and determining the reasonableness of results

a. Represents and uses numbers with exponents

b. Uses computation, estimation, and proportions to solve word problems involving

scientific notation

c. Uses computation, estimation, and proportions to solve word problems involving

integers, exponents, and square roots

Indicator B: Applies data collection, data analysis, and probability to interpret, predict,

and/or solve real-life problems

1. Constructs, reads, analyzes, and interprets tables, charts, and graphs

a. Chooses an appropriate graphic format to organize and represent data

b. Organizes collections of data into frequency charts, stem-and-leaf plots, scatter plots and matrices

2. Makes valid inferences and predictions based on statistical analysis

a. Formulates predictions from a given set of data and justifies predictions

b. Compares a given prediction with the results

c. Differentiates between a sampling and a census

3. Uses measures of mean, median, mode and range applied to a data set

a. Finds the mean, mode, range, median, and quartile of a data set

b. Applies the concepts of mean, mode, and median to draw conclusions about data

4. Determines probabilities through experiments and/or simulations and compares the results

with prediction

a. Expresses probability as a fraction or percent

Indicator C: Applies algebraic concepts and methods to explore, analyze or solve real life

problems

1. Solves problems with formulas

a. Uses formulas on GED Math test (i.e., simple interest, distance, total cost) to solve word problems

2. Solves equations using addition, subtraction, multiplication, and division and checks by substituting the solution into the original equation

a. Solves a one-step equation and uses substitution to check answer

b. Solves a two-step equation and uses substitution to check answer

c. Analyzes and solves story problems involving one- and two-step equations

d. Solves ratio and proportion problems

e. Solves computations of cost, distance, and simple interest word problems

f. Determines slope of a line

Indicator D: Uses geometric properties, relationships, and methods to identify, analyze

and solve real-life problems

1. Demonstrates an ability to recognize, define and apply geometric formulas and characteristics of rectangular coordinate planes, solid figures and linear measurements in solving problems

a. Applies the appropriate geometric formula (i.e., area, perimeter, volume, Pythagorean relationship, distance between two points in a plane) from the GED Math test for problem solving

b. Solves problems using similarity and proportion

c. Solves problems using alternate interior angles

d. Defines and graphs ordered pairs on rectangular coordinate plane

Indicator E: Applies knowledge of standard measurements to real-life situations

1. Describes and converts complex measurement units

a. Converts units of measurement into equivalent units of measurement using proportion (e.g., 3 feet: 1 yard; 18 feet: 6 yards)

^b.Uses scientific notation to express units of measurement in large scales (e.g., distance of sun from earth = 93,678,912 miles = 93.678912 x 10⁶⁾

c. Uses scientific notation to express units of measurement in small scales using negative exponents

d. Demonstrates change of placement in converting measurement units in the metric system (e.g., 353mm = 35.3cm, 2.5km = 25,000cm)

ASE II

Indicator A: Develops and applies number sense to solve a variety of real-life problems

and to determine if the results are reasonable

1. Develops concepts, number sense, and number relationships relating to integers and rational numbers (e.g., whole numbers, decimals, fractions)

a. Explains the meaning of absolute value, e.g., |-8| = 8

b. Uses positive and negative exponents

2. Selects and uses appropriate techniques while solving problems and determining the reasonableness of results

3. Compares and contrasts the real number system and its various subsystems with regard to

their structural characteristics

a. Classifies numbers as members of the sets (natural, whole, integers, rationals, and

irrationals)

b. Compares subsets of the real number system with regard to their properties

(commutative, associative, distributive, identity, inverse and closure properties)

Indicator B: Applies data collection, data analysis, and probability to interpret, predict,

and/or solve real-life problems

1. Constructs, reads, analyzes, and interprets tables, charts, and graphs

a. Evaluates the reasonableness of conclusions drawn from interpretation of data in a graphic format

2. Constructs and draws inferences including measures of central tendency, from charts, tables, graphs and data plots that summarize data from real-world situations

a. Organizes collections of data into frequency charts, stem-and-leaf plots, scatter plots and matrices and determines outliers

b. Constructs histograms, line graphs, circle graphs and box-and-whisker plots

c. Uses mode, quartiles and range as a means for effective decision making in analyzing the data

3. Applies curve fitting to make predictions from data

a. Draws a line or a curve which closely fits a scatter plot

4. Explains the effects of sampling on statistical claims and recognizes misuses of statistics

a. Differentiates between a biased and an unbiased sample

b. Recognizes the impact of interpreting data from a biased sample

5. Determines probabilities through experiments and/or simulations and compares the results

with prediction

a. Uses simulations to estimate probabilities of real-life situations

b. Designs a statistical experiment based on a given hypothesis

6. Describes, in general terms, the normal curve and uses its properties to answer questions about sets of data that are assumed to be normally distributed

a. Determines if data gathered from a real-world situation fit a normal curve

b Describes the central tendency characteristics of the normal curve

c. Makes simple predictions from data represented on the graph

7. Explains the concept of a random variable

a Distinguishes situations where a random variable is needed or used

b. Uses a random number table or technology to generate random numbers in modeling real-life situations (e.g., select randomly who belongs in what group)

8. Applies measures of central tendency, variability, and correlation

a. Draws conclusions about the “spread” of data given the variance and standard deviation (e.g., compare sets of data with the same central tendency but with different variance)

b. Determines, from a given plot of data, whether it has strong or weak, positive or negative correlation

Indicator C: Applies algebraic concepts and methods to explore, analyze or solve real life

problems

1. Models real-world phenomena using functions and relations

a. Identifies the independent and dependent variables from a real-life situation

b. Expresses the relationship between two variables using a table, equation, graph, and matrix

c. Describes the relationship suggested by two or more graphs of related real-world

situations

2. Interprets algebraic equations and inequalities geometrically and describes

geometric relationships algebraically

a. Graphs a linear equation in two variables

b. Graphs a linear inequality in two variables

c. Determines slope and intercepts of a linear equation

d. Writes an equation of the line that passes through two given points

e. Determines from two linear equations whether the lines are parallel, are perpendicular or coincide

3. Applies trigonometry to real-life problem situations (e.g., investigates how to find the distance across the river using similar triangles and trigonometric ratios; compares the sine and cosine curves to the curves of sound waves and tide variations)

a. Uses the definitions of trigonometric functions to find the sine, cosine and tangent of the acute angles of a right triangle

b. Solves simple right-triangle trigonometric equations involving sine, cosine and tangent

c. Uses an appropriate right-triangle trigonometric model to solve a real-life problem

4. Performs mathematical operations on expressions and matrices, and solves

equations and inequalities

a. Simplifies numerical expressions using the order of operations including exponents

b. Evaluates algebraic expressions using substitution

c. Simplifies square roots and cube roots with monomial radicands that are perfect

squares or perfect cubes

d. Evaluates numerical and algebraic absolute value expressions

e. Multiplies and divides monomial expressions with integer exponents

f. Solves linear equations and inequalities in one variable

g. Solves quadratic equations

h. Solves radical equations involving one radical

i. Solves proportions which generate linear or quadratic equations

j. Solves absolute value equations containing a single absolute value expression

k. Solves systems of linear equations in two variables

5. Translates among tabular, symbolic and graphical representations of functions

a. Creates a linear equation from a table of values

b. Creates a graph from a table of values

c. Determines the solution to a system of inequalities in two variables, from a given

graph (e.g., "Which of the shaded regions represents the solution to the system?")

d. Determines the solution to a system of equations in two variables, from a given graph

Indicator D: Uses geometric properties, relationships, and methods to identify, analyze

and solve real-life problems

1. Interprets and draws three-dimensional objects

a. Sketches prisms, pyramids, cones, and spheres

b. Classifies prisms, pyramids, cones, cylinders and spheres by base shape, lateral

surface shape, related surface area and volume formulas

2. Represents problem situations with geometric models and applies properties of figures

a. Calculates surface areas and volumes of three-dimensional geometric figures

given the required formulas

3. Deduces properties of figures using transformations in coordinate systems, identifying

congruency and similarity

a. Determines whether a figure is symmetric with respect to a line or a point

b. Gives the new coordinates of a transformed geometric figure

c. Determines the effects of a transformation on linear and area measurements of the

original figure

d. Sketches the figure that is the result of a given transformation

4. Deduces properties of and relationships between figures from given assumptions

a. Finds similarities and differences among geometric shapes and designs using a given

attribute (e.g., height, area, perimeter, diagonals, angle measurements)

b. Identifies arcs, chords, tangents and secants of a circle

c. States valid conclusions using informal deductive reasoning

5. Translates between synthetic and coordinate representations (e.g., a straight line is

represented by the algebraic equation Ax + By = C)

a. Verifies characteristics of a given geometric figure using coordinate formulas such

as distance, mid-point, and slope to confirm parallelism, perpendicularity, and

congruency

Recognizes and analyzes Euclidean transformations (e.g., reflections, rotations, dilations and translations)

a. Classifies transformations based on whether they produce congruent or similar non-

congruent figures

b. Determines whether a given pair of figures on a coordinate plane represents a

translation, reflection, rotation and/or dilation

c. Applies transformational principles to practical situations (e.g., enlarge a photograph)

Indicator E: Uses both inductive and deductive reasoning in making conjectures and

testing the validity of arguments

1. Uses inductive and deductive logic to construct simple valid arguments

a. Constructs a simple informal

deductive proof (e.g., write a proof of the statement: "You can fly from Bombay

to Mexico City, given an airline schedule”)

b. Produces a valid conjecture using inductive reasoning by generalizing from a

pattern of observations (e.g., if 10' = 10, 10'= 100, 10'= 1000, make a conjecture)

2. Determines the validity of arguments

a. Determines if the converse of a given statement is true or false

b. Draws a simple valid conclusion from a given if ... then statement and a minor

premise

c Lists related if….then statements in logical order

d. Distinguishes valid arguments from invalid arguments

e. Analyzes assertions about everyday life by using principles of logic (e.g., examine

the fallacies of advertising)

f.. Uses Venn diagrams to determine the validity of an argument

g. Recognizes the difference between a statement verified by mathematical proof

(i.e., a theorem) and one verified by empirical data (e.g., women score higher than

men on vocabulary tests)

3. Formulates counterexamples and uses indirect proof

4. Develops and analyzes algorithms

a. Constructs a counterexample to show that a given invalid conjecture is false (e.g.,

Nina makes a conjecture that x' > x' for all values of x. Find a counterexample.)

b. Writes an algorithm that explains a particular mathematical process (e.g., tell a younger child how to find the average of two numbers)

c. Determines the purpose of a given algorithm

d. Determines whether given algorithms are equivalent

Math Performance Standards

Definition of terms

Familiar situation: Context in which the performance of a skill is assessed under routine

circumstances similar to those in which instruction has taken

place

Unfamiliar situation: Context in which the performance of a skill is assessed under non-routine

circumstances which:

· are not necessarily similar to those in which instruction has taken place (e.g., different vocabulary, reordering of information)

· necessitate application of the skill to a real life situation

· necessitate the use of analytic reasoning skills to distinguish relevant and non-relevant information and/or situations where there is not enough information to solve the problem.

Note: In all areas of the Math Performance Standards, it is recommended that students be involved in:

· Problem solving opportunities based on the students’ experiences at home, at work, and in the community

· Estimating answers to problems

· Checking answers for reasonableness

· Looking for alternative solution strategies

Pre-Literacy

Beginning

At this level, the student performs the following tasks with a rudimentary understanding of the concepts and basic reasoning skills. The student’s explanations are minimal and presented without a lot of supporting information.

Sometimes in familiar, routine situations, the student:

Number Sense:

· reads and writes numerals between 0 and 20

· recognizes American currency symbols (e.g., $ and ¢)

· performs the operations of addition and subtraction of one digit numbers

Data Analysis:

· collects, records, and organizes data from a simple survey of at least five respondents

Algebra:

· replicates a three-item single-attribute pattern, (e.g., red square, blue square, yellow square, red square…)

· determines the next number in a given sequence of numbers up to 20

Geometry:

· names simple polygons using the student’s own vocabulary

Measurement:

· selects the appropriate device for measuring an object

Approaching

At this level, the student performs the following tasks with a basic understanding of the concepts and reasoning skills; however, explanations about how and why problems were solved are minimal.

Often in familiar, routine situations and sometimes in unfamiliar, non-routine situations, the student:

Number Sense:

· recognizes relationships among real life representations, number names, and symbolic representation of numbers between 0 and 50

· performs the operations of addition and subtraction of one digit numbers

· identifies coins and currency using pennies, nickels, dimes, quarters, half-dollars, and dollar bills

Data Analysis:

· in a simple survey of at least five respondents:

- collects and records data accurately

- organizes data according to choice

- identifies choices receiving largest and smallest number of responses

- constructs a display of data indicating responses

Algebra:

· replicates a five item one attribute pattern (e.g., red square, blue square, yellow square, green square, purple square, red square….)

· determines the next number in a given sequence of numbers up to 50

Geometry:

· names simple polygons and solid geometric forms using the student’s own vocabulary

Measurement:

· selects the appropriate device for measuring an object or event using United States customary units

Met

At this level, the student demonstrates some conceptual understanding while performing the following tasks. The student provides organized solutions complete with supporting information and explanations about how they were achieved.

Most of the time in both familiar and unfamiliar, non-routine situations, the student:

Number Sense:

· writes, reads, and places in correct sequence whole numbers between 0 and 100

· performs the operations of addition and subtraction of one digit numbers

· recognizes symbols for coins and currency

· identifies American currency up to and including dollar bills

Data Analysis:

· in a simple survey of at least five respondents:

- collects and records data accurately

- organizes data according to choice

- identifies choices receiving largest and smallest number of responses

- constructs a data display indicating responses

Algebra:

· replicates a five-item two-attribute pattern (e.g., large red square, small blue square, large yellow square, small red, large red square….)

· determines the next number in a given sequence of numbers up to 100

Geometry:

· identifies characteristics of simple polygons and solid geometric forms using the student’s own vocabulary

Measurement:

· selects the appropriate device for measuring an object or event using U.S. customary units

Exceeds

At this level, the student:

· consistently performs all the above tasks by applying both procedural knowledge and conceptual understanding to both familiar, routine and unfamiliar, non-routine situations

· provides solutions that are clear, logical, and go beyond the obvious in the interpretations

· justifies solutions by explaining how, as well as why, the answer was achieved

ABE I

Beginning

Sometimes in familiar situations the student:

Number Sense:

· adds and subtracts whole numbers, without regrouping, up to 100

· expresses equal relationships of coins using dimes, nickels, and pennies up to $.50

Data Analysis:

· reads and interprets most pictographs

· describes many events that have a probability of 100% or 0%

Algebra:

· creates three-item, single-attribute patterns and at times, is able to explain the logic of the sequence

Geometry:

· identifies a few attributes of simple polygons .

Measurement:

· describes how the attributes of some objects and events can be measured using different units of measurement

Approaching

At this level, the student performs the following tasks with some understanding of the concepts. The student is able to employ problem-solving strategies such as identifying and using appropriate information. Although reasoning skills are evident and supporting information is present, explanations are not always complete.

Often in familiar, routine situations and sometimes in unfamiliar, non-routine situations, the student:

Number Sense:

· places numbers between 0 and 1000 on a number line

· expresses, reads, and writes whole numbers between 0 and 1000 as numerals and number words

· matches a fraction and number word to a pictorial representation of halves, thirds, and fourths

· adds and subtracts whole numbers up to 500 with regrouping

· distinguishes between odd and even numbers

· explains place value up to the tenth’s

· counts specific amounts of money using coins and bills up to $1.00

Data Analysis:

· collects and records data

· reads and interprets bar graphs

· identifies outcomes that are likely to occur in one-step probability experiments

Algebra:

· creates a five-item, single-attribute pattern and explains the logic of the sequence

· skip counts by 2’s, 5’s, and 10’s up to 20

· finds the missing element in a number sentence involving addition and subtraction

Geometry:

· identifies the characteristics of simple polygons (i.e., side, leg, angle, right angle)

Measurement:

· chooses the appropriate tool and unit to measure an object or event

Met

At this level, the student makes sound decisions about how to set up a problem and performs the following tasks by applying both procedural knowledge and conceptual understanding. The student explains the reasoning used and justifies the procedures selected with concrete objects and pictorial representations. The student notes connections between one problem and another.

Most of the time in both familiar, routine and unfamiliar, non-routine situations, the student:

Number Sense:

· adds and subtracts whole numbers up to 500

· multiplies and divides single digit numbers

· distinguishes between odd and even numbers

· identifies and describes models of common fractions

· makes a model to represent a fractional representation of halves, thirds, and fourths

· expresses whole numbers between 0 and 1000 in expanded notation

· selects appropriate operations to solve single-step word problems involving whole numbers between 0 and 500 for addition and subtraction and single digits for multiplication and division

· rounds whole numbers to the hundredths

· uses estimation to check the reasonableness of results in solving single step word problems

· expresses equal relationships of coins and currency up to $5.00

· demonstrates the meaning of addition and subtraction

Data Analysis:

· collects, records, and organizes data

· constructs, reads, analyzes, and interprets pictographs, circle graphs and bar graphs

· predicts the likelihood of events in any one-step probability experiment and compares the outcome of an experiment to the predictions

Algebra:

· creates, extends, and describes the logic of a variety of patterns

· skip counts up to 100 by 2’s, 5’s, and 10’s

Geometry:

· identifies the characteristics of simple polygons

· identifies the characteristics of simple solid geometric figures

Measurement:

· makes reasonable estimates and measures various attributes of objects and events with appropriate tools and measuring units

· solves real life problems involving measurements using U.S. customary units

· identifies the appropriate measurement of an object or event with U.S. customary units (length, capacity, weight, area, volume, time, and temperature)

Exceeds

At this level, the student:

· consistently performs all the above tasks in both familiar, routine and unfamiliar, non-routine situations

· identifies relationships, discriminate relevant from irrelevant information, sequences, prioritizes, and observes patterns

· shows mathematical reasoning in solutions in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models

· expresses solutions clearly and logically using appropriate mathematical notation and terms and clear language, and supports solutions with evidence, in both oral and written work

ABE II

Beginning

At this level, the student performs the following tasks with basic understanding of the concepts and limited reasoning skills. The student’s explanations are often minimal and presented without much supporting information.

Sometimes in familiar, routine situations the student:

Number Sense:

· adds and subtracts whole numbers up to 500

· multiplies and divides double digit numbers

· expresses equal relationships of coins using dimes, nickels, and pennies up to $5.00

· places numbers between 0 and 10,000 on a number line

· expresses, reads and writes whole numbers between 0 and 10,000 as numerals and numbers

· identifies models of mixed numbers

· uses estimation to check the reasonableness of results and rounds whole numbers to hundredths

Data Analysis:

· describes events that have a probability of 100% or 0%

· collects, records, and organizes data

· constructs, reads, analyzes, and interprets pictographs, circle graphs and bar graphs

Algebra:

· finds the missing element in some number sentences involving addition, subtraction, and multiplication

· sorts and classifies objects according to many observable attributes

· creates a five-item, single-attribute pattern and explains the logic of the sequence

· skip counts by 2’s, 5’s, and 10’s

Geometry:

· contrasts some of the attributes of simple polygons

· contrasts some of the attributes of simple solid geometric figures

Measurement:

· describes how the attributes of objects and events can be measured using different units of measurement

Approaching

Often in familiar, routine situations and sometimes in unfamiliar, non-routine situations, the student:

Number Sense:

· places numbers between 0 and 10,000 on a number line

· adds, subtracts, and multiplies whole numbers up to 1000 with regrouping

· explains place value up to the thousand’s place

· counts specific amounts of money using any coin or bill

· describes mixed numbers as parts of a whole

· expresses, reads and writes whole numbers between 0 and 10,000 as numerals and numbers

· identifies models of mixed numbers

· uses estimation to check the reasonableness of results and rounds whole numbers to thousandths

Data Analysis:

· organizes the data and constructs and reads pictographs, circle graphs and bar graphs

· identifies outcomes that are more likely or less likely to occur in one-step probability experiment

· describes events that have 100% or 0% probability

Algebra:

· finds the missing element in a number sentence involving addition, subtraction, multiplication, and division

· sorts and classifies objects according to many observable attributes

· extends and describes in writing the logic of a variety of geometric and numeric patterns

· uses words such as all and none to make reasonable statements about the probability of events

Geometry:

· contrasts many of the characteristics of simple polygons (i.e., side, leg, angle, right angle)

· contrasts many of the characteristics of simple solid geometric figures (i.e., edge, face),

Measurement:

· measures various attributes of objects and events with appropriate tools and customary and metric measuring units

· using U.S. customary or metric units, estimates a measurement of a given object or event and compares the estimation to actual measurement and justifies the reasonableness of the answer

Met

Most of the time in both familiar, routine and unfamiliar, non-routine situations, the student:

Number Sense:

· performs operations, estimates, and recognizes relationships with whole numbers up to 10,000

· expresses, reads and writes whole numbers between 0 and 10,000 as numerals and numbers

· expresses whole numbers between 0 and 10,000 in expanded notation

· identifies models of mixed numbers

· matches mixed numbers to pictorial representations and makes a model to represent a fractional representation of mixed numbers

· expresses equal relationships of coins and currency using pennies, nickels, dimes, quarters, half-dollars, and bills up to $100.00

· explains the meaning of multiplication and division and use one operation to check the answers of the other

· adds, subtracts, multiplies, and divides whole numbers between 0 and 1,000 correctly

· selects appropriate operation to solve one-step word problems involving whole numbers between 0 and 1,000

· uses estimation to check the reasonableness of results and rounds whole numbers to thousandths

Data Analysis:

· organizes the data and constructs, reads, analyzes, and interprets pictographs, circle graphs and bar graphs representing one unit and multiple units

· describes events that have 100% or 0% probability

· identifies outcomes that are more likely, less likely, or equally likely to occur

· describes the concept of sample

Algebra:

· sorts and classifies objects according to observable attributes

· creates, extends, and describes in writing the logic of a variety of geometric and numeric patterns

· uses words such as all, none, some, and many to make reasonable statements about the probability of events

· describes a rule for a simple pattern

Geometry:

· contrasts the characteristics of simple polygons

· contrasts the characteristics of simple, solid geometric figures

· identifies the characteristics of intersecting, parallel, and perpendicular lines

Measurement:

· measures various attributes of objects and events with appropriate tools and customary and metric measuring units

· solves real life problems involving measurements using U.S. customary and metric units

· using U.S. customary or metric units, estimates a measurement of a given object or event and compares the estimation to actual measurement and justifies and judges the reasonableness of the answer

· compares units of measurement to determine more or less relationships using U.S. customary and metric units (e.g., 2 cups = 1 pint, 3 cups > 1 pint)

Exceeds

At this level, the student:

· consistently performs all the above tasks in both familiar, routine and unfamiliar, non-routine situations

· analyzes problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing, and observing patterns

· applies strategies and results from simpler problems to more complex situations

· shows mathematical reasoning in solutions in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models

· expresses the solution clearly and logically using appropriate mathematical notation and terms and clear language, and supports solutions with evidence, in both oral and written work

· indicates the relative advantages of exact and approximate solutions to problems and gives answers to a specified degree of accuracy

ABE III

Beginning

At this level, the student exhibits some evidence of conceptual and procedural understanding of the following tasks in routine situations. Generally, the student is able to determine which of the available data are necessary and sufficient for correct solutions although the student shows limited skill in communicating mathematically.

Sometimes in familiar, routine situations, the student:

Number Sense:

· performs operations on whole numbers up to 10,000 and decimals to the tenths place

· solves one-operation word problems containing some irrelevant information

· expresses a quantity as an equivalent fraction, decimal, and percent

· reads and writes fractions, decimals, and percents as numerals and number words

· reads and writes numerals between 1000 and 1,000,000,000

· reads and writes whole numbers between 1000 and 1,000,000,000 as number words

· writes whole numbers between 1000 and 1,000,000,000 in expanded notation

· places numbers between 1000 and 1,000,000,000 in correct sequence

Data Analysis:

· reads and interprets a chart

Algebra:

· when given word problems using one variable and a constant, identifies the variable and the constant

· translates the word problem into a one-operation expression using correct mathematical symbolism (e.g., <, >,¹ , = )

Geometry:

· using the student’s vocabulary, identifies and draws an angle

· using the student’s own vocabulary, identifies the attributes of:

- similarity, congruence, and symmetry in geometric figures

- alternate interior, corresponding, complementary, and supplementary angles

- equilateral, acute, and obtuse triangles

- circle, cylinder, parallelogram and pentagon

Measurement:

· converts common U.S. linear and time measurements into equivalent measurements

Approaching

This level of performance signifies an understanding of arithmetic operations and some ability to use fundamental algebraic and informal geometric concepts in problem solving. The student is able to solve problems through the appropriate selection and use of strategies and tools and by distinguishing between relevant and irrelevant information. The student recognizes the degree of precision needed in the answer. Written solutions are organized and presented with some supporting information.

Often in familiar, routine situations and sometimes in unfamiliar, non-routine situations, the student:

Number Sense:

· describes a fraction of any quantity as the relationship between the given numerator part(s) related to the entire number of part(s) in the whole denominator

· describes a decimal as the fractional representation of the quantity expressed as a whole number and/or tenths, hundredths, thousandths, etc.

· describes percents as a fraction or as parts out of 100

· performs operations on whole numbers up to 100,000, decimals to the hundredths place, and any simple fraction

· rounds any whole number to specified equivalent, any decimal to nearest hundredth, and any fraction to nearest half or whole

· solves two-operation word problems containing whole numbers up to 100,000, decimals up to hundredths, and any simple fraction

· identifies the whole, part, and percent in problems involving percents

· solves word problems involving averaging of whole numbers up to 100,000, decimals up to the hundredths place, and any simple fraction

· places in correct sequence fractions, decimals, and percents in same groups or mixed groups

· selects and uses correctly the operations of addition, subtraction, multiplication, and division in story problems involving whole numbers, fractions and decimals

· defines prime and composite numbers

Data Analysis:

· constructs, reads, and interprets a table and a line graph

Algebra:

· when given word problems with one variable and a constant, translates the facts of the situation into algebraic terms

· simplifies an expression by combining like terms

· solves a one-operation algebraic equation requiring addition and subtraction

Geometry:

· describes with appropriate vocabulary, draws, and accurately measures right, acute, obtuse, straight, and reflex angles

· using appropriate vocabulary, describes the attributes of:

- similarity, congruence, and symmetry in geometric figures

- alternate interior, corresponding, complementary, and supplementary angles

- equilateral, acute, obtuse, isosceles, and scalene triangles

- a circle, cylinder, parallelogram, pentagon, hexagon, octagon, decagon, rhombus, and trapezoid

Measurement:

· converts measurement units to equivalent units within a given system

· solves problems involving the perimeter of objects

· using own vocabulary, differentiates between perimeter, area, and volume

· differentiates between weight and mass

· differentiates between capacity and volume

· estimates and records measurements for circumference, angles, and distance in scale drawings

Met

At this level, the student has a thorough understanding of the concepts – an understanding sufficient for problem solving in practical situations. The student is able to convey underlying reasoning skills beyond the level of arithmetic operations to fundamental algebraic and geometric concepts in problem solving. The student is able to compare and contrast mathematical ideas and generate examples, distinguish between relevant and irrelevant information; sequence, prioritize, and observe patterns; and recognize the degree of precision needed in the answer. Written solutions are organized and presented both with supporting information and explanations of how they were achieved.

Most of the time in both familiar, routine and unfamiliar, non-routine situations, the student:

Number Sense:

· performs operations on whole numbers up to 1,000,000,000, decimals to the thousandths place, any simple fraction or mixed number, and percents

· represents any rational number as a numeral, number word, or expanded notation

· expresses a quantity as an equivalent fraction, decimal, and percent

· places in correct sequence whole numbers up to 1,000,000,000

· places in correct sequence fractions, decimals, and percents in same groups or mixed groups

· solves multiple-operation word problems involving whole numbers, fractions, decimals,

· identifies the whole, part, and percent in problems involving percents

· solves word problems involving averaging of whole numbers, fractions, or decimals

· solves word problems involving the order of operations

· places in correct sequence fractions, decimals, and percents in same groups or mixed groups

· selects and uses correctly the operations of addition, subtraction, multiplication, and division in story problems involving whole numbers, fractions and decimals

· defines prime and composite numbers

· identifies and defines multiples, factors, and square roots of numbers using own vocabulary

· sorts and defines numbers by their properties

Data Analysis:

· constructs, reads, analyzes, interprets, and solves word problems using tables, charts, circle graphs, and line graphs

· formulates questions from graphs, tables, and charts

· predicts outcomes in a two-step probability experiment and compares the outcomes to the predictions

Algebra:

· when given a word problem with one variable and a constant, translates the facts of the situation into algebraic terms

· constructs and solves a one-operation equation requiring addition, subtraction, multiplication, or division

· describes and uses a variable and a constant in a real life situation

· represents and describes how changing the value of one variable in a relationship results in a change in another

· uses correct order of operations in solving algebraic equations

· solves simple ratio and proportion problems

· translates word problems into algebraic terms

· defines a term, expression, equation, and inequality

· simplifies an expression by combining like terms

· uses mathematical symbols (e.g., <, >, ¹, =)

Geometry:

· draws, classifies, and measures right, acute, obtuse, straight, and reflex angles

· using appropriate vocabulary, identifies and describes the attributes of:

- similarity, congruence, and symmetry in geometric figures

- alternate interior, corresponding, complementary, and supplementary angles

- equilateral, acute, obtuse, isosceles, and scalene triangles

- a circle, cylinder, parallelogram, pentagon, hexagon, octagon, decagon, rhombus, and trapezoid

Measurement:

· solves problems involving the perimeter of any polygon

· differentiates between perimeter, area, and volume of any object

· uses formulas to find:

- area of simple polygon

- surface area of rectangular containers

- volume of rectangular containers

· converts measurement units to equivalent units within a given system

· compares estimated measurements between U.S. customary and metric systems

· compares estimated measurements between Fahrenheit and Celsius systems

· differentiates between weight and mass

· differentiates between capacity and volume

· estimates and records measurements for circumference, angles, and distance in scale drawings

Exceeds

At this level, the student:

· applies mathematical concepts and procedures consistently to solve complex problems in the various strands as noted above

· provides solutions that are clear, logical, and go beyond the obvious in their interpretations to identify significant connections

· moves beyond a particular problem by probing examples and counterexamples, making general conclusions, summary statements and posing new, related questions and comments

· creates unique problem-solving techniques and explains the reasoning process underlying the conclusions

· analyzes problems by identifying relationships, discriminating relevant from irrelevant information, sequencing and prioritizing, and observing patterns

· shows mathematical reasoning in solutions in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams and models

· expresses the solution clearly and logically using appropriate mathematical notation and terms and clear language, and supports solutions with evidence, in both oral and written work

· indicates the relative advantages of exact and approximate solutions to problems and give answers to a specified degree of accuracy

ASE I/GED

Beginning

At this level, the student exhibits some evidence of conceptual and procedural understanding when solving problems and performing the tasks below. The student is able to determine which of the available data are necessary and sufficient for correct solutions and use them in problem solving; however, the student’s skill in communicating mathematically about these concepts is limited.

Sometimes in familiar, routine situations, the student:

Number Sense:

· explains the concept of positive and negative numbers

· sets up a ratio/proportion problem

Data Analysis:

· organizes and represents data

· formulates predictions based on a data set

· expresses probability as a simple fraction or percent

Algebra:

· recognizes whether positive or negative numbers are to be used in creating algebraic expressions

· solves word problems involving computation of cost, distance, and simple interest

Geometry:

· applies the appropriate geometric formula from the GED Math test

· uses similarity and proportionality for problem solving

· locates an ordered pair of positive numbers on a rectangular coordinate plane

Measurement:

· converts units of measurement into equivalent units of measurement using proportion

· converts units of measurement into equivalent units in the metric system by the movement

of the decimal point one place value in either direction (e.g., 45 mm = 4.5 cm,

or 4.5 m = 450 cm)

Approaching

The student is able to apply reasoning and generalize from some patterns and examples in the areas of algebra, geometry, and statistics. The student is able to use correct mathematical language and symbols to communicate many mathematical relationships and reasoning processes. Written solutions are organized and presented with some supporting information.

Often in familiar, routine situations and sometimes in unfamiliar, non-routine situations, the student:

Number Sense:

· uses computation and estimation to solve word problems involving integers, exponents, square roots, and scientific notation

· places positive and negative numbers on a number line

· adds, subtracts, multiplies, and divides positive and negative numbers

· estimates the square root of any whole number to the nearest whole number

Data Analysis:

· organizes and represents a given data set graphically

· formulates predictions based on a given data set

· finds the mean, median, and mode of a data set

· expresses probability as a fraction or percent

Algebra:

· solves computations of cost, distance, and simple interest word problems

· determines slope of a line

· when given word problems, solves algebraic equations

· solves multi-operational equations

Geometry:

· applies the appropriate geometric formula (i.e., area, perimeter, volume, Pythagorean relationship, distance between two points in a plane) from the GED Math test

· uses similarity and proportionality for problem solving

· defines and graphs ordered pairs of positive numbers on a rectangular coordinate plane

Measurement:

· converts units of measurement into equivalent units of measurement using proportion

· converts units of measurement into equivalent units in the metric system by the movement of the decimal point in either direction any number of place values

· uses scientific notation to express whole numbers and fractions

Met

At this level, the student has a thorough understanding of the concepts listed below – an understanding sufficient for problem solving in practical situations. The student is able to apply reasoning and generalize from some patterns and examples as well as integrate mathematical concepts and procedures in the areas of algebra, geometry, and statistics. The student is able to judge and defend the reasonableness of answers, make conjectures, defend ideas, and give supporting examples. The student is able to compare and contrast mathematical ideas and generate examples; distinguish between relevant and irrelevant information; sequence, prioritize, and observe patterns; and recognize the degree of precision needed in the answer. Written solutions are organized and presented both with supporting information and explanations of how they were achieved.

Most of the time in both familiar, routine and unfamiliar, non-routine situations, the student:

Number Sense:

· uses computation, estimation, and/or proportions to solve word problems involving integers, rational numbers, exponents, square roots, and scientific notation

· estimates the square root of any whole number to the nearest whole number

· places integers in correct sequence

· adds, subtracts, multiplies, and divides positive and negative numbers and explains the effect

· represents and uses numbers with exponents

Data Analysis:

· expresses probability as a fraction or percent

· finds the mean, median, mode, quartile, and range of a data set

· chooses an appropriate graphic format to organize and represent data

· makes valid inferences and evaluates the reasonableness of conclusions drawn from data

· formulates and justifies predictions from a given set of data

· differentiates between a sampling and a census

· uses simulations to determine probabilities of real-world situations

Algebra:

· when given word problems, solves multi-operation equations

· solves algebraic equations using substitutions

· sets up and solves ratio and proportion problems

· solves computations of cost, distance, and simple interest word problems

· determines slope of a line

Geometry:

· recognizes, defines, applies the appropriate geometric formula (i.e., area, perimeter, volume, Pythagorean relationship, distance between two points in a plane) from the GED Math test

· uses similarity, proportionality, and alternate interior angles for problem solving

· defines and graphs any ordered pair on a rectangular coordinate plane

Measurement:

· using proportion method, converts units of measurement into equivalent units of measurement

· converts units of measurement to equivalent units of measurement in the metric system

· uses scientific notation to express whole numbers, fractions, and units of measurement

Exceeds

At this level, the student:

· applies mathematical concepts and procedures consistently to solve complex problems in the various strands

· applies strategies and results from simpler problems to more complex situations and integrates concepts and techniques from different areas of mathematics to solve problems

· express the solution clearly and logically using appropriate mathematical notation and terms and clear language, and supports solutions with evidence, in both oral and written work

· formulate generalizations of the results obtained and extends them to other areas of mathematics and other circumstances, including expressing the solution as a general rule

· creates models through probing examples and counterexamples

· communicate mathematical reasoning through the clear, concise, and correct use of mathematical symbolism and logical thinking

· explain the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation

ASE II

Beginning

At this level, the student exhibits some evidence of conceptual and procedural understanding when solving problems and performing the tasks below. The student is able to determine which of the available data are necessary and sufficient for correct solutions and use them in problem solving; however, the student’s skill in communicating mathematically about these concepts is limited.

Sometimes in familiar, routine situations, the student:

Number Sense

· Explains the meaning of absolute value

· Uses positive and negative exponents

Data Analysis

· Evaluates the reasonableness of conclusions drawn from interpretation of data in a graphic format

· Constructs histograms, line graphs, circle graphs and box-and-whisker plots

· Uses mode, quartiles and range as a means for effective decision making in analyzing the data

· Explains the effects of sampling on statistical claims and recognizes misuses of statistics

· Determines probabilities through experiments and/or simulations and compares the results with predictions

Algebra

· Identifies the independent and dependent variables from a real-life situation

· Expresses the relationship between two variables using a table, equation, graph, and matrix and describes the relationship suggested by two or more graphs

· Creates a graph from a table of values

· Writes an equation of the line that passes through two given points

· Evaluates algebraic expressions using substitution

· Multiplies and divides monomial expressions with integer exponents

· Solves linear equation and inequalities in one variable

Geometry

· Sketches prisms, pyramids, cones, and spheres

· Calculates surface areas and volumes of three- dimensional geometric figures given the required formulas

· Identifies arcs, chords, tangents and secants of a circle

· Classifies transformations based on whether they produce congruent or similar non-congruent figures

· Determines whether a given pair of figures on a coordinate plane represents a translation, reflection, rotation and/or dilation

Approaching

At this level, the student demonstrates some procedural and conceptual knowledge in solving problems in the following areas. The student is able to apply reasoning and generalize from some patterns and examples in the areas of algebra, geometry, and statistics. The student is able to use the correct mathematical language and symbols to communicate many mathematical relationships and reasoning processes.

Often in familiar, routine situations and sometimes in unfamiliar, non-routine situations, the student:

Number Sense

· Explains the meaning of absolute value

· Uses positive and negative exponents

· Compares and contrasts the real number system and its various subsystems with regard to their structural characteristics

Data Analysis

· Evaluates the reasonableness of conclusions drawn from interpretation of data in a graphic format

· Constructs histograms, line graphs, circle graphs and box-and-whisker plots

· Uses mode, quartiles and range as a means for effective decision making in analyzing the data

· Explains the effects of sampling on statistical claims and recognizes misuses of statistics

· Determines probabilities through experiments and/or simulations and compares the results with predictions

· Determines, from a given plot of data, whether it has strong or weak, positive or negative correlation

Algebra

· Identifies the independent and dependent variables from a real-life situation

· Expresses the relationship between two variables using a table, equation, graph, and matrix and describes the relationship suggested by two or more graphs

· Creates a graph from a table of values

· Writes an equation of the line that passes through two given points

· Determines from two linear equations whether the lines are parallel, are perpendicular or coincide

· Uses the definitions of trigonometric functions to find the sine, cosine and tangent of the acute angles of a right triangle

· Evaluates algebraic expressions using substitution

· Multiplies and divides monomial expressions with integer exponents

· Solves linear equation and inequalities in one variable, and systems of linear equations in two variables

Geometry

· Sketches prisms, pyramids, cones, and spheres

· Calculates surface areas and volumes of three- dimensional geometric figures given the required formulas

· Identifies arcs, chords, tangents and secants of a circle

· Verifies characteristics of a given geometric figure using coordinate formulas such as distance, mid-point, and slope to confirm parallelism, perpendicularity, and congruency

· Classifies transformations based on whether they produce congruent or similar non-congruent figures

· Determines whether a given pair of figures on a coordinate plane represents a translation, reflection, rotation and/or dilation

Logic & Reasoning

· Determines the validity of arguments

· Draws a simple valid conclusion from a given if ... then statement and a minor premise and places the statements in logical order

· Uses Venn diagrams to determine the validity of an argument

· Recognizes the difference between a statement verified by mathematical proof (i.e., a theorem) and one verified by empirical data

Met

At this level, the student has a thorough understanding of the concepts listed below – an understanding sufficient for problem solving in practical situations. The student is able to apply reasoning and generalize from some patterns and examples as well as integrate mathematical concepts and procedures in the areas of algebra, geometry, and statistics. The student is able to judge and defend the reasonableness of answers, make conjectures, defend ideas, and give supporting examples. The student is able to analyze problems by identifying relationships, discriminating relevant from irrelevant information, identifying missing information, sequencing and prioritizing information, and observing patterns. Written solutions are organized and presented both with supporting information and explanations of how they were achieved.

Most of the time in both familiar, routine and unfamiliar, non-routine situations, the student:

Number Sense

· Explains the meaning of absolute value

· Uses positive and negative exponents

· Compares and contrasts the real number system and its various subsystems with regard to their structural characteristics

Data Analysis

· Evaluates the reasonableness of conclusions drawn from interpretation of data in a graphic format

· Organizes collections of data into frequency charts, stem-and-leaf plots, scatter plots and matrices and determine outliers

· Applies curve fitting to make predictions from data

· Explains the effects of sampling on statistical claims and recognizes misuses of statistics

· Determines probabilities through experiments and/or simulations and compares the results with predictions

· Designs a statistical experiment based on a given hypothesis

· Describes, in general terms, the normal curve and uses its properties to answer questions about sets of data that are assumed to be normally distributed

· Explains and uses the concept of a random variable

· Draws conclusions about the “spread” of data given the variance and standard deviation

Algebra

· Expresses the relationship between two variables using a table, equation, graph, and matrix and describes the relationship suggested by two or more graphs

· Creates a linear equation from a table of values and graphs a linear equation and linear inequality in two variables

· Determines slope and intercepts of a linear equation

· Writes an equation of the line that passes through two given points

· Determines from two linear equations whether the lines are parallel, are perpendicular or coincide

· Solves simple right-triangle trigonometric equations involving sine, cosine and tangent and uses an appropriate right-triangle trigonometric model to solve a real-life problem

· Simplifies numerical expressions using the order of operations including exponents

· Simplifies square roots and cube roots with monomial radicands that are perfect squares or perfect cubes

· Evaluates numerical and algebraic absolute value expressions and algebraic expressions using substitution

· Multiplies and divides monomial expressions with integer exponents

· Solves linear equation and inequalities in one variable and two variables, quadratic equations, radical equations involving one radical, absolute value equations, systems of linear equations in two variables

· Solves proportions which generate linear or quadratic equations

Geometry

· Sketches prisms, pyramids, cones, cylinders and spheres and classifies them by base shape, lateral surface shape, related surface area and volume formulas

· Calculates surface areas and volumes of three-dimensional geometric figures given the required formulas

· Deduces properties of, comparisons of, and relationships between geometric figures from given assumptions using informal deductive reasoning

· Identifies arcs, chords, tangents and secants of a circle

· Translates between synthetic and coordinate representations (e.g., a straight line is represented by the algebraic equation Ax + By = C)

· Verifies characteristics of a given geometric figure using coordinate formulas such as distance, mid-point, and slope to confirm parallelism, perpendicularity, and congruency

· Applies transformational principles to practical situations (e.g., enlarge a photograph) and gives the new coordinates of a transformed geometric figure

· Deduces properties of figures using transformations in coordinate systems, identifying congruency and similarity

· Determines the effects of a transformation on linear and area measurements of the original figure and sketches the figure that is the result of a given transformation

Logic & Reasoning

· Uses inductive and deductive logic to construct simple valid arguments

· Determines the validity of arguments and if the converse of a given statement is true or false

· Draws a simple valid conclusion from a given if ... then statement and a minor premise and places the statements in logical order

· Analyzes assertions about everyday life by using principles of logic

· Uses Venn diagrams to determine the validity of an argument

· Recognizes the difference between a statement verified by mathematical proof (i.e., a theorem) and one verified by empirical data

· Formulates counterexamples and uses indirect proof to show that a given invalid conjecture is false

· Determines the purpose of and writes an algorithm that explains a particular mathematical process

Exceeds

At this level, the student:

· applies mathematical concepts and procedures consistently to solve complex problems in the various strands

· applies strategies and results from simpler problems to more complex situations and integrates concepts and techniques from different areas of mathematics to solve problems

· expresses the solution clearly and logically using appropriate mathematical notation and terms and clear language, and supports solutions with evidence, in both oral and written work

· formulates generalizations of the results obtained and extends them to other areas of mathematics and other circumstances, including expressing the solution as a general rule

· creates models through probing examples and counterexamples

· communicates mathematical reasoning through the clear, concise, and correct use of mathematical symbolism and logical thinking

· explains the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation

· employs forms of mathematical reasoning and proof appropriate to the solution of the problem at hand, including deductive and inductive reasoning, making and testing conjectures and using counterexamples and indirect proof