Activity that - Newark Public Schoolscontent.nps.k12.nj.us/wp-content/uploads/sites/46/2014/0… · Web viewDefine and create tessellations (3.8) Secure. 4a. Divide decimal numbers

A Curriculum Guide forMathematics

GRADE 5

Newark Public SchoolsOffice of Mathematics

2004-2005

NEWARK PUBLIC SCHOOLS

NEWARK PUBLIC SCHOOLSADMINISTRATION

2004-2005

District Superintendent………………………........……..……………Ms. Marion A. Bolden

District Deputy Superintendent……..………….........….…………….Ms. Anzella K. Nelms

Chief of Staff……..………….........….………………………………….....Ms. Bessie H. White

Chief Financial Officer………………......………………...………………...…Mr. Ronald LeeHuman Resource Services

Assistant Superintendent………….……….…..................…….....Ms. Joanne C. BergamottoSchool Leadership Team I

Assistant Superintendent………….……….…....................……………..Dr. J. Russell GarrisSchool Leadership Team II

Assistant Superintendent……………………..…………..........…Dr. Glenda Johnson-GreenSchool Leadership Team

Assistant Superintendent…………………............……….......……………... Ms. Lydia SilvaSchool Leadership Team IV

Assistant Superintendent……..…………........................……………….…Dr. Don MarinaroSchool Leadership Team V

Assistant Superintendent…………………………….........….………….Dr. Gayle W. GriffinDepartment of Teaching and Learning

Associate Superintendent………………………………………………...Ms. Alyson BarillariDepartment of Special Education

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Associate Superintendent…………………………………..…………...Mr. Benjamin O'NealDepartment of Special Programs

Departmentof

Teaching and Learning

Dr. Gayle W. GriffinAssistant Superintendent

Office of Mathematics

May L. SamuelsDirector

GRADE 5 MATHEMATICSCURRICULUM

GUIDE2

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Table of Contents

Mission Statement...........................................................................................4

Philosophy........................................................................................................5

To the Teacher.................................................................................................6

Content Emphasis............................................................................................7

Suggested Timeline.........................................................................................9

Suggested Pacing and Objectives (with New Jersey Core Content Standards).......................................................................................................................10

Open Ended Problem Solving and Scoring.....................................................20

Grade 5 Sample Open Ended Problem....................................................................................21

Reference:Instructional Technology (Web Sources)........................................................27

NJCCCS and Cumulative Progress Indicators.................................................29

Holistic Scoring Guide for Math Open-Ended Items.......................................43

National Council of Teachers of Mathematics Principles and Standards........44

Glossary.........................................................................................................45

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Mission Statement

The Newark Public Schools recognizes that each child is a unique individual possessing talents, abilities, goals, and dreams. We further recognize that each child can be successful only when we acknowledge all aspects of that child’s life: addressing their needs; enhancing their intellect; developing their character; and uplifting their spirit. Finally, we recognize that individuals learn, grow, and achieve differently; and it is therefore critical that, as a district, we provide a diversity of programs based on student needs.

As a district we recognize that education does not exist in a vacuum. In recognizing the rich diversity of our student population, we also acknowledge the richness of the diverse environment that surrounds us. The numerous cultural, educational, and economic institutions that are part of the greater Newark community play a critical role in the lives of our children. It is equally essential that these institutions become an integral part of our educational program.

To this end, the Newark Public Schools is dedicated to providing a quality education, embodying a philosophy of critical and creative thinking and designed to equip each graduate with the knowledge and skills needed to be a productive citizen. Our educational program is informed by high academic standards, high expectations, and equal access to programs that provide and motivate a variety of interests and abilities for every student based on his or her needs. Accountability at every level is an integral part of our approach. As a result of the conscientious, committed, and coordinated efforts of teachers, administrators, parents, and the community, all children will learn.

Adapted from: The Newark Public Schools Strategic Plan

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Philosophy

“Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. There are ambitious expectations for all, with accommodation for those who need it. Knowledgeable teachers have adequate resources to support their work and are continually growing as professionals. The curriculum is mathematically rich, offering students opportunities to learn important mathematical concepts and procedures with understanding. Technology is an essential component of the environment. Students confidently engage in complex mathematical tasks chosen carefully by teachers. They draw on knowledge from a wide variety of mathematical topics, sometimes approaching the same problem from different mathematical perspectives or representing the mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine, and explore conjectures on the basis of evidence and use a variety of reasoning and proof techniques to confirm or disprove those conjectures. Students are flexible and resourceful problem solvers. Alone or in groups and with access to technology, they work productively and reflectively, with the skilled guidance of their teachers. Orally and in writing, students communicate their ideas and results effectively. They value mathematics and engage actively in learning it.” *

This model, envisioned in the NCTM Standards 2000, is the ideal, which Newark Public Schools hopes to achieve in all mathematics classrooms. We believe the classroom described above is attainable through the cooperative efforts of all Newark Public Schools stakeholders.

*A Vision for School Mathematics National Council of Teachers of

Mathematics Standards 2000

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To the Teacher

The Everyday Mathematics Program is a standards-based program that is a complete K-6 mathematics curriculum that embraces many of the traditional goals of school mathematics as well as two ambitious new goals:

To substantially raise expectations with respect to the amount and range of mathematics that children can learn

To provide materials for children and support for teachers that enable them to meet these higher expectations.

Everyday Mathematics introduces children to all the major mathematical content domains - number sense, algebra, measurement, geometry, data analysis and probability - beginning in Kindergarten. The program helps teachers move beyond basic arithmetic and nurture higher order and critical thinking skills in their students, using everyday, real-world problems and situations – while also building and maintaining basic skills, including automatic fact recall.

The Everyday Mathematics program features a spiraling curriculum in which mathematical content is taught in a repeated fashion, beginning with concrete experiences. Children learn best when new topics are presented briskly and in an interesting way. Most children will not master a new topic the first time it is presented, so Everyday Mathematics allows children to revisit content in varied contexts, integrating new learning with previous knowledge. Everyday Mathematics periodically reviews, practices, and applies newly learned concepts and skills in wide variety of contexts.

It is important to note how the differences between Everyday Mathematics and other programs may effect day-to-day planning and teaching. Daily routines and games are a necessary part of the program, not optional extensions. Routines and games are designed to build conceptual understanding and ensure mastery of basic skills. The Everyday Mathematics program is designed for the teacher, offering materials that provide students with a rich variety of experiences across mathematical content strands. Everyday Mathematics employs cooperative learning activities, explorations, problem solving, and projects. The classroom needs to be set up to accommodate group work, and students must be able to work together without direct supervision.

Assessment is closely linked with instruction. While some formal assessment is necessary (district and state-mandated tests), a balanced approach, including less formal, ongoing methods, will provide a more complete picture of student progress. A number of assessment tools are built into the program to help create an assessment program that will give feedback about students’ instructional needs.

Everyday Mathematics assumes that virtually all students are capable of a much greater understanding of and proficiency in mathematics than has been traditionally expected. The program establishes high expectations for

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all students and gives teachers the tools they need to help students meet, and often exceed, these expectations.

Grade 5 Everyday Mathematics Content Emphasis

Numeration

Reading, writing, and comparing negative numbers, fractions, whole numbers through billions, and decimals through thousandths; reading, writing and interpreting whole-number powers of 10; translating between exponential and standard notation; understanding and identifying prime numbers, composite numbers, and square numbers.

Operations and Computation

Using paper-and-pencil algorithms to add, subtract, multiply, and divide multidigit

whole numbers and decimals; using mental arithmetic to compute exact answers and to estimate; rounding from billions to hundredths; translating among fractions, decimals, and percents; prime factoring; converting between fractions and mixed numbers; adding and subtracting fractions and mixed numbers with unlike denominators; finding least common multiples and greatest common factors; multiplying and dividing fractions.

Data and Chance

Comparing probabilities for different outcomes; comparing theoretical and experimental probabilities; expressing probabilities as fractions, decimals, and percents; drawing justifiable conclusions from data; displaying data in more than one way; formulating a question, carrying out a survey or experiment, recording data, and communicating results; drawing and interpreting circle graphs and stem-and-leaf plots; understanding measures of central tendency (mean, median, mode).

Measurement and Reference Frames

Geometry

Constructing a circle with a given radius or diameter; defining and creating tessellations; measuring and drawing angles, including reflex and straight angles; identifying and defining right, isosceles, and equilateral triangles; plotting points in four quadrants; using translations, reflections, and rotations; solving perimeter, area, and volume problems; understanding the relationship between the volumes of cones/pyramids and cylinders/prisms; finding the surface area of a cube and the area of a circle; identifying angle relationships in triangles and in quadrilaterals.

Measurement and Reference Frames

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Measuring and estimating length, area, volume, weight, and capacity; converting and computing with common units of measure; creating scale drawings.

Patterns, Functions, and Algebra

Evaluating simple algebraic expressions; finding rules for patterns; finding the nth term in a sequence; solving simple open number sentences and simple rate problems; working with equations by doing the same thing to both sides; understanding simple direct proportion; using variables and equations to represent situations; graphing ordered pairs; translating among verbal, numerical, and graphical representations

Problem Solving

Within these content strands, Everyday Mathematics emphasizes: A problem-solving approach based on everyday situations that

develops critical thinking. Mathematical communication, including understanding and evaluating

the mathematical thinking and strategies of others. Frequent practice of basic skills through ongoing program routines and

mathematical games. An instructional approach that revisits topics regularly to ensure full

concept development. Activities that explore a wide variety of mathematical content and offer

opportunities for students to apply their knowledge.

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Suggested TimelineThis guide provides a full description of the mathematics objectives for Everyday Mathematics Grade 5 and correlates them to the New Jersey Core Curriculum Content Standards for Mathematics (NJCCC) for Grade 5. The Mathematical Process Standards: Problem Solving, Communication, Connections, Reasoning, Representations, and Technology, although not explicitly referenced, are integrated throughout the mathematics program.

PACING GUIDEMonth Lessons

September Lessons 1.1 - 2.2 12 Lessons

October Lessons 2.3 - 3.5 14 Lessons

November Lessons 3.6 - 4.5 11 Lessons

December Lessons 4.6 - 5.8 10 Lessons

January Lessons 5.9 - 6.9 14 Lessons

February Lessons 6.10 - 7.11 13 Lessons

March Lessons 8.1 - 8.13 13 Lessons

April Lessons 9.1 - 10.2 13 Lessons

May Lessons 10.3 - 11.6 14 Lessons

June Lessons 11.7 - 12.10 12 Lessons

Please Note: The NJCCCS for grade 5 requires student proficiency in division (Numerical Operations, B3). The Everyday Math curriculum identifies the objectives concerning division introduced in November (4a to 4h) as developing skills. Students must be secure in these skills as a pre-requisite for the grade 6 Connected Mathematics curriculum.

TEACHERS MUST PROVIDE INSTRUCTION AND PRACTICE IN DIVISION SKILLS THROUGHOUT THE YEAR SO THAT STUDENTS BECOME SECURE IN THESE SKILLS.

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Suggested Pacing

Objectives/Everyday Mathematics NJCCC Standard and Strand

September

1a. Find the prime factorization of numbers (1.9) Beginning

1b. Rename numbers written in exponential notation (1.7-1.9) Beginning/Developing

1c. Use a divisibility test to determine if a number is divisible by another number (1.5) Developing/Secure

1d. Identify prime and composite numbers (1.6, 1.9) Developing/Secure

1e. Understand how square numbers and their square roots are related

(1.8) Developing/Secure

1f. Draw arrays to model multiplication

(1.2, 1.7) Secure

1g. Know multiplication facts (1.2-1.9) Secure

1h. Identify even and odd numbers (1.4, 1.5) Secure

1i. Find the factors of numbers (1.3, 1.4, 1.6, 1.9) Secure

2f. Find the sum and difference of multidigit whole numbers and decimals (2.2, 2.3) Secure

5.4.1 Number and Numerical Operations

Number Sense - A5

6.4.3 Algebra and Patterns Procedures - D4 (NJCCCS Gr. 6)5.4.1 Number and Numerical Numerical Operations B3, B5

Number Sense - A5


Numerical Operations

B5 (NJCCCS Gr. 6)4.4.1 Number and

Numerical Operations Numerical Operations

B1, B2 (NJCCCS Grade 4)


Number Sense - A7 (NJCCCS Gr. 6)


Number Sense A1, A2, A4 Numerical

Operations B1

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*Learning Goals that should be secure by the end of each unit appear in bold type.

Suggested Pacing


October 2a. Write and solve open sentences for number stories (2.4) Beginning

2b. Round numbers to designated places (2.7) Developing

2c. Make magnitude estimates (2.7) Developing/Secure

2d. Find the product of multidigit whole numbers and decimals (2.8, 2.9) Developing/Secure

2e. Know place value to billions (2.10) Developing/Secure

2f. Find the sum and difference of multidigit whole numbers and decimals (2.3) Secure

2g. Identify the maximum, minimum, median, mode, and mean for a data set (2.5) Secure

3a. Determine angle measures based on relationships between angles (3.3-3.5) Developing

3c. Measure an angle to within 2º (3.4) Developing/Secure

3d. Identify types of angles (3.4, 3.5) Developing/Secure

3f. Identify place value in numbers to billions (3.2) Secure

5.4.3 Patterns and Algebra Modeling - C1 Procedures - D1


Estimation - C1-4 Number Sense A3

Number Sense A1, A2, A4

5.4.4 Data Analysis, Probability, and Discrete Mathematics

Data Analysis - A2

5.4.2 Geometry and Measurement

Geometric Properties

A1, A2, A3, A4

5.4.1 (NJCCCS grade 4) Number and

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Number Sense - A1, 2, 6


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Suggested Pacing


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November

Provideample practice in divisionskillsthroughout the year.

3a. Determine angle measures based on relationships between angles

(3.9) Developing

3b. Estimate the measure of an angle (3.6, 3.8) Developing/Secure

3c. .Measure an angle to within 2º (3.9) Developing/Secure

3e. Identify types of triangles (3.6) Developing/Secure

3g. Know properties of polygons (3.7) Secure

3h. Define and create tessellations (3.8) Secure

4a. Divide decimal numbers by whole numbers with no remainders (4.4, 4.5) Beginning

4b. Write and solve number sentences with variables for division number stories (4.5) Beginning/Developing

4c. Find the quotient and remainder of a whole number divided by a 1-digit whole number (4.1, 4.2, 4.4, 4.5) Developing

4d. Find the quotient and remainder of a whole number divided by a 2-digit whole number (4.2, 4.4, 4.5) Developing

4e. Make magnitude estimates for quotients of whole and decimal numbers divided by whole numbers (4.4, 4.5) Developing

4f. Interpret the remainder in division number stories (4.5) Developing

4h. Know place value to hundredths (4.1) Secure



A1, A2, A3, A4

Coordinate Geometry

C1


A2 Transforming

Shapes B15.4.1 Number and


Number Sense - A1, A2, A4

5.4.3 Patterns and Algebra Modeling - C1


Numerical Operations -

B1-6

Estimation - C1-4


Numerical Operations -

B1-6

Number Sense - A1-6

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16

Suggested Pacing


December 4b. Write and solve number sentences with variables for division number stories

(4.6) Beginning/Developing

4g. Determine the value of a variable; use this value to complete a number sentence

(4.6) Developing

5a. Add fractions with like denominators (5.3) Beginning/Developing

5b. Order and compare fractions (5.3) Developing

5c. Convert between fractions and percents

(5.8) Developing

5f. Convert between fractions and mixed numbers (5.2) Developing/Secure

5g. Find equivalent fractions (5.4) Developing/Secure

5.4.3 Patterns and Algebra Modeling - C1

5.4.1 Number and Numerical

Operations Number Sense A1, A4, A6 Numerical

Operations B1, B2


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Suggested Pacing


January 5d. Draw a circle graph for a set of data (5.11) Developing

5e. Measure pieces of a circle graph; interpret a circle graph (5.10) Developing

6a. Construct stem-and-leaf plots (6.3) Beginning/Developing

6b. Read and interpret stem-and-leaf plots (6.4) Beginning/Developing

6c. Add and subtract fractions with like denominators (6.8, 6.9) Developing

6d. Add and subtract fractions with unlike denominators (6.8, 6.9) Developing

6e. Understand how sample size affects results (6.5) Developing

6f. Find common denominators (6.9) Developing

6g. Convert between fractions, decimals, and percents (6.5, 6.8) Developing/Secure

6h. Identify and use data landmarks (6.1, 6.5, 6.6) Secure


Data Analysis - A2


Data Analysis - A2 (NJCCCS Gr.6)



Operations B1, B2


Data Analysis - A1-3



Operations B1, B2


Data Analysis - A1-3

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19

Suggested Pacing


February 6c. Add and subtract fractions with like denominators (6.10) Developing

6d. Add and subtract fractions with unlike denominators (6.10) Developing

6f. Find common denominators (6.10) Developing

6g. Convert between fractions, decimals, and percents (6.10) Developing/Secure

7a. Understand and apply scientific notation (7.3) Beginning/Developing

7b. Understand and apply powers of 10 (7.2) Developing

7c. Understand and apply order of operations to evaluate expressions and solve number sentences (7.5) Developing

7d. Add and subtract integers (7.7-7.10) Developing

7e. Understand and apply exponential notation (7.1, 7.2) Developing/Secure

7f. Determine whether number sentences are true or false (7.4) Developing/Secure

7g. Understand the function and placement of parentheses in number sentences


7h. Compare and order integers (7.6) Developing/Secure



Operations B1, B2


A 4, 5, 6 (NJCCCS Gr. 6)6.4.3 Patterns and

Algebra Procedures - D4 (NJCCCS Gr. 6) 4.4.1 Number and


A1, 2, 5 (NJCCCS Gr. 4)

Procedures - D2


Number Sense - A1 Numerical

Operations B1 Numerical

Operations B2 (NJCCCS Grade

7)

5.4.3 Patterns and Algebra

Modeling - C1 Procedures - D1

6.4.1 Number and Numerical Operation


B85.4.1 Number and


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Number Sense - A6


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Suggested Pacing


March 8a. Use an algorithm to multiply mixed numbers (8.8) Beginning

8b. Use an algorithm to multiply fractions (8.5-8.7, 8.9) Developing

8c. Use an algorithm to subtract mixed numbers with like denominators (8.3) Developing

8d. Find a percent of a number (8.9-8.11) Developing

8e. Use an algorithm to add mixed numbers (8.2, 8.4) Developing/Secure

8f. Order and compare fractions (8.1, 8.2, 8.12) Developing/Secure

8g. Convert among fractions, decimals, and percents (8.8, 8.9) Secure

8h. Convert between fractions and mixed or whole numbers (8.2, 8.3, 8.8) Secure

8i. Find common denominators (8.1, 8.2, 8.4, 8.12) Secure


Numerical Operations B2 (NJCCCS Grade 6)


Numerical Operations B26.4.1 Number and


Number Sense - A5 (NJCCCS Grade 6)


Numerical Operations B2 Number Sense - A6


Numerical Operations B25.4.1 Number and


Number Sense - A6

Numerical Operations B2


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Suggested Pacing


April 9a. Plot ordered pairs on a four-quadrant coordinate grid (9.3) Developing

9b. Understand the concept of volume of a

figure (9.8-9.10) Developing/Secure

9c. Use a formula to find the volume of prisms

(9.8, 9.9) Developing

9d. Plot ordered pairs on a one-quadrant

coordinate grid (9.1-9.3) Secure

9e. Identify the base and height of triangles

and parallelograms (9.4-9.6) Developing

9f. Use a formula to find the area of triangles and parallelograms


9g. Understand the concept of area of a figure (9.4-9.6) Secure

9h. Use a formula to find the area of rectangles (9.4) Secure

10a. Solve two-step pan-balance problems (10.2) Beginning

10f. Solve one-step pan-balance problems (10.1) Developing/Secure


Coordinate Geometry C1


Units of Measurement D2 (NJCCCS Grade 4)


Measuring Geometric Objects - E3

(NJCCCS Grade 6) Coordinate Geometry A2, E2, 3, 4


Units of Measurement

D2, 3, 4

5.4.3 Patterns and Algebra Modeling C1 Procedures D1


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Suggested Pacing


May 10a. Solve two-step pan-balance problems (10.5) Beginning

10b. Write algebraic expressions to represent situations (10.3-10.5, 10.7) Developing

10c. Represent rate problems as formulas, tables, and graphs (10.4-10.7) Developing

10d. Use formulas to find circumference and area of a circle (10.8, 10.9) Developing

10e. Distinguish between circumference and area of a circle (10.9) Developing

10f. Solve one-step pan-balance problems (10.5) Developing/Secure

10g. Interpret mystery line plots and graphs (10.7) Developing

11a. Understand the relationship between the volume of pyramids and prisms, and the volume of cones and cylinders

(11.4) Beginning

11d. Understand the concept of capacity and how to calculate it (11.6) Beginning

11e. Use formulas to find the volume of prisms and cylinders


11f. Use formulas to find the area of polygons and circles (11.1, 11.4) Secure

11g. Know the properties of geometric solids (11.1, 11.2) Secure

5.4.3 Patterns and Algebra Modeling - C1 Procedures -D 1

Functions and Relationships - B1

Modeling - C1, 2


Measuring Geometric Objects - E2

(NJCCCS Gr. 6)

5.4.3 Patterns and Algebra Modeling - C1 Procedures -D 1

Functions and Relationships - B2



A5 (NJCCCS Gr. 6)


D1 (NJCCCS Gr. 6)



A 2 (NJCCCS Gr. 4)

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25

Suggested Pacing


June 11b. Find the surface area of prisms (11.7) Beginning

11c. Understand how to find the surface area of cylinders (11.7) Beginning

12a. Understand and use tree diagrams to solve problems (12.2) Beginning

12b. Compute the probability of outcomes when choices are equally likely (12.2) Beginning

12c. Use the Multiplication Counting Principle to find the total number of possible outcomes of a sequence of choices

(12.2) Beginning/Developing

12d. Find the least common multiple of two numbers (12.1) Developing

12e. Find the least common multiple of two numbers (12.1) Developing

12f. Solve ratio and rate number stories (12.1, 12.3-12.8) Developing/Secure

12g. Find the factors of numbers (12.1) Secure

12h. Find the prime factorizations of numbers (12.1) Secure



D1 (NJCCCS Gr. 6)

5.4.4 Data Analysis, Probability and Discrete Mathematics

Discrete Mathematics Systematic Listing and Counting - C

Probability B1, 2, 3


Number Sense - A5

5.4.3 Patterns and Algebra

Modeling - C1, 2


Number Sense - A5


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Open-Ended Problem Solving and Scoring

The material on the following pages provides students with experience in solving open-ended tasks and in rubric-based scoring. Students should score the open-ended work samples using the scoring rubric on the following page and the New Jersey Holistic Scoring Guide, which is included in the Reference section of this document. Two purposes of these activities are: (1) to give students experience in formulating complete and accurate responses to open-ended questions, and (2) to generate greater understanding of the process by which they will be assessed. The content of the items is less significant than the answering and scoring process.

A suggested procedure for presenting each task is given below. Time limits are not set, because the nature of the task and the characteristics of the class should be taken into account.

Component 1Introduce the problem. (For the first problem, you may want to conduct a discussion of what constitutes a complete solution.) Students work in groups to read and discuss the solution of the problem.

Component 2 Groups write a solution for scoring.

Component 3

Present the sample solution and scoring rubric. Discuss given solution and alternate methods of solution, and make sure students understand the rubric. Groups self assess using the scoring rubric and/or exchange work to assess each other.

Component 4Discuss and justify scores, and discuss how scores could be improved. Class generates a “perfect” solution, which can be used as a model.

Component 5 Assess understanding through quiz, individual work, or other method of choice.

It is important that students have a thorough understanding of the scoring process. Be as specific as possible in validating scores and suggesting improvements to solutions.

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GRADE 5 OPEN-ENDED PROBLEM

Our class designed stamps to be displayed at our local post office. The postmaster said he had other things to display and wanted to know how much space was needed to hang up all our stamps.

If the stamps are 8 inches by 11 inches and there are 17 of them, what would you tell him?

Explain your solution. Show all your work.

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The rubric below was used to score students’ solutions.

Scoring Rubric for The Post Office Display Problem

3 points The student shows a strong understanding of the task and verifies his/her solution. He/she acknowledges the fact that there is more than one solution to this task. The student gives a clear explanation of how the problem was solved, and applies mathematical procedures correctly.

2 points

The student shows a broad understanding of the problem and major concepts needed to solve it. He/she attempts to use reasoning, but is not entirely successful. There is a clear explanation and mathematical procedures are used appropriately, but minor mathematical errors have been made.

1 points

0 points

The student uses a strategy that is partially useful (e.g.: marking off on graph paper where the stamps are). The students seems to understand that the sides of the stamps need to be added together, but may not compute accurately. There is some evidence of reasoning, and an attempt at using mathematical language and representation is made.It is unclear what this student has done. There may be an attempt at an explanation, but no solution is found. There is no evidence of a strategy.

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Grade 5 Student Sample 1

30


31


32


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Reference

Instructional Technology (Web Sources)

New Jersey Core Curriculum Content Standards

Cumulative Progress Indicators

Holistic Scoring Guide for Math Open-Ended Items

National Council of Teachers of Mathematics

Principles and Standards

Glossary for Selected Mathematics Terms

Instructional Technology (Web Sources)

34

Teachers and students who have access to the Internet, and the World Wide Web, can take advantage of a variety of useful sites. These have been selected because they are well -established and impressive.

National Council of Teachers of MathematicsContains news and information of interest to math teachers.

http://www.nctm.org

New Jersey State Department of EducationInformation on the department’s proposals and regulations, including state testing program.

http://www.state.nj.us/education/

Everyday Mathematics Home PageInformation, teacher resources, professional development opportunities, and more for the Everyday Mathematics Program.

http://everyday math.uchicago.edu

Math GoodiesInteractive math lessons, as well as homework help, puzzles, calculators.

http://www.mathgoodies.com

The Math ForumA center for teachers, students, parents, and citizens at all levels who have an interest in mathematics education (includes lesson plans, open-ended problems with multiple solutions, homework helper ask Dr. Math, and more).

http://mathforum.org/

Math Stories.comOffers over 5,000 word problems for K-8 that help students improve their math problem solving and critical-thinking skills. An excellent resource for teachers who need additional, creative problems.

http://www.mathstories.com

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http://everyday/

http://everyday/

New Jersey Core Curriculum Content Standards for Mathematics

The following pages contain the New Jersey Core Standards for mathematics. The vision of these standards revolves around what takes place in classrooms and is focused on achieving one crucial goal:

GOAL: To enable ALL of New Jersey’s children to move into the twenty-first century with the mathematical skills, understandings, and attitudes that they will need to be successful in their careers and daily lives.

The use of the term "all students" in the content standards is intended to convey the idea that these standards are universally achievable.

As more and more teachers incorporate the recommendations of the Mathematics Standards into their teaching, we should be able to see the following results (as described in Mathematics to Prepare Our Children for the 21st Century: A Guide for New Jersey Parents, published by the New Jersey Mathematics Coalition in September 1994.).

Students who are excited by and interested in their activities. Students who are learning important mathematical concepts rather

than simply memorizing and practicing procedures. Students who are posing and solving meaningful problems. Students who are working together to learn mathematics. Students who write and talk about math topics every day. Calculators and computers being used as important tools of learning. Teachers who have high expectations for ALL of their students. A variety of assessment strategies rather than sole reliance on

traditional short-answer tests.

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New Jersey Mathematics Core Standards

4.1 NUMBER AND NUMERICAL OPERATIONSAll students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.

4.2 GEOMETRY AND MEASUREMENTAll students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe, and analyze phenomena.

4.3 PATTERNS AND ALGEBRAAll students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.

4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICSAll students will develop an understanding of the concepts and techniques of data analysis, probability, and discrete mathematics, and will use them to model situations, solve problems, and analyze and draw appropriate inferences from data.

4.5 MATHEMATICAL PROCESSESAll students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.

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Cumulative Progress Indicators for Grade 3

Bulleted items that appear below expectations indicate terminology, concepts, or content material addressed in that expectation. When an indicator is followed by bulleted content material, the list provided is intended to be exhaustive; content material not mentioned is therefore not included in the expectation at that grade level. When examples are provided, they are always introduced with "e.g." and are not intended to be exhaustive.

Items presented at one grade level are not repeated at subsequent grade levels. Teachers will need to refer to the standards at earlier grade levels to know what topics their students should be assumed to have learned at earlier grades.

Each cumulative progress indicator is assigned a value consisting of the standard, grade level, macro, and indicator. For example, 4.1.3 A3 represents Mathematics Standard 4.1 (Number and Numerical Operations), grade 3, macro A (Number Sense), and cumulative progress indicator 3. In the suggested pacing and objectives section of this document, each mathematics unit is aligned to the New Jersey Core Curriculum Content Standards. The Cumulative Progress Indicators are listed in the third column of the table.

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NEW JERSEY CORE CURRICULUM CONTENT STANDARDS FOR MATHEMATICS GRADE 5

Standard 4.1 Number and Numerical Operation

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:Macro A Number Sense

1. Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 5 pertain to these sets of numbers as well). All fractions as part of a whole, as subset of a

set, as a location on a number line, and as divisions of whole numbers

All decimals 2. Recognize the decimal nature of United States currency

and compute with money.3. Demonstrate a sense of the relative magnitudes of

numbers.4. Use whole numbers, fractions, and decimals to represent

equivalent forms of the same number. 5. Develop and apply number theory concepts in problem

solving situations. Primes, factors, multiples

6. Compare and order numbers. Macro B Numerical Operations

1. Recognize the appropriate use of each arithmetic operation in problem situations.

2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with: Pencil-and-paper Mental math Calculator

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3. Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.

4. Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.

5. Check the reasonableness of results of computations.6. Understand and use the various relationships among

operations and properties of operations.Macro C Estimation

1. Use a variety of estimation strategies for both number and computation.

2. Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

3. Determine the reasonableness of an answer by estimating the result of operations.

4. Determine whether a given estimate is an overestimate or an underestimate.

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Standard 4.2 Geometry and Measurement

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:Macro A Geometric Properties

1. Understand and apply concepts involving lines and angles. Notation for line, ray, angle, line segment Properties of parallel, perpendicular, and

intersecting lines Sum of the measures of the interior angles of a

triangle is 180°2. Identify, describe, compare, and classify polygons.

Triangles by angles and sides Quadrilaterals, including squares, rectangles,

parallelograms, trapezoids, rhombi Polygons by number of sides. Equilateral, equiangular, regular All points equidistant from a given point form a

circle3. Identify similar figures.4. Understand and apply the concepts of congruence and

symmetry (line and rotational).Macro B Transforming Shapes

1. Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

2. Recognize, identify, and describe geometric relationships and properties, as they exist in nature, art, and other real-world settings.

Macro C Coordinate Geometry

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1. Create geometric shapes with specified properties in the first quadrant on a coordinate grid.

Macro D Units of Measurement

1. Select and use appropriate units to measure angles and area.

2. Convert measurement units within a system (e.g., 3 feet = ___ inches).

3. Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).

4. Use measurements and estimates to describe and compare phenomena.

Macro E Measuring Geometric Objects

1. Use a protractor to measure angles.2. Develop and apply strategies and formulas for finding

perimeter and area. Square Rectangle

3. Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.

4. Develop informal ways of approximating the measures of familiar objects (e.g., use a grid to approximate the area of the bottom of one’s foot).

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Standard 4.3 Patterns and Algebra

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:Macro A Patterns

1. Recognize, describe, extend, and create patterns involving whole numbers. Descriptions using tables, verbal rules, simple

equations, and graphsMacro B Functions & Relationships

1. Describe arithmetic operations as functions, including combining operations and reversing them.

2. Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.

Macro C Modeling

1. Use number sentences to model situations. Using variables to represent unknown

quantities Using concrete materials, tables, graphs,

verbal rules, algebraic expressions/equations2. Draw freehand sketches of graphs that model real

phenomena and use such graphs to predict and interpret events. Changes over time Rates of change (e.g., when is plant growing

slowly/rapidly, when is temperature dropping most rapidly/slowly)

Macro D Procedures

1. Solve simple linear equations with manipulatives and informally Whole-number coefficients only, answers also

whole numbers. Variables on one side of equation

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Standard 4.4 Data Analysis, Probability, and Discrete Mathematics

Building upon knowledge and skills gained in preceding grades, by the end of Grade 5, students will:Macro A Data Analysis

1. Collect, generate, organize, and display data. Data generated from surveys

2. Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Bar graph, line graph, circle graph, table Range, median, and mean

3. Respond to questions about data and generate their own questions and hypotheses.

Macro B Probability

1. Determine probabilities of events. Event, probability of an event Probability of certain event is 1 and of

impossible event is 02. Determine probability using intuitive, experimental, and

theoretical methods (e.g., using model of picking items of different colors from a bag). Given numbers of various types of items in a

bag, what is the probability that an item of one type will be picked

Given data obtained experimentally, what is the likely distribution of items in the bag

3. Model situations involving probability using simulations (with spinners, dice) and theoretical models.

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Macro C Discrete Mathematics—Systematic Listing and Counting

1. Solve counting problems and justify that all possibilities have been enumerated without duplication. Organized lists, charts, tree diagrams, tables

2. Explore the multiplication principle of counting in simple situations by representing all possibilities in an organized way (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).

Macro D Discrete Mathematics—Vertex-Edge Graphs and Algorithms

1. Devise strategies for winning simple games (e.g., start with two piles of objects, each of two players in turn removes any number of objects from a single pile, and the person to take the last group of objects wins) and express those strategies as sets of directions.

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Standard 4.5 Mathematical Processes

Problem Solving

Problem posing and problem solving involve examining situations that arise in mathematics and other disciplines and in common experiences, describing these

situations mathematically, formulating appropriate mathematical questions, and using a variety of strategies to find solutions. Through problem solving, students experience the power and usefulness of mathematics. Problem solving is interwoven throughout the grades to provide a context for learning and applying mathematical ideas.

Communication

Communication of mathematical ideas involves students’ sharing their mathematical understandings in oral and written form with their classmates, teachers, and parents. Such communication helps students clarify and solidify their understanding of mathematics and develop confidence in themselves as mathematics learners. It also enables teachers to better monitor student progress.

Connections

Making connections involves seeing relationships between different topics, and drawing on those relationships in future study. This applies within mathematics, so that students can translate readily between fractions and decimals, or between algebra and geometry; to other content areas, so that students understand how mathematics is used in the sciences, the social sciences, and the arts; and to the everyday world, so that students can connect school mathematics to daily life.

Reasoning

Mathematical reasoning is the critical skill that enables a student to make use of all other mathematical skills. With the development of mathematical reasoning, students recognize that mathematics makes sense and can be understood. They learn how to evaluate situations, select problem-solving strategies, draw logical conclusions, develop and describe solutions, and recognize how those solutions can be applied.

Representations

Representations refers to the use of physical objects, drawings, charts, graphs, and symbols to represent mathematical concepts and problem situations. By using various representations, students will be better able to communicate their thinking and solve problems. Using multiple representations will enrich the

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problem solver with alternative perspectives on the problem. Historically, people have developed and successfully used manipulatives (concrete representations such as fingers, base ten blocks, geoboards, and algebra tiles) and other representations (such as coordinate systems) to help them understand and develop mathematics.

Technology

Calculators and computers need to be used along with other mathematical tools by students in both instructional and assessment activities. These tools should be used, not to replace mental math and paper-and-pencil computational skills, but to enhance understanding of mathematics and the power to use mathematics. Students should explore both new and familiar concepts with calculators and computers and should also become proficient in using technology as it is used by adults (e.g., for assistance in solving real-world problems).Cumulative Progress Indicators

At each grade level, with respect to content appropriate for that grade level, students will:

Macro A Problem Solving

1. Learn mathematics through problem solving, inquiry, and discovery. 2. Solve problems that arise in mathematics and in other contexts

(cf. workplace readiness standard 8.3). Open-ended problems Non-routine problems Problems with multiple solutions Problems that can be solved in several ways

3. Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.

4. Pose problems of various types and levels of difficulty. 5. Monitor their progress and reflect on the process of their

problem solving activity. Macro B Communication

1. Use communication to organize and clarify their mathematical thinking.

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Reading and writing Discussion, listening, and questioning

2. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.

3. Analyze and evaluate the mathematical thinking and strategies of others. 4. Use the language of mathematics to express mathematical ideas

precisely.

Macro C Connections

1. Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

2. Use connections among mathematical ideas to explain concepts (e.g., two linear equations have a unique solution because the lines they represent intersect at a single point).

3. Recognize that mathematics is used in a variety of contexts outside of mathematics.

4. Apply mathematics in practical situations and in other disciplines. 5. Trace the development of mathematical concepts over time and

across cultures (cf. world languages and social studies standards). 6. Understand how mathematical ideas interconnect and build on

one another to produce a coherent whole.Macro D Reasoning

1. Recognize that mathematical facts, procedures, and claims must be justified.

2. Use reasoning to support their mathematical conclusions and problem solutions.

3. Select and use various types of reasoning and methods of proof. 4. Rely on reasoning, rather than answer keys, teachers, or peers,

to check the correctness of their problem solutions.5. Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjectures Verifying conjectures using informal reasoning or proofs.6. Evaluate examples of mathematical reasoning and determine

whether they are valid.48

Macro E Representations

1. Create and use representations to organize, record, and communicate mathematical ideas.

Concrete representations (e.g., base-ten blocks or algebra tiles)

Pictorial representations (e.g., diagrams, charts, or tables) Symbolic representations (e.g., a formula) Graphical representations (e.g., a line graph)2. Select, apply, and translate among mathematical representations

to solve problems. 3. Use representations to model and interpret physical, social, and

mathematical phenomena. Macro F Technology

1. Use technology to gather, analyze, and communicate mathematical information.

2. Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information (cf. workplace readiness standard 8.4-D).

3. Use graphing calculators and computer software to investigate properties of functions and their graphs.

4. Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

5. Use computer software to make and verify conjectures about geometric objects.

6. Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).

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Holistic Scoring Guide for Mathematics Open-Ended Items(Generic Rubric)

The generic rubric below is used as a guide to develop specific scoring guides or rubrics for each of the open-ended items, which appear on the grade three and four proficiency assessment (ASK) in mathematics. The generic rubric helps insure that students are scored in the same way for the same demonstration of knowledge and skills regardless of the test question.

3-Point ResponseThe response shows complete understanding of the problem’s essential mathematical concepts. The student executes procedures completely and gives relevant responses to all parts of the task. The response contains few minor errors, if any. The response contains a clear, effective explanation detailing how the problem was solved so that the reader does not need to infer how and why decisions were made.

2-Point ResponseThe response shows nearly complete understanding of the problem’s essential mathematical concepts. The student executes nearly all procedures and gives relevant responses to most parts of the task. The response may have minor errors. The explanation detailing how the problem was solved may not be clear, causing the reader to make some inferences.

1-Point ResponseThe response shows limited understanding of the problem’s essential mathematical concepts. The response and procedures may be incomplete and/or may contain major errors. An incomplete explanation of how the problem was solved may contribute to questions as to how and why decisions were made.

0-Point ResponseThe response shows insufficient understanding of the problem’s essential mathematical concepts. The procedures, if any, contain major errors. There may be no explanation of the solution or the reader may not be able to understand the explanation. The reader may not be able to understand how and why decisions were made.

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National Council of Teachers of MathematicsPrinciples and Standards

In the National Council of Teachers of Mathematics document Principles and Standards for School Mathematics, six principles are identified as overarching themes:

The Equity PrincipleExcellence in mathematics education requires equity – high expectations and strong support for all students.

The Curriculum PrincipleA curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.

The Teaching PrincipleEffective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well.

The Learning PrincipleStudents must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

The Assessment PrincipleAssessment should support the learning of important mathematics and furnish useful information to both teachers and students.

The Technology PrincipleTechnology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning.

Visit the Standards 2000 web site at http://nctm .org/standards/

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http://nctm/

Glossary for Selected Mathematics Terms

ACCOUNT BALANCE

An amount of money that you have or that you owe. See "in the black" and "in the red."

ACUTE ANGLE An angle with a measure greater than 0º and less than 90ºADJACENT ANGLES Two angles with a common side and vertex that do not

otherwise overlap.

2 1 3 4

APEX In a pyramid or cone, the vertex opposite the base. In a pyramid, all the non-base faces meet at the apex.

AREA A measure of a bounded surface. The boundary might be a triangle or rectangle in a plane or the boundaries of a state or country on Earth's surface. Area is expressed in square units such as square miles, square inches, or square centimeters, and can be thought of as the approximate number of non-overlapping squares that will "tile" or "cover" the surface within the boundary.

AREA MODEL 1. A model for multiplication in which the length and width of a rectangle represent the factors and the area of the rectangle represents the product.

2. A model for fractions that represents parts of a whole. The whole is a region, such as a circle or a rectangle, representing the number ONE.

AXIS 1. Either of the two number lines use to form a coordinate grid. Plural axes.

2. A line about which a solid figure rotates.BAR GRAPH A graph that shows relationships in data by the use of bars to

represent quantities.BASE 1. Geometry: A side of a polygon, usually used for area

computations along with the "altitude," or height, perpendicular to it. See base of a parallelogram and base of a triangle.

2. Geometry: Either of two parallel and congruent faces that define the shape of a prism or cylinder, or the face that defines the shape of a cone or pyramid. see base of a prism or cylinder and base of a pyramid or cone.

3. Arithmetic: See exponential notation.4. Arithmetic: The foundation number for a numeration system.

For example, our ordinary system for writing numbers is a base-ten place value system, with 1, 10, 100, 1,000, and other powers of 10 as the values of the places in whole numbers. In computers, bases of two, eight, or sixteen are usual, instead of base ten.

356 = 300 + 50 + 6expanded notation for a base-ten number

BASE OF A One of the sides of a parallelogram; also, the length of this side. 53

Angles 1 and 2, 2 and 3, 3 and 4, and 4 and 1 are pairs of adjacent angles.

PARALLELOGRAM In calculating area, the base is used along with the height, or altitude, which is measured on a perpendicular to the side opposite this base. See height of a parallelogram.

BASE OF A PRISM OR CYLINDER

Either of the two parallel, congruent faces of a prism or cylinder that define its shape and are used to determine its name and classification. In a cylinder, the base is a circle or ellipse. See height of a prism or cylinder.

BASE OF A PYRAMID OR CONE

The face of a pyramid that defines its shape and is used to name and classify the pyramid. The base of a pyramid is the face opposite the apex, which is the vertex where all the other faces meet. The base of a cone is a circle. See height of a pyramid or cone.

BASE OF A TRIANGLE

The side of a triangle to which an altitude is drawn, also, the length of this side. The height, or altitude, is the length of the shortest line segment between the line containing the base and the vertex opposite the base. See height of a triangle.

CALIBRATE to divide or mark something, such as a thermometer, with gradations.

CAPACITY 1. A measure of how much a container can hold, usually in such units as quart, gallon, cup, or liter. See volume.2. The maximum weight a scale can measure.

CENSUS An official count of population and the recording of such data as age, sex, income, education, and so on.

CENTER Of a circle: The point in the plane of a circle equally distant from all points on the circle. Of a sphere: The point equally distant from all points on the sphere.

CIRCLE GRAPH A graph in which a circle and its interior are divided into parts to represent the parts of a set of data. The whole circle represents the whole set of data. Same as pie graph.

CIRCUMFERENCE The distance around a circle or maximum distance around a sphere.

COMMON DENOMINATOR

Any nonzero number that is a multiple of the denominators of two or more fractions. For example, the fractions ½ and 2/3 have common denominators 6, 12, 18, and so on. See denominator.

COMMON FACTOR Any number that is a factor of two or more numbers. The common factors of 18 and 24 are 1, 2, 3, and 6. See factor.

COMMUTATIVE PROPERTY

A property of addition and multiplication (but not division or subtraction) that says that changing the order of the elements being added or multiplied will not change the sum or product. For addition: a + b = b + a , so 5 + 10 = 10 + 5; For multiplication: a x b = b x a, so 5 x 10 - 10 x 5.

COMPOSITE NUMBER

A whole number that has more than two factors. For example, 10 is a composite number because it has more than two factors: 1, 2, 5, and 10. A composite number is divisible by at least three whole numbers. Compare to prime number.

CONE A 3-dimensional shape having a circular base, a curved lateral surface, and one vertex, called the apex.

CONGRUENT Having the same size and shape. Two figures are congruent if a combination of slides, flips, and turns can be used to move one

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of the figures so that it exactly fits "on top of" the other figure. In diagrams of congruent figures, the congruent sides may be marked with the same number of tick marks. The symbol means "is congruent to."

CONTOUR LINE A curve on a map through places where a measurement (such as temperature, elevation, air pressure, or growing season) is constant.

COUNTOUR MAP A map that uses contour lines to delineate areas according to a particular feature, such as elevation or climate. See contour line.

COORDINATE A number used to locate a point on a number line, or one of two numbers used to locate a point on a coordinate grid.

COORDINATE GRID A device for locating points in a plane by means of ordered pairs of numbers. A rectangular coordinate grid is formed by two number lines that intersect at right angles at their zero points.

CUBE A polyhedron with six square faces. One of the 5 regular polyhedra.

CUBIC CENTIMETER (cc or cm3)

A metric unit of volume; the volume of a cube that is 1 centimeter on a side. 1 cubic centimeter is equal to 1 milliliter.

CUBIC UNIT A unit used in measuring volume. Common cubic units include cubic centimeters, cubic inches, cubic feet, and cubic meters.

CUBIT An ancient unit of length, measured from the point of the elbow to the end of the middle finger. The cubit has been standardized at various times to be between 18 and 22 inches. The Latin work cubitum means "elbow."

CUP In the US customary system, a unit of capacity equal to 8 fluid ounces; ½ pint

CURVED SURFACE 1. A surface that does not lie in a plane; for example, a sphere or the lateral surface of a cylinder.

2. A non-base surface of a cone or cylinder.CYLINDER A 3-dimensional shape having a curved surface and parallel,

congruent, circular or elliptical bases. A can is a common object shaped like a cylinder.

DENOMINATOR In a fraction, the number written below the line or to the right of the slash. In the fraction or a/b, b is the denominator. In a part-whole fraction, the denominator is the number of equal parts into which the whole (or ONE) has been divided

DIAMETER A line segment that passes through the center of a circle or sphere and has endpoints on the circle or sphere; also, the length of such a line segment. The diameter of a circle or sphere is twice the length of the radius.

DISCOUNT The amount by which the regular price of an item is reduced, expressed as a fraction or percent of the original price. For example, a $4.00 item that is on sale for $2.00 is discounted by 50 percent or by ½. Or, when a $10.00 item has a discount

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percent of 10% (or the equivalent discount fraction of ) its sale price is $9.00.

DISPLACE To move something from one position to another.DIVIDEND In division, the number that is being divided. For example, in

35 ÷ 5 = 7, the dividend is 35.DIVISIBLE One whole number is divisible by another whole number if the

result of the division is a whole number with remainder 0.DIVISOR In division, the number that divides another number (the

dividend). In 40 ÷ 8 = 5, the divisor is 8.

EDGE A line segment where two faces of a polyhedron meet.EQUAL CHANCE When none of the possible outcomes of an event is more likely

to occur than any other, it is an equal chance situation.EQUATION A mathematical sentence that asserts the equality of two

quantities.EQUILATERAL POLYGON

A polygon in which all sides are the same length.

EUQILATERAL TRIANGLE

A triangle in which all three sides are the same length and all three angles are the same measure.

EQUIVALENT FRACTIONS

Fractions that have different denominators but represent the same number.

EVEN NUMBER A whole number that can be evenly divided by 2.EXPONENTIAL NOTATION

A way of representing repeated multiplication by the same factor. For example, 23 is exponential notation for 2 x 2 x 2. The small, raised 3, called the exponent, indicates how many times the number 2, called the base, is used as a factor.

EXPRESSION A group of mathematical symbols (numbers, operation signs, variables, grouping symbols) that represent or can represent a number if values are assigned to any variables that the expression contains.

FACE 1. Any of the polygonal regions that form 3-dimensional prisms, pyramids, or polyhedra. Some special faces are called bases.

2. Any flat surface of a cylinder, cone, or other geometric solid.FACTOR 1. A number being multiplied in a multiplication number model.

In the number model 6 x 0.5 = 3, 6 and 0.5 are factors and 3 is the product.

2. A whole number that can divide another whole number without a remainder. For example, 4 and 7 are both factors of 28 because 28 is divisible by both 4 and 7.

3. To represent a number as a product of factors. To factor 21, for example, is to write it as 7 x 3.

FACTOR PAIR Two whole-number factors of a number whose product is the number. A number may have more than one factor pair. For example, the factor pairs for 24 are 1 and 24, 2 and 12, 3 and 8, and 4 and 6.

FACTOR RAINBOW A way to show factor pairs in a list of all the factors of a number. A factor rainbow can be used to check whether a list of factors is correct.

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1 2 3 4 6 8 12 24 factor rainbow for 24.

FACTOR STRING A name for a number written as a product of at least two whole-number factors other than 1. For example, a factor string for the number 24 is 2 x 3 x 4. This factor string has three factors, so its length is 3. By convention, the number 1 is not allowed in factor strings.

FACTOR TREE A method used to obtain the prime factorization of a number. The original number is represented as a product of factors, and each of those factors is represented as a product of factors, and so on, until the factors are all prime numbers. Factor trees are drawn upside down, with the root at the top and the leaves at the bottom.

FAIR GAME A game in which every player has the same chance of winning. If any player has an advantage or disadvantage at the beginning (for example, by playing first), then the game is not fair.

FATHOM A unit used mainly by people who work with boats and ships to measure depth of water and lengths of cables. A fathom is 6 feet, or 2 yards. Same as arm span.

FORMULA A general rule for finding the value of something. A formula is often written symbolically using letters, called variables, to stand for the quantities involved. For example, a formula for distance traveled can be written as d = s x t, where d stands for distance, s for speed, and t for time.

FREQUENCY 1. The number of times a value occurs in a set of data.2. The number of vibrations per second of a sound wave; more generally, the number of repetitions per unit of time.

FREQUENCY GRAPH

A graph showing how often each value in a data set occurs.

FREQUENCY TABLE A chart on which data is tallied to find the frequency of given events or values.

GEOMETRIC SOLID A 3-dimensional shape bounded by surfaces. Common geometric solids include the rectangular prism, square-based pyramid, cylinder, cone, and sphere.

GREAT SPAN The distance from the tip of the thumb to the tip of the little finger (pinkie), when the hand is stretched as far as possible. The great span averages about 9 inches for adults.

HEIGHT A measure of how tall something is.HORIZONTAL Positioned in a left-to-right orientation. Parallel to the line of

the horizon.IMPROPER FRACTION

A term for a fraction whose numerator is greater than or equal to its denominator. An improper fraction names a number greater than or equal to 1.

INEQUALITY A number-sentence stating that two quantities are not equal, or might not be equal. Relation symbols for inequalities include ,

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<, >, , and .IRRATIONAL NUMBER

A number that cannot be written as a fraction where both the numerator and denominator are integers and the denominator is not zero. An irrational number can be represented by a nonterminating, nonrepeating decimal.

IRREGULAR POLYGON

A polygon with sides of different lengths or angles of different measures.

ISOSCELES TRIANGLE

A triangle with at least two sides that are the same length and at least two angles that are the same measure.

KITE A quadrilateral with exactly two pairs of adjacent sides that are the same length. (A rhombus is not a kite.)

LANDMARK A notable feature of a data set. Landmarks include median, mode, maximum, minimum, and range.

LATITUDE The angular distance of a point on Earth's surface, north or south from the equator measured on the meridian of the point. Compare to longitude.

LATITUDE LINES Lines of constant latitude drawn on a map or globe. Lines of latitude are used to indicate the location of a place with reference to the equator. Latitude is measured in degrees, from 0º to 90º, north or south of the equator. Lines of latitude are also called "parallels," because they are parallel to the equator and to each other. Compare to longitude lines.

LEAST COMMON MULTIPLE (LCM)

The smallest number that is a multiple of two or more given numbers. For example, while some common multiples of 6 and 8 are 24, 48, and 72, the least common multiple of 6 and 8 is 24.

LINE GRAPH A graph in which data points are connected by a line or line segments.

LINE PLOT A sketch of data in which check marks, X's, or other symbols above a labeled line show the frequency of each value.

LITER (L) In the metric system, a unit of capacity equal to the volume of a cube that measures 10 centimeters on a side. 1L=1,000cm3. A liter is a little larger than a quart.

LONGITUDE A measure of how far east or west of the prime meridian a location on Earth is. Longitude is the measure, usually in degrees, of the angle formed by the plane containing the meridian of a particular place and the plane containing the Prime Meridian. Compare to latitude.

LONGITUDE LINES Lines of constant longitude; semicircles connecting the North and South Poles. Longitude lines are used to locate places with reference to the prime Meridian. Lines of longitude are also called meridians. Compare to latitude lines.

MAGNITUDE ESTIMATE

A rough estimate of the size of a numerical result-whether it is in the 1s, 10s, 100s, 1,000s, and so on.

MAJORITY More than half of a total amount.MAP DIRECTION SYMBOL

A symbol on a map that identifies north, south, east, and west. Sometimes only north is indicated.

MAP LEGEND a diagram that explains the symbols, markings, and colors on a map. Also called a map key.

MAP SCALE A device for relating distances on a map to corresponding 58

distances in the real world. One inch on a map, for example, might correspond to 1 mile in the real world. A map scale is often represented by a labeled line segment, similar to a ruler; by a ratio of distances; or by ;an incorrect use of the = symbol (as in "1 inch = 1 mile").

MAXIMUM The largest amount; the greatest number in a set of data. Compare to minimum.

MEAN A measure of central tendency. It is found by adding the numbers in the set and dividing the sum by the number of numbers. It is often referred to as the average.

MEDIAN The middle value in a set of data when the data are listed in order from least to greatest (or greatest to least). If there is an even number of data points, the median is the mean of the two middle values. The median is also known as the middle value. Compare to mean and mode.

MILLILITER (ml) In the metric system, a unit of capacity equal to of a liter; 1 cubic centimeter.

MINIMUM The smallest amount; the smallest number in a set of data. Compare to maximum.

MINUEND The number that is reduced in subtraction. For example, in 19 - 5 = 14, the minuend is 19.

MIXED NUMBER A number that is written using both a whole number and a fraction. For example, 2 is a mixed number equal to 2 + .

MODE The value or values that occur most often in a set of data. Compare to median and mean. In the data set 3, 4, 4, 4, 5, 6, the mode is 4.

NEGATIVE NUMBER

A number less than 0; a number to the left of 0 on a horizontal number line or below 0 on a thermometer or other vertical number line.

NORMAL SPAN The distance from the end of the thumb to the end of the index (first) finger of an outstretched hand. For estimating lengths, many people can adjust this distance to approximately 6 inches or 20 centimeters. Compare to great span.

NUMBER MODEL A number sentence that models or fits a situation. For example, the situations "Sally had $5 and then she earned $8," "A young plant 5cm high grew 8cm," and Harry is 8 years older than his 5-year-old sister Sally" can all be modeled by the number sentence 5 + 8 = 13

NUMBER SENTENCE

A sentence made up of at least two numbers or expressions and a single relation symbol (=, <, >, , , or ). Number sentences usually contain at least one operation symbol. They may also have grouping symbols, such as parentheses. If a number sentence contains one or more variables, it is called an open sentence.

NUMERATOR In a fraction, the number written above the line or to the left of the slash. In a part-whole fraction, where the whole is divided into a number of equal parts, the numerator names the number

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of equal parts being considered. In the fraction or a/b, a is the numerator.

OBTUSE ANGLE An angle measuring more than 90º and less than 180º.ODD NUMBER A whole number that cannot be evenly divided by 2. Compare

to even number.OPEN NUMBER SENTENCE

A number sentence that is neither true nor false because one or more variables hold the place of missing numbers.

OPPOSITE ANGLES 1. Of a quadrilateral: angles that do not share a common side.2. Of a triangle: an angle is opposite the side of a triangle that

is not one of the sides of the angle.3. When two lines intersect, the angles that do not share a

common side are opposite angles. Opposite angles have equal measures. Also called vertical angles.

OPPOSITE OF A NUMBER

A number that is the same distance from zero on the number line as the given number, but on the opposite side of zero. The opposite of any number n is written as (op)n or -n. If n is a negative number, (op)n or -n will be a positive number. The sum of a number and its opposite is zero.

ORDERED PAIR 1. A pair of numbers used to locate a point on a coordinate grid. The first number corresponds to position along the horizontal axis, and the second number corresponds to position along the vertical axis.

2. Any pair of objects or numbers in a particular order.PARALLEL Lines, rays, line segments, and planes that are equidistant at all

points, no matter how far extended.PARALLELOGRAM A quadrilateral that has two pairs of parallel sides. All

rectangles are parallelograms, but not all parallelograms are rectangles because parallelograms do not need to have right angles.

PENTAGON a 5-sided polygon.PERCENT (%) Per hundred, or out of a hundred. 1% means or 0.01. For

example, "48% of the students in the school are boys" means that out of every 100 students in the school, 48 are boys.

PERIMETER The distance around a closed plane figure or region. Peri- comes from the Greek word for "around," and meter comes from the Greek word for "measure"; perimeter means "around measure."

PERPENDICULAR Rays, lines, line segments, or planes that form right angles are perpendicular to each other.

PI () The ratio of the circumference of a circle to its diameter, Pi, which is approximately 3.14, is the same for every circle. Pi is also the ratio of a circle's area to the square of its radius. The first twenty digits of Pi are: 3.1415926535897932384

PIE GRAPH Same as circle graph.PLACE VALUE The relative worth of each digit in a number, which is

determined by its position. Each place has a value ten times

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that of the place to its right and one-tenth of the value of the place to its left.

POLYGON A closed plane figure formed by three or more line segments that meet only at their endpoints. Exactly two sides come together at each corner of a polygon.

POLYHEDRON A closed 3-dimensional shape, all of whose surfaces (faces) are flat. Each face consists of a polygon and its interior.

POPULATION 1. The total number of people living within a certain geographical area.2. In data collection, the collection of people or objects that is

the focus of study. The population is often larger than the target audience for a given survey, in which case a smaller, representative sample is considered.

POWER 1. The exponent to which a "base" number is raised in exponential notation; the number a in na, where n is the base. If n is any number and a is a positive whole number, a tells how many times to use n as a factor in a product.

2. The result of a "powering" or "exponential" operation xy.POWER OF 10 1. A whole number that can be written as a product using only

10 as a factor; also called a positive power of 10. 2. More generally, any number that can be written as a product

using only 10s or s as factors.PREDICT To tell what will happen ahead of time; to make an educated

guess about what might happen.PRIME FACTORIZATION

A whole number expressed as a product of prime factors. For example, the prime factorization of 24 is 2 x 2 x 2 x 3.

PRISM A polyhedron with two parallel faces (bases) that are the same size and shape and other faces that are bounded by parallelograms. In a right prism, the non-base faces are rectangles. Prisms are classified according to the shape of the two parallel bases.

PROBABILITY A number from 0 to 1 that indicates the likelihood that an event will happen. The closer a probability is to 1, the more likely that the event will happen. The closer a probability is to 0, the less likely that the event will happen. For example, the probability that a fair coin will show heads is ½..

PRODUCT The result of a multiplication. In the number model 4 x 3 = 12, the product is 12.

PYRAMID A polyhedron in which one face (the base) is a polygon and all other faces are triangles with a common vertex called the apex. Pyramids are classified according to the shapes of their bases.

QUART In the US customary system, a unit of capacity equal to 32 fluid ounces, 2 pints, or 4 cups.

QUOTIENT The result of dividing one number by another number. In the division model 10 5 = 2, the quotient is 2.

RADIUS A line segment from the center of a circle (or sphere) to any point on the circle (or sphere); also, the length of such a line segment.

RANGE The difference between the greatest and least values in a set of 61

data.RATE A comparison by division of two quantities with different units.

For example, traveling 100 miles in 2 hours can be expressed as 100 mi/2 hr or 50 miles per hour. In this case, the rate compares distance (miles) to time (hours). Compare to ratio.

RATIO A comparison by division of two quantities with the same units. Ratios can be expressed as fractions, decimals, or percents, as well as in words. Ratios can also be written with a colon between the two numbers being compared. Compare to rate.

RECTANGULAR ARRAY

A rectangular arrangement of objects in rows and columns such that each row has the same number of objects and each column has the same number of objects.

RECTANGULAR PRISM

1. In common usage, a prism whose faces (including the bases) are all rectangles. Man;y packing boxes have the shape of rectangular prisms.

2. More generally, any prism with rectangular bases, some of the faces of which might be non-rectangle parallelograms.

REFLEX ANGLE An angle with a measure between 180 and 360.REGULAR POLYGON

A polygon whose sides are the same length and whose angles are all equal.

REGULAR POLYHEDRON

A polyhedron whose faces are all congruent regular polygons and with the same number of faces meeting at every vertex, all at the same angle. There are five regular polyhedra known as the Platonic solids.tetrahedron: 4 faces, each an equilateral trianglecube: 6 faces, each a squareoctahedron: 8 faces, each an equilateral triangledodecahedron: 12 faces, each a regular pentagonicosahedron: 20 faces, each an equilateral triangle

REGULAR TESSELATION

A tessellation made up of only one kind of regular polygon. There are only three regular tessellations.

REMAINDER An amount left over when one number is divided by another. In the division number model 16 3 5 R1, the remainder is 1.

REPEATING DECIMAL

A decimal in which one digit, or a group of digits, is repeated without end. For example, 0.333...and 0.147 are repeating decimals.

RIGHT 1. Of an angle: An angle whose measure is 90.2. Of a prism or cylinder: Having lateral faces or surfaces that

are all perpendicular to their bases.3. Of a pyramid or cone: Having an apex directly above the

center of its base.4. Of a triangle: Having a right angle.

ROUND 1. Arithmetic: To express a number in a simplified way. Examples of rounding include expressing a measure of weight to the nearest pound and expressing an amount of money to the nearest dollar.

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2. Geometry: Circular in shape.SAMPLE A part of a population intended to represent the whole.SCALENE TRIANGLE

A triangle with sides of three different lengths and angles of three different sizes.

SCIENTIFIC NOTATION

A system for representing numbers in which a number is written as the product of a power of 10 and number that is at least 1 and less than 10. Scientific notation allows writing big and small numbers with only a few symbols. For example, 4,300,000 in scientific notation is 4.3 x 106, and 0.00001 in scientific notation is 1 x 10-3. Compare to standard notation.

SIMPLEST FORM 1. Of proper fractions: Having numerator and denominator with no common factors (other than 1). For example, and

are equivalent fractions. However, is not in simplest form because the numerator and denominator can each be divided by 5; is in simplest form because 2 and 3 have no common factore (other than 1).

2. Of mixed numbers and improper fractions: Being in mixed number form in; which the fraction part is proper and in simplest form. For example, 1 is not in simplest form because the fraction part is not proper.

SLIDE RULE 1. A mechanical tool composed of a ruler and a sliding insert. Slide rules can be used to do may types of calculations, but have been rendered obsolete by electronic calculators.

SOLUTION 1. Of an open sentence: A value or values for the variable(s) which make the sentence true. For example, the open sentence 4 + ___ = 10 has the solution 6.

2. Of a problem: The answer or the method by which the answer was obtained.

SPAN Same as normal span.SPHERE A 3-dimensional shape whose curved surface is, at all points, a

given distance from its center point. A ball is shaped like a shpere. A shpere is hollow; it does not include the points in its interior.

SQUARE NUMBER A number that is the product of a whole number and itself; a whole number to the second power. For example, 25 is a square number, because 25 = 5 x 5. A square number can be represented by a square array. See rectangular array.

SQUARE OF A NUMBER

The product of a number and itself. The square of a number is symbolized by a raised 2. For example, 3.52= 3.5 x 3.5 = 12.25

SQUARE ROOT KEY The key on certain calculators. The key undoes the result of squaring a non-negative number.

SQUARE ROOT OF A NUMBER

The square root of a number n is a number which, when multiplies by itself, results in the number n. For example, 4 is a square root of 16, because 4 x 4 = 16. Normally, squre root refers to the positive square root, but the opposite of a positive

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squre root is also a square root, for example, -4 is also a square root of 16 because (-4) x (-4) = 16.

SQUARE UNIT A unit used to measure area. A square unit represents a square with the measure of each side being a related unit of length. For example, a square inch is the area of a square that measures one inch on each side.

STANDARD NOTATION

The most familiar way of representing whole numbers, integers, and decimals. Standard notation is base-ten place-value numberation. For example, standard notation for three hundred fifty-six is 356. Compare to scientific notation.

STEM-AND-LEAF PLOT

A display of data in which digits with larger place values are "stems" and digits with smaller place values are "leaves."

stem-and-leaf-plot

STEP GRAPH A graph that looks like steps because the values are the same for an interval and then change (or "step") for another interval. The horizontal axis of a step graph often represents time.

STRAIGHT ANGLE An angle measureing 180. straight angle

SURFACE AREA A measure of the surface of a 3-dimensional figure. The surface area of a polyhedron is the sum of the areas of its faces. See area.

SURVEY A study that collects data. Surveys are used to find out about people's characteristics, behaviors, interests, opinions, and so on. Surveys are used to generate data for graphing and analysis.

TESSELLATE 1. To make a tessellation or tiling. See tessellation.2. To fit into a tessellation. Any quadrilateral will tessellate.

TESSELLATION An arrangement of closed shapes that covers a surface completely without overlaps or gaps.

TRANSFORMATION An operation on a geometric figure that produces a new figure, called the image, from the original figure, called the preimage. Transformations are sometimes thought of as moving a figure from one place to another and sometimes changing its size or shape. The study of transformations is called transformation geometry.

TREE DIAGRAM A network of points connected by line segments and containing no closed loops. One special point is the root of the tree. Tree diagrams can be used to factor numbers and to represent probability situations in whi;ch there is a series of events.

TRUE NUMBER SENTENCE

A number sentence in which the relations symbol accurately reflects the relation between the two sides of the sentence. 75 = 25 + 50 is a true number sentence.

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Stems 10's

Leaves 1's

2 4 4 5 6 7 73 1 1 2 2 6 6 4 1 1 3 5 85 0 2

TURN-AROUND RULE

A principle, based on the commutative property, for solving math fact problems. If you know, for example, that 6 x 8 = 48, then by the turn-around rule you also know that 8 x 6 = 48.

UNIT A label, descriptive word, or unit of measure used to put a number in context. Using units with numbers reinforces the idea that numbers refer to something. Fingers, snowballs, miles, and cents are examples of units.

UNIT FRACTION A fraction whose numerator is 1. For example, and are all unit fractions.

UNLIKE 1. Of denomintaors: Being unequal, the fractions and have unlike denominators.

2. Of fractions: Having different denominators. Compare to like fractions.

VALUE A specific number or quantity represented by a variable. In the equation y = 4x + 3, if the value of x is 7, then that value of y is 31.

VARIABLE A letter or other symbol that represents a number. A variable need not represent one specific number; it can stand for many different values. For example, in the expression 2x + 3y x and y are variables, and in the equation a + 12 = 2b + 6, a and b are variables.

VERTEX The point at which the rays or line segments of an angle, the sides of a polygon, or the edges of a polyhedron meet.

VERTICAL Upright; perpendicular to the horizon.VERTICAL ANGLES When two lines intersect; the angles that do not share a

common side; the angles opposite each other. Vertical angles have equal measures. Same as opposite angles.

VOLUME A measure of the amount of space occupied by a 3-dimensional shape; generally expressed in "cubic" units, such as cm3, cubic inches, or cubic feet.

WHOLE The entire object, collection or objects, or quantity being considered; the unit, 100%.

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Activity that - Newark Public Schoolscontent.nps.k12.nj.us/wp-content/uploads/sites/46/2014/0… · Web viewDefine and create tessellations (3.8) Secure. 4a. Divide decimal numbers