new jersey standards alignment grades two through sixdoc.renlearn.com/kmnet/r004309720gj0ab8.pdf ·...

New Jersey Standards Alignment Grades Two through Six

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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 ....................................................20

Grade 4 ....................................................37

Grade 5 ....................................................62

Grade 6 ....................................................77

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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 Domain Description Skillset 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

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)NJ 4.1.2 A.2 - By the end of Grade 2, students will: Demonstrate an understanding of whole number place value concepts.NJ 4.1.2 A.3 - By the end of Grade 2, students will: Understand that numbers have a variety of uses.NJ 4.1.2 A.4 - By the end of Grade 2, students will: Count and perform simple computations with coins.

Standards List with Aligned Product SkillsProduct NameMathFacts in a Flash

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Standard Domain Description Skillset Description


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.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.

Domain 1 - Overall Product (OP) (student focus)

Skillset 1 - OP01 Developing computational fluency with basic math facts

Skillset 3 - OP03 Solving math facts in horizontal and vertical forms

Skillset 6 - OP06 Memorizing math factsSkillset 7 - OP07 Developing automaticity with addition, subtraction, multiplication, and division facts through repeated practice

Domain 3 - Math Skills - Addition (MS)

Skillset 1 - MS01 Addition of 0,1 (Adding 0 or 1 to numbers 0 - 10)Skillset 2 - MS02 Addition of 2,3 (Adding 2 or 3 to numbers 0 - 10)Skillset 3 - MS03 Addition of 4,5 (Adding 4 or 5 to numbers 0 - 10)

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Skillset 4 - MS04 Alternate Forms: Addition of 0 to 5 (Adding 0 - 5 to numbers 0 - 10 with the unknown in the operand or answer position)

Skillset 5 - MS05 Addition of 6,7 (Adding 6 or 7 to numbers 0 - 10)Skillset 6 - MS06 Addition of 8,9 (Adding 8 or 9 to numbers 0 - 10)Skillset 7 - MS07 Addition of 10 (Adding 10 to numbers 0 - 10)Skillset 8 - MS08 Alternate Forms: Addition of 6 to 10 (Adding 6 - 10 to numbers 0 - 10 with the unknown in the operand or answer position)

Domain 4 - Math Skills - Subtraction (MS)

Skillset 1 - MS09 Subtraction of 0,1 (Subtracting 0 or 1 from numbers 0 - 11)Skillset 2 - MS10 Subtraction of 2,3 (Subtracting 2 or 3 from numbers 2 - 13)Skillset 3 - MS11 Subtraction of 4,5 (Subtracting 4 or 5 from numbers 4 - 15)Skillset 4 - MS12 Alternate Forms: Subtraction of 0 to 5 (Subtracting 0 - 5 from numbers 0 - 15 with the unknown in the operand or answer position)Skillset 5 - MS13 Subtraction of 6,7 (Subtracting 6 or 7 from numbers 6 - 17)Skillset 6 - MS14 Subtraction of 8,9 (Subtracting 8 or 9 from numbers 8 - 19)Skillset 7 - MS15 Subtraction of 10 (Subtracting 10 from numbers 10 - 20)Skillset 8 - MS16 Alternate Forms: Subtraction of 6 to 10 (Subtracting 6 - 10 from numbers 6 - 20 with the unknown in the operand or answer position)

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:

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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-paperNJ 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 mathNJ 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.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 numbersNJ 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 numbersNJ 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.


Skillset 4 - OP04 Supplying a missing operand in a math facts sentence

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Skillset 4 - MS12 Alternate Forms: Subtraction of 0 to 5 (Subtracting 0 - 5 from numbers 0 - 15 with the unknown in the operand or answer position)Skillset 8 - MS16 Alternate Forms: Subtraction of 6 to 10 (Subtracting 6 - 10 from numbers 6 - 20 with the unknown in the operand or answer position)

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).

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.

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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

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

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.

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

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

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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

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)

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

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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, kilogramNJ 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

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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

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 - PatternsNJ 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

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 patterns

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NJ 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.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)




Skillset 7 - OP07 Developing automaticity with addition, subtraction, multiplication, and division facts through repeated practice


Skillset 1 - MS01 Addition of 0,1 (Adding 0 or 1 to numbers 0 - 10)Skillset 2 - MS02 Addition of 2,3 (Adding 2 or 3 to numbers 0 - 10)Skillset 3 - MS03 Addition of 4,5 (Adding 4 or 5 to numbers 0 - 10)Skillset 4 - MS04 Alternate Forms: Addition of 0 to 5 (Adding 0 - 5 to numbers 0 - 10 with the unknown in the operand or answer position)

Skillset 5 - MS05 Addition of 6,7 (Adding 6 or 7 to numbers 0 - 10)

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Skillset 6 - MS06 Addition of 8,9 (Adding 8 or 9 to numbers 0 - 10)Skillset 7 - MS07 Addition of 10 (Adding 10 to numbers 0 - 10)Skillset 8 - MS08 Alternate Forms: Addition of 6 to 10 (Adding 6 - 10 to numbers 0 - 10 with the unknown in the operand or answer position)


Skillset 1 - MS09 Subtraction of 0,1 (Subtracting 0 or 1 from numbers 0 - 11)Skillset 2 - MS10 Subtraction of 2,3 (Subtracting 2 or 3 from numbers 2 - 13)Skillset 3 - MS11 Subtraction of 4,5 (Subtracting 4 or 5 from numbers 4 - 15)Skillset 5 - MS13 Subtraction of 6,7 (Subtracting 6 or 7 from numbers 6 - 17)Skillset 6 - MS14 Subtraction of 8,9 (Subtracting 8 or 9 from numbers 8 - 19)Skillset 7 - MS15 Subtraction of 10 (Subtracting 10 from numbers 10 - 20)Skillset 8 - MS16 Alternate Forms: Subtraction of 6 to 10 (Subtracting 6 - 10 from numbers 6 - 20 with the unknown in the operand or answer position)

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)






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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)

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

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

NJ 4.4.2 B.2 - By the end of Grade 2, students will: Provide probability of specific outcomes.

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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

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

NJ 4.1.3 A.5 - By the end of Grade 3, students will: Understand the various uses of numbers.


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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)

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

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").

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:

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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-paperNJ 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 mathNJ 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 numbersNJ 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 numbersNJ 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

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 (¢)

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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.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.

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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 sizesNJ 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.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

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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, endpointNJ 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 ShapesNJ 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, kilometer

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NJ 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

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

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.

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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,...)

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).





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Skillset 7 - OP07 Developing automaticity with addition, subtraction, multiplication, and division facts through repeated practice


Skillset 1 - MS01 Addition of 0,1 (Adding 0 or 1 to numbers 0 - 10)Skillset 2 - MS02 Addition of 2,3 (Adding 2 or 3 to numbers 0 - 10)Skillset 3 - MS03 Addition of 4,5 (Adding 4 or 5 to numbers 0 - 10)Skillset 4 - MS04 Alternate Forms: Addition of 0 to 5 (Adding 0 - 5 to numbers 0 - 10 with the unknown in the operand or answer position)

Skillset 5 - MS05 Addition of 6,7 (Adding 6 or 7 to numbers 0 - 10)Skillset 6 - MS06 Addition of 8,9 (Adding 8 or 9 to numbers 0 - 10)Skillset 7 - MS07 Addition of 10 (Adding 10 to numbers 0 - 10)Skillset 8 - MS08 Alternate Forms: Addition of 6 to 10 (Adding 6 - 10 to numbers 0 - 10 with the unknown in the operand or answer position)


Skillset 1 - MS09 Subtraction of 0,1 (Subtracting 0 or 1 from numbers 0 - 11)Skillset 2 - MS10 Subtraction of 2,3 (Subtracting 2 or 3 from numbers 2 - 13)Skillset 3 - MS11 Subtraction of 4,5 (Subtracting 4 or 5 from numbers 4 - 15)Skillset 4 - MS12 Alternate Forms: Subtraction of 0 to 5 (Subtracting 0 - 5 from numbers 0 - 15 with the unknown in the operand or answer position)Skillset 5 - MS13 Subtraction of 6,7 (Subtracting 6 or 7 from numbers 6 - 17)Skillset 6 - MS14 Subtraction of 8,9 (Subtracting 8 or 9 from numbers 8 - 19)Skillset 7 - MS15 Subtraction of 10 (Subtracting 10 from numbers 10 - 20)

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Skillset 8 - MS16 Alternate Forms: Subtraction of 6 to 10 (Subtracting 6 - 10 from numbers 6 - 20 with the unknown in the operand or answer position)

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 zeroNJ 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.

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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, tableNJ 4.4.3 B - ProbabilityNJ 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).

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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 CountingNJ 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, charts

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NJ 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.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

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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.

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.

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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).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.

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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 conjecturesNJ 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)

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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.

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 Domain Description Skillset 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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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.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.

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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.




Skillset 6 - OP06 Memorizing math factsSkillset 7 - OP07 Developing automaticity with addition, subtraction, multiplication, and division facts through repeated practice

Domain 5 - Math Skills - Multiplication (MS)

Skillset 1 - MS17 Multiplication by 0,1 (Multiplying 0 or 1 by numbers 0 - 10)Skillset 2 - MS18 Multiplication by 2,3 (Multiplying 2 or 3 by numbers 0 - 10)Skillset 3 - MS19 Multiplication by 4,5 (Multiplying 4 or 5 by numbers 0 - 10)

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Skillset 4 - MS20 Alternate Forms: Multiplication by 0 to 5 (Multiplying 0 - 5 by numbers 0 - 10 with the unknown in the operand or answer position)Skillset 5 - MS21 Multiplication by 6,7 (Multiplying 6 or 7 by numbers 0 - 10)Skillset 6 - MS22 Multiplication by 8,9 (Multiplying 8 - 9 by numbers 0 - 10)Skillset 7 - MS23 Multiplication by 10 (Multiplying 10 by numbers 0 - 10)

Skillset 8 - MS24 Alternate Forms: Multiplication by 6 to 10 (Multiplying 6 - 10 by numbers 0 - 10 with the unknown in the operand or answer position)Skillset 9 - MS25 Multiplication by 11,12 (Multiplying 11 or 12 by numbers 0 - 12)Skillset 10 - MS26 Alternate Forms: Multiplication by 11 to 12 (Multiplying 11 or 12 by numbers 0 - 12 with the unknown in the operand or answer position)

Domain 7 - Math Skills - Division (MS)

Skillset 1 - MS27 Division by 1,2 (Dividing numbers 0 - 20 by 1 or 2)Skillset 2 - MS28 Division by 3,4 (Dividing numbers 0 - 40 by 3 or 4)Skillset 3 - MS29 Division by 5,6 (Dividing numbers 0 - 60 by 5 or 6)Skillset 4 - MS30 Alternate Forms: Division by 1 - 6 (Dividing numbers 0 - 60 by 1 - 6 with the unknown in the operand or answer position)

Skillset 5 - MS31 Division by 7,8 (Dividing numbers 0 - 80 by 7 or 8)Skillset 6 - MS32 Division by 9,10 (Dividing numbers 0 - 100 by 9 or 10)Skillset 7 - MS33 Alternate Forms: Division by 7 - 10 (Dividing numbers 0 - 100 by 7 - 10 with the unknown in the operand or answer position)

Skillset 8 - MS34 Division by 11,12 (Dividing numbers 0 - 144 by 11 or 12)

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Skillset 9 - MS35 Alternate Forms: Division by 11 to 12 (Dividing numbers 0 - 144 by 11 or 12 with unknown in operand position)

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

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 mathNJ 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 numbersNJ 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 numbersNJ 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

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






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Skillset 4 - MS20 Alternate Forms: Multiplication by 0 to 5 (Multiplying 0 - 5 by numbers 0 - 10 with the unknown in the operand or answer position)Skillset 8 - MS24 Alternate Forms: Multiplication by 6 to 10 (Multiplying 6 - 10 by numbers 0 - 10 with the unknown in the operand or answer position)Skillset 10 - MS26 Alternate Forms: Multiplication by 11 to 12 (Multiplying 11 or 12 by numbers 0 - 12 with the unknown in the operand or answer position)


Skillset 4 - MS30 Alternate Forms: Division by 1 - 6 (Dividing numbers 0 - 60 by 1 - 6 with the unknown in the operand or answer position)



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.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.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.

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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

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.

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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, endpointNJ 4.2.4 A.4.b - By the end of grade 4, students will: Understand and apply concepts involving lines, angles, and circles. Parallel, perpendicularNJ 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, obtuseNJ 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, centerNJ 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 Measurement

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NJ 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.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).

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NJ 4.2.4 D.4 - By the end of grade 4, students will: Incorporate estimation in measurement activities (e.g., estimate before measuring).

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 ObjectsNJ 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 infinitely

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NJ 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,...)

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)

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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)

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).




Skillset 4 - OP04 Supplying a missing operand in a math facts sentenceSkillset 7 - OP07 Developing automaticity with addition, subtraction, multiplication, and division facts through repeated practice




Skillset 8 - MS16 Alternate Forms: Subtraction of 6 to 10 (Subtracting 6 - 10 from numbers 6 - 20 with the unknown in the operand or answer position)


Skillset 1 - MS17 Multiplication by 0,1 (Multiplying 0 or 1 by numbers 0 - 10)Skillset 2 - MS18 Multiplication by 2,3 (Multiplying 2 or 3 by numbers 0 - 10)Skillset 3 - MS19 Multiplication by 4,5 (Multiplying 4 or 5 by numbers 0 - 10)Skillset 4 - MS20 Alternate Forms: Multiplication by 0 to 5 (Multiplying 0 - 5 by numbers 0 - 10 with the unknown in the operand or answer position)

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Skillset 5 - MS21 Multiplication by 6,7 (Multiplying 6 or 7 by numbers 0 - 10)Skillset 6 - MS22 Multiplication by 8,9 (Multiplying 8 - 9 by numbers 0 - 10)Skillset 7 - MS23 Multiplication by 10 (Multiplying 10 by numbers 0 - 10)

Skillset 8 - MS24 Alternate Forms: Multiplication by 6 to 10 (Multiplying 6 - 10 by numbers 0 - 10 with the unknown in the operand or answer position)Skillset 9 - MS25 Multiplication by 11,12 (Multiplying 11 or 12 by numbers 0 - 12)Skillset 10 - MS26 Alternate Forms: Multiplication by 11 to 12 (Multiplying 11 or 12 by numbers 0 - 12 with the unknown in the operand or answer position)


Skillset 1 - MS27 Division by 1,2 (Dividing numbers 0 - 20 by 1 or 2)Skillset 2 - MS28 Division by 3,4 (Dividing numbers 0 - 40 by 3 or 4)Skillset 3 - MS29 Division by 5,6 (Dividing numbers 0 - 60 by 5 or 6)Skillset 4 - MS30 Alternate Forms: Division by 1 - 6 (Dividing numbers 0 - 60 by 1 - 6 with the unknown in the operand or answer position)

Skillset 5 - MS31 Division by 7,8 (Dividing numbers 0 - 80 by 7 or 8)Skillset 6 - MS32 Division by 9,10 (Dividing numbers 0 - 100 by 9 or 10)Skillset 7 - MS33 Alternate Forms: Division by 7 - 10 (Dividing numbers 0 - 100 by 7 - 10 with the unknown in the operand or answer position)

Skillset 8 - MS34 Division by 11,12 (Dividing numbers 0 - 144 by 11 or 12)Skillset 9 - MS35 Alternate Forms: Division by 11 to 12 (Dividing numbers 0 - 144 by 11 or 12 with unknown in operand position)

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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 (=, <, >)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 Analysis

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NJ 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

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)

NJ 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.

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

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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.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)

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)

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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 diagramsNJ 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)

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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.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.

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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 - CommunicationNJ 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 writing

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NJ 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.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.

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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 - RepresentationsNJ 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)

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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.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).

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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 Domain Description Skillset 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.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.


Skillset 8 - OP08 Developing automaticity converting decimals to fractions and fractions to decimals through repeated practice


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Domain 8 - Math Skills - Fractions Decimal Conversion (MS)

Skillset 1 - MS37 Fractions to Decimals (Converting fractions with numerators 1-9 and denominator 10 or 100; unit fractions (1/n) with denominators of 2, 3, 4, 5, and 3/4, 2/5, 3/5, 4/5)Skillset 2 - MS38 Decimals to Fractions (Converting decimals to fractions with values 0.5, 0.333..., 0.25, 0.75, 0.1, 0.3, 0.7, 0.9., 0.2, 0.4, 0.6, and 0.8)

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, multiplesNJ 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-paperNJ 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: Calculator

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NJ 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.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.




Skillset 8 - MS24 Alternate Forms: Multiplication by 6 to 10 (Multiplying 6 - 10 by numbers 0 - 10 with the unknown in the operand or answer position)Skillset 10 - MS26 Alternate Forms: Multiplication by 11 to 12 (Multiplying 11 or 12 by numbers 0 - 12 with the unknown in the operand or answer position)





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.

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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.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 segmentNJ 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

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°

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

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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 GeometryNJ 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.

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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).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.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

NJ 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

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 - Patterns

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NJ 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 quantitiesNJ 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 time

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NJ 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.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.

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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

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 meanNJ 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 eventNJ 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 0NJ 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

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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

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.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.

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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

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.

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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.

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).

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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.

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 conjectures

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NJ 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)

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.

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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.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.

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.


Skillset 8 - OP08 Developing automaticity converting decimals to fractions and fractions to decimals through repeated practice

Domain 8 - Math Skills - Fractions Decimal Conversion (MS)

Skillset 1 - MS37 Fractions to Decimals (Converting fractions with numerators 1-9 and denominator 10 or 100; unit fractions (1/n) with denominators of 2, 3, 4, 5, and 3/4, 2/5, 3/5, 4/5)Skillset 2 - MS38 Decimals to Fractions (Converting decimals to fractions with values 0.5, 0.333..., 0.25, 0.75, 0.1, 0.3, 0.7, 0.9., 0.2, 0.4, 0.6, and 0.8)

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, multiplesNJ 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

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 Operations

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NJ 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

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

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.


Skillset 9 - OP09 Developing automaticity converting squares to numbers through repeated practice

Domain 6 - Math Skills - Squares (MS)

Skillset 1 - MS36 Squares to 15,20 (Identifying Squares of numbers 0 - 15, and 20)

NJ 4.1.6 B.6 - By the end of Grade 6, students will: Check the reasonableness of results of computations.

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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.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

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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

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°

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).

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NJ 4.2.6 A.5 - By the end of Grade 6, students will: Compare properties of cylinders, prisms, cones, pyramids, and spheres.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.

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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).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 trapezoidNJ 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 circleNJ 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.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.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).

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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 4.3.6 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 RelationshipsNJ 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.

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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/inequalitiesNJ 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

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 numbers

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NJ 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.

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 surveys

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NJ 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 meanNJ 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

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 eventNJ 4.4.6 B.1.b - By the end of Grade 6, students will: Determine probabilities of events. Multiplication rule for probabilities

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NJ 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

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.

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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.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)

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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.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 travel

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NJ 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

NJ 4.5 A.2.b - By the end of Grade 6, 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 6, 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 6, 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 6, 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 6, students will: Pose problems of various types and levels of difficulty.

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NJ 4.5 A.5 - By the end of Grade 6, 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 6, 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 6, students will: Use communication to organize and clarify their mathematical thinking.NJ 4.5 B.1.a - By the end of Grade 6, students will: Use communication to organize and clarify their mathematical thinking. Reading and writingNJ 4.5 B.1.b - By the end of Grade 6, 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 6, 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 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).

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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 - ReasoningNJ 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.3 - By the end of Grade 6, students will: Select and use various types of reasoning and methods of proof.NJ 4.5 D.4 - By the end of Grade 6, 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 6, students will: Make and investigate mathematical conjectures.

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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.

NJ 4.5 E - RepresentationsNJ 4.5 E.1 - By the end of Grade 6, students will: Create and use representations to organize, record, and communicate mathematical ideas.NJ 4.5 E.1.a - By the end of Grade 6, 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 6, 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 6, 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 6, 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 6, students will: Select, apply, and translate among mathematical representations to solve problems.

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NJ 4.5 E.3 - By the end of Grade 6, 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 6, students will: Use technology to gather, analyze, and communicate mathematical information.NJ 4.5 F.2 - By the end of Grade 6, 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 6, 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 6, 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 6, students will: Use computer software to make and verify conjectures about geometric objects.

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

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new jersey standards alignment grades two through sixdoc.renlearn.com/kmnet/r004309720gj0ab8.pdf ·...

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