new york city state standards alignment kindergarten...

New York City State Standards Alignment Kindergarten through Grade Ten

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New York City State 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.

Introduction Letter .....................................................iii

Kindergarten - Grade 4 ................................................. 1

Grades 5 - 8 ....................................................81

Grades 9 - 10 ..................................................157

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 standard 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 that 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 (877) 204-5058. Sincerely, Renaissance Sales and Funding Staff

Helping educators improve schools through technology, professional development and research

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Agency Tag Set NameNew York City, Math, Adopted 1998, Revised 2001, Grade K-4, New Standards Performance Standards--Mathematics, produced by City of New York Department of Education

Standard Topic Description Objective DescriptionNYC M1 - ARITHMETIC AND NUMBER CONCEPTS/NUMBER AND OPERATION CONCEPTSNYC M1-1 - The student produces evidence that demonstrates understanding of arithmetic and number concepts; that is, the student:

NYC M1.a - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is:NYC M1.a.i - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: adds, i.e., joins things together, increases.

Topic 2 - Whole Numbers: Addition

Obj. 10 - Add using pictures to 12

Obj. 11 - Addition sentences to 12Obj. 12 - Add using number line to 12

Obj. 13 - Addition facts to 12, vertical

Obj. 14 - Addition facts to 12, horizontal

Topic 5 - Whole Numbers: Addition and Subtraction

Obj. 37 - Addition facts from 11 to 18, verticalObj. 38 - Addition facts from 11 to 18, horizontalObj. 43 - Add three 1-digit #sObj. 44 - Adding in any orderObj. 45 - Add 2-digit #s, no regroup

NYC M1.a.ii - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: subtracts, i.e., takes away, compares, finds the difference.

Topic 3 - Whole Numbers: Subtraction

Obj. 16 - Subtract using pictures from 12

Obj. 17 - Subtraction sentences from 12Obj. 18 - Finish subtraction sentences from 12Obj. 19 - Subtract using number line from 12Obj. 20 - Subtraction facts from 12, verticalObj. 21 - Subtraction facts from 12, horizontal

Standards List with Aligned Product SkillsProduct NameAM NY Custom Grade 1

Accelerated Math Kindergarten - Grade 4

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Standard Topic Description Objective Description



Obj. 39 - Subtraction facts from 11 to 18, verticalObj. 40 - Subtraction facts from 11 to 18, horizontalObj. 46 - Subtract 2-digit #s, no regroup

NYC M1.a.iii - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: multiplies, i.e., uses repeated addition, counts by multiples, combines things that come in groups, makes arrays, uses area models, computes simple scales, uses simple rates.

Topic 11 - Exploring Multiplication and Division of Whole Numbers

Obj. 99 - Make equal groups of objects

Obj. 100 - Skip-counting & repeated additionObj. 102 - Use multiplication patterns to complete tables

NYC M1.a.iv - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: divides, i.e., puts things into groups, shares equally; calculates simple rates.


Obj. 103 - Divide into equal groups

Obj. 104 - Divide into equal groups with left over

NYC M1.a.v - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide.


Obj. 15 - WP: Add, 1-step


Obj. 22 - WP: Subtract, 1-step


Obj. 41 - Determine add & subt in # sentenceObj. 47 - WP: Write number sentence, add & subtract (to 18)Obj. 48 - WP: Add & subtract, 1-step

Obj. 49 - WP: Add & subtract, 2-step

Topic 9 - Graphs, Data, and Probability

Obj. 80 - Add with graphs & tables (to 12)Obj. 81 - Subtract with graphs & tables (to 12)


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Obj. 82 - Add & subtract with graphs (to 18)


Obj. 101 - WP: Multiplication strategies

Obj. 105 - WP: Division strategiesNYC M1.a.vi - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.

NYC M1.a.vii - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: computes answers mentally, e.g., 27 + 45, 30 x 4.


Obj. 13 - Addition facts to 12, vertical

Obj. 14 - Addition facts to 12, horizontal


Obj. 20 - Subtraction facts from 12, verticalObj. 21 - Subtraction facts from 12, horizontal


Obj. 37 - Addition facts from 11 to 18, verticalObj. 38 - Addition facts from 11 to 18, horizontalObj. 39 - Subtraction facts from 11 to 18, verticalObj. 40 - Subtraction facts from 11 to 18, horizontalObj. 43 - Add three 1-digit #s

NYC M1.a.viii - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: uses simple concepts of negative numbers, e.g., on a number line, in counting, in temperature, "owing."NYC M1.b - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is:


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NYC M1.b.i - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000.NYC M1.b.ii - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.

NYC M1.c - Estimates, approximates, rounds off, uses landmark numbers, or uses exact numbers, as appropriate, in calculations.

Topic 6 - Money Obj. 57 - Estimate coin value

NYC M1.d - Describes and compares quantities by using concrete and real world models of simple fractions; that is:NYC M1.d.i - Describes and compares quantities by using concrete and real world models of simple fractions; that is: finds simple parts of wholes.

Topic 10 - Geometry and Fractions

Obj. 95 - Equal parts

Obj. 96 - 1/2, 1/3, & 1/4 as parts of a whole

NYC M1.d.ii - Describes and compares quantities by using concrete and real world models of simple fractions; that is: recognizes simple fractions as instructions to divide, e.g., 1/4 of something is the same as dividing something by 4.


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NYC M1.d.iii - Describes and compares quantities by using concrete and real world models of simple fractions; that is: recognizes the place of fractions on number lines, e.g., in measurement.

NYC M1.d.iv - Describes and compares quantities by using concrete and real world models of simple fractions; that is: uses drawings, diagrams, or models to show what the numerator and denominator mean, including when adding like fractions, e.g., 1/8 + 5/8, or when showing that 3/4 is more than 3/8.


Obj. 97 - Equivalent fractions

NYC M1.d.v - Describes and compares quantities by using concrete and real world models of simple fractions; that is: uses beginning proportional reasoning and simple ratios, e.g., "about half of the people."NYC M1.e - Describes and compares quantities by using simple decimals; that is:NYC M1.e.i - Describes and compares quantities by using simple decimals; that is: adds, subtracts, multiplies, and divides money amounts.NYC M1.e.ii - Describes and compares quantities by using simple decimals; that is: recognizes relationships among simple fractions, decimals, and percents, i.e., that 1/2 is the same as 0.5, and 1/2 is the same as 50%, with concrete materials, diagrams, and in real world situations, e.g., when discovering the chance of a coin landing on heads or tails.

NYC M1.f - Describes and compares quantities by using whole numbers up to 10,000; that is:


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NYC M1.f.i - Describes and compares quantities by using whole numbers up to 10,000; that is: connects ideas of quantities to the real world, e.g., how many people fit in the school’s cafeteria; how far away is a kilometer.NYC M1.f.ii - Describes and compares quantities by using whole numbers up to 10,000; that is: finds, identifies, and sorts numbers by their properties, e.g., odd, even, multiple, square.

Topic 4 - Whole Number Concepts II

Obj. 34 - Odd & even numbers

NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:NYC M2.a - Gives and responds to directions about location, e.g., by using words such as "in front of," "right," and "above."


Obj. 92 - Proximity in space

NYC M2.b - Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes.NYC M2.c - Uses simple two dimensional coordinate systems to find locations on a map and to represent points and simple figures.


Obj. 98 - Read a coordinate grid

NYC M2.d - Uses many types of figures (angles, triangles, squares, rectangles, rhombi, parallelograms, quadrilaterals, polygons, prisms, pyramids, cubes, circles, and spheres) and identifies the figures by their properties, e.g., symmetry, number of faces, two- or three-dimensionality, no right angles.

Topic 1 - Whole Number Concepts I

Obj. 7 - Classify by attributes


Obj. 86 - Identify plane figures

Obj. 87 - Count sides & corners on a plane figure


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Obj. 89 - Identify solidsObj. 90 - Lines of symmetry

NYC M2.e - Solves problems by showing relationships between and among figures, e.g., using congruence and similarity, and using transformations including flips, slides, and rotations.


Obj. 91 - Congruent figures

Obj. 93 - Slides, flips, and turnsObj. 94 - Combine & take apart shapes

NYC M2.f - Extends and creates geometric patterns using concrete and pictorial models.


Obj. 8 - Picture patterns

NYC M2.g - Uses basic ways of estimating and measuring the size of figures and objects in the real world, including length, width, perimeter, and area.

Topic 8 - Measurement Obj. 74 - Use tiles to estimate area

NYC M2.h - Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations.NYC M2.i - Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time.

NYC M2.j - Carries out simple unit conversions, such as between cm and m, and between hours and minutes.NYC M2.k - Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.NYC M3 - FUNCTION AND ALGEBRA CONCEPTSNYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

NYC M3.a - Uses linear patterns to solve problems; that is:


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NYC M3.a.i - Uses linear patterns to solve problems; that is: shows how one quantity determines another in a linear ("repeating") pattern, i.e., describes, extends, and recognizes the linear pattern by its rule, such as, the total number of legs on a given number of horses can be calculated by counting by fours.


Obj. 9 - Number patterns

NYC M3.a.ii - Uses linear patterns to solve problems; that is: shows how one quantity determines another quantity in a functional relationship based on a linear pattern, e.g., for the "number of people and total number of eyes," figure out how many eyes 100 people have all together.

Topic 12 - Problem Solving Obj. 109 - WP: Patterns

NYC M3.b - Builds iterations of simple non-linear patterns, including multiplicative and squaring patterns (e.g., "growing" patterns) with concrete materials, and recognizes that these patterns are not linear.

NYC M3.c - Uses the understanding that an equality relationship between two quantities remains the same as long as the same change is made to both quantities.

NYC M3.d - Uses letters, boxes, or other symbols to stand for any number, measured quantity, or object in simple situations with concrete materials, i.e., demonstrates understanding and use of a beginning concept of a variable.


Obj. 42 - Solve equations for symbol

NYC M4 - STATISTICS AND PROBABILITY CONCEPTSNYC M4-1 - The student produces evidence that demonstrates understanding of statistics and probability concepts in the following areas; that is, the student:


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NYC M4.a - Collects and organizes data to answer a question or test a hypothesis by comparing sets of data.


Obj. 79 - Record data and answer questions

NYC M4.b - Displays data in line plots, graphs, tables, and charts.NYC M4.c - Makes statements and draws simple conclusions based on data; that is:NYC M4.c.i - Makes statements and draws simple conclusions based on data; that is: reads data in line plots, graphs, tables, and charts.


Obj. 76 - Tally charts

Obj. 77 - PictographsObj. 78 - Bar graphsObj. 83 - Money with graphs

NYC M4.c.ii - Makes statements and draws simple conclusions based on data; that is: compares data in order to make true statements, e.g., "seven plants grew at least 5 cm".

NYC M4.c.iii - Makes statements and draws simple conclusions based on data; that is: identifies and uses the mode necessary for making true statements, e.g., "more people chose red".NYC M4.c.iv - Makes statements and draws simple conclusions based on data; that is: makes true statements based on a simple concept of average (median and mean), for a small sample size and where the situation is made evident with concrete materials or clear representations.NYC M4.c.v - Makes statements and draws simple conclusions based on data; that is: interprets data to determine the reasonableness of statements about the data, e.g., "twice as often," "three times faster".


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NYC M4.c.vi - Makes statements and draws simple conclusions based on data; that is: uses data, including statements about the data, to make a simple concluding statement about a situation, e.g., "This kind of plant grows better near sunlight because the seven plants that were near the window grew at least 5 cm."

NYC M4.d - Gathers data about an entire group or by sampling group members to understand the concept of sample, i.e., that a large sample leads to more reliable information, e.g., when flipping coins.

NYC M4.e - Predicts results, analyzes data, and finds out why some results are more likely, less likely, or equally likely.


Obj. 85 - Identify more likely events

NYC M4.f - Finds all possible combinations and arrangements within certain constraints involving a limited number of variables.NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING

NYC M5-1 - The student demonstrates logical reasoning throughout work in mathematics, i.e., concepts and skills, problem solving, and projects; demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow; and solves problems that make demands on all three aspects of the solution process--formulation, implementation, and conclusion.

- FORMULATIONNYC M5.a - Given the basic statement of a problem situation, the student:


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NYC M5.a.i - Given the basic statement of a problem situation, the student: makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.NYC M5.a.ii - Given the basic statement of a problem situation, the student: uses previously learned strategies, skills, knowledge, and concepts to make decisions.


Obj. 47 - WP: Write number sentence, add & subtract (to 18)



Obj. 50 - WP: Too much or little info, 1- & 2-dig add & subt

Topic 6 - Money Obj. 58 - WP: Add & subtract money expressions



Obj. 105 - WP: Division strategiesNYC M5.a.iii - Given the basic statement of a problem situation, the student: uses strategies, such as using manipulatives or drawing sketches, to model problems.





Topic 12 - Problem Solving Obj. 106 - Draw pictures to solve problemsObj. 107 - Guess and check to solve problemsObj. 108 - Use lists to solve problems

- IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:


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NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: makes up and uses a variety of strategies and approaches to solving problems and uses or learns approaches that other people use, as appropriate.

NYC M5.b.ii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: makes connections among concepts in order to solve problems.

NYC M5.b.iii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: solves problems in ways that make sense and explains why these ways make sense, e.g., defends the reasoning, explains the solution.

- CONCLUSIONNYC M5.c - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student:NYC M5.c.i - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains a pattern that can be used in similar situations.

NYC M5.c.ii - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains how the problem is similar to other problems he or she has solved.


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NYC M5.c.iii - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains how the mathematics used in the problem is like other concepts in mathematics.

NYC M5.c.iv - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains how the problem solution can be applied to other school subjects and in real world situations.NYC M5.c.v - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: makes the solution into a general rule that applies to other circumstances.NYC M6 - MATHEMATICAL SKILLS AND TOOLSNYC M6-1 - The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task, perhaps referring to notes, books, or other students, perhaps working to reconstruct a method; that is, the student:

NYC M6.a - Adds, subtracts, multiplies, and divides whole numbers correctly; that is:NYC M6.a.i - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: knows single digit addition, subtraction, multiplication, and division facts.




Obj. 43 - Add three 1-digit #s


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NYC M6.a.ii - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: adds and subtracts numbers with several digits.


Obj. 45 - Add 2-digit #s, no regroup

Obj. 46 - Subtract 2-digit #s, no regroup

NYC M6.a.iii - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: multiplies and divides numbers with one or two digits.NYC M6.a.iv - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: multiplies and divides three digit numbers by one digit numbers.NYC M6.b - Estimates numerically and spatially.


Obj. 33 - Estimate # in group (2 dig)

Topic 6 - Money Obj. 57 - Estimate coin valueTopic 8 - Measurement Obj. 66 - Estimate length in non-

standard unitsNYC M6.c - Measures length, area, perimeter, circumference, diameter, height, weight, and volume accurately in both the customary and metric systems.

Topic 8 - Measurement Obj. 68 - Measure length in cm with ruler

NYC M6.d - Computes time (in hours and minutes) and money (in dollars and cents).

Topic 6 - Money Obj. 55 - Add & subtract with coins

NYC M6.e - Refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle, square, rectangle, side, edge, face, cube, point, line, perimeter, area, and circle; and refers with assistance to rhombus, parallelogram, quadrilateral, polygon, polyhedron, angle, vertex, volume, diameter, circumference, sphere, prism, and pyramid.


Obj. 86 - Identify plane figures

Obj. 89 - Identify solidsNYC M6.f - Uses +, -, x, ÷, /, long division symbol, $, ¢, %, and . (decimal point) correctly in number sentences and expressions.


Obj. 41 - Determine add & subt in # sentence


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NYC M6.g - Reads, creates, and represents data on line plots, charts, tables, diagrams, bar graphs, simple circle graphs, and coordinate graphs.

NYC M6.h - Uses recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, and advice from peers, as appropriate, to achieve solutions; that is, uses measuring devices, graded appropriately for given situations, such as rulers (customary to the 1/8 inch; metric to the millimeter), graph paper (customary to the inch or half-inch; metric to the centimeter), measuring cups (customary to the ounce; metric to the milliliter), and scales (customary to the pound or ounce; metric to the kilogram or gram).

Topic 8 - Measurement Obj. 68 - Measure length in cm with ruler

Obj. 70 - Capacity in cups, pints, quarts, & gallons

NYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.NYC M7.b - Shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.


Obj. 10 - Add using pictures to 12


Obj. 16 - Subtract using pictures from 12


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Topic 12 - Problem Solving Obj. 106 - Draw pictures to solve problems

NYC M7.c - Explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.NYC M7.d - Considers purpose and audience when communicating about mathematics.NYC M7.e - Comprehends mathematics from reading assignments and from other sources.

NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale project each year, beginning in fourth grade, drawn from the following kinds and, over the course of elementary school, conducts projects drawn from at least two of the kinds. A single project may draw on more than one kind.

NYC M8.a - Data study, in which the student:NYC M8.a.i - Data study, in which the student: develops a question and a hypothesis in a situation where data could help make a decision or recommendation.NYC M8.a.ii - Data study, in which the student: decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers.NYC M8.a.iii - Data study, in which the student: collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions.


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NYC M8.a.iv - Data study, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.

NYC M8.b - Science study, in which the student:NYC M8.b.i - Science study, in which the student: decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement.NYC M8.b.ii - Science study, in which the student: develops a prediction (a hypothesis) and develops procedures to test the hypothesis.NYC M8.b.iii - Science study, in which the student: collects and records data, represents and displays data, and compares results with predictions.NYC M8.b.iv - Science study, in which the student: writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers.NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: decides on a structure to design, the size and budget constraints, and the scale of design.NYC M8.c.ii - Design of a physical structure, in which the student: makes a first draft of the design, and revises and improves the design in response to input from peers and teachers.


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NYC M8.c.iii - Design of a physical structure, in which the student: makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: decides on what to manage or plan, and the criteria to be used to see if the plan worked.NYC M8.d.ii - Management and planning, in which the student: identifies unexpected events that could disrupt the plan and further plans for such contingencies.NYC M8.d.iii - Management and planning, in which the student: identifies resources needed, e.g., materials, money, time, space, and other people.NYC M8.d.iv - Management and planning, in which the student: writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers.

NYC M8.d.v - Management and planning, in which the student: carries out the plan (optional).NYC M8.d.vi - Management and planning, in which the student: writes a report on the plan that includes resources, budget, and schedule, and acknowledges assistance received from parents, peers, and teachers.

NYC M8.d.vii - Management and planning, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.


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NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns.NYC M8.e.ii - Pure mathematics investigation, in which the student: describes a question or concept to investigate.NYC M8.e.iii - Pure mathematics investigation, in which the student: decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models.NYC M8.e.iv - Pure mathematics investigation, in which the student: carries out the investigation.NYC M8.e.v - Pure mathematics investigation, in which the student: writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.


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Topic 2 - Whole Number Addition and Subtraction I

Obj. 11 - Addition facts to 10

Obj. 12 - Addition facts from 11 to 18

Obj. 13 - Add three 1-digit #sObj. 14 - Add multiples of tenObj. 15 - Add 1- & 2-digit #s, no regroupObj. 16 - Add 1- & 2-digit #s, regroup

Obj. 17 - Add 2-digit #s, regroupObj. 18 - Add three #s, regroup (1 & 2 dig)

Topic 10 - Whole Number Addition and Subtraction II

Obj. 115 - Use doubles to add

Obj. 116 - Use compensation to add 2-digit numbersObj. 117 - Add 3-digit #s, no regroup

Obj. 118 - Add 3-digit #s, regroupObj. 119 - Add 3-digit money expressions


Obj. 128 - Repeated addition



Obj. 19 - Subtraction facts from 10

Obj. 20 - Subtraction facts from 11 to 18Obj. 21 - Subtract multiples of ten



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Obj. 22 - Subtract 1- & 2-digit #s, no regroupObj. 23 - Subtract 1-dig from 2-dig #s, regroupObj. 24 - Subtract 2-digit #s, regroup


Obj. 44 - Compare #s to 1000 using <, >, =


Obj. 120 - Subtract 3-digit #s, no regroupObj. 121 - Subtract 3-digit #s, regroupObj. 122 - Subtract 3-digit money expressions



Obj. 127 - Identify equal groups

Obj. 128 - Repeated additionObj. 130 - Multiplication facts, to 5s

Obj. 133 - Multiply with graphs (1 dig)




Obj. 135 - Divide into equal groups with left overObj. 137 - Divide money



Obj. 29 - WP: Add 1-digit #s, one-step

Obj. 30 - WP: Subtract 1-digit #s, one-stepObj. 31 - WP: Add 2-digit #s, one-step


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Obj. 32 - WP: Subtract 2-digit #s, one-step




Obj. 132 - WP: Multiplication

Obj. 138 - WP: DivisionNYC M1.a.vi - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.





Obj. 13 - Add three 1-digit #sObj. 14 - Add multiples of ten


Obj. 130 - Multiplication facts, to 5s



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NYC M1.b.i - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000.


Obj. 116 - Use compensation to add 2-digit numbers

NYC M1.b.ii - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.




Obj. 36 - Estimate sum & diff, front digits only (2 dig)Obj. 37 - Round to nearest ten (2 dig)

Obj. 38 - Estimate sum & diff, round (2 dig)


Topic 9 - Fractions Obj. 105 - Equal parts

Obj. 106 - Meaning of a fraction's partsObj. 107 - Fractional parts of a whole

Obj. 108 - Fractional parts equal to a wholeObj. 109 - Fractional parts of a set


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Topic 9 - Fractions Obj. 111 - Compare & order fractions

NYC M1.d.v - Describes and compares quantities by using concrete and real world models of simple fractions; that is: uses beginning proportional reasoning and simple ratios, e.g., "about half of the people."NYC M1.e - Describes and compares quantities by using simple decimals; that is:NYC M1.e.i - Describes and compares quantities by using simple decimals; that is: adds, subtracts, multiplies, and divides money amounts.


Obj. 119 - Add 3-digit money expressions

Obj. 122 - Subtract 3-digit money expressions


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NYC M1.e.ii - Describes and compares quantities by using simple decimals; that is: recognizes relationships among simple fractions, decimals, and percents, i.e., that 1/2 is the same as 0.5, and 1/2 is the same as 50%, with concrete materials, diagrams, and in real world situations, e.g., when discovering the chance of a coin landing on heads or tails.


NYC M1.f.i - Describes and compares quantities by using whole numbers up to 10,000; that is: connects ideas of quantities to the real world, e.g., how many people fit in the school’s cafeteria; how far away is a kilometer.

Topic 7 - Measurement Obj. 79 - Estimate capacity in customary & metric units

NYC M1.f.ii - Describes and compares quantities by using whole numbers up to 10,000; that is: finds, identifies, and sorts numbers by their properties, e.g., odd, even, multiple, square.



NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:NYC M2.a - Gives and responds to directions about location, e.g., by using words such as "in front of," "right," and "above."

Topic 8 - Geometry Obj. 102 - Use positional words

NYC M2.b - Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes.


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NYC M2.c - Uses simple two dimensional coordinate systems to find locations on a map and to represent points and simple figures.

Topic 8 - Geometry Obj. 103 - Coordinate graphs


Topic 8 - Geometry Obj. 91 - Count faces, edges, & corners


Topic 8 - Geometry Obj. 97 - Identify symmetry

Obj. 99 - Flips, slides, and turnsNYC M2.f - Extends and creates geometric patterns using concrete and pictorial models.

Topic 4 - Graphs and Patterns Obj. 51 - Picture patterns

Topic 8 - Geometry Obj. 104 - Geometric patternsNYC M2.g - Uses basic ways of estimating and measuring the size of figures and objects in the real world, including length, width, perimeter, and area.

Topic 7 - Measurement Obj. 73 - Measure length in cm & m

NYC M2.h - Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations.NYC M2.i - Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time.

Topic 7 - Measurement Obj. 71 - Measure length in non-standard units

Obj. 72 - Measure length in in., ft, & ydObj. 73 - Measure length in cm & m

Obj. 74 - Estimate length & heightObj. 78 - Capacity in litersObj. 82 - Weight in grams & kilograms


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Obj. 83 - Estimate weight in cust & metric unitsObj. 84 - Compare weights

Topic 8 - Geometry Obj. 94 - Find area by counting squaresObj. 96 - Find volume of a box by counting cubes

NYC M2.j - Carries out simple unit conversions, such as between cm and m, and between hours and minutes.NYC M2.k - Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.NYC M3 - FUNCTION AND ALGEBRA CONCEPTSNYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

NYC M3.a - Uses linear patterns to solve problems; that is:NYC M3.a.i - Uses linear patterns to solve problems; that is: shows how one quantity determines another in a linear ("repeating") pattern, i.e., describes, extends, and recognizes the linear pattern by its rule, such as, the total number of legs on a given number of horses can be calculated by counting by fours.

Topic 4 - Graphs and Patterns Obj. 52 - Number patterns

Obj. 53 - Complete patterns in tables

Obj. 54 - WP: Number patternsNYC M3.a.ii - Uses linear patterns to solve problems; that is: shows how one quantity determines another quantity in a functional relationship based on a linear pattern, e.g., for the "number of people and total number of eyes," figure out how many eyes 100 people have all together.


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Obj. 26 - Find missing addend in addition sentence

Obj. 27 - Find missing parts in subtraction sentence


NYC M4.a - Collects and organizes data to answer a question or test a hypothesis by comparing sets of data.NYC M4.b - Displays data in line plots, graphs, tables, and charts.NYC M4.c - Makes statements and draws simple conclusions based on data; that is:NYC M4.c.i - Makes statements and draws simple conclusions based on data; that is: reads data in line plots, graphs, tables, and charts.

Topic 4 - Graphs and Patterns Obj. 45 - Tally charts

Obj. 46 - PictographsObj. 47 - Bar graphs


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Topic 4 - Graphs and Patterns Obj. 48 - Compare graphs

NYC M4.c.iii - Makes statements and draws simple conclusions based on data; that is: identifies and uses the mode necessary for making true statements, e.g., "more people chose red".NYC M4.c.iv - Makes statements and draws simple conclusions based on data; that is: makes true statements based on a simple concept of average (median and mean), for a small sample size and where the situation is made evident with concrete materials or clear representations.NYC M4.c.v - Makes statements and draws simple conclusions based on data; that is: interprets data to determine the reasonableness of statements about the data, e.g., "twice as often," "three times faster".




Topic 12 - Problem Solving Obj. 140 - Identify most likely/least likely event


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NYC M4.f - Finds all possible combinations and arrangements within certain constraints involving a limited number of variables.

Topic 12 - Problem Solving Obj. 145 - Use lists to solve problems

NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING


Topic 12 - Problem Solving Obj. 146 - Explain reasoning, evaluate reasonableness

- FORMULATIONNYC M5.a - Given the basic statement of a problem situation, the student:NYC M5.a.i - Given the basic statement of a problem situation, the student: makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.NYC M5.a.ii - Given the basic statement of a problem situation, the student: uses previously learned strategies, skills, knowledge, and concepts to make decisions.


Obj. 28 - WP: Find the correct equation

Obj. 33 - WP: Too much or little info, 2-dig add or subt

Topic 4 - Graphs and Patterns Obj. 54 - WP: Number patternsTopic 5 - Money Obj. 63 - WP: Add & subt money

expressionsTopic 7 - Measurement Obj. 76 - WP: Find length using cust

& metric unitsTopic 8 - Geometry Obj. 93 - WP: Perimeter

Obj. 95 - WP: Area


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NYC M5.a.iii - Given the basic statement of a problem situation, the student: uses strategies, such as using manipulatives or drawing sketches, to model problems.

Topic 9 - Fractions Obj. 112 - WP: Fractions, draw pictures

Topic 12 - Problem Solving Obj. 141 - Draw a picture to solve a problem

- IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: makes up and uses a variety of strategies and approaches to solving problems and uses or learns approaches that other people use, as appropriate.



- CONCLUSIONNYC M5.c - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student:


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NYC M5.c.i - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains a pattern that can be used in similar situations.

NYC M5.c.ii - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains how the problem is similar to other problems he or she has solved.NYC M5.c.iii - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains how the mathematics used in the problem is like other concepts in mathematics.

NYC M5.c.iv - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: explains how the problem solution can be applied to other school subjects and in real world situations.NYC M5.c.v - The student moves beyond a particular problem by making connections, extensions, and/or generalizations; for example, the student: makes the solution into a general rule that applies to other circumstances.NYC M6 - MATHEMATICAL SKILLS AND TOOLS


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NYC M6-1 - The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task, perhaps referring to notes, books, or other students, perhaps working to reconstruct a method; that is, the student:




Obj. 13 - Add three 1-digit #sObj. 19 - Subtraction facts from 10


Obj. 128 - Repeated addition


Obj. 131 - Multiply money expressions




Obj. 14 - Add multiples of tenObj. 15 - Add 1- & 2-digit #s, no regroupObj. 16 - Add 1- & 2-digit #s, regroup

Obj. 17 - Add 2-digit #s, regroupObj. 18 - Add three #s, regroup (1 & 2 dig)Obj. 20 - Subtraction facts from 11 to 18Obj. 21 - Subtract multiples of tenObj. 23 - Subtract 1-dig from 2-dig #s, regroup


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Obj. 24 - Subtract 2-digit #s, regroup

NYC M6.a.iii - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: multiplies and divides numbers with one or two digits.



Obj. 131 - Multiply money expressions

NYC M6.a.iv - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: multiplies and divides three digit numbers by one digit numbers.NYC M6.b - Estimates numerically and spatially.



Obj. 36 - Estimate sum & diff, front digits only (2 dig)Obj. 38 - Estimate sum & diff, round (2 dig)Obj. 39 - WP: Estimate sum & diff, round (2 dig)

Topic 7 - Measurement Obj. 74 - Estimate length & heightTopic 10 - Whole Number Addition and Subtraction II

Obj. 125 - Estimate sum & diff, front digits only (3 dig)Obj. 126 - WP: Estimate sum & diff, front digits (2-3 dig)

NYC M6.c - Measures length, area, perimeter, circumference, diameter, height, weight, and volume accurately in both the customary and metric systems.

Topic 7 - Measurement Obj. 72 - Measure length in in., ft, & yd

Obj. 73 - Measure length in cm & m

Topic 8 - Geometry Obj. 96 - Find volume of a box by counting cubes


Topic 5 - Money Obj. 61 - Add money expressions

Obj. 62 - Subtract money expressionsObj. 63 - WP: Add & subt money expressions

Topic 6 - Time Obj. 69 - Calculate elapsed time


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Topic 8 - Geometry Obj. 88 - Identify plane figures

Obj. 89 - Count sides & cornersObj. 90 - Identify solidsObj. 91 - Count faces, edges, & corners

NYC M6.f - Uses +, -, x, ÷, /, long division symbol, $, ¢, %, and . (decimal point) correctly in number sentences and expressions.

Topic 5 - Money Obj. 56 - Cent, dollar, & decimal symbols


Obj. 129 - Multiplication sentences


Topic 4 - Graphs and Patterns Obj. 45 - Tally charts

Obj. 47 - Bar graphsObj. 49 - Add with graphs (2 dig)Obj. 50 - Subtract with graphs (2 dig)


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Topic 7 - Measurement Obj. 72 - Measure length in in., ft, & yd

Obj. 80 - Read scales & estimate in poundsObj. 86 - Read thermometers

NYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.NYC M7.b - Shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.

Topic 9 - Fractions Obj. 112 - WP: Fractions, draw pictures


NYC M7.c - Explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.


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NYC M7.d - Considers purpose and audience when communicating about mathematics.NYC M7.e - Comprehends mathematics from reading assignments and from other sources.


NYC M8.a - Data study, in which the student:NYC M8.a.i - Data study, in which the student: develops a question and a hypothesis in a situation where data could help make a decision or recommendation.NYC M8.a.ii - Data study, in which the student: decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers.NYC M8.a.iii - Data study, in which the student: collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions.NYC M8.a.iv - Data study, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.

NYC M8.b - Science study, in which the student:


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NYC M8.b.i - Science study, in which the student: decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement.NYC M8.b.ii - Science study, in which the student: develops a prediction (a hypothesis) and develops procedures to test the hypothesis.NYC M8.b.iii - Science study, in which the student: collects and records data, represents and displays data, and compares results with predictions.NYC M8.b.iv - Science study, in which the student: writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers.NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: decides on a structure to design, the size and budget constraints, and the scale of design.NYC M8.c.ii - Design of a physical structure, in which the student: makes a first draft of the design, and revises and improves the design in response to input from peers and teachers.NYC M8.c.iii - Design of a physical structure, in which the student: makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers.NYC M8.d - Management and planning, in which the student:


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NYC M8.d.i - Management and planning, in which the student: decides on what to manage or plan, and the criteria to be used to see if the plan worked.NYC M8.d.ii - Management and planning, in which the student: identifies unexpected events that could disrupt the plan and further plans for such contingencies.NYC M8.d.iii - Management and planning, in which the student: identifies resources needed, e.g., materials, money, time, space, and other people.NYC M8.d.iv - Management and planning, in which the student: writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers.


NYC M8.d.vii - Management and planning, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns.NYC M8.e.ii - Pure mathematics investigation, in which the student: describes a question or concept to investigate.


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NYC M8.e.iii - Pure mathematics investigation, in which the student: decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models.NYC M8.e.iv - Pure mathematics investigation, in which the student: carries out the investigation.NYC M8.e.v - Pure mathematics investigation, in which the student: writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.


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Topic 2 - Whole Numbers: Add and Subtract

Obj. 15 - Addition facts

Obj. 16 - Add 2 whole #s, no regroup (1-3 dig)Obj. 17 - Add 3 whole #s, no regroup (1-3 dig)Obj. 19 - Add 2 whole #s, regroup (1-4 dig)Obj. 21 - Add 3 whole #s, regroup (1-4 dig)Obj. 28 - Add & subtract multiples of ten & hundred



Obj. 6 - Compare whole #s (to 4 dig)


Obj. 23 - Subtraction facts

Obj. 24 - Subtract whole #s, no regroup (1-4 dig)Obj. 26 - Subtract whole #s, regroup (2-4 dig)Obj. 28 - Add & subtract multiples of ten & hundred



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Topic 3 - Whole Numbers: Multiply and Divide

Obj. 40 - Use strategies to solve mult problems (to 12 x 12)

Obj. 41 - Multiplication as repeated additionObj. 42 - Arrays and multiplicationObj. 45 - DoublesObj. 46 - Multiply by multiples of ten (1 dig)Obj. 47 - Multiply by 1-digit, no regroup (3 dig)Obj. 48 - Multiply by 1-digit, regroup (2-3 dig)


Obj. 67 - Count with multiples




Obj. 53 - Use manipulatives to solve division problemsObj. 54 - Divide by 1-digit, 1-digit quotient, remainderObj. 55 - Divide multiple of ten or hundred by 1-digitObj. 56 - Divide by 1-digit, 2-digit quotient, no remainderObj. 57 - Divide by 1-digit, 2-digit quotient, remainder



Obj. 18 - WP: Add whole #s, no regroup (1-2 dig)

Obj. 20 - WP: Add 2 whole #s, regroup (2-3 dig)Obj. 22 - WP: Add 3 whole #s, regroup (1-3 dig)


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Obj. 25 - WP: Subtract whole #s, no regroup (1-3 dig)Obj. 27 - WP: Subtract whole #s, regroup (1-4 dig)


Obj. 49 - WP: Multiply by 1-digit (2-3 dig)Obj. 58 - WP: Divide by 1-digit divisor, 2-digit dividendObj. 65 - WP: Solve multi-step problems

NYC M1.a.vi - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.




Obj. 23 - Subtraction factsTopic 3 - Whole Numbers: Multiply and Divide

Obj. 38 - Multiplication basic facts to 5sObj. 39 - Multiplication facts from 6s to 9sObj. 50 - Division facts, horizontalObj. 51 - Division facts, vertical



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NYC M1.b.i - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000.NYC M1.b.ii - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.


Obj. 46 - Multiply by multiples of ten (1 dig)

Obj. 55 - Divide multiple of ten or hundred by 1-digit



Obj. 29 - Estimate sums, round (2-4 dig)

Obj. 31 - Estimate sums, front digits only (2-4 dig)Obj. 33 - Estimate differences, round (2-4 dig)


Obj. 59 - Estimate products, round (1-3 dig)Obj. 60 - Estimate products, front digits only (1-3 dig)Obj. 63 - Estimate quotients, compatible #s (1-2 dig)


Topic 5 - Fractions Obj. 76 - Name unit fractions

Obj. 77 - Model & name proper fractions as part of a wholeObj. 78 - Model & name proper fractions as part of a set


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Obj. 79 - Model fractions on number linesObj. 80 - Compare unit fractions (1/2 to 1/6 & 1/10)Obj. 83 - Equivalent proper fractions with models



Topic 5 - Fractions Obj. 79 - Model fractions on number lines


Topic 5 - Fractions Obj. 82 - Compare & order proper fractions

Obj. 83 - Equivalent proper fractions with models

NYC M1.d.v - Describes and compares quantities by using concrete and real world models of simple fractions; that is: uses beginning proportional reasoning and simple ratios, e.g., "about half of the people."NYC M1.e - Describes and compares quantities by using simple decimals; that is:NYC M1.e.i - Describes and compares quantities by using simple decimals; that is: adds, subtracts, multiplies, and divides money amounts.

Topic 7 - Money Obj. 98 - WP: Add money expressions


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Obj. 101 - WP: Subtract money expressions



NYC M1.f.i - Describes and compares quantities by using whole numbers up to 10,000; that is: connects ideas of quantities to the real world, e.g., how many people fit in the school’s cafeteria; how far away is a kilometer.

Topic 8 - Measurement, Time, and Temperature

Obj. 105 - Estimate customary length

Obj. 106 - Estimate customary weight

Obj. 111 - Estimate metric lengthObj. 112 - Estimate metric weight

NYC M1.f.ii - Describes and compares quantities by using whole numbers up to 10,000; that is: finds, identifies, and sorts numbers by their properties, e.g., odd, even, multiple, square.


Obj. 1 - Count with base-ten number blocks

Obj. 4 - Expanded to standard form (2-6 dig)Obj. 12 - Odd & even numbers


Obj. 67 - Count with multiples

NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:


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NYC M2.a - Gives and responds to directions about location, e.g., by using words such as "in front of," "right," and "above."

Topic 9 - Geometry Obj. 129 - Use compass directions

NYC M2.b - Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes.NYC M2.c - Uses simple two dimensional coordinate systems to find locations on a map and to represent points and simple figures.



Topic 9 - Geometry Obj. 124 - Classify shapes & count sides & corners

Obj. 132 - Faces of 3-D solidsNYC M2.e - Solves problems by showing relationships between and among figures, e.g., using congruence and similarity, and using transformations including flips, slides, and rotations.NYC M2.f - Extends and creates geometric patterns using concrete and pictorial models.

Topic 11 - Patterns, Expressions and Equations

Obj. 153 - Complete geometric patterns


Topic 9 - Geometry Obj. 135 - WP: Perimeter

NYC M2.h - Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations.


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NYC M2.i - Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time.


Obj. 104 - Appropriate unit of customary length


Obj. 106 - Estimate customary weight

Obj. 107 - Measure cust length with rulerObj. 108 - Measure customary weight

Obj. 109 - Customary units of capacityObj. 110 - Appropriate unit of metric lengthObj. 112 - Estimate metric weightObj. 113 - Measure metric length with rulerObj. 115 - Tell time to hour & minute

Obj. 116 - Tell time to quarter hourTopic 9 - Geometry Obj. 136 - Area with a grid

NYC M2.j - Carries out simple unit conversions, such as between cm and m, and between hours and minutes.NYC M2.k - Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.

Topic 9 - Geometry Obj. 138 - Use scales in maps & drawings

NYC M3 - FUNCTION AND ALGEBRA CONCEPTSNYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

NYC M3.a - Uses linear patterns to solve problems; that is:


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NYC M3.a.i - Uses linear patterns to solve problems; that is: shows how one quantity determines another in a linear ("repeating") pattern, i.e., describes, extends, and recognizes the linear pattern by its rule, such as, the total number of legs on a given number of horses can be calculated by counting by fours.


Obj. 68 - Missing terms in number patterns


Topic 10 - Graphs, Probability, and Statistics

Obj. 148 - Make predictions and inferences from data display






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NYC M4.a - Collects and organizes data to answer a question or test a hypothesis by comparing sets of data.NYC M4.b - Displays data in line plots, graphs, tables, and charts.


Obj. 143 - Construct frequency tables

Obj. 144 - Create pictographs & bar graphs


NYC M4.c - Makes statements and draws simple conclusions based on data; that is:NYC M4.c.i - Makes statements and draws simple conclusions based on data; that is: reads data in line plots, graphs, tables, and charts.



Obj. 140 - PictographsObj. 141 - Bar graphsObj. 142 - Line plotsObj. 145 - Compare pictographs & bar graphs



Obj. 148 - Make predictions and inferences from data display

NYC M4.c.iii - Makes statements and draws simple conclusions based on data; that is: identifies and uses the mode necessary for making true statements, e.g., "more people chose red".NYC M4.c.iv - Makes statements and draws simple conclusions based on data; that is: makes true statements based on a simple concept of average (median and mean), for a small sample size and where the situation is made evident with concrete materials or clear representations.


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NYC M4.c.v - Makes statements and draws simple conclusions based on data; that is: interprets data to determine the reasonableness of statements about the data, e.g., "twice as often," "three times faster".





Obj. 146 - Identify likelihood of events

NYC M4.f - Finds all possible combinations and arrangements within certain constraints involving a limited number of variables.





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- FORMULATIONNYC M5.a - Given the basic statement of a problem situation, the student:NYC M5.a.i - Given the basic statement of a problem situation, the student: makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.

Topic 12 - Problem Solving Obj. 155 - Eliminate possiblities to solve problems

Obj. 156 - Draw a picture to solve a problemObj. 159 - Evaluate reasonableness of solution from context

NYC M5.a.ii - Given the basic statement of a problem situation, the student: uses previously learned strategies, skills, knowledge, and concepts to make decisions.

Topic 12 - Problem Solving Obj. 155 - Eliminate possiblities to solve problems

Obj. 156 - Draw a picture to solve a problemObj. 159 - Evaluate reasonableness of solution from context

NYC M5.a.iii - Given the basic statement of a problem situation, the student: uses strategies, such as using manipulatives or drawing sketches, to model problems.


Obj. 157 - Use lists to solve problems

- IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:


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Obj. 23 - Subtraction factsTopic 3 - Whole Numbers: Multiply and Divide

Obj. 38 - Multiplication basic facts to 5s


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Obj. 39 - Multiplication facts from 6s to 9sObj. 50 - Division facts, horizontalObj. 51 - Division facts, vertical



Obj. 16 - Add 2 whole #s, no regroup (1-3 dig)

Obj. 17 - Add 3 whole #s, no regroup (1-3 dig)Obj. 19 - Add 2 whole #s, regroup (1-4 dig)Obj. 21 - Add 3 whole #s, regroup (1-4 dig)Obj. 24 - Subtract whole #s, no regroup (1-4 dig)Obj. 26 - Subtract whole #s, regroup (2-4 dig)Obj. 28 - Add & subtract multiples of ten & hundred



Obj. 46 - Multiply by multiples of ten (1 dig)

Obj. 47 - Multiply by 1-digit, no regroup (3 dig)Obj. 48 - Multiply by 1-digit, regroup (2-3 dig)Obj. 54 - Divide by 1-digit, 1-digit quotient, remainderObj. 55 - Divide multiple of ten or hundred by 1-digitObj. 56 - Divide by 1-digit, 2-digit quotient, no remainderObj. 57 - Divide by 1-digit, 2-digit quotient, remainder

NYC M6.a.iv - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: multiplies and divides three digit numbers by one digit numbers.NYC M6.b - Estimates numerically and spatially.


Obj. 59 - Estimate products, round (1-3 dig)Obj. 60 - Estimate products, front digits only (1-3 dig)Obj. 63 - Estimate quotients, compatible #s (1-2 dig)


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Topic 7 - Money Obj. 102 - Estimate sum & diff of money



Obj. 107 - Measure cust length with ruler

Obj. 108 - Measure customary weight

Obj. 113 - Measure metric length with ruler



Obj. 117 - Calculate elapsed time

Obj. 118 - Read calendarsNYC M6.e - Refers to geometric shapes and terms correctly with concrete objects or drawings, including triangle, square, rectangle, side, edge, face, cube, point, line, perimeter, area, and circle; and refers with assistance to rhombus, parallelogram, quadrilateral, polygon, polyhedron, angle, vertex, volume, diameter, circumference, sphere, prism, and pyramid.

Topic 9 - Geometry Obj. 121 - Points, lines, segments, rays, & angles

Obj. 124 - Classify shapes & count sides & cornersObj. 130 - SolidsObj. 132 - Faces of 3-D solidsObj. 133 - Perimeter of rectangles & squaresObj. 134 - Perimeter of polygons

NYC M6.f - Uses +, -, x, ÷, /, long division symbol, $, ¢, %, and . (decimal point) correctly in number sentences and expressions.


Obj. 43 - Write multiplication sentences





Obj. 141 - Bar graphsObj. 142 - Line plotsObj. 143 - Construct frequency tables


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Obj. 107 - Measure cust length with ruler

Obj. 108 - Measure customary weight

Obj. 113 - Measure metric length with ruler

NYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.



NYC M7.b - Shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.


Obj. 5 - Read & write whole #s (3-6 dig)

Obj. 6 - Compare whole #s (to 4 dig)


Obj. 43 - Write multiplication sentences


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Topic 5 - Fractions Obj. 77 - Model & name proper fractions as part of a wholeObj. 78 - Model & name proper fractions as part of a setObj. 79 - Model fractions on number linesObj. 80 - Compare unit fractions (1/2 to 1/6 & 1/10)Obj. 83 - Equivalent proper fractions with models

Topic 6 - Decimals Obj. 88 - Model decimals (tenths)Topic 9 - Geometry Obj. 122 - Intersecting, parallel, &

perpendicular linesObj. 124 - Classify shapes & count sides & cornersObj. 126 - Lines of symmetryObj. 127 - Flips, turns, & slides


Obj. 143 - Construct frequency tables


NYC M7.c - Explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.NYC M7.d - Considers purpose and audience when communicating about mathematics.NYC M7.e - Comprehends mathematics from reading assignments and from other sources.


NYC M8.a - Data study, in which the student:


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NYC M8.a.i - Data study, in which the student: develops a question and a hypothesis in a situation where data could help make a decision or recommendation.NYC M8.a.ii - Data study, in which the student: decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers.NYC M8.a.iii - Data study, in which the student: collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions.NYC M8.a.iv - Data study, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.

NYC M8.b - Science study, in which the student:NYC M8.b.i - Science study, in which the student: decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement.NYC M8.b.ii - Science study, in which the student: develops a prediction (a hypothesis) and develops procedures to test the hypothesis.NYC M8.b.iii - Science study, in which the student: collects and records data, represents and displays data, and compares results with predictions.NYC M8.b.iv - Science study, in which the student: writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers.


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NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: decides on a structure to design, the size and budget constraints, and the scale of design.NYC M8.c.ii - Design of a physical structure, in which the student: makes a first draft of the design, and revises and improves the design in response to input from peers and teachers.NYC M8.c.iii - Design of a physical structure, in which the student: makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: decides on what to manage or plan, and the criteria to be used to see if the plan worked.NYC M8.d.ii - Management and planning, in which the student: identifies unexpected events that could disrupt the plan and further plans for such contingencies.NYC M8.d.iii - Management and planning, in which the student: identifies resources needed, e.g., materials, money, time, space, and other people.NYC M8.d.iv - Management and planning, in which the student: writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers.

NYC M8.d.v - Management and planning, in which the student: carries out the plan (optional).


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NYC M8.d.vi - Management and planning, in which the student: writes a report on the plan that includes resources, budget, and schedule, and acknowledges assistance received from parents, peers, and teachers.

NYC M8.d.vii - Management and planning, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns.NYC M8.e.ii - Pure mathematics investigation, in which the student: describes a question or concept to investigate.NYC M8.e.iii - Pure mathematics investigation, in which the student: decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models.NYC M8.e.iv - Pure mathematics investigation, in which the student: carries out the investigation.NYC M8.e.v - Pure mathematics investigation, in which the student: writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.


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Obj. 13 - WP: Add 2 whole #s, no regroup (1-3 dig)

Obj. 15 - WP: Add 3 whole #s, no regroup (1-3 dig)Obj. 17 - WP: Add 2 whole #s (2-4 dig)Obj. 19 - WP: Add 3 whole #s (2-4 dig)



Obj. 27 - WP: Subtract whole #s, no regroup (1-3 dig)

Obj. 29 - WP: Subtract whole #s (2-5 dig)



Obj. 38 - WP: Multiply by 2-digit (2-3 dig)



Obj. 48 - WP: Divide whole #s by 1-digit



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NYC M1.a.v - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: analyzes problem situations and contexts in order to figure out when to add, subtract, multiply, or divide.NYC M1.a.vi - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: solves arithmetic problems by relating addition, subtraction, multiplication, and division to one another.

NYC M1.a.vii - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: computes answers mentally, e.g., 27 + 45, 30 x 4.NYC M1.a.viii - Adds, subtracts, multiplies, and divides whole numbers, with and without calculators; that is: uses simple concepts of negative numbers, e.g., on a number line, in counting, in temperature, "owing."NYC M1.b - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is:NYC M1.b.i - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: counts 1, 10, 100, or 1,000 more than or less than, e.g., 1 less than 10,000, 10 more than 380, 1,000 more than 23,000, 100 less than 9,000.


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NYC M1.b.ii - Demonstrates understanding of the base ten place value system and uses this knowledge to solve arithmetic tasks; that is: uses knowledge about ones, tens, hundreds, and thousands to figure out answers to multiplication and division tasks, e.g., 36 x 10, 18 x 100, 7 x 1,000, 4,000 ÷ 4.


Obj. 39 - Multiply by multiples of ten (2-4 dig)

Obj. 40 - Multiply by powers of ten (2-3 dig)Obj. 51 - Divide by a multiple of ten (2-4 dig)



Obj. 23 - Estimate sums, 2 #s: front digits only (2-4 dig)

Obj. 24 - Estimate sums, 3 #s: front digits only (2-4 dig)Obj. 25 - WP: Estimate sums, front digits only (2-3 dig)Obj. 30 - Estimate differences, round (2-4 dig)Obj. 31 - WP: Estimate differences, round (2-4 dig)


Obj. 41 - Estimate products, round (1-4 dig)Obj. 42 - WP: Estimate products, round (1-4 dig)Obj. 55 - Estimate quotients, compatible #s (2-3 dig)Obj. 56 - WP: Estimate whole quotients (2-3 dig)


Topic 4 - Fractions Obj. 63 - Model & name proper fractions as part of a whole

Obj. 64 - Model & name proper fractions as part of a setObj. 67 - Model equivalent fractions


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Topic 4 - Fractions Obj. 65 - Model fractions & mixed #s on number lines

NYC M1.d.iv - Describes and compares quantities by using concrete and real world models of simple fractions; that is: uses drawings, diagrams, or models to show what the numerator and denominator mean, including when adding like fractions, e.g., 1/8 + 5/8, or when showing that 3/4 is more than 3/8.NYC M1.d.v - Describes and compares quantities by using concrete and real world models of simple fractions; that is: uses beginning proportional reasoning and simple ratios, e.g., "about half of the people."NYC M1.e - Describes and compares quantities by using simple decimals; that is:NYC M1.e.i - Describes and compares quantities by using simple decimals; that is: adds, subtracts, multiplies, and divides money amounts.


Obj. 95 - Subtract money expressionsObj. 100 - Multiply money expressions by whole #s


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NYC M1.f.i - Describes and compares quantities by using whole numbers up to 10,000; that is: connects ideas of quantities to the real world, e.g., how many people fit in the school’s cafeteria; how far away is a kilometer.NYC M1.f.ii - Describes and compares quantities by using whole numbers up to 10,000; that is: finds, identifies, and sorts numbers by their properties, e.g., odd, even, multiple, square.

Topic 1 - Whole Number Concepts


NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:NYC M2.a - Gives and responds to directions about location, e.g., by using words such as "in front of," "right," and "above."NYC M2.b - Visualizes and represents two dimensional views of simple rectangular three dimensional shapes, e.g., by showing the front view and side view of a building made of cubes.


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NYC M2.c - Uses simple two dimensional coordinate systems to find locations on a map and to represent points and simple figures.

Topic 8 - Geometry Obj. 137 - Plot & name ordered pairs (first quadrant)


Topic 8 - Geometry Obj. 129 - Identify polygons


Topic 8 - Geometry Obj. 130 - Congruent & similar figures

Obj. 134 - Flips, turns, & slidesNYC M2.f - Extends and creates geometric patterns using concrete and pictorial models.

Topic 11 - Patterns, Expressions, and Equations

Obj. 165 - Complete geometric patterns


Topic 9 - Perimeter, Area, and Volume

Obj. 142 - Area with a grid

NYC M2.h - Uses models to reason about the relationship between the perimeter and area of rectangles in simple situations.


Obj. 138 - Model perimeter of rectangles

NYC M2.i - Selects and uses units, both formal and informal as appropriate, for estimating and measuring quantities such as weight, length, area, volume, and time.



Obj. 109 - Select appropriate unit to measure mass, kg/gObj. 110 - Select appropriate units to measure capacity, mL/L

NYC M2.j - Carries out simple unit conversions, such as between cm and m, and between hours and minutes.


Obj. 107 - Convert customary units of length


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Obj. 117 - Convert metric units of length

NYC M2.k - Uses scales in maps, and uses, measures, and creates scales for rectangular scale drawings based on work with concrete models and graph paper.


Obj. 118 - Use scales in maps and scale drawings


NYC M3.a - Uses linear patterns to solve problems; that is:NYC M3.a.i - Uses linear patterns to solve problems; that is: shows how one quantity determines another in a linear ("repeating") pattern, i.e., describes, extends, and recognizes the linear pattern by its rule, such as, the total number of legs on a given number of horses can be calculated by counting by fours.


Obj. 172 - Find the rule, given table or input/output box



Obj. 157 - Predictions from graphs




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NYC M4.a - Collects and organizes data to answer a question or test a hypothesis by comparing sets of data.NYC M4.b - Displays data in line plots, graphs, tables, and charts.NYC M4.c - Makes statements and draws simple conclusions based on data; that is:NYC M4.c.i - Makes statements and draws simple conclusions based on data; that is: reads data in line plots, graphs, tables, and charts.



Obj. 149 - PictographsObj. 150 - Bar graphsObj. 151 - Line graphsObj. 152 - Circle graphsObj. 153 - Frequency tablesObj. 155 - WP: Graphs


NYC M4.c.iii - Makes statements and draws simple conclusions based on data; that is: identifies and uses the mode necessary for making true statements, e.g., "more people chose red".


Obj. 159 - Modes of data sets


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NYC M4.c.iv - Makes statements and draws simple conclusions based on data; that is: makes true statements based on a simple concept of average (median and mean), for a small sample size and where the situation is made evident with concrete materials or clear representations.


Obj. 160 - Averages (means) of data sets

Obj. 161 - Medians of data setsNYC M4.c.v - Makes statements and draws simple conclusions based on data; that is: interprets data to determine the reasonableness of statements about the data, e.g., "twice as often," "three times faster".



NYC M4.e - Predicts results, analyzes data, and finds out why some results are more likely, less likely, or equally likely.NYC M4.f - Finds all possible combinations and arrangements within certain constraints involving a limited number of variables.NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING


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Topic 12 - Problem solving Obj. 173 - Guess & check to solve problems

Obj. 174 - Eliminate possibilities to solve problems

- FORMULATIONNYC M5.a - Given the basic statement of a problem situation, the student:NYC M5.a.i - Given the basic statement of a problem situation, the student: makes the important decisions about the approach, materials, and strategies to use, i.e., does not merely fill in a given chart, use a pre-specified manipulative, or go through a predetermined set of steps.NYC M5.a.ii - Given the basic statement of a problem situation, the student: uses previously learned strategies, skills, knowledge, and concepts to make decisions.

NYC M5.a.iii - Given the basic statement of a problem situation, the student: uses strategies, such as using manipulatives or drawing sketches, to model problems. - IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:


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Obj. 34 - Multiplication facts

Obj. 43 - Division facts


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Obj. 12 - Add 2 whole #s, no regroup (1-3 dig)

Obj. 14 - Add 3 whole #s, no regroup (1-3 dig)Obj. 16 - Add 2 whole #s (2-6 dig)Obj. 18 - Add 3 or more whole #s (2-5 dig)Obj. 26 - Subtract whole #s, no regroup (1-3 dig)Obj. 28 - Subtract whole #s (2-5 dig)



Obj. 36 - WP: Multiply by 1-digit (2-4 dig)

Obj. 39 - Multiply by multiples of ten (2-4 dig)

NYC M6.a.iv - Adds, subtracts, multiplies, and divides whole numbers correctly; that is: multiplies and divides three digit numbers by one digit numbers.


Obj. 35 - Multiply by 1-digit (2-4 dig)

Obj. 45 - Divide 3-digit by 1-digit, no borrow, no remainderObj. 46 - Divide 3-digit by 1-digit, borrow, no remainderObj. 47 - Divide 3-digit by 1-digitObj. 48 - WP: Divide whole #s by 1-digit

NYC M6.b - Estimates numerically and spatially.


Obj. 20 - Estimate sums, round (2-4 dig)Obj. 21 - Estimate sums, round to 10 or 100Obj. 23 - Estimate sums, 2 #s: front digits only (2-4 dig)Obj. 24 - Estimate sums, 3 #s: front digits only (2-4 dig)Obj. 25 - WP: Estimate sums, front digits only (2-3 dig)Obj. 30 - Estimate differences, round (2-4 dig)Obj. 31 - WP: Estimate differences, round (2-4 dig)


Obj. 41 - Estimate products, round (1-4 dig)


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Obj. 42 - WP: Estimate products, round (1-4 dig)Obj. 55 - Estimate quotients, compatible #s (2-3 dig)Obj. 56 - WP: Estimate whole quotients (2-3 dig)

Topic 6 - Money Obj. 97 - Estimate sums & differences of moneyObj. 98 - WP: Estimate sums & differences of money



Obj. 105 - Estimate customary capacityObj. 111 - Estimate metric lengthObj. 113 - Estimate metric capacity



Obj. 106 - Measure customary length

Obj. 114 - Measure metric lengthNYC M6.d - Computes time (in hours and minutes) and money (in dollars and cents).


Obj. 95 - Subtract money expressionsObj. 99 - Count money & figure changeObj. 100 - Multiply money expressions by whole #s


Obj. 123 - Calculate elapsed time


Topic 8 - Geometry Obj. 126 - Points, lines, segments, rays, & angles

Obj. 128 - Classify quadrilateralsObj. 135 - Solids


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NYC M6.f - Uses +, -, x, ÷, /, long division symbol, $, ¢, %, and . (decimal point) correctly in number sentences and expressions.NYC M6.g - Reads, creates, and represents data on line plots, charts, tables, diagrams, bar graphs, simple circle graphs, and coordinate graphs.



Obj. 149 - PictographsObj. 150 - Bar graphsObj. 151 - Line graphsObj. 152 - Circle graphsObj. 153 - Frequency tablesObj. 154 - Stem-and-leaf plotsObj. 158 - Organize data to answer a question




Obj. 114 - Measure metric lengthNYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:


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NYC M7.a - Uses appropriate mathematical terms, vocabulary, and language, based on prior conceptual work.


Obj. 167 - Equations to word sentences

NYC M7.b - Shows mathematical ideas in a variety of ways, including words, numbers, symbols, pictures, charts, graphs, tables, diagrams, and models.NYC M7.c - Explains solutions to problems clearly and logically, and supports solutions with evidence, in both oral and written work.NYC M7.d - Considers purpose and audience when communicating about mathematics.NYC M7.e - Comprehends mathematics from reading assignments and from other sources.


NYC M8.a - Data study, in which the student:NYC M8.a.i - Data study, in which the student: develops a question and a hypothesis in a situation where data could help make a decision or recommendation.NYC M8.a.ii - Data study, in which the student: decides on a group or groups to be sampled and makes predictions of the results, with specific percents, fractions, or numbers.


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NYC M8.a.iii - Data study, in which the student: collects, represents, and displays data in order to help make the decision or recommendation; compares the results with the predictions.NYC M8.a.iv - Data study, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.

NYC M8.b - Science study, in which the student:NYC M8.b.i - Science study, in which the student: decides on a specific science question to study and identifies the mathematics that will be used, e.g., measurement.NYC M8.b.ii - Science study, in which the student: develops a prediction (a hypothesis) and develops procedures to test the hypothesis.NYC M8.b.iii - Science study, in which the student: collects and records data, represents and displays data, and compares results with predictions.NYC M8.b.iv - Science study, in which the student: writes a report that compares the results with the hypothesis; supports the results with diagrams, charts, and graphs; acknowledges assistance received from parents, peers, and teachers.NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: decides on a structure to design, the size and budget constraints, and the scale of design.


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NYC M8.c.ii - Design of a physical structure, in which the student: makes a first draft of the design, and revises and improves the design in response to input from peers and teachers.NYC M8.c.iii - Design of a physical structure, in which the student: makes a final draft and report of the design, drawn and written so that another person could make the structure; acknowledges assistance received from parents, peers, and teachers.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: decides on what to manage or plan, and the criteria to be used to see if the plan worked.NYC M8.d.ii - Management and planning, in which the student: identifies unexpected events that could disrupt the plan and further plans for such contingencies.NYC M8.d.iii - Management and planning, in which the student: identifies resources needed, e.g., materials, money, time, space, and other people.NYC M8.d.iv - Management and planning, in which the student: writes a detailed plan and revises and improves the plan in response to feedback from peers and teachers.



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NYC M8.d.vii - Management and planning, in which the student: writes a report that includes recommendations supported by diagrams, charts, and graphs, and acknowledges assistance received from parents, peers, and teachers.NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: decides on the area of mathematics to investigate, e.g., numbers, shapes, patterns.NYC M8.e.ii - Pure mathematics investigation, in which the student: describes a question or concept to investigate.NYC M8.e.iii - Pure mathematics investigation, in which the student: decides on representations that will be used, e.g., numbers, symbols, diagrams, shapes, or physical models.NYC M8.e.iv - Pure mathematics investigation, in which the student: carries out the investigation.NYC M8.e.v - Pure mathematics investigation, in which the student: writes a report that includes any generalizations drawn from the investigation, and acknowledges assistance received from parents, peers, and teachers.


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Agency Tag Set NameNew York City, Math, Adopted 1998, Revised 2001, Grade 5-8, New Standards Performance Standards--Mathematics, produced by City of New York Department of Education

Standard Topic Description Objective DescriptionNYC M1 - ARITHMETIC AND NUMBER CONCEPTS/NUMBER AND OPERATION CONCEPTSNYC M1-1 - The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:NYC M1.a - Consistently and accurately adds, subtracts, multiplies, and divides rational numbers using appropriate methods (e.g., the student can add (1/2) + (5/6) mentally or on paper but may opt to add (13/24) + (57/68) on a calculator) and raises rational numbers to whole number powers. (Students should have facility with the different kinds and forms of rational numbers, i.e., integers, both whole numbers and negative integers; and other positive and negative rationals, written as decimals, as percents, or as proper, improper, or mixed fractions. Irrational numbers, i.e., those that cannot be written as a ratio of two integers, are not required content but are suitable for introduction, especially since the student should be familiar with the irrational number (pi).)


Obj. 17 - Add 2 whole #s

Obj. 18 - Add 3 or more whole #sObj. 26 - Subtract whole #s (3-6 dig)

Obj. 27 - Subtract 3 or more whole #s


Obj. 33 - Multiply by 1-digit

Obj. 34 - Multiply by 2 or more digObj. 36 - Multiply by a multiple of 10

Obj. 37 - Multiply by a power of 10Obj. 42 - Divide by 1-digit, no remainderObj. 43 - Divide by 1-digit, remainder

Obj. 45 - Divide by 2-digitObj. 47 - Divide by multiple of 10


Accelerated Math Grades 5 - 8

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Obj. 51 - Use calculator to solve problems

Topic 5 - Decimals: Add and Subtract

Obj. 59 - Add money expressions

Obj. 61 - Add same-place decimals (to thousandths)Obj. 62 - Add varied-place decimals (to thousandths)Obj. 64 - Subtract money expressionsObj. 66 - Subtract same-place decimals (to thousandths)Obj. 67 - Subtract varied-place decimals (to thousandths)

Topic 6 - Decimals: Multiply and Divide

Obj. 73 - Multiply money expressions by whole # (1-2 dig)Obj. 75 - Multiply decimals by whole #s (1-2 dig)Obj. 76 - Multiply decimals (to thousandths)Obj. 77 - Multiply decimals by powers of 10Obj. 81 - Divide money expressions by whole # (1-2 dig)Obj. 83 - Divide decimals by 1-digit whole #s

Topic 8 - Fractions: Add and Subtract

Obj. 98 - Add like fractions

Obj. 99 - Add like mixed #sObj. 101 - Add unlike fractionsObj. 102 - Add unlike mixed #sObj. 104 - Subtract like fractionsObj. 105 - Subtract like mixed #sObj. 106 - Subtract unlike fractionsObj. 107 - Subtract mixed #s from whole #sObj. 108 - Subtract unlike mixed #s, no regroupObj. 109 - Subtract mixed #s, regroup

Topic 9 - Fractions: Multiply and Divide

Obj. 111 - Multiply fractions

Obj. 112 - Multiply fractions by whole #sObj. 114 - Multiply mixed #s by fractionsObj. 116 - Multiply mixed #sObj. 117 - Divide whole #s by fractions


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Topic 14 - Fractions, Decimals, Percents, and Ratios

Obj. 184 - Percent of a number

NYC M1.b - Uses and understands the inverse relationships between addition and subtraction, multiplication and division, and exponentiation and root-extraction (e.g., squares and square roots, cubes and cube roots); uses the inverse operation to determine unknown quantities in equations.NYC M1.c - Consistently and accurately applies and converts the different kinds and forms of rational numbers.

Topic 7 - Introduction to Fractions

Obj. 92 - Mixed #s to improper fractions

Obj. 93 - Improper fractions to mixed #s


Obj. 180 - Fractions to decimals

Obj. 181 - Decimals to fractionsObj. 182 - Mixed #s to decimalsObj. 186 - Convert decimals & percentsObj. 187 - Fractions to percentsObj. 189 - Write ratios as fractions

NYC M1.d - Is familiar with characteristics of numbers (e.g., divisibility, prime factorization) and with properties of operations (e.g., commutativity and associativity), short of formal statements.


Obj. 11 - Prime & composite numbers

Obj. 12 - Factors of composite numbersObj. 13 - Prime factorizationObj. 14 - Greatest common factorObj. 15 - Multiples of whole #sObj. 16 - Least common multiple


Obj. 19 - Properties of addition


Obj. 38 - Properties of multiplication

NYC M1.e - Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.


Obj. 183 - Find percents with models


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NYC M1.f - Uses ratios and rates to express "part-to-part" and "whole-to-whole" relationships, and reasons proportionally to solve problems involving equivalent fractions, equal ratios, or constant rates, recognizing the multiplicative nature of these problems in the constant factor of change.


Obj. 188 - Identify ratios

Obj. 191 - Equivalent ratios, same form

NYC M1.g - Orders numbers with the > and < relationships and by location on a number line; estimates and compares rational numbers using sense of the magnitudes and relative magnitudes of numbers and of base-ten place values (e.g., recognizes relationships to "benchmark" numbers 1/2 and 1 to conclude that the sum 1/2)+ 5/6 must be between 1 and 1 (1/2) (likewise, (13/24) + (1/4)).


Obj. 6 - Compare whole #s (2+ dig)

Topic 4 - Introduction to Decimals

Obj. 57 - Compare decimals using <, >, =


Obj. 86 - Model fractions & mixed #s on number linesObj. 94 - Compare fractions

NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts in the following areas; that is, the student:

NYC M2.a - Is familiar with assorted two- and three-dimensional objects, including squares, triangles, other polygons, circles, cubes, rectangular prisms, pyramids, spheres, and cylinders.


Obj. 141 - Classify quadrilateralsObj. 148 - SolidsObj. 149 - Count faces, edges, & verticesObj. 150 - Parts of a circle


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NYC M2.b - Identifies similar and congruent shapes and uses transformations in the coordinate plane, i.e., translations, rotations, and reflections.


Obj. 145 - Similar trianglesObj. 147 - Flips, turns, & slides

NYC M2.c - Identifies three dimensional shapes from two dimensional perspectives; draws two dimensional sketches of three dimensional objects that preserve significant features.


Obj. 163 - Identify nets (patterns) of solids

NYC M2.d - Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.

Topic 10 - Measurement, Time, Money, and Temperature


Obj. 125 - Measure metric lengthObj. 126 - Measure to nearest centimeter


Obj. 154 - Perimeter of rectangles

Obj. 155 - Perimeter of polygonsObj. 157 - Area by countingObj. 158 - Area of rectanglesObj. 159 - Area of parallelogramsObj. 161 - Volume of rectangular prisms & cubes

NYC M2.e - Recognizes similarity and rotational and bilateral symmetry in two- and three-dimensional figures.

Topic 11 - Geometry Obj. 146 - Lines of symmetry

NYC M2.f - Analyzes and generalizes geometric patterns, such as tessellations and sequences of shapes.

Topic 15 - Number Patterns Obj. 194 - Analyze geometric patterns

NYC M2.g - Measures angles, weights, capacities, times, and temperatures using appropriate units.


Obj. 132 - Read thermometers

Topic 11 - Geometry Obj. 139 - Measure anglesTopic 13 - Graphs, Probability, and Statistics

Obj. 171 - Time schedules


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NYC M2.h - Chooses appropriate units of measure and converts with ease between like units, e.g., inches and miles, within a customary or metric system. (Conversions between customary and metric are not required.)


Obj. 121 - Convert customary length

Obj. 122 - Convert customary weight

Obj. 123 - Convert customary capacityObj. 127 - Convert metric lengthObj. 128 - Convert metric weightObj. 129 - Convert metric capacity

NYC M2.i - Reasons proportionally in situations with similar figures.

Topic 11 - Geometry Obj. 145 - Similar triangles

NYC M2.j - Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings.NYC M2.k - Models situations geometrically to formulate and solve problems.


Obj. 156 - WP: Perimeter

Obj. 160 - WP: AreaObj. 162 - WP: Volume of rectangular solids



NYC M3.a - Discovers, describes, and generalizes patterns, including linear, exponential, and simple quadratic relationships, i.e., those of the form f(n)=n^2 or f(n)=cn^2, for constant c, including A = (pi)r^2, and represents them with variables and expressions.

Topic 15 - Number Patterns Obj. 192 - Missing terms in number patterns

Obj. 193 - Terms in described number patternsObj. 194 - Analyze geometric patternsObj. 195 - Determine function rule


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Obj. 197 - Translate verbal expressions

NYC M3.b - Represents relationships with tables, graphs in the coordinate plane, and verbal or symbolic rules.

Topic 16 - Problem Solving Obj. 203 - Everyday situations to math sentences

NYC M3.c - Analyzes tables, graphs, and rules to determine functional relationships.

Topic 15 - Number Patterns Obj. 195 - Determine function rule

NYC M3.d - Finds solutions for unknown quantities in linear equations and in simple equations and inequalities.

Topic 15 - Number Patterns Obj. 199 - Solve add/subtr equations, whole # facts

Obj. 200 - Solve mult/div equations, whole # factsObj. 201 - Solve 1-step equations, whole numbersObj. 202 - Determine perimeter, given formula

NYC M4 - STATISTICS AND PROBABILITY CONCEPTSNYC M4-1 - The student produces evidence that demonstrates understanding of statistics and probability concepts; that is, the student:NYC M4.a - Collects data, organizes data, and displays data with tables, charts, and graphs that are appropriate, i.e., consistent with the nature of the data.

NYC M4.b - Analyzes data with respect to characteristics of frequency and distribution, including mode and range.


Obj. 173 - Means of data sets

Obj. 174 - Medians of data setsObj. 175 - Modes of data setsObj. 176 - Ranges of data sets

NYC M4.c - Analyzes appropriately central tendencies of data by considering mean and median.



Obj. 174 - Medians of data setsNYC M4.d - Makes conclusions and recommendations based on data analysis.NYC M4.e - Critiques the conclusions and recommendations of others’ statistics.


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NYC M4.f - Considers the effects of missing or incorrect information.NYC M4.g - Formulates hypotheses to answer a question and uses data to test hypotheses.NYC M4.h - Represents and determines probability as a fraction of a set of equally likely outcomes; recognizes equally likely outcomes, and constructs sample spaces (including those described by numerical combinations and permutations).


Obj. 177 - Find possible outcomes

Obj. 178 - Probability of single events

NYC M4.i - Makes predictions based on experimental or theoretical probabilities.


Obj. 179 - Predict based on experimental probability

NYC M4.j - Predicts the result of a series of trials once the probability for one trial is known.NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING

NYC M5-1 - The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process--formulation, implementation, and conclusion. - FORMULATIONNYC M5.a - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student:

NYC M5.a.i - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: formulates and solves a variety of meaningful problems.


Obj. 20 - WP: Add whole #s

Obj. 28 - WP: Subtract whole #sTopic 3 - Whole Numbers: Multiply and Divide

Obj. 35 - WP: Multiply whole #s


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Obj. 44 - WP: Divide by 1-digitObj. 46 - WP: Divide by 2-digitObj. 50 - WP: 2-step with whole #s


Obj. 60 - WP: Add money expressionsObj. 63 - WP: Add decimalsObj. 65 - WP: Subtract money expressionsObj. 68 - WP: Subtract decimals


Obj. 74 - WP: Multiply money expressions by whole #sObj. 78 - WP: Multiply decimalsObj. 82 - WP: Divide money expressionsObj. 84 - WP: Divide decimals by whole #s


Obj. 87 - WP: Name fraction of a whole or set


Obj. 100 - WP: Add like fractions & mixed #sObj. 103 - WP: Add unlike fractions & mixed #sObj. 110 - WP: Subtract fractions


Obj. 113 - WP: Multiply fractions by whole #sObj. 115 - WP: Multiply mixed #s by whole #s, fractionsObj. 118 - WP: Divide whole #s by fractions


Obj. 134 - WP: 2-step with money


Obj. 156 - WP: Perimeter

Obj. 160 - WP: AreaObj. 162 - WP: Volume of rectangular solids


Obj. 185 - WP: Percent of a number

NYC M5.a.ii - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: extracts pertinent information from situations and figures out what additional information is needed.


Obj. 31 - WP: Too much information


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Obj. 32 - WP: Not enough information

- IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: uses and invents a variety of approaches and understands and evaluates those of others.NYC M5.b.ii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table.

NYC M5.b.iii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: determines, where helpful, how to break a problem into simpler parts.NYC M5.b.iv - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: solves for unknown or undecided quantities using algebra, graphing, sound reasoning, and other strategies.

Topic 16 - Problem Solving Obj. 215 - Arguments and conclusions

Obj. 216 - Think visually to solve a problemObj. 218 - Use diagrams/tables to solve a problem

NYC M5.b.v - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: integrates concepts and techniques from different areas of mathematics.


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NYC M5.b.vi - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: works effectively in teams when the nature of the task or the allotted time makes this an appropriate strategy.

- CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: verifies and interprets results with respect to the original problem situation.NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: generalizes solutions and strategies to new problem situations.

- MATHEMATICAL REASONINGNYC M5.d - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student:NYC M5.d.i - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: formulates conjectures and argues why they must be or seem true.


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NYC M5.d.ii - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: makes sensible, reasonable estimates.


Obj. 23 - WP: Estimate sums, round (3-5 dig)

Obj. 25 - WP: Estimate sums, front-end with adjustmentObj. 30 - WP: Estimate differences, round


Obj. 40 - WP: Estimate products, round (2-4 dig)Obj. 49 - WP: Estimate quotients, compatible #s (2-4 dig)


Obj. 70 - WP: Estimate decimal sums, roundObj. 72 - WP: Estimate decimal differences, round


Obj. 80 - WP: Estimate decimal products, round

Topic 16 - Problem Solving Obj. 206 - Evaluate reasonableness using estimationObj. 208 - Guess and check to solve a problem

NYC M5.d.iii - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: makes justified, logical statements.

Topic 15 - Number Patterns Obj. 195 - Determine function rule

NYC M6 - MATHEMATICAL SKILLS AND TOOLS


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NYC M6-1 - The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task (perhaps referring to notes, or books, perhaps working to reconstruct a method); that is, the student:

NYC M6.a - Computes accurately with arithmetic operations on rational numbers.


Obj. 17 - Add 2 whole #s

Obj. 18 - Add 3 or more whole #sObj. 26 - Subtract whole #s (3-6 dig)

Obj. 27 - Subtract 3 or more whole #s


Obj. 33 - Multiply by 1-digit

Obj. 34 - Multiply by 2 or more digObj. 36 - Multiply by a multiple of 10

Obj. 37 - Multiply by a power of 10Obj. 42 - Divide by 1-digit, no remainderObj. 43 - Divide by 1-digit, remainder

Obj. 45 - Divide by 2-digitObj. 47 - Divide by multiple of 10



Obj. 61 - Add same-place decimals (to thousandths)Obj. 62 - Add varied-place decimals (to thousandths)Obj. 64 - Subtract money expressionsObj. 66 - Subtract same-place decimals (to thousandths)Obj. 67 - Subtract varied-place decimals (to thousandths)


Obj. 73 - Multiply money expressions by whole # (1-2 dig)Obj. 75 - Multiply decimals by whole #s (1-2 dig)


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Obj. 76 - Multiply decimals (to thousandths)Obj. 77 - Multiply decimals by powers of 10Obj. 81 - Divide money expressions by whole # (1-2 dig)Obj. 83 - Divide decimals by 1-digit whole #s



Obj. 99 - Add like mixed #sObj. 101 - Add unlike fractionsObj. 102 - Add unlike mixed #sObj. 104 - Subtract like fractionsObj. 105 - Subtract like mixed #sObj. 106 - Subtract unlike fractionsObj. 107 - Subtract mixed #s from whole #sObj. 108 - Subtract unlike mixed #s, no regroupObj. 109 - Subtract mixed #s, regroup



Obj. 112 - Multiply fractions by whole #sObj. 114 - Multiply mixed #s by fractionsObj. 116 - Multiply mixed #sObj. 117 - Divide whole #s by fractions

NYC M6.b - Knows and uses the correct order of operations for arithmetic computations.NYC M6.c - Estimates numerically and spatially.


Obj. 21 - Estimate sums, round to greatest (3-5 dig)Obj. 22 - Estimate sums, round to 10,100,1000Obj. 24 - Estimate sums, front-end with adjustmentObj. 29 - Estimate differences, round


Obj. 39 - Estimate products, round (2-4 dig)Obj. 48 - Estimate quotients, compatible #s (2-4 dig)


Obj. 69 - Estimate decimal sums, round


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Obj. 71 - Estimate decimal differences, round


Obj. 79 - Estimate decimal products, round


Obj. 119 - Estimate customary measure

NYC M6.d - Measures length, area, volume, weight, time, and temperature accurately.



Obj. 125 - Measure metric lengthObj. 132 - Read thermometers


Obj. 157 - Area by counting

Obj. 158 - Area of rectanglesObj. 159 - Area of parallelogramsObj. 161 - Volume of rectangular prisms & cubes

NYC M6.e - Refers to geometric shapes and terms correctly.

Topic 11 - Geometry Obj. 137 - Classify angles

Obj. 140 - Identify polygonsObj. 141 - Classify quadrilateralsObj. 148 - SolidsObj. 150 - Parts of a circle

NYC M6.f - Uses equations, formulas, and simple algebraic notation appropriately.

Topic 15 - Number Patterns Obj. 196 - Identify algebraic terms

Obj. 197 - Translate verbal expressions

NYC M6.g - Reads and organizes data on charts and graphs, including scatter plots, bar, line, and circle graphs, and Venn diagrams; calculates mean and median.


Obj. 165 - Bar graphs

Obj. 166 - Line graphsObj. 169 - Circle graphsObj. 173 - Means of data setsObj. 174 - Medians of data sets

NYC M6.h - Uses recall, mental computations, pencil and paper, measuring devices, mathematics texts, manipulatives, calculators, computers, and advice from peers, as appropriate, to achieve solutions.

NYC M7 - MATHEMATICAL COMMUNICATION


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NYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Uses mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.NYC M7.b - Organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.NYC M7.c - Uses mathematical language to make complex situations easier to understand.NYC M7.d - Exhibits developing reasoning abilities by justifying statements and defending work.NYC M7.e - Shows understanding of concepts by explaining ideas not only to teachers and assessors but to fellow students or younger children.

NYC M7.f - Comprehends mathematics from reading assignments and from other sources.

NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds. A single investigation or project may draw on more than one kind.


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NYC M8.a - Data study based on civic, economic, or social issues, in which the student:NYC M8.a.i - Data study based on civic, economic, or social issues, in which the student: selects an issue to investigate.NYC M8.a.ii - Data study based on civic, economic, or social issues, in which the student: makes a hypothesis on an expected finding, if appropriate.NYC M8.a.iii - Data study based on civic, economic, or social issues, in which the student: gathers data.

NYC M8.a.iv - Data study based on civic, economic, or social issues, in which the student: analyzes the data using concepts from Standard 4, e.g., considering mean and median, and the frequency and distribution of the data.

NYC M8.a.v - Data study based on civic, economic, or social issues, in which the student: shows how the study’s results compare with the hypothesis.NYC M8.a.vi - Data study based on civic, economic, or social issues, in which the student: uses pertinent statistics to summarize.NYC M8.a.vii - Data study based on civic, economic, or social issues, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.NYC M8.b - Mathematical model of physical phenomena, often used in science studies, in which the student:


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NYC M8.b.i - Mathematical model of physical phenomena, often used in science studies, in which the student: carries out a study of a physical system using a mathematical representation of the structure.

NYC M8.b.ii - Mathematical model of physical phenomena, often used in science studies, in which the student: uses understanding from Standard 3, particularly with respect to the determination of the function governing behavior in the model.

NYC M8.b.iii - Mathematical model of physical phenomena, often used in science studies, in which the student: generalizes about the structure with a rule, i.e., a function, that clearly applies to the phenomenon and goes beyond statistical analysis of a pattern of numbers generated by the situation.

NYC M8.b.iv - Mathematical model of physical phenomena, often used in science studies, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.

NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: generates a plan to build something of value, not necessarily monetary value.NYC M8.c.ii - Design of a physical structure, in which the student: uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes.


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NYC M8.c.iii - Design of a physical structure, in which the student: summarizes the important features of the structure.NYC M8.c.iv - Design of a physical structure, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling.NYC M8.d.ii - Management and planning, in which the student: notes any constraints that will affect the plan.NYC M8.d.iii - Management and planning, in which the student: determines a plan.NYC M8.d.iv - Management and planning, in which the student: uses concepts from any of Standards 1 to 4, depending on the nature of the project.NYC M8.d.v - Management and planning, in which the student: considers the possibility of a more efficient solution.NYC M8.d.vi - Management and planning, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.NYC M8.e - Pure mathematics investigation, in which the student:


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NYC M8.e.i - Pure mathematics investigation, in which the student: extends or "plays with," as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers.

NYC M8.e.ii - Pure mathematics investigation, in which the student: uses concepts from any of Standards 1 to 4, e.g., an investigation of Pascal’s triangle would have roots in Standard 1 but could tie in concepts from geometry, algebra, and probability; investigations of derivations of geometric formulas would be rooted in Standard 2 but could require algebra.

NYC M8.e.iii - Pure mathematics investigation, in which the student: determines and expresses generalizations from patterns.NYC M8.e.iv - Pure mathematics investigation, in which the student: makes conjectures on apparent properties and argues, short of formal proof, why they seem true.NYC M8.e.v - Pure mathematics investigation, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.


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Topic 2 - Whole Number Concepts Part 2

Obj. 15 - Add whole #s

Obj. 18 - Subtract whole #sObj. 21 - Multiply by 2 or more digObj. 23 - Multiply by a multiple of 10

Obj. 24 - Multiply by a power of 10Obj. 27 - Exponential notationObj. 28 - Evaluate exponentsObj. 29 - Multiply with exponentsObj. 30 - Divide whole #s (2+ dig divisor)


Obj. 50 - Add decimals

Obj. 54 - Subtract decimalsTopic 5 - Decimals: Multiply and Divide

Obj. 64 - Multiply decimals by whole #s (2-3 dig)Obj. 65 - Multiply decimals (to thousandths)



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Obj. 67 - Multiply decimals by a power of 10Obj. 71 - Divide decimals by 1- & 2-digit whole #sObj. 73 - Divide whole #s by powers of 10 (decimal answer)Obj. 74 - Divide decimals by powers of 10Obj. 76 - Divide whole #s by decimalsObj. 78 - Divide decimals by decimals



Obj. 98 - Add like fractions & mixed #sObj. 100 - Add unlike fractionsObj. 101 - Add unlike mixed #sObj. 105 - Subtract like fractionsObj. 106 - Subtract like mixed #sObj. 107 - Subtract unlike fractionsObj. 108 - Subtract unlike mixed #s, no regroupObj. 109 - Subtract mixed #s from whole #sObj. 110 - Subtract mixed #s, regroup

Topic 8 - Fractions: Multiply, Divide, and Combined Operations


Obj. 115 - Multiply fractions by whole #sObj. 117 - Multiply mixed #s by fractionsObj. 118 - Multiply mixed #sObj. 120 - Divide fractionsObj. 121 - Divide mixed #s by fractionsObj. 122 - Divide mixed #sObj. 123 - Divide whole #s & fractions

Topic 15 - Integers Obj. 232 - Add integersObj. 234 - Subtract positive integers

Obj. 235 - Subtract integersTopic 17 - Problem Solving Obj. 249 - Solve problems using

calculator


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NYC M1.b - Uses and understands the inverse relationships between addition and subtraction, multiplication and division, and exponentiation and root-extraction (e.g., squares and square roots, cubes and cube roots); uses the inverse operation to determine unknown quantities in equations.


Obj. 243 - Solve addition & subtraction equations

NYC M1.c - Consistently and accurately applies and converts the different kinds and forms of rational numbers.

Topic 13 - Fractions, Percents, and Decimals

Obj. 205 - Fractions to decimals, terminating

Obj. 206 - Decimals to fractionsObj. 207 - Decimals to mixed #sObj. 208 - Mixed #s to decimalsObj. 209 - Decimals to percentsObj. 210 - Percents to decimalsObj. 211 - Percents to fractionsObj. 212 - Fractions to percentsObj. 213 - Fractions to decimals, repeating




Obj. 11 - Prime factorizationTopic 2 - Whole Number Concepts Part 2

Obj. 16 - Properties of addition

Obj. 34 - Divisibility rulesNYC M1.e - Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.

Topic 14 - Percents, Ratios, and Proportions

Obj. 218 - Ratio as percent



Obj. 215 - Ratio as fraction, lowest terms

Obj. 216 - WP: RatiosObj. 224 - Ratios as proportionsObj. 225 - Solve proportions


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Obj. 226 - Write proportion representing situationObj. 227 - WP: Solve proportions



Obj. 7 - Compare whole #s (4+ dig)


Obj. 88 - Model fractions & mixed #s on number linesObj. 95 - Compare fractions

Topic 15 - Integers Obj. 228 - Graph integers on number linesObj. 229 - Compare & order integers




Obj. 148 - Classify quadrilateralsObj. 154 - Name 3-dimensional object



Obj. 158 - Flips, turns, & slides


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Topic 10 - Geometry Obj. 156 - Nets and 3-d figures

Obj. 157 - Orthographic drawingsNYC M2.d - Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.


Obj. 164 - Perimeter of rectangles

Obj. 165 - Perimeter of polygonsObj. 166 - WP: PerimeterObj. 170 - Area of rectangles & squaresObj. 171 - WP: Area of rectangles & squaresObj. 173 - Area of trianglesObj. 174 - WP: Area of trianglesObj. 176 - Area of circlesObj. 179 - Find volume of rectangular prismsObj. 180 - Volume of rectangular prisms & cubesObj. 181 - WP: VolumeObj. 182 - Surface area of rectangular prisms



Obj. 153 - Rotational symmetryNYC M2.f - Analyzes and generalizes geometric patterns, such as tessellations and sequences of shapes.


Obj. 240 - Extend & generalize geometric patterns



Obj. 129 - Measure length (customary & metric)

Topic 10 - Geometry Obj. 146 - Measure angles


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Obj. 130 - Convert customary length


Obj. 132 - Convert customary capacityObj. 133 - WP: Customary measurementObj. 134 - Convert metric lengthObj. 135 - Convert metric weightObj. 136 - Convert metric capacityObj. 137 - Mixed units of lengthObj. 138 - WP: Metric measurement

NYC M2.i - Reasons proportionally in situations with similar figures.NYC M2.j - Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings.NYC M2.k - Models situations geometrically to formulate and solve problems.

Topic 17 - Problem Solving Obj. 248 - Model situations geometrically





Obj. 239 - Terms in described number patternsObj. 240 - Extend & generalize geometric patterns


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Obj. 247 - Equations and graphs





Topic 12 - Graphs, Probability and Statistics

Obj. 194 - Choose best type of graph



Obj. 203 - Modes of data sets

Obj. 204 - Ranges of data setsNYC M4.c - Analyzes appropriately central tendencies of data by considering mean and median.



Obj. 202 - Medians of data setsNYC M4.d - Makes conclusions and recommendations based on data analysis.NYC M4.e - Critiques the conclusions and recommendations of others’ statistics.NYC M4.f - Considers the effects of missing or incorrect information.NYC M4.g - Formulates hypotheses to answer a question and uses data to test hypotheses.


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NYC M4.h - Represents and determines probability as a fraction of a set of equally likely outcomes; recognizes equally likely outcomes, and constructs sample spaces (including those described by numerical combinations and permutations).


Obj. 195 - Find possible outcomes


Obj. 198 - List outcomes for compound events

NYC M4.i - Makes predictions based on experimental or theoretical probabilities.


Obj. 197 - Predict using probability, theor/exper



NYC M5.a.i - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: formulates and solves a variety of meaningful problems.


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NYC M5.a.ii - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: extracts pertinent information from situations and figures out what additional information is needed.


Obj. 37 - WP: Not enough information

Obj. 38 - WP: Too much information

- IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: uses and invents a variety of approaches and understands and evaluates those of others.NYC M5.b.ii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table.

Topic 17 - Problem Solving Obj. 248 - Model situations geometrically



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- CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: verifies and interprets results with respect to the original problem situation.NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: generalizes solutions and strategies to new problem situations.

- MATHEMATICAL REASONINGNYC M5.d - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student:


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NYC M5.d.i - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: formulates conjectures and argues why they must be or seem true.NYC M5.d.ii - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: makes sensible, reasonable estimates.


Obj. 26 - WP: Estimate products, round (2+ dig)

Obj. 33 - WP: Estimate quotients, compatible #s


Obj. 57 - WP: Round decimal sum & diff to nearest whole #Obj. 59 - WP: Round decimal sum & diff to greatest placeObj. 61 - WP: Front-end estimate decimal sums w/adjustment


Obj. 81 - WP: Estimate decimal products, roundObj. 83 - WP: Estimate decimal quotients, compatible #s


Obj. 104 - WP: Estimate fraction sums, roundObj. 113 - WP: Estimate fraction differences, round

Topic 17 - Problem Solving Obj. 258 - Make/identify reasonable estimates




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Obj. 15 - Add whole #s

Obj. 18 - Subtract whole #sObj. 21 - Multiply by 2 or more digObj. 23 - Multiply by a multiple of 10

Obj. 24 - Multiply by a power of 10Obj. 27 - Exponential notationObj. 28 - Evaluate exponentsObj. 29 - Multiply with exponentsObj. 30 - Divide whole #s (2+ dig divisor)



Obj. 50 - Add decimalsObj. 52 - Subtract money expressionsObj. 54 - Subtract decimals


Obj. 62 - Multiply money expressions by whole #sObj. 64 - Multiply decimals by whole #s (2-3 dig)Obj. 65 - Multiply decimals (to thousandths)Obj. 67 - Multiply decimals by a power of 10Obj. 69 - Divide money expressions by 2-digit whole #sObj. 71 - Divide decimals by 1- & 2-digit whole #sObj. 73 - Divide whole #s by powers of 10 (decimal answer)Obj. 74 - Divide decimals by powers of 10Obj. 76 - Divide whole #s by decimals


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Obj. 78 - Divide decimals by decimals



Obj. 98 - Add like fractions & mixed #sObj. 100 - Add unlike fractionsObj. 101 - Add unlike mixed #sObj. 105 - Subtract like fractionsObj. 106 - Subtract like mixed #sObj. 107 - Subtract unlike fractionsObj. 108 - Subtract unlike mixed #s, no regroupObj. 109 - Subtract mixed #s from whole #sObj. 110 - Subtract mixed #s, regroup



Obj. 115 - Multiply fractions by whole #sObj. 117 - Multiply mixed #s by fractionsObj. 118 - Multiply mixed #sObj. 120 - Divide fractionsObj. 121 - Divide mixed #s by fractionsObj. 122 - Divide mixed #sObj. 123 - Divide whole #s & fractions

Topic 15 - Integers Obj. 232 - Add integersObj. 234 - Subtract positive integers

Obj. 235 - Subtract integersNYC M6.b - Knows and uses the correct order of operations for arithmetic computations.


Obj. 29 - Multiply with exponents

Obj. 35 - Order of operationsObj. 36 - Simplify with grouping symbols


Obj. 84 - Order of operations with decimals

NYC M6.c - Estimates numerically and spatially.


Obj. 20 - Estimate whole # sums & differences, roundObj. 25 - Estimate products, round (3+ dig)Obj. 32 - Estimate quotients, compatible #s


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Obj. 60 - Front-end estimate decimal sums w/adjustment


Obj. 80 - Estimate decimal products, roundObj. 82 - Estimate decimal quotients, compatible #s


Obj. 103 - Estimate fraction sums, roundObj. 112 - Estimate fraction differences, round


Obj. 125 - Estimate fraction products & quotients

Topic 17 - Problem Solving Obj. 258 - Make/identify reasonable estimates

NYC M6.d - Measures length, area, volume, weight, time, and temperature accurately.NYC M6.e - Refers to geometric shapes and terms correctly.

Topic 10 - Geometry Obj. 143 - Points, lines, segments, rays, & anglesObj. 144 - Intersecting, parallel, & perpendicular linesObj. 145 - Classify anglesObj. 147 - Identify polygonsObj. 148 - Classify quadrilateralsObj. 149 - Classify triangles by angles & sidesObj. 154 - Name 3-dimensional objectObj. 159 - Parts of a circle



Obj. 245 - Translate sentence to expression/equation




Obj. 185 - Line graphsObj. 186 - Double bar graphsObj. 187 - Double line graphsObj. 188 - Circle graphsObj. 201 - Means of data setsObj. 202 - Medians of data sets

Topic 17 - Problem Solving Obj. 259 - Create Venn diagrams to solve problems


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Topic 17 - Problem Solving Obj. 249 - Solve problems using calculator

NYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Uses mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.


Obj. 241 - Variable expressions to word phrases

Obj. 242 - Equations to word sentencesObj. 245 - Translate sentence to expression/equation

NYC M7.b - Organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.NYC M7.c - Uses mathematical language to make complex situations easier to understand.NYC M7.d - Exhibits developing reasoning abilities by justifying statements and defending work.NYC M7.e - Shows understanding of concepts by explaining ideas not only to teachers and assessors but to fellow students or younger children.


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NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds. A single investigation or project may draw on more than one kind.NYC M8.a - Data study based on civic, economic, or social issues, in which the student:NYC M8.a.i - Data study based on civic, economic, or social issues, in which the student: selects an issue to investigate.NYC M8.a.ii - Data study based on civic, economic, or social issues, in which the student: makes a hypothesis on an expected finding, if appropriate.NYC M8.a.iii - Data study based on civic, economic, or social issues, in which the student: gathers data.


NYC M8.a.v - Data study based on civic, economic, or social issues, in which the student: shows how the study’s results compare with the hypothesis.NYC M8.a.vi - Data study based on civic, economic, or social issues, in which the student: uses pertinent statistics to summarize.


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NYC M8.a.vii - Data study based on civic, economic, or social issues, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.NYC M8.b - Mathematical model of physical phenomena, often used in science studies, in which the student:





NYC M8.c - Design of a physical structure, in which the student:


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NYC M8.c.i - Design of a physical structure, in which the student: generates a plan to build something of value, not necessarily monetary value.NYC M8.c.ii - Design of a physical structure, in which the student: uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes.NYC M8.c.iii - Design of a physical structure, in which the student: summarizes the important features of the structure.NYC M8.c.iv - Design of a physical structure, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling.NYC M8.d.ii - Management and planning, in which the student: notes any constraints that will affect the plan.NYC M8.d.iii - Management and planning, in which the student: determines a plan.NYC M8.d.iv - Management and planning, in which the student: uses concepts from any of Standards 1 to 4, depending on the nature of the project.NYC M8.d.v - Management and planning, in which the student: considers the possibility of a more efficient solution.


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NYC M8.d.vi - Management and planning, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: extends or "plays with," as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers.




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Obj. 12 - Divide whole #s by a power of 10

Topic 2 - Decimals Obj. 33 - Multiply decimalsObj. 34 - Multiply decimals by a power of 10Obj. 35 - Multiply money expressions by whole #sObj. 37 - Divide decimals by whole #sObj. 38 - Divide decimals by a power of 10Obj. 39 - Divide money expressions by whole #sObj. 41 - Divide decimals by decimalsObj. 42 - Divide whole #s by decimals



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Obj. 46 - Add pos and neg decimals

Obj. 47 - Subtract pos and neg decimalsObj. 48 - Multiply pos and neg decimalsObj. 49 - Divide pos and neg decimals

Topic 3 - Fractions Obj. 56 - Add unlike fractionsObj. 57 - Add unlike mixed #sObj. 59 - Subtract unlike fractionsObj. 60 - Subtract unlike mixed #sObj. 61 - Subtract mixed #s from whole #sObj. 65 - Multiply fractionsObj. 66 - Multiply mixed #s by fractionsObj. 67 - Multiply mixed #sObj. 68 - Multiply fractions by whole #sObj. 70 - Divide fractionsObj. 71 - Divide mixed #s

Topic 10 - Integers Obj. 194 - Add integers using models

Obj. 195 - Add integersObj. 196 - Subtract integers using modelsObj. 197 - Subtract positive integers

Obj. 198 - Subtract integersObj. 200 - Multiply integersObj. 201 - Divide integers



Obj. 10 - Square root of perfect squares

Topic 12 - Patterns, Functions, Expressions, and Equations



Topic 8 - Fractions, Percents, and Decimals

Obj. 161 - Decimals to mixed #s & fractions

Obj. 162 - Mixed #s to decimals


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Obj. 163 - Fractions to repeating decimals

Topic 11 - Number Representation

Obj. 204 - Negative/zero exponents as fraction/decimal




Obj. 2 - Factors of composite numbersObj. 3 - Divisibility rulesObj. 4 - Prime factorizationObj. 7 - Properties of addition & multiplication

NYC M1.e - Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.


Obj. 171 - Ratios as percents

Obj. 181 - Percent of decreaseObj. 182 - Percent of increaseObj. 183 - Percent of a number



Obj. 169 - Ratios as fractions, lowest terms

Obj. 170 - Ratios as rates, lowest termsObj. 171 - Ratios as percentsObj. 173 - Unit ratesObj. 175 - Ratios as proportionsObj. 176 - Write ratios


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Topic 2 - Decimals Obj. 17 - Compare & order decimals

Topic 3 - Fractions Obj. 50 - Compare & order fractions

Topic 11 - Number Representation

Obj. 207 - Compare #s in scientific notation




Obj. 90 - Classify quadrilateralsObj. 91 - Classify triangles by angles & sidesObj. 99 - Classify plane or solid figuresObj. 101 - Solids



Obj. 107 - TranslationsObj. 108 - ReflectionsObj. 109 - Rotations


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NYC M2.c - Identifies three dimensional shapes from two dimensional perspectives; draws two dimensional sketches of three dimensional objects that preserve significant features.NYC M2.d - Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.


Obj. 113 - Perimeter of polygons

Obj. 114 - WP: Perimeter of polygons

Obj. 119 - Area of rectangles & squaresObj. 120 - WP: Areas of rectangles & squaresObj. 121 - Area of trapezoidsObj. 122 - Area of trianglesObj. 123 - Area of parallelogramsObj. 124 - Area of circlesObj. 125 - WP: Area of circlesObj. 127 - Volume of rectangular prisms & cubesObj. 128 - WP: Volume of rectangular prisms & cubesObj. 129 - Volume of triangular prismsObj. 132 - Surface area of triangular prismsObj. 133 - Surface area of cylinders

Obj. 134 - Surface area of rectangular prisms





Obj. 210 - Extend geometric patterns


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Topic 5 - Geometry Obj. 87 - Measure angles


Obj. 143 - Time schedules


Topic 4 - Measurement Obj. 78 - Convert customary length


Obj. 80 - Convert customary capacity

Obj. 81 - Convert metric lengthObj. 82 - Convert metric weightObj. 83 - Convert metric capacity


Obj. 178 - Convert Currency

NYC M2.i - Reasons proportionally in situations with similar figures.NYC M2.j - Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings.


Obj. 179 - Scale drawings

Obj. 180 - Use map scale to calculate distance

NYC M2.k - Models situations geometrically to formulate and solve problems.NYC M3 - FUNCTION AND ALGEBRA CONCEPTSNYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:



Obj. 216 - Interpret graphs of linear functions


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Obj. 212 - Function tables

NYC M3.c - Analyzes tables, graphs, and rules to determine functional relationships.NYC M3.d - Finds solutions for unknown quantities in linear equations and in simple equations and inequalities.


Obj. 219 - Identify solutions to equations

Obj. 220 - Solve addition & subtraction equationsObj. 221 - Solve multiplication & division equationsObj. 222 - Solve 2-step equationsObj. 223 - Solve multi-step equations

Obj. 224 - Solve inequalities, pos integer coefficientsObj. 225 - Solve inequalities, pos rational coefficientsObj. 226 - Solve inequalities, neg coefficients





Obj. 156 - Medians of data setsObj. 157 - Modes of data setsObj. 158 - Ranges of data sets



Obj. 159 - Identify best measure of central tendency


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NYC M4.d - Makes conclusions and recommendations based on data analysis.


Obj. 160 - Critique conclusions

NYC M4.e - Critiques the conclusions and recommendations of others’ statistics.


Obj. 160 - Critique conclusions

NYC M4.f - Considers the effects of missing or incorrect information.NYC M4.g - Formulates hypotheses to answer a question and uses data to test hypotheses.NYC M4.h - Represents and determines probability as a fraction of a set of equally likely outcomes; recognizes equally likely outcomes, and constructs sample spaces (including those described by numerical combinations and permutations).NYC M4.i - Makes predictions based on experimental or theoretical probabilities.



Obj. 149 - Probability of complements of eventsObj. 150 - Probability of independent & dependent eventsObj. 151 - Probability of compound eventsObj. 152 - Odds of an event


NYC M5-1 - The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process--formulation, implementation, and conclusion.


Obj. 15 - WP: 2-step with whole #s

Topic 2 - Decimals Obj. 24 - WP: Round decimal sum & diff to nearest whole #Obj. 26 - WP: Round decimal sum & diff to greatest place


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Obj. 28 - WP: Front-end estimate decimal sums w/adjustmentObj. 30 - WP: Estimate decimal products, round


Obj. 184 - WP: Percent of a number

Obj. 186 - WP: Find total when percent is knownObj. 188 - WP: Find percent when total is knownObj. 189 - WP: DiscountsObj. 190 - WP: Simple interest

Topic 10 - Integers Obj. 199 - WP: Add & subtract integersObj. 202 - WP: Multiply & divide integers

- FORMULATIONNYC M5.a - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student:

NYC M5.a.i - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: formulates and solves a variety of meaningful problems.NYC M5.a.ii - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: extracts pertinent information from situations and figures out what additional information is needed. - IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: uses and invents a variety of approaches and understands and evaluates those of others.


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NYC M5.b.ii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table.




- CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:


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NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: verifies and interprets results with respect to the original problem situation.NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: generalizes solutions and strategies to new problem situations.

- MATHEMATICAL REASONINGNYC M5.d - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student:NYC M5.d.i - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: formulates conjectures and argues why they must be or seem true.NYC M5.d.ii - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: makes sensible, reasonable estimates.





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Obj. 210 - Extend geometric patterns

Topic 13 - Problem Solving Obj. 231 - Support conjecturesNYC M6 - MATHEMATICAL SKILLS AND TOOLSNYC M6-1 - The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task (perhaps referring to notes, or books, perhaps working to reconstruct a method); that is, the student:




Obj. 12 - Divide whole #s by a power of 10

Topic 2 - Decimals Obj. 33 - Multiply decimalsObj. 34 - Multiply decimals by a power of 10Obj. 35 - Multiply money expressions by whole #sObj. 37 - Divide decimals by whole #sObj. 38 - Divide decimals by a power of 10Obj. 39 - Divide money expressions by whole #sObj. 40 - WP: Divide money expressionsObj. 41 - Divide decimals by decimalsObj. 42 - Divide whole #s by decimalsObj. 43 - WP: Divide decimalsObj. 46 - Add pos and neg decimals

Obj. 47 - Subtract pos and neg decimalsObj. 48 - Multiply pos and neg decimalsObj. 49 - Divide pos and neg decimals

Topic 3 - Fractions Obj. 56 - Add unlike fractions


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Obj. 57 - Add unlike mixed #sObj. 58 - WP: Add fractions & mixed #sObj. 59 - Subtract unlike fractionsObj. 60 - Subtract unlike mixed #sObj. 61 - Subtract mixed #s from whole #sObj. 62 - WP: Subtract fractions & mixed #sObj. 65 - Multiply fractionsObj. 66 - Multiply mixed #s by fractionsObj. 67 - Multiply mixed #sObj. 68 - Multiply fractions by whole #sObj. 69 - WP: Multiply fractions & mixed #sObj. 70 - Divide fractionsObj. 71 - Divide mixed #sObj. 72 - WP: Divide fractions

Topic 10 - Integers Obj. 195 - Add integersObj. 196 - Subtract integers using modelsObj. 197 - Subtract positive integers

Obj. 198 - Subtract integersObj. 199 - WP: Add & subtract integersObj. 200 - Multiply integersObj. 201 - Divide integersObj. 202 - WP: Multiply & divide integers

NYC M6.b - Knows and uses the correct order of operations for arithmetic computations.

Topic 2 - Decimals Obj. 45 - Order of operations with decimals


Topic 2 - Decimals Obj. 27 - Front-end estimate decimal sums w/adjustmentObj. 28 - WP: Front-end estimate decimal sums w/adjustmentObj. 29 - Estimate decimal products, roundObj. 30 - WP: Estimate decimal products, roundObj. 31 - Estimate decimal quotients, compatible #sObj. 32 - WP: Estimate decimal quotients, compatible #s

Topic 3 - Fractions Obj. 53 - Estimate fraction sum & diff, round


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Obj. 54 - WP: Estimate fraction sums, roundObj. 55 - WP: Estimate fraction differences, roundObj. 63 - Estimate fraction products & quotientsObj. 64 - WP: Estimate fraction products & quotients


Topic 5 - Geometry Obj. 86 - Classify angles

Obj. 89 - Identify polygonsObj. 90 - Classify quadrilateralsObj. 91 - Classify triangles by angles & sidesObj. 100 - Identify parts of right triangleObj. 103 - Count faces, edges, & vertices

NYC M6.f - Uses equations, formulas, and simple algebraic notation appropriately.NYC M6.g - Reads and organizes data on charts and graphs, including scatter plots, bar, line, and circle graphs, and Venn diagrams; calculates mean and median.



Obj. 136 - Double line graphsObj. 137 - Circle graphsObj. 138 - Frequency tablesObj. 139 - Line plotsObj. 140 - HistogramsObj. 141 - Stem-and-leaf plotsObj. 142 - Box-and-whisker plotsObj. 155 - Means of data setsObj. 156 - Medians of data sets




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NYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Uses mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.NYC M7.b - Organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.NYC M7.c - Uses mathematical language to make complex situations easier to understand.NYC M7.d - Exhibits developing reasoning abilities by justifying statements and defending work.NYC M7.e - Shows understanding of concepts by explaining ideas not only to teachers and assessors but to fellow students or younger children.


NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds. A single investigation or project may draw on more than one kind.


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NYC M8.a - Data study based on civic, economic, or social issues, in which the student:NYC M8.a.i - Data study based on civic, economic, or social issues, in which the student: selects an issue to investigate.NYC M8.a.ii - Data study based on civic, economic, or social issues, in which the student: makes a hypothesis on an expected finding, if appropriate.NYC M8.a.iii - Data study based on civic, economic, or social issues, in which the student: gathers data.




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NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: generates a plan to build something of value, not necessarily monetary value.NYC M8.c.ii - Design of a physical structure, in which the student: uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes.


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NYC M8.c.iii - Design of a physical structure, in which the student: summarizes the important features of the structure.NYC M8.c.iv - Design of a physical structure, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling.NYC M8.d.ii - Management and planning, in which the student: notes any constraints that will affect the plan.NYC M8.d.iii - Management and planning, in which the student: determines a plan.NYC M8.d.iv - Management and planning, in which the student: uses concepts from any of Standards 1 to 4, depending on the nature of the project.NYC M8.d.v - Management and planning, in which the student: considers the possibility of a more efficient solution.NYC M8.d.vi - Management and planning, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.NYC M8.e - Pure mathematics investigation, in which the student:


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NYC M8.e.i - Pure mathematics investigation, in which the student: extends or "plays with," as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers.




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Topic 1 - Integers Obj. 5 - Add integers

Obj. 7 - Subtract integersObj. 11 - Multiply integersObj. 12 - Divide integers

Topic 3 - Rational Number Operations


Obj. 31 - Add fractions & mixed #sObj. 32 - Subtract decimalsObj. 33 - Subtract fractions & mixed #sObj. 34 - Multiply decimalsObj. 35 - Multiply fractions & mixed #sObj. 36 - Divide decimalsObj. 37 - Divide fractions & mixed #s

Obj. 38 - Operations w/fractions & mixed-numbers, pos/neg



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Topic 7 - Roots and Exponents Obj. 94 - Square integersObj. 95 - Raise integers to power


Topic 1 - Integers Obj. 6 - Write a difference as a sum

Topic 5 - Linear Equations Obj. 59 - Solve 1-step eqns, add & subtract (integers)Obj. 60 - Solve 1-step eqns, add & subt (decimal & fraction)Obj. 63 - Solve 1-step eqns, multiply & divide by reciprocalObj. 64 - Solve 1-step eqns, multiply & divide (decimals)Obj. 66 - Solve 2-step eqns (integers)

Obj. 67 - Solve 2-step eqns (decimal & fraction)Obj. 69 - Solve multi-step equations

Topic 7 - Roots and Exponents Obj. 110 - Use square roots to solve equations


Topic 1 - Integers Obj. 10 - WP: Add & subtract integers

Obj. 13 - WP: Multiply & divide integers


Obj. 41 - WP: Multiply & divide rational #sObj. 43 - WP: Add & subtract rational #s

Topic 8 - Ratio, Proportion, and Percent

Obj. 114 - WP: Proportions

Obj. 115 - Decimals to percentsObj. 116 - Fractions to percentsObj. 117 - Percents to decimalsObj. 118 - Percents to fractionsObj. 119 - Greater and lesser percentsObj. 121 - WP: Percents and taxes

Obj. 125 - WP: Percent


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Obj. 126 - WP: Discount or markup

Obj. 127 - WP: Percent changeObj. 128 - WP: Simple interestObj. 129 - WP: Commission, Interest, Gratuities


Topic 1 - Integers Obj. 3 - Opposites of integers

Topic 2 - Number Theory Obj. 15 - Divisibility rulesObj. 16 - Prime & composite numbersObj. 17 - Prime factorizationObj. 18 - Greatest common factorObj. 19 - Least common multiple

Topic 4 - Variables, Expressions, and Functions

Obj. 46 - Properties of addition & multiplicationObj. 47 - Distributive property with variables

NYC M1.e - Interprets percent as part of 100 and as a means of comparing quantities of different sizes or changing sizes.NYC M1.f - Uses ratios and rates to express "part-to-part" and "whole-to-whole" relationships, and reasons proportionally to solve problems involving equivalent fractions, equal ratios, or constant rates, recognizing the multiplicative nature of these problems in the constant factor of change.


Obj. 111 - Ratios as fractions, lowest terms

Obj. 112 - Word statements to proportionsObj. 113 - Solve proportionsObj. 114 - WP: Proportions


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Topic 1 - Integers Obj. 1 - Graph integers on number lines

Obj. 2 - Compare & order integersTopic 3 - Rational Number Operations

Obj. 27 - Compare & order decimals

Obj. 29 - Compare & order fractions


Obj. 123 - WP: Estimate the percent of a quantity



Topic 11 - Geometry Obj. 162 - Identify polygons & quadrilaterals

Obj. 163 - SolidsObj. 164 - Nets for solidsObj. 173 - Classify angles & triangles


Topic 11 - Geometry Obj. 166 - Transl, rot, refl, & dilations

Obj. 174 - Similarity & congruence


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Topic 11 - Geometry Obj. 164 - Nets for solids

Obj. 165 - Isometric & orthographic drawings

NYC M2.d - Determines and understands length, area, and volume (as well as the differences among these measurements), including perimeter and surface area; uses units, square units, and cubic units of measure correctly; computes areas of rectangles, triangles, and circles; computes volumes of prisms.


Obj. 183 - Perimeter of quadrilaterals

Obj. 184 - WP: PerimeterObj. 185 - Circumference of circlesObj. 187 - Area of rectangles & squaresObj. 188 - Area of parallelogramsObj. 189 - Area of trianglesObj. 191 - Area of circlesObj. 192 - Surface area of prismsObj. 195 - Volume of rectangular solids



Topic 2 - Number Theory Obj. 23 - Write expression describing a pattern


Topic 11 - Geometry Obj. 157 - Measure, draw, & name angles


Topic 12 - Measurement Obj. 177 - Convert customary units

Obj. 178 - Convert metric unitsObj. 179 - Convert complex units


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NYC M2.i - Reasons proportionally in situations with similar figures.

Topic 11 - Geometry Obj. 175 - Similar triangles & proportions

NYC M2.j - Reasons proportionally with measurements to interpret maps and to make smaller and larger scale drawings.

Topic 12 - Measurement Obj. 182 - Scale drawings

NYC M2.k - Models situations geometrically to formulate and solve problems.

Topic 11 - Geometry Obj. 176 - WP: Similar triangles & proportions





Obj. 53 - Function tables



Obj. 44 - Word phrases to algebraic expressions

Obj. 56 - Multiple representations of linear equations

Topic 5 - Linear Equations Obj. 74 - Graph lines from standard formObj. 77 - Determine graph from equation or situation

Topic 6 - Linear Inequalities Obj. 92 - Graph linear inequalities (coordinate plane)



Obj. 54 - Determine if a relation is a function


Topic 5 - Linear Equations Obj. 58 - Verify solutions to equations

Obj. 59 - Solve 1-step eqns, add & subtract (integers)Obj. 60 - Solve 1-step eqns, add & subt (decimal & fraction)


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Obj. 63 - Solve 1-step eqns, multiply & divide by reciprocalObj. 64 - Solve 1-step eqns, multiply & divide (decimals)Obj. 66 - Solve 2-step eqns (integers)

Obj. 67 - Solve 2-step eqns (decimal & fraction)Obj. 82 - Verify solutions to linear equations

Topic 6 - Linear Inequalities Obj. 85 - Solve 1-step inequalitiesNYC M4 - STATISTICS AND PROBABILITY CONCEPTSNYC M4-1 - The student produces evidence that demonstrates understanding of statistics and probability concepts; that is, the student:NYC M4.a - Collects data, organizes data, and displays data with tables, charts, and graphs that are appropriate, i.e., consistent with the nature of the data.

Topic 9 - Statistics and Graphs Obj. 136 - Histograms


Topic 9 - Statistics and Graphs Obj. 139 - Box-and-whisker plots


Topic 9 - Statistics and Graphs Obj. 130 - Means, medians, modes, & ranges for data sets

NYC M4.d - Makes conclusions and recommendations based on data analysis.

Topic 9 - Statistics and Graphs Obj. 141 - Analyze conclusions from data, graphs

NYC M4.e - Critiques the conclusions and recommendations of others’ statistics.NYC M4.f - Considers the effects of missing or incorrect information.

Topic 9 - Statistics and Graphs Obj. 140 - Misleading graphs

NYC M4.g - Formulates hypotheses to answer a question and uses data to test hypotheses.

Topic 9 - Statistics and Graphs Obj. 142 - Use data to test hypotheses


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NYC M4.h - Represents and determines probability as a fraction of a set of equally likely outcomes; recognizes equally likely outcomes, and constructs sample spaces (including those described by numerical combinations and permutations).

Topic 10 - Probability Obj. 150 - Probability of single events

NYC M4.i - Makes predictions based on experimental or theoretical probabilities.NYC M4.j - Predicts the result of a series of trials once the probability for one trial is known.

Topic 10 - Probability Obj. 155 - Experimental and sample-based probability



NYC M5.a.i - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: formulates and solves a variety of meaningful problems.NYC M5.a.ii - The student participates in the formulation of problems; that is, given the basic statement of a problem situation, the student: extracts pertinent information from situations and figures out what additional information is needed. - IMPLEMENTATION


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NYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: uses and invents a variety of approaches and understands and evaluates those of others.NYC M5.b.ii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: invokes problem solving strategies, such as illustrating with sense-making sketches to clarify situations or organizing information in a table.



Obj. 41 - WP: Multiply & divide rational #s

Topic 5 - Linear Equations Obj. 68 - WP: Solve multi-step equations

Topic 6 - Linear Inequalities Obj. 89 - WP: InequalitiesTopic 8 - Ratio, Proportion, and Percent

Obj. 114 - WP: Proportions

Obj. 121 - WP: Percents and taxes

Obj. 125 - WP: PercentObj. 126 - WP: Discount or markup

Obj. 127 - WP: Percent changeObj. 128 - WP: Simple interestObj. 129 - WP: Commission, Interest, Gratuities

Topic 10 - Probability Obj. 148 - WP: Permutations


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Obj. 149 - WP: CombinationsTopic 11 - Geometry Obj. 161 - WP: Angles formed by

transversalObj. 176 - WP: Similar triangles & proportions


Obj. 194 - WP: Area

Obj. 200 - WP: VolumeTopic 14 - Right Triangles Obj. 203 - WP: Pythagorean

TheoremTopic 16 - Problem Solving Obj. 223 - Use similar problem or

partial steps to solve probObj. 224 - Work backward to solve a problemObj. 225 - Use guess & check, drawing, patterns, simpler probObj. 226 - Solve problems with given constraints



- CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: verifies and interprets results with respect to the original problem situation.


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NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: generalizes solutions and strategies to new problem situations.

- MATHEMATICAL REASONINGNYC M5.d - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student:NYC M5.d.i - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: formulates conjectures and argues why they must be or seem true.NYC M5.d.ii - The student demonstrates mathematical reasoning by generalizing patterns, making conjectures and explaining why they seem true, and by making sensible, justifiable statements; that is, the student: makes sensible, reasonable estimates.


Obj. 42 - WP: Estimate rational #s






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Topic 1 - Integers Obj. 5 - Add integers

Obj. 7 - Subtract integersObj. 10 - WP: Add & subtract integersObj. 11 - Multiply integersObj. 12 - Divide integersObj. 13 - WP: Multiply & divide integers



Obj. 31 - Add fractions & mixed #sObj. 32 - Subtract decimalsObj. 33 - Subtract fractions & mixed #sObj. 34 - Multiply decimalsObj. 35 - Multiply fractions & mixed #sObj. 36 - Divide decimalsObj. 37 - Divide fractions & mixed #s

Obj. 38 - Operations w/fractions & mixed-numbers, pos/negObj. 41 - WP: Multiply & divide rational #sObj. 43 - WP: Add & subtract rational #s


Obj. 48 - Evaluate expressions

Topic 7 - Roots and Exponents Obj. 95 - Raise integers to powerObj. 96 - Evaluate expressions with exponents

NYC M6.b - Knows and uses the correct order of operations for arithmetic computations.

Topic 1 - Integers Obj. 14 - Order of operations with integers


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Obj. 40 - Estimate rational #s

Obj. 42 - WP: Estimate rational #sTopic 8 - Ratio, Proportion, and Percent



Topic 11 - Geometry Obj. 162 - Identify polygons & quadrilateralsObj. 163 - SolidsObj. 173 - Classify angles & triangles




Obj. 44 - Word phrases to algebraic expressions

Topic 6 - Linear Inequalities Obj. 84 - Translate with inequalities



Obj. 134 - Bar graphsObj. 136 - HistogramsObj. 137 - Circle graphsObj. 138 - Stem-and-leaf plots


NYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:


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NYC M7.a - Uses mathematical language and representations with appropriate accuracy, including numerical tables and equations, simple algebraic equations and formulas, charts, graphs, and diagrams.NYC M7.b - Organizes work, explains facets of a solution orally and in writing, labels drawings, and uses other techniques to make meaning clear to the audience.NYC M7.c - Uses mathematical language to make complex situations easier to understand.NYC M7.d - Exhibits developing reasoning abilities by justifying statements and defending work.NYC M7.e - Shows understanding of concepts by explaining ideas not only to teachers and assessors but to fellow students or younger children.


NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of middle school, conducts investigations or projects drawn from three of the kinds. A single investigation or project may draw on more than one kind.NYC M8.a - Data study based on civic, economic, or social issues, in which the student:NYC M8.a.i - Data study based on civic, economic, or social issues, in which the student: selects an issue to investigate.


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NYC M8.a.ii - Data study based on civic, economic, or social issues, in which the student: makes a hypothesis on an expected finding, if appropriate.NYC M8.a.iii - Data study based on civic, economic, or social issues, in which the student: gathers data.



Obj. 133 - Frequency distributions & line plots




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NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: generates a plan to build something of value, not necessarily monetary value.NYC M8.c.ii - Design of a physical structure, in which the student: uses mathematics from Standard 2 to make the design realistic or appropriate, e.g., areas and volumes in general and of specific geometric shapes.NYC M8.c.iii - Design of a physical structure, in which the student: summarizes the important features of the structure.


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NYC M8.c.iv - Design of a physical structure, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.NYC M8.d - Management and planning, in which the student:NYC M8.d.i - Management and planning, in which the student: determines the needs of the event to be managed or planned, e.g., cost, supply, scheduling.NYC M8.d.ii - Management and planning, in which the student: notes any constraints that will affect the plan.NYC M8.d.iii - Management and planning, in which the student: determines a plan.NYC M8.d.iv - Management and planning, in which the student: uses concepts from any of Standards 1 to 4, depending on the nature of the project.NYC M8.d.v - Management and planning, in which the student: considers the possibility of a more efficient solution.NYC M8.d.vi - Management and planning, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the plan.NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: extends or "plays with," as with mathematical puzzles, some mathematical feature, e.g., properties and patterns in numbers.


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NYC M8.e.iii - Pure mathematics investigation, in which the student: determines and expresses generalizations from patterns.


NYC M8.e.iv - Pure mathematics investigation, in which the student: makes conjectures on apparent properties and argues, short of formal proof, why they seem true.NYC M8.e.v - Pure mathematics investigation, in which the student: prepares a presentation or report that includes the question investigated, a detailed description of how the project was carried out, and an explanation of the findings.


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Standard Topic Description Objective DescriptionNYC M1 - ARITHMETIC AND NUMBER CONCEPTS/NUMBER AND OPERATION CONCEPTSNYC M1-1 - The student produces evidence that demonstrates understanding of number and operation concepts; that is, the student:NYC M1.a - Uses addition, subtraction, multiplication, division, exponentiation, and root-extraction in forming and working with numerical and algebraic expressions.

Topic 3 - Exponents Obj. 41 - Evaluate expressions, positive exponents

Topic 6 - Polynomial Operations Obj. 92 - Add polynomials

Obj. 93 - Subtract polynomialsObj. 94 - Add/subtract 3 polynomials

Obj. 96 - Multiply monomials by polynomialsObj. 97 - Multiply 2 binomials, (x + a)(x + b)Obj. 98 - Multiply 2 binomials, (ax + b)(cx + d)Obj. 99 - Multiply 2 binomials, (ax + by)(cx + dy)Obj. 100 - Square a binomialObj. 102 - Multiply binomials by trinomialsObj. 103 - Cube binomials, (x + a)Obj. 104 - Expand binomialsObj. 105 - Simplify rational expr containing polynomialObj. 106 - Multiply, divide monomials w/ common base (variables, integers exponents)Obj. 107 - Divide polynomials by monomialsObj. 108 - Divide polynomials by binomialsObj. 109 - Divide polynomials by quadratics

Topic 8 - Relations and Functions

Obj. 132 - Add, sub, mult, & div functions

Topic 12 - Rational Expressions and Equations

Obj. 177 - Rational expr, rewrite with given denominatorObj. 179 - Multiply rational expressions

Standards List with Aligned Product SkillsProduct NameAM NY Custom Algebra1


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Obj. 180 - Divide rational expressions

Obj. 181 - Multiply & divide 3 rational expressionsObj. 182 - Add rational expressions (like denominators)Obj. 183 - Add rational expressions

Obj. 184 - Subtract rational expressions (like denominators)Obj. 185 - Subtract rational expressionsObj. 187 - Solve equations containing rational expressionsObj. 189 - Algebraic proportions (linear, quadratic solns)

NYC M1.b - Understands and uses operations such as opposite, reciprocal, raising to a power, taking a root, and taking a logarithm.

Topic 2 - Introduction to Algebra Obj. 30 - Additive inverses of rational #s

Obj. 31 - Reciprocals of rational #sTopic 9 - Quadratic Equations Obj. 135 - Solve quadratic eqns,

square root ruleNYC M1.c - Has facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications.

Topic 1 - Integers and Rational Numbers

Obj. 10 - Distributive property

Obj. 11 - Order of operationsObj. 12 - Order of operations with grouping

Topic 2 - Introduction to Algebra Obj. 29 - Absolute values of real #s

Topic 3 - Exponents Obj. 40 - Write repeated mult as powerObj. 44 - Simplify expressions using properties of exponentsObj. 50 - Division & power properties of exponentsObj. 51 - WP: Evaluate exponential expressions

NYC M1.d - Understands and uses number systems: natural, integer, rational, and real.

Topic 2 - Introduction to Algebra Obj. 28 - Order real #s


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NYC M1.e - Represents numbers in decimal or fraction form and in scientific notation, and graphs numbers on the number line and number pairs in the coordinate plane.


Obj. 1 - Graph integers on number lines

Obj. 2 - Interpret scientific calculator displays

Topic 5 - Graph and Write Linear Equations

Obj. 66 - Graph ordered pairs

NYC M1.f - Compares numbers using order relations, differences, ratios, proportions, percents, and proportional change.


Topic 14 - Ratio and Proportion Obj. 201 - Ratios as fractions, lowest termsObj. 202 - Equivalent ratiosObj. 205 - Ratios as proportions

NYC M1.g - Carries out proportional reasoning in cases involving part-whole relationships and in cases involving expansions and contractions.

Topic 14 - Ratio and Proportion Obj. 201 - Ratios as fractions, lowest terms

Obj. 202 - Equivalent ratiosObj. 203 - Unit ratesObj. 205 - Ratios as proportionsObj. 206 - Word statements to proportionsObj. 207 - Solve proportionsObj. 208 - WP: Solve proportionsObj. 209 - Solve direct & indirect variation problems

NYC M1.h - Understands dimensionless numbers, such as proportions, percents, and multiplicative factors, as well as numbers with specific units of measure, such as numbers with length, time, and rate units.

Topic 14 - Ratio and Proportion Obj. 201 - Ratios as fractions, lowest terms

Obj. 203 - Unit ratesObj. 205 - Ratios as proportions

Topic 15 - Data and Statistics Obj. 232 - Percentile rankNYC M1.i - Carries out counting procedures such as those involving sets (unions and intersections) and arrangements (permutations and combinations).


Obj. 24 - Complement of a given subset

Obj. 25 - Union and intersection of sets


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Topic 15 - Data and Statistics Obj. 234 - Determine number of outcomes/eventsObj. 235 - Permutations

NYC M1.j - Uses concepts such as prime, relatively prime, factor, divisor, multiple, and divisibility in solving problems involving integers.


Obj. 7 - WP: Multiply integers

NYC M1.k - Uses a scientific calculator effectively and efficiently in carrying out complex calculations.



NYC M1.l - Recognizes and represents basic number patterns, such as patterns involving multiples, squares, or cubes.


Obj. 133 - Patterns and their equations

NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:NYC M2.a - Models situations geometrically to formulate and solve problems.


Obj. 73 - Plot points to write or graph eqns


Obj. 134 - Identify function type from graph

Topic 9 - Quadratic Equations Obj. 142 - Graph to find quadratic roots

Topic 11 - Inequalities and Absolute Value

Obj. 168 - Solve & graph inequalities

Obj. 169 - Solve & graph compound inequalities

NYC M2.b - Works with two- and three-dimensional figures and their properties, including polygons and circles, cubes and pyramids, and cylinders, cones, and spheres.

Topic 9 - Quadratic Equations Obj. 146 - WP: Pythagorean problems

Topic 16 - Measurement Obj. 240 - Find angle measure in right triangle, given sidesObj. 241 - Find side of right triangle, given 1 angle/1 side

NYC M2.c - Uses congruence and similarity in describing relationships between figures.


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NYC M2.d - Visualizes objects, paths, and regions in space, including intersections and cross sections of three dimensional figures, and describes these using geometric language.NYC M2.e - Knows, uses, and derives formulas for perimeter, circumference, area, surface area, and volume of many types of figures.

Topic 9 - Quadratic Equations Obj. 148 - WP: Area & perimeter

NYC M2.f - Uses the Pythagorean Theorem in many types of situations, and works through more than one proof of this theorem.

Topic 9 - Quadratic Equations Obj. 146 - WP: Pythagorean problems

NYC M2.g - Works with similar triangles, and extends the ideas to include simple uses of the three basic trigonometric functions.NYC M2.h - Analyzes figures in terms of their symmetries using, for example, concepts of reflection, rotation, and translation.NYC M2.i - Compares slope (rise over run) and angle of elevation as measures of steepness.


Obj. 68 - Find slopes from 2 points

Obj. 69 - Find slopes from a graphNYC M2.j - Investigates geometric patterns, including sequences of growing shapes.



NYC M2.k - Works with geometric measures of length, area, volume, and angle; and non-geometric measures such as weight and time.

Topic 16 - Measurement Obj. 236 - Enlargement: Surface area and volume

Obj. 237 - Conversion within meas systemsObj. 238 - Relative error in square, cubic unitsObj. 239 - Find sin/cos/tan ratiosObj. 240 - Find angle measure in right triangle, given sidesObj. 241 - Find side of right triangle, given 1 angle/1 side

NYC M2.l - Uses quotient measures, such as speed and density, that give "per unit" amounts; and uses product measures, such as person-hours.

Topic 14 - Ratio and Proportion Obj. 203 - Unit rates


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NYC M2.m - Understands the structure of standard measurement systems, both SI and customary, including unit conversions and dimensional analysis.

Topic 16 - Measurement Obj. 237 - Conversion within meas systems

NYC M2.n - Solves problems involving scale, such as in maps and diagrams.NYC M2.o - Represents geometric curves and graphs of functions in standard coordinate systems.NYC M2.p - Analyzes geometric figures and proves simple things about them using deductive methods.

NYC M2.q - Explores geometry using computer programs such as CAD software, Sketchpad programs, or LOGO.NYC M3 - FUNCTION AND ALGEBRA CONCEPTSNYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:

NYC M3.a - Models given situations with formulas and functions, and interprets given formulas and functions in terms of situations.

Topic 9 - Quadratic Equations Obj. 141 - Graph quadratic equations

Obj. 146 - WP: Pythagorean problemsObj. 147 - WP: Number problemsObj. 148 - WP: Area & perimeter

Topic 10 - Systems of Equations Obj. 156 - WP: Solve systems, number problemsObj. 157 - WP: Solve systems, motion problemsObj. 158 - WP: Solve systems, age problemsObj. 159 - WP: Solve systems, investment problemsObj. 160 - WP: Solve systems, mixture problems


Obj. 164 - WP: Solve 1-var inequalitiesObj. 166 - WP: Identify absolute value inequalities


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Obj. 175 - WP: Solve 2-var ineq using linear programming


Obj. 188 - WP: Rational expressions

Topic 13 - Radical Expressions and Equations

Obj. 199 - WP: Solve radical equations

Topic 14 - Ratio and Proportion Obj. 204 - WP: Calculate ratesNYC M3.b - Describes, generalizes, and uses basic types of functions: linear, exponential, power, rational, square and square root, and cube and cube root.


Obj. 123 - Domain & range, relations & functions

Obj. 125 - Determine if functions are one-to-oneObj. 130 - Find specified values of functionsObj. 134 - Identify function type from graph


Obj. 174 - Max or min functions with constraints

NYC M3.c - Utilizes the concepts of slope, evaluation, and inverse in working with functions.


Obj. 70 - Find slopes from equations

Obj. 74 - Write eqns in standard/slope-intercept formObj. 75 - Write eqns given slope & y-intObj. 82 - Graph lines, slope-intercept form


Obj. 129 - Find inverses of relations

NYC M3.d - Works with rates of many kinds, expressed numerically, symbolically, and graphically.


Obj. 71 - Slope as rate of change btw indep./dep. variables

Topic 14 - Ratio and Proportion Obj. 203 - Unit ratesObj. 204 - WP: Calculate ratesObj. 209 - Solve direct & indirect variation problems

NYC M3.e - Represents constant rates as the slope of a straight line graph, and interprets slope as the amount of one quantity (y) per unit amount of another (x).



Obj. 71 - Slope as rate of change btw indep./dep. variables


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NYC M3.f - Understands and uses linear functions as a mathematical representation of proportional relationships.

Topic 4 - Solving Linear Equations

Obj. 64 - WP: Mixture problems

NYC M3.g - Uses arithmetic sequences and geometric sequences and their sums, and sees these as the discrete forms of linear and exponential functions, respectively.



NYC M3.h - Defines, uses, and manipulates expressions involving variables, parameters, constants, and unknowns in work with formulas, functions, equations, and inequalities.

Topic 2 - Introduction to Algebra Obj. 35 - Algebraic expressions vs. algebraic equations

Topic 3 - Exponents Obj. 41 - Evaluate expressions, positive exponentsObj. 44 - Simplify expressions using properties of exponentsObj. 47 - Divide expressions with positive exponents


Obj. 186 - Simplify complex fractions


Obj. 190 - Radical expressions, simplifyObj. 193 - Radical expressions, multiply then simplifyObj. 194 - Radical expressions, multiply 2 binomialsObj. 195 - Radical expressions, divide

NYC M3.i - Represents functional relationships in formulas, tables, and graphs, and translates between pairs of these.


Obj. 87 - Graph absolute value equations




NYC M3.j - Solves equations symbolically, graphically, and numerically, especially linear, quadratic, and exponential equations; and knows how to use the quadratic formula for solving quadratic equations.

Topic 3 - Exponents Obj. 51 - WP: Evaluate exponential expressions


Obj. 57 - Solve 1-step eqns, add & subtract


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Obj. 58 - Solve 1-step eqns, multiply & divideObj. 59 - Solve multi-step equations

Obj. 60 - Solve eqns, combine like termsObj. 61 - Solve eqns, combine like terms with parenthesesObj. 62 - Solve eqns, multiply by LCD

Obj. 63 - Solve formulas for specified variables


Obj. 86 - Solve absolute value equations

Topic 9 - Quadratic Equations Obj. 135 - Solve quadratic eqns, square root ruleObj. 136 - Solve quadratic equations (integer)Obj. 137 - Solve quadratic equations (rational)Obj. 139 - Solve quadratic eqns, complete the squareObj. 140 - Solve quadratic eqns, quadratic formula


Obj. 187 - Solve equations containing rational expressions


Obj. 198 - Solve equations containing radicalsObj. 200 - Solve equations containing multiple radicals

Topic 14 - Ratio and Proportion Obj. 207 - Solve proportionsNYC M3.k - Makes predictions by interpolating or extrapolating from given data or a given graph.

Topic 15 - Data and Statistics Obj. 223 - Use line of best fit to make predictions

NYC M3.l - Understands the basic algebraic structure of number systems.



Obj. 3 - Add integersObj. 4 - Subtract integersObj. 6 - Multiply integersObj. 8 - Divide integersObj. 13 - Raise integers to powersObj. 14 - Add rational #s (fraction)Obj. 15 - Subtract rational #s (decimal)Obj. 17 - Multiply rational #s (decimal)Obj. 18 - Divide rational #s (fraction)


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Obj. 29 - Absolute values of real #s

Obj. 30 - Additive inverses of rational #sObj. 31 - Reciprocals of rational #sObj. 36 - Combine like termsObj. 37 - Combine like terms with parentheses

Topic 3 - Exponents Obj. 52 - Standard notation to scientific notationObj. 53 - Scientific notation to standard notationObj. 54 - Scientific notation, multiply & divide


Obj. 176 - Rational expressions, domains


Obj. 190 - Radical expressions, simplify

NYC M3.m - Uses equations to represent curves such as lines, circles, and parabolas.


Obj. 75 - Write eqns given slope & y-int

Obj. 76 - Write eqns given point & slopeObj. 77 - Write eqns given 2 pointsObj. 79 - Write eqns parallel or perp to a lineObj. 81 - Describe graphs of equations

NYC M3.n - Uses technology such as graphics calculators to represent and analyze functions and their graphs.

NYC M3.o - Uses functions to analyze patterns and represent their structure.




NYC M4 - STATISTICS AND PROBABILITY CONCEPTSNYC M4-1 - The student demonstrates understanding of statistics and probability concepts; that is, the student:


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NYC M4.a - Organizes, analyzes, and displays single-variable data, choosing appropriate frequency distributions, circle graphs, line plots, histograms, and summary statistics.

Topic 15 - Data and Statistics Obj. 210 - Qualitative vs. quantitative data

Obj. 211 - Univariate vs. bivariate dataObj. 212 - Determine if sample population is biasedObj. 214 - Stem-and-leaf plotsObj. 216 - Box-and-whisker plotsObj. 217 - Cumulative frequency histogramsObj. 222 - Scatter plotsObj. 224 - Construct line of best fit; determine eqnObj. 227 - Evaluate data and conclusionsObj. 228 - Means of data setsObj. 229 - Medians of data setsObj. 230 - Modes of data setsObj. 232 - Percentile rank

NYC M4.b - Organizes, analyzes, and displays two-variable data using scatter plots, estimated regression lines, and computer generated regression lines and correlation coefficients.

Topic 15 - Data and Statistics Obj. 222 - Scatter plots

Obj. 224 - Construct line of best fit; determine eqnObj. 225 - Pos. & neg. correlations and causal relationships

NYC M4.c - Uses sampling techniques to draw inferences about large populations.

Topic 15 - Data and Statistics Obj. 212 - Determine if sample population is biased

Obj. 226 - Determine if sample question is biased

NYC M4.d - Understands that making an inference about a population from a sample always involves uncertainty and that the role of statistics is to estimate the size of that uncertainty.NYC M4.e - Formulates hypotheses to answer a question and uses data to test hypotheses.

Topic 15 - Data and Statistics Obj. 227 - Evaluate data and conclusions


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NYC M4.f - Interprets representations of data, compares distributions of data, and critiques conclusions and the use of statistics, both in school materials and in public documents.

Topic 15 - Data and Statistics Obj. 215 - Interpret elements of box-and-whisker plots

Obj. 223 - Use line of best fit to make predictionsObj. 225 - Pos. & neg. correlations and causal relationshipsObj. 227 - Evaluate data and conclusionsObj. 232 - Percentile rank

NYC M4.g - Explores questions of experimental design, use of control groups, and reliability.

Topic 15 - Data and Statistics Obj. 212 - Determine if sample population is biased

Obj. 226 - Determine if sample question is biased

NYC M4.h - Creates and uses models of probabilistic situations and understands the role of assumptions in this process.

Topic 15 - Data and Statistics Obj. 234 - Determine number of outcomes/events

NYC M4.i - Uses concepts such as equally likely, sample space, outcome, and event in analyzing situations involving chance.

Topic 15 - Data and Statistics Obj. 213 - Probability of single events

Obj. 218 - Conditional probabilityObj. 219 - Empirical probabilityObj. 220 - Probability of independent/dependent eventsObj. 221 - Probability of compound eventsObj. 234 - Determine number of outcomes/eventsObj. 235 - Permutations

NYC M4.j - Constructs appropriate sample spaces, and applies the addition and multiplication principles for probabilities.

Topic 15 - Data and Statistics Obj. 234 - Determine number of outcomes/events

Obj. 235 - PermutationsNYC M4.k - Uses the concept of a probability distribution to discuss whether an event is rare or reasonably likely.NYC M4.l - Chooses an appropriate probability model and uses it to arrive at a theoretical probability for a chance event.

Topic 15 - Data and Statistics Obj. 218 - Conditional probability


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NYC M4.m - Uses relative frequencies based on empirical data to arrive at an experimental probability for a chance event.

Topic 15 - Data and Statistics Obj. 219 - Empirical probability

NYC M4.n - Designs simulations including Monte Carlo simulations to estimate probabilities.NYC M4.o - Works with the normal distribution in some of its basic applications.NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING

NYC M5-1 - The student demonstrates problem solving by using mathematical concepts and skills to solve non-routine problems that do not lay out specific and detailed steps to follow, and solves problems that make demands on all three aspects of the solution process--formulation, implementation, and conclusion. - FORMULATIONNYC M5.a - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student:

NYC M5.a.i - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: fills out the formulation of a definite problem that is to be solved.


Obj. 85 - WP: Linear equations

Topic 6 - Polynomial Operations Obj. 95 - WP: Add & subtract polynomialsObj. 101 - WP: Multiply binomials


Obj. 128 - Verbal sentences and eqns/inequalities

Topic 10 - Systems of Equations Obj. 155 - Write systems of equations


Obj. 166 - WP: Identify absolute value inequalities


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NYC M5.a.ii - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: extracts pertinent information from the situation as a basis for working on the problem.

Topic 3 - Exponents Obj. 40 - Write repeated mult as power



Topic 6 - Polynomial Operations Obj. 95 - WP: Add & subtract polynomialsObj. 101 - WP: Multiply binomials




Obj. 166 - WP: Identify absolute value inequalities

NYC M5.a.iii - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: asks and answers a series of appropriate questions in pursuit of a solution and does so with minimal "scaffolding" in the form of detailed guiding questions. - IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: chooses and employs effective problem solving strategies in dealing with non-routine and multi-step problems.



Obj. 65 - WP: Motion problemsTopic 9 - Quadratic Equations Obj. 146 - WP: Pythagorean

problemsObj. 147 - WP: Number problemsObj. 148 - WP: Area & perimeter



Topic 14 - Ratio and Proportion Obj. 204 - WP: Calculate rates


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NYC M5.b.ii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: selects appropriate mathematical concepts and techniques from different areas of mathematics and applies them to the solution of the problem.

Topic 6 - Polynomial Operations Obj. 101 - WP: Multiply binomials



Topic 9 - Quadratic Equations Obj. 146 - WP: Pythagorean problemsObj. 148 - WP: Area & perimeter

NYC M5.b.iii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: applies mathematical concepts to new situations within mathematics and uses mathematics to model real world situations involving basic applications of mathematics in the physical and biological sciences, the social sciences, and business.


Obj. 5 - WP: Add & subtract integers

Obj. 16 - WP: Add & subtract rational #s (fraction)Obj. 19 - WP: Multiply or divide rational #s (fraction)

Topic 3 - Exponents Obj. 55 - WP: Scientific notationTopic 4 - Solving Linear Equations


Obj. 65 - WP: Motion problemsTopic 5 - Graph and Write Linear Equations


Topic 10 - Systems of Equations Obj. 159 - WP: Solve systems, investment problems

Topic 14 - Ratio and Proportion Obj. 204 - WP: Calculate ratesObj. 208 - WP: Solve proportions

- CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:


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NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: concludes a solution process with a useful summary of results.

NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: evaluates the degree to which the results obtained represent a good response to the initial problem.NYC M5.c.iii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: formulates generalizations of the results obtained.

NYC M5.c.iv - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: carries out extensions of the given problem to related problems.

- MATHEMATICAL REASONINGNYC M5.d - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student:


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NYC M5.d.i - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student: employs forms of mathematical reasoning and proof appropriate to the solution of the problem at hand, including deductive and inductive reasoning, making and testing conjectures, and using counterexamples and indirect proof.


Obj. 22 - Venn diagrams and logic

Topic 2 - Introduction to Algebra Obj. 38 - Evaluate written arguments for validity

NYC M5.d.ii - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student: differentiates clearly between giving examples that support a conjecture and giving proof of the conjecture.

Topic 2 - Introduction to Algebra Obj. 39 - Missing steps in algebraic proofs



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NYC M6-1 - The student demonstrates fluency with basic and important skills by using these skills accurately and automatically, and demonstrates practical competence and persistence with other skills by using them effectively to accomplish a task, perhaps referring to notes, or books, perhaps working to reconstruct a method; that is, the student:

NYC M6.a - Carries out numerical calculations and symbol manipulations effectively, using mental computations, pencil and paper, or other technological aids, as appropriate.


Obj. 3 - Add integers

Obj. 4 - Subtract integersObj. 6 - Multiply integersObj. 8 - Divide integersObj. 9 - Write products as quotients & vice versaObj. 10 - Distributive propertyObj. 11 - Order of operationsObj. 12 - Order of operations with groupingObj. 13 - Raise integers to powersObj. 14 - Add rational #s (fraction)Obj. 15 - Subtract rational #s (decimal)Obj. 17 - Multiply rational #s (decimal)Obj. 18 - Divide rational #s (fraction)

Topic 2 - Introduction to Algebra Obj. 29 - Absolute values of real #s

Topic 3 - Exponents Obj. 41 - Evaluate expressions, positive exponentsObj. 42 - Evaluate expressions, negative/zero exponentsObj. 43 - Evaluate expressions, fractional exponentsObj. 54 - Scientific notation, multiply & divide


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NYC M6.b - Uses a variety of methods to estimate the values, in appropriate units, of quantities met in applications, and rounds numbers used in applications to an appropriate degree of accuracy.NYC M6.c - Evaluates and analyzes formulas and functions of many kinds, using both pencil and paper and more advanced technology.


Obj. 123 - Domain & range, relations & functions

Obj. 124 - Determine if relations are functionsObj. 125 - Determine if functions are one-to-oneObj. 126 - Identify constant, linear, & quad polynomialsObj. 130 - Find specified values of functionsObj. 132 - Add, sub, mult, & div functionsObj. 134 - Identify function type from graph




Obj. 200 - Solve equations containing multiple radicals

NYC M6.d - Uses basic geometric terminology accurately, and deduces information about basic geometric figures in solving problems.

Topic 9 - Quadratic Equations Obj. 148 - WP: Area & perimeter

NYC M6.e - Makes and uses rough sketches, schematic diagrams, or precise scale diagrams to enhance a solution.


Obj. 73 - Plot points to write or graph eqns

Topic 9 - Quadratic Equations Obj. 144 - Find vertex/axis of symm of parabola, given graph

NYC M6.f - Uses the number line and Cartesian coordinates in the plane and in space.





Obj. 69 - Find slopes from a graphObj. 73 - Plot points to write or graph eqnsObj. 81 - Describe graphs of equations


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Obj. 82 - Graph lines, slope-intercept formObj. 83 - Graph lines, standard form

Obj. 84 - Graph lines, horizontal & verticalObj. 87 - Graph absolute value equations




Obj. 142 - Graph to find quadratic rootsObj. 144 - Find vertex/axis of symm of parabola, given graph

Topic 10 - Systems of Equations Obj. 149 - Graph systems, real solutionsObj. 150 - Graph systems, no solutions


Obj. 167 - Graph 1-var inequalities


Obj. 169 - Solve & graph compound inequalitiesObj. 170 - Graph absolute value inequalitiesObj. 171 - Graph 2-var inequalities

NYC M6.g - Creates and interprets graphs of many kinds, such as function graphs, circle graphs, scatter plots, regression lines, and histograms.


Obj. 69 - Find slopes from a graph

Obj. 73 - Plot points to write or graph eqnsObj. 81 - Describe graphs of equationsObj. 82 - Graph lines, slope-intercept formObj. 83 - Graph lines, standard form

Obj. 84 - Graph lines, horizontal & verticalObj. 87 - Graph absolute value equations




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Obj. 142 - Graph to find quadratic rootsObj. 144 - Find vertex/axis of symm of parabola, given graph




Obj. 169 - Solve & graph compound inequalitiesObj. 170 - Graph absolute value inequalitiesObj. 171 - Graph 2-var inequalities

Topic 15 - Data and Statistics Obj. 214 - Stem-and-leaf plotsObj. 216 - Box-and-whisker plotsObj. 217 - Cumulative frequency histogramsObj. 222 - Scatter plotsObj. 223 - Use line of best fit to make predictionsObj. 224 - Construct line of best fit; determine eqn

NYC M6.h - Sets up and solves equations symbolically (when possible) and graphically.


Obj. 57 - Solve 1-step eqns, add & subtract

Obj. 58 - Solve 1-step eqns, multiply & divideObj. 59 - Solve multi-step equations

Obj. 60 - Solve eqns, combine like termsObj. 61 - Solve eqns, combine like terms with parenthesesObj. 62 - Solve eqns, multiply by LCD

Obj. 63 - Solve formulas for specified variablesObj. 64 - WP: Mixture problemsObj. 65 - WP: Motion problems



Obj. 86 - Solve absolute value equations

Topic 9 - Quadratic Equations Obj. 135 - Solve quadratic eqns, square root ruleObj. 136 - Solve quadratic equations (integer)


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Obj. 137 - Solve quadratic equations (rational)Obj. 139 - Solve quadratic eqns, complete the squareObj. 140 - Solve quadratic eqns, quadratic formulaObj. 146 - WP: Pythagorean problemsObj. 147 - WP: Number problemsObj. 148 - WP: Area & perimeter




Obj. 198 - Solve equations containing radicalsObj. 199 - WP: Solve radical equationsObj. 200 - Solve equations containing multiple radicals

Topic 14 - Ratio and Proportion Obj. 207 - Solve proportionsObj. 208 - WP: Solve proportionsObj. 209 - Solve direct & indirect variation problems

NYC M6.i - Knows how to use algorithms in mathematics, such as the Euclidean Algorithm.NYC M6.j - Uses technology to create graphs or spreadsheets that contribute to the understanding of a problem.NYC M6.k - Writes a simple computer program to carry out a computation or simulation to be repeated many times.NYC M6.l - Uses tools such as rulers, tapes, compasses, and protractors in solving problems.NYC M6.m - Knows standard methods to solve basic problems and uses these methods in approaching more complex problems.



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NYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Is familiar with basic mathematical terminology, standard notation and use of symbols, common conventions for graphing, and general features of effective mathematical communication styles.



Obj. 23 - Set builder and interval notation

Topic 3 - Exponents Obj. 40 - Write repeated mult as powerObj. 52 - Standard notation to scientific notationObj. 53 - Scientific notation to standard notationObj. 54 - Scientific notation, multiply & divide



Obj. 67 - Identify quadrants & axes of pointsObj. 68 - Find slopes from 2 pointsObj. 69 - Find slopes from a graphObj. 72 - Find lines' interceptsObj. 73 - Plot points to write or graph eqnsObj. 74 - Write eqns in standard/slope-intercept form

NYC M7.b - Uses mathematical representations with appropriate accuracy, including numerical tables, formulas, functions, equations, charts, graphs, and diagrams.





Obj. 67 - Identify quadrants & axes of pointsObj. 68 - Find slopes from 2 points


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Obj. 69 - Find slopes from a graphObj. 70 - Find slopes from equations

Obj. 71 - Slope as rate of change btw indep./dep. variablesObj. 72 - Find lines' interceptsObj. 73 - Plot points to write or graph eqnsObj. 74 - Write eqns in standard/slope-intercept formObj. 75 - Write eqns given slope & y-intObj. 76 - Write eqns given point & slopeObj. 77 - Write eqns given 2 pointsObj. 78 - Determine parallel lines, given equationsObj. 79 - Write eqns parallel or perp to a lineObj. 80 - Equations parallel to x- or y-axisObj. 82 - Graph lines, slope-intercept formObj. 83 - Graph lines, standard form

Obj. 84 - Graph lines, horizontal & verticalObj. 85 - WP: Linear equationsObj. 86 - Solve absolute value equationsObj. 87 - Graph absolute value equations


Obj. 128 - Verbal sentences and eqns/inequalitiesObj. 133 - Patterns and their equationsObj. 134 - Identify function type from graph




Obj. 170 - Graph absolute value inequalitiesObj. 171 - Graph 2-var inequalities

Topic 15 - Data and Statistics Obj. 214 - Stem-and-leaf plotsObj. 215 - Interpret elements of box-and-whisker plotsObj. 216 - Box-and-whisker plots


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Obj. 217 - Cumulative frequency histogramsObj. 222 - Scatter plotsObj. 223 - Use line of best fit to make predictions

NYC M7.c - Organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.NYC M7.d - Communicates logical arguments clearly, showing why a result makes sense and why the reasoning is valid.


Obj. 77 - Write eqns given 2 points

NYC M7.e - Presents mathematical ideas effectively both orally and in writing.


Obj. 23 - Set builder and interval notation


Obj. 73 - Plot points to write or graph eqnsObj. 74 - Write eqns in standard/slope-intercept formObj. 75 - Write eqns given slope & y-intObj. 76 - Write eqns given point & slopeObj. 79 - Write eqns parallel or perp to a line


Obj. 127 - Write verbal expressions


NYC M7.f - Explains mathematical concepts clearly enough to be of assistance to those who may be having difficulty with them.NYC M7.g - Writes narrative accounts of the history and process of work on a mathematical problem or extended project.NYC M7.h - Writes succinct accounts of the mathematical results obtained in a mathematical problem or extended project, with diagrams, graphs, tables, and formulas integrated into the text.NYC M7.i - Keeps narrative accounts of process separate from succinct accounts of results, and realizes that doing so can enhance the effectiveness of each.


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NYC M7.j - Reads mathematics texts and other writing about mathematics with understanding.NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of high school, conducts investigations or projects drawn from at least three of the kinds. A single investigation or project may draw on more than one kind.

NYC M8.a - Data study, in which the student:NYC M8.a.i - Data study, in which the student: carries out a study of data relevant to current civic, economic, scientific, health, or social issues.

NYC M8.a.ii - Data study, in which the student: uses methods of statistical inference to generalize from the data.NYC M8.a.iii - Data study, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.b - Mathematical model of a physical system or phenomenon, in which the student:NYC M8.b.i - Mathematical model of a physical system or phenomenon, in which the student: carries out a study of a physical system or phenomenon by constructing a mathematical model based on functions to make generalizations about the structure of the system.


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NYC M8.b.ii - Mathematical model of a physical system or phenomenon, in which the student: uses structural analysis (a direct analysis of the structure of the system) rather than numerical or statistical analysis (an analysis of data about the system).

NYC M8.b.iii - Mathematical model of a physical system or phenomenon, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.

NYC M8.c - Design of a physical structure, in which the student:NYC M8.c.i - Design of a physical structure, in which the student: creates a design for a physical structure.NYC M8.c.ii - Design of a physical structure, in which the student: uses general mathematical ideas and techniques to discuss specifications for building the structure.

NYC M8.c.iii - Design of a physical structure, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.d - Management and planning analysis, in which the student:NYC M8.d.i - Management and planning analysis, in which the student: carries out a study of a business or public policy situation involving issues such as optimization, cost-benefit projections, and risks.


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NYC M8.d.ii - Management and planning analysis, in which the student: uses decision rules and strategies both to analyze options and balance trade-offs; and brings in mathematical ideas that serve to generalize the analysis across different conditions.NYC M8.d.iii - Management and planning analysis, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.e - Pure mathematics investigation, in which the student:NYC M8.e.i - Pure mathematics investigation, in which the student: carries out a mathematical investigation of a phenomenon or concept in pure mathematics.NYC M8.e.ii - Pure mathematics investigation, in which the student: uses methods of mathematical reasoning and justification to make generalizations about the phenomenon.NYC M8.e.iii - Pure mathematics investigation, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.f - History of a mathematical idea, in which the student:

NYC M8.f.i - History of a mathematical idea, in which the student: carries out a historical study tracing the development of a mathematical concept and the people who contributed to it.


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NYC M8.f.ii - History of a mathematical idea, in which the student: includes a discussion of the actual mathematical content and its place in the curriculum of the present day.NYC M8.f.iii - History of a mathematical idea, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.


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NYC M1.c - Has facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications.NYC M1.d - Understands and uses number systems: natural, integer, rational, and real.NYC M1.e - Represents numbers in decimal or fraction form and in scientific notation, and graphs numbers on the number line and number pairs in the coordinate plane.

Topic 2 - Introduction to the Coordinate Plane

Obj. 16 - Ordered pairs

NYC M1.f - Compares numbers using order relations, differences, ratios, proportions, percents, and proportional change.NYC M1.g - Carries out proportional reasoning in cases involving part-whole relationships and in cases involving expansions and contractions.

Topic 9 - Similarity Obj. 115 - Solve proportions

NYC M1.h - Understands dimensionless numbers, such as proportions, percents, and multiplicative factors, as well as numbers with specific units of measure, such as numbers with length, time, and rate units.

Topic 3 - Transformational Geometry

Obj. 41 - Dilations

Standards List with Aligned Product SkillsProduct NameAM NY Custom Geometry


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NYC M1.i - Carries out counting procedures such as those involving sets (unions and intersections) and arrangements (permutations and combinations).NYC M1.j - Uses concepts such as prime, relatively prime, factor, divisor, multiple, and divisibility in solving problems involving integers.NYC M1.k - Uses a scientific calculator effectively and efficiently in carrying out complex calculations.

NYC M1.l - Recognizes and represents basic number patterns, such as patterns involving multiples, squares, or cubes.

Topic 4 - Reasoning and Proofs Obj. 45 - Visual patterns

Obj. 46 - Numerical patternsNYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:NYC M2.a - Models situations geometrically to formulate and solve problems.NYC M2.b - Works with two- and three-dimensional figures and their properties, including polygons and circles, cubes and pyramids, and cylinders, cones, and spheres.

Topic 6 - Triangles Obj. 70 - Interior & exterior angles of triangles

Obj. 73 - Angle measures in isosceles trianglesObj. 76 - Use segment joining midpoints of 2 sides of triangleObj. 77 - Use centroid for indirect measure

Topic 7 - Right Triangles and Trigonometry

Obj. 88 - 30-60-90 triangles

Obj. 89 - 45-45-90 trianglesObj. 90 - Apply special right triangles

Obj. 91 - WP: Special right triangles

Topic 8 - Polygons Obj. 103 - Polygon propertiesObj. 104 - Sum of interior angles of polygons


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Obj. 105 - Interior or exterior angles of polygonsObj. 106 - WP: PolygonsObj. 110 - Use properties of trapezoids to solve problems

Topic 10 - Circles Obj. 126 - Parts of circlesObj. 132 - Perpendicular line to tangent at point of tangencyObj. 137 - Two intersecting chords of a circle

NYC M2.c - Uses congruence and similarity in describing relationships between figures.

Topic 6 - Triangles Obj. 79 - Rule that proves triangle congruence

Topic 9 - Similarity Obj. 117 - Similar polygon characteristics

NYC M2.d - Visualizes objects, paths, and regions in space, including intersections and cross sections of three dimensional figures, and describes these using geometric language.

Topic 10 - Circles Obj. 149 - Solve compound loci problems using coordinates

Obj. 150 - WP: LociNYC M2.e - Knows, uses, and derives formulas for perimeter, circumference, area, surface area, and volume of many types of figures.

Topic 10 - Circles Obj. 140 - Circumference

Topic 11 - Area Obj. 153 - Area & perimeter of rectangles & squaresObj. 154 - Area of parallelogramsObj. 155 - Area of trianglesObj. 156 - Area of trapezoidsObj. 157 - Area of rhombuses & kites

Obj. 158 - Area of circlesObj. 159 - Area of sector or segment

Obj. 161 - Area of regular polygonsObj. 162 - Area of composite shapes

Obj. 163 - Perimeter of composite shapes

Topic 12 - Surface Area and Volume

Obj. 169 - Surface area of rectangular prismsObj. 170 - Surface area of non-rectangular prismsObj. 171 - Volume of rectangular prisms & cubes


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Obj. 172 - Volume of non-rectangular prismsObj. 173 - Surface area of rectangular pyramidsObj. 175 - Volume of pyramidsObj. 176 - Surface area of cylinders

Obj. 177 - Surface area of conesObj. 178 - Volume of cylindersObj. 179 - Volume of conesObj. 180 - Volume or surface area of spheresObj. 181 - Volume of composite solidsObj. 182 - Surface area of composite solidsObj. 184 - Solve area, surface area, volume problems



Obj. 82 - Pythagorean Theorem to solve sides

Obj. 83 - Pythagorean Theorem to determine right triangleObj. 84 - Pythagorean triplesObj. 87 - WP: Pythagorean Theorem

NYC M2.g - Works with similar triangles, and extends the ideas to include simple uses of the three basic trigonometric functions.


Obj. 92 - Sine & cosine ratios

Obj. 93 - Tangent ratioObj. 94 - Trig to solve sides in triangleObj. 96 - WP: Trigonometry

Topic 9 - Similarity Obj. 118 - Proofs: Similar trianglesObj. 120 - Similar triangles & proportionsObj. 121 - WP: Similar triangles & proportions

NYC M2.h - Analyzes figures in terms of their symmetries using, for example, concepts of reflection, rotation, and translation.


Obj. 29 - Translations

Obj. 30 - ReflectionsObj. 31 - Glide reflectionsObj. 32 - RotationsObj. 39 - Lines of symmetry


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NYC M2.i - Compares slope (rise over run) and angle of elevation as measures of steepness.


Obj. 18 - Slope of a line

NYC M2.j - Investigates geometric patterns, including sequences of growing shapes.

Topic 4 - Reasoning and Proofs Obj. 45 - Visual patterns

NYC M2.k - Works with geometric measures of length, area, volume, and angle; and non-geometric measures such as weight and time.

Topic 1 - The Basics of Geometry

Obj. 9 - Angle definitions, notation, & properties

Obj. 10 - Classify angles as acute, right, or obtuseObj. 11 - Estimate & draw angles


NYC M2.m - Understands the structure of standard measurement systems, both SI and customary, including unit conversions and dimensional analysis.NYC M2.n - Solves problems involving scale, such as in maps and diagrams.



NYC M2.o - Represents geometric curves and graphs of functions in standard coordinate systems.


Obj. 19 - Graph line on coordinate plane

Topic 10 - Circles Obj. 143 - Equations of circlesNYC M2.p - Analyzes geometric figures and proves simple things about them using deductive methods.

Topic 4 - Reasoning and Proofs Obj. 57 - Proofs: Angle & segment addition

Obj. 58 - Proofs: AnglesTopic 5 - Parallel Lines and Line Relationships

Obj. 63 - Proofs: Parallel lines

Topic 6 - Triangles Obj. 80 - Proofs: TrianglesObj. 81 - Proofs: Direct & indirect


Obj. 100 - Proofs: Right triangle congruenceObj. 101 - Proofs: Flow & paragraph

Topic 8 - Polygons Obj. 109 - Proofs: QuadrilateralsObj. 113 - Proofs: Coordinate geometryObj. 114 - Proofs: Coordinate with quadrilaterals


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Topic 9 - Similarity Obj. 118 - Proofs: Similar trianglesNYC M2.q - Explores geometry using computer programs such as CAD software, Sketchpad programs, or LOGO.NYC M3 - FUNCTION AND ALGEBRA CONCEPTSNYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:


NYC M3.b - Describes, generalizes, and uses basic types of functions: linear, exponential, power, rational, square and square root, and cube and cube root.NYC M3.c - Utilizes the concepts of slope, evaluation, and inverse in working with functions.




NYC M3.e - Represents constant rates as the slope of a straight line graph, and interprets slope as the amount of one quantity (y) per unit amount of another (x).



NYC M3.f - Understands and uses linear functions as a mathematical representation of proportional relationships.NYC M3.g - Uses arithmetic sequences and geometric sequences and their sums, and sees these as the discrete forms of linear and exponential functions, respectively.


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Obj. 25 - Solve system of equations

Topic 10 - Circles Obj. 143 - Equations of circlesObj. 146 - WP: Equations of circles

NYC M3.i - Represents functional relationships in formulas, tables, and graphs, and translates between pairs of these.NYC M3.j - Solves equations symbolically, graphically, and numerically, especially linear, quadratic, and exponential equations; and knows how to use the quadratic formula for solving quadratic equations.NYC M3.k - Makes predictions by interpolating or extrapolating from given data or a given graph.NYC M3.l - Understands the basic algebraic structure of number systems.NYC M3.m - Uses equations to represent curves such as lines, circles, and parabolas.


Obj. 22 - Write equation of a line

Obj. 23 - Write eqn of line given point & parallel/perp line eqnObj. 24 - Write eqn of perp bisector, given segment endpts

Topic 10 - Circles Obj. 143 - Equations of circlesObj. 144 - Find eqn of circle, given graphObj. 146 - WP: Equations of circles


NYC M3.o - Uses functions to analyze patterns and represent their structure.NYC M4 - STATISTICS AND PROBABILITY CONCEPTS


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NYC M4-1 - The student demonstrates understanding of statistics and probability concepts; that is, the student:NYC M4.a - Organizes, analyzes, and displays single-variable data, choosing appropriate frequency distributions, circle graphs, line plots, histograms, and summary statistics.

NYC M4.b - Organizes, analyzes, and displays two-variable data using scatter plots, estimated regression lines, and computer generated regression lines and correlation coefficients.NYC M4.c - Uses sampling techniques to draw inferences about large populations.NYC M4.d - Understands that making an inference about a population from a sample always involves uncertainty and that the role of statistics is to estimate the size of that uncertainty.NYC M4.e - Formulates hypotheses to answer a question and uses data to test hypotheses.NYC M4.f - Interprets representations of data, compares distributions of data, and critiques conclusions and the use of statistics, both in school materials and in public documents.

NYC M4.g - Explores questions of experimental design, use of control groups, and reliability.NYC M4.h - Creates and uses models of probabilistic situations and understands the role of assumptions in this process.NYC M4.i - Uses concepts such as equally likely, sample space, outcome, and event in analyzing situations involving chance.NYC M4.j - Constructs appropriate sample spaces, and applies the addition and multiplication principles for probabilities.


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NYC M4.k - Uses the concept of a probability distribution to discuss whether an event is rare or reasonably likely.NYC M4.l - Chooses an appropriate probability model and uses it to arrive at a theoretical probability for a chance event.

Topic 11 - Area Obj. 164 - Geometric probability using area

NYC M4.m - Uses relative frequencies based on empirical data to arrive at an experimental probability for a chance event.NYC M4.n - Designs simulations including Monte Carlo simulations to estimate probabilities.NYC M4.o - Works with the normal distribution in some of its basic applications.NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING




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NYC M5.a.ii - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: extracts pertinent information from the situation as a basis for working on the problem.NYC M5.a.iii - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: asks and answers a series of appropriate questions in pursuit of a solution and does so with minimal "scaffolding" in the form of detailed guiding questions. - IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: chooses and employs effective problem solving strategies in dealing with non-routine and multi-step problems.


NYC M5.b.iii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: applies mathematical concepts to new situations within mathematics and uses mathematics to model real world situations involving basic applications of mathematics in the physical and biological sciences, the social sciences, and business.


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- CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: concludes a solution process with a useful summary of results.

NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: evaluates the degree to which the results obtained represent a good response to the initial problem.NYC M5.c.iii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: formulates generalizations of the results obtained.


- MATHEMATICAL REASONING


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NYC M5.d - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student:


Topic 4 - Reasoning and Proofs Obj. 47 - Apply inductive reasoning in conjectures

Obj. 48 - Conditionals & logical statementsObj. 49 - Determine the negation of a statementObj. 50 - Converses, inverses, & contrapositivesObj. 51 - CounterexamplesObj. 52 - Biconditional statements & definitionsObj. 53 - Conjunctions & disjunctions

Obj. 54 - Symbolic notation of logical statements


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Obj. 55 - Form conclusions by applying laws of logicObj. 56 - Proofs: Algebra & properties of equalityObj. 57 - Proofs: Angle & segment additionObj. 58 - Proofs: Angles

Topic 5 - Parallel Lines and Line Relationships

Obj. 63 - Proofs: Parallel lines

Topic 6 - Triangles Obj. 80 - Proofs: TrianglesObj. 81 - Proofs: Direct & indirect


Obj. 100 - Proofs: Right triangle congruenceObj. 101 - Proofs: Flow & paragraph

Topic 8 - Polygons Obj. 113 - Proofs: Coordinate geometryObj. 114 - Proofs: Coordinate with quadrilaterals

Topic 9 - Similarity Obj. 118 - Proofs: Similar trianglesNYC M5.d.ii - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student: differentiates clearly between giving examples that support a conjecture and giving proof of the conjecture.



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NYC M6.a - Carries out numerical calculations and symbol manipulations effectively, using mental computations, pencil and paper, or other technological aids, as appropriate.NYC M6.b - Uses a variety of methods to estimate the values, in appropriate units, of quantities met in applications, and rounds numbers used in applications to an appropriate degree of accuracy.NYC M6.c - Evaluates and analyzes formulas and functions of many kinds, using both pencil and paper and more advanced technology.


Obj. 17 - Midpoints in the coordinate plane

Obj. 18 - Slope of a lineTopic 7 - Right Triangles and Trigonometry

Obj. 82 - Pythagorean Theorem to solve sidesObj. 86 - Distance formulaObj. 97 - Law of Sines & Cosines

Topic 10 - Circles Obj. 140 - CircumferenceObj. 141 - Arc length

Topic 12 - Surface Area and Volume

Obj. 169 - Surface area of rectangular prismsObj. 170 - Surface area of non-rectangular prismsObj. 171 - Volume of rectangular prisms & cubesObj. 172 - Volume of non-rectangular prismsObj. 173 - Surface area of rectangular pyramidsObj. 175 - Volume of pyramidsObj. 176 - Surface area of cylinders


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Obj. 177 - Surface area of conesObj. 178 - Volume of cylindersObj. 179 - Volume of conesObj. 180 - Volume or surface area of spheres



Obj. 1 - Collinear & coplanar points

Obj. 3 - Points, lines, planes, & space

Obj. 9 - Angle definitions, notation, & propertiesObj. 10 - Classify angles as acute, right, or obtuse

Topic 6 - Triangles Obj. 67 - Name parts of trianglesObj. 68 - Classify triangles as scalene, isos, or equilObj. 69 - Classify triangles as acute, obtuse, or right

Topic 8 - Polygons Obj. 102 - Identify polygons by # of sides

NYC M6.e - Makes and uses rough sketches, schematic diagrams, or precise scale diagrams to enhance a solution.



NYC M6.f - Uses the number line and Cartesian coordinates in the plane and in space.


Obj. 5 - Find distances & midpoints on number lines


Obj. 17 - Midpoints in the coordinate planeObj. 18 - Slope of a lineObj. 19 - Graph line on coordinate planeObj. 21 - Betweenness & collinearity

Obj. 26 - Graph to solve system w/one linear, one quadratic eqnObj. 28 - Area with Coordinates


Obj. 36 - Find coordinates under transformations



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Obj. 25 - Solve system of equations

NYC M6.i - Knows how to use algorithms in mathematics, such as the Euclidean Algorithm.NYC M6.j - Uses technology to create graphs or spreadsheets that contribute to the understanding of a problem.NYC M6.k - Writes a simple computer program to carry out a computation or simulation to be repeated many times.NYC M6.l - Uses tools such as rulers, tapes, compasses, and protractors in solving problems.

Topic 5 - Parallel Lines and Line Relationships

Obj. 65 - Constructions: Angles & triangles

Obj. 66 - Constructions: Bisectors, lines, equi. triangle

Topic 8 - Polygons Obj. 111 - Constructions: Quadrilaterals

Topic 10 - Circles Obj. 147 - Constructions: Circles, arcs, tangents, & chords

NYC M6.m - Knows standard methods to solve basic problems and uses these methods in approaching more complex problems.

NYC M7 - MATHEMATICAL COMMUNICATIONNYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Is familiar with basic mathematical terminology, standard notation and use of symbols, common conventions for graphing, and general features of effective mathematical communication styles.


Obj. 1 - Collinear & coplanar points

Obj. 3 - Points, lines, planes, & space


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Obj. 9 - Angle definitions, notation, & propertiesObj. 10 - Classify angles as acute, right, or obtuse

Topic 6 - Triangles Obj. 67 - Name parts of trianglesObj. 68 - Classify triangles as scalene, isos, or equilObj. 69 - Classify triangles as acute, obtuse, or right

Topic 8 - Polygons Obj. 102 - Identify polygons by # of sides


NYC M7.c - Organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.NYC M7.d - Communicates logical arguments clearly, showing why a result makes sense and why the reasoning is valid.NYC M7.e - Presents mathematical ideas effectively both orally and in writing.NYC M7.f - Explains mathematical concepts clearly enough to be of assistance to those who may be having difficulty with them.NYC M7.g - Writes narrative accounts of the history and process of work on a mathematical problem or extended project.NYC M7.h - Writes succinct accounts of the mathematical results obtained in a mathematical problem or extended project, with diagrams, graphs, tables, and formulas integrated into the text.NYC M7.i - Keeps narrative accounts of process separate from succinct accounts of results, and realizes that doing so can enhance the effectiveness of each.


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NYC M7.j - Reads mathematics texts and other writing about mathematics with understanding.NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of high school, conducts investigations or projects drawn from at least three of the kinds. A single investigation or project may draw on more than one kind.


NYC M8.a.ii - Data study, in which the student: uses methods of statistical inference to generalize from the data.NYC M8.a.iii - Data study, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.b - Mathematical model of a physical system or phenomenon, in which the student:NYC M8.b.i - Mathematical model of a physical system or phenomenon, in which the student: carries out a study of a physical system or phenomenon by constructing a mathematical model based on functions to make generalizations about the structure of the system.


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NYC M8.c.iii - Design of a physical structure, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.d - Management and planning analysis, in which the student:NYC M8.d.i - Management and planning analysis, in which the student: carries out a study of a business or public policy situation involving issues such as optimization, cost-benefit projections, and risks.


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NYC M8.f.i - History of a mathematical idea, in which the student: carries out a historical study tracing the development of a mathematical concept and the people who contributed to it.


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NYC M8.f.ii - History of a mathematical idea, in which the student: includes a discussion of the actual mathematical content and its place in the curriculum of the present day.NYC M8.f.iii - History of a mathematical idea, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.


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Topic 1 - The Real Numbers Obj. 6 - Operations with rational #s

Obj. 7 - Order of operationsObj. 8 - Evaluate expressions for given values

Topic 6 - Roots, Radicals, and Complex Numbers

Obj. 72 - Square roots

Obj. 73 - Write square roots as exponential expressionsObj. 74 - Write expression w/fraction exp as a radicalObj. 75 - Simplify expressions w/fractional exponentsObj. 76 - Simplify nth rootsObj. 78 - Simplify expressions w/rational exponentsObj. 79 - Evaluate exponential expressionsObj. 80 - Rewrite expr using only positive exponentsObj. 81 - Add & subtract radical expressionsObj. 82 - Rationalize denominators

Topic 8 - Polynomials Obj. 119 - Evaluate polynomials for given valuesObj. 120 - Add polynomialsObj. 121 - Subtract polynomialsObj. 122 - Add & subtract polynomialsObj. 123 - Square a binomial, ax + by

Obj. 124 - Multiply 2 binomialsObj. 125 - Multiply monomials by polynomialsObj. 126 - Multiply binomials by trinomialsObj. 127 - Square trinomials



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Obj. 128 - Simplify polynomial expressions using multiplicationObj. 129 - Divide polynomials by monomialsObj. 130 - Divide polynomialsObj. 131 - Long division, factor higher term polynomials


Obj. 155 - Simplify rational expressionsObj. 156 - Simplify rational expressions by factoringObj. 158 - Multiply & simplify rational expressionsObj. 159 - Divide & simplify rational expressionsObj. 160 - Add rational expressions

Obj. 161 - Subtract rational expressionsObj. 162 - Add & subtract rational expressionsObj. 164 - Simplify complex rational expressions




Obj. 76 - Simplify nth rootsObj. 77 - WP: Roots

Topic 9 - Exponents and Logarithms

Obj. 141 - Find logs by converting to exponential formObj. 142 - Find natural logs of numbers

NYC M1.c - Has facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications.NYC M1.d - Understands and uses number systems: natural, integer, rational, and real.

Topic 1 - The Real Numbers Obj. 1 - Determine if #s are irrational

Obj. 2 - Classify Real #sObj. 3 - Properties of real #s

NYC M1.e - Represents numbers in decimal or fraction form and in scientific notation, and graphs numbers on the number line and number pairs in the coordinate plane.


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NYC M1.f - Compares numbers using order relations, differences, ratios, proportions, percents, and proportional change.

Topic 1 - The Real Numbers Obj. 5 - Order rational #s

NYC M1.g - Carries out proportional reasoning in cases involving part-whole relationships and in cases involving expansions and contractions.NYC M1.h - Understands dimensionless numbers, such as proportions, percents, and multiplicative factors, as well as numbers with specific units of measure, such as numbers with length, time, and rate units.NYC M1.i - Carries out counting procedures such as those involving sets (unions and intersections) and arrangements (permutations and combinations).

Topic 13 - Probability Obj. 206 - Fundamental Counting Principle

Obj. 209 - Combinations & permutations

NYC M1.j - Uses concepts such as prime, relatively prime, factor, divisor, multiple, and divisibility in solving problems involving integers.NYC M1.k - Uses a scientific calculator effectively and efficiently in carrying out complex calculations.

NYC M1.l - Recognizes and represents basic number patterns, such as patterns involving multiples, squares, or cubes.NYC M2 - GEOMETRY AND MEASUREMENT CONCEPTSNYC M2-1 - The student produces evidence that demonstrates understanding of geometry and measurement concepts; that is, the student:NYC M2.a - Models situations geometrically to formulate and solve problems.


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NYC M2.b - Works with two- and three-dimensional figures and their properties, including polygons and circles, cubes and pyramids, and cylinders, cones, and spheres.NYC M2.c - Uses congruence and similarity in describing relationships between figures.NYC M2.d - Visualizes objects, paths, and regions in space, including intersections and cross sections of three dimensional figures, and describes these using geometric language.NYC M2.e - Knows, uses, and derives formulas for perimeter, circumference, area, surface area, and volume of many types of figures.

Topic 15 - Trigonometry Obj. 263 - Areas of parallelograms and triangles


Topic 7 - Quadratic Obj. 110 - WP: Pythagorean Theorem

NYC M2.g - Works with similar triangles, and extends the ideas to include simple uses of the three basic trigonometric functions.

Topic 15 - Trigonometry Obj. 260 - Find a side given side & angle

Obj. 261 - Find unknown sides of right triangles

NYC M2.h - Analyzes figures in terms of their symmetries using, for example, concepts of reflection, rotation, and translation.NYC M2.i - Compares slope (rise over run) and angle of elevation as measures of steepness.NYC M2.j - Investigates geometric patterns, including sequences of growing shapes.NYC M2.k - Works with geometric measures of length, area, volume, and angle; and non-geometric measures such as weight and time.

Topic 15 - Trigonometry Obj. 237 - Degrees to radians

Obj. 238 - Radians to degreesObj. 239 - Manipulate degrees, minutes, & seconds


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NYC M2.m - Understands the structure of standard measurement systems, both SI and customary, including unit conversions and dimensional analysis.NYC M2.n - Solves problems involving scale, such as in maps and diagrams.NYC M2.o - Represents geometric curves and graphs of functions in standard coordinate systems.

Topic 3 - Relations, Functions, and Graphs

Obj. 35 - Graph linear equations, ax + by = c

Obj. 36 - Graph linear equations, y = ax + bObj. 46 - Sketch the graph of a functionObj. 47 - Graph/write eqn that is result of transformation


Obj. 85 - Graph square roots

Obj. 86 - Graph nth rootsTopic 7 - Quadratic Obj. 106 - Graph quadratic functions


Obj. 146 - Graph log functions

Obj. 152 - Graph exponential functions


Obj. 157 - Graph rational functions & equations

Topic 15 - Trigonometry Obj. 248 - Graph sine & cosine functionsObj. 249 - Graph trig functions

NYC M2.p - Analyzes geometric figures and proves simple things about them using deductive methods.

NYC M2.q - Explores geometry using computer programs such as CAD software, Sketchpad programs, or LOGO.NYC M3 - FUNCTION AND ALGEBRA CONCEPTS


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NYC M3-1 - The student produces evidence that demonstrates understanding of function and algebra concepts; that is, the student:



Obj. 38 - WP: Function problems

Obj. 41 - WP: Linear equationsTopic 7 - Quadratic Obj. 107 - WP: Graph quadratic

functionsNYC M3.b - Describes, generalizes, and uses basic types of functions: linear, exponential, power, rational, square and square root, and cube and cube root.


Obj. 26 - Use function notation


Obj. 77 - WP: Roots


Obj. 150 - WP: Growth & decay problemsObj. 151 - WP: Earthquake problems

NYC M3.c - Utilizes the concepts of slope, evaluation, and inverse in working with functions.


Obj. 34 - Find inverses of functions


NYC M3.e - Represents constant rates as the slope of a straight line graph, and interprets slope as the amount of one quantity (y) per unit amount of another (x).NYC M3.f - Understands and uses linear functions as a mathematical representation of proportional relationships.NYC M3.g - Uses arithmetic sequences and geometric sequences and their sums, and sees these as the discrete forms of linear and exponential functions, respectively.

Topic 12 - Sequences and Series Obj. 190 - Common differences of arithmetic sequences

Obj. 191 - Find terms of arith seq (1st term & common diff)


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Obj. 192 - Find specified term of arithmetic sequenceObj. 193 - Find terms of arith seq (formula for nth term)Obj. 194 - WP: Arithmetic sequences

Obj. 195 - Find sums of arithmetic seriesObj. 196 - Find terms of geo seqObj. 197 - Find ratios of geo seq given 3 termsObj. 198 - Find specified term of geo seq given first 3 termsObj. 199 - Find general terms of geo seq given first 4 termsObj. 200 - WP: Geometric sequencesObj. 201 - Find sums of finite geometric seriesObj. 202 - Find sums of infinite geometric series


Topic 2 - Linear Equations and Inequalities

Obj. 11 - Solve linear equations


Obj. 84 - Solve formulas for specified variables

Topic 7 - Quadratic Obj. 98 - Factor quadratics, real roots

Obj. 100 - Factor quadratics, roots with radicalsObj. 105 - Solve quad eqns containing rational expr

NYC M3.i - Represents functional relationships in formulas, tables, and graphs, and translates between pairs of these.


Obj. 35 - Graph linear equations, ax + by = c

Obj. 36 - Graph linear equations, y = ax + bObj. 46 - Sketch the graph of a function





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Topic 8 - Polynomials Obj. 137 - Graph polynomial functions






Topic 14 - Statistics Obj. 224 - Write function for regression model, given data

NYC M3.j - Solves equations symbolically, graphically, and numerically, especially linear, quadratic, and exponential equations; and knows how to use the quadratic formula for solving quadratic equations.


Obj. 11 - Solve linear equations


Obj. 83 - Solve equations containing radicals

Topic 7 - Quadratic Obj. 95 - Solve quad eqns, square root rule (real roots)Obj. 96 - Solve quad eqns, square root rule (complex roots)Obj. 101 - Quadratic formula, 2 real rootsObj. 114 - Solve cubic equations


Obj. 145 - Solve log equations

Obj. 147 - Solve exponential equationsObj. 148 - Solve equations involving natural logs



NYC M3.k - Makes predictions by interpolating or extrapolating from given data or a given graph.

Topic 7 - Quadratic Obj. 107 - WP: Graph quadratic functions

NYC M3.l - Understands the basic algebraic structure of number systems.

Topic 1 - The Real Numbers Obj. 3 - Properties of real #s

Obj. 4 - Reflex, symm, trans, & subst properties

NYC M3.m - Uses equations to represent curves such as lines, circles, and parabolas.


Obj. 39 - Write equation from slope & point, ax + by = c

Obj. 45 - Write equation of circle given graph

Topic 7 - Quadratic Obj. 104 - Write quad eqns given solutions


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Topic 11 - Conics and Second-Degree Equations

Obj. 170 - Circles, write eqns given centers & radiiObj. 171 - Circles, write eqns using given factsObj. 175 - Ellipses, write eqns using given factsObj. 176 - Graph hyperbolas (standard equation)Obj. 177 - Graph hyperbolas (general equation)Obj. 180 - Parabolas, find vertex from equationObj. 184 - Identify eqns of conics


NYC M3.o - Uses functions to analyze patterns and represent their structure.NYC M4 - STATISTICS AND PROBABILITY CONCEPTSNYC M4-1 - The student demonstrates understanding of statistics and probability concepts; that is, the student:NYC M4.a - Organizes, analyzes, and displays single-variable data, choosing appropriate frequency distributions, circle graphs, line plots, histograms, and summary statistics.

Topic 14 - Statistics Obj. 230 - Group frequency distributions

NYC M4.b - Organizes, analyzes, and displays two-variable data using scatter plots, estimated regression lines, and computer generated regression lines and correlation coefficients.

Topic 14 - Statistics Obj. 223 - Name type of regression model, given scatterplot

NYC M4.c - Uses sampling techniques to draw inferences about large populations.NYC M4.d - Understands that making an inference about a population from a sample always involves uncertainty and that the role of statistics is to estimate the size of that uncertainty.


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NYC M4.e - Formulates hypotheses to answer a question and uses data to test hypotheses.NYC M4.f - Interprets representations of data, compares distributions of data, and critiques conclusions and the use of statistics, both in school materials and in public documents.

NYC M4.g - Explores questions of experimental design, use of control groups, and reliability.NYC M4.h - Creates and uses models of probabilistic situations and understands the role of assumptions in this process.NYC M4.i - Uses concepts such as equally likely, sample space, outcome, and event in analyzing situations involving chance.NYC M4.j - Constructs appropriate sample spaces, and applies the addition and multiplication principles for probabilities.

Topic 13 - Probability Obj. 215 - Probability of independent events

Obj. 216 - Conditional probabilityObj. 217 - Probability of dependent events

NYC M4.k - Uses the concept of a probability distribution to discuss whether an event is rare or reasonably likely.NYC M4.l - Chooses an appropriate probability model and uses it to arrive at a theoretical probability for a chance event.NYC M4.m - Uses relative frequencies based on empirical data to arrive at an experimental probability for a chance event.

Topic 13 - Probability Obj. 219 - WP: Calculate empirical probability

NYC M4.n - Designs simulations including Monte Carlo simulations to estimate probabilities.NYC M4.o - Works with the normal distribution in some of its basic applications.NYC M5 - PROBLEM SOLVING AND MATHEMATICAL REASONING


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NYC M5.a.ii - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: extracts pertinent information from the situation as a basis for working on the problem.NYC M5.a.iii - The student participates in the formulation of problems; that is, given the statement of a problem situation, the student: asks and answers a series of appropriate questions in pursuit of a solution and does so with minimal "scaffolding" in the form of detailed guiding questions. - IMPLEMENTATIONNYC M5.b - The student makes the basic choices involved in planning and carrying out a solution; that is, the student:


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NYC M5.b.i - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: chooses and employs effective problem solving strategies in dealing with non-routine and multi-step problems.


NYC M5.b.iii - The student makes the basic choices involved in planning and carrying out a solution; that is, the student: applies mathematical concepts to new situations within mathematics and uses mathematics to model real world situations involving basic applications of mathematics in the physical and biological sciences, the social sciences, and business. - CONCLUSIONNYC M5.c - The student provides closure to the solution process through summary statements and general conclusions; that is, the student:NYC M5.c.i - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: concludes a solution process with a useful summary of results.

NYC M5.c.ii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: evaluates the degree to which the results obtained represent a good response to the initial problem.


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NYC M5.c.iii - The student provides closure to the solution process through summary statements and general conclusions; that is, the student: formulates generalizations of the results obtained.


- MATHEMATICAL REASONINGNYC M5.d - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student:


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Topic 16 - Problem solving and reasoning

Obj. 266 - Proof by induction

Obj. 267 - Indirect proofNYC M5.d.ii - The student demonstrates mathematical reasoning by using logic to prove specific conjectures, by explaining the logic inherent in a solution process, by making generalizations and showing that they are valid, and by revealing mathematical patterns inherent in a situation. The student not only makes observations and states results but also justifies or proves why the results hold in general; that is, the student: differentiates clearly between giving examples that support a conjecture and giving proof of the conjecture.



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NYC M6.a - Carries out numerical calculations and symbol manipulations effectively, using mental computations, pencil and paper, or other technological aids, as appropriate.

Topic 1 - The Real Numbers Obj. 6 - Operations with rational #s

Obj. 7 - Order of operationsObj. 8 - Evaluate expressions for given valuesObj. 9 - WP: Evaluate expressions



Obj. 75 - Simplify expressions w/fractional exponentsObj. 76 - Simplify nth rootsObj. 78 - Simplify expressions w/rational exponentsObj. 79 - Evaluate exponential expressionsObj. 80 - Rewrite expr using only positive exponentsObj. 81 - Add & subtract radical expressions

Topic 8 - Polynomials Obj. 119 - Evaluate polynomials for given valuesObj. 120 - Add polynomialsObj. 121 - Subtract polynomialsObj. 122 - Add & subtract polynomialsObj. 123 - Square a binomial, ax + by

Obj. 124 - Multiply 2 binomialsObj. 125 - Multiply monomials by polynomialsObj. 126 - Multiply binomials by trinomialsObj. 127 - Square trinomials


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Obj. 128 - Simplify polynomial expressions using multiplicationObj. 129 - Divide polynomials by monomialsObj. 130 - Divide polynomialsObj. 131 - Long division, factor higher term polynomialsObj. 132 - Factor trinomials, ax(x + b)(x - c)Obj. 133 - Factor difference of squaresObj. 134 - Factor sum or difference of 2 cubesObj. 135 - Factor polynomials into binomials & trinomialsObj. 136 - Factor 4-term expressions by grouping


Obj. 155 - Simplify rational expressionsObj. 156 - Simplify rational expressions by factoringObj. 158 - Multiply & simplify rational expressionsObj. 159 - Divide & simplify rational expressionsObj. 160 - Add rational expressions

Obj. 161 - Subtract rational expressionsObj. 162 - Add & subtract rational expressionsObj. 164 - Simplify complex rational expressions

NYC M6.b - Uses a variety of methods to estimate the values, in appropriate units, of quantities met in applications, and rounds numbers used in applications to an appropriate degree of accuracy.NYC M6.c - Evaluates and analyzes formulas and functions of many kinds, using both pencil and paper and more advanced technology.


Obj. 141 - Find logs by converting to exponential form

Obj. 142 - Find natural logs of numbersObj. 145 - Solve log equationsObj. 146 - Graph log functions


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Obj. 147 - Solve exponential equationsObj. 148 - Solve equations involving natural logs


Obj. 153 - Rational expressions, domains


NYC M6.e - Makes and uses rough sketches, schematic diagrams, or precise scale diagrams to enhance a solution.NYC M6.f - Uses the number line and Cartesian coordinates in the plane and in space.



Obj. 45 - Write equation of circle given graph


Obj. 167 - Find distances between 2 pointsObj. 170 - Circles, write eqns given centers & radiiObj. 171 - Circles, write eqns using given facts



Obj. 19 - Graph inequalities (number line)

Obj. 20 - Graph inequalities, 1-variable (coordinate plane)Obj. 21 - Graph linear inequalities in 2-variablesObj. 22 - Graph absolute value inequalities (number line)Obj. 23 - Graph absolute value ineq (coordinate plane)Obj. 24 - Solve & graph compound inequalities


Obj. 35 - Graph linear equations, ax + by = cObj. 36 - Graph linear equations, y = ax + bObj. 47 - Graph/write eqn that is result of transformation




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Obj. 107 - WP: Graph quadratic functionsObj. 116 - Graph quadratic inequalities

Topic 8 - Polynomials Obj. 137 - Graph polynomial functions







Obj. 168 - Graph circles

Obj. 173 - Graph ellipsesObj. 176 - Graph hyperbolas (standard equation)Obj. 177 - Graph hyperbolas (general equation)Obj. 179 - Hyperbolas, graph inequalitiesObj. 181 - Graph parabolas (y = x^2)

Obj. 182 - Graph parabolas (x = y^2)


Topic 1 - The Real Numbers Obj. 9 - WP: Evaluate expressions


Obj. 12 - WP: Solve linear equations

Obj. 18 - WP: Solve linear inequalitiesObj. 25 - WP: Absolute value & inequalities


Obj. 38 - WP: Function problems

Obj. 41 - WP: Linear equationsTopic 4 - Systems of Equations and Inequalities

Obj. 55 - WP: Solve systems, number problemsObj. 56 - WP: Solve systems, motion & distance problemsObj. 57 - WP: Solve systems, age problemsObj. 58 - WP: Solve systems, coin problems


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Obj. 59 - WP: Solve systems, mixture & investment problems

Topic 7 - Quadratic Obj. 107 - WP: Graph quadratic functionsObj. 109 - WP: Variation problemsObj. 110 - WP: Pythagorean TheoremObj. 111 - WP: Number problems using quad eqnsObj. 112 - WP: Area & perimeterObj. 113 - WP: Other problems using quad eqns

Topic 8 - Polynomials Obj. 138 - WP: PolynomialsTopic 9 - Exponents and Logarithms

Obj. 149 - WP: Interest problems

Obj. 150 - WP: Growth & decay problemsObj. 151 - WP: Earthquake problems




Obj. 172 - WP: Circles

Obj. 183 - WP: Production & profit problems

NYC M6.i - Knows how to use algorithms in mathematics, such as the Euclidean Algorithm.NYC M6.j - Uses technology to create graphs or spreadsheets that contribute to the understanding of a problem.NYC M6.k - Writes a simple computer program to carry out a computation or simulation to be repeated many times.NYC M6.l - Uses tools such as rulers, tapes, compasses, and protractors in solving problems.NYC M6.m - Knows standard methods to solve basic problems and uses these methods in approaching more complex problems.



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NYC M7-1 - The student uses the language of mathematics, its symbols, notation, graphs, and expressions, to communicate through reading, writing, speaking, and listening, and communicates about mathematics by describing mathematical ideas and concepts and explaining reasoning and results; that is, the student:NYC M7.a - Is familiar with basic mathematical terminology, standard notation and use of symbols, common conventions for graphing, and general features of effective mathematical communication styles.


NYC M7.c - Organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.NYC M7.d - Communicates logical arguments clearly, showing why a result makes sense and why the reasoning is valid.NYC M7.e - Presents mathematical ideas effectively both orally and in writing.NYC M7.f - Explains mathematical concepts clearly enough to be of assistance to those who may be having difficulty with them.NYC M7.g - Writes narrative accounts of the history and process of work on a mathematical problem or extended project.NYC M7.h - Writes succinct accounts of the mathematical results obtained in a mathematical problem or extended project, with diagrams, graphs, tables, and formulas integrated into the text.


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NYC M7.i - Keeps narrative accounts of process separate from succinct accounts of results, and realizes that doing so can enhance the effectiveness of each.NYC M7.j - Reads mathematics texts and other writing about mathematics with understanding.NYC M8 - PUTTING MATHEMATICS TO WORKNYC M8-1 - The student conducts at least one large scale investigation or project each year drawn from the following kinds and, over the course of high school, conducts investigations or projects drawn from at least three of the kinds. A single investigation or project may draw on more than one kind.


NYC M8.a.ii - Data study, in which the student: uses methods of statistical inference to generalize from the data.NYC M8.a.iii - Data study, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.b - Mathematical model of a physical system or phenomenon, in which the student:


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NYC M8.b.i - Mathematical model of a physical system or phenomenon, in which the student: carries out a study of a physical system or phenomenon by constructing a mathematical model based on functions to make generalizations about the structure of the system.




NYC M8.c.iii - Design of a physical structure, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.NYC M8.d - Management and planning analysis, in which the student:


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NYC M8.d.i - Management and planning analysis, in which the student: carries out a study of a business or public policy situation involving issues such as optimization, cost-benefit projections, and risks.



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NYC M8.f.i - History of a mathematical idea, in which the student: carries out a historical study tracing the development of a mathematical concept and the people who contributed to it.NYC M8.f.ii - History of a mathematical idea, in which the student: includes a discussion of the actual mathematical content and its place in the curriculum of the present day.NYC M8.f.iii - History of a mathematical idea, in which the student: prepares a report that explains the purpose of the project, the organizational plan, and conclusions, and uses an appropriate balance of different ways of presenting information.


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new york city state standards alignment kindergarten...

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