new jersey standards alignment grades two through...

New Jersey Standards Alignment Grades Two through Twelve

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New Jersey Standards Alignment

Standards list with Aligned Product Skills

The Standards list with Aligned Product Skills Report is a standards-oriented document showing the entire list of standards for the subject and grade on the left side of the report with the aligning product objectives on the right side. This alignment report shows the breadth of standards coverage for the purpose and focus of this product.

Note to Educator .....................................................iii

Grade 2 ..................................................... 1

Grade 3 ....................................................18

Grade 4 ....................................................36

Grade 5 ....................................................59

Grade 6 ....................................................75

Grade 7 ....................................................97

Grade 8 ..................................................114

Grades 9 - 12 ..................................................137

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Note to Educator: Thank you for your interest in Renaissance Learning technology. The attached document contains the alignment between the software and/or instructional materials and the skills described in the state standards documentation. At Renaissance Learning, we recognize the impact that the standards-based reform movement and high-stakes standardized testing have on schools, and we share the concerns of educators and administrators that students perform well on high-stakes assessments. We hope this report answers your questions regarding the alignment of Renaissance Learning technology and materials to your state standards. If you have any questions about the attached document, please feel free to call us at (800) 338-4204. Sincerely, Renaissance Sales and Funding Staff

P.O. Box 8036 Wisconsin Rapids, WI 54495-8036

Phone: (800) 338-4204 Fax: (715) 424-4242

www.renlearn.com

Agency Tag Set NameNew Jersey, Math, 2008, Grade 2, Core Curriculum Content Standards, State Department of Education

Standard Skill Area Description Skill DescriptionNJ 4.1 - All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.NJ 4.1.2 A - Number SenseNJ 4.1.2 A.1 - By the end of Grade 2, students will: 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).

NJ 4.1.2 A.1.a - By the end of Grade 2, students will: 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

NJ 4.1.2 A.1.b - By the end of Grade 2, students will: 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). Ordinals

Skill Area 3 - Numeration Concepts

Skill 4 - NA4 - Ones: Use ordinal numbers

NJ 4.1.2 A.1.c - By the end of Grade 2, students will: 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). Proper fractions (denominators of 2, 3, 4, 8, 10)

Skill Area 6 - Geometry Skill 8 - G00 - Identify fraction parts of common plane shapes

NJ 4.1.2 A.2 - By the end of Grade 2, students will: Demonstrate an understanding of whole number place value concepts.


Skill 13 - N08 - Hundreds: Identify place value of digits

NJ 4.1.2 A.3 - By the end of Grade 2, students will: Understand that numbers have a variety of uses.


Skill 2 - NA2 - Ones: Using numerals to indicate quantity

Skill 7 - N02 - Tens: Associate numeral with group of objects

Standards List with Aligned Product SkillsProduct NameSTAR Math

STAR Math Grade 2

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Standard Skill Area Description Skill Description


NJ 4.1.2 A.4 - By the end of Grade 2, students will: Count and perform simple computations with coins.NJ 4.1.2 A.4.a - Amounts up to $1.00 (using cents notation)NJ 4.1.2 A.5 - By the end of Grade 2, students will: Compare and order whole numbers.


Skill 1 - NA1 - Ones: Placing numerals in order

Skill 6 - N01 - Tens: Place numerals (10-99) in order of valueSkill 11 - N06 - Hundreds: Place numerals in order of value

NJ 4.1.2 B - Numerical OperationsNJ 4.1.2 B.1 - By the end of Grade 2, students will: Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems.

NJ 4.1.2 B.1.a - By the end of Grade 2, students will: Develop the meanings of addition and subtraction by concretely modeling and discussing a large variety of problems. Joining, separating, and comparingNJ 4.1.2 B.2 - By the end of Grade 2, students will: Explore the meanings of multiplication and division by modeling and discussing problems.

NJ 4.1.2 B.3 - By the end of Grade 2, students will: 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.

Skill Area 4 - Computation Processes

Skill 1 - C01 - Addition of basic facts to 10

Skill 2 - C02 - Subtraction of basic facts to 10Skill 3 - C03 - Addition of basic facts to 18Skill 4 - C04 - Subtraction of basic facts to 18Skill 5 - C05 - Addition of three single digit addends

Skill Area 9 - Word Problems Skill 1 - W03 - Word problems: Addition of basic factsSkill 2 - W04 - Word problems: Subtraction of basic facts

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NJ 4.1.2 B.4 - By the end of Grade 2, students will: Construct, use, and explain procedures for performing addition and subtraction calculations with:NJ 4.1.2 B.4.a - By the end of Grade 2, students will: Construct, use, and explain procedures for performing addition and subtraction calculations with: Pencil-and-paper


Skill 5 - C05 - Addition of three single digit addends

Skill 6 - C06 - Addition beyond basic facts, no regrouping (2d+1d)Skill 7 - C07 - Subtraction beyond basic facts, no regrouping (2d-1d)Skill 8 - C08 - Addition beyond basic facts with regrouping (2d+1d, 2d+2d)

Skill Area 9 - Word Problems Skill 3 - W06 - Word problems: Addition beyond basic facts, no regrouping (2d+1d)Skill 4 - W08 - Word problems: Addition beyond basic facts with regrouping (2d+1d, 2d+2d)

NJ 4.1.2 B.4.b - By the end of Grade 2, students will: Construct, use, and explain procedures for performing addition and subtraction calculations with: Mental math



Skill 2 - C02 - Subtraction of basic facts to 10Skill 3 - C03 - Addition of basic facts to 18Skill 4 - C04 - Subtraction of basic facts to 18

Skill Area 9 - Word Problems Skill 1 - W03 - Word problems: Addition of basic factsSkill 2 - W04 - Word problems: Subtraction of basic facts

NJ 4.1.2 B.4.c - By the end of Grade 2, students will: Construct, use, and explain procedures for performing addition and subtraction calculations with: CalculatorNJ 4.1.2 B.5 - By the end of Grade 2, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.

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NJ 4.1.2 B.5.a - By the end of Grade 2, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Addition of 2-digit numbers


Skill 6 - C06 - Addition beyond basic facts, no regrouping (2d+1d)

Skill 8 - C08 - Addition beyond basic facts with regrouping (2d+1d, 2d+2d)

Skill Area 9 - Word Problems Skill 3 - W06 - Word problems: Addition beyond basic facts, no regrouping (2d+1d)Skill 4 - W08 - Word problems: Addition beyond basic facts with regrouping (2d+1d, 2d+2d)

NJ 4.1.2 B.5.b - By the end of Grade 2, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Subtraction of 2-digit numbers


Skill 7 - C07 - Subtraction beyond basic facts, no regrouping (2d-1d)

NJ 4.1.2 B.6 - By the end of Grade 2, students will: Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.NJ 4.1.2 B.7 - By the end of Grade 2, students will: Check the reasonableness of results of computations.NJ 4.1.2 B.8 - By the end of Grade 2, students will: Understand and use the inverse relationship between addition and subtraction.NJ 4.1.2 C - EstimationNJ 4.1.2 C.1 - By the end of Grade 2, students will: Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

NJ 4.1.2 C.2 - By the end of Grade 2, students will: Determine the reasonableness of an answer by estimating the result of computations (e.g., 15 + 16 is not 211).

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NJ 4.1.2 C.3 - By the end of Grade 2, students will: Explore a variety of strategies for estimating both quantities (e.g., the number of marbles in a jar) and results of computation.NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.2 A - Geometric PropertiesNJ 4.2.2 A.1 - By the end of Grade 2, students will: Identify and describe spatial relationships among objects in space and their relative shapes and sizes.NJ 4.2.2 A.1.a - By the end of Grade 2, students will: Identify and describe spatial relationships among objects in space and their relative shapes and sizes. Inside/outside, left/right, above/below, between

Skill Area 6 - Geometry Skill 1 - GA1 - Use basic terms to describe position

NJ 4.2.2 A.1.b - By the end of Grade 2, students will: Identify and describe spatial relationships among objects in space and their relative shapes and sizes. Smaller/larger/same size, wider/narrower, longer/shorter

Skill Area 7 - Measurement Skill 1 - MA1 - Use simple vocabulary of measurement

NJ 4.2.2 A.1.c - By the end of Grade 2, students will: Identify and describe spatial relationships among objects in space and their relative shapes and sizes. Congruence (i.e., same size and shape)

NJ 4.2.2 A.2 - By the end of Grade 2, students will: Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes.

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NJ 4.2.2 A.2.a - By the end of Grade 2, students will: Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes. Vertex, edge, face, side

Skill Area 6 - Geometry Skill 6 - GA6 - Recognize elements of basic shapes

NJ 4.2.2 A.2.b - By the end of Grade 2, students will: Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes. 3D figures - cube, rectangular prism, sphere, cone, cylinder, and pyramid

Skill Area 6 - Geometry Skill 7 - GA7 - Identify common solid shapes

NJ 4.2.2 A.2.c - By the end of Grade 2, students will: Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes. 2D figures - square, rectangle, circle, triangle

Skill Area 6 - Geometry Skill 2 - GA2 - Identify common plane shapes

Skill 6 - GA6 - Recognize elements of basic shapes

NJ 4.2.2 A.2.d - By the end of Grade 2, students will: Use concrete objects, drawings, and computer graphics to identify, classify, and describe standard three-dimensional and two-dimensional shapes. Relationships between three- and two-dimensional shapes (i.e., the face of a 3D shape is a 2D shape)

Skill Area 6 - Geometry Skill 4 - GA4 - Compare common objects to basic shapes

NJ 4.2.2 A.3 - By the end of Grade 2, students will: Describe, identify and create instances of line symmetry.

Skill Area 6 - Geometry Skill 5 - GA5 - Understand basic symmetry

NJ 4.2.2 A.4 - By the end of Grade 2, students will: Recognize, describe, extend and create designs and patterns with geometric objects of different shapes and colors.

NJ 4.2.2 B - Transforming Shapes

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NJ 4.2.2 B.1 - By the end of Grade 2, students will: Use simple shapes to make designs, patterns, and pictures.

NJ 4.2.2 B.2 - By the end of Grade 2, students will: Combine and subdivide simple shapes to make other shapes.

NJ 4.2.2 C - Coordinate GeometryNJ 4.2.2 C.1 - By the end of Grade 2, students will: Give and follow directions for getting from one point to another on a map or grid.NJ 4.2.2 D - Units of MeasurementNJ 4.2.2 D.1 - By the end of Grade 2, students will: Directly compare and order objects according to measurable attributes.NJ 4.2.2 D.1.a - By the end of Grade 2, students will: Directly compare and order objects according to measurable attributes. Attributes - length, weight, capacity, time, temperature

NJ 4.2.2 D.2 - By the end of Grade 2, students will: Recognize the need for a uniform unit of measure.NJ 4.2.2 D.3 - By the end of Grade 2, students will: Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems.NJ 4.2.2 D.3.a - By the end of Grade 2, students will: 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, meterNJ 4.2.2 D.3.b - By the end of Grade 2, students will: Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems. Weight - pound, gram, kilogram

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NJ 4.2.2 D.3.c - By the end of Grade 2, students will: Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems. Capacity - pint, quart, literNJ 4.2.2 D.3.d - By the end of Grade 2, students will: Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems. Time - second, minute, hour, day, week, month, year

Skill Area 7 - Measurement Skill 5 - MA5 - Tell time to the hour and half hour

Skill 7 - MA7 - Order days of the weekSkill 8 - M00 - Order months of the year

NJ 4.2.2 D.3.e - By the end of Grade 2, students will: Select and use appropriate standard and non-standard units of measure and standard measurement tools to solve real-life problems. Temperature - degrees Celsius, degrees Fahrenheit

Skill Area 7 - Measurement Skill 6 - MA6 - Read a thermometer

NJ 4.2.2 D.4 - By the end of Grade 2, students will: Estimate measures.

NJ 4.2.2 E - Measuring Geometric ObjectsNJ 4.2.2 E.1 - By the end of Grade 2, students will: Directly measure the perimeter of simple two-dimensional shapes.NJ 4.2.2 E.2 - By the end of Grade 2, students will: Directly measure the area of simple two-dimensional shapes by covering them with squares.NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.2 A - Patterns

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NJ 4.3.2 A.1 - By the end of Grade 2, students will: Recognize, describe, extend, and create patterns.

NJ 4.3.2 A.1.a - By the end of Grade 2, students will: Recognize, describe, extend, and create patterns. Using concrete materials (manipulatives), pictures, rhythms, & whole numbers

Skill Area 6 - Geometry Skill 9 - G01 - Identify numeric patterns

Skill Area 10 - Algebra Skill 1 - A00 - Can skip count by 2, 5, 10 in ascending order

NJ 4.3.2 A.1.b - By the end of Grade 2, students will: Recognize, describe, extend, and create patterns. Descriptions using words and symbols (e.g., "add two" or "+ 2")

NJ 4.3.2 A.1.c - By the end of Grade 2, students will: Recognize, describe, extend, and create patterns. Repeating patternsNJ 4.3.2 A.1.d - By the end of Grade 2, students will: Recognize, describe, extend, and create 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)



NJ 4.3.2 B - Functions and RelationshipsNJ 4.3.2 B.1 - By the end of Grade 2, students will: Use concrete and pictorial models of function machines to explore the basic concept of a function.NJ 4.3.2 C - ModelingNJ 4.3.2 C.1 - By the end of Grade 2, students will: Recognize and describe changes over time (e.g., temperature, height).NJ 4.3.2 C.2 - By the end of Grade 2, students will: Construct and solve simple open sentences involving addition or subtraction.

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NJ 4.3.2 C.2.a - By the end of Grade 2, students will: Construct and solve simple open sentences involving addition or subtraction. Result unknown (e.g., 6 - 2 = __ or n = 3 + 5)NJ 4.3.2 C.2.b - By the end of Grade 2, students will: Construct and solve simple open sentences involving addition or subtraction. Part unknown (e.g., 3 + [] = 8)

Skill Area 10 - Algebra Skill 2 - A01 - Simple number sentence

NJ 4.3.2 D - ProceduresNJ 4.3.2 D.1 - By the end of Grade 2, students will: Understand and apply (but don't name) the following properties of addition:NJ 4.3.2 D.1.a - By the end of Grade 2, students will: Understand and apply (but don't name) the following properties of addition: Commutative (e.g., 5 + 3 = 3 + 5)NJ 4.3.2 D.1.b - By the end of Grade 2, students will: Understand and apply (but don't name) the following properties of addition: Zero as the identity element (e.g., 7 + 0 = 7)


Skill 10 - N05 - Tens: Understand the concept of zero

NJ 4.3.2 D.1.c - By the end of Grade 2, students will: Understand and apply (but don't name) the following properties of addition: Associative (e.g., 7 + 3 + 2 can be found by first adding either 7 + 3 or 3 + 2)

NJ 4.4 - 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.NJ 4.4.2 A - Data AnalysisNJ 4.4.2 A.1 - By the end of Grade 2, students will: Collect, generate, record, and organize data in response to questions, claims, or curiosity.

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NJ 4.4.2 A.1.a - By the end of Grade 2, students will: Collect, generate, record, and organize data in response to questions, claims, or curiosity. Data collected from students' everyday experiences

NJ 4.4.2 A.1.b - By the end of Grade 2, students will: Collect, generate, record, and organize data in response to questions, claims, or curiosity. Data generated from chance devices, such as spinners and diceNJ 4.4.2 A.2 - By the end of Grade 2, students will: Read, interpret, construct, and analyze displays of data.NJ 4.4.2 A.2.a - By the end of Grade 2, students will: Read, interpret, construct, and analyze displays of data. Pictures, tally chart, pictograph, bar graph, Venn diagram

Skill Area 8 - Data Analysis and Statistics

Skill 1 - SA1 - Read tally charts

Skill 2 - S00 - Read simple pictographsSkill 4 - S02 - Read bar graphSkill 7 - S05 - Interpret bar graph

NJ 4.4.2 A.2.b - By the end of Grade 2, students will: Read, interpret, construct, and analyze displays of data. Smallest to largest, most frequent (mode)NJ 4.4.2 B - ProbabilityNJ 4.4.2 B.1 - By the end of Grade 2, students will: Use chance devices like spinners and dice to explore concepts of probability.NJ 4.4.2 B.1.a - By the end of Grade 2, students will: Use chance devices like spinners and dice to explore concepts of probability. Certain, impossibleNJ 4.4.2 B.1.b - By the end of Grade 2, students will: Use chance devices like spinners and dice to explore concepts of probability. More likely, less likely, equally likely

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NJ 4.4.2 B.2 - By the end of Grade 2, students will: Provide probability of specific outcomes.NJ 4.4.2 B.2.a - By the end of Grade 2, students will: 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 five equal sectors, then probability of getting a particular sector is one out of five)

NJ 4.4.2 B.2.b - By the end of Grade 2, students will: Provide probability of specific outcomes. When picking a marble from a bag with three red marbles and four blue marbles, the probability of getting a red marble is three out of seven

NJ 4.4.2 C - Discrete Mathematics-Systematic Listing and CountingNJ 4.4.2 C.1 - By the end of Grade 2, students will: Sort and classify objects according to attributes.NJ 4.4.2 C.1.a - By the end of Grade 2, students will: Sort and classify objects according to attributes. Venn diagramsNJ 4.4.2 C.2 - By the end of Grade 2, students will: Generate all possibilities in simple counting situations (e.g., all outfits involving two shirts and three pants).NJ 4.4.2 D - Discrete Mathematics-Vertex-Edge Graphs and Algorithms

NJ 4.4.2 D.1 - By the end of Grade 2, students will: Follow simple sets of directions (e.g., from one location to another, or from a recipe).

NJ 4.4.2 D.2 - By the end of Grade 2, students will: Color simple maps with a small number of colors.NJ 4.4.2 D.3 - By the end of Grade 2, students will: Play simple two-person games (e.g., tic-tac-toe) and informally explore the idea of what the outcome should be.

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NJ 4.4.2 D.4 - By the end of Grade 2, students will: Explore concrete models of vertex-edge graphs (e.g. vertices as "islands" and edges as "bridges").NJ 4.4.2 D.4.a - By the end of Grade 2, students will: Explore concrete models of vertex-edge graphs (e.g. vertices as "islands" and edges as "bridges"). Paths from one vertex to anotherNJ 4.5 - All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.

NJ 4.5.2 A - Problem SolvingNJ 4.5.2 A.1 - By the end of Grade 2, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5.2 A.2 - By the end of Grade 2, students will: Solve problems that arise in mathematics and in other contexts.NJ 4.5.2 A.2.a - By the end of Grade 2, students will: Solve problems that arise in mathematics and in other contexts. Open-ended problems

NJ 4.5.2 A.2.b - By the end of Grade 2, students will: Solve problems that arise in mathematics and in other contexts. Non-routine problems

NJ 4.5.2 A.2.c - By the end of Grade 2, students will: Solve problems that arise in mathematics and in other contexts. Problems with multiple solutions

NJ 4.5.2 A.2.d - By the end of Grade 2, students will: Solve problems that arise in mathematics and in other contexts. Problems that can be solved in several ways

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NJ 4.5.2 A.3 - By the end of Grade 2, students will: Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.NJ 4.5.2 A.4 - By the end of Grade 2, students will: Pose problems of various types and levels of difficulty.

NJ 4.5.2 A.5 - By the end of Grade 2, students will: Monitor their progress and reflect on the process of their problem solving activity.

NJ 4.5.2 A.6 - By the end of Grade 2, students will: Distinguish relevant from irrelevant information, and identify missing information.NJ 4.5.2 B - CommunicationNJ 4.5.2 B.1 - By the end of Grade 2, students will: Use communication to organize and clarify their mathematical thinking.NJ 4.5.2 B.1.a - By the end of Grade 2, students will: Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5.2 B.1.b - By the end of Grade 2, students will: Use communication to organize and clarify their mathematical thinking. Discussion, listening, and questioning

NJ 4.5.2 B.2 - By the end of Grade 2, students will: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.

NJ 4.5.2 B.3 - By the end of Grade 2, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5.2 B.4 - By the end of Grade 2, students will: Use the language of mathematics to express mathematical ideas precisely.NJ 4.5.2 C - Connections

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NJ 4.5.2 C.1 - By the end of Grade 2, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

NJ 4.5.2 C.2 - By the end of Grade 2, students will: 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).NJ 4.5.2 C.3 - By the end of Grade 2, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.NJ 4.5.2 C.4 - By the end of Grade 2, students will: Apply mathematics in practical situations and in other disciplines.NJ 4.5.2 C.5 - By the end of Grade 2, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5.2 C.6 - By the end of Grade 2, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5.2 D - ReasoningNJ 4.5.2 D.1 - By the end of Grade 2, students will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5.2 D.2 - By the end of Grade 2, students will: Use reasoning to support their mathematical conclusions and problem solutions.

NJ 4.5.2 D.3 - By the end of Grade 2, students will: Select and use various types of reasoning and methods of proof.NJ 4.5.2 D.4 - By the end of Grade 2, students will: Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.

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NJ 4.5.2 D.5 - By the end of Grade 2, students will: Make and investigate mathematical conjectures.

NJ 4.5.2 D.5.a - By the end of Grade 2, students will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjecturesNJ 4.5.2 D.5.b - By the end of Grade 2, students will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5.2 D.6 - By the end of Grade 2, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.

NJ 4.5.2 E - RepresentationsNJ 4.5.2 E.1 - By the end of Grade 2, students will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5.2 E.1.a - By the end of Grade 2, students will: Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles)

NJ 4.5.2 E.1.b - By the end of Grade 2, students will: Create and use representations to organize, record, and communicate mathematical ideas. Pictorial representations (e.g., diagrams, charts, or tables)

NJ 4.5.2 E.1.c - By the end of Grade 2, students will: Create and use representations to organize, record, and communicate mathematical ideas. Symbolic representations (e.g., a formula)NJ 4.5.2 E.1.d - By the end of Grade 2, students will: Create and use representations to organize, record, and communicate mathematical ideas. Graphical representations (e.g., a line graph)

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NJ 4.5.2 E.2 - By the end of Grade 2, students will: Select, apply, and translate among mathematical representations to solve problems.NJ 4.5.2 E.3 - By the end of Grade 2, students will: Use representations to model and interpret physical, social, and mathematical phenomena.

NJ 4.5.2 F - TechnologyNJ 4.5.2 F.1 - By the end of Grade 2, students will: Use technology to gather, analyze, and communicate mathematical information.NJ 4.5.2 F.2 - By the end of Grade 2, students will: Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.NJ 4.5.2 F.3 - By the end of Grade 2, students will: Use graphing calculators and computer software to investigate properties of functions and their graphs.NJ 4.5.2 F.4 - By the end of Grade 2, students will: Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

NJ 4.5.2 F.5 - By the end of Grade 2, students will: Use computer software to make and verify conjectures about geometric objects.

NJ 4.5.2 F.6 - By the end of Grade 2, students will: Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).

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NJ 4.1.3 A.1.a - By the end of Grade 3, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 3 pertain to these sets of numbers as well). Whole numbers through hundred thousands

NJ 4.1.3 A.1.b - By the end of Grade 3, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 3 pertain to these sets of numbers as well). Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10) as part of a whole, as a subset of a set, and as a location on a number line


Skill 29 - N27 - Fractions and decimals: Identify position of fractions on number line


NJ 4.1.3 A.2 - By the end of Grade 3, students will: Demonstrate an understanding of whole number place value concepts.



Skill 14 - N09 - Hundreds: Write numerals in expanded formSkill 17 - N13 - Thousands: Identify place value of digitsSkill 18 - N14 - Thousands: Write numerals in expanded formSkill 21 - N18 - Ten thousands, hundred thousands, millions, billions: Identify place value of digits


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Skill 22 - N19 - Ten thousands, hundred thousands, millions, billions: Write numerals in expanded form

NJ 4.1.3 A.3 - By the end of Grade 3, students will: Identify whether any whole number is odd or even.NJ 4.1.3 A.4 - By the end of Grade 3, students will: Explore the extension of the place value system to decimals through hundredths.


Skill 27 - N25 - Fractions and decimals: Identify place value of digits in decimals

NJ 4.1.3 A.5 - By the end of Grade 3, students will: Understand the various uses of numbers.NJ 4.1.3 A.5.a - By the end of Grade 3, students will: Understand the various uses of numbers. Counting, measuring, labeling (e.g., numbers on baseball uniforms)


Skill 2 - NA2 - Ones: Using numerals to indicate quantity

Skill 7 - N02 - Tens: Associate numeral with group of objects

NJ 4.1.3 A.6 - By the end of Grade 3, students will: Compare and order numbers.


Skill 11 - N06 - Hundreds: Place numerals in order of value

Skill 15 - N11 - Thousands: Place numerals in order of valueSkill 19 - N16 - Ten thousands, hundred thousands, millions, billions: Place numerals in order of value

NJ 4.1.3 B - Numerical OperationsNJ 4.1.3 B.1 - By the end of Grade 3, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems.

NJ 4.1.3 B.1.a - By the end of Grade 3, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. Addition and subtraction: joining, separating, comparing

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NJ 4.1.3 B.1.b - By the end of Grade 3, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. Multiplication: repeated addition, area/arrayNJ 4.1.3 B.1.c - By the end of Grade 3, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. Division: repeated subtraction, sharingNJ 4.1.3 B.2 - By the end of Grade 3, students will: Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as "skip counting" and "repeated subtraction").


Skill 12 - C12 - Multiplication basic facts

Skill 13 - C13 - Division basic facts


Skill Area 9 - Word Problems Skill 6 - W12 - Word problems: Multiplication of basic factsSkill 7 - W13 - Word problems: Division of basic facts


NJ 4.1.3 B.3 - By the end of Grade 3, students will: Construct, use, and explain procedures for performing whole number calculations with:

NJ 4.1.3 B.3.a - By the end of Grade 3, students will: Construct, use, and explain procedures for performing whole number calculations with: Pencil-and-paper


Skill 8 - C08 - Addition beyond basic facts with regrouping (2d+1d, 2d+2d)

Skill 9 - C09 - Subtraction beyond basic facts with regrouping (2d-1d, 2d-2d)Skill 10 - C10 - Addition beyond basic facts with double regrouping (3d+2d, 3d+3d)Skill 11 - C11 - Subtraction beyond basic facts with double regrouping (3d-2d, 3d-3d)

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Skill 14 - C14 - Multiplication beyond basic facts, no regrouping (2dx1d)

Skill Area 9 - Word Problems Skill 4 - W08 - Word problems: Addition beyond basic facts with regrouping (2d+1d, 2d+2d)Skill 5 - W09 - Word problems: Subtraction beyond basic facts with regrouping (2d-1d, 2d-2d)Skill 8 - W14 - Word problems: Multiplication beyond basic facts, no regrouping (2dx1d)

NJ 4.1.3 B.3.b - By the end of Grade 3, students will: Construct, use, and explain procedures for performing whole number calculations with: Mental math



Skill 2 - C02 - Subtraction of basic facts to 10Skill 3 - C03 - Addition of basic facts to 18Skill 4 - C04 - Subtraction of basic facts to 18Skill 6 - C06 - Addition beyond basic facts, no regrouping (2d+1d)Skill 7 - C07 - Subtraction beyond basic facts, no regrouping (2d-1d)

Skill Area 9 - Word Problems Skill 1 - W03 - Word problems: Addition of basic factsSkill 2 - W04 - Word problems: Subtraction of basic factsSkill 3 - W06 - Word problems: Addition beyond basic facts, no regrouping (2d+1d)

NJ 4.1.3 B.3.c - By the end of Grade 3, students will: Construct, use, and explain procedures for performing whole number calculations with: CalculatorNJ 4.1.3 B.4 - By the end of Grade 3, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.NJ 4.1.3 B.4.a - By the end of Grade 3, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Addition of 3-digit numbers


Skill 10 - C10 - Addition beyond basic facts with double regrouping (3d+2d, 3d+3d)

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NJ 4.1.3 B.4.b - By the end of Grade 3, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Subtraction of 3-digit numbers


Skill 11 - C11 - Subtraction beyond basic facts with double regrouping (3d-2d, 3d-3d)

NJ 4.1.3 B.4.c - By the end of Grade 3, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Multiplication of 2-digit numbers by 1-digit numbers



Skill Area 9 - Word Problems Skill 8 - W14 - Word problems: Multiplication beyond basic facts, no regrouping (2dx1d)

NJ 4.1.3 B.5 - By the end of Grade 3, students will: Count and perform simple computations with money.NJ 4.1.3 B.5.a - By the end of Grade 3, students will: Count and perform simple computations with money. Cents notation (¢)

Skill Area 7 - Measurement Skill 4 - MA4 - Understand the value of groups of coins to $1.00

NJ 4.1.3 B.6 - By the end of Grade 3, students will: Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.NJ 4.1.3 B.7 - By the end of Grade 3, students will: Check the reasonableness of results of computations.NJ 4.1.3 C - EstimationNJ 4.1.3 C.1 - By the end of Grade 3, students will: Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

NJ 4.1.3 C.2 - By the end of Grade 3, students will: Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the result of computations.

Skill Area 5 - Estimation Skill 1 - E06 - Estimation problems: Addition beyond basic facts, no regrouping (2d+1d)

Skill 2 - E07 - Estimation problems: Subtraction beyond basic facts, no regrouping (2d-1d)

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Skill 3 - E14 - Estimation problems: Multiplication beyond basic facts, no regrouping (2dx1d)Skill 5 - E18 - Estimation problems: Addition of whole numbers, any difficultySkill 6 - E19 - Estimation problems: Subtraction of whole numbers, any difficulty

NJ 4.1.3 C.3 - By the end of Grade 3, students will: Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.NJ 4.1.3 C.4 - By the end of Grade 3, students will: Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.3 A - Geometric PropertiesNJ 4.2.3 A.1 - By the end of Grade 3, students will: Identify and describe spatial relationships of two or more objects in space.NJ 4.2.3 A.1.a - By the end of Grade 3, students will: Identify and describe spatial relationships of two or more objects in space. Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)

NJ 4.2.3 A.1.b - By the end of Grade 3, students will: Identify and describe spatial relationships of two or more objects in space. Relative shapes and sizes

Skill Area 7 - Measurement Skill 1 - MA1 - Use simple vocabulary of measurement

NJ 4.2.3 A.2 - By the end of Grade 3, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.

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NJ 4.2.3 A.2.a - By the end of Grade 3, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. Vertex, edge, face, side, angle


NJ 4.2.3 A.2.b - By the end of Grade 3, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. 3D figures - cube, rectangular prism, sphere, cone, cylinder, and pyramid

NJ 4.2.3 A.2.c - By the end of Grade 3, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. 2D figures - square, rectangle, circle, triangle, pentagon, hexagon, octagon


NJ 4.2.3 A.3 - By the end of Grade 3, students will: Identify and describe relationships among two-dimensional shapes.NJ 4.2.3 A.3.a - By the end of Grade 3, students will: Identify and describe relationships among two-dimensional shapes. Same size, same shape

NJ 4.2.3 A.3.b - By the end of Grade 3, students will: Identify and describe relationships among two-dimensional shapes. Lines of symmetry


NJ 4.2.3 A.4 - By the end of Grade 3, students will: Understand and apply concepts involving lines, angles, and circles.NJ 4.2.3 A.4.a - By the end of Grade 3, students will: Understand and apply concepts involving lines, angles, and circles. Line, line segment, endpoint

Skill Area 6 - Geometry Skill 20 - G13 - Identify line segments

NJ 4.2.3 A.5 - By the end of Grade 3, students will: Recognize, describe, extend, and create space-filling patterns.NJ 4.2.3 B - Transforming Shapes

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NJ 4.2.3 B.1 - By the end of Grade 3, students will: Describe and use geometric transformations (slide, flip, turn).NJ 4.2.3 B.2 - By the end of Grade 3, students will: Investigate the occurrence of geometry in nature and art.NJ 4.2.3 C - Coordinate GeometryNJ 4.2.3 C.1 - By the end of Grade 3, students will: Locate and name points in the first quadrant on a coordinate grid.NJ 4.2.3 D - Units of MeasurementNJ 4.2.3 D.1 - By the end of Grade 3, students will: Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.NJ 4.2.3 D.2 - By the end of Grade 3, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems.NJ 4.2.3 D.2.a - By the end of Grade 3, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Length - fractions of an inch (1/4, 1/2), mile, decimeter, kilometerNJ 4.2.3 D.2.b - By the end of Grade 3, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Area - square inch, square centimeterNJ 4.2.3 D.2.c - By the end of Grade 3, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Weight - ounceNJ 4.2.3 D.2.d - By the end of Grade 3, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Capacity - fluid ounce, cup, gallon, milliliter

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NJ 4.2.3 D.3 - By the end of Grade 3, students will: Incorporate estimation in measurement activities (e.g., estimate before measuring).

Skill Area 7 - Measurement Skill 10 - M02 - Customary measures: Estimating linear measures

Skill 15 - M08 - Estimating linear measure in metric units.

NJ 4.2.3 E - Measuring Geometric ObjectsNJ 4.2.3 E.1 - By the end of Grade 3, students will: Determine the area of simple two-dimensional shapes on a square grid.NJ 4.2.3 E.2 - By the end of Grade 3, students will: Determine the perimeter of simple shapes by measuring all of the sides.NJ 4.2.3 E.3 - By the end of Grade 3, students will: Measure and compare the volume of three-dimensional objects using materials such as rice or cubes.NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.3 A - PatternsNJ 4.3.3 A.1 - By the end of Grade 3, students will: Recognize, describe, extend, and create patterns.

NJ 4.3.3 A.1.a - By the end of Grade 3, students will: Recognize, describe, extend, and create patterns. Descriptions using words and number sentences/expressions

NJ 4.3.3 A.1.b - By the end of Grade 3, students will: Recognize, describe, extend, and create patterns. Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11,... or 800, 400, 200,...)



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NJ 4.3.3 B - Functions and RelationshipsNJ 4.3.3 B.1 - By the end of Grade 3, students will: Use concrete and pictorial models to explore the basic concept of a function.NJ 4.3.3 B.1.a - By the end of Grade 3, students will: Use concrete and pictorial models to explore the basic concept of a function. Input/output tables, T-charts

NJ 4.3.3 C - ModelingNJ 4.3.3 C.1 - By the end of Grade 3, students will: Recognize and describe change in quantities.NJ 4.3.3 C.1.a - By the end of Grade 3, students will: Recognize and describe change in quantities. Graphs representing change over time (e.g., temperature, height)NJ 4.3.3 C.2 - By the end of Grade 3, students will: Construct and solve simple open sentences involving addition or subtraction (e.g., 3 + 6 = __, n = 15 - 3, 3 +__= 3, 16 - c = 7).


NJ 4.3.3 D - ProceduresNJ 4.3.3 D.1 - By the end of Grade 3, students will: Understand and apply the properties of operations and numbers.NJ 4.3.3 D.1.a - By the end of Grade 3, students will: Understand and apply the properties of operations and numbers. Commutative (e.g., 3 x 7 = 7 x 3)NJ 4.3.3 D.1.b - By the end of Grade 3, students will: Understand and apply the properties of operations and numbers. Identity element for multiplication is 1 (e.g., 1 x 8 = 8)

NJ 4.3.3 D.1.c - By the end of Grade 3, students will: Understand and apply the properties of operations and numbers. Any number multiplied by zero is zero

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NJ 4.3.3 D.2 - By the end of Grade 3, students will: Understand and use the concepts of equals, less than, and greater than to describe relations between numbers.NJ 4.3.3 D.2.a - By the end of Grade 3, students will: Understand and use the concepts of equals, less than, and greater than to describe relations between numbers. Symbols (=, <, >)

NJ 4.4 - 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.NJ 4.4.3 A - Data AnalysisNJ 4.4.3 A.1 - By the end of Grade 3, students will: Collect, generate, organize, and display data in response to questions, claims, or curiosity.NJ 4.4.3 A.1.a - By the end of Grade 3, students will: Collect, generate, organize, and display data in response to questions, claims, or curiosity. Data collected from the classroom environmentNJ 4.4.3 A.2 - By the end of Grade 3, students will: Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data.NJ 4.4.3 A.2.a - By the end of Grade 3, students will: Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data. Pictograph, bar graph, table


Skill 2 - S00 - Read simple pictographs

Skill 3 - S01 - Read tableSkill 4 - S02 - Read bar graphSkill 6 - S04 - Interpret tableSkill 7 - S05 - Interpret bar graph

NJ 4.4.3 B - Probability

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NJ 4.4.3 B.1 - By the end of Grade 3, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.

NJ 4.4.3 B.1.a - By the end of Grade 3, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability. Likely, unlikely, certain, impossible

NJ 4.4.3 B.1.b - By the end of Grade 3, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability. More likely, less likely, equally likely

NJ 4.4.3 B.2 - By the end of Grade 3, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).NJ 4.4.3 B.2.a - By the end of Grade 3, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color). What students think will happen (intuitive)

NJ 4.4.3 B.2.b - By the end of Grade 3, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color). Collect data and use that data to predict the probability (experimental)

NJ 4.4.3 C - Discrete Mathematics-Systematic Listing and Counting

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NJ 4.4.3 C.1 - By the end of Grade 3, students will: Represent and classify data according to attributes, such as shape or color, and relationships.

NJ 4.4.3 C.1.a - By the end of Grade 3, students will: Represent and classify data according to attributes, such as shape or color, and relationships. Venn diagramsNJ 4.4.3 C.1.b - By the end of Grade 3, students will: Represent and classify data according to attributes, such as shape or color, and relationships. Numerical and alphabetical orderNJ 4.4.3 C.2 - By the end of Grade 3, students will: Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation.NJ 4.4.3 C.2.a - By the end of Grade 3, students will: Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation. Organized lists, chartsNJ 4.4.3 D - Discrete Mathematics-Vertex-Edge Graphs and Algorithms

NJ 4.4.3 D.1 - By the end of Grade 3, students will: Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).

NJ 4.4.3 D.2 - By the end of Grade 3, students will: Explore vertex-edge graphsNJ 4.4.3 D.2.a - By the end of Grade 3, students will: Explore vertex-edge graphs Vertex, edgeNJ 4.4.3 D.2.b - By the end of Grade 3, students will: Explore vertex-edge graphs PathNJ 4.4.3 D.3 - By the end of Grade 3, students will: Find the smallest number of colors needed to color a map.

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NJ 4.5 - All students will use mathematical processes of problem solving, communications, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.NJ 4.5 A - Problem SolvingNJ 4.5 A.1 - By the end of Grade 3, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5 A.2 - By the end of Grade 3, students will: Solve problems that arise in mathematics and in other contexts.NJ 4.5 A.2.a - By the end of Grade 3, students will: Solve problems that arise in mathematics and in other contexts. Open-ended problems

NJ 4.5 A.2.b - By the end of Grade 3, students will: Solve problems that arise in mathematics and in other contexts. Non-routine problems

NJ 4.5 A.2.c - By the end of Grade 3, students will: Solve problems that arise in mathematics and in other contexts. Problems with multiple solutionsNJ 4.5 A.2.d - By the end of Grade 3, students will: Solve problems that arise in mathematics and in other contexts. Problems that can be solved in several waysNJ 4.5 A.3 - By the end of Grade 3, students will: Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.NJ 4.5 A.4 - By the end of Grade 3, students will: Pose problems of various types and levels of difficulty.

NJ 4.5 A.5 - By the end of Grade 3, students will: Monitor their progress and reflect on the process of their problem solving activity.

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NJ 4.5 A.6 - By the end of Grade 3, students will: Distinguish relevant from irrelevant information, and identify missing information.NJ 4.5 B - CommunicationNJ 4.5 B.1 - By the end of Grade 3, students will: Use communication to organize and clarify their mathematical thinking.NJ 4.5 B.1.a - By the end of Grade 3, students will: Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5 B.1.b - By the end of Grade 3, students will: Use communication to organize and clarify their mathematical thinking. Discussion, listening, and questioning

NJ 4.5 B.2 - By the end of Grade 3, students will: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.

NJ 4.5 B.3 - By the end of Grade 3, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5 B.4 - By the end of Grade 3, students will: Use the language of mathematics to express mathematical ideas precisely.NJ 4.5 C - ConnectionsNJ 4.5 C.1 - By the end of Grade 3, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

NJ 4.5 C.2 - By the end of Grade 3, students will: 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).

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NJ 4.5 C.3 - By the end of Grade 3, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.NJ 4.5 C.4 - By the end of Grade 3, students will: Apply mathematics in practical situations and in other disciplines.NJ 4.5 C.5 - By the end of Grade 3, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5 C.6 - By the end of Grade 3, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5 D - ReasoningNJ 4.5 D.1 - By the end of Grade 3, students will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5 D.2 - By the end of Grade 3, students will: Use reasoning to support their mathematical conclusions and problem solutions.

NJ 4.5 D.3 - By the end of Grade 3, students will: Select and use various types of reasoning and methods of proof.NJ 4.5 D.4 - By the end of Grade 3, students will: Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.NJ 4.5 D.5 - By the end of Grade 3, students will: Make and investigate mathematical conjectures.

NJ 4.5 D.5.a - By the end of Grade 3, students will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjectures

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NJ 4.5 D.5.b - By the end of Grade 3, students will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5 D.6 - By the end of Grade 3, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.

NJ 4.5 E - RepresentationsNJ 4.5 E.1 - By the end of Grade 3, students will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5 E.1.a - By the end of Grade 3, students will: Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles)

NJ 4.5 E.1.b - By the end of Grade 3, students will: Create and use representations to organize, record, and communicate mathematical ideas. Pictorial representations (e.g., diagrams, charts, or tables)

NJ 4.5 E.1.c - By the end of Grade 3, students will: Create and use representations to organize, record, and communicate mathematical ideas. Symbolic representations (e.g., a formula)NJ 4.5 E.1.d - By the end of Grade 3, students will: Create and use representations to organize, record, and communicate mathematical ideas. Graphical representations (e.g., a line graph)NJ 4.5 E.2 - By the end of Grade 3, students will: Select, apply, and translate among mathematical representations to solve problems.NJ 4.5 E.3 - By the end of Grade 3, students will: Use representations to model and interpret physical, social, and mathematical phenomena.

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NJ 4.5 F - TechnologyNJ 4.5 F.1 - By the end of Grade 3, students will: Use technology to gather, analyze, and communicate mathematical information.NJ 4.5 F.2 - By the end of Grade 3, students will: Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.NJ 4.5 F.3 - By the end of Grade 3, students will: Use graphing calculators and computer software to investigate properties of functions and their graphs.NJ 4.5 F.4 - By the end of Grade 3, students will: Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

NJ 4.5 F.5 - By the end of Grade 3, students will: Use computer software to make and verify conjectures about geometric objects.

NJ 4.5 F.6 - By the end of Grade 3, students will: Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).

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Standard Skill Area Description Skill DescriptionNJ 4.1 - All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.NJ 4.1.4 A - Number SenseNJ 4.1.4 A.1 - By the end of grade 4, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well).

NJ 4.1.4 A.1.a - By the end of grade 4, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well). Whole numbers through millions

NJ 4.1.4 A.1.b - By the end of grade 4, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well). Commonly used fractions (denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 16) as part of a whole, as a subset of a set, and as a location on a number line




NJ 4.1.4 A.1.c - By the end of grade 4, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 4 pertain to these sets of numbers as well). Decimals through hundredths

NJ 4.1.4 A.2 - By the end of grade 4, students will: Demonstrate an understanding of place value concepts.




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Skill 17 - N13 - Thousands: Identify place value of digitsSkill 21 - N18 - Ten thousands, hundred thousands, millions, billions: Identify place value of digits

Skill 27 - N25 - Fractions and decimals: Identify place value of digits in decimals

NJ 4.1.4 A.3 - By the end of grade 4, students will: Demonstrate a sense of the relative magnitudes of numbers.




NJ 4.1.4 A.4 - By the end of grade 4, students will: Understand the various uses of numbers.NJ 4.1.4 A.4.a - By the end of grade 4, students will: Understand the various uses of numbers. Counting, measuring, labeling (e.g., numbers on baseball uniforms), locating (e.g., Room 235 is on the second floor)

NJ 4.1.4 A.5 - By the end of grade 4, students will: Use concrete and pictorial models to relate whole numbers, commonly used fractions, and decimals to each other, and to represent equivalent forms of the same number.NJ 4.1.4 A.6 - By the end of grade 4, students will: Compare and order numbers.




NJ 4.1.4 A.7 - By the end of grade 4, students will: Explore settings that give rise to negative numbers.

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NJ 4.1.4 A.7.a - By the end of grade 4, students will: Explore settings that give rise to negative numbers. Temperatures below 0°, debts

NJ 4.1.4 A.7.b - By the end of grade 4, students will: Explore settings that give rise to negative numbers. Extension of the number line

NJ 4.1.4 B - Numerical OperationsNJ 4.1.4 B.1 - By the end of grade 4, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems.

NJ 4.1.4 B.1.a - By the end of grade 4, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. Addition and subtraction: joining, separating, comparing

NJ 4.1.4 B.1.b - By the end of grade 4, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. Multiplication: repeated addition, area/arrayNJ 4.1.4 B.1.c - By the end of grade 4, students will: Develop the meanings of the four basic arithmetic operations by modeling and discussing a large variety of problems. Division: repeated subtraction, sharingNJ 4.1.4 B.2 - By the end of grade 4, students will: Develop proficiency with basic multiplication and division number facts using a variety of fact strategies (such as "skip counting" and "repeated subtraction") and then commit them to memory.


Skill 12 - C12 - Multiplication basic facts

Skill 13 - C13 - Division basic facts

Skill Area 9 - Word Problems Skill 6 - W12 - Word problems: Multiplication of basic facts

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Skill 7 - W13 - Word problems: Division of basic facts

NJ 4.1.4 B.3 - By the end of grade 4, students will: Construct, use, and explain procedures for performing whole number calculations and with:

NJ 4.1.4 B.3.a - By the end of grade 4, students will: Construct, use, and explain procedures for performing whole number calculations and with: Pencil-and-paper


Skill 9 - C09 - Subtraction beyond basic facts with regrouping (2d-1d, 2d-2d)

Skill 10 - C10 - Addition beyond basic facts with double regrouping (3d+2d, 3d+3d)Skill 11 - C11 - Subtraction beyond basic facts with double regrouping (3d-2d, 3d-3d)Skill 14 - C14 - Multiplication beyond basic facts, no regrouping (2dx1d)

Skill 15 - C15 - Division beyond basic facts, no remainders (2d/1d)Skill 16 - C16 - Multiplication with regrouping (2dx1d, 2dx2d)Skill 17 - C17 - Division with remainders (2d/1d, 3d/1d)

Skill Area 9 - Word Problems Skill 4 - W08 - Word problems: Addition beyond basic facts with regrouping (2d+1d, 2d+2d)Skill 5 - W09 - Word problems: Subtraction beyond basic facts with regrouping (2d-1d, 2d-2d)Skill 8 - W14 - Word problems: Multiplication beyond basic facts, no regrouping (2dx1d)

NJ 4.1.4 B.3.b - By the end of grade 4, students will: Construct, use, and explain procedures for performing whole number calculations and with: Mental math



Skill 15 - C15 - Division beyond basic facts, no remainders (2d/1d)

Skill Area 9 - Word Problems Skill 11 - W17 - Word problems: Division with remainders (2d/1d, 3d/1d)

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NJ 4.1.4 B.3.c - By the end of grade 4, students will: Construct, use, and explain procedures for performing whole number calculations and with: CalculatorNJ 4.1.4 B.4 - By the end of grade 4, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers.NJ 4.1.4 B.4.a - By the end of grade 4, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Addition of 3-digit numbers


Skill 10 - C10 - Addition beyond basic facts with double regrouping (3d+2d, 3d+3d)

NJ 4.1.4 B.4.b - By the end of grade 4, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Subtraction of 3-digit numbers


Skill 11 - C11 - Subtraction beyond basic facts with double regrouping (3d-2d, 3d-3d)

Skill Area 9 - Word Problems Skill 5 - W09 - Word problems: Subtraction beyond basic facts with regrouping (2d-1d, 2d-2d)

NJ 4.1.4 B.4.c - By the end of grade 4, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Multiplication of 2-digit numbers



Skill 16 - C16 - Multiplication with regrouping (2dx1d, 2dx2d)

Skill Area 9 - Word Problems Skill 8 - W14 - Word problems: Multiplication beyond basic facts, no regrouping (2dx1d)Skill 10 - W16 - Word problems: Multiplication with regrouping (2dx1d, 2dx2d)

NJ 4.1.4 B.4.d - By the end of grade 4, students will: Use efficient and accurate pencil-and-paper procedures for computation with whole numbers. Division of 3-digit numbers by 1-digit numbers


Skill 15 - C15 - Division beyond basic facts, no remainders (2d/1d)

Skill 17 - C17 - Division with remainders (2d/1d, 3d/1d)

Skill Area 9 - Word Problems Skill 9 - W15 - Word problems: Division beyond basic facts, no remainders (2d/1d)

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Skill 11 - W17 - Word problems: Division with remainders (2d/1d, 3d/1d)

NJ 4.1.4 B.5 - By the end of grade 4, students will: Construct and use procedures for performing decimal addition and subtraction.


Skill 31 - C33 - Addition of decimals, place change (e.g. 2 + .45)

Skill 32 - C35 - Subtraction of decimals, place change (e.g. 5-.4)

Skill Area 9 - Word Problems Skill 23 - W33 - Word problems: Addition of decimals, place change (e.g. 2+.45)

NJ 4.1.4 B.6 - By the end of grade 4, students will: Count and perform simple computations with money.NJ 4.1.4 B.6.a - By the end of grade 4, students will: Count and perform simple computations with money. Standard dollars and cents notation

NJ 4.1.4 B.7 - By the end of grade 4, students will: Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.NJ 4.1.4 B.8 - By the end of grade 4, students will: Check the reasonableness of results of computations.NJ 4.1.4 B.9 - By the end of grade 4, students will: Use concrete models to explore addition and subtraction with fractions.NJ 4.1.4 B.10 - By the end of grade 4, students will: Understand and use the inverse relationships between addition and subtraction and between multiplication and division.

NJ 4.1.4 C - EstimationNJ 4.1.4 C.1 - By the end of grade 4, students will: Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

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NJ 4.1.4 C.2 - By the end of grade 4, students will: Construct and use a variety of estimation strategies (e.g., rounding and mental math) for estimating both quantities and the results of computations.

Skill Area 5 - Estimation Skill 1 - E06 - Estimation problems: Addition beyond basic facts, no regrouping (2d+1d)

Skill 2 - E07 - Estimation problems: Subtraction beyond basic facts, no regrouping (2d-1d)Skill 3 - E14 - Estimation problems: Multiplication beyond basic facts, no regrouping (2dx1d)Skill 4 - E15 - Estimation problems: Division beyond basic facts, no remainders (2d/1d)Skill 5 - E18 - Estimation problems: Addition of whole numbers, any difficultySkill 6 - E19 - Estimation problems: Subtraction of whole numbers, any difficultySkill 7 - E20 - Estimation problems: Multiplication of whole numbers, any difficultySkill 8 - E21 - Estimation problems: Division of whole numbers, any difficultySkill 13 - E32 - Estimation problems: Addition of decimals, no place change (e.g. 2.34+10.32)Skill 14 - E33 - Estimation problems: Addition of decimals, place change (e.g. 2+.45)Skill 15 - E34 - Estimation problems: Subtraction of decimals, no place change (e.g. .53 - .42)

NJ 4.1.4 C.3 - By the end of grade 4, students will: Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.NJ 4.1.4 C.4 - By the end of grade 4, students will: Use estimation to determine whether the result of a computation (either by calculator or by hand) is reasonable.

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NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.4 A - Geometric PropertiesNJ 4.2.4 A.1 - By the end of grade 4, students will: Identify and describe spatial relationships of two or more objects in space.NJ 4.2.4 A.1.a - By the end of grade 4, students will: Identify and describe spatial relationships of two or more objects in space. Direction, orientation, and perspectives (e.g., which object is on your left when you are standing here?)

NJ 4.2.4 A.1.b - By the end of grade 4, students will: Identify and describe spatial relationships of two or more objects in space. Relative shapes and sizesNJ 4.2.4 A.1.c - By the end of grade 4, students will: Identify and describe spatial relationships of two or more objects in space. Shadows (projections) of everyday objects

NJ 4.2.4 A.2 - By the end of grade 4, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them.NJ 4.2.4 A.2.a - By the end of grade 4, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. Vertex, edge, face, side, angle

NJ 4.2.4 A.2.b - By the end of grade 4, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. 3D figures - cube, rectangular prism, sphere, cone, cylinder, and pyramid

Skill Area 6 - Geometry Skill 7 - GA7 - Identify common solid shapes

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NJ 4.2.4 A.2.c - By the end of grade 4, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. 2D figures - square, rectangle, circle, triangle, quadrilateral, pentagon, hexagon, octagon

NJ 4.2.4 A.2.d - By the end of grade 4, students will: Use properties of standard three-dimensional and two-dimensional shapes to identify, classify, and describe them. Inclusive relationships - squares are rectangles, cubes are rectangular prisms

NJ 4.2.4 A.3 - By the end of grade 4, students will: Identify and describe relationships among two-dimensional shapes.NJ 4.2.4 A.3.a - By the end of grade 4, students will: Identify and describe relationships among two-dimensional shapes. CongruenceNJ 4.2.4 A.3.b - By the end of grade 4, students will: Identify and describe relationships among two-dimensional shapes. Lines of symmetry


NJ 4.2.4 A.4 - By the end of grade 4, students will: Understand and apply concepts involving lines, angles, and circles.NJ 4.2.4 A.4.a - By the end of grade 4, students will: Understand and apply concepts involving lines, angles, and circles. Point, line, line segment, endpoint

Skill Area 6 - Geometry Skill 20 - G13 - Identify line segments

NJ 4.2.4 A.4.b - By the end of grade 4, students will: Understand and apply concepts involving lines, angles, and circles. Parallel, perpendicular

Skill Area 6 - Geometry Skill 21 - G14 - Identify parallel lines

Skill 23 - G16 - Identify perpendicular lines

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NJ 4.2.4 A.4.c - By the end of grade 4, students will: Understand and apply concepts involving lines, angles, and circles. Angles - acute, right, obtuse

Skill Area 6 - Geometry Skill 28 - G21 - Classify angles (obtuse, etc.)

NJ 4.2.4 A.4.d - By the end of grade 4, students will: Understand and apply concepts involving lines, angles, and circles. Circles - diameter, radius, center

Skill Area 6 - Geometry Skill 10 - G02 - Circle terms

NJ 4.2.4 A.5 - By the end of grade 4, students will: Recognize, describe, extend, and create space-filling patterns.NJ 4.2.4 B - Transforming ShapesNJ 4.2.4 B.1 - By the end of grade 4, students will: Use simple shapes to cover an area (tessellations).

NJ 4.2.4 B.2 - By the end of grade 4, students will: Describe and use geometric transformations (slide, flip, turn).NJ 4.2.4 B.3 - By the end of grade 4, students will: Investigate the occurrence of geometry in nature and art.NJ 4.2.4 C - Coordinate GeometryNJ 4.2.4 C.1 - By the end of grade 4, students will: Locate and name points in the first quadrant on a coordinate grid.NJ 4.2.4 C.2 - By the end of grade 4, students will: Use coordinates to give or follow directions from one point to another on a map or grid.NJ 4.2.4 D - Units of MeasurementNJ 4.2.4 D.1 - By the end of grade 4, students will: Understand that everyday objects have a variety of attributes, each of which can be measured in many ways.NJ 4.2.4 D.2 - By the end of grade 4, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems.

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NJ 4.2.4 D.2.a - By the end of grade 4, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Length - fractions of an inch (1/8, 1/4, 1/2), mile, decimeter, kilometerNJ 4.2.4 D.2.b - By the end of grade 4, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Area - square inch, square centimeterNJ 4.2.4 D.2.c - By the end of grade 4, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Volume - cubic inch, cubic centimeterNJ 4.2.4 D.2.d - By the end of grade 4, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Weight - ounceNJ 4.2.4 D.2.e - By the end of grade 4, students will: Select and use appropriate standard units of measure and measurement tools to solve real-life problems. Capacity - fluid ounce, cup, gallon, milliliter

NJ 4.2.4 D.3 - By the end of grade 4, students will: Develop and use personal referents to approximate standard units of measure (e.g., a common paper clip is about an inch long).NJ 4.2.4 D.4 - By the end of grade 4, students will: Incorporate estimation in measurement activities (e.g., estimate before measuring).

Skill Area 7 - Measurement Skill 15 - M08 - Estimating linear measure in metric units.

NJ 4.2.4 D.5 - By the end of grade 4, students will: Solve problems involving elapsed time.NJ 4.2.4 E - Measuring Geometric Objects

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NJ 4.2.4 E.1 - By the end of grade 4, students will: Determine the area of simple two-dimensional shapes on a square grid.NJ 4.2.4 E.2 - By the end of grade 4, students will: Distinguish between perimeter and area and use each appropriately in problem-solving situations.NJ 4.2.4 E.3 - By the end of grade 4, students will: Measure and compare the volume of three-dimensional objects using materials such as rice or cubes.NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.4 A - PatternsNJ 4.3.4 A.1 - By the end of grade 4, students will: Recognize, describe, extend, and create patterns.

NJ 4.3.4 A.1.a - By the end of grade 4, students will: Recognize, describe, extend, and create patterns. Descriptions using words, number sentences/expressions, graphs, tables, variables (e.g., shape, blank, or letter)

NJ 4.3.4 A.1.b - By the end of grade 4, students will: Recognize, describe, extend, and create patterns. Sequences that stop or that continue infinitelyNJ 4.3.4 A.1.c - By the end of grade 4, students will: Recognize, describe, extend, and create patterns. Whole number patterns that grow or shrink as a result of repeatedly adding, subtracting, multiplying by, or dividing by a fixed number (e.g., 5, 8, 11,... or 800, 400, 200,...)



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NJ 4.3.4 A.1.d - By the end of grade 4, students will: Recognize, describe, extend, and create patterns. Sequences can often be extended in more than one way (e.g., the next term after 1, 2, 4,... could be 8, or 7, or ...)NJ 4.3.4 B - Functions and RelationshipsNJ 4.3.4 B.1 - By the end of grade 4, students will: Use concrete and pictorial models to explore the basic concept of a function.NJ 4.3.4 B.1.a - By the end of grade 4, students will: Use concrete and pictorial models to explore the basic concept of a function. Input/output tables, T-charts

NJ 4.3.4 B.1.b - By the end of grade 4, students will: Use concrete and pictorial models to explore the basic concept of a function. Combining two function machines

NJ 4.3.4 B.1.c - By the end of grade 4, students will: Use concrete and pictorial models to explore the basic concept of a function. Reversing a function machine

NJ 4.3.4 C - ModelingNJ 4.3.4 C.1 - By the end of grade 4, students will: Recognize and describe change in quantities.NJ 4.3.4 C.1.a - By the end of grade 4, students will: Recognize and describe change in quantities. Graphs representing change over time (e.g., temperature, height)NJ 4.3.4 C.1.b - By the end of grade 4, students will: Recognize and describe change in quantities. How change in one physical quantity can produce a corresponding change in another (e.g., pitch of a sound depends on the rate of vibration)

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NJ 4.3.4 C.2 - By the end of grade 4, students will: Construct and solve simple open sentences involving any one operation (e.g., 3 x 6 = __, n = 15 ÷ 3, 3 x __ = 0, 16 - c = 7).


NJ 4.3.4 D - ProceduresNJ 4.3.4 D.1 - By the end of grade 4, students will: Understand, name, and apply the properties of operations and numbers.NJ 4.3.4 D.1.a - By the end of grade 4, students will: Understand, name, and apply the properties of operations and numbers. Commutative (e.g., 3 x 7 = 7 x 3)NJ 4.3.4 D.1.b - By the end of grade 4, students will: Understand, name, and apply the properties of operations and numbers. Identity element for multiplication is 1 (e.g., 1 x 8 = 8)NJ 4.3.4 D.1.c - By the end of grade 4, students will: Understand, name, and apply the properties of operations and numbers. Associative (e.g., 2 x 4 x 25 can be found by first multiplying either 2 x 4 or 4 x 25)

NJ 4.3.4 D.1.d - By the end of grade 4, students will: Understand, name, and apply the properties of operations and numbers. Division by zero is undefinedNJ 4.3.4 D.1.e - By the end of grade 4, students will: Understand, name, and apply the properties of operations and numbers. Any number multiplied by zero is zero.NJ 4.3.4 D.2 - By the end of grade 4, students will: Understand and use the concepts of equals, less than, and greater than in simple number sentences.NJ 4.3.4 D.2.a - Symbols (=, <, >)

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NJ 4.4 - 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.NJ 4.4.4 A - Data AnalysisNJ 4.4.4 A.1 - By the end of grade 4, students will: Collect, generate, organize, and display data in response to questions, claims, or curiosity.NJ 4.4.4 A.1.a - By the end of grade 4, students will: Collect, generate, organize, and display data in response to questions, claims, or curiosity. Data collected from the school environmentNJ 4.4.4 A.2 - By the end of grade 4, students will: Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data.NJ 4.4.4 A.2.a - By the end of grade 4, students will: Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data. Pictograph, bar graph, line plot, line graph, table


Skill 3 - S01 - Read table

Skill 4 - S02 - Read bar graphSkill 6 - S04 - Interpret tableSkill 7 - S05 - Interpret bar graph

NJ 4.4.4 A.2.b - By the end of grade 4, students will: Read, interpret, construct, analyze, generate questions about, and draw inferences from displays of data. Average (mean), most frequent (mode), middle term (median)


Skill 9 - S07 - Statistics: Mean

Skill 10 - S08 - Statistics: MedianNJ 4.4.4 B - ProbabilityNJ 4.4.4 B.1 - By the end of grade 4, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability.

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NJ 4.4.4 B.1.a - By the end of grade 4, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability. Likely, unlikely, certain, impossible, improbable, fair, unfair

NJ 4.4.4 B.1.b - By the end of grade 4, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability. More likely, less likely, equally likely

NJ 4.4.4 B.1.c - By the end of grade 4, students will: Use everyday events and chance devices, such as dice, coins, and unevenly divided spinners, to explore concepts of probability. Probability of tossing "heads" does not depend on outcomes of previous tosses

NJ 4.4.4 B.2 - By the end of grade 4, students will: Determine probabilities of simple events based on equally likely outcomes and express them as fractions.


Skill 11 - S11 - Probability: Simple

NJ 4.4.4 B.3 - By the end of grade 4, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color).NJ 4.4.4 B.3.a - By the end of grade 4, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color). What students think will happen (intuitive)

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NJ 4.4.4 B.3.b - By the end of grade 4, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color). Collect data and use that data to predict the probability (experimental)

NJ 4.4.4 B.3.c - By the end of grade 4, students will: Predict probabilities in a variety of situations (e.g., given the number of items of each color in a bag, what is the probability that an item picked will have a particular color). Analyze all possible outcomes to find the probability (theoretical)

NJ 4.4.4 C - Discrete Mathematics-Systematic Listing and CountingNJ 4.4.4 C.1 - By the end of grade 4, students will: Represent and classify data according to attributes, such as shape or color, and relationships.

NJ 4.4.4 C.1.a - By the end of grade 4, students will: Represent and classify data according to attributes, such as shape or color, and relationships. Venn diagramsNJ 4.4.4 C.1.b - By the end of grade 4, students will: Represent and classify data according to attributes, such as shape or color, and relationships. Numerical and alphabetical orderNJ 4.4.4 C.2 - By the end of grade 4, students will: Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation.NJ 4.4.4 C.2.a - By the end of grade 4, students will: Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation. Organized lists, charts, tree diagrams

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NJ 4.4.4 C.2.b - By the end of grade 4, students will: Represent all possibilities for a simple counting situation in an organized way and draw conclusions from this representation. Dividing into categories (e.g., to find the total number of rectangles in a grid, find the number of rectangles of each size and add the results)NJ 4.4.4 D - Discrete Mathematics-Vertex-Edge Graphs and Algorithms

NJ 4.4.4 D.1 - By the end of grade 4, students will: Follow, devise, and describe practical sets of directions (e.g., to add two 2-digit numbers).

NJ 4.4.4 D.2 - By the end of grade 4, students will: Play two-person games and devise strategies for winning the games (e.g., "make 5" where players alternately add 1 or 2 and the person who reaches 5, or another designated number, is the winner).

NJ 4.4.4 D.3 - By the end of grade 4, students will: Explore vertex-edge graphs and tree diagrams.NJ 4.4.4 D.3.a - By the end of grade 4, students will: Explore vertex-edge graphs and tree diagrams. Vertex, edge, neighboring/adjacent, number of neighbors

NJ 4.4.4 D.3.b - By the end of grade 4, students will: Explore vertex-edge graphs and tree diagrams. Path, circuit (i.e., path that ends at its starting point)NJ 4.4.4 D.4 - By the end of grade 4, students will: Find the smallest number of colors needed to color a map or a graph.NJ 4.5 - All students will use mathematical processes of problem solving, communications, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.

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NJ 4.5 A - Problem SolvingNJ 4.5 A.1 - By the end of grade 4, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5 A.2 - By the end of grade 4, students will: Solve problems that arise in mathematics and in other contexts.NJ 4.5 A.2.a - By the end of grade 4, students will: Solve problems that arise in mathematics and in other contexts. Open-ended problems

NJ 4.5 A.2.b - By the end of grade 4, students will: Solve problems that arise in mathematics and in other contexts. Non-routine problems

NJ 4.5 A.2.c - By the end of grade 4, students will: Solve problems that arise in mathematics and in other contexts. Problems with multiple solutionsNJ 4.5 A.2.d - By the end of grade 4, students will: Solve problems that arise in mathematics and in other contexts. Problems that can be solved in several waysNJ 4.5 A.3 - By the end of grade 4, students will: Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.NJ 4.5 A.4 - By the end of grade 4, students will: Pose problems of various types and levels of difficulty.

NJ 4.5 A.5 - By the end of grade 4, students will: Monitor their progress and reflect on the process of their problem solving activity.

NJ 4.5 A.6 - By the end of grade 4, students will: Distinguish relevant from irrelevant information, and identify missing information.NJ 4.5 B - Communication

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NJ 4.5 B.1 - By the end of grade 4, students will: Use communication to organize and clarify their mathematical thinking.NJ 4.5 B.1.a - By the end of grade 4, students will: Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5 B.1.b - By the end of grade 4, students will: Use communication to organize and clarify their mathematical thinking. Discussion, listening, and questioning

NJ 4.5 B.2 - By the end of grade 4, students will: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.NJ 4.5 B.3 - By the end of grade 4, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5 B.4 - By the end of grade 4, students will: Use the language of mathematics to express mathematical ideas precisely.NJ 4.5 C - ConnectionsNJ 4.5 C.1 - By the end of grade 4, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).NJ 4.5 C.2 - By the end of grade 4, students will: 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).NJ 4.5 C.3 - By the end of grade 4, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.NJ 4.5 C.4 - By the end of grade 4, students will: Apply mathematics in practical situations and in other disciplines.

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NJ 4.5 C.5 - By the end of grade 4, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5 C.6 - By the end of grade 4, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5 D - ReasoningNJ 4.5 D.1 - By the end of grade 4, students will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5 D.2 - By the end of grade 4, students will: Use reasoning to support their mathematical conclusions and problem solutions.

NJ 4.5 D.3 - By the end of grade 4, students will: Select and use various types of reasoning and methods of proof.NJ 4.5 D.4 - By the end of grade 4, students will: Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.NJ 4.5 D.5 - By the end of grade 4, students will: Make and investigate mathematical conjectures.NJ 4.5 D.5.a - By the end of grade 4, students will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjecturesNJ 4.5 D.5.b - By the end of grade 4, students will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5 D.6 - By the end of grade 4, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.NJ 4.5 E - Representations

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NJ 4.5 E.1 - By the end of grade 4, students will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5 E.1.a - By the end of grade 4, students will: Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles)

NJ 4.5 E.1.b - By the end of grade 4, students will: Create and use representations to organize, record, and communicate mathematical ideas. Pictorial representations (e.g., diagrams, charts, or tables)

NJ 4.5 E.1.c - By the end of grade 4, students will: Create and use representations to organize, record, and communicate mathematical ideas. Symbolic representations (e.g., a formula)NJ 4.5 E.1.d - By the end of grade 4, students will: Create and use representations to organize, record, and communicate mathematical ideas. Graphical representations (e.g., a line graph)NJ 4.5 E.2 - By the end of grade 4, students will: Select, apply, and translate among mathematical representations to solve problems.NJ 4.5 E.3 - By the end of grade 4, students will: Use representations to model and interpret physical, social, and mathematical phenomena.

NJ 4.5 F - TechnologyNJ 4.5 F.1 - By the end of grade 4, students will: Use technology to gather, analyze, and communicate mathematical information.NJ 4.5 F.2 - By the end of grade 4, students will: Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.

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NJ 4.5 F.3 - By the end of grade 4, students will: Use graphing calculators and computer software to investigate properties of functions and their graphs.NJ 4.5 F.4 - By the end of grade 4, students will: Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

NJ 4.5 F.5 - By the end of grade 4, students will: Use computer software to make and verify conjectures about geometric objects.

NJ 4.5 F.6 - By the end of grade 4, students will: Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).

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Standard Skill Area Description Skill DescriptionNJ 4.1 - All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.NJ 4.1.5 A - Number SenseNJ 4.1.5 A.1 - By the end of grade 5, the student will: 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).

NJ 4.1.5 A.1.a - By the end of grade 5, the student will: 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



NJ 4.1.5 A.1.b - By the end of grade 5, the student will: 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 decimals

NJ 4.1.5 A.2 - By the end of grade 5, the student will: Recognize the decimal nature of United States currency and compute with money.NJ 4.1.5 A.3 - By the end of grade 5, the student will: Demonstrate a sense of the relative magnitudes of numbers.





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NJ 4.1.5 A.4 - By the end of grade 5, the student will: Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.


Skill 23 - N21 - Fractions and decimals: Convert fraction to equivalent fraction

Skill 24 - N22 - Fractions and decimals: Convert fraction to decimal

Skill 25 - N23 - Fractions and decimals: Convert decimal to fraction

NJ 4.1.5 A.5 - By the end of grade 5, the student will: Develop and apply number theory concepts in problem solving situations.NJ 4.1.5 A.5.a - By the end of grade 5, the student will: Develop and apply number theory concepts in problem solving situations. Primes, factors, multiples


Skill 40 - N38 - Advanced concepts: Knows meaning of primes and composites

NJ 4.1.5 A.6 - By the end of grade 5, the student will: Compare and order numbers.




NJ 4.1.5 B - Numerical OperationsNJ 4.1.5 B.1 - By the end of grade 5, the student will: Recognize the appropriate use of each arithmetic operation in problem situations.NJ 4.1.5 B.2 - By the end of grade 5, the student will: Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with:NJ 4.1.5 B.2.a - By the end of grade 5, the student will: Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with: Pencil-and-paper


Skill 21 - C22 - Addition of fractions: like single digit denominators

Skill 22 - C23 - Subtraction of fractions: like single digit denominatorsSkill 23 - C24 - Addition of fractions: unlike single digit denominators

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Skill 24 - C25 - Subtraction of fractions: unlike single digit denominatorsSkill 27 - C28 - Addition of mixed numbersSkill 28 - C29 - Subtraction of mixed numbersSkill 31 - C33 - Addition of decimals, place change (e.g. 2 + .45)

Skill 32 - C35 - Subtraction of decimals, place change (e.g. 5-.4)

Skill Area 9 - Word Problems Skill 16 - W22 - Word problems: Addition of fractions, like single digit denominatorsSkill 17 - W23 - Word problems: Subtraction of fractions, like single digit denominatorsSkill 18 - W24 - Word problems: Addition of fractions, unlike single digit denominatorsSkill 19 - W25 - Word problems: Subtraction of fractions, unlike single digit denominatorsSkill 20 - W28 - Word problems: Addition of mixed numbersSkill 21 - W29 - Word problems: Subtraction of mixed numbersSkill 23 - W33 - Word problems: Addition of decimals, place change (e.g. 2+.45)Skill 24 - W35 - Word problems: Subtraction of decimals, place change (e.g. 5-.4)

NJ 4.1.5 B.2.b - By the end of grade 5, the student will: Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with: Mental mathNJ 4.1.5 B.2.c - By the end of grade 5, the student will: Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with: CalculatorNJ 4.1.5 B.3 - By the end of grade 5, the student will: Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.

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NJ 4.1.5 B.4 - By the end of grade 5, the student will: Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.NJ 4.1.5 B.5 - By the end of grade 5, the student will: Check the reasonableness of results of computations.NJ 4.1.5 B.6 - By the end of grade 5, the student will: Understand and use the various relationships among operations and properties of operations.NJ 4.1.5 C - EstimationNJ 4.1.5 C.1 - By the end of grade 5, the student will: Use a variety of estimation strategies for both number and computation.

Skill Area 5 - Estimation Skill 8 - E21 - Estimation problems: Division of whole numbers, any difficulty

Skill 9 - E24 - Estimation problems: Addition of fractions, unlike single digit denominatorsSkill 10 - E25 - Estimation problems: Subtraction of fractions, unlike single digit denominatorsSkill 11 - E28 - Estimation problems: Addition of mixed numbers

Skill 12 - E29 - Estimation problems: Subtraction of mixed numbers

Skill 13 - E32 - Estimation problems: Addition of decimals, no place change (e.g. 2.34+10.32)Skill 14 - E33 - Estimation problems: Addition of decimals, place change (e.g. 2+.45)Skill 15 - E34 - Estimation problems: Subtraction of decimals, no place change (e.g. .53 - .42)Skill 16 - E35 – Estimation problems: Subtraction of decimals, place change (e.g. 5 - .4)

NJ 4.1.5 C.2 - By the end of grade 5, the student will: Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

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NJ 4.1.5 C.3 - By the end of grade 5, the student will: Determine the reasonableness of an answer by estimating the result of operations.NJ 4.1.5 C.4 - By the end of grade 5, the student will: Determine whether a given estimate is an overestimate or an underestimate.NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.5 A - Geometric PropertiesNJ 4.2.5 A.1 - By the end of grade 5, the student will: Understand and apply concepts involving lines and angles.NJ 4.2.5 A.1.a - By the end of grade 5, the student will: Understand and apply concepts involving lines and angles. Notation for line, ray, angle, line segment

Skill Area 6 - Geometry Skill 19 - G12 - Identify rays

Skill 20 - G13 - Identify line segments

NJ 4.2.5 A.1.b - By the end of grade 5, the student will: Understand and apply concepts involving lines and angles. Properties of parallel, perpendicular, and intersecting lines

Skill Area 6 - Geometry Skill 21 - G14 - Identify parallel lines

Skill 22 - G15 - Identify intersecting linesSkill 23 - G16 - Identify perpendicular linesSkill 24 - G17 - Use properties of parallel linesSkill 25 - G18 - Use properties of intersecting linesSkill 26 - G19 - Use properties of perpendicular lines

NJ 4.2.5 A.1.c - By the end of grade 5, the student will: Understand and apply concepts involving lines and angles. Sum of the measures of the interior angles of a triangle is 180°

Skill Area 6 - Geometry Skill 29 - G22 - Using parts of a triangle

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NJ 4.2.5 A.2 - By the end of grade 5, the student will: Identify, describe, compare, and classify polygons.

NJ 4.2.5 A.2.a - By the end of grade 5, the student will: Identify, describe, compare, and classify polygons. Triangles by angles and sides

NJ 4.2.5 A.2.b - By the end of grade 5, the student will: Identify, describe, compare, and classify polygons. Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombiNJ 4.2.5 A.2.c - By the end of grade 5, the student will: Identify, describe, compare, and classify polygons. Polygons by number of sides.

NJ 4.2.5 A.2.d - By the end of grade 5, the student will: Identify, describe, compare, and classify polygons. Equilateral, equiangular, regular

NJ 4.2.5 A.2.e - By the end of grade 5, the student will: Identify, describe, compare, and classify polygons. All points equidistant from a given point form a circleNJ 4.2.5 A.3 - By the end of grade 5, the student will: Identify similar figures.NJ 4.2.5 A.4 - By the end of grade 5, the student will: Understand and apply the concepts of congruence and symmetry (line and rotational).NJ 4.2.5 B - Transforming ShapesNJ 4.2.5 B.1 - By the end of grade 5, the student will: Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

NJ 4.2.5 B.2 - By the end of grade 5, the student will: Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.

NJ 4.2.5 C - Coordinate Geometry

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NJ 4.2.5 C.1 - By the end of grade 5, the student will: Create geometric shapes with specified properties in the first quadrant on a coordinate grid.NJ 4.2.5 D - Units of MeasurementNJ 4.2.5 D.1 - By the end of grade 5, the student will: Select and use appropriate units to measure angles and area.NJ 4.2.5 D.2 - By the end of grade 5, the student will: Convert measurement units within a system (e.g., 3 feet = ___ inches).

Skill Area 7 - Measurement Skill 9 - M01 - Customary measures: Inches, feet, yards

Skill 11 - M04 - Customary measures: Pints, quarts, gallonsSkill 12 - M05 - Metric prefixes

NJ 4.2.5 D.3 - By the end of grade 5, the student will: Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).NJ 4.2.5 D.4 - By the end of grade 5, the student will: Use measurements and estimates to describe and compare phenomena.NJ 4.2.5 E - Measuring Geometric ObjectsNJ 4.2.5 E.1 - By the end of grade 5, the student will: Use a protractor to measure angles.

Skill Area 7 - Measurement Skill 14 - M07 - Measures of angles

NJ 4.2.5 E.2 - By the end of grade 5, the student will: Develop and apply strategies and formulas for finding perimeter and area.NJ 4.2.5 E.2.a - By the end of grade 5, the student will: Develop and apply strategies and formulas for finding perimeter and area. Square

Skill Area 6 - Geometry Skill 11 - G03 - Perimeter: square

Skill 14 - G06 - Area: squareNJ 4.2.5 E.2.b - By the end of grade 5, the student will: Develop and apply strategies and formulas for finding perimeter and area. Rectangle

Skill Area 6 - Geometry Skill 12 - G04 - Perimeter: rectangle

Skill 15 - G07 - Area: rectangle

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NJ 4.2.5 E.3 - By the end of grade 5, the student will: Recognize that rectangles with the same perimeter do not necessarily have the same area and vice versa.NJ 4.2.5 E.4 - By the end of grade 5, the student will: 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).NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.5 A - PatternsNJ 4.3.5 A.1 - By the end of grade 5, the student will: Recognize, describe, extend, and create patterns involving whole numbers.NJ 4.3.5 A.1.a - By the end of grade 5, the student will: Recognize, describe, extend, and create patterns involving whole numbers. Descriptions using tables, verbal rules, simple equations, and graphs

NJ 4.3.5 B - Functions & RelationshipsNJ 4.3.5 B.1 - By the end of grade 5, the student will: Describe arithmetic operations as functions, including combining operations and reversing them.NJ 4.3.5 B.2 - By the end of grade 5, the student will: Graph points satisfying a function from T-charts, from verbal rules, and from simple equations.NJ 4.3.5 C - ModelingNJ 4.3.5 C.1 - By the end of grade 5, the student will: Use number sentences to model situations.NJ 4.3.5 C.1.a - By the end of grade 5, the student will: Use number sentences to model situations. Using variables to represent unknown quantities

Skill Area 10 - Algebra Skill 3 - A02 - Translate word problem to equation

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NJ 4.3.5 C.1.b - By the end of grade 5, the student will: Use number sentences to model situations. Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations


NJ 4.3.5 C.2 - By the end of grade 5, the student will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.NJ 4.3.5 C.2.a - By the end of grade 5, the student will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. Changes over timeNJ 4.3.5 C.2.b - By the end of grade 5, the student will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

NJ 4.3.5 D - ProceduresNJ 4.3.5 D.1 - By the end of grade 5, the student will: Solve simple linear equations with manipulatives and informallyNJ 4.3.5 D.1.a - By the end of grade 5, the student will: Solve simple linear equations with manipulatives and informally Whole-number coefficients only, answers also whole numbers

NJ 4.3.5 D.1.b - By the end of grade 5, the student will: Solve simple linear equations with manipulatives and informally Variables on one side of equationNJ 4.4 - 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.

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NJ 4.4.5 A - Data AnalysisNJ 4.4.5 A.1 - By the end of grade 5, the student will: Collect, generate, organize, and display data.

NJ 4.4.5 A.1.a - By the end of grade 5, the student will: Collect, generate, organize, and display data. Data generated from surveysNJ 4.4.5 A.2 - By the end of grade 5, the student will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.

NJ 4.4.5 A.2.a - By the end of grade 5, the student will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Bar graph, line graph, circle graph, table


Skill 3 - S01 - Read table

Skill 5 - S03 - Read pie graphSkill 6 - S04 - Interpret tableSkill 7 - S05 - Interpret bar graphSkill 8 - S06 - Interpret pie graph

NJ 4.4.5 A.2.b - By the end of grade 5, the student will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Range, median, and mean



Skill 10 - S08 - Statistics: MedianNJ 4.4.5 A.3 - By the end of grade 5, the student will: Respond to questions about data and generate their own questions and hypotheses.

NJ 4.4.5 B - ProbabilityNJ 4.4.5 B.1 - By the end of grade 5, the student will: Determine probabilities of events.NJ 4.4.5 B.1.a - By the end of grade 5, the student will: Determine probabilities of events. Event, probability of an event



NJ 4.4.5 B.1.b - By the end of grade 5, the student will: Determine probabilities of events. Probability of certain event is 1 and of impossible event is 0

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NJ 4.4.5 B.2 - By the end of grade 5, the student will: Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).

NJ 4.4.5 B.2.a - By the end of grade 5, the student will: 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

NJ 4.4.5 B.2.b - By the end of grade 5, the student will: Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag). Given data obtained experimentally, what is the likely distribution of items in the bag

NJ 4.4.5 B.3 - By the end of grade 5, the student will: Model situations involving probability using simulations (with spinners, dice) and theoretical models.NJ 4.4.5 C - Discrete Mathematics-Systematic Listing and CountingNJ 4.4.5 C.1 - By the end of grade 5, the student will: Solve counting problems and justify that all possibilities have been enumerated without duplication.NJ 4.4.5 C.1.a - By the end of grade 5, the student will: Solve counting problems and justify that all possibilities have been enumerated without duplication. Organized lists, charts, tree diagrams, tables

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NJ 4.4.5 C.2 - By the end of grade 5, the student will: 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).

NJ 4.4.5 D - Discrete Mathematics-Vertex-Edge Graphs and Algorithms

NJ 4.4.5 D.1 - By the end of grade 5, the student will: 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.

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

NJ 4.5 A - Problem SolvingNJ 4.5 A.1 - By the end of grade 5, the student will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5 A.2 - By the end of grade 5, the student will: Solve problems that arise in mathematics and in other contexts.NJ 4.5 A.2.a - By the end of grade 5, the student will: Solve problems that arise in mathematics and in other contexts. Open-ended problems

NJ 4.5 A.2.b - By the end of grade 5, the student will: Solve problems that arise in mathematics and in other contexts. Non-routine problems

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NJ 4.5 A.2.c - By the end of grade 5, the student will: Solve problems that arise in mathematics and in other contexts. Problems with multiple solutionsNJ 4.5 A.2.d - By the end of grade 5, the student will: Solve problems that arise in mathematics and in other contexts. Problems that can be solved in several waysNJ 4.5 A.3 - By the end of grade 5, the student will: Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.NJ 4.5 A.4 - By the end of grade 5, the student will: Pose problems of various types and levels of difficulty.

NJ 4.5 A.5 - By the end of grade 5, the student will: Monitor their progress and reflect on the process of their problem solving activity.

NJ 4.5 A.6 - By the end of grade 5, the student will: Distinguish relevant from irrelevant information, and identify missing information.NJ 4.5 B - CommunicationNJ 4.5 B.1 - By the end of grade 5, the student will: Use communication to organize and clarify their mathematical thinking.NJ 4.5 B.1.a - By the end of grade 5, the student will: Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5 B.1.b - By the end of grade 5, the student will: Use communication to organize and clarify their mathematical thinking. Discussion, listening, and questioning

NJ 4.5 B.2 - By the end of grade 5, the student will: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.

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NJ 4.5 B.3 - By the end of grade 5, the student will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5 B.4 - By the end of grade 5, the student will: Use the language of mathematics to express mathematical ideas precisely.


NJ 4.5 C - ConnectionsNJ 4.5 C.1 - By the end of grade 5, the student will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

NJ 4.5 C.2 - By the end of grade 5, the student will: 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).NJ 4.5 C.3 - By the end of grade 5, the student will: Recognize that mathematics is used in a variety of contexts outside of mathematics.NJ 4.5 C.4 - By the end of grade 5, the student will: Apply mathematics in practical situations and in other disciplines.NJ 4.5 C.5 - By the end of grade 5, the student will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5 C.6 - By the end of grade 5, the student will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5 D - ReasoningNJ 4.5 D.1 - By the end of grade 5, the student will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5 D.2 - By the end of grade 5, the student will: Use reasoning to support their mathematical conclusions and problem solutions.

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NJ 4.5 D.3 - By the end of grade 5, the student will: Select and use various types of reasoning and methods of proof.NJ 4.5 D.4 - By the end of grade 5, the student will: Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.NJ 4.5 D.5 - By the end of grade 5, the student will: Make and investigate mathematical conjectures.

NJ 4.5 D.5.a - By the end of grade 5, the student will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjecturesNJ 4.5 D.5.b - By the end of grade 5, the student will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5 D.6 - By the end of grade 5, the student will: Evaluate examples of mathematical reasoning and determine whether they are valid.

NJ 4.5 E - RepresentationsNJ 4.5 E.1 - By the end of grade 5, the student will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5 E.1.a - By the end of grade 5, the student will: Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles)

NJ 4.5 E.1.b - By the end of grade 5, the student will: Create and use representations to organize, record, and communicate mathematical ideas. Pictorial representations (e.g., diagrams, charts, or tables)

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NJ 4.5 E.1.c - By the end of grade 5, the student will: Create and use representations to organize, record, and communicate mathematical ideas. Symbolic representations (e.g., a formula)NJ 4.5 E.1.d - By the end of grade 5, the student will: Create and use representations to organize, record, and communicate mathematical ideas. Graphical representations (e.g., a line graph)NJ 4.5 E.2 - By the end of grade 5, the student will: Select, apply, and translate among mathematical representations to solve problems.NJ 4.5 E.3 - By the end of grade 5, the student will: Use representations to model and interpret physical, social, and mathematical phenomena.NJ 4.5 F - TechnologyNJ 4.5 F.1 - By the end of grade 5, the student will: Use technology to gather, analyze, and communicate mathematical information.NJ 4.5 F.2 - By the end of grade 5, the student will: Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.NJ 4.5 F.3 - By the end of grade 5, the student will: Use graphing calculators and computer software to investigate properties of functions and their graphs.NJ 4.5 F.4 - By the end of grade 5, the student will: Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

NJ 4.5 F.5 - By the end of grade 5, the student will: Use computer software to make and verify conjectures about geometric objects.

NJ 4.5 F.6 - By the end of grade 5, the student will: Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).

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NJ 4.1.6 A.1.a - By the end of Grade 6, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 6 pertain to these sets of numbers as well). All integers

NJ 4.1.6 A.1.b - By the end of Grade 6, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 6 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



NJ 4.1.6 A.1.c - By the end of Grade 6, students will: Use real-life experiences, physical materials, and technology to construct meanings for numbers (unless otherwise noted, all indicators for grade 6 pertain to these sets of numbers as well). All decimals

NJ 4.1.6 A.2 - By the end of Grade 6, students will: Recognize the decimal nature of United States currency and compute with money.NJ 4.1.6 A.3 - By the end of Grade 6, students will: Demonstrate a sense of the relative magnitudes of numbers.


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NJ 4.1.6 A.4 - By the end of Grade 6, students will: Explore the use of ratios and proportions in a variety of situations.


Skill 39 - C42 - Ratios

Skill Area 9 - Word Problems Skill 30 - W41 - Word problems: ProportionsSkill 31 - W42 - Word problems: Ratios

NJ 4.1.6 A.5 - By the end of Grade 6, students will: Understand and use whole-number percents between 1 and 100 in a variety of situations.


Skill 35 - C38 - Percent A (10 is what % of 40)

Skill 36 - C39 - Percent B (20% of 50 is what)Skill 37 - C40 - Percent C (30 is 50% of what)

Skill Area 9 - Word Problems Skill 27 - W38 - Word problems: Percent A (10 is what % of 40)Skill 28 - W39 - Word problems: Percent B (20% of 50 is what)Skill 29 - W40 - Word problems: Percent C (30 is 50% of what)

NJ 4.1.6 A.6 - By the end of Grade 6, students will: Use whole numbers, fractions, and decimals to represent equivalent forms of the same number.





Skill 30 - N28 - Fractions and decimals: Convert improper fraction to mixed number

NJ 4.1.6 A.7 - By the end of Grade 6, students will: Develop and apply number theory concepts in problem solving situations.NJ 4.1.6 A.7.a - By the end of Grade 6, students will: Develop and apply number theory concepts in problem solving situations. Primes, factors, multiples


Skill 40 - N38 - Advanced concepts: Knows meaning of primes and composites

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NJ 4.1.6 A.7.b - By the end of Grade 6, students will: Develop and apply number theory concepts in problem solving situations. Common multiples, common factors


Skill 41 - N39 - Advanced concepts: Can determine greatest common factor

Skill 42 - N40 - Advanced concepts: Can determine least common multiple

NJ 4.1.6 A.7.c - By the end of Grade 6, students will: Develop and apply number theory concepts in problem solving situations. Least common multiple, greatest common factor

NJ 4.1.6 A.8 - By the end of Grade 6, students will: Compare and order numbers.NJ 4.1.6 B - Numerical OperationsNJ 4.1.6 B.1 - By the end of Grade 6, students will: Recognize the appropriate use of each arithmetic operation in problem situations.NJ 4.1.6 B.2 - By the end of Grade 6, students will: Construct, use, and explain procedures for performing calculations with fractions and decimals with:NJ 4.1.6 B.2.a - By the end of Grade 6, students will: Construct, use, and explain procedures for performing calculations with fractions and decimals with: Pencil-and-paper


Skill 21 - C22 - Addition of fractions: like single digit denominators

Skill 22 - C23 - Subtraction of fractions: like single digit denominatorsSkill 23 - C24 - Addition of fractions: unlike single digit denominators

Skill 24 - C25 - Subtraction of fractions: unlike single digit denominatorsSkill 25 - C26 - Multiplication of fractions: single digit denominatorsSkill 26 - C27 - Division of fractions: single digit denominatorsSkill 27 - C28 - Addition of mixed numbersSkill 28 - C29 - Subtraction of mixed numbers

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Skill 29 - C30 - Multiplication of mixed numbersSkill 30 - C31 - Division of mixed numbersSkill 31 - C33 - Addition of decimals, place change (e.g. 2 + .45)

Skill 32 - C35 - Subtraction of decimals, place change (e.g. 5-.4)Skill 33 - C36 - Multiplication of decimalsSkill 34 - C37 - Division of decimals

Skill Area 9 - Word Problems Skill 16 - W22 - Word problems: Addition of fractions, like single digit denominatorsSkill 17 - W23 - Word problems: Subtraction of fractions, like single digit denominatorsSkill 18 - W24 - Word problems: Addition of fractions, unlike single digit denominatorsSkill 19 - W25 - Word problems: Subtraction of fractions, unlike single digit denominatorsSkill 20 - W28 - Word problems: Addition of mixed numbersSkill 21 - W29 - Word problems: Subtraction of mixed numbersSkill 23 - W33 - Word problems: Addition of decimals, place change (e.g. 2+.45)Skill 24 - W35 - Word problems: Subtraction of decimals, place change (e.g. 5-.4)Skill 25 - W36 - Word problems: Multiplication of decimalsSkill 26 - W37 - Word problems: Division decimals

NJ 4.1.6 B.2.b - By the end of Grade 6, students will: Construct, use, and explain procedures for performing calculations with fractions and decimals with: Mental math

NJ 4.1.6 B.2.c - By the end of Grade 6, students will: Construct, use, and explain procedures for performing calculations with fractions and decimals with: Calculator

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NJ 4.1.6 B.3 - By the end of Grade 6, students will: Use an efficient and accurate pencil-and-paper procedure for division of a 3-digit number by a 2-digit number.NJ 4.1.6 B.4 - By the end of Grade 6, students will: Select pencil-and-paper, mental math, or a calculator as the appropriate computational method in a given situation depending on the context and numbers.NJ 4.1.6 B.5 - By the end of Grade 6, students will: Find squares and cubes of whole numbers.NJ 4.1.6 B.6 - By the end of Grade 6, students will: Check the reasonableness of results of computations.NJ 4.1.6 B.7 - By the end of Grade 6, students will: Understand and use the various relationships among operations and properties of operations.NJ 4.1.6 B.8 - By the end of Grade 6, students will: Understand and apply the standard algebraic order of operations for the four basic operations, including appropriate use of parentheses.NJ 4.1.6 C - EstimationNJ 4.1.6 C.1 - By the end of Grade 6, students will: Use a variety of strategies for estimating both quantities and the results of computations.


Skill 31 - N29 - Fractions and decimals: Round decimals to tenths, hundredths

Skill Area 5 - Estimation Skill 9 - E24 - Estimation problems: Addition of fractions, unlike single digit denominatorsSkill 10 - E25 - Estimation problems: Subtraction of fractions, unlike single digit denominatorsSkill 11 - E28 - Estimation problems: Addition of mixed numbers

Skill 12 - E29 - Estimation problems: Subtraction of mixed numbers

Skill 13 - E32 - Estimation problems: Addition of decimals, no place change (e.g. 2.34+10.32)

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Skill 14 - E33 - Estimation problems: Addition of decimals, place change (e.g. 2+.45)Skill 15 - E34 - Estimation problems: Subtraction of decimals, no place change (e.g. .53 - .42)Skill 16 - E35 – Estimation problems: Subtraction of decimals, place change (e.g. 5 - .4)

NJ 4.1.6 C.2 - By the end of Grade 6, students will: Recognize when an estimate is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.NJ 4.1.6 C.3 - By the end of Grade 6, students will: Determine the reasonableness of an answer by estimating the result of operations.NJ 4.1.6 C.4 - By the end of Grade 6, students will: Determine whether a given estimate is an overestimate or an underestimate.NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.6 A - Geometric PropertiesNJ 4.2.6 A.1 - By the end of Grade 6, students will: Understand and apply concepts involving lines and angles.

NJ 4.2.6 A.1.a - By the end of Grade 6, students will: Understand and apply concepts involving lines and angles. Notation for line, ray, angle, line segment

Skill Area 6 - Geometry Skill 19 - G12 - Identify rays

NJ 4.2.6 A.1.b - By the end of Grade 6, students will: Understand and apply concepts involving lines and angles. Properties of parallel, perpendicular, and intersecting lines

Skill Area 6 - Geometry Skill 24 - G17 - Use properties of parallel lines

Skill 25 - G18 - Use properties of intersecting linesSkill 26 - G19 - Use properties of perpendicular lines

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NJ 4.2.6 A.1.c - By the end of Grade 6, students will: Understand and apply concepts involving lines and angles. Sum of the measures of the interior angles of a triangle is 180°

Skill Area 6 - Geometry Skill 29 - G22 - Using parts of a triangle

NJ 4.2.6 A.2 - By the end of Grade 6, students will: Identify, describe, compare, and classify polygons and circles.NJ 4.2.6 A.2.a - By the end of Grade 6, students will: Identify, describe, compare, and classify polygons and circles. Triangles by angles and sides

NJ 4.2.6 A.2.b - By the end of Grade 6, students will: Identify, describe, compare, and classify polygons and circles. Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombi

NJ 4.2.6 A.2.c - By the end of Grade 6, students will: Identify, describe, compare, and classify polygons and circles. Polygons by number of sides.

NJ 4.2.6 A.2.d - By the end of Grade 6, students will: Identify, describe, compare, and classify polygons and circles. Equilateral, equiangular, regularNJ 4.2.6 A.2.e - By the end of Grade 6, students will: Identify, describe, compare, and classify polygons and circles. All points equidistant from a given point form a circle

NJ 4.2.6 A.3 - By the end of Grade 6, students will: Identify similar figures.

NJ 4.2.6 A.4 - By the end of Grade 6, students will: Understand and apply the concepts of congruence and symmetry (line and rotational).NJ 4.2.6 A.5 - By the end of Grade 6, students will: Compare properties of cylinders, prisms, cones, pyramids, and spheres.

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NJ 4.2.6 A.6 - By the end of Grade 6, students will: Identify, describe, and draw the faces or shadows (projections) of three-dimensional geometric objects from different perspectives.NJ 4.2.6 A.7 - By the end of Grade 6, students will: Identify a three-dimensional shape with given projections (top, front and side views).NJ 4.2.6 A.8 - By the end of Grade 6, students will: Identify a three-dimensional shape with a given net (i.e., a flat pattern that folds into a 3D shape).NJ 4.2.6 B - Transforming ShapesNJ 4.2.6 B.1 - By the end of Grade 6, students will: Use a translation, a reflection, or a rotation to map one figure onto another congruent figure.

NJ 4.2.6 B.2 - By the end of Grade 6, students will: Recognize, identify, and describe geometric relationships and properties as they exist in nature, art, and other real-world settings.

NJ 4.2.6 C - Coordinate GeometryNJ 4.2.6 C.1 - By the end of Grade 6, students will: Create geometric shapes with specified properties in the first quadrant on a coordinate grid.NJ 4.2.6 D - Units of MeasurementNJ 4.2.6 D.1 - By the end of Grade 6, students will: Select and use appropriate units to measure angles, area, surface area, and volume.

NJ 4.2.6 D.2 - By the end of Grade 6, students will: Use a scale to find a distance on a map or a length on a scale drawing.NJ 4.2.6 D.3 - By the end of Grade 6, students will: Convert measurement units within a system (e.g., 3 feet = ___ inches).

Skill Area 7 - Measurement Skill 9 - M01 - Customary measures: Inches, feet, yards

Skill 11 - M04 - Customary measures: Pints, quarts, gallonsSkill 12 - M05 - Metric prefixes

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NJ 4.2.6 D.4 - By the end of Grade 6, students will: Know approximate equivalents between the standard and metric systems (e.g., one kilometer is approximately 6/10 of a mile).NJ 4.2.6 D.5 - By the end of Grade 6, students will: Use measurements and estimates to describe and compare phenomena.NJ 4.2.6 E - Measuring Geometric ObjectsNJ 4.2.6 E.1 - By the end of Grade 6, students will: Use a protractor to measure angles.NJ 4.2.6 E.2 - By the end of Grade 6, students will: Develop and apply strategies and formulas for finding perimeter and area.NJ 4.2.6 E.2.a - By the end of Grade 6, students will: Develop and apply strategies and formulas for finding perimeter and area. Triangle, square, rectangle, parallelogram, and trapezoid

Skill Area 6 - Geometry Skill 11 - G03 - Perimeter: square

Skill 12 - G04 - Perimeter: rectangle

Skill 13 - G05 - Perimeter: triangleSkill 14 - G06 - Area: squareSkill 15 - G07 - Area: rectangleSkill 16 - G08 - Area: right triangle

NJ 4.2.6 E.2.b - By the end of Grade 6, students will: Develop and apply strategies and formulas for finding perimeter and area. Circumference and area of a circle

Skill Area 6 - Geometry Skill 17 - G09 - Area: circle

NJ 4.2.6 E.3 - By the end of Grade 6, students will: Develop and apply strategies and formulas for finding the surface area and volume of rectangular prisms and cylinders.

Skill Area 6 - Geometry Skill 18 - G10 - Volume: rectangular prism

NJ 4.2.6 E.4 - By the end of Grade 6, students will: Recognize that shapes with the same perimeter do not necessarily have the same area and vice versa.

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NJ 4.2.6 E.5 - By the end of Grade 6, students will: 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).NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.6 A - PatternsNJ 4.3.6 A.1 - By the end of Grade 6, students will: Recognize, describe, extend, and create patterns involving whole numbers and rational numbers.NJ 4.3.6 A.1.a - By the end of Grade 6, students will: Recognize, describe, extend, and create patterns involving whole numbers and rational numbers. Descriptions using tables, verbal rules, simple equations, and graphsNJ 4.3.6 A.1.b - By the end of Grade 6, students will: Recognize, describe, extend, and create patterns involving whole numbers and rational numbers. Formal iterative formulas (e.g., NEXT = NOW * 3)

NJ 4.3.6 A.1.c - By the end of Grade 6, students will: Recognize, describe, extend, and create patterns involving whole numbers and rational numbers. Recursive patterns, including Pascal's Triangle (where each entry is the sum of the entries above it) and the Fibonacci Sequence: 1, 1, 2, 3, 5, 8,... (where NEXT = NOW + PREVIOUS)

NJ 4.3.6 B - Functions and Relationships

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NJ 4.3.6 B.1 - By the end of Grade 6, students will: Describe the general behavior of functions given by formulas or verbal rules (e.g., graph to determine whether increasing or decreasing, linear or not).

NJ 4.3.6 C - ModelingNJ 4.3.6 C.1 - By the end of Grade 6, students will: Use patterns, relations, and linear functions to model situations.NJ 4.3.6 C.1.a - By the end of Grade 6, students will: Use patterns, relations, and linear functions to model situations. Using variables to represent unknown quantities


NJ 4.3.6 C.1.b - By the end of Grade 6, students will: Use patterns, relations, and linear functions to model situations. Using concrete materials, tables, graphs, verbal rules, algebraic expressions/equations/inequalities


NJ 4.3.6 C.2 - By the end of Grade 6, students will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events.NJ 4.3.6 C.2.a - By the end of Grade 6, students will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. Changes over timeNJ 4.3.6 C.2.b - By the end of Grade 6, students will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. Relations between quantities

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NJ 4.3.6 C.2.c - By the end of Grade 6, students will: Draw freehand sketches of graphs that model real phenomena and use such graphs to predict and interpret events. Rates of change (e.g., when is plant growing slowly/rapidly, when is temperature dropping most rapidly/slowly)

NJ 4.3.6 D - ProceduresNJ 4.3.6 D.1 - By the end of Grade 6, students will: Solve simple linear equations with manipulatives and informally.NJ 4.3.6 D.1.a - By the end of Grade 6, students will: Solve simple linear equations with manipulatives and informally. Whole-number coefficients only, answers also whole numbersNJ 4.3.6 D.1.b - By the end of Grade 6, students will: Solve simple linear equations with manipulatives and informally. Variables on one or both sides of equation

NJ 4.3.6 D.2 - By the end of Grade 6, students will: Understand and apply the properties of operations and numbers.NJ 4.3.6 D.2.a - By the end of Grade 6, students will: Understand and apply the properties of operations and numbers. Distributive property

NJ 4.3.6 D.2.b - By the end of Grade 6, students will: Understand and apply the properties of operations and numbers. The product of a number and its reciprocal is 1

NJ 4.3.6 D.3 - By the end of Grade 6, students will: Evaluate numerical expressions.NJ 4.3.6 D.4 - By the end of Grade 6, students will: Extend understanding and use of inequality.

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NJ 4.3.6 D.4.a - By the end of Grade 6, students will: Extend understanding and use of inequality. Symbols (greater than or equal to, not equal to, less than or equal to)

NJ 4.4 - 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.NJ 4.4.6 A - Data AnalysisNJ 4.4.6 A.1 - By the end of Grade 6, students will: Collect, generate, organize, and display data.NJ 4.4.6 A.1.a - By the end of Grade 6, students will: Collect, generate, organize, and display data. Data generated from surveysNJ 4.4.6 A.2 - By the end of Grade 6, students will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data.

NJ 4.4.6 A.2.a - By the end of Grade 6, students will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Bar graph, line graph, circle graph, table, histogram

NJ 4.4.6 A.2.b - By the end of Grade 6, students will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Range, median, and mean



NJ 4.4.6 A.2.c - By the end of Grade 6, students will: Read, interpret, select, construct, analyze, generate questions about, and draw inferences from displays of data. Calculators and computers used to record and process information

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NJ 4.4.6 A.3 - By the end of Grade 6, students will: Respond to questions about data, generate their own questions and hypotheses, and formulate strategies for answering their questions and testing their hypotheses.

NJ 4.4.6 B - ProbabilityNJ 4.4.6 B.1 - By the end of Grade 6, students will: Determine probabilities of events.NJ 4.4.6 B.1.a - By the end of Grade 6, students will: Determine probabilities of events. Event, complementary event, probability of an event



NJ 4.4.6 B.1.b - By the end of Grade 6, students will: Determine probabilities of events. Multiplication rule for probabilitiesNJ 4.4.6 B.1.c - By the end of Grade 6, students will: Determine probabilities of events. Probability of certain event is 1 and of impossible event is 0NJ 4.4.6 B.1.d - By the end of Grade 6, students will: Determine probabilities of events. Probabilities of event and complementary event add up to 1NJ 4.4.6 B.2 - By the end of Grade 6, students will: Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag).

NJ 4.4.6 B.2.a - By the end of Grade 6, students will: 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

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NJ 4.4.6 B.2.b - By the end of Grade 6, students will: Determine probability using intuitive, experimental, and theoretical methods (e.g., using model of picking items of different colors from a bag). Given data obtained experimentally, what is the likely distribution of items in the bag

NJ 4.4.6 B.3 - By the end of Grade 6, students will: Explore compound events.NJ 4.4.6 B.4 - By the end of Grade 6, students will: Model situations involving probability using simulations (with spinners, dice) and theoretical models.NJ 4.4.6 B.5 - By the end of Grade 6, students will: Recognize and understand the connections among the concepts of independent outcomes, picking at random, and fairness.NJ 4.4.6 C - Discrete Mathematics-Systematic Listing and CountingNJ 4.4.6 C.1 - By the end of Grade 6, students will: Solve counting problems and justify that all possibilities have been enumerated without duplication.NJ 4.4.6 C.1.a - By the end of Grade 6, students will: Solve counting problems and justify that all possibilities have been enumerated without duplication. Organized lists, charts, tree diagrams, tables

NJ 4.4.6 C.1.b - By the end of Grade 6, students will: Solve counting problems and justify that all possibilities have been enumerated without duplication. Venn diagrams

NJ 4.4.6 C.2 - By the end of Grade 6, students will: Apply the multiplication principle of counting.

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NJ 4.4.6 C.2.a - By the end of Grade 6, students will: Apply the multiplication principle of counting. Simple situations (e.g., you can make 3 x 4 = 12 outfits using 3 shirts and 4 skirts).NJ 4.4.6 C.2.b - By the end of Grade 6, students will: Apply the multiplication principle of counting. Number of ways a specified number of items can be arranged in order (concept of permutation)NJ 4.4.6 C.2.c - By the end of Grade 6, students will: Apply the multiplication principle of counting. Number of ways of selecting a slate of officers from a class (e.g., if there are 23 students and 3 officers, the number is 23 x 22 x 21)

NJ 4.4.6 C.3 - By the end of Grade 6, students will: List the possible combinations of two elements chosen from a given set (e.g., forming a committee of two from a group of 12 students, finding how many handshakes there will be among ten people if everyone shakes each other person's hand once).


NJ 4.4.6 D.1 - By the end of Grade 6, students will: 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.

NJ 4.4.6 D.2 - By the end of Grade 6, students will: Analyze vertex-edge graphs and tree diagrams.

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NJ 4.4.6 D.2.a - By the end of Grade 6, students will: Analyze vertex-edge graphs and tree diagrams. Can a picture or a vertex-edge graph be drawn with a single line? (degree of vertex)NJ 4.4.6 D.2.b - By the end of Grade 6, students will: Analyze vertex-edge graphs and tree diagrams. Can you get from any vertex to any other vertex? (connectedness)

NJ 4.4.6 D.3 - By the end of Grade 6, students will: Use vertex-edge graphs to find solutions to practical problems.NJ 4.4.6 D.3.a - By the end of Grade 6, students will: Use vertex-edge graphs to find solutions to practical problems. Delivery route that stops at specified sites but involves least travelNJ 4.4.6 D.3.b - By the end of Grade 6, students will: Use vertex-edge graphs to find solutions to practical problems. Shortest route from one site on a map to anotherNJ 4.5 - All students will use mathematical processes of problem solving, communication, connections, reasoning, representations, and technology to solve problems and communicate mathematical ideas.

NJ 4.5 A - Problem SolvingNJ 4.5 A.1 - By the end of Grade 6, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5 A.2 - By the end of Grade 6, students will: Solve problems that arise in mathematics and in other contexts.NJ 4.5 A.2.a - By the end of Grade 6, students will: Solve problems that arise in mathematics and in other contexts. Open-ended problems

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NJ 4.5 B.3 - By the end of Grade 6, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5 B.4 - By the end of Grade 6, students will: Use the language of mathematics to express mathematical ideas precisely.


NJ 4.5 C - ConnectionsNJ 4.5 C.1 - By the end of Grade 6, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

NJ 4.5 C.2 - By the end of Grade 6, students will: 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).NJ 4.5 C.3 - By the end of Grade 6, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.NJ 4.5 C.4 - By the end of Grade 6, students will: Apply mathematics in practical situations and in other disciplines.NJ 4.5 C.5 - By the end of Grade 6, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5 C.6 - By the end of Grade 6, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5 D - Reasoning

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NJ 4.5 D.1 - By the end of Grade 6, students will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5 D.2 - By the end of Grade 6, students will: Use reasoning to support their mathematical conclusions and problem solutions.


NJ 4.5 D.5.a - By the end of Grade 6, students will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjecturesNJ 4.5 D.5.b - By the end of Grade 6, students will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5 D.6 - By the end of Grade 6, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.


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Standard Skill Area Description Skill DescriptionNJ 4.1 - All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.NJ 4.1.7 A - Number SenseNJ 4.1.7 A.1 - By the end of Grade 7, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well):

NJ 4.1.7 A.1.a - By the end of Grade 7, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well): Rational numbersNJ 4.1.7 A.1.b - By the end of Grade 7, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well): Percents

NJ 4.1.7 A.1.c - By the end of Grade 7, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 7 pertain to these sets of numbers as well): Whole numbers with exponents

NJ 4.1.7 A.2 - By the end of Grade 7, students will: Demonstrate a sense of the relative magnitudes of numbers.

NJ 4.1.7 A.3 - By the end of Grade 7, students will: Understand and use ratios, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.



Skill 36 - C39 - Percent B (20% of 50 is what)


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Skill 37 - C40 - Percent C (30 is 50% of what)Skill 38 - C41 - ProportionsSkill 39 - C42 - Ratios

Skill Area 9 - Word Problems Skill 27 - W38 - Word problems: Percent A (10 is what % of 40)Skill 28 - W39 - Word problems: Percent B (20% of 50 is what)Skill 29 - W40 - Word problems: Percent C (30 is 50% of what)Skill 30 - W41 - Word problems: ProportionsSkill 31 - W42 - Word problems: Ratios

NJ 4.1.7 A.4 - By the end of Grade 7, students will: Compare and order numbers of all named types.NJ 4.1.7 A.5 - By the end of Grade 7, students will: Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.





Skill 30 - N28 - Fractions and decimals: Convert improper fraction to mixed numberSkill 32 - N30 - Fractions and decimals: Relate decimals to percents

NJ 4.1.7 A.6 - By the end of Grade 7, students will: Understand that all fractions can be represented as repeating or terminating decimals.NJ 4.1.7 B - Numerical OperationsNJ 4.1.7 B.1 - By the end of Grade 7, students will: Use and explain procedures for performing calculations with integers and all number types named above with:NJ 4.1.7 B.1.a - By the end of Grade 7, students will: Use and explain procedures for performing calculations with integers and all number types named above with: Pencil-and-paper


Skill 43 - N41 - Advanced concepts: Recognizes use of negative numbers

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Skill 23 - C24 - Addition of fractions: unlike single digit denominators

Skill 24 - C25 - Subtraction of fractions: unlike single digit denominatorsSkill 25 - C26 - Multiplication of fractions: single digit denominatorsSkill 26 - C27 - Division of fractions: single digit denominatorsSkill 27 - C28 - Addition of mixed numbersSkill 28 - C29 - Subtraction of mixed numbersSkill 29 - C30 - Multiplication of mixed numbersSkill 30 - C31 - Division of mixed numbersSkill 32 - C35 - Subtraction of decimals, place change (e.g. 5-.4)Skill 33 - C36 - Multiplication of decimalsSkill 34 - C37 - Division of decimals

Skill Area 9 - Word Problems Skill 18 - W24 - Word problems: Addition of fractions, unlike single digit denominatorsSkill 19 - W25 - Word problems: Subtraction of fractions, unlike single digit denominatorsSkill 20 - W28 - Word problems: Addition of mixed numbersSkill 21 - W29 - Word problems: Subtraction of mixed numbersSkill 24 - W35 - Word problems: Subtraction of decimals, place change (e.g. 5-.4)Skill 25 - W36 - Word problems: Multiplication of decimalsSkill 26 - W37 - Word problems: Division decimals

NJ 4.1.7 B.1.b - By the end of Grade 7, students will: Use and explain procedures for performing calculations with integers and all number types named above with: Mental math

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NJ 4.1.7 B.1.c - By the end of Grade 7, students will: Use and explain procedures for performing calculations with integers and all number types named above with: CalculatorNJ 4.1.7 B.2 - By the end of Grade 7, students will: Use exponentiation to find whole number powers of numbers.


Skill 36 - N34 - Advanced concepts: Recognize meaning of exponents (2-10)

NJ 4.1.7 B.3 - By the end of Grade 7, students will: Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.NJ 4.1.7 C - EstimationNJ 4.1.7 C.1 - By the end of Grade 7, students will: Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.

Skill Area 5 - Estimation Skill 17 - E38 - Estimation problems: Percent A (10 is what % of 40)

Skill 18 - E39 - Estimation problems: Percent B (20% of 50 is what)

Skill 19 - E40 - Estimation problems: Percent C (30 is 50% of what)

NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.7 A - Geometric PropertiesNJ 4.2.7 A.1 - By the end of Grade 7, students will: Understand and apply properties of polygons.NJ 4.2.7 A.1.a - By the end of Grade 7, students will: Understand and apply properties of polygons. Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombiNJ 4.2.7 A.1.b - By the end of Grade 7, students will: Understand and apply properties of polygons. Regular polygonsNJ 4.2.7 A.2 - By the end of Grade 7, students will: Understand and apply the concept of similarity.

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NJ 4.2.7 A.2.a - By the end of Grade 7, students will: Understand and apply the concept of similarity. Using proportions to find missing measures

NJ 4.2.7 A.2.b - By the end of Grade 7, students will: Understand and apply the concept of similarity. Scale drawingsNJ 4.2.7 A.2.c - By the end of Grade 7, students will: Understand and apply the concept of similarity. Models of 3D objectsNJ 4.2.7 A.3 - By the end of Grade 7, students will: Use logic and reasoning to make and support conjectures about geometric objects.

NJ 4.2.7 B - Transforming ShapesNJ 4.2.7 B.1 - By the end of Grade 7, students will: Understand and apply transformations.NJ 4.2.7 B.1.a - By the end of Grade 7, students will: Understand and apply transformations. Finding the image, given the pre-image, and vice-versaNJ 4.2.7 B.1.b - By the end of Grade 7, students will: Understand and apply transformations. Sequence of transformations needed to map one figure onto another

NJ 4.2.7 B.1.c - By the end of Grade 7, students will: Understand and apply transformations. Reflections, rotations, and translations result in images congruent to the pre-image

NJ 4.2.7 B.1.d - By the end of Grade 7, students will: Understand and apply transformations. Dilations (stretching/shrinking) result in images similar to the pre-image

NJ 4.2.7 C - Coordinate GeometryNJ 4.2.7 C.1 - By the end of Grade 7, students will: Use coordinates in four quadrants to represent geometric concepts.

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NJ 4.2.7 C.2 - By the end of Grade 7, students will: Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).

NJ 4.2.7 D - Units of MeasurementNJ 4.2.7 D.1 - By the end of Grade 7, students will: Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4'3" plus 7'10" equals 12'1").NJ 4.2.7 D.2 - By the end of Grade 7, students will: Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.NJ 4.2.7 D.3 - By the end of Grade 7, students will: Recognize that all measurements of continuous quantities are approximations.NJ 4.2.7 E - Measuring Geometric ObjectsNJ 4.2.7 E.1 - By the end of Grade 7, students will: Develop and apply strategies for finding perimeter and area.NJ 4.2.7 E.1.a - By the end of Grade 7, students will: Develop and apply strategies for finding perimeter and area. Geometric figures made by combining triangles, rectangles and circles or parts of circles

NJ 4.2.7 E.1.b - By the end of Grade 7, students will: Develop and apply strategies for finding perimeter and area. Estimation of area using grids of various sizesNJ 4.2.7 E.2 - By the end of Grade 7, students will: Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).

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NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.7 A - PatternsNJ 4.3.7 A.1 - By the end of Grade 7, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.NJ 4.3.7 A.1.a - By the end of Grade 7, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressionsNJ 4.3.7 A.1.b - By the end of Grade 7, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Finite and infinite sequencesNJ 4.3.7 A.1.c - By the end of Grade 7, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Generating sequences by using calculators to repeatedly apply a formula

NJ 4.3.7 B - Functions and RelationshipsNJ 4.3.7 B.1 - By the end of Grade 7, students will: Graph functions, and understand and describe their general behavior.NJ 4.3.7 B.1.a - By the end of Grade 7, students will: Graph functions, and understand and describe their general behavior. Equations involving two variables

Skill Area 10 - Algebra Skill 7 - A06 - Graph of linear equation (integers add, subtract)

NJ 4.3.7 C - Modeling

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NJ 4.3.7 C.1 - By the end of Grade 7, students will: Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

NJ 4.3.7 C.2 - By the end of Grade 7, students will: Use patterns, relations, symbolic algebra, and linear functions to model situations.

NJ 4.3.7 C.2.a - By the end of Grade 7, students will: Use patterns, relations, symbolic algebra, and linear functions to model situations. Using manipulatives, tables, graphs, verbal rules, algebraic expressions/equations/inequalities


NJ 4.3.7 C.2.b - By the end of Grade 7, students will: Use patterns, relations, symbolic algebra, and linear functions to model situations. Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

NJ 4.3.7 D - ProceduresNJ 4.3.7 D.1 - By the end of Grade 7, students will: Use graphing techniques on a number line.NJ 4.3.7 D.1.a - By the end of Grade 7, students will: Use graphing techniques on a number line. Absolute valueNJ 4.3.7 D.1.b - By the end of Grade 7, students will: Use graphing techniques on a number line. Arithmetic operations represented by vectors (arrows) (e.g., "-3 + 6" is "left 3, right 6")NJ 4.3.7 D.2 - By the end of Grade 7, students will: Solve simple linear equations informally and graphically.

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NJ 4.3.7 D.2.a - By the end of Grade 7, students will: Solve simple linear equations informally and graphically. Multi-step, integer coefficients only (although answers may not be integers)NJ 4.3.7 D.2.b - By the end of Grade 7, students will: Solve simple linear equations informally and graphically. Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology

NJ 4.3.7 D.3 - By the end of Grade 7, students will: Create, evaluate, and simplify algebraic expressions involving variables.NJ 4.3.7 D.3.a - By the end of Grade 7, students will: Create, evaluate, and simplify algebraic expressions involving variables. Order of operations, including appropriate use of parenthesesNJ 4.3.7 D.3.b - By the end of Grade 7, students will: Create, evaluate, and simplify algebraic expressions involving variables. Substitution of a number for a variable

NJ 4.3.7 D.4 - By the end of Grade 7, students will: Understand and apply the properties of operations, numbers, equations, and inequalities.

NJ 4.3.7 D.4.a - By the end of Grade 7, students will: Understand and apply the properties of operations, numbers, equations, and inequalities. Additive inverseNJ 4.3.7 D.4.b - By the end of Grade 7, students will: Understand and apply the properties of operations, numbers, equations, and inequalities. Multiplicative inverse

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NJ 4.4 - 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.NJ 4.4.7 A - Data AnalysisNJ 4.4.7 A.1 - By the end of Grade 7, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).

NJ 4.4.7 A.1.a - By the end of Grade 7, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Type of display most appropriate for given dataNJ 4.4.7 A.1.b - By the end of Grade 7, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Box-and-whisker plot, upper quartile, lower quartileNJ 4.4.7 A.1.c - By the end of Grade 7, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Scatter plotNJ 4.4.7 A.1.d - By the end of Grade 7, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Calculators and computer used to record and process information

NJ 4.4.7 A.2 - By the end of Grade 7, students will: Make inferences and formulate and evaluate arguments based on displays and analysis of data.NJ 4.4.7 B - Probability

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NJ 4.4.7 B.1 - By the end of Grade 7, students will: Interpret probabilities as ratios, percents, and decimals.

NJ 4.4.7 B.2 - By the end of Grade 7, students will: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.

NJ 4.4.7 B.2.a - By the end of Grade 7, students will: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models. Frequency, relative frequency

NJ 4.4.7 B.3 - By the end of Grade 7, students will: Estimate probabilities and make predictions based on experimental and theoretical probabilities.NJ 4.4.7 B.4 - By the end of Grade 7, students will: Play and analyze probability-based games, and discuss the concepts of fairness and expected value.NJ 4.4.7 C - Discrete Mathematics-Systematic Listing and CountingNJ 4.4.7 C.1 - By the end of Grade 7, students will: Apply the multiplication principle of counting.NJ 4.4.7 C.1.a - By the end of Grade 7, students will: Apply the multiplication principle of counting. Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

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NJ 4.4.7 C.2 - By the end of Grade 7, students will: Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).

NJ 4.4.7 C.3 - By the end of Grade 7, students will: Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.NJ 4.4.7 D - Discrete Mathematics-Vertex-Edge Graphs and Algorithms

NJ 4.4.7 D.1 - By the end of Grade 7, students will: Use vertex-edge graphs to represent and find solutions to practical problems.NJ 4.4.7 D.1.a - By the end of Grade 7, students will: Use vertex-edge graphs to represent and find solutions to practical problems. Finding the shortest network connecting specified sitesNJ 4.4.7 D.1.b - By the end of Grade 7, students will: Use vertex-edge graphs to represent and find solutions to practical problems. Finding the shortest route on a map from one site to anotherNJ 4.4.7 D.1.c - By the end of Grade 7, students will: Use vertex-edge graphs to represent and find solutions to practical problems. Finding the shortest circuit on a map that makes a tour of specified sites

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NJ 4.5 A - Problem SolvingNJ 4.5 A.1 - By the end of Grade 7, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5 A.2 - By the end of Grade 7, students will: Solve problems that arise in mathematics and in other contexts.NJ 4.5 A.2.a - By the end of Grade 7, students will: Solve problems that arise in mathematics and in other contexts. Open-ended problems




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NJ 4.5 B.3 - By the end of Grade 7, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5 B.4 - By the end of Grade 7, students will: Use the language of mathematics to express mathematical ideas precisely.NJ 4.5 C - ConnectionsNJ 4.5 C.1 - By the end of Grade 7, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

NJ 4.5 C.2 - By the end of Grade 7, students will: 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).

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NJ 4.5 C.3 - By the end of Grade 7, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.NJ 4.5 C.4 - By the end of Grade 7, students will: Apply mathematics in practical situations and in other disciplines.NJ 4.5 C.5 - By the end of Grade 7, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5 C.6 - By the end of Grade 7, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5 D - ReasoningNJ 4.5 D.1 - By the end of Grade 7, students will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5 D.2 - By the end of Grade 7, students will: Use reasoning to support their mathematical conclusions and problem solutions.


NJ 4.5 D.5.a - By the end of Grade 7, students will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjectures

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NJ 4.5 D.5.b - By the end of Grade 7, students will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5 D.6 - By the end of Grade 7, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.




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Standard Skill Area Description Skill DescriptionNJ 4.1 - All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.NJ 4.1.8 A - Number SenseNJ 4.1.8 A.1 - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well):

NJ 4.1.8 A.1.a - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well): Rational numbersNJ 4.1.8 A.1.b - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well): Percents

NJ 4.1.8 A.1.c - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well): Exponents

NJ 4.1.8 A.1.d - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well): Roots


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NJ 4.1.8 A.1.e - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well): Absolute valuesNJ 4.1.8 A.1.f - By the end of grade 8, students will: Extend understanding of the number system by constructing meanings for the following (unless otherwise noted, all indicators for grade 8 pertain to these sets of numbers as well): Numbers represented in scientific notation


Skill 39 - N37 - Advanced concepts: Can use scientific notation

NJ 4.1.8 A.2 - By the end of grade 8, students will: Demonstrate a sense of the relative magnitudes of numbers.

NJ 4.1.8 A.3 - By the end of grade 8, students will: Understand and use ratios, rates, proportions, and percents (including percents greater than 100 and less than 1) in a variety of situations.



Skill 36 - C39 - Percent B (20% of 50 is what)Skill 37 - C40 - Percent C (30 is 50% of what)Skill 38 - C41 - ProportionsSkill 39 - C42 - Ratios

Skill Area 9 - Word Problems Skill 27 - W38 - Word problems: Percent A (10 is what % of 40)Skill 28 - W39 - Word problems: Percent B (20% of 50 is what)Skill 29 - W40 - Word problems: Percent C (30 is 50% of what)Skill 30 - W41 - Word problems: ProportionsSkill 31 - W42 - Word problems: Ratios

NJ 4.1.8 A.4 - By the end of grade 8, students will: Compare and order numbers of all named types.NJ 4.1.8 A.5 - By the end of grade 8, students will: Use whole numbers, fractions, decimals, and percents to represent equivalent forms of the same number.



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Skill 32 - N30 - Fractions and decimals: Relate decimals to percents

NJ 4.1.8 A.6 - By the end of grade 8, students will: Recognize that repeating decimals correspond to fractions and determine their fractional equivalents.NJ 4.1.8 A.6.a - By the end of grade 8, students will: Recognize that repeating decimals correspond to fractions and determine their fractional equivalents. 5/7 = 0. 714285714285... = 0.714285 repeatingNJ 4.1.8 A.7 - By the end of grade 8, students will: Construct meanings for common irrational numbers, such as pi and the square root of 2.

NJ 4.1.8 B - Numerical OperationsNJ 4.1.8 B.1 - By the end of grade 8, students will: Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with:

NJ 4.1.8 B.1.a - By the end of grade 8, students will: Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with: Pencil-and-paperNJ 4.1.8 B.1.b - By the end of grade 8, students will: Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with: Mental math

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NJ 4.1.8 B.1.c - By the end of grade 8, students will: Use and explain procedures for performing calculations involving addition, subtraction, multiplication, division, and exponentiation with integers and all number types named above with: CalculatorNJ 4.1.8 B.2 - By the end of grade 8, students will: Use exponentiation to find whole number powers of numbers.


Skill 36 - N34 - Advanced concepts: Recognize meaning of exponents (2-10)

NJ 4.1.8 B.3 - By the end of grade 8, students will: Find square and cube roots of numbers and understand the inverse nature of powers and roots.


Skill 33 - N31 - Advanced concepts: Determine square roots of perfect squares

Skill 35 - N33 - Advanced concepts: Recognize meaning of nth root

NJ 4.1.8 B.4 - By the end of grade 8, students will: Solve problems involving proportions and percents.



Skill 36 - C39 - Percent B (20% of 50 is what)Skill 37 - C40 - Percent C (30 is 50% of what)Skill 38 - C41 - Proportions

Skill Area 9 - Word Problems Skill 27 - W38 - Word problems: Percent A (10 is what % of 40)Skill 28 - W39 - Word problems: Percent B (20% of 50 is what)Skill 29 - W40 - Word problems: Percent C (30 is 50% of what)Skill 30 - W41 - Word problems: Proportions

NJ 4.1.8 B.5 - By the end of grade 8, students will: Understand and apply the standard algebraic order of operations, including appropriate use of parentheses.NJ 4.1.8 C - EstimationNJ 4.1.8 C.1 - By the end of grade 8, students will: Estimate square and cube roots of numbers.


Skill 34 - N32 - Advanced concepts: Give approximate square roots of a number

NJ 4.1.8 C.2 - By the end of grade 8, students will: Use equivalent representations of numbers such as fractions, decimals, and percents to facilitate estimation.

Skill Area 5 - Estimation Skill 17 - E38 - Estimation problems: Percent A (10 is what % of 40)

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Skill 18 - E39 - Estimation problems: Percent B (20% of 50 is what)

Skill 19 - E40 - Estimation problems: Percent C (30 is 50% of what)

NJ 4.1.8 C.3 - By the end of grade 8, students will: Recognize the limitations of estimation and assess the amount of error resulting from estimation.NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.8 A - Geometric PropertiesNJ 4.2.8 A.1 - By the end of grade 8, students will: Understand and apply concepts involving lines, angles, and planes.NJ 4.2.8 A.1.a - By the end of grade 8, students will: Understand and apply concepts involving lines, angles, and planes. Complementary and supplementary angles

Skill Area 6 - Geometry Skill 27 - G20 - Vertical and supplementary angles

NJ 4.2.8 A.1.b - By the end of grade 8, students will: Understand and apply concepts involving lines, angles, and planes. Vertical anglesNJ 4.2.8 A.1.c - By the end of grade 8, students will: Understand and apply concepts involving lines, angles, and planes. Bisectors and perpendicular bisectorsNJ 4.2.8 A.1.d - By the end of grade 8, students will: Understand and apply concepts involving lines, angles, and planes. Parallel, perpendicular, and intersecting planesNJ 4.2.8 A.1.e - By the end of grade 8, students will: Understand and apply concepts involving lines, angles, and planes. Intersection of plane with cube, cylinder, cone, and sphereNJ 4.2.8 A.2 - By the end of grade 8, students will: Understand and apply the Pythagorean theorem.

Skill Area 6 - Geometry Skill 30 - G23 - Pythagorean theorem

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NJ 4.2.8 A.3 - By the end of grade 8, students will: Understand and apply properties of polygons.NJ 4.2.8 A.3.a - By the end of grade 8, students will: Understand and apply properties of polygons. Quadrilaterals, including squares, rectangles, parallelograms, trapezoids, rhombiNJ 4.2.8 A.3.b - By the end of grade 8, students will: Understand and apply properties of polygons. Regular polygonsNJ 4.2.8 A.3.c - By the end of grade 8, students will: Understand and apply properties of polygons. Sum of measures of interior angles of a polygonNJ 4.2.8 A.3.d - By the end of grade 8, students will: Understand and apply properties of polygons. Which polygons can be used alone to generate a tessellation and whyNJ 4.2.8 A.4 - By the end of grade 8, students will: Understand and apply the concept of similarity.NJ 4.2.8 A.4.a - By the end of grade 8, students will: Understand and apply the concept of similarity. Using proportions to find missing measures

NJ 4.2.8 A.4.b - By the end of grade 8, students will: Understand and apply the concept of similarity. Scale drawingsNJ 4.2.8 A.4.c - By the end of grade 8, students will: Understand and apply the concept of similarity. Models of 3D objectsNJ 4.2.8 A.5 - By the end of grade 8, students will: Use logic and reasoning to make and support conjectures about geometric objects.

NJ 4.2.8 A.6 - By the end of grade 8, students will: Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology).

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NJ 4.2.8 A.6.a - By the end of grade 8, students will: Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology). Congruent angles or line segmentsNJ 4.2.8 A.6.b - By the end of grade 8, students will: Perform basic geometric constructions using a variety of methods (e.g., straightedge and compass, patty/tracing paper, or technology). Midpoint of a line segmentNJ 4.2.8 A.7 - By the end of grade 8, students will: Create two-dimensional representations (e.g., nets or projective views) for the surfaces of three-dimensional objects.

NJ 4.2.8 B - Transforming ShapesNJ 4.2.8 B.1 - By the end of grade 8, students will: Understand and apply transformations.NJ 4.2.8 B.1.a - By the end of grade 8, students will: Understand and apply transformations. Finding the image, given the pre-image, and vice-versaNJ 4.2.8 B.1.b - By the end of grade 8, students will: Understand and apply transformations. Sequence of transformations needed to map one figure onto another

NJ 4.2.8 B.1.c - By the end of grade 8, students will: Understand and apply transformations. Reflections, rotations, and translations result in images congruent to the pre-image

NJ 4.2.8 B.1.d - By the end of grade 8, students will: Understand and apply transformations. Dilations (stretching/shrinking) result in images similar to the pre-image

NJ 4.2.8 B.2 - By the end of grade 8, students will: Use iterative procedures to generate geometric patterns.

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NJ 4.2.8 B.2.a - By the end of grade 8, students will: Use iterative procedures to generate geometric patterns. Fractals (e.g., the Koch Snowflake)NJ 4.2.8 B.2.b - By the end of grade 8, students will: Use iterative procedures to generate geometric patterns. Self-similarityNJ 4.2.8 B.2.c - By the end of grade 8, students will: Use iterative procedures to generate geometric patterns. Construction of initial stages

NJ 4.2.8 B.2.d - By the end of grade 8, students will: Use iterative procedures to generate geometric patterns. Patterns in successive stages (e.g., number of triangles in each stage of Sierpinski's Triangle)NJ 4.2.8 C - Coordinate GeometryNJ 4.2.8 C.1 - By the end of grade 8, students will: Use coordinates in four quadrants to represent geometric concepts.NJ 4.2.8 C.2 - By the end of grade 8, students will: Use a coordinate grid to model and quantify transformations (e.g., translate right 4 units).

NJ 4.2.8 D - Units of MeasurementNJ 4.2.8 D.1 - By the end of grade 8, students will: Solve problems requiring calculations that involve different units of measurement within a measurement system (e.g., 4'3" plus 7'10" equals 12'1").NJ 4.2.8 D.2 - By the end of grade 8, students will: Use approximate equivalents between standard and metric systems to estimate measurements (e.g., 5 kilometers is about 3 miles).NJ 4.2.8 D.3 - By the end of grade 8, students will: Recognize that the degree of precision needed in calculations depends on how the results will be used and the instruments used to generate the measurements.

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NJ 4.2.8 D.4 - By the end of grade 8, students will: Select and use appropriate units and tools to measure quantities to the degree of precision needed in a particular problem-solving situation.NJ 4.2.8 D.5 - By the end of grade 8, students will: Recognize that all measurements of continuous quantities are approximations.NJ 4.2.8 D.6 - By the end of grade 8, students will: Solve problems that involve compound measurement units, such as speed (miles per hour), air pressure (pounds per square inch), and population density (persons per square mile).

NJ 4.2.8 E - Measuring Geometric ObjectsNJ 4.2.8 E.1 - By the end of grade 8, students will: Develop and apply strategies for finding perimeter and area.NJ 4.2.8 E.1.a - By the end of grade 8, students will: Develop and apply strategies for finding perimeter and area. Geometric figures made by combining triangles, rectangles and circles or parts of circles

NJ 4.2.8 E.1.b - By the end of grade 8, students will: Develop and apply strategies for finding perimeter and area. Estimation of area using grids of various sizesNJ 4.2.8 E.1.c - By the end of grade 8, students will: Develop and apply strategies for finding perimeter and area. Impact of a dilation on the perimeter and area of a 2-dimensional figureNJ 4.2.8 E.2 - By the end of grade 8, students will: Recognize that the volume of a pyramid or cone is one-third of the volume of the prism or cylinder with the same base and height (e.g., use rice to compare volumes of figures with same base and height).

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NJ 4.2.8 E.3 - By the end of grade 8, students will: Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure.NJ 4.2.8 E.3.a - By the end of grade 8, students will: Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure. Volume - prism, cone, pyramid

Skill Area 6 - Geometry Skill 18 - G10 - Volume: rectangular prism

NJ 4.2.8 E.3.b - By the end of grade 8, students will: Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure. Surface area - prism (triangular or rectangular base), pyramid (triangular or rectangular base)NJ 4.2.8 E.3.c - By the end of grade 8, students will: Develop and apply strategies and formulas for finding the surface area and volume of a three-dimensional figure. Impact of a dilation on the surface area and volume of a three-dimensional figure

NJ 4.2.8 E.4 - By the end of grade 8, students will: Use formulas to find the volume and surface area of a sphere.

NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.8 A - PatternsNJ 4.3.8 A.1 - By the end of grade 8, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers.NJ 4.3.8 A.1.a - By the end of grade 8, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Descriptions using tables, verbal and symbolic rules, graphs, simple equations or expressions

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NJ 4.3.8 A.1.b - By the end of grade 8, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Finite and infinite sequencesNJ 4.3.8 A.1.c - By the end of grade 8, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Arithmetic sequences (i.e., sequences generated by repeated addition of a fixed number, positive or negative)

Skill Area 10 - Algebra Skill 23 - A22 - Sequences and series: Find specified term of arithmetic sequences

NJ 4.3.8 A.1.d - By the end of grade 8, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Geometric sequences (i.e., sequences generated by repeated multiplication by a fixed positive ratio, greater than 1 or less than 1)NJ 4.3.8 A.1.e - By the end of grade 8, students will: Recognize, describe, extend, and create patterns involving whole numbers, rational numbers, and integers. Generating sequences by using calculators to repeatedly apply a formula

NJ 4.3.8 B - Functions and RelationshipsNJ 4.3.8 B.1 - By the end of grade 8, students will: Graph functions, and understand and describe their general behavior.NJ 4.3.8 B.1.a - By the end of grade 8, students will: Graph functions, and understand and describe their general behavior. Equations involving two variables


NJ 4.3.8 B.1.b - By the end of grade 8, students will: Graph functions, and understand and describe their general behavior. Rates of change (informal notion of slope)

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NJ 4.3.8 B.2 - By the end of grade 8, students will: Recognize and describe the difference between linear and exponential growth, using tables, graphs, and equations.

NJ 4.3.8 C - ModelingNJ 4.3.8 C.1 - By the end of grade 8, students will: Analyze functional relationships to explain how a change in one quantity can result in a change in another, using pictures, graphs, charts, and equations.

NJ 4.3.8 C.2 - By the end of grade 8, students will: Use patterns, relations, symbolic algebra, and linear functions to model situations.

NJ 4.3.8 C.2.a - By the end of grade 8, students will: Use patterns, relations, symbolic algebra, and linear functions to model situations. Using concrete materials (manipulatives), tables, graphs, verbal rules, algebraic expressions/equations/inequalitiesNJ 4.3.8 C.2.b - By the end of grade 8, students will: Use patterns, relations, symbolic algebra, and linear functions to model situations. Growth situations, such as population growth and compound interest, using recursive (e.g., NOW-NEXT) formulas (cf. science standard 5.5 and social studies standard 6.6)

NJ 4.3.8 D - ProceduresNJ 4.3.8 D.1 - By the end of grade 8, students will: Use graphing techniques on a number line.NJ 4.3.8 D.1.a - By the end of grade 8, students will: Use graphing techniques on a number line. Absolute valueNJ 4.3.8 D.1.b - By the end of grade 8, students will: Use graphing techniques on a number line. Arithmetic operations represented by vectors (arrows) (e.g., "-3 + 6" is "left 3, right 6")

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NJ 4.3.8 D.2 - By the end of grade 8, students will: Solve simple linear equations informally, graphically, and using formal algebraic methods.

NJ 4.3.8 D.2.a - By the end of grade 8, students will: Solve simple linear equations informally, graphically, and using formal algebraic methods. Multi-step, integer coefficients only (although answers may not be integers)

Skill Area 10 - Algebra Skill 4 - A03 - Linear equations: 1 unknown

NJ 4.3.8 D.2.b - By the end of grade 8, students will: Solve simple linear equations informally, graphically, and using formal algebraic methods. Simple literal equations (e.g., A = lw)

NJ 4.3.8 D.2.c - By the end of grade 8, students will: Solve simple linear equations informally, graphically, and using formal algebraic methods. Using paper-and-pencil, calculators, graphing calculators, spreadsheets, and other technology


NJ 4.3.8 D.3 - By the end of grade 8, students will: Solve simple linear inequalities.

Skill Area 10 - Algebra Skill 8 - A07 - Linear inequalities: 1 unknown

Skill 9 - A08 - Linear inequalities: 2 unknown

NJ 4.3.8 D.4 - By the end of grade 8, students will: Create, evaluate, and simplify algebraic expressions involving variables.NJ 4.3.8 D.4.a - By the end of grade 8, students will: Create, evaluate, and simplify algebraic expressions involving variables. Order of operations, including appropriate use of parenthesesNJ 4.3.8 D.4.b - By the end of grade 8, students will: Create, evaluate, and simplify algebraic expressions involving variables. Distributive property

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NJ 4.3.8 D.4.c - By the end of grade 8, students will: Create, evaluate, and simplify algebraic expressions involving variables. Substitution of a number for a variable

NJ 4.3.8 D.4.d - By the end of grade 8, students will: Create, evaluate, and simplify algebraic expressions involving variables. Translation of a verbal phrase or sentence into an algebraic expression, equation, or inequality, and vice versa


NJ 4.3.8 D.5 - By the end of grade 8, students will: Understand and apply the properties of operations, numbers, equations, and inequalities.

NJ 4.3.8 D.5.a - By the end of grade 8, students will: Understand and apply the properties of operations, numbers, equations, and inequalities. Additive inverseNJ 4.3.8 D.5.b - By the end of grade 8, students will: Understand and apply the properties of operations, numbers, equations, and inequalities. Multiplicative inverse

NJ 4.3.8 D.5.c - By the end of grade 8, students will: Understand and apply the properties of operations, numbers, equations, and inequalities. Addition and multiplication properties of equalityNJ 4.3.8 D.5.d - By the end of grade 8, students will: Understand and apply the properties of operations, numbers, equations, and inequalities. Addition and multiplication properties of inequalities

NJ 4.4 - 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.

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NJ 4.4.8 A - Data AnalysisNJ 4.4.8 A.1 - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode).

NJ 4.4.8 A.1.a - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Type of display most appropriate for given dataNJ 4.4.8 A.1.b - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Box-and-whisker plot, upper quartile, lower quartileNJ 4.4.8 A.1.c - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Scatter plotNJ 4.4.8 A.1.d - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Calculators and computer used to record and process information

NJ 4.4.8 A.1.e - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Finding the median and mean (weighted average) using frequency data.


Skill 10 - S08 - Statistics: Median

NJ 4.4.8 A.1.f - By the end of grade 8, students will: Select and use appropriate representations for sets of data, and measures of central tendency (mean, median, and mode). Effect of additional data on measures of central tendency

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NJ 4.4.8 A.2 - By the end of grade 8, students will: Make inferences and formulate and evaluate arguments based on displays and analysis of data sets.NJ 4.4.8 A.3 - By the end of grade 8, students will: Estimate lines of best fit and use them to interpolate within the range of the data.NJ 4.4.8 A.4 - By the end of grade 8, students will: Use surveys and sampling techniques to generate data and draw conclusions about large groups.NJ 4.4.8 B - ProbabilityNJ 4.4.8 B.1 - By the end of grade 8, students will: Interpret probabilities as ratios, percents, and decimals.

NJ 4.4.8 B.2 - By the end of grade 8, students will: Determine probabilities of compound events.NJ 4.4.8 B.3 - By the end of grade 8, students will: Explore the probabilities of conditional events (e.g., if there are seven marbles in a bag, three red and four green, what is the probability that two marbles picked from the bag, without replacement, are both red).

NJ 4.4.8 B.4 - By the end of grade 8, students will: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models.

NJ 4.4.8 B.4.a - By the end of grade 8, students will: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models. Frequency, relative frequency

NJ 4.4.8 B.5 - By the end of grade 8, students will: Estimate probabilities and make predictions based on experimental and theoretical probabilities.

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NJ 4.4.8 B.6 - By the end of grade 8, students will: Play and analyze probability-based games, and discuss the concepts of fairness and expected value.NJ 4.4.8 C - Discrete Mathematics-Systematic Listing and CountingNJ 4.4.8 C.1 - By the end of grade 8, students will: Apply the multiplication principle of counting.NJ 4.4.8 C.1.a - By the end of grade 8, students will: Apply the multiplication principle of counting. Permutations: ordered situations with replacement (e.g., number of possible license plates) vs. ordered situations without replacement (e.g., number of possible slates of 3 class officers from a 23 student class)

NJ 4.4.8 C.1.b - By the end of grade 8, students will: Apply the multiplication principle of counting. Factorial notationNJ 4.4.8 C.1.c - By the end of grade 8, students will: Apply the multiplication principle of counting. Concept of combinations (e.g., number of possible delegations of 3 out of 23 students)NJ 4.4.8 C.2 - By the end of grade 8, students will: Explore counting problems involving Venn diagrams with three attributes (e.g., there are 15, 20, and 25 students respectively in the chess club, the debating team, and the engineering society; how many different students belong to the three clubs if there are 6 students in chess and debating, 7 students in chess and engineering, 8 students in debating and engineering, and 2 students in all three?).

NJ 4.4.8 C.3 - By the end of grade 8, students will: Apply techniques of systematic listing, counting, and reasoning in a variety of different contexts.

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NJ 4.4.8 D.1 - By the end of grade 8, students will: Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems.NJ 4.4.8 D.1.a - By the end of grade 8, students will: Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems. Finding the shortest network connecting specified sitesNJ 4.4.8 D.1.b - By the end of grade 8, students will: Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems. Finding a minimal route that includes every street (e.g., for trash pick-up)

NJ 4.4.8 D.1.c - By the end of grade 8, students will: Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems. Finding the shortest route on a map from one site to anotherNJ 4.4.8 D.1.d - By the end of grade 8, students will: Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems. Finding the shortest circuit on a map that makes a tour of specified sites

NJ 4.4.8 D.1.e - By the end of grade 8, students will: Use vertex-edge graphs and algorithmic thinking to represent and find solutions to practical problems. Limitations of computers (e.g., the number of routes for a delivery truck visiting n sites is n!, so finding the shortest circuit by examining all circuits would overwhelm the capacity of any computer, now or in the future, even if n is less than 100)

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NJ 4.5 A - Problem SolvingNJ 4.5 A.1 - By the end of grade 8, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5 A.2 - By the end of grade 8, students will: Solve problems that arise in mathematics and in other contexts.NJ 4.5 A.2.a - By the end of grade 8, students will: Solve problems that arise in mathematics and in other contexts. Open-ended problems

NJ 4.5 A.2.b - By the end of grade 8, students will: Solve problems that arise in mathematics and in other contexts. Non-routine problems

NJ 4.5 A.2.c - By the end of grade 8, students will: Solve problems that arise in mathematics and in other contexts. Problems with multiple solutionsNJ 4.5 A.2.d - By the end of grade 8, students will: Solve problems that arise in mathematics and in other contexts. Problems that can be solved in several waysNJ 4.5 A.3 - By the end of grade 8, students will: Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.NJ 4.5 A.4 - By the end of grade 8, students will: Pose problems of various types and levels of difficulty.

NJ 4.5 A.5 - By the end of grade 8, students will: Monitor their progress and reflect on the process of their problem solving activity.

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NJ 4.5 A.6 - By the end of grade 8, students will: Distinguish relevant from irrelevant information, and identify missing information.NJ 4.5 B - CommunicationNJ 4.5 B.1 - By the end of grade 8, students will: Use communication to organize and clarify their mathematical thinking.NJ 4.5 B.1.a - By the end of grade 8, students will: Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5 B.1.b - By the end of grade 8, students will: Use communication to organize and clarify their mathematical thinking. Discussion, listening, and questioning

NJ 4.5 B.2 - By the end of grade 8, students will: Communicate their mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.NJ 4.5 B.3 - By the end of grade 8, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5 B.4 - By the end of grade 8, students will: Use the language of mathematics to express mathematical ideas precisely.


NJ 4.5 C - ConnectionsNJ 4.5 C.1 - By the end of grade 8, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).NJ 4.5 C.2 - By the end of grade 8, students will: 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).NJ 4.5 C.3 - By the end of grade 8, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.

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NJ 4.5 C.4 - By the end of grade 8, students will: Apply mathematics in practical situations and in other disciplines.NJ 4.5 C.5 - By the end of grade 8, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5 C.6 - By the end of grade 8, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.NJ 4.5 D - ReasoningNJ 4.5 D.1 - By the end of grade 8, students will: Recognize that mathematical facts, procedures, and claims must be justified.NJ 4.5 D.2 - By the end of grade 8, students will: Use reasoning to support their mathematical conclusions and problem solutions.

NJ 4.5 D.3 - By the end of grade 8, students will: Select and use various types of reasoning and methods of proof.NJ 4.5 D.4 - By the end of grade 8, students will: Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.NJ 4.5 D.5 - By the end of grade 8, students will: Make and investigate mathematical conjectures.NJ 4.5 D.5.a - By the end of grade 8, students will: Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjecturesNJ 4.5 D.5.b - By the end of grade 8, students will: Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5 D.6 - By the end of grade 8, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.

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NJ 4.5 E - RepresentationsNJ 4.5 E.1 - By the end of grade 8, students will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5 E.1.a - By the end of grade 8, students will: Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles)

NJ 4.5 E.1.b - By the end of grade 8, students will: Create and use representations to organize, record, and communicate mathematical ideas. Pictorial representations (e.g., diagrams, charts, or tables)

NJ 4.5 E.1.c - By the end of grade 8, students will: Create and use representations to organize, record, and communicate mathematical ideas. Symbolic representations (e.g., a formula)NJ 4.5 E.1.d - By the end of grade 8, students will: Create and use representations to organize, record, and communicate mathematical ideas. Graphical representations (e.g., a line graph)NJ 4.5 E.2 - By the end of grade 8, students will: Select, apply, and translate among mathematical representations to solve problems.NJ 4.5 E.3 - By the end of grade 8, students will: Use representations to model and interpret physical, social, and mathematical phenomena.

NJ 4.5 F - TechnologyNJ 4.5 F.1 - By the end of grade 8, students will: Use technology to gather, analyze, and communicate mathematical information.NJ 4.5 F.2 - By the end of grade 8, students will: Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.

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NJ 4.5 F.3 - By the end of grade 8, students will: Use graphing calculators and computer software to investigate properties of functions and their graphs.NJ 4.5 F.4 - By the end of grade 8, students will: Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

NJ 4.5 F.5 - By the end of grade 8, students will: Use computer software to make and verify conjectures about geometric objects.

NJ 4.5 F.6 - By the end of grade 8, students will: Use computer-based laboratory technology for mathematical applications in the sciences (cf. science standards).

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Agency Tag Set NameNew Jersey, Math, 2002, Grades: 9-12, Core Curriculum Content Standards, State Department of EducationStandard Skill Area Description Skill DescriptionNJ 4.1 - All students will develop number sense and will perform standard numerical operations and estimations on all types of numbers in a variety of ways.NJ 4.1.12 A - Number SenseNJ 4.1.12 A.1 - By the end of grade 12, students will: Extend understanding of the number system to all real numbers.NJ 4.1.12 A.2 - By the end of grade 12, students will: Compare and order rational and irrational numbers.

NJ 4.1.12 A.3 - By the end of grade 12, students will: Develop conjectures and informal proofs of properties of number systems and sets of numbers.NJ 4.1.12 B - Numerical Operations

NJ 4.1.12 B.1 - By the end of grade 12, students will: Extend understanding and use of operations to real numbers and algebraic procedures.NJ 4.1.12 B.2 - By the end of grade 12, students will: Develop, apply, and explain methods for solving problems involving rational and negative exponents.


Skill 37 - N35 - Advanced concepts: Recognize meaning of negative exponents

Skill 38 - N36 - Advanced concepts: Recognize meaning of fractional exponents

NJ 4.1.12 B.3 - By the end of grade 12, students will: Perform operations on matrices.NJ 4.1.12 B.3.a - Perform operations on matrices. Addition and subtraction

NJ 4.1.12 B.3.b - Perform operations on matrices. Scalar multiplication

NJ 4.1.12 B.4 - By the end of grade 12, students will: Understand and apply the laws of exponents to simplify expressions involving numbers raised to powers.

NJ 4.1.12 C - Estimation


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Agency Tag Set NameNew Jersey, Math, 2002, Grades: 9-12, Core Curriculum Content Standards, State Department of EducationStandard Skill Area Description Skill Description


NJ 4.1.12 C.1 - By the end of grade 12, students will: Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.NJ 4.2 - All students will develop spatial sense and the ability to use geometric properties, relationships, and measurement to model, describe and analyze phenomena.NJ 4.2.12 A - Geometric Properties

NJ 4.2.12 A.1 - By the end of grade 12, students will: Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).NJ 4.2.12 A.2 - By the end of grade 12, students will: Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).

NJ 4.2.12 A.3 - By the end of grade 12, students will: Apply the properties of geometric shapes.NJ 4.2.12 A.3.a - Apply the properties of geometric shapes. Parallel lines - transversal, alternate interior angles, corresponding angles

Skill Area 6 - Geometry Skill 24 - G17 - Use properties of parallel lines

NJ 4.2.12 A.3.b - Apply the properties of geometric shapes. TrianglesNJ 4.2.12 A.3.b.a1 - Apply the properties of geometric shapes. Triangles Conditions for congruence

NJ 4.2.12 A.3.b.b2 - Apply the properties of geometric shapes. Triangles Segment joining midpoints of two sides is parallel to and half the length of the third sideNJ 4.2.12 A.3.b.c3 - Apply the properties of geometric shapes. Triangles Triangle Inequality


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NJ 4.2.12 A.3.c - Apply the properties of geometric shapes. Minimal conditions for a shape to be a special quadrilateralNJ 4.2.12 A.3.d - Apply the properties of geometric shapes. Circles - arcs, central and inscribed angles, chords, tangentsNJ 4.2.12 A.3.e - Apply the properties of geometric shapes. Self-similarity

NJ 4.2.12 A.4 - By the end of grade 12, students will: Use reasoning and some form of proof to verify or refute conjectures and theorems.

NJ 4.2.12 A.4.a - Use reasoning and some form of proof to verify or refute conjectures and theorems. Verification or refutation of proposed proofsNJ 4.2.12 A.4.b - Use reasoning and some form of proof to verify or refute conjectures and theorems. Simple proofs involving congruent triangles

NJ 4.2.12 A.4.c - Use reasoning and some form of proof to verify or refute conjectures and theorems. Counterexamples to incorrect conjecturesNJ 4.2.12 B - Transforming ShapesNJ 4.2.12 B.1 - By the end of grade 12, students will: Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.NJ 4.2.12 B.2 - By the end of grade 12, students will: Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.


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NJ 4.2.12 B.3 - By the end of grade 12, students will: Determine whether two or more given shapes can be used to generate a tessellation.

NJ 4.2.12 B.4 - By the end of grade 12, students will: Generate and analyze iterative geometric patterns.

NJ 4.2.12 B.4.a - Generate and analyze iterative geometric patterns. Fractals (e.g., Sierpinski's Triangle)

NJ 4.2.12 B.4.b - Generate and analyze iterative geometric patterns. Patterns in areas and perimeters of self-similar figuresNJ 4.2.12 B.4.c - Generate and analyze iterative geometric patterns. Outcome of extending iterative process indefinitelyNJ 4.2.12 C - Coordinate Geometry

NJ 4.2.12 C.1 - By the end of grade 12, students will: Use coordinate geometry to represent and verify properties of lines.NJ 4.2.12 C.1.a - Use coordinate geometry to represent and verify properties of lines. Distance between two pointsNJ 4.2.12 C.1.b - Use coordinate geometry to represent and verify properties of lines. Midpoint and slope of a line segmentNJ 4.2.12 C.1.c - Use coordinate geometry to represent and verify properties of lines. Finding the intersection of two linesNJ 4.2.12 C.1.d - Use coordinate geometry to represent and verify properties of lines. Lines with the same slope are parallelNJ 4.2.12 C.1.e - Use coordinate geometry to represent and verify properties of lines. Lines that are perpendicular have slopes whose product is -1


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NJ 4.2.12 C.2 - By the end of grade 12, students will: Show position and represent motion in the coordinate plane using vectors.NJ 4.2.12 C.2.a - Show position and represent motion in the coordinate plane using vectors. Addition and subtraction of vectorsNJ 4.2.12 D - Units of Measurement

NJ 4.2.12 D.1 - By the end of grade 12, students will: Understand and use the concept of significant digits.

NJ 4.2.12 D.2 - By the end of grade 12, students will: Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation.

NJ 4.2.12 D.2.a - Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation. Degree of accuracy of a given measurement toolNJ 4.2.12 D.2.b - Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation. Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements

NJ 4.2.12 E - Measuring Geometric ObjectsNJ 4.2.12 E.1 - By the end of grade 12, students will: Use techniques of indirect measurement to represent and solve problems.

NJ 4.2.12 E.1.a - Use techniques of indirect measurement to represent and solve problems. Similar triangles

NJ 4.2.12 E.1.b - Use techniques of indirect measurement to represent and solve problems. Pythagorean theorem

Skill Area 6 - Geometry Skill 30 - G23 - Pythagorean theorem


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NJ 4.2.12 E.1.c - Use techniques of indirect measurement to represent and solve problems. Right triangle trigonometry (sine, cosine, tangent)

NJ 4.2.12 E.2 - By the end of grade 12, students will: Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures.

NJ 4.2.12 E.2.a - Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures. Approximation of area using grids of different sizesNJ 4.2.12 E.2.b - Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures. Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets

NJ 4.2.12 E.2.c - Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures. Estimation of area, perimeter, volume, and surface area

NJ 4.3 - All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes.NJ 4.3.12 A - PatternsNJ 4.3.12 A.1 - By the end of grade 12, students will: Use models and algebraic formulas to represent and analyze sequences and series.

NJ 4.3.12 A.1.a - Use models and algebraic formulas to represent and analyze sequences and series. Explicit formulas for nth terms


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NJ 4.3.12 A.1.b - Use models and algebraic formulas to represent and analyze sequences and series. Sums of finite arithmetic seriesNJ 4.3.12 A.1.c - Use models and algebraic formulas to represent and analyze sequences and series. Sums of finite and infinite geometric series

NJ 4.3.12 A.2 - By the end of grade 12, students will: Develop an informal notion of limit.NJ 4.3.12 A.3 - By the end of grade 12, students will: Use inductive reasoning to form generalizations.

NJ 4.3.12 B - Functions and RelationshipsNJ 4.3.12 B.1 - By the end of grade 12, students will: Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.


Skill 10 - A09 - Graph linear inequalities

NJ 4.3.12 B.2 - By the end of grade 12, students will: Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

NJ 4.3.12 B.2.a - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Slope of a line or curve

NJ 4.3.12 B.2.b - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Domain and rangeNJ 4.3.12 B.2.c - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Intercepts


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NJ 4.3.12 B.2.d - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. ContinuityNJ 4.3.12 B.2.e - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Maximum/minimumNJ 4.3.12 B.2.f - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Estimating roots of equations

NJ 4.3.12 B.2.g - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Intersecting points as solutions of systems of equations

NJ 4.3.12 B.2.h - Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Rates of changeNJ 4.3.12 B.3 - By the end of grade 12, students will: Understand and perform transformations on commonly-used functions.

NJ 4.3.12 B.3.a - Understand and perform transformations on commonly-used functions. Translations, reflections, dilationsNJ 4.3.12 B.3.b - Understand and perform transformations on commonly-used functions. Effects on linear and quadratic graphs of parameter changes in equationsNJ 4.3.12 B.3.c - Understand and perform transformations on commonly-used functions. Using graphing calculators or computers for more complex functions


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NJ 4.3.12 B.4 - By the end of grade 12, students will: Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.

NJ 4.3.12 B.4.a - Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions. Linear vs. non-linearNJ 4.3.12 B.4.b - Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions. SymmetryNJ 4.3.12 B.4.c - Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions. Increasing/decreasing on an interval

NJ 4.3.12 C - ModelingNJ 4.3.12 C.1 - By the end of grade 12, students will: Use functions to model real-world phenomena and solve problems that involve varying quantities.NJ 4.3.12 C.1.a - Use functions to model real-world phenomena and solve problems that involve varying quantities. Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)NJ 4.3.12 C.1.b - Use functions to model real-world phenomena and solve problems that involve varying quantities. Direct and inverse variationNJ 4.3.12 C.1.c - Use functions to model real-world phenomena and solve problems that involve varying quantities. Absolute value


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NJ 4.3.12 C.1.d - Use functions to model real-world phenomena and solve problems that involve varying quantities. Expressions, equations and inequalitiesNJ 4.3.12 C.1.e - Use functions to model real-world phenomena and solve problems that involve varying quantities. Same function can model variety of phenomenaNJ 4.3.12 C.1.f - Use functions to model real-world phenomena and solve problems that involve varying quantities. Growth/decay and change in the natural worldNJ 4.3.12 C.1.g - Use functions to model real-world phenomena and solve problems that involve varying quantities. Applications in mathematics, biology, and economics (including compound interest)

NJ 4.3.12 C.2 - By the end of grade 12, students will: Analyze and describe how a change in an independent variable leads to change in a dependent one.NJ 4.3.12 C.3 - By the end of grade 12, students will: Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).NJ 4.3.12 D - ProceduresNJ 4.3.12 D.1 - By the end of grade 12, students will: Evaluate and simplify expressions.NJ 4.3.12 D.1.a - Evaluate and simplify expressions. Add and subtract polynomials

Skill Area 10 - Algebra Skill 13 - A12 - Polynomials: Addition and subtraction

NJ 4.3.12 D.1.b - Evaluate and simplify expressions. Multiply a polynomial by a monomial or binomial

Skill Area 10 - Algebra Skill 14 - A13 - Polynomials: Multiplication and division

NJ 4.3.12 D.1.c - Evaluate and simplify expressions. Divide a polynomial by a monomialNJ 4.3.12 D.2 - By the end of grade 12, students will: Select and use appropriate methods to solve equations and inequalities.


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NJ 4.3.12 D.2.a - Select and use appropriate methods to solve equations and inequalities. Linear equations - algebraically


NJ 4.3.12 D.2.b - Select and use appropriate methods to solve equations and inequalities. Quadratic equations - factoring (when the coefficient of x² is 1) and using the quadratic formula

Skill Area 10 - Algebra Skill 17 - A16 - Quadratic equations: Solve by factoring

NJ 4.3.12 D.2.c - Select and use appropriate methods to solve equations and inequalities. All types of equations using graphing, computer, and graphing calculator techniquesNJ 4.3.12 D.3 - By the end of grade 12, students will: Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.NJ 4.4 - 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.NJ 4.4.12 A - Data AnalysisNJ 4.4.12 A.1 - By the end of grade 12, students will: Use surveys and sampling techniques to generate data and draw conclusions about large groups.NJ 4.4.12 A.1.a - Use surveys and sampling techniques to generate data and draw conclusions about large groups. Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)

NJ 4.4.12 A.2 - By the end of grade 12, students will: Evaluate the use of data in real-world contexts.


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NJ 4.4.12 A.2.a - Evaluate the use of data in real-world contexts. Accuracy and reasonableness of conclusions drawnNJ 4.4.12 A.2.b - Evaluate the use of data in real-world contexts. Bias in conclusions drawn (e.g., influence of how data is displayed)NJ 4.4.12 A.2.c - Evaluate the use of data in real-world contexts. Statistical claims based on sampling

NJ 4.4.12 A.3 - By the end of grade 12, students will: Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome.NJ 4.4.12 A.4 - By the end of grade 12, students will: Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data.NJ 4.4.12 A.5 - By the end of grade 12, students will: Analyze data using technology, and use statistical terminology to describe conclusions.

NJ 4.4.12 A.5.a - Analyze data using technology, and use statistical terminology to describe conclusions. Measures of dispersion: variance, standard deviation, outliers

NJ 4.4.12 A.5.b - Analyze data using technology, and use statistical terminology to describe conclusions. Correlation coefficientNJ 4.4.12 A.5.c - Analyze data using technology, and use statistical terminology to describe conclusions. Normal distribution (e.g., approximately 95% of the sample lies between two standard deviations on either side of the mean)

NJ 4.4.12 B - Probability


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NJ 4.4.12 B.1 - By the end of grade 12, students will: Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair.

NJ 4.4.12 B.2 - By the end of grade 12, students will: Use concepts and formulas of area to calculate geometric probabilities.NJ 4.4.12 B.3 - By the end of grade 12, students will: Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.NJ 4.4.12 B.4 - By the end of grade 12, students will: Determine probabilities in complex situations.NJ 4.4.12 B.4.a - Determine probabilities in complex situations. Conditional eventsNJ 4.4.12 B.4.b - Determine probabilities in complex situations. Complementary eventsNJ 4.4.12 B.4.c - Determine probabilities in complex situations. Dependent and independent events


Skill 12 - S12 - Probability: Joint

NJ 4.4.12 B.5 - By the end of grade 12, students will: Estimate probabilities and make predictions based on experimental and theoretical probabilities.NJ 4.4.12 B.6 - By the end of grade 12, students will: Understand and use the "law of large numbers" (that experimental results tend to approach theoretical probabilities after a large number of trials).

NJ 4.4.12 C - Discrete Mathematics-Systematic Listing and Counting


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NJ 4.4.12 C.1 - By the end of grade 12, students will: Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).

NJ 4.4.12 C.2 - By the end of grade 12, students will: Apply the multiplication rule of counting in complex situations, recognize the difference between situations with replacement and without replacement, and recognize the difference between ordered and unordered counting situations.NJ 4.4.12 C.3 - By the end of grade 12, students will: Justify solutions to counting problems.NJ 4.4.12 C.4 - By the end of grade 12, students will: Recognize and explain relationships involving combinations and Pascal's Triangle, and apply those methods to situations involving probability.NJ 4.4.12 D - Discrete Mathematics-Vertex-Edge Graphs and Algorithms

NJ 4.4.12 D.1 - By the end of grade 12, students will: Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems.NJ 4.4.12 D.1.a - Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems. Circuits that include every edge in a graphNJ 4.4.12 D.1.b - Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems. Circuits that include every vertex in a graph


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NJ 4.4.12 D.1.c - Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems. Scheduling problems (e.g., when project meetings should be scheduled to avoid conflicts) using graph coloringNJ 4.4.12 D.1.d - Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems. Applications to science (e.g., who-eats-whom graphs, genetic trees, molecular structures)

NJ 4.4.12 D.2 - By the end of grade 12, students will: Explore strategies for making fair decisions.

NJ 4.4.12 D.2.a - Explore strategies for making fair decisions. Combining individual preferences into a group decision (e.g., determining winner of an election or selection process)

NJ 4.4.12 D.2.b - Explore strategies for making fair decisions. Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment)


NJ 4.5.12 A - Problem SolvingNJ 4.5.12 A.1 - By the end of grade 12, students will: Learn mathematics through problem solving, inquiry, and discovery.NJ 4.5.12 A.2 - By the end of grade 12, students will: Solve problems that arise in mathematics and in other contexts.


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NJ 4.5.12 A.2.a - Solve problems that arise in mathematics and in other contexts. Open-ended problems

NJ 4.5.12 A.2.b - Solve problems that arise in mathematics and in other contexts. Non-routine problems

NJ 4.5.12 A.2.c - Solve problems that arise in mathematics and in other contexts. Problems with multiple solutionsNJ 4.5.12 A.2.d - Solve problems that arise in mathematics and in other contexts. Problems that can be solved in several waysNJ 4.5.12 A.3 - By the end of grade 12, students will: Select and apply a variety of appropriate problem-solving strategies (e.g., "try a simpler problem" or "make a diagram") to solve problems.NJ 4.5.12 A.4 - By the end of grade 12, students will: Pose problems of various types and levels of difficulty.

NJ 4.5.12 A.5 - By the end of grade 12, students will: Monitor their progress and reflect on the process of their problem solving activity.

NJ 4.5.12 B - CommunicationNJ 4.5.12 B.1 - By the end of grade 12, students will: Use communication to organize and clarify their mathematical thinking.NJ 4.5.12 B.1.a - Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5.12 B.1.b - Use communication to organize and clarify their mathematical thinking. Discussion, listening, and questioning

NJ 4.5.12 B.2 - By the end of grade 12, students will: Communicate mathematical thinking coherently and clearly to peers, teachers, and others, both orally and in writing.


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NJ 4.5.12 B.3 - By the end of grade 12, students will: Analyze and evaluate the mathematical thinking and strategies of others.NJ 4.5.12 B.4 - By the end of grade 12, students will: Use the language of mathematics to express mathematical ideas precisely.

NJ 4.5.12 C - ConnectionsNJ 4.5.12 C.1 - By the end of grade 12, students will: Recognize recurring themes across mathematical domains (e.g., patterns in number, algebra, and geometry).

NJ 4.5.12 C.2 - By the end of grade 12, students will: 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).

NJ 4.5.12 C.3 - By the end of grade 12, students will: Recognize that mathematics is used in a variety of contexts outside of mathematics.

NJ 4.5.12 C.4 - By the end of grade 12, students will: Apply mathematics in practical situations and in other disciplines.NJ 4.5.12 C.5 - By the end of grade 12, students will: Trace the development of mathematical concepts over time and across cultures (cf. world languages and social studies standards).NJ 4.5.12 C.6 - By the end of grade 12, students will: Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

NJ 4.5.12 D - ReasoningNJ 4.5.12 D.1 - By the end of grade 12, students will: Recognize that mathematical facts, procedures, and claims must be justified.


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NJ 4.5.12 D.2 - By the end of grade 12, students will: Use reasoning to support their mathematical conclusions and problem solutions.

NJ 4.5.12 D.3 - By the end of grade 12, students will: Select and use various types of reasoning and methods of proof.NJ 4.5.12 D.4 - By the end of grade 12, students will: Rely on reasoning, rather than answer keys, teachers, or peers, to check the correctness of their problem solutions.

NJ 4.5.12 D.5 - By the end of grade 12, students will: Make and investigate mathematical conjectures.

NJ 4.5.12 D.5.a - Make and investigate mathematical conjectures. Counterexamples as a means of disproving conjecturesNJ 4.5.12 D.5.b - Make and investigate mathematical conjectures. Verifying conjectures using informal reasoning or proofs.NJ 4.5.12 D.6 - By the end of grade 12, students will: Evaluate examples of mathematical reasoning and determine whether they are valid.

NJ 4.5.12 E - RepresentationsNJ 4.5.12 E.1 - By the end of grade 12, students will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5.12 E.1.a - Create and use representations to organize, record, and communicate mathematical ideas. Concrete representations (e.g., base-ten blocks or algebra tiles)

NJ 4.5.12 E.1.b - Create and use representations to organize, record, and communicate mathematical ideas. Pictorial representations (e.g., diagrams, charts, or tables)


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NJ 4.5.12 E.1.c - Create and use representations to organize, record, and communicate mathematical ideas. Symbolic representations (e.g., a formula)NJ 4.5.12 E.1.d - Create and use representations to organize, record, and communicate mathematical ideas. Graphical representations (e.g., a line graph)NJ 4.5.12 E.2 - By the end of grade 12, students will: Select, apply, and translate among mathematical representations to solve problems.

NJ 4.5.12 E.3 - By the end of grade 12, students will: Use representations to model and interpret physical, social, and mathematical phenomena.NJ 4.5.12 F - TechnologyNJ 4.5.12 F.1 - By the end of grade 12, students will: Use technology to gather, analyze, and communicate mathematical information.

NJ 4.5.12 F.2 - By the end of grade 12, students will: Use computer spreadsheets, software, and graphing utilities to organize and display quantitative information.

NJ 4.5.12 F.3 - By the end of grade 12, students will: Use graphing calculators and computer software to investigate properties of functions and their graphs.NJ 4.5.12 F.4 - By the end of grade 12, students will: Use calculators as problem-solving tools (e.g., to explore patterns, to validate solutions).

NJ 4.5.12 F.5 - By the end of grade 12, students will: Use computer software to make and verify conjectures about geometric objects.


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NJ 4.5.12 F.6 - By the end of grade 12, students will: Use computer-based laboratory technology for mathematical applications in the sciences.


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new jersey standards alignment grades two through...

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