A Curriculum Guide for - Newark Public Schools



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|A Curriculum Guide for |

|Mathematics |

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|Grade 2 |

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|Newark Public Schools |

|Office of Mathematics |

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|2004-2005 |

|NEWARK PUBLIC SCHOOLS |

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NEWARK PUBLIC SCHOOLS

Administration

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 Lee

Human Resource Services

Assistant Superintendent………….……….…..................…….....Ms. Joanne C. Bergamotto

School Leadership Team I

Assistant Superintendent………….……….…....................……………..Dr. J. Russell Garris

School Leadership Team II

Assistant Superintendent……………………..…………..........…Dr. Glenda Johnson-Green

School Leadership Team

Assistant Superintendent…………………............……….......……………... Ms. Lydia Silva

School Leadership Team IV

Assistant Superintendent……..…………........................……………….…Dr. Don Marinaro

School Leadership Team V

Assistant Superintendent…………………………….........….………….Dr. Gayle W. Griffin

Department of Teaching and Learning

Associate Superintendent………………………………………………...Ms. Alyson Barillari

Department of Special Education

Associate Superintendent…………………………………..…………...Mr. Benjamin O'Neal

Department of Special Programs

Department

of

Teaching and Learning

Dr. Gayle W. Griffin

Assistant Superintendent

Office of Mathematics

May L. Samuels

Director

GRADE 2 MATHEMATICS

CURRICULUM

GUIDE

Table of Contents

Mission Statement 4

Philosophy 5

To the Teacher 6

Content Emphasis 7

Suggested Timeline 8

Suggested Pacing and Objectives (with New Jersey Core Content Standards) 9

Open Ended Problem Solving and Scoring 20

Reference:

Instructional Technology (Web Sources) 24

NJCCCS and Cumulative Progress Indicators 25

National Council of Teachers of Mathematics Principles and Standards 34

Glossary 35

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

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

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

Grade 2 Everyday Mathematics Content Emphasis

Numeration

Counting; reading and writing numbers, identifying place-value; comparing numbers; working with fractions; using money to develop place-value and decimal concepts.

Operations and Computation

Recalling addition and subtraction facts; exploring fact families; adding and subtracting with tens and hundreds; beginning multiplication and division; exchanging money.

Data and Chance

Collecting, organizing, and interpreting data using tables, charts and graphs; exploring concepts of chance.

Measurement and Reference Frames

Using tools to measure length, capacity, weight, and volume; using U.S. customary and metric measurement units.

Patterns, Functions, and Algebra

Exploring number patterns, rules for number sequences, relations between numbers, and attributes.

Within the content of Everyday Mathematics, emphasis is placed on :

• A problem-solving approach based on everyday situations that develop critical thinking.

• Frequent practice of basic skills through ongoing program routines and mathematical games.

• An instructional approach that revisits topics regularly to ensure full conceptual development.

• Activities that explore a wide variety of mathematical content and offer opportunities for students to apply their basic fact skills to geometry, measurement, and algebra.

Teachers and students will incorporate mathematical processes as a part of everyday work and play. These processes will gradually shape children’s ways of thin king about mathematics and foster the development of mathematical intuition and understanding.

Suggested Timeline

This guide provides a full description of the mathematics objectives for Everyday Mathematics Grade 2 and correlates them to the New Jersey Core Curriculum Content Standards for Mathematics (NJCCC) for Grade 2.

The Mathematical Process Standards: Problem Solving, Communication, Connections, Reasoning, Representations, and Technology, although not explicitly referenced, are integrated throughout the mathematics program.

|PACING GUIDE |

|Month | Lessons |

|September/October |Lessons 1.1 – 3.6 |

|November |Lessons 3.7 – 4.7 |

|December |Lessons 4.8 – 5.10 |

|January |Lessons 6.1 – 7.3 |

|February |Lessons 7.4 – 8.5 |

|March |Lessons 8.6 – 9.11 |

|April |Lessons 10.1 – 11.3 |

|May/June |Lessons 11.4 – 12.8 |

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|September /October | |Number and Numerical Operations |

| |1a. Calculate the values of coin and bill combinations |Number Sense |

| |(Lessons 1.2, 1.6) Developing |A2. Demonstrate an understanding of whole number place |

| | |value concepts. |

| | |A4. Count and perform simple computations with coins. |

| | |Amounts up to $1.00 (using cents notation) |

| | |A5. Compare and order whole numbers. |

| |1b. Know addition facts for sums to 10 (Lesson 1.2) | |

| |Developing/Secure |Numerical Operations |

| | |B1. Develop the meanings of addition and subtraction by |

| |1c. Identify place value for 1s, 10s, and 100s (Lesson |concretely modeling and discussing a large variety of |

| |1.9) Developing |problems |

| | |Joining, separating, and comparing |

| | |B3. Develop proficiency with basic addition and |

| | |subtraction number facts using a variety of fact strategies|

| | |(such as "counting on" and "near doubles") and then commit |

| | |them to memory. |

| | |B8. Understand and use the inverse relationship between |

| |1d. Complete number sequences; Identify and use number |addition and subtraction. |

| |patterns to solve problems (Lessons 1.1, 1.8) Developing | |

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| |1e. Find equivalent names for numbers (Lessons 1.10, 1.11)| |

| |Developing | |

| | |Patterns and Algebra |

| |1f. Compare numbers; Write the symbol , or = |Patterns |

| |(Lesson 1.12) Developing |A1. Recognize, describe, extend, and create patterns. Whole|

| | |number patterns that grow or shrink as a result of |

| |1g. Count by 2s, 5s, and 10s (Lesson 1.11) Secure |repeatedly adding or subtracting a fixed number (e.g. skip |

| | |counting forward or backward). |

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| |1h. Make tallies and give the total (Lesson 1.5) Secure | |

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|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|September-October |2a. Know harder subtraction facts (Lesson 2.13) | Number and Numerical Operations |

| |Developing |Numerical Operations |

| | |B1. Develop the meanings of addition and subtraction by |

| |2b. Know harder addition facts (Lessons 2.4, 2.5) |concretely modeling and discussing a large variety of |

| |Developing/Secure |problems |

| | |Joining, separating, and comparing |

| |2c. Know easier subtraction facts (Lessons 2.8, 2.12) |B3. Develop proficiency with basic addition and |

| |Developing/Secure |subtraction number facts using a variety of fact strategies|

| | |(such as "counting on" and "near doubles") and then commit |

| | |them to memory. |

| | |B8. Understand and use the inverse relationship between |

| |2d. Complete "What's My Rule?" tables (Lesson 2.11) |addition and |

| |Developing/Secure |subtraction. |

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| |2e. Solve subtraction number stories (Lesson 2.6) |Patterns and Algebra |

| |Developing/Secure |Functions and Relationships |

| | |B1. Use concrete and pictorial models of function machines|

| |2f. Know easier addition facts (Lessons 2.2, 2.3, 2.8) |to explore the basic concept of a function |

| |Secure |Modeling |

| | |C2. Construct and solve simple open sentences involving |

| |2g. Construct fact families for addition and subtraction |addition or subtraction |

| |(Lessons 2.6, 2.8) |Result unknown (e.g., 6 – 2 = ___ or n = 3 + 5) |

| |Developing /Secure |Part unknown (e.g., 3 + ___ = 8) |

| | |Procedures |

| |2h. Complete Simple Frames-and Arrows diagrams (lesson |D1. Understand and apply (but don’t name) the following |

| |2.10) Secure |properties of addition: |

| | |Commutative (e.g., 5 + 3 = 3 + 5) |

| |2i. Solve addition number stories (Lesson 2.1) Secure |Zero as the identity element (e.g., 7 + 0 = 7) |

| | |Associative (e.g., 7 + 3 + 2 can be found by adding either |

| |2j. Find equivalent names for numbers (Lesson 2.9) |7 + 3 or 3 + 2) |

| |Secure | |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|November |3a. Solve Frames-and-Arrows problems having 2 rules |Patterns and Algebra |

| |(Lesson 3.6) Developing |Functions and Relationships |

| | |B1. Use concrete and pictorial models of function machines|

| |3b. Make change (Lessons 3.2, 3.7, 3.8) Developing |to explore the basic concept of a function |

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| |3c. Know harder subtraction facts (Lesson 3.5) Developing | |

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| |3d. Tell time to 5-minute intervals (Lessons 3.3, 3.4) | |

| |Developing/Secure | |

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| | |Geometry and Measurement |

| | |Units of Measurement |

| |3e. Identify place value in 2-digit and 3-digit numbers |D3. Select and use appropriate standard and non-standard |

| |(Lessons 3.1, 3.4) Developing/Secure |units of measure and standard measurement tools to solve |

| | |real-life problems |

| |3f. Show "P", "N", "D," and "Q" for a given amount (Lesson|Time – second, minute, hour, day, week, month, year |

| |3.2) Secure | |

| | |Number and Numerical Operation |

| |3g. Know addition facts (Lesson 3.5) Secure |Number Sense |

| | |A2. Demonstrate an understanding of whole number place |

| |3h. Know easier subtraction facts (Lesson 3.5) Secure |value concepts |

| | |A4. Count and perform simple computations with coins |

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| | |Numerical Operations |

| | |B1. Develop the meanings of addition and subtraction by |

| | |concretely modeling and discussing a large variety of |

| | |problems |

| | |Joining, separating, and comparing |

| | |B3. Develop proficiency with basic addition and |

| | |subtraction number facts using a variety of fact strategies|

| | |(such as "counting on" and "near doubles") and then commit |

| | |them to memory. |

| | |B8. Understand and use the inverse relationship between |

| | |addition and |

| | |subtraction. |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|December |4a. Devise and use strategies for finding sums of 2-digit |Number and Numerical Operation |

| |numbers (Lessons 4.1, 4.6-4.9) Developing |Numerical Operations |

| | |B5. Use efficient and accurate pencil-and-paper |

| |4b. Devise and use strategies for finding differences of |procedures for computation with whole numbers |

| |2-digit numbers (Lesson 4.4) Developing |Addition and subtraction of 2-digit numbers |

| | |Estimation |

| |4c. Estimate approximate costs and sums (Lessons 4.5, 4.8)|C2. Determine the reasonableness of an answer by |

| |Developing |estimating the result of computations (e.g., 15 + 16|

| | |is not 211) |

| |4d. Read ºF on a thermometer (Lessons 4.3, 4.4) Developing| |

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| |4e. Add and subtract with multiples of 10 (Lessons | |

| |4.1-4.3, 4.6-4.9) Developing/Secure |Geometry and Measurement |

| | |Units of Measurement |

| | |D3. Select and use appropriate standard and |

| | |non-standard units of measure |

| | |Temperature - ºF |

|December |5a. Identify 3-dimensional shapes such as rectangular |Geometry and Measurement |

| |prisms, cylinders, pyramids, cones, and spheres (Lessons |Geometric Properties |

| |5.7, 5.8) Developing |A2. Use concrete objects, drawings, and computer |

| | |graphics to identify, classify, and describe |

| |5b. Identify symmetrical figures (Lesson 5.9) Developing |standard 2- and 3-dimensional shapes |

| | |3-D figures: cube, rectangular prism, sphere, cone, |

| |5c. Find common attributes of shapes (Lessons 5.1, 5.2) |cylinder, pyramid |

| |Developing |2-D figures: square, rectangle, circle, triangle |

| | |A3. Describe, identify, and create instances of |

| |5d. Identify parallel and non-parallel line segments |line symmetry |

| |(Lesson 5.5) Developing | |

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| |5e. Identify 2-dimensional shapes (Lessons 5.2, 5.3, 5.6) | |

| |Secure | |

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|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|January |6a. Solve stories about multiples of equal groups (Lessons|Number and Numerical Operation |

| |6.8, 6.9, 6.11) Beginning/Developing |Numerical Operations |

| | |B1. Develop the meanings of addition and subtraction|

| |6b. Solve equal-grouping and equal-sharing division |by concretely modeling and discussing a large |

| |problems (Lessons 6.11, 6.12) Beginning/Developing |variety of problems. |

| | |Joining, separating, and comparing |

| |6c. Use the trade-first method to solve 2-digit |B2. Explore the meanings of multiplication and |

| |subtraction problems (Lessons 6.6, 6.10, 6.12) |division by modeling and discussing problems. |

| |Developing |B5. Use efficient and accurate pencil-and-paper |

| | |procedures for computation with whole numbers. |

| |6d. Model ballpark estimates of exact answers (Lessons |Subtraction of 2-digit numbers |

| |6.1,6.4, 6.6, 6.7, 6.10, 6.11) Developing |B6. Select pencil-and-paper, mental math, or a |

| | |calculator as the appropriate computational method |

| |6e. Model multiplication problems with arrays (Lessons |in a given situation depending on the context and |

| |6.,9, 6.10) Developing |numbers. |

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| |6f. Add three 2-digit numbers mentally (Lessons 6.1, 6.5, | |

| |6.10) Developing |Estimation |

| | |C2. Determine the reasonableness of an answer by |

| |6g. Add and subtract with multiples of 10: Lessons 6.1, |estimating the result of computations (e.g., 15 + 16|

| |6.5, 6.10) Developing/Secure |is not 211). |

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| |6h. Solve addition and subtraction number stories (Lessons| |

| |6.2-6.4, 6.7) Developing/Secure | |

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| |6i. Add three 1-digit numbers mentally: Lessons 6.,1, 6.4,| |

| |6.7) Secure | |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|February |7a. Find missing addends for any multiple of 10 and Know |Number and Numerical Operation |

| |complements of 10 (Lesson 7.3) Developing |Numerical Operations |

| | |B4. Construct, use, and explain procedures for |

| |7b. Find the median (middle value) of a data set (Lesson |performing addition and subtraction calculations |

| |7.8) Developing | |

| | |Data Analysis, Probability, and Discrete |

| |7c. Add three 2-digit numbers mentally (Lesson 7.4) |Mathematics |

| |Developing |Data Analysis |

| | |A2. Read, interpret, construct, and analyze |

| |7d,e. Measure to the nearest inch and centimeter (Lesson |displays of data |

| |7.7) Developing/Secure | |

| | |Geometry and Measurement |

| |7f. Know complements of 10 (Lesson 7.3) Secure |Units of Measurement |

| | |D1. Directly compare and order objects according |

| |7g. Count by 2s, 5s, and 10s and describe the patterns |to measurable attributes |

| |(Lesson 7.1) Secure |Attributes – length, weight, capacity, time, |

| | |temperature |

| |7h. Find missing addends for the next multiple of 10 |D3. Select and use appropriate standard and |

| |(Lesson 7.2) Secure |non-standard units of measure and standard |

| | |measurement tools to solve real-life problems |

| |7i. Solve number-grid puzzles (Lesson 7.2) Secure |Length – inch, foot, yard, centimeter, meter |

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| |7j. Plot data on a bar graph (Lesson 7.9) Secure |Patterns and Algebra |

| | |Patterns |

| | |A1. Recognize, describe, extend, and create |

| | |patterns. |

| | |Repeating patterns |

| | |Whole number patterns that grow or shrink as a |

| | |result of repeatedly adding or subtracting a fixed |

| | |number (e.g., skip counting forward or backward) |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|March |8a. Compare fractions less than one (Lessons 8.5-8.7) |Number and Numerical Operations |

| |Beginning/Developing |Number Sense |

| | |A1. Use real life experiences, physical materials, |

| |8b. Understand fractions as names for equal parts of a |and technology to construct meanings for numbers |

| |region or set (Lessons 8.1-8.3, 8.7) Developing |Proper fractions (denominators of 2, 3, 4, 8, and |

| | |10) |

| |8c. Understand that the amount represented by a fraction; | |

| |depends on the size of the whole (ONE) (Lessons 8.2, 8.3, | |

| |8.7) Developing | |

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| |8d. Shade a specified fractional part of a set (Lessons | |

| |8.3, 8.5 Developing | |

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| |8e. Give the fraction name for the shaded part of a set | |

| |(Lesson 8.3 Developing | |

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| |8f. Find the equivalent fractions for given fractions | |

| |(Lessons 8.4-8.6) Developing | |

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| |8g. Shade a specified fractional part of a region (Lessons | |

| |8.1, 8.4) Developing | |

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| |8h. Give the fraction name for the shaded part of a region | |

| |(Lessons 8.1, 8.3, 8.5) Developing | |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|March |9a. Identify equivalencies for mm, cm, dm, and m (Lessons |Geometry and Measurement |

| |9.2, 9.3, 9.5, 9.9) Beginning/Developing |Units of Measurement |

| | |D2. Recognize the need for a uniform unit of |

| |9b. Measure to the nearest ½ inch (Lesson 9.3) Developing |measure. |

| | |D3. Select and use appropriate standard and |

| |9c. Measure to the nearest ½ cm (Lesson 9.3) Developing |non-standard units of measure and standard |

| | |measurement tools to solve real-life problems. |

| |9d. Use appropriate units for measurement and recognize |Length - inch, foot, yard, centimeter, meter |

| |sensible measurements (Lessons 9.1-9.6, 9.9, 9.10) | |

| |Developing |Measuring Geometric Objects |

| | |E1. Directly measure the perimeter of simple |

| |9e. Find area concretely (Lessons 9.7, 9.8) Developing |two-dimensional shapes |

| | |E2. Directly measure the area of simple |

| |9f. Find perimeter concretely (Lessons 9.4, 9.5, 9.8) |two-dimensional shapes by covering them with |

| |Developing |squares |

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| |9g. Identify equivalencies for inches, feet, and yards | |

| |(Lessons 9.2, 9.9) Developing | |

| | | |

| |9h. Use a ruler, tape measure, and meter/yardstick | |

| |correctly (Lessons 9.1-9.4) Developing/Secure | |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|April |10a. Use parentheses in number models (Lesson 10.11) |Number and Numerical Operations |

| |Beginning |Number Sense |

| | |A2. Demonstrate an understanding of whole number |

| |10b. Solve money stories involving change (Lessons 10.6, |place value concepts |

| |10.8) Developing |A4. Count and perform simple computations with |

| | |coins |

| |10c. Estimate totals for a ballpark check for exact answers|Amounts up to one dollar (using cents notation) |

| |(Lessons 10.5, 10.6, 10.8, 10.9) Developing | |

| | |Estimation |

| |10d. Know and express automatically the values of digits in|C2. Determine the reasonableness of an answer by |

| |5-digit numbers (Lessons 10.10, 10.11) |estimating the result of computations (e.g., 15 + |

| |Developing |16 is not 211) |

| | | |

| |10e. Read and write money amounts in decimal notation |Numerical Operations |

| |(Lessons 10.2-10.4, 10.6) Developing/Secure |B1. Develop the meanings of addition and |

| | |subtraction by concretely modeling and discussing |

| |10f. Use equivalent coins to show money amounts in |a large variety of problems |

| |different ways (lesson 10.1) Secure |Joining, separating, and comparing |

| | | |

| |10g. Use a calculator to compute money amounts (Lessons |B4. Construct, use, and explain procedures for |

| |10.3, 10.4, 10.7) |performing addition and subtraction calculations |

| | |with: |

| |10h. Exchange pennies, nickels, dimes, ad quarters (Lessons|Pencil and paper |

| |0.2, 10.8, 10.10) Secure |Mental math |

| | |Calculator |

| |10i. Know and express automatically the values of digits in| |

| |2-, 3-, and 4-digit numbers (Lessons 10.8-10.11) | |

| |Developing/ Secure | |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|May-June |11a. Estimate and solve addition and subtraction number |Number and Numerical Operations |

| |stories with dollars and cents (Lessons 11.1, 11.2) |Number Sense |

| |Developing |A1. Use real-life experiences, physical materials,|

| | |and technology to construct meanings for numbers |

| |11b. Solve 1-digit multiplication stories (Multiples of |A4. Count and perform simple computations with |

| |equal groups) (Lessons 11.3, 11.7) Developing |coins |

| | |Amounts up to one dollar (using cents notation) |

| |11c. Solve simple division stories (equal sharing and | |

| |grouping) (Lessons 11.4, 11.7) Developing |Numerical Operations |

| | |B2. Explore the meanings of multiplication and |

| |11d. Multiply numbers with 2, 5, or 10 as a factor (Lessons|division by modeling and discussing problems |

| |11.5-11.9) Developing | |

| | |Estimation |

| |11e. Construct multiplication/division fact families |C2. Determine the reasonableness of an answer by |

| |(Lessons 11.7, 11.8) Developing |estimating the result of computations (e.g., 15 + |

| | |16 is not 211) |

| | | |

| |11f. Make difference and ratio comparisons (Lessons 11.2, | |

| |11.9) Developing | |

| | | |

| |11g. Multiply numbers with 0 or 1 as a factor (Lesson 11.6)| |

| |Secure | |

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

|Month |Objectives/Everyday Mathematics |NJCCC Standard and Strands |

|May-June |12a. Use alternate names for times of day (Lesson 12.2) |Number and Numerical Operations |

| |Beginning |Numerical Operations |

| | |B2. Explore the meanings of multiplication and |

| |12b. Know "harder" multiplication facts (Lesson 12.5) |division by modeling and discussing problems |

| |Beginning | |

| | |Data Analysis, Probability, and Discrete |

| |12c. Determine the mode of a data set (Lesson 12.7) |Mathematics |

| |Beginning |Data Analysis |

| | |A1. Collect, generate, record, and organize data |

| |12d. Determine the median, maximum, minimum, and range of a|in response to questions, claims, or curiosity |

| |data set (Lesson 12.4) Developing |A2. Read, interpret, construct, and analyze |

| | |displays of data |

| |12e. Construct multiplication/division fact families |Pictures, tally chart, pictograph, bar graph, Venn |

| |(Lessons 12.1, 12.5) Developing/Secure |diagram |

| | |Smallest to largest, most frequent (mode) |

| |12f. Multiply numbers with 2, 5, and 10 as a factor (Lesson| |

| |12.4) Developing/Secure |Geometry and Measurement |

| | |Units of Measurement |

| |12g. Tell time to 5-minute intervals (Lessons 12.1, 12.2) |D3. Select and use appropriate standard and |

| |Secure |non-standard units of measure and standard |

| | |measurement tools to solve real-life problems |

| |12h. Demonstrate calendar concepts and skills (Lesson 12.1)| |

| |Secure |Time – second, minute, hour, day, week, month, year|

| | | |

| |12i. Compare quantities from a bar graph (Lessons 12.6, | |

| |12.7) Secure | |

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

Open-Ended Problem Solving and Scoring

The material on the following pages provides a process for posing and solving open-ended tasks and rubric-based scoring.

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.

| |Pose a problem and solve it as a group activity. Brainstorm for possible solutions |

|1 |and approaches. Model/have students model several strategies and record solutions. |

| | |

| |Present a problem. Read the problem with the group, check student understanding by |

| |asking students to restate the problem in their own words. Brainstorm about possible |

|2 |strategies and then set the task. |

| |Encourage students to use manipulatives or act out the problem before attempting to |

|3 |write about the solution. Allow students to works as partners or in small groups to |

| |arrive at and record solutions. |

| |Ask a few students to share strategies they have used. It is useful for students to |

| |model the implementation of their strategy and how solutions were recorded. Teachers |

|4 |may choose to use a simplified version of the Generic Problem Solving Rubric with |

| |students. |

Generic Problem-Solving Rubric

Level 3

The child arrives at a correct solution. The solution is correct. The solution shows reasoning, use of problem solving strategies, and uses pictures, words, and/or numbers to explain the solution. The reader does not need to make inferences to understand the explanation provided.

Level 2

The child arrives at a solution. The solution may show reasoning and use of the correct processes, but the answer may be incorrect. The use of problem solving strategies may be inconsistent. The solution may be correct with no evidence of the thought process (words, diagrams, pictures, symbols, numbers, etc.) or the reader needs to make inferences to understand any explanation provided.

Level 1

The child attempts to solve the problem and provides a partial solution. The child’s knowledge of the mathematical concept(s) or understanding of the problem is limited. The use of problem-solving strategies is not apparent. No explanation is provided.

Level 0

The child may or may not attempt to solve the problem. The response shows little evidence of understanding of the mathematical concept or procedures.

Teachers may use this Generic Problem Solving Rubric when examining student work. If desired, specific rubrics can be developed for each problem using the generic rubric as a guide. Be as specific as possible in validating scores and suggesting improvements to solutions. It is important that students begin to develop an understanding of the scoring process. Teachers may use the Student Self Evaluation Rubric to help them understand the purpose of a rubric.

Student Self Evaluation Rubric for Mathematics

|Understanding |Strategies |Communication |

| | | |

|I got it. I can prove I’m right. I could do|I have a plan to solve it. I can give |I laid out the problem clearly. I explained |

|other problems like this. |reasons for what I did. |everything I did. Someone else can easily |

| | |see what I did. |

| | | |

|Not perfect, But I understand most of it. |I have a plan, but I haven’t figured out how |I explained what I did, but some parts of my |

|There is still a part I’m not sure about. |to solve one part. I can explain most of my |explanation are not easy to follow. |

| |strategy. | |

| | | |

|I think I understand part of the problem. |I have the beginning of a plan. I know |I explained only a little, or my explanation |

| |something I should do, but I’m not sure why. |is hard to understand. |

| | | |

|I don’t understand. |I’m not sure what to do. |I can’t explain what should be done. |

| | | |

| | | |

Reference

Instructional Technology (Web Sources)

New Jersey Core Curriculum Content Standards

Cumulative Progress Indicators

National Council of Teachers of Mathematics

Principles and Standards

Glossary for Selected Mathematics Terms

Instructional Technology (Web Sources)

Instructional Technology (Web Sources)

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 Mathematics | |

|Contains news and information of interest to math teachers. | |

| | |

|New Jersey State Department of Education | |

|Information on the department’s proposals and regulations, including state| |

|testing program. | |

| | |

|Everyday Mathematics Home Page | |

|Information, teacher resources, professional development opportunities, | |

|and more for the Everyday Mathematics Program. | |

| | |

|Math Goodies | |

|Interactive math lessons, as well as homework help, puzzles, calculators. | |

| | |

|The Math Forum | |

|A 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). | |

| | |

|Math | |

|Offers 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. | |

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.

New Jersey Mathematics Core Standards

|4.1 |NUMBER AND NUMERICAL OPERATIONS |

| |All 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 MEASUREMENT |

| |All 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 ALGEBRA |

| |All 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 MATHEMATICS |

| |All 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 PROCESSES |

| |All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and |

| |technology to solve problems and communicate mathematical ideas. |

NEW JERSEY CORE CURRICULUM CONTENT STANDARDS FOR MATHEMATICS GRADE 2

STANDARD 4.1 Number and Numerical Operations

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 2 pertain to these sets of numbers as well).

• Whole numbers through hundreds

• Ordinals

• Proper fractions (denominators of 2, 3, 4, 8, 10)

2. Demonstrate an understanding of whole number place value concepts.

3. Understand that numbers have a variety of uses.

4. Count and perform simple computations with coins.

• Amounts up to $1.00 (using cents notation)

5. Compare and order whole numbers.

Macro B Numerical Operations

1. Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems.

• Joining, separating, and comparing

2. Explore the meanings of multiplication and division by modeling and discussing problems.

3. Develop proficiency with basic addition and subtraction number facts using a variety of fact strategies (such as "counting on" and "near doubles "0 and then commit them to memory. Construct, use, and explain procedures for performing addition and subtraction calculations with:

• Pencil-and-paper

• Mental math

• Calculator

4. Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

• Addition of 2-digit numbers

• Subtraction of 2-digit numbers

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

6. Check the reasonableness of results of computations.

7. Understand and use the inverse relationship between addition and subtraction.

STANDARD 4.1 Number and Numerical Operations

Macro C Estimation

1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

2. Determine the reasonableness of an answer by estimating the result of computations (e.g., 15 + 16 is not 211).

3. Explore a variety of strategies for estimating both quantities (e.g., the number of marbles in a jar) and results of computation.

STANDARD 4.2 Geometry and Measurement

Macro A Geometric Properties

1. Identify and describe spatial relationships among objects in space and their relative shapes and sizes.

• Inside/outside, left/right, above/below, between

• Smaller/larger/same size, wider/narrower, longer/shorter

• Congruence (ie., same size and shape)

2. Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes.

• Vertex, edge, face, side

• 3-D figures-cube, rectangular prism, sphere, cone, cylinder, and pyramid

• 2-D figures-square, rectangle, circle, and triangle

• Relationships between three- and two- dimensional shapes (i.e., the face of a 3-D shape is a 2-D shape)

3. Describe, identify, and create instances of line symmetry.

4. Recognize, describe, extend, and create designs and patterns with geometric objects of different shapes and colors.

Macro B Transforming Shapes

1. Use simple shapes to make designs, patterns, and pictures.

2. Combine and subdivide simple shapes to make other shapes.

Macro C Coordinate Geometry

1. Give and follow directions for getting from one point to another on a map or grid.

STANDARD 4.2 Geometry and Measurement

Macro D Units of Measurement

1. Directly compare and order objects according to measurable attributes.

• Attributes-length, weight, capacity, time, temperature

2. Recognize the need for a uniform unit of measure.

3. Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems.

• Length-inch, foot, yard, centimeter, and meter

• Weight-pound, gram, kilogram

• Capacity-pint, quart, liter

• Time-Second, minute, hour, day, week, month, year

• Temperature-degrees Celsius, degrees Fahrenheit

4. Estimate measures.

Macro E Measuring Geometric Objects

1. Directly measure the perimeter of simple 2-dimensional shapes.

2. Directly measure the area of simple 2-dimensional shapes by covering them with squares.

STANDARD 4.3 PATTERNS AND ALGEBRA

Macro A Patterns and Relationships

1. Recognize, describe, extend, and create patterns.

• Using concrete materials (manipulatives, pictures, rhythms, and whole numbers)

• Descriptions using words and symbols (e.g., "add to" or "+2")

• Repeating patterns

2. Whole number patterns that grow or shrink as a result of repeatedly adding or subtracting a fixed number (e.g., skip counting forward or backward)

Macro B Functions

1. Use concrete and pictorial models of function machines to explore the basic concept of a function.

STANDARD 4.3 PATTERNS AND ALGEBRA

Macro C Modeling

1. Recognize and describe changes over time (e.g., temperature and height).

2. Construct and solve simple open sentences involving addition or subtraction.

• Result unknown (e.g., 6 - 2 = __ or n = 3 + 5)

• Part unknown (e.g., 3 + = 8)

Macro D Procedures

1. Understand and apply (but don't name) the following properties of addition:

• Commutative (e.g., 5 + 3 = 3 + 5)

• Zero as the identity element (e.g., 7 + 0 = 7)

• Associative (e.g., 7 + 3 + 2 can be found either by first adding 7 + 3 or 3 + 2)

STANDARD 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS

Macro A Data Analysis (Statistics)

1. Collect, generate, record, and organize data in response to questions, claims, or curiosity.

• Data collected from students everyday experiences

• Data generated from chance devices such as spinners and dice.

2. Read, interpret, construct, and analyze displays of data.

• Pictures, tally chart, pictograph, bar graph, Venn diagram

3. Smallest to largest, most frequent (mode)

Macro B Probability

1. Use chance devices like spinners and dice to explore concepts of probability.

• Certain, impossible

• More likely, less likely, equally likely

2. Provide probability of specific outcomes.

• Probability of getting specific outcome when coin is tossed, when die is rolled, when spinner is spun (e.g., if spinner has 5 equal sectors, then probability of getting a particular sector is one out of five)

3. When picking a marble from a bag with three red marbles and four blue marbles, the probability of getting a red marble is 3 out of 7)

STANDARD 4.4 DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS

Macro C Discrete Mathematics-Systemic Listing and Counting

1. Sort and classify objects according to attributes.

• Venn diagrams

2. Generate all possibilities in simple counting situations (e.g., all outfits involving two shirts and three pants)

Macro D Discrete Mathematics-Vertex-Edge Graphs and Algorithms

1. Follow simple sets of directions (e.g., from one location to another, or from a recipe)

2. Color simple maps with a small number of colors.

3. Play simple two-person games (e.g., Tic-tac-toe) and informally explore the idea of what the outcome should be.

4. Explore concrete models of vertex-edge graphs (e.g., vertices as "islands" and edges as "bridges")

• Paths from one vertex to another

|Standard 4.5 Mathematical Processes |

|Macro A Problem Solving |

|1. |Learn mathematics through problem solving, inquiry, and discovery. |

|2. |Solve problems that arise in mathematics and in other contexts |

| |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. |

| |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 (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. |

|Standard 4.5 Mathematical Processes |

| |

|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. |

|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. |

National Council of Teachers of Mathematics

Principles 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 Principle

Excellence in mathematics education requires equity – high expectations and strong support for all students.

The Curriculum Principle

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

The Teaching Principle

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

The Learning Principle

Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

The Assessment Principle

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

The Technology Principle

Technology 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

Glossary for Selected Mathematics Terms

|ADDEND |One of two or more numbers that are added. |

|addition fact |Two one-digit numbers and their sum, such as 9 + 7 = 16. |

|algorithm |A set of step-by-step instructions for doing something, such as carrying out a computation or solving a problem. |

|a. m. |The abbreviation for ante meridiem, which means “before the middle of the day”; from midnight to noon. |

|analog clock |A clock that shows the time by the positions of the hour and minute hands. |

|angle |A figure that is formed by two rays or two line segments with a common endpoint. |

|area |The measure of a bounded surface. |

|arm span |The distance from fingertip to fingertip of a person’s outstretched arms. |

|array |A rectangular arrangement of objects in rows and columns. |

|associative property |A property of addition and multiplication (but not of subtraction or division) that says that changing the grouping of the |

| |elements being added or multiplied will not change the sum or product. |

|attribute |A feature of an object or a common feature of a set of objects. Examples of attributes include size, shape, color, and |

| |number of sides. See Property. |

|ballpark estimate |A rough estimate used as a check on the reasonableness of an answer or when an exact figure is not necessary. |

|bank draft |A written order for the exchange of money. |

|bar graph |A graph that shows the relationships among variables by the use of bars to represent quantities. |

|base |1. Any side of a polygon, usually used, along with the altitude perpendicular to it, for computing area. |

| |2. The flat face or faces that define the shape when classifying polyhedrons. |

|capacity |A measure of how much a container can hold, usually in units such as quarts, gallons, cups, or liters. |

|celsius |The temperature scale on which 0( is the temperature at which pure water, at sea level, freezes and 100( is the temperature |

| |at which it boils. The Celsius scale is used in the metric system. |

|centimeter (cm) |In the metric system, a unit of length equivalent to 10 millimeters, 1/10 of a decimeter, and 1/100 of a meter. |

|circle |The set of all points in a plane that are equally distant from a given point in a plane called the center of a circle. |

|column |A vertical arrangement of objects or numbers in an array or table. |

|commutative property |A property of addition and multiplication (but not subtraction or division) that says that changing the order of the |

| |elements being added or multiplied will not change the sum or product. |

|cone |A 3-dimensional shape having a circular base, a curved surface, and 1 vertex, called the apex. |

|consecutive |Following one another in an uninterrupted order such as A, B, C, D or 6, 7, 8, 9. |

|corner |See Vertex |

|counting numbers |The numbers used to count things. The set of counting numbers is {, 2, 3, 4…} Sometimes 0 is included with the counting |

| |numbers. |

|cube |A polyhedron with 6 square faces. One of the five regular polyhedra. See Regular Polyhedron |

|cubit |An ancient unit of length, measured from the point of the elbow to the end of the middle finger. |

|cup |In the U.S. customary system, a unit of capacity equal to 8 fluid ounces; ½ pint. |

|Curved Surface or face |A surface that does not lie in a plane; for example, a sphere or cylindrical surface. Also, a nonbase face of a cone or |

| |cylinder. |

|Customary system |The measuring system most often used in the United States, in contrast to the metric system used nearly everywhere else. |

|cylinder |A 3-dimensional shape having a curved surface and parallel circular or elliptical bases that are the same size. A can is an|

| |object shaped like a cylinder. |

|data |Information gathered by observing, counting, or measuring. Data is the plural of datum. |

|Deci- |Prefix meaning one tenth. |

|decimal point |The mark that separates the whole number from the fraction in decimal notation; in expressing money, it separates the |

| |dollars from the cents. |

|decimeter (dm) |In the metric system, a unit of length equivalent to one tenth of a meter or ten centimeters. |

|degree |A unit for measuring temperature. Also a unit of measure for angles based on dividing one complete circle (rotation) into |

| |360 equal parts. The small raised symbol (() is called the degree symbol. |

|denominator |The number written below the line in a fraction. In a part-whole fraction, the number of equal parts into which the whole |

| |(ONE) is divided. Compare to numerator. |

|diagonal (Of a table) |A line of objects or numbers from upper left to lower right or from upper right to lower left in an array or a table. |

|difference |The amount by which one number is greater or less than another number. |

|digit |In the base-ten numeration system, one of the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 which can be used to write any number.|

|digital clock |A clock that uses numbers to show the time in hours and minutes, with a colon used to separate them. |

|division |The operation used to solve equal sharing problems. It is used to find how a total amount can be separated into an equal |

| |number of groups, or into groups of equal size. |

|doubles fact |The addition and multiplication facts without turn-around partners (4 + 4 = 8). |

|EDGE |A line segment where two faces of a polyhedron meet. |

|endpoint |A point at the end of a line segment or a ray. A line segment is named by its two endpoints. “Segment LT or segment TL” is |

| |the line segment between points or L and T. |

|equal grouping story |A number story that involves separating something into equal groups. In such a problem, the total and the number in each |

| |group are known. Division can often be used to solve equal grouping stories. |

|equal groups |Sets with the same number of elements. |

|equal sharing story |A number story that involves sharing something equally. In such a problem, the total and the number of groups are known. |

| |Division can often be used to solve equal sharing stories. |

|equivalent names |Different ways of naming the same number. For example, 2 + 6, 4 + 4, 12 – 4, 18 – 10, 100 – 92, 5 + 1 + 2, eight, VIII, llll|

| |lll, are all equivalent names for 8. |

|estimate |1. n. A close, rather than exact answer. A number close to another number. |

| |2. v. To make an estimate. |

|even number |A whole number that can be evenly divided by 2. It has 0, 2, 4, 6, or 8 in the ones place. Compare to odd number. |

|extended fact |A variation of basic arithmetic facts involving multiples of 10, 100, and so on. For example, 30 + 70 = 100, 40 x 5 = 200, |

| |560 ( 7 = 80, are extended facts. |

|face |A surface that bounds a 3-dimensional shape. It may be curved (as on a cylinder or a cone) or flat (as on a prism). |

|fact extensions |Calculations of larger numbers using knowledge of basic facts. For example, knowing the basic fact 5 + 8 = 13 makes it easy |

| |to solve problems such as 50 + 80 = ? and 65 + ? = 73. Fact extensions can also be applied to subtraction, multiplication,|

| |and division facts. |

|fact family |A collection of related addition and subtraction facts, or multiplication and division facts, made from the same numbers. |

| |For 5, 6, and 11 the addition/subtraction family consists of 5 + 6 = 11, 6 + 5 = 11, 11 – 5 = 6, 11 – 6 = 5. |

|factors |The numbers being multiplied in a multiplication number model. In the number model, 4 x 3 = 12, 4 and 3 are factors. |

|fact family |A collection of related addition and subtraction facts, or multiplication and division facts, made from the same numbers. |

| |For 5, 6, and 11 the addition/subtraction family consists of 5 + 6 = 11, 6 + 5 = 11, 11 – 5 = 6, 11 – 6 = 5. |

|factors |The numbers being multiplied in a multiplication number model. In the number model, 4 x 3 = 12, 4 and 3 are factors. |

|fahrenheit |The temperature scale on which pure water at sea level freezes at 32 degrees and boils at 212 degrees. The Fahrenheit scale |

| |is used in the U. S. customary system. |

|foot (ft.) |In the U. S. customary system, a unit of length equivalent to 12 inches or 1/3 of a yard. |

|fraction |A number in the form of a/b that names part of an object or collection of objects, compares two quantities, or represents |

| |division. A fraction names equal parts of a whole. |

|frequency |The number of times a value occurs in a set of data. |

|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. |

|gallon (gal.) |In the U. S. customary system a unit of capacity equal to four quarts. |

|geometric solid |A 3-dimensional shape bounded by surfaces. Common geometric solids include the rectangular prism, square based pyramid, |

| |cylinder, cone, and sphere. |

|geometry |The study of spatial objects and their properties and relationships. The word geometry is derived from the Greek words for |

| |earth and measure. |

|gram (g) |In the metric system, a unit of mass equal to 1/1000 of a kilogram. |

|height |A measure of how tall something is. |

|heptagon |A seven-sided polygon. |

|hexagon |A six-sided polygon. |

|Inch (in.) |In the U. S. customary system a unit of length equal to 1/12 of a foot and equivalent to 2.54 centimeters. |

|input |A number inserted into an imaginary function machine, which processes numbers according to a designated rule. |

|Kilogram (kg) |In the metric system, the fundamental unit of mass; it is equal to 1,000 grams. One kilogram equals about 2.2 pounds. |

|kilometer (km) |In the metric system, a unit of length equal to 1,000 meters. One kilometer equals about 0.62 miles. |

|kite |A quadrilateral with 2 adjacent sides that are the same length (a rhombus is not a kite). |

|label |Descriptive word or phrase used to put numbers in context. Using a label reinforces the idea that numbers refer to |

| |something. Flags, snowballs, and scary monsters are example of labels. |

|length |Usually, but not necessarily, the longer dimension of a rectangle or a rectangular object. |

|line |A straight path that extends infinitely in opposite directions. |

|line of symmetry |A line that divides a figure into two halves that are mirror images. |

|line plot |A sketch of data in which checkmarks, Xs, or stick on notes above a number line show the frequency of each value. |

|line segment |A straight path joining two points. |

|liter (l) |In the metric system, a unit of capacity equal to 1,000 milliliters. A liter is a little larger than a quart. |

|mathematics |A study of relationships among numbers, shapes, systems, and patterns. It is used to count and measure things, to |

| |discover similarities and differences between them, to solve problems, and to learn about an organize the world. |

|measurement unit |The reference unit used when measuring length, weight, capacity, time, or temperature. Ounces, degrees, and centimeters|

| |are examples of measurement units. |

|median |The middle value in a set of data when the data are listed in order from least to greatest (or greatest to least). See |

| |middle value. |

|memory |The mechanical or electronic storage of information for later recall as in computers or calculators. |

|meter (m) |In the metric system, the fundamental unit of length equal to 10 decimeters, 100 centimeters, or 1,000 millimeters. |

|metric system |A measurement system based on the base-10 numeration system and used in most countries in the world. Units for linear |

| |measure (length, distance) include millimeter, centimeter, meter, and kilometer; units for mass include gram and |

| |kilogram; units for capacity include milliliter and liter and the unit for temperature change is degrees Celsius. |

|middle value |The number in the middle when a set of data is organized in sequential order. Also called the median. |

|mile (mi.) |In the U. S. customary system a unit of length equivalent to 5,280 feet, 1,760 yards, or about 1,609 meters. |

|milliliter (mL) |In the metric system, a unit of capacity equal to 1/1000 of a liter; one cubic centimeter. |

|millimeter (mm) |In the metric system, a unit of length equivalent to 1/10 of a centimeter, or 1/1000 of a meter. |

|mode |The value or values that occur most often in a set of data. For example, in the data set 3, 4, 4, 4, 5, 5, 6, 4 is the |

| |mode. |

|multiples |Repeated groups of the same amount. Multiples of a number are the products of that number and whole numbers. |

|multiplication |The operation used with whole numbers, fractions, or decimals to find the total number of things in several equal |

| |groups. |

|multiplication fact |The product of two 1-digit numbers, such as 6 x 7 = 42 |

|number family |A triplet of numbers consisting of two addends and their sum or two factors and their product. |

|number grid |A table in which consecutive numbers are arranged in rows of ten. A move from one number to the next number within a |

| |row is a change of one; a move from one number to the next number within a column is a change of ten. |

|number line |A line on which equidistant points correspond to numbers in order. Used as a frame of reference for counting and |

| |numeration activities. |

|number model |A number sentence that models or fits a situation. For example, the situation Sally had $5 and then she earned $8, can |

| |be modeled as 5 + 8 = 13. |

|number story |A story that contains a problem that can be solved using one or more of the four basic arithmetic operations or by |

| |sorting out relations such as equals, is less than, or is greater than. |

|numerator |The number written above the line in a fraction. In a part-whole fraction, it names the number of equal parts of the |

| |whole being considered. |

|octagon |An 8-sided polygon. |

|odd number |A number that cannot be evenly divided by 2. It has 1, 3, 5, 7, or 9 in the ones place. Compare to even number. |

|operation |An action performed on 1 or 2 numbers producing a single number result. |

|ordinal number |A number used to express position or order in a series, such as first, third, and tenth. |

|ounce (oz.) |In the U. S. customary system, a unit of weight equal to 1/16 of a pound, one ounce is 31.103 grams. |

|outcome |A possible result of a random process. Heads and tails are the two outcomes of tossing a coin. |

|output |The number resulting from the application of a rule used by an imaginary function machine to process numbers. |

|pan balance |A device used to weigh objects or to compare their weights. |

|parallel |Lines, rays, line segments, or planes that are equidistant at all points, no matter how far extended; never meeting |

|parallelogram |A quadrilateral that has two pairs of parallel sides and opposite sides that are congruent. |

|pattern |A model or plan by which objects or numbers can be arranged so that what comes next can be predicted. |

|pentagon |A 5-sided polygon. |

|per |In each or for each, as in 10 chairs per row or 6 tickets per family. |

|percent (%) |Per hundred, or out of a hundred. Times 1/100; times 0.01; 1 one hundredth. 15% means 15/100 or 0.15 of a number. |

|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 metron that means “measure”; perimeter means “around measure”. |

|pictograph |A graph constructed with pictures or symbols. A pictograph makes it possible to compare, at a glance, the relative |

| |amounts of two or more counts or measures. |

|pint |In the U. S. customary system, a unit of capacity equal to 2 cups or 16 fluid ounces. |

|place value |The relative worth of each digit in a number, which is determined by its position. Each place has a value ten times that|

| |of the place to its right, and 1/10 of the value of the place to its left. |

|point |An exact location in space. Points are usually labeled with capital letters. |

|polygon |A closed plane figure formed by 3 or more line segments that meet only at their endpoints. The word comes from Greek: |

| |poly means “many” and gon (from gonia) means “angle.” |

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

| |of the polygon. |

|pound (lb.) |In the U. S. customary system, a unit of weight equal to 16 ounces and defined as 0.45359237 kilograms. |

|prism |A polyhedron with 2 parallel flat faces (bases) with the same size and shape. The other faces are bounded by |

| |parallelograms. Prisms are classified according to the shape of the 2 parallel bases. |

|probability |A number from 0 to 1 that indicates the likelihood that an event will happen. The closer the probability is to 1, the |

| |more likely it is that the event will happen. The closer a probability is to 0, the less likely it is that an event will|

| |happen. For example, the probability that a fair coin will shows heads is ½. |

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

|property |A feature of an object. For example, size, shape, color, and number of parts are all properties. Same as attribute. |

|Pyramid |A polyhedron (3-dimensional shape) in which 1 face (the base) is a polygon and the other faces are triangles with a |

| |common vertex called the apex. A pyramid is classified according to the shape of its base. |

|quadrangle |A 4-sided polygon. Same as quadrilateral. |

|quadrilateral |A 4-sided polygon. Same as quadrangle. |

|quart |In the U. S. 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; the number of equal shares. In the division number model 15 ( 3 = 5, 5|

| |is the quotient. |

|range |The difference between the greatest and least numbers in a set of data. |

|rectangle |A parallelogram whose angles are all right angles. Seeparallelogram. |

|rectangular prism |A prism whose bases are rectangles. |

|rectangular pyramid |A pyramid whose base is a rectangle. |

|regular polygon |A polygon whose sides are all 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. |

|relation symbol |A symbol used to express a relationship between 2 quantities. Some relation symbols used in number sentences are = (is |

| |equal to), < (is less than), and > (is greater than). |

|remainder |The amount left over when things are divided into equal shares. In the division number model 16 ( 3 = 5 R1, the remainder |

| |is 1. |

|rhombus |A parallelogram with sides that are all the same length. The angles may be right angles, in which case the rhombus is a |

| |square. |

|right angle |A square corner; a 90( angle. See angle. |

|round |1. v. 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. |

| |2. adj. Circular in shape |

|scale |The ratio of the distance on a map, globe, or drawing to the actual distance. On a thermometer a vertical number line used |

| |for measuring temperature. |

|set |A collection or group of objects, numbers, or other items. |

|side |Any one of the line segments that make up a polygon. Sometimes a face of a 3-dimensional figure is called a side. |

|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 sphere. A sphere is hollow; it does not include the points in it s interior. |

|square |A rectangle whose sides are all the same length. |

|square number |A number that is the product of a whole number and itself. A square number can be represented by a square array. |

|square of a number |The product of a number and itself. |

|square pyramid |A pyramid with a square base. |

|square units |The units used to measure area. A square unit represents a square with the measure of each side being one of that unit. |

| |For example, a square inch represents a square that measures one inch on each side. |

|straightedge |A tool, such as a ruler, used to draw line segments. A straightedge does not need to have measure marks on it. |

|subtraction |A mathematical operation based on taking away one quantity from another or decreasing a quantity. |

|sum |The result of adding two or more numbers. |

|survey |A study that collects data. |

|symmetry |The property of exact balance in a figure; having the same size and shape across a dividing line or around a point. |

|tally marks |Marks (llll llll) used to keep track of a count. |

|tens |The place value position equal to 10 times the unit value. |

|tetrahedron |A polyhedron with 4 sides. |

|3-dimensional |Objects that are not completely within a single flat surface; objects with thickness, as well as length and width. |

|(3-D) | |

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

|time line |A device for showing in sequence when events took place. A time line is a number line with numbers naming years, days, |

| |and so on. |

|trapezoid |A quadrilateral that has one pair of parallel sides. No two sides need be the same length. |

|triangle |A 3-sided polygon. |

|triangular prism |A prism whose bases are triangles. |

|triangular pyramid |A pyramid in which all faces are triangles, any one of which can be called the base. |

|2-dimensional |Objects completely within a plane; objects with length and width, but no thickness. |

|(2-D) | |

|unit |A label, descriptive word, or unit of measure, used to put a number in context. Using a unit with a number reinforces the|

| |idea that numbers refer to something. Fingers, snowballs, miles, and cents are examples of units. |

|U. S. Customary system |The measuring system most frequently used in the United States. Units for linear measure (length, distance) include inch,|

| |foot, yard, mile; units for weight include ounce and pound; units for capacity include cup, pint, quart, and gallon; for |

| |temperature change, degrees Fahrenheit. |

|vertex (vertices) |The point at which the rays or line segments of an angle, sides of a polygon, or the edges of a polyhedron meet. Same as |

| |corner. |

|weight |A measure of how heavy something is. |

|width of a rectangle |Length of one side of a rectangle or rectangular object; often the shorter side. |

|x-by-y array |An arrangement have x rows of y per row, representing x sets of y objects in each set. |

|yard (yd) |Historically, the distance from the tip of the nose to the tip of the longest finger. In the U. S. customary system, a |

| |unit of length eqivalent to 3 feet or 36 inches. |

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

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

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