NEW JERSEY
STUDENT LEARNING STANDARDS
FOR
Mathematics
New Jersey Student Learning Standards for Mathematics
2
Table of Contents
Standards for Mathematical Practice
3
Standards for Mathematical Content
Kindergarten 7
Grade 1 11
Grade 2 16
Grade 3 21
Grade 4 27
Grade 5 33
Grade 6 39
Grade 7 46
Grade 8 53
High School — Introduction 59
High School — Number and Quantity 60
High School — Algebra 64
High School — Functions 70
High School — Modeling 75
High School — Geometry 77
High School — Statistics and Probability 84
Notes on Courses and Transitions
89
Glossary
90
Sample of Works Consulted
96
New Jersey Student Learning Standards for Mathematics
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Mathematics | Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on important
“processes and proficiencies” with longstanding importance in mathematics education. The first
of these are the NCTM process standards of problem solving, reasoning and proof,
communication, representation, and connections. The second are the strands of mathematical
proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning,
strategic competence, conceptual understanding (comprehension of mathematical concepts,
operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately,
efficiently and appropriately), and productive disposition (habitual inclination to see mathematics
as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem
and looking for entry points to its solution. They analyze givens, constraints, relationships, and
goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous problems,
and try special cases and simpler forms of the original problem in order to gain insight into its
solution. They monitor and evaluate their progress and change course if necessary. Older
students might, depending on the context of the problem, transform algebraic expressions or
change the viewing window on their graphing calculator to get the information they need.
Mathematically proficient students can explain correspondences between equations, verbal
descriptions, tables, and graphs or draw diagrams of important features and relationships, graph
data, and search for regularity or trends. Younger students might rely on using concrete objects
or pictures to help conceptualize and solve a problem. Mathematically proficient students check
their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems
and identify correspondences between different approaches.
2 Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it
symbolically and manipulate the representing symbols as if they have a life of their own, without
necessarily attending to their referents—and the ability to contextualize, to pause as needed
during the manipulation process in order to probe into the referents for the symbols involved.
Quantitative reasoning entails habits of creating a coherent representation of the problem at
hand; considering the units involved; attending to the meaning of quantities, not just how to
compute them; and knowing and flexibly using different properties of operations and objects.
3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and
previously established results in constructing arguments. They make conjectures and build a
logical progression of statements to explore the truth of their conjectures. They are able to
analyze situations by breaking them into cases, and can recognize and use counterexamples.
They justify their conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that take into account
New Jersey Student Learning Standards for Mathematics
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the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary
students can construct arguments using concrete referents such as objects, drawings, diagrams,
and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to
which an argument applies. Students at all grades can listen or read the arguments of others,
decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace. In early grades, this might be as simple as
writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high
school, a student might use geometry to solve a design problem or use a function to describe
how one quantity of interest depends on another. Mathematically proficient students who can
apply what they know are comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to identify
important quantities in a practical situation and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the
context of the situation and reflect on whether the results make sense, possibly improving the
model if it has not served its purpose.
5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry
software. Proficient students are sufficiently familiar with tools appropriate for their grade or
course to make sound decisions about when each of these tools might be helpful, recognizing
both the insight to be gained and their limitations. For example, mathematically proficient high
school students analyze graphs of functions and solutions generated using a graphing calculator.
They detect possible errors by strategically using estimation and other mathematical knowledge.
When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data.
Mathematically proficient students at various grade levels are able to identify relevant external
mathematical resources, such as digital content located on a website, and use them to pose or
solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the
symbols they choose, including using the equal sign consistently and appropriately. They are
careful about specifying units of measure, and labeling axes to clarify the correspondence with
quantities in a problem. They calculate accurately and efficiently, express numerical answers with
a degree of precision appropriate for the problem context. In the elementary grades, students
give carefully formulated explanations to each other. By the time they reach high school they
have learned to examine claims and make explicit use of definitions.
New Jersey Student Learning Standards for Mathematics
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7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students,
for example, might notice that three and seven more is the same amount as seven and three
more, or they may sort a collection of shapes according to how many sides the shapes have.
Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for
learning about the distributive property. In the expression x
2
+ 9x + 14, older students can see the
14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric
figure and can use the strategy of drawing an auxiliary line for solving problems. They also can
step back for an overview and shift perspective. They can see complicated things, such as some
algebraic expressions, as single objects or as being composed of several objects. For example,
they can see 5 – 3(x – y)
2
as 5 minus a positive number times a square and use that to realize that
its value cannot be more than 5 for any real numbers x and y.
8 Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general
methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that
they are repeating the same calculations over and over again, and conclude they have a
repeating decimal. By paying attention to the calculation of slope as they repeatedly check
whether points are on the line through (1, 2) with slope 3, middle school students might abstract
the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x
– 1)(x + 1), (x – 1)(x
2
+ x + 1), and (x – 1)(x
3
+ x
2
+ x + 1) might lead them to the general formula for
the sum of a geometric series. As they work to solve a problem, mathematically proficient
students maintain oversight of the process, while attending to the details. They continually
evaluate the reasonableness of their intermediate results.
New Jersey Student Learning Standards for Mathematics
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Connecting the Standards for Mathematical Practice to the Standards for
Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners
of the discipline of mathematics increasingly ought to engage with the subject matter as they
grow in mathematical maturity and expertise throughout the elementary, middle and high school
years. Designers of curricula, assessments, and professional development should all attend to the
need to connect the mathematical practices to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and
understanding. Expectations that begin with the word “understand” are often especially good
opportunities to connect the practices to the content. Students who lack understanding of a topic
may rely on procedures too heavily. Without a flexible base from which to work, they may be less
likely to consider analogous problems, represent problems coherently, justify conclusions, apply
the mathematics to practical situations, use technology mindfully to work with the mathematics,
explain the mathematics accurately to other students, step back for an overview, or deviate from
a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a
student from engaging in the mathematical practices.
In this respect, those content standards, which set an expectation of understanding, are potential
“points of intersection” between the Standards for Mathematical Content and the Standards for
Mathematical Practice. These points of intersection are intended to be weighted toward central
and generative concepts in the school mathematics curriculum that most merit the time,
resources, innovative energies, and focus necessary to qualitatively improve the curriculum,
instruction, assessment, professional development, and student achievement in mathematics.
New Jersey Student Learning Standards for Mathematics
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Mathematics | Kindergarten
In Kindergarten, instructional time should focus on two critical areas: (1) representing
and comparing whole numbers, initially with sets of objects; (2) describing shapes and
space. More learning time in Kindergarten should be devoted to number than to other
topics.
(1) Students use numbers, including written numerals, to represent quantities and to
solve quantitative problems, such as counting objects in a set; counting out a given
number of objects; comparing sets or numerals; and modeling simple joining and
separating situations with sets of objects, or eventually with equations such as 5 + 2 = 7
and 7 – 2 = 5. (Kindergarten students should see addition and subtraction equations, and
student writing of equations in kindergarten is encouraged, but it is not required.)
Students choose, combine, and apply effective strategies for answering quantitative
questions, including quickly recognizing the cardinalities of small sets of objects, counting
and producing sets of given sizes, counting the number of objects in combined sets, or
counting the number of objects that remain in a set after some are taken away.
(2) Students describe their physical world using geometric ideas (e.g., shape, orientation,
spatial relations) and vocabulary. They identify, name, and describe basic two-
dimensional shapes, such as squares, triangles, circles, rectangles, and hexagons,
presented in a variety of ways (e.g., with different sizes and orientations), as well as
three-dimensional shapes such as cubes, cones, cylinders, and spheres. They use basic
shapes and spatial reasoning to model objects in their environment and to construct
more complex shapes.
New Jersey Student Learning Standards for Mathematics
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade K Overview
Counting and Cardinality
• Know number names and the count
sequence.
• Count to tell the number of objects.
• Compare numbers.
Operations and Algebraic Thinking
• Understand addition as putting together and
adding to, and understand subtraction as
taking apart and taking from.
Number and Operations in Base Ten
• Work with numbers 11–19 to gain
foundations for place value.
Measurement and Data
• Describe and compare measurable attributes.
• Classify objects and count the number of objects in categories.
Geometry
• Identify and describe shapes.
• Analyze, compare, create, and compose shapes.
New Jersey Student Learning Standards for Mathematics
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Counting and Cardinality K.CC
A. Know number names and the count sequence.
1. Count to 100 by ones and by tens.
2. Count forward beginning from a given number within the known sequence (instead of having
to begin at 1).
3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20
(with 0 representing a count of no objects).
B. Count to tell the number of objects.
4. Understand the relationship between numbers and quantities; connect counting to
cardinality.
a. When counting objects, say the number names in the standard order, pairing each object
with one and only one number name and each number name with one and only one
object.
b. Understand that the last number name said tells the number of objects counted. The
number of objects is the same regardless of their arrangement or the order in which they
were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
5. Count to answer “how many?” questions about as many as 20 things arranged in a line, a
rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a
number from 1–20, count out that many objects.
C. Compare numbers.
6. Identify whether the number of objects in one group is greater than, less than, or equal to
the number of objects in another group, e.g., by using matching and counting strategies.
1
7. Compare two numbers between 1 and 10 presented as written numerals.
Operations and Algebraic Thinking K.OA
A. Understand addition as putting together and adding to, and understand subtraction as
taking apart and taking from.
1. Represent addition and subtraction up to 10 with objects, fingers, mental images, drawings
2
,
sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using
objects or drawings to represent the problem.
3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using
objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3
and 5 = 4 + 1).
4. For any number from 1 to 9, find the number that makes 10 when added to the given
number, e.g., by using objects or drawings, and record the answer with a drawing or
equation.
5. Demonstrate fluency for addition and subtraction within 5.
1
Include groups with up to ten objects.
2
Drawings need not show details, but should show the mathematics in the problem.
(This applies wherever drawings are mentioned in the Standards.)
New Jersey Student Learning Standards for Mathematics
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Number and Operations in Base Ten K.NBT
A. Work with numbers 11–19 to gain foundations for place value.
1. Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g.,
by using objects or drawings, and record each composition or decomposition by a drawing or
equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and
one, two, three, four, five, six, seven, eight, or nine ones.
Measurement and Data K.MD
A. Describe and compare measurable attributes.
1. Describe measurable attributes of objects, such as length or weight. Describe several
measurable attributes of a single object.
2. Directly compare two objects with a measurable attribute in common, to see which object
has “more of”/“less of” the attribute, and describe the difference. For example, directly
compare the heights of two children and describe one child as taller/shorter.
B. Classify objects and count the number of objects in each category.
3. Classify objects into given categories; count the numbers of objects in each category and sort
the categories by count.
3
Geometry K.G
A. Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres).
1. Describe objects in the environment using names of shapes, and describe the relative
positions of these objects using terms such as above, below, beside, in front of, behind, and
next to.
2. Correctly name shapes regardless of their orientations or overall size.
3. Identify shapes as two-dimensional (lying in a plane, “flat”) or three-dimensional (“solid”).
B. Analyze, compare, create, and compose shapes.
4. Analyze and compare two- and three-dimensional shapes, in different sizes and orientations,
using informal language to describe their similarities, differences, parts (e.g., number of sides
and vertices/“corners”) and other attributes (e.g., having sides of equal length).
5. Model shapes in the world by building shapes from components (e.g., sticks and clay balls)
and drawing shapes.
6. Compose simple shapes to form larger shapes. For example, “Can you join these two
triangles with full sides touching to make a rectangle?”
3
Limit category counts to be less than or equal to 10.
New Jersey Student Learning Standards for Mathematics
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Mathematics | Grade 1
In Grade 1, instructional time should focus on four critical areas: (1) developing
understanding of addition, subtraction, and strategies for addition and subtraction within
20; (2) developing understanding of whole number relationships and place value,
including grouping in tens and ones; (3) developing understanding of linear measurement
and measuring lengths as iterating length units; and (4) reasoning about attributes of,
and composing and decomposing geometric shapes.
(1) Students develop strategies for adding and subtracting whole numbers based on their
prior work with small numbers. They use a variety of models, including discrete objects
and length-based models (e.g., cubes connected to form lengths), to model add-to, take-
from, put-together, take-apart, and compare situations to develop meaning for the
operations of addition and subtraction, and to develop strategies to solve arithmetic
problems with these operations. Students understand connections between counting
and addition and subtraction (e.g., adding two is the same as counting on two). They use
properties of addition to add whole numbers and to create and use increasingly
sophisticated strategies based on these properties (e.g., “making tens”) to solve addition
and subtraction problems within 20. By comparing a variety of solution strategies,
children build their understanding of the relationship between addition and subtraction.
(2) Students develop, discuss, and use efficient, accurate, and generalizable methods to
add within 100 and subtract multiples of 10. They compare whole numbers (at least to
100) to develop understanding of and solve problems involving their relative sizes. They
think of whole numbers between 10 and 100 in terms of tens and ones (especially
recognizing the numbers 11 to 19 as composed of a ten and some ones). Through
activities that build number sense, they understand the order of the counting numbers
and their relative magnitudes.
(3) Students develop an understanding of the meaning and processes of measurement,
including underlying concepts such as iterating (the mental activity of building up the
length of an object with equal-sized units) and the transitivity principle for indirect
measurement.
1
(4) Students compose and decompose plane or solid figures (e.g., put two triangles
together to make a quadrilateral) and build understanding of part-whole relationships as
well as the properties of the original and composite shapes. As they combine shapes,
they recognize them from different perspectives and orientations, describe their
geometric attributes, and determine how they are alike and different, to develop the
background for measurement and for initial understandings of properties such as
congruence and symmetry.
1
Students should apply the principle of transitivity of measurement to make indirect comparisons, but they need not
use this technical term.
New Jersey Student Learning Standards for Mathematics
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 1 Overview
Operations and Algebraic Thinking
• Represent and solve problems involving
addition and subtraction.
• Understand and apply properties of
operations and the relationship between
addition and subtraction.
• Add and subtract within 20.
• Work with addition and subtraction
equations.
Number and Operations in Base Ten
• Extend the counting sequence.
• Understand place value.
• Use place value understanding and
properties of operations to add and subtract.
Measurement and Data
• Measure lengths indirectly and by iterating length units.
• Tell and write time.
• Represent and interpret data.
Geometry
• Reason with shapes and their attributes.
New Jersey Student Learning Standards for Mathematics
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Operations and Algebraic Thinking 1.OA
A. Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 20 to solve word problems involving situations of adding
to, taking from, putting together, taking apart, and comparing, with unknowns in all
positions, e.g., by using objects, drawings, and equations with a symbol for the unknown
number to represent the problem.
2
2. Solve word problems that call for addition of three whole numbers whose sum is less than or
equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown
number to represent the problem.
B. Understand and apply properties of operations and the relationship between addition and
subtraction.
3. Apply properties of operations as strategies to add and subtract.
3
Examples: If 8 + 3 = 11 is
known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4,
the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative
property of addition.)
{Students need not use formal terms for these properties}
4. Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8.
C. Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.
Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14);
decomposing a number leading to a ten (e.g., 13 – 4 = 13 – 3 – 1 = 10 – 1 = 9); using the
relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 –
8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the
known equivalent 6 + 6 + 1 = 12 + 1 = 13).
D. Work with addition and subtraction equations.
7. Understand the meaning of the equal sign, and determine if equations involving addition and
subtraction are true or false. For example, which of the following equations are true and
which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
8. Determine the unknown whole number in an addition or subtraction equation relating to
three whole numbers. For example, determine the unknown number that makes the equation
true in each of the equations 8 + ? = 11, 5 =
�
– 3, 6 + 6 =
�
.
Number and Operations in Base Ten 1.NBT
A. Extend the counting sequence.
1. Count to 120, starting at any number less than 120. In this range, read and write numerals
and represent a number of objects with a written numeral.
2
See Glossary, Table 1.
3
Students need not use formal terms for these properties.
New Jersey Student Learning Standards for Mathematics
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B. Understand place value.
2. Understand that the two digits of a two-digit number represent amounts of tens and ones.
Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six,
seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven,
eight, or nine tens (and 0 ones).
3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording
the results of comparisons with the symbols >, =, and <.
C. Use place value understanding and properties of operations to add and subtract.
4. Add within 100, including adding a two-digit number and a one-digit number, and adding a
two-digit number and a multiple of 10, using concrete models (e.g., base ten blocks) or
drawings and strategies based on place value, properties of operations, and/or the
relationship between addition and subtraction; relate the strategy to a written method and
explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and
tens, ones and ones; and sometimes it is necessary to compose a ten.
5. Given a two-digit number, mentally find 10 more or 10 less than the number, without having
to count; explain the reasoning used.
6. Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive
or zero differences), using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate
the strategy to a written method and explain the reasoning used.
Measurement and Data 1.MD
A. Measure lengths indirectly and by iterating length units.
1. Order three objects by length; compare the lengths of two objects indirectly by using a third
object.
2. Express the length of an object as a whole number of length units, by laying multiple copies
of a shorter object (the length unit) end to end; understand that the length measurement of
an object is the number of same-size length units that span it with no gaps or overlaps. Limit
to contexts where the object being measured is spanned by a whole number of length units
with no gaps or overlaps.
B. Tell and write time.
3. Tell and write time in hours and half-hours using analog and digital clocks.
C. Represent and interpret data.
4. Organize, represent, and interpret data with up to three categories; ask and answer
questions about the total number of data points, how many in each category, and how many
more or less are in one category than in another.
New Jersey Student Learning Standards for Mathematics
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Geometry 1.G
A. Reason with shapes and their attributes.
1. Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus
non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to
possess defining attributes.
2. Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and
quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular
cones, and right circular cylinders) to create a composite shape, and compose new shapes
from the composite shape.
4
3. Partition circles and rectangles into two and four equal shares, describe the shares using the
words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of.
Describe the whole as two of, or four of the shares. Understand for these examples that
decomposing into more equal shares creates smaller shares.
4
Students do not need to learn formal names such as “right rectangular prism.”
New Jersey Student Learning Standards for Mathematics
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Mathematics | Grade 2
In Grade 2, instructional time should focus on four critical areas: (1) extending
understanding of base-ten notation; (2) building fluency with addition and subtraction;
(3) using standard units of measure; and (4) describing and analyzing shapes.
(1) Students extend their understanding of the base-ten system. This includes ideas of
counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number
relationships involving these units, including comparing. Students understand multi-digit
numbers (up to 1000) written in base-ten notation, recognizing that the digits in each
place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds +
5 tens + 3 ones).
(2) Students use their understanding of addition to develop fluency with addition and
subtraction within 100. They solve problems within 1000 by applying their understanding
of models for addition and subtraction, and they develop, discuss, and use efficient,
accurate, and generalizable methods to compute sums and differences of whole numbers
in base-ten notation, using their understanding of place value and the properties of
operations. They select and accurately apply methods that are appropriate for the
context and the numbers involved to mentally calculate sums and differences for
numbers with only tens or only hundreds.
(3) Students recognize the need for standard units of measure (centimeter and inch) and
they use rulers and other measurement tools with the understanding that linear measure
involves an iteration of units. They recognize that the smaller the unit, the more
iterations they need to cover a given length.
(4) Students describe and analyze shapes by examining their sides and angles. Students
investigate, describe, and reason about decomposing and combining shapes to make
other shapes. Through building, drawing, and analyzing two- and three-dimensional
shapes, students develop a foundation for understanding area, volume, congruence,
similarity, and symmetry in later grades.
New Jersey Student Learning Standards for Mathematics
17
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 2 Overview
Operations and Algebraic Thinking
• Represent and solve problems involving
addition and subtraction.
• Add and subtract within 20.
• Work with equal groups of objects to gain
foundations for multiplication.
Number and Operations in Base Ten
• Understand place value.
• Use place value understanding and
properties of operations to add and subtract.
Measurement and Data
• Measure and estimate lengths in standard
units.
• Relate addition and subtraction to length.
• Work with time and money.
• Represent and interpret data.
Geometry
• Reason with shapes and their attributes.
New Jersey Student Learning Standards for Mathematics
18
Operations and Algebraic Thinking 2.OA
A. Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 100 to solve one- and two-step word problems involving
situations of adding to, taking from, putting together, taking apart, and comparing, with
unknowns in all positions, e.g., by using drawings and equations with a symbol for the
unknown number to represent the problem.
1
B. Add and subtract within 20.
2. Fluently add and subtract within 20 using mental strategies.
2
By end of Grade 2, know from
memory all sums of two one-digit numbers.
C. Work with equal groups of objects to gain foundations for multiplication.
3. Determine whether a group of objects (up to 20) has an odd or even number of members,
e.g., by pairing objects or counting them by 2s; write an equation to express an even number
as a sum of two equal addends.
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5
rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Number and Operations in Base Ten 2.NBT
A. Understand place value.
1. Understand that the three digits of a three-digit number represent amounts of hundreds,
tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as
special cases:
a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four,
five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2. Count within 1000; skip-count by 5s, 10s, and 100s.
3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded
form.
4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits,
using >, =, and < symbols to record the results of comparisons.
B. Use place value understanding and properties of operations to add and subtract.
5. Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
6. Add up to four two-digit numbers using strategies based on place value and properties of
operations.
7. Add and subtract within 1000, using concrete models or drawings and strategies based on
place value, properties of operations, and/or the relationship between addition and
subtraction; relate the strategy to a written method. Understand that in adding or
subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and
tens, ones and ones; and sometimes it is necessary to compose or decompose tens or
hundreds.
1
See Glossary, Table 1.
2
See standard 1.OA.6 for a list of mental strategies.
New Jersey Student Learning Standards for Mathematics
19
8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a
given number 100–900.
9. Explain why addition and subtraction strategies work, using place value and the properties of
operations.
3
Measurement and Data 2.MD
A. Measure and estimate lengths in standard units.
1. Measure the length of an object by selecting and using appropriate tools such as rulers,
yardsticks, meter sticks, and measuring tapes.
2. Measure the length of an object twice, using length units of different lengths for the two
measurements; describe how the two measurements relate to the size of the unit chosen.
3. Estimate lengths using units of inches, feet, centimeters, and meters.
4. Measure to determine how much longer one object is than another, expressing the length
difference in terms of a standard length unit.
B. Relate addition and subtraction to length.
5. Use addition and subtraction within 100 to solve word problems involving lengths that are
given in the same units, e.g., by using drawings (such as drawings of rulers) and equations
with a symbol for the unknown number to represent the problem.
6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced
points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and
differences within 100 on a number line diagram.
C. Work with time and money.
7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and
p.m.
8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and
¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you
have?
D. Represent and interpret data.
9. Generate measurement data by measuring lengths of several objects to the nearest whole
unit, or by making repeated measurements of the same object. Show the measurements by
making a line plot, where the horizontal scale is marked off in whole-number units.
10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up
to four categories. Solve simple put together, take-apart, and compare problems
4
using
information presented in a bar graph.
3
Explanations may be supported by drawings or objects.
4
See Glossary, Table 1.
New Jersey Student Learning Standards for Mathematics
20
Geometry 2.G
A. Reason with shapes and their attributes.
1. Recognize and draw shapes having specified attributes, such as a given number of angles or a
given number of equal faces.
5
Identify triangles, quadrilaterals, pentagons, hexagons, and
cubes.
2. Partition a rectangle into rows and columns of same-size squares and count to find the total
number of them.
3. Partition circles and rectangles into two, three, or four equal shares, describe the shares
using the words halves, thirds, half of, a third of, etc., and desc`ibe the whole as two halves,
three thirds, four fourths. Recognize that equal shares of identical wholes need not have the
same shape.
5
Sizes are compared directly or visually, not compared by measuring.
New Jersey Student Learning Standards for Mathematics
21
Mathematics | Grade 3
In Grade 3, instructional time should focus on four critical areas: (1) developing
understanding of multiplication and division and strategies for multiplication and division
within 100; (2) developing understanding of fractions, especially unit fractions (fractions
with numerator 1); (3) developing understanding of the structure of rectangular arrays
and of area; and (4) describing and analyzing two-dimensional shapes.
(1) Students develop an understanding of the meanings of multiplication and division of
whole numbers through activities and problems involving equal-sized groups, arrays, and
area models; multiplication is finding an unknown product, and division is finding an
unknown factor in these situations. For equal-sized group situations, division can require
finding the unknown number of groups or the unknown group size. Students use
properties of operations to calculate products of whole numbers, using increasingly
sophisticated strategies based on these properties to solve multiplication and division
problems involving single-digit factors. By comparing a variety of solution strategies,
students learn the relationship between multiplication and division.
(2) Students develop an understanding of fractions, beginning with unit fractions.
Students view fractions in general as being built out of unit fractions, and they use
fractions along with visual fraction models to represent parts of a whole. Students
understand that the size of a fractional part is relative to the size of the whole. For
example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a
larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when
the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is
divided into 5 equal parts. Students are able to use fractions to represent numbers equal
to, less than, and greater than one. They solve problems that involve comparing fractions
by using visual fraction models and strategies based on noticing equal numerators or
denominators.
(3) Students recognize area as an attribute of two-dimensional regions. They measure the
area of a shape by finding the total number of same size units of area required to cover
the shape without gaps or overlaps, a square with sides of unit length being the standard
unit for measuring area. Students understand that rectangular arrays can be decomposed
into identical rows or into identical columns. By decomposing rectangles into rectangular
arrays of squares, students connect area to multiplication, and justify using multiplication
to determine the area of a rectangle.
(4) Students describe, analyze, and compare properties of two-dimensional shapes. They
compare and classify shapes by their sides and angles, and connect these with definitions
of shapes. Students also relate their fraction work to geometry by expressing the area of
part of a shape as a unit fraction of the whole.
New Jersey Student Learning Standards for Mathematics
22
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 3 Overview
Operations and Algebraic Thinking
• Represent and solve problems involving
multiplication and division.
• Understand properties of multiplication and
the relationship between multiplication and
division.
• Multiply and divide within 100.
• Solve problems involving the four
operations, and identify and explain patterns
in arithmetic.
Number and Operations in Base Ten
• Use place value understanding and
properties of operations to perform multi-
digit arithmetic.
Number and Operations—Fractions
• Develop understanding of fractions as numbers.
Measurement and Data
• Solve problems involving measurement and estimation of intervals of
time, liquid volumes, and masses of objects.
• Represent and interpret data.
• Geometric measurement: understand concepts of area and relate area to
multiplication and to addition.
• Geometric measurement: recognize perimeter as an attribute of plane
figures and distinguish between linear and area measures.
Geometry
• Reason with shapes and their attributes.
New Jersey Student Learning Standards for Mathematics
23
Operations and Algebraic Thinking 3.OA
A. Represent and solve problems involving multiplication and division.
1. Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5
groups of 7 objects each. For example, describe and/or represent a context in which a total
number of objects can be expressed as 5 × 7.
2. Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of
objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of
shares when 56 objects are partitioned into equal shares of 8 objects each. For example,
describe and/or represent a context in which a number of shares or a number of groups can
be expressed as 56 ÷ 8.
3. Use multiplication and division within 100 to solve word problems in situations involving
equal groups, arrays, and measurement quantities, e.g., by using drawings and equations
with a symbol for the unknown number to represent the problem.
1
4. Determine the unknown whole number in a multiplication or division equation relating three
whole numbers. For example, determine the unknown number that makes the equation true
in each of the equations 8 × ? = 48, 5 =
�
÷ 3, 6 × 6 = ?.
B. Understand properties of multiplication and the relationship between multiplication and
division.
5. Apply properties of operations as strategies to multiply and divide.
2
Examples: If 6 × 4 = 24 is
known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can
be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative
property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5
+ 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
6. Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the
number that makes 32 when multiplied by 8.
C. Multiply and divide within 100.
7. Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties
of operations. By the end of Grade 3, know from memory all products of two one-digit
numbers.
D. Solve problems involving the four operations, and identify and explain patterns in
arithmetic.
8. Solve two-step word problems using the four operations. Represent these problems using
equations with a letter standing for the unknown quantity. Assess the reasonableness of
answers using mental computation and estimation strategies including rounding.
3
9. Identify arithmetic patterns (including patterns in the addition table or multiplication table),
and explain them using properties of operations. For example, observe that 4 times a number
is always even, and explain why 4 times a number can be decomposed into two equal
addends.
1
See Glossary, Table 2.
2
Students need not use formal terms for these properties.
3
This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to
perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).
New Jersey Student Learning Standards for Mathematics
24
Number and Operations in Base Ten 3.NBT
A. Use place value understanding and properties of operations to perform multi-digit
arithmetic.
4
1. Use place value understanding to round whole numbers to the nearest 10 or 100.
2. Fluently add and subtract within 1000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction.
3. Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 × 60)
using strategies based on place value and properties of operations.
Number and Operations—Fractions
5
3.NF
A. Develop understanding of fractions as numbers.
1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into
b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
2. Understand a fraction as a number on the number line; represent fractions on a number line
diagram.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as
the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and
that the endpoint of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0.
Recognize that the resulting interval has size a/b and that its endpoint locates the number
a/b on the number line.
3. Explain equivalence of fractions in special cases, and compare fractions by reasoning about
their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why
the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1
at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by reasoning
about their size. Recognize that comparisons are valid only when the two fractions refer to
the same whole. Record the results of comparisons with the symbols >, =, or <, and justify
the conclusions, e.g., by using a visual fraction model.
Measurement and Data 3.MD
A. Solve problems involving measurement and estimation of intervals of time, liquid volumes,
and masses of objects.
1. Tell and write time to the nearest minute and measure time intervals in minutes. Solve word
problems involving addition and subtraction of time intervals in minutes, e.g., by
representing the problem on a number line diagram.
4
A range of algorithms may be used.
5
Grade 3 expectations in this domain are limited to fractions with denominators 2, 3, 4, 6, and 8.
New Jersey Student Learning Standards for Mathematics
25
2. Measure and estimate liquid volumes and masses of objects using standard units of grams
(g), kilograms (kg), and liters (l).
6
Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using
drawings (such as a beaker with a measurement scale) to represent the problem.
7
B. Represent and interpret data.
3. Draw a scaled picture graph and a scaled bar graph to represent a data set with several
categories. Solve one- and two-step “how many more” and “how many less” problems using
information presented in scaled bar graphs. For example, draw a bar graph in which each
square in the bar graph might represent 5 pets.
4. Generate measurement data by measuring lengths using rulers marked with halves and
fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked
off in appropriate units— whole numbers, halves, or quarters.
C. Geometric measurement: understand concepts of area and relate area to multiplication and
to addition.
5. Recognize area as an attribute of plane figures and understand concepts of area
measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of
area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to
have an area of n square units.
6. Measure areas by counting unit squares (square cm, square m, square in, square ft, and non-
standard units).
7. Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the
area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in the
context of solving real world and mathematical problems, and represent whole-number
products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side
lengths a and b + c is the sum of a × b and a × c. Use area models to represent the
distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-
overlapping rectangles and adding the areas of the non-overlapping parts, applying this
technique to solve real world problems.
D. Geometric measurement: recognize perimeter as an attribute of plane figures and
distinguish between linear and area measures.
8. Solve real world and mathematical problems involving perimeters of polygons, including
finding the perimeter given the side lengths, finding an unknown side length, and exhibiting
rectangles with the same perimeter and different areas or with the same area and different
perimeters.
6
Excludes compound units such as cm3 and finding the geometric volume of a container.
7
Excludes multiplicative comparison problems (problems involving notions of “times as much”; see Glossary, Table 2).
New Jersey Student Learning Standards for Mathematics
26
Geometry 3.G
A. Reason with shapes and their attributes.
1. Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may
share attributes (e.g., having four sides), and that the shared attributes can define a larger
category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of
quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these
subcategories.
2. Partition shapes into parts with equal areas. Express the area of each part as a unit fraction
of the whole. For example, partition a shape into 4 parts with equal area, and describe the
area of each part as 1/4 of the area of the shape.
New Jersey Student Learning Standards for Mathematics
27
Mathematics | Grade 4
In Grade 4, instructional time should focus on three critical areas: (1) developing
understanding and fluency with multi-digit multiplication, and developing understanding
of dividing to find quotients involving multi-digit dividends; (2) developing an
understanding of fraction equivalence, addition and subtraction of fractions with like
denominators, and multiplication of fractions by whole numbers; (3) understanding that
geometric figures can be analyzed and classified based on their properties, such as
having parallel sides, perpendicular sides, particular angle measures, and symmetry.
(1) Students generalize their understanding of place value to 1,000,000, understanding
the relative sizes of numbers in each place. They apply their understanding of models for
multiplication (equal-sized groups, arrays, area models), place value, and properties of
operations, in particular the distributive property, as they develop, discuss, and use
efficient, accurate, and generalizable methods to compute products of multi-digit whole
numbers. Depending on the numbers and the context, they select and accurately apply
appropriate methods to estimate or mentally calculate products. They develop fluency
with efficient procedures for multiplying whole numbers; understand and explain why
the procedures work based on place value and properties of operations; and use them to
solve problems. Students apply their understanding of models for division, place value,
properties of operations, and the relationship of division to multiplication as they
develop, discuss, and use efficient, accurate, and generalizable procedures to find
quotients involving multi-digit dividends. They select and accurately apply appropriate
methods to estimate and mentally calculate quotients, and interpret remainders based
upon the context.
(2) Students develop understanding of fraction equivalence and operations with
fractions. They recognize that two different fractions can be equal (e.g., 15/9 = 5/3), and
they develop methods for generating and recognizing equivalent fractions. Students
extend previous understandings about how fractions are built from unit fractions,
composing fractions from unit fractions, decomposing fractions into unit fractions, and
using the meaning of fractions and the meaning of multiplication to multiply a fraction by
a whole number.
(3) Students describe, analyze, compare, and classify two-dimensional shapes. Through
building, drawing, and analyzing two-dimensional shapes, students deepen their
understanding of properties of two-dimensional objects and the use of them to solve
problems involving symmetry.
New Jersey Student Learning Standards for Mathematics
28
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 4 Overview
Operations and Algebraic Thinking
• Use the four operations with whole numbers
to solve problems.
• Gain familiarity with factors and multiples.
• Generate and analyze patterns.
Number and Operations in Base Ten
• Generalize place value understanding for
multi-digit whole numbers.
• Use place value understanding and properties
of operations to perform multi-digit
arithmetic.
Number and Operations—Fractions
• Extend understanding of fraction equivalence
and ordering.
• Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
• Understand decimal notation for fractions, and compare decimal fractions.
Measurement and Data
• Solve problems involving measurement and conversion of measurements
from a larger unit to a smaller unit.
• Represent and interpret data.
• Geometric measurement: understand concepts of angle and measure
angles.
Geometry
• Draw and identify lines and angles, and classify shapes by properties of
their lines and angles.
New Jersey Student Learning Standards for Mathematics
29
Operations and Algebraic Thinking 4.OA
A. Use the four operations with whole numbers to solve problems.
1. Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement
that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of
multiplicative comparisons as multiplication equations.
2. Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using
drawings and equations with a symbol for the unknown number to represent the problem,
distinguishing multiplicative comparison from additive comparison.
1
3. Solve multistep word problems posed with whole numbers and having whole-number
answers using the four operations, including problems in which remainders must be
interpreted. Represent these problems using equations with a letter standing for the
unknown quantity. Assess the reasonableness of answers using mental computation and
estimation strategies including rounding.
B. Gain familiarity with factors and multiples.
4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number
is a multiple of each of its factors. Determine whether a given whole number in the range 1–
100 is a multiple of a given one-digit number. Determine whether a given whole number in
the range 1–100 is prime or composite.
C. Generate and analyze patterns.
5. Generate a number or shape pattern that follows a given rule. Identify apparent features of
the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and
the starting number 1, generate terms in the resulting sequence and observe that the terms
appear to alternate between odd and even numbers. Explain informally why the numbers will
continue to alternate in this way.
Number and Operations in Base Ten
2
4.NBT
A. Generalize place value understanding for multi-digit whole numbers.
1. Recognize that in a multi-digit whole number, a digit in one place represents ten times what
it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying
concepts of place value and division.
2. Read and write multi-digit whole numbers using base-ten numerals, number names, and
expanded form. Compare two multi-digit numbers based on meanings of the digits in each
place, using >, =, and < symbols to record the results of comparisons.
3. Use place value understanding to round multi-digit whole numbers to any place.
B. Use place value understanding and properties of operations to perform multi-digit
arithmetic.
4. Fluently add and subtract multi-digit whole numbers using the standard algorithm.
5. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two
two-digit numbers, using strategies based on place value and the properties of operations.
Illustrate and explain the calculation by using equations, rectangular arrays, and/or area
models.
1
See Glossary, Table 2.
2
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
New Jersey Student Learning Standards for Mathematics
30
6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit
divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.
Number and Operations—Fractions
3
4.NF
A. Extend understanding of fraction equivalence and ordering.
1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction
models, with attention to how the number and size of the parts differ even though the two
fractions themselves are the same size. Use this principle to recognize and generate
equivalent fractions.
2. Compare two fractions with different numerators and different denominators, e.g., by
creating common denominators or numerators, or by comparing to a benchmark fraction
such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the
same whole. Record the results of comparisons with symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
B. Build fractions from unit fractions by applying and extending previous understandings of
operations on whole numbers.
3. Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating parts referring
to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more than
one way, recording each decomposition by an equation. Justify decompositions, e.g., by
using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 +
1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed
number with an equivalent fraction, and/or by using properties of operations and the
relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the same
whole and having like denominators, e.g., by using visual fraction models and equations to
represent the problem.
4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole
number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction model to
represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 ×
(1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to multiply a
fraction by a whole number. For example, use a visual fraction model to express 3 × (2/5) as
6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g., by
using visual fraction models and equations to represent the problem. For example, if each
person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the
party, how many pounds of roast beef will be needed? Between what two whole numbers
does your answer lie?
3
Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
New Jersey Student Learning Standards for Mathematics
31
C. Understand decimal notation for fractions, and compare decimal fractions.
5. Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and
use this technique to add two fractions with respective denominators 10 and 100.
4
For
example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
6. Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62
as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
7. Compare two decimals to hundredths by reasoning about their size. Recognize that
comparisons are valid only when the two decimals refer to the same whole. Record the
results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a
visual model.
Measurement and Data 4.MD
A. Solve problems involving measurement and conversion of measurements from a larger unit
to a smaller unit.
1. Know relative sizes of measurement units within one system of units including km, m, cm.
mm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express
measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in
a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length
of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number
pairs (1, 12), (2, 24), (3, 36), ...
2. Use the four operations to solve word problems involving distances, intervals of time, liquid
volumes, masses of objects, and money, including problems involving simple fractions or
decimals, and problems that require expressing measurements given in a larger unit in terms
of a smaller unit. Represent measurement quantities using diagrams such as number line
diagrams that feature a measurement scale.
3. Apply the area and perimeter formulas for rectangles in real world and mathematical
problems. For example, find the width of a rectangular room given the area of the flooring
and the length, by viewing the area formula as a multiplication equation with an unknown
factor.
B. Represent and interpret data.
4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).
Solve problems involving addition and subtraction of fractions by using information
presented in line plots. For example, from a line plot find and interpret the difference in
length between the longest and shortest specimens in an insect collection.
C. Geometric measurement: understand concepts of angle and measure angles.
5. Recognize angles as geometric shapes that are formed wherever two rays share a common
endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common endpoint of
the rays, by considering the fraction of the circular arc between the points where the two
rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-
degree angle,” and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of n
degrees.
4
Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But
addition and subtraction with unlike denominators in general is not a requirement at this grade.
New Jersey Student Learning Standards for Mathematics
32
6. Measure angles in whole-number degrees using a protractor. Sketch angles of specified
measure.
7. Recognize angle measure as additive. When an angle is decomposed into non-overlapping
parts, the angle measure of the whole is the sum of the angle measures of the parts. Solve
addition and subtraction problems to find unknown angles on a diagram in real world and
mathematical problems, e.g., by using an equation with a symbol for the unknown angle
measure.
Geometry 4.G
A. Draw and identify lines and angles, and classify shapes by properties of their lines and
angles.
1. Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Identify these in two-dimensional figures.
2. Classify two-dimensional figures based on the presence or absence of parallel or
perpendicular lines, or the presence or absence of angles of a specified size. Recognize right
triangles as a category, and identify right triangles.
3. Recognize a line of symmetry for a two-dimensional figure as a line across the figure such
that the figure can be folded along the line into matching parts. Identify line-symmetric
figures and draw lines of symmetry.
New Jersey Student Learning Standards for Mathematics
33
Mathematics | Grade 5
In Grade 5, instructional time should focus on three critical areas: (1) developing fluency
with addition and subtraction of fractions, and developing understanding of the
multiplication of fractions and of division of fractions in limited cases (unit fractions
divided by whole numbers and whole numbers divided by unit fractions); (2) extending
division to 2-digit divisors, integrating decimal fractions into the place value system and
developing understanding of operations with decimals to hundredths, and developing
fluency with whole number and decimal operations; and (3) developing understanding of
volume.
(1) Students apply their understanding of fractions and fraction models to represent the
addition and subtraction of fractions with unlike denominators as equivalent calculations
with like denominators. They develop fluency in calculating sums and differences of
fractions, and make reasonable estimates of them. Students also use the meaning of
fractions, of multiplication and division, and the relationship between multiplication and
division to understand and explain why the procedures for multiplying and dividing
fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole
numbers and whole numbers by unit fractions.)
(2) Students develop understanding of why division procedures work based on the
meaning of base-ten numerals and properties of operations. They finalize fluency with
multi-digit addition, subtraction, multiplication, and division. They apply their
understandings of models for decimals, decimal notation, and properties of operations to
add and subtract decimals to hundredths. They develop fluency in these computations,
and make reasonable estimates of their results. Students use the relationship between
decimals and fractions, as well as the relationship between finite decimals and whole
numbers (i.e., a finite decimal multiplied by an appropriate power of 10 is a whole
number), to understand and explain why the procedures for multiplying and dividing
finite decimals make sense. They compute products and quotients of decimals to
hundredths efficiently and accurately.
(3) Students recognize volume as an attribute of three-dimensional space. They
understand that volume can be measured by finding the total number of same-size units
of volume required to fill the space without gaps or overlaps. They understand that a 1-
unit by 1-unit by 1-unit cube is the standard unit for measuring volume. They select
appropriate units, strategies, and tools for solving problems that involve estimating and
measuring volume. They decompose three-dimensional shapes and find volumes of right
rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They
measure necessary attributes of shapes in order to determine volumes to solve real
world and mathematical problems.
New Jersey Student Learning Standards for Mathematics
34
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 5 Overview
Operations and Algebraic Thinking
• Write and interpret numerical expressions.
• Analyze patterns and relationships.
Number and Operations in Base Ten
• Understand the place value system.
• Perform operations with multi-digit whole
numbers and with decimals to hundredths.
Number and Operations—Fractions
• Use equivalent fractions as a strategy to add
and subtract fractions.
• Apply and extend previous understandings
of multiplication and division to multiply and
divide fractions.
Measurement and Data
• Convert like measurement units within a given measurement system.
• Represent and interpret data.
• Geometric measurement: understand concepts of volume and relate
volume to multiplication and to addition.
Geometry
• Graph points on the coordinate plane to solve real-world and
mathematical problems.
• Classify two-dimensional figures into categories based on their properties.
New Jersey Student Learning Standards for Mathematics
35
Operations and Algebraic Thinking 5.OA
A. Write and interpret numerical expressions.
1. Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions
with these symbols.
2. Write simple expressions that record calculations with numbers, and interpret numerical
expressions without evaluating them. For example, express the calculation “add 8 and 7, then
multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932
+ 921, without having to calculate the indicated sum or product.
B. Analyze patterns and relationships.
3. Generate two numerical patterns using two given rules. Identify apparent relationships
between corresponding terms. Form ordered pairs consisting of corresponding terms from
the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the
rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number
0, generate terms in the resulting sequences, and observe that the terms in one sequence are
twice the corresponding terms in the other sequence. Explain informally why this is so.
Number and Operations in Base Ten 5.NBT
A. Understand the place value system.
1. Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it
represents in the place to its right and 1/10 of what it represents in the place to its left.
2. Explain patterns in the number of zeros of the product when multiplying a number by
powers of 10, and explain patterns in the placement of the decimal point when a decimal is
multiplied or divided by a power of 10. Use whole-number exponents to denote powers of
10.
3. Read, write, and compare decimals to thousandths.
a. Read and write decimals to thousandths using base-ten numerals, number names, and
expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 ×
(1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each place, using
>, =, and < symbols to record the results of comparisons.
4. Use place value understanding to round decimals to any place.
B. Perform operations with multi-digit whole numbers and with decimals to hundredths.
5. Fluently multiply multi-digit whole numbers using the standard algorithm.
6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-
digit divisors, using strategies based on place value, the properties of operations, and/or the
relationship between multiplication and division. Illustrate and explain the calculation by
using equations, rectangular arrays, and/or area models.
7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or
drawings and strategies based on place value, properties of operations, and/or the
relationship between addition and subtraction; relate the strategy to a written method and
explain the reasoning used.
New Jersey Student Learning Standards for Mathematics
36
Number and Operations—Fractions 5.NF
A. Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing
given fractions with equivalent fractions in such a way as to produce an equivalent sum or
difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 =
23/12. (In general, a/b + c/d = (ad + bc)/bd.)
2. Solve word problems involving addition and subtraction of fractions referring to the same
whole, including cases of unlike denominators, e.g., by using visual fraction models or
equations to represent the problem. Use benchmark fractions and number sense of fractions
to estimate mentally and assess the reasonableness of answers. For example, recognize an
incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2.
B. Apply and extend previous understandings of multiplication and division to multiply and
divide fractions.
3. Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve
word problems involving division of whole numbers leading to answers in the form of
fractions or mixed numbers, e.g., by using visual fraction models or equations to represent
the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4
multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each
person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by
weight, how many pounds of rice should each person get? Between what two whole numbers
does your answer lie?
4. Apply and extend previous understandings of multiplication to multiply a fraction or whole
number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently,
as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model
to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3)
× (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the
appropriate unit fraction side lengths, and show that the area is the same as would be
found by multiplying the side lengths. Multiply fractional side lengths to find areas of
rectangles, and represent fraction products as rectangular areas.
5. Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the
other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product
greater than the given number (recognizing multiplication by whole numbers greater than
1 as a familiar case); explaining why multiplying a given number by a fraction less than 1
results in a product smaller than the given number; and relating the principle of fraction
equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
6. Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by
using visual fraction models or equations to represent the problem.
New Jersey Student Learning Standards for Mathematics
37
7. Apply and extend previous understandings of division to divide unit fractions by whole
numbers and whole numbers by unit fractions.
1
a. Interpret division of a unit fraction by a non-zero whole number,
and compute such
quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model
to show the quotient. Use the relationship between multiplication and division to explain
that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For
example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the
quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) =
20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers
and division of whole numbers by unit fractions, e.g., by using visual fraction models and
equations to represent the problem. For example, how much chocolate will each person get
if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of
raisins?
Measurement and Data 5.MD
A. Convert like measurement units within a given measurement system.
1. Convert among different-sized standard measurement units within a given measurement
system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real
world problems.
B. Represent and interpret data.
2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).
Use operations on fractions for this grade to solve problems involving information presented
in line plots. For example, given different measurements of liquid in identical beakers, find the
amount of liquid each beaker would contain if the total amount in all the beakers were
redistributed equally.
C. Geometric measurement: understand concepts of volume and relate volume to
multiplication and to addition.
3. Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit” of
volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is said to
have a volume of n cubic units.
4. Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-standard
units.
5. Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by packing it
with unit cubes, and show that the volume is the same as would be found by multiplying
the edge lengths, equivalently by multiplying the height by the area of the base. Represent
threefold whole-number products as volumes, e.g., to represent the associative property of
multiplication.
1
Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the
relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.
New Jersey Student Learning Standards for Mathematics
38
b. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of
right rectangular prisms with whole number edge lengths in the context of solving real
world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two non-
overlapping right rectangular prisms by adding the volumes of the non-overlapping parts,
applying this technique to solve real world problems.
Geometry 5.G
A. Graph points on the coordinate plane to solve real-world and mathematical problems.
1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with
the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a
given point in the plane located by using an ordered pair of numbers, called its coordinates.
Understand that the first number indicates how far to travel from the origin in the direction
of one axis, and the second number indicates how far to travel in the direction of the second
axis, with the convention that the names of the two axes and the coordinates correspond
(e.g., x-axis and x-coordinate, y-axis and y-coordinate).
2. Represent real world and mathematical problems by graphing points in the first quadrant of
the coordinate plane, and interpret coordinate values of points in the context of the
situation.
B. Classify two-dimensional figures into categories based on their properties.
3. Understand that attributes belonging to a category of two-dimensional figures also belong to
all subcategories of that category. For example, all rectangles have four right angles and
squares are rectangles, so all squares have four right angles.
4. Classify two-dimensional figures in a hierarchy based on properties.
New Jersey Student Learning Standards for Mathematics
39
Mathematics | Grade 6
In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and
rate to whole number multiplication and division and using concepts of ratio and rate to
solve problems; (2) completing understanding of division of fractions and extending the
notion of number to the system of rational numbers, which includes negative numbers;
(3) writing, interpreting, and using expressions and equations; and (4) developing
understanding of statistical thinking.
(1) Students use reasoning about multiplication and division to solve ratio and
rate problems about quantities. By viewing equivalent ratios and rates as deriving
from, and extending, pairs of rows (or columns) in the multiplication table, and by
analyzing simple drawings that indicate the relative size of quantities, students
connect their understanding of multiplication and division with ratios and rates.
Thus students expand the scope of problems for which they can use
multiplication and division to solve problems, and they connect ratios and
fractions. Students solve a wide variety of problems involving ratios and rates.
(2) Students use the meaning of fractions, the meanings of multiplication and
division, and the relationship between multiplication and division to understand
and explain why the procedures for dividing fractions make sense. Students use
these operations to solve problems. Students extend their previous
understandings of number and the ordering of numbers to the full system of
rational numbers, which includes negative rational numbers, and in particular
negative integers. They reason about the order and absolute value of rational
numbers and about the location of points in all four quadrants of the coordinate
plane.
(3) Students understand the use of variables in mathematical expressions. They
write expressions and equations that correspond to given situations, evaluate
expressions, and use expressions and formulas to solve problems. Students
understand that expressions in different forms can be equivalent, and they use
the properties of operations to rewrite expressions in equivalent forms. Students
know that the solutions of an equation are the values of the variables that make
the equation true. Students use properties of operations and the idea of
maintaining the equality of both sides of an equation to solve simple one-step
equations. Students construct and analyze tables, such as tables of quantities that
are in equivalent ratios, and they use equations (such as 3x = y) to describe
relationships between quantities.
(4) Building on and reinforcing their understanding of number, students begin to
develop their ability to think statistically. Students recognize that a data
New Jersey Student Learning Standards for Mathematics
40
distribution may not have a definite center and that different ways to measure
center yield different values. The median measures center in the sense that it is
roughly the middle value. The mean measures center in the sense that it is the
value that each data point would take on if the total of the data values were
redistributed equally, and also in the sense that it is a balance point. Students
recognize that a measure of variability (interquartile range or mean absolute
deviation) can also be useful for summarizing data because two very different
sets of data can have the same mean and median yet be distinguished by their
variability. Students learn to describe and summarize numerical data sets,
identifying clusters, peaks, gaps, and symmetry, considering the context in which
the data were collected.
Students in Grade 6 also build on their work with area in elementary school by
reasoning about relationships among shapes to determine area, surface area, and
volume. They find areas of right triangles, other triangles, and special
quadrilaterals by decomposing these shapes, rearranging or removing pieces, and
relating the shapes to rectangles. Using these methods, students discuss, develop,
and justify formulas for areas of triangles and parallelograms. Students find areas
of polygons and surface areas of prisms and pyramids by decomposing them into
pieces whose area they can determine. They reason about right rectangular
prisms with fractional side lengths to extend formulas for the volume of a right
rectangular prism to fractional side lengths. They prepare for work on scale
drawings and constructions in Grade 7 by drawing polygons in the coordinate
plane.
New Jersey Student Learning Standards for Mathematics
41
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 6 Overview
Ratios and Proportional Relationships
• Understand ratio concepts and use ratio
reasoning to solve problems.
The Number System
• Apply and extend previous understandings
of multiplication and division to divide
fractions by fractions.
• Compute fluently with multi-digit numbers
and find common factors and multiples.
• Apply and extend previous understandings
of numbers to the system of rational
numbers.
Expressions and Equations
• Apply and extend previous understandings of arithmetic to algebraic
expressions.
• Reason about and solve one-variable equations and inequalities.
• Represent and analyze quantitative relationships between dependent and
independent variables.
Geometry
• Solve real-world and mathematical problems involving area, surface area,
and volume.
Statistics and Probability
• Develop understanding of statistical variability.
• Summarize and describe distributions.
New Jersey Student Learning Standards for Mathematics
42
Ratios and Proportional Relationships 6.RP
A. Understand ratio concepts and use ratio reasoning to solve problems.
1. Understand the concept of a ratio and use ratio language to describe a ratio relationship
between two quantities. For example, “The ratio of wings to beaks in the bird house at the
zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A
received, candidate C received nearly three votes.”
2. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use
rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3
cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid
$75 for 15 hamburgers, which is a rate of $5 per hamburger.”
1
3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by
reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or
equations.
a. Make tables of equivalent ratios relating quantities with whole number measurements,
find missing values in the tables, and plot the pairs of values on the coordinate plane. Use
tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For
example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times
the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
The Number System 6.NS
A. Apply and extend previous understandings of multiplication and division to divide fractions
by fractions.
1. Interpret and compute quotients of fractions, and solve word problems involving division of
fractions by fractions, e.g., by using visual fraction models and equations to represent the
problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model
to show the quotient; use the relationship between multiplication and division to explain that
(2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc). How much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-
cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length
3/4 mi and area 1/2 square mi?
B. Compute fluently with multi-digit numbers and find common factors and multiples.
2. Fluently divide multi-digit numbers using the standard algorithm.
3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm
for each operation.
4. Find the greatest common factor of two whole numbers less than or equal to 100 and the
least common multiple of two whole numbers less than or equal to 12. Use the distributive
property to express a sum of two whole numbers 1–100 with a common factor as a multiple
of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 +
2).
1
Expectations for unit rates in this grade are limited to non-complex fractions.
New Jersey Student Learning Standards for Mathematics
43
C. Apply and extend previous understandings of numbers to the system of rational numbers.
5. Understand that positive and negative numbers are used together to describe quantities
having opposite directions or values (e.g., temperature above/below zero, elevation
above/below sea level, credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts, explaining the meaning of 0
in each situation.
6. Understand a rational number as a point on the number line. Extend number line diagrams
and coordinate axes familiar from previous grades to represent points on the line and in the
plane with negative number coordinates.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the
number line; recognize that the opposite of the opposite of a number is the number itself,
e.g., –(–3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the
coordinate plane; recognize that when two ordered pairs differ only by signs, the locations
of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical number
line diagram; find and position pairs of integers and other rational numbers on a
coordinate plane.
7. Understand ordering and absolute value of rational numbers.
a. Interpret statements of inequality as statements about the relative position of two
numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3
is located to the right of –7 on a number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world
contexts. For example, write –3
o
C > –7
o
C to express the fact that
–3
o
C is warmer than –7
o
C.
c. Understand the absolute value of a rational number as its distance from 0 on the number
line; interpret absolute value as magnitude for a positive or negative quantity in a real-
world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to
describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For example,
recognize that an account balance less than –30 dollars represents a debt greater than 30
dollars.
8. Solve real-world and mathematical problems by graphing points in all four quadrants of the
coordinate plane. Include use of coordinates and absolute value to find distances between
points with the same first coordinate or the same second coordinate.
Expressions and Equations 6.EE
A. Apply and extend previous understandings of arithmetic to algebraic expressions.
1. Write and evaluate numerical expressions involving whole-number exponents.
2. Write, read, and evaluate expressions in which letters stand for numbers.
a. Write expressions that record operations with numbers and with letters standing for
numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product, factor,
quotient, coefficient); view one or more parts of an expression as a single entity. For
example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a
single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that arise
from formulas used in real-world problems. Perform arithmetic operations, including those
New Jersey Student Learning Standards for Mathematics
44
involving whole number exponents, in the conventional order when there are no
parentheses to specify a particular order (Order of Operations). For example, use the
formulas V = s
3
and A = 6s
2
to find the volume and surface area of a cube with sides of
length s = 1/2.
3. Apply the properties of operations to generate equivalent expressions. For example, apply
the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 +
3x; apply the distributive property to the expression 24x + 18y to produce the equivalent
expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent
expression 3y.
4. Identify when two expressions are equivalent (i.e., when the two expressions name the same
number regardless of which value is substituted into them). For example, the expressions y +
y + y and 3y are equivalent because they name the same number regardless of which number
y stands for.
B. Reason about and solve one-variable equations and inequalities.
5. Understand solving an equation or inequality as a process of answering a question: which
values from a specified set, if any, make the equation or inequality true? Use substitution to
determine whether a given number in a specified set makes an equation or inequality true.
6. Use variables to represent numbers and write expressions when solving a real-world or
mathematical problem; understand that a variable can represent an unknown number, or,
depending on the purpose at hand, any number in a specified set.
7. Solve real-world and mathematical problems by writing and solving equations of the form x +
p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-
world or mathematical problem. Recognize that inequalities of the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number line diagrams.
C. Represent and analyze quantitative relationships between dependent and independent
variables.
9. Use variables to represent two quantities in a real-world problem that change in relationship
to one another; write an equation to express one quantity, thought of as the dependent
variable, in terms of the other quantity, thought of as the independent variable. Analyze the
relationship between the dependent and independent variables using graphs and tables, and
relate these to the equation. For example, in a problem involving motion at constant speed,
list and graph ordered pairs of distances and times, and write the equation d = 65t to
represent the relationship between distance and time.
Geometry 6.G
A. Solve real-world and mathematical problems involving area, surface area, and volume.
1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by
composing into rectangles or decomposing into triangles and other shapes; apply these
techniques in the context of solving real-world and mathematical problems.
2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with
unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the
same as would be found by multiplying the edge lengths of the prism. Apply the formulas V =
l w h and V = B h to find volumes of right rectangular prisms with fractional edge lengths in
the context of solving real-world and mathematical problems.
New Jersey Student Learning Standards for Mathematics
45
3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to
find the length of a side joining points with the same first coordinate or the same second
coordinate. Apply these techniques in the context of solving real-world and mathematical
problems.
4. Represent three-dimensional figures using nets made up of rectangles and triangles, and use
the nets to find the surface area of these figures. Apply these techniques in the context of
solving real-world and mathematical problems.
Statistics and Probability 6.SP
A. Develop understanding of statistical variability.
1. Recognize a statistical question as one that anticipates variability in the data related to the
question and accounts for it in the answers. For example, “How old am I?” is not a statistical
question, but “How old are the students in my school?” is a statistical question because one
anticipates variability in students’ ages.
2. Understand that a set of data collected to answer a statistical question has a distribution
which can be described by its center, spread, and overall shape.
3. Recognize that a measure of center for a numerical data set summarizes all of its values with
a single number, while a measure of variation describes how its values vary with a single
number.
B. Summarize and describe distributions.
4. Display numerical data in plots on a number line, including dot plots, histograms, and box
plots.
5. Summarize numerical data sets in relation to their context, such as by:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was measured
and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability (interquartile
range and/or mean absolute deviation), as well as describing any overall pattern and any
striking deviations from the overall pattern with reference to the context in which the data
were gathered.
d. Relating the choice of measures of center and variability to the shape of the data
distribution and the context in which the data were gathered.
New Jersey Student Learning Standards for Mathematics
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Mathematics | Grade 7
In Grade 7, instructional time should focus on four critical areas: (1) developing
understanding of and applying proportional relationships; (2) developing understanding
of operations with rational numbers and working with expressions and linear equations;
(3) solving problems involving scale drawings and informal geometric constructions, and
working with two- and three-dimensional shapes to solve problems involving area,
surface area, and volume; and (4) drawing inferences about populations based on
samples.
(1) Students extend their understanding of ratios and develop understanding of
proportionality to solve single- and multi-step problems. Students use their
understanding of ratios and proportionality to solve a wide variety of percent
problems, including those involving discounts, interest, taxes, tips, and percent
increase or decrease. Students solve problems about scale drawings by relating
corresponding lengths between the objects or by using the fact that relationships
of lengths within an object are preserved in similar objects. Students graph
proportional relationships and understand the unit rate informally as a measure
of the steepness of the related line, called the slope. They distinguish
proportional relationships from other relationships.
(2) Students develop a unified understanding of number, recognizing fractions,
decimals (that have a finite or a repeating decimal representation), and percents
as different representations of rational numbers. Students extend addition,
subtraction, multiplication, and division to all rational numbers, maintaining the
properties of operations and the relationships between addition and subtraction,
and multiplication and division. By applying these properties, and by viewing
negative numbers in terms of everyday contexts (e.g., amounts owed or
temperatures below zero), students explain and interpret the rules for adding,
subtracting, multiplying, and dividing with negative numbers. They use the
arithmetic of rational numbers as they formulate expressions and equations in
one variable and use these equations to solve problems.
(3) Students continue their work with area from Grade 6, solving problems
involving the area and circumference of a circle and surface area of three-
dimensional objects. In preparation for work on congruence and similarity in
Grade 8 they reason about relationships among two-dimensional figures using
scale drawings and informal geometric constructions, and they gain familiarity
with the relationships between angles formed by intersecting lines. Students
work with three-dimensional figures, relating them to two-dimensional figures by
examining cross-sections. They solve real-world and mathematical problems
New Jersey Student Learning Standards for Mathematics
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involving area, surface area, and volume of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons, cubes and right prisms.
(4) Students build on their previous work with single data distributions to
compare two data distributions and address questions about differences between
populations. They begin informal work with random sampling to generate data
sets and learn about the importance of representative samples for drawing
inferences.
New Jersey Student Learning Standards for Mathematics
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 7 Overview
Ratios and Proportional Relationships
• Analyze proportional relationships and use
them to solve real-world and mathematical
problems.
The Number System
• Apply and extend previous understandings
of operations with fractions to add, subtract,
multiply, and divide rational numbers.
Expressions and Equations
• Use properties of operations to generate
equivalent expressions.
• Solve real-life and mathematical problems
using numerical and algebraic expressions and equations.
Geometry
• Draw, construct and describe geometrical figures and describe the
relationships between them.
• Solve real-life and mathematical problems involving angle measure, area,
surface area, and volume.
Statistics and Probability
• Use random sampling to draw inferences about a population.
• Draw informal comparative inferences about two populations.
• Investigate chance processes and develop, use, and evaluate probability
models.
New Jersey Student Learning Standards for Mathematics
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Ratios and Proportional Relationships 7.RP
A. Analyze proportional relationships and use them to solve real-world and mathematical
problems.
1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and
other quantities measured in like or different units. For example, if a person walks 1/2 mile in
each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour,
equivalently 2 miles per hour.
2. Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing whether the
graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams,
and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is
proportional to the number n of items purchased at a constant price p, the relationship
between the total cost and the number of items can be expressed as t = pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of
the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
3. Use proportional relationships to solve multistep ratio and percent problems. Examples:
simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent
increase and decrease, percent error.
The Number System 7.NS
A. Apply and extend previous understandings of operations with fractions to add, subtract,
multiply, and divide rational numbers.
1. Apply and extend previous understandings of addition and subtraction to add and subtract
rational numbers; represent addition and subtraction on a horizontal or vertical number line
diagram.
a. Describe situations in which opposite quantities combine to make 0. For example, in the
first round of a game, Maria scored 20 points. In the second round of the same game, she
lost 20 points. What is her score at the end of the second round?
b. Understand p + q as the number located a distance |q| from p, in the positive or negative
direction depending on whether q is positive or negative. Show that a number and its
opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by
describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q).
Show that the distance between two rational numbers on the number line is the absolute
value of their difference, and apply this principle in real-world contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
2. Apply and extend previous understandings of multiplication and division and of fractions to
multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by requiring
that operations continue to satisfy the properties of operations, particularly the distributive
property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed
numbers. Interpret products of rational numbers by describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and every
quotient of integers (with non-zero divisor) is a rational number. If p and q are integers,
New Jersey Student Learning Standards for Mathematics
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then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real
world contexts.
c. Apply properties of operations as strategies to multiply and divide rational numbers.
d. Convert a rational number to a decimal using long division; know that the decimal form of
a rational number terminates in 0s or eventually repeats.
3. Solve real-world and mathematical problems involving the four operations with rational
numbers.
1
Expressions and Equations 7.EE
A. Use properties of operations to generate equivalent expressions.
1. Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
2. Understand that rewriting an expression in different forms in a problem context can shed
light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a
means that “increase by 5%” is the same as “multiply by 1.05.”
B. Solve real-life and mathematical problems using numerical and algebraic expressions and
equations.
3. Solve multi-step real-life and mathematical problems posed with positive and negative
rational numbers in any form (whole numbers, fractions, and decimals), using tools
strategically. Apply properties of operations to calculate with numbers in any form; convert
between forms as appropriate; and assess the reasonableness of answers using mental
computation and estimation strategies. For example: If a woman making $25 an hour gets a
10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary
of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27
1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate
can be used as a check on the exact computation.
4. Use variables to represent quantities in a real-world or mathematical problem, and construct
simple equations and inequalities to solve problems by reasoning about the quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p,
q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an
algebraic solution to an arithmetic solution, identifying the sequence of the operations
used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6
cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q,
and r are specific rational numbers. Graph the solution set of the inequality and interpret it
in the context of the problem. For example: As a salesperson, you are paid $50 per week
plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for
the number of sales you need to make, and describe the solutions.
Geometry 7.G
A. Draw, construct, and describe geometrical figures and describe the relationships between
them.
1. Solve problems involving scale drawings of geometric figures, including computing actual
lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.
1
Computations with rational numbers extend the rules for manipulating fractions to complex fractions.
New Jersey Student Learning Standards for Mathematics
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2. Draw (with technology, with ruler and protractor, as well as freehand) geometric shapes with
given conditions. Focus on constructing triangles from three measures of angles or sides,
noticing when the conditions determine a unique triangle, more than one triangle, or no
triangle.
3. Describe the two-dimensional figures that result from slicing three-dimensional figures, as in
plane sections of right rectangular prisms and right rectangular pyramids.
B. Solve real-life and mathematical problems involving angle measure, area, surface area, and
volume.
4. Know the formulas for the area and circumference of a circle and use them to solve
problems; give an informal derivation of the relationship between the circumference and
area of a circle.
5. Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step
problem to write and solve simple equations for an unknown angle in a figure.
6. Solve real-world and mathematical problems involving area, volume and surface area of two-
and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and
right prisms.
Statistics and Probability 7.SP
A. Use random sampling to draw inferences about a population.
1. Understand that statistics can be used to gain information about a population by examining
a sample of the population; generalizations about a population from a sample are valid only if
the sample is representative of that population. Understand that random sampling tends to
produce representative samples and support valid inferences.
2. Use data from a random sample to draw inferences about a population with an unknown
characteristic of interest. Generate multiple samples (or simulated samples) of the same size
to gauge the variation in estimates or predictions. For example, estimate the mean word
length in a book by randomly sampling words from the book; predict the winner of a school
election based on randomly sampled survey data. Gauge how far off the estimate or
prediction might be.
B. Draw informal comparative inferences about two populations.
3. Informally assess the degree of visual overlap of two numerical data distributions with
similar variabilities, measuring the difference between the centers by expressing it as a
multiple of a measure of variability. For example, the mean height of players on the
basketball team is 10 cm greater than the mean height of players on the soccer team, about
twice the variability (mean absolute deviation) on either team; on a dot plot, the separation
between the two distributions of heights is noticeable.
4. Use measures of center and measures of variability for numerical data from random samples
to draw informal comparative inferences about two populations. For example, decide
whether the words in a chapter of a seventh-grade science book are generally longer than the
words in a chapter of a fourth-grade science book.
C. Investigate chance processes and develop, use, and evaluate probability models.
5. Understand that the probability of a chance event is a number between 0 and 1 that
expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A
probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event
that is neither unlikely nor likely, and a probability near 1 indicates a likely event.
New Jersey Student Learning Standards for Mathematics
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6. Approximate the probability of a chance event by collecting data on the chance process that
produces it and observing its long-run relative frequency, and predict the approximate
relative frequency given the probability. For example, when rolling a number cube 600 times,
predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
7. Develop a probability model and use it to find probabilities of events. Compare probabilities
from a model to observed frequencies; if the agreement is not good, explain possible sources
of the discrepancy.
a. Develop a uniform probability model by assigning equal probability to all outcomes, and
use the model to determine probabilities of events. For example, if a student is selected at
random from a class, find the probability that Jane will be selected and the probability that
a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies in data
generated from a chance process. For example, find the approximate probability that a
spinning penny will land heads up or that a tossed paper cup will land open-end down. Do
the outcomes for the spinning penny appear to be equally likely based on the observed
frequencies?
8. Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation.
a. Understand that, just as with simple events, the probability of a compound event is the
fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized lists,
tables and tree diagrams. For an event described in everyday language (e.g., “rolling double
sixes”), identify the outcomes in the sample space which compose the event.
c. Design and use a simulation to generate frequencies for compound events. For example,
use random digits as a simulation tool to approximate the answer to the question: If 40% of
donors have type A blood, what is the probability that it will take at least 4 donors to find
one with type A blood?
New Jersey Student Learning Standards for Mathematics
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Mathematics | Grade 8
In Grade 8, instructional time should focus on three critical areas: (1) formulating and
reasoning about expressions and equations, including modeling an association in
bivariate data with a linear equation, and solving linear equations and systems of linear
equations; (2) grasping the concept of a function and using functions to describe
quantitative relationships; (3) analyzing two- and three-dimensional space and figures
using distance, angle, similarity, and congruence, and understanding and applying the
Pythagorean Theorem.
(1) Students use linear equations and systems of linear equations to represent, analyze,
and solve a variety of problems. Students recognize equations for proportions (y/x = m or
y = mx) as special linear equations (y = mx + b), understanding that the constant of
proportionality (m) is the slope, and the graphs are lines through the origin. They
understand that the slope (m) of a line is a constant rate of change, so that if the input or
x-coordinate changes by an amount A, the output or y-coordinate changes by the
amount m·A. Students also use a linear equation to describe the association between
two quantities in bivariate data (such as arm span vs. height for students in a classroom).
At this grade, fitting the model, and assessing its fit to the data are done informally.
Interpreting the model in the context of the data requires students to express a
relationship between the two quantities in question and to interpret components of the
relationship (such as slope and y-intercept) in terms of the situation.
Students strategically choose and efficiently implement procedures to solve linear
equations in one variable, understanding that when they use the properties of equality
and the concept of logical equivalence, they maintain the solutions of the original
equation. Students solve systems of two linear equations in two variables and relate the
systems to pairs of lines in the plane; these intersect, are parallel, or are the same line.
Students use linear equations, systems of linear equations, linear functions, and their
understanding of slope of a line to analyze situations and solve problems.
(2) Students grasp the concept of a function as a rule that assigns to each input exactly
one output. They understand that functions describe situations where one quantity
determines another. They can translate among representations and partial
representations of functions (noting that tabular and graphical representations may be
partial representations), and they describe how aspects of the function are reflected in
the different representations.
(3) Students use ideas about distance and angles, how they behave under translations,
rotations, reflections, and dilations, and ideas about congruence and similarity to
describe and analyze two-dimensional figures and to solve problems. Students show that
the sum of the angles in a triangle is the angle formed by a straight line, and that various
configurations of lines give rise to similar triangles because of the angles created when a
transversal cuts parallel lines. Students understand the statement of the Pythagorean
Theorem and its converse, and can explain why the Pythagorean Theorem holds, for
New Jersey Student Learning Standards for Mathematics
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example, by decomposing a square in two different ways. They apply the Pythagorean
Theorem to find distances between points on the coordinate plane, to find lengths, and
to analyze polygons. Students complete their work on volume by solving problems
involving cones, cylinders, and spheres.
New Jersey Student Learning Standards for Mathematics
55
Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Grade 8 Overview
The Number System
• Know that there are numbers that are not
rational, and approximate them by rational
numbers.
Expressions and Equations
• Work with radicals and integer exponents.
• Understand the connections between
proportional relationships, lines, and linear
equations.
• Analyze and solve linear equations and pairs of
simultaneous linear equations.
Functions
• Define, evaluate, and compare functions.
• Use functions to model relationships between quantities.
Geometry
• Understand congruence and similarity using physical models,
transparencies, or geometry software.
• Understand and apply the Pythagorean Theorem.
• Solve real-world and mathematical problems involving volume of
cylinders, cones and spheres.
Statistics and Probability
• Investigate patterns of association in bivariate data.
New Jersey Student Learning Standards for Mathematics
56
The Number System 8.NS
A. Know that there are numbers that are not rational, and approximate them by rational
numbers.
1. Know that numbers that are not rational are called irrational. Understand informally that
every number has a decimal expansion; for rational numbers show that the decimal
expansion repeats eventually, and convert a decimal expansion which repeats eventually into
a rational number.
2. Use rational approximations of irrational numbers to compare the size of irrational numbers,
locate them approximately on a number line diagram, and estimate the value of expressions
(e.g., π
2
). For example, by truncating the decimal expansion of
√
2, show that
√
2 is between 1
and 2, then between 1.4 and 1.5, and explain how to continue on to get better
approximations.
Expressions and Equations 8.EE
A. Work with radicals and integer exponents.
1. Know and apply the properties of integer exponents to generate equivalent numerical
expressions. For example, 3
2
× 3
–5
= 3
–3
= 1/3
3
= 1/27.
2. Use square root and cube root symbols to represent solutions to equations of the form x
2
= p
and x
3
= p, where p is a positive rational number. Evaluate square roots of small perfect
squares and cube roots of small perfect cubes. Know that √2 is irrational.
3. Use numbers expressed in the form of a single digit times an integer power of 10 to estimate
very large or very small quantities, and to express how many times as much one is than the
other. For example, estimate the population of the United States as 3 × 10
8
and the
population of the world as 7 × 10
9
, and determine that the world population is more than 20
times larger.
4. Perform operations with numbers expressed in scientific notation, including problems where
both decimal and scientific notation are used. Use scientific notation and choose units of
appropriate size for measurements of very large or very small quantities (e.g., use millimeters
per year for seafloor spreading). Interpret scientific notation that has been generated by
technology.
B. Understand the connections between proportional relationships, lines, and linear equations.
5. Graph proportional relationships, interpreting the unit rate as the slope of the graph.
Compare two different proportional relationships represented in different ways. For
example, compare a distance-time graph to a distance-time equation to determine which of
two moving objects has greater speed.
6. Use similar triangles to explain why the slope m is the same between any two distinct points
on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through
the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
C. Analyze and solve linear equations and pairs of simultaneous linear equations.
7. Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely many
solutions, or no solutions. Show which of these possibilities is the case by successively
transforming the given equation into simpler forms, until an equivalent equation of the
form x = a, a = a, or a = b results (where a and b are different numbers).
New Jersey Student Learning Standards for Mathematics
57
b. Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and collecting like
terms.
8. Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond
to points of intersection of their graphs, because points of intersection satisfy both
equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions
by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and
3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two
variables. For example, given coordinates for two pairs of points, determine whether the
line through the first pair of points intersects the line through the second pair.
Functions 8.F
A. Define, evaluate, and compare functions.
1. Understand that a function is a rule that assigns to each input exactly one output. The graph
of a function is the set of ordered pairs consisting of an input and the corresponding output.
1
2. Compare properties (e.g. rate of change, intercepts, domain and range) of two functions
each represented in a different way (algebraically, graphically, numerically in tables, or by
verbal descriptions). For example, given a linear function represented by a table of values and
a linear function represented by an algebraic expression, determine which function has the
greater rate of change.
3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line;
give examples of functions that are not linear. For example, the function A = s
2
giving the area
of a square as a function of its side length is not linear because its graph contains the points
(1,1), (2,4) and (3,9), which are not on a straight line.
B. Use functions to model relationships between quantities.
4. Construct a function to model a linear relationship between two quantities. Determine the
rate of change and initial value of the function from a description of a relationship or from
two (x, y) values, including reading these from a table or from a graph. Interpret the rate of
change and initial value of a linear function in terms of the situation it models, and in terms
of its graph or a table of values.
5. Describe qualitatively the functional relationship between two quantities by analyzing a
graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a
graph that exhibits the qualitative features of a function that has been described verbally.
Geometry 8.G
A. Understand congruence and similarity using physical models, transparencies, or geometry
software.
1. Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are transformed to lines, and line segments to line segments of the same length.
b. Angles are transformed to angles of the same measure.
c. Parallel lines are transformed to parallel lines.
1
Function notation is not required in Grade 8.
New Jersey Student Learning Standards for Mathematics
58
2. Understand that a two-dimensional figure is congruent to another if the second can be
obtained from the first by a sequence of rotations, reflections, and translations; given two
congruent figures, describe a sequence that exhibits the congruence between them.
3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional
figures using coordinates.
4. Understand that a two-dimensional figure is similar to another if the second can be obtained
from the first by a sequence of rotations, reflections, translations, and dilations; given two
similar two-dimensional figures, describe a sequence that exhibits the similarity between
them.
5. Use informal arguments to establish facts about the angle sum and exterior angle of
triangles, about the angles created when parallel lines are cut by a transversal, and the angle-
angle criterion for similarity of triangles. For example, arrange three copies of the same
triangle so that the sum of the three angles appears to form a line, and give an argument in
terms of transversals why this is so.
B. Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-
world and mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate
system.
C. Solve real-world and mathematical problems involving volume of cylinders, cones, and
spheres.
9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
Statistics and Probability 8.SP
A. Investigate patterns of association in bivariate data.
1. Construct and interpret scatter plots for bivariate measurement data to investigate patterns
of association between two quantities. Describe patterns such as clustering, outliers, positive
or negative association, linear association, and nonlinear association.
2. Know that straight lines are widely used to model relationships between two quantitative
variables. For scatter plots that suggest a linear association, informally fit a straight line, and
informally assess the model fit (e.g. line of best fit) by judging the closeness of the data
points to the line.
3. Use the equation of a linear model to solve problems in the context of bivariate
measurement data, interpreting the slope and intercept. For example, in a linear model for a
biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of
sunlight each day is associated with an additional 1.5 cm in mature plant height.
4. Understand that patterns of association can also be seen in bivariate categorical data by
displaying frequencies and relative frequencies in a two-way table. Construct and interpret a
two-way table summarizing data on two categorical variables collected from the same
subjects. Use relative frequencies calculated for rows or columns to describe possible
association between the two variables. For example, collect data from students in your class
on whether or not they have a curfew on school nights and whether or not they have assigned
chores at home. Is there evidence that those who have a curfew also tend to have chores?
New Jersey Student Learning Standards for Mathematics
59
Mathematics Standards for High School
The high school standards specify the mathematics that all students should study in order
to be college and career ready. Additional mathematics that students should learn in
order to take advanced courses such as calculus, advanced statistics, or discrete
mathematics is indicated by (+), as in this example:
(+) Represent complex numbers on the complex plane in rectangular and polar
form (including real and imaginary numbers).
All standards without a (+) symbol should be in the common mathematics curriculum for
all college and career ready students. Standards without a (+) symbol may also appear in
courses intended for all students.
The high school standards are listed in conceptual categories:
• Number and Quantity
• Algebra
• Functions
• Modeling
• Geometry
• Statistics and Probability
Conceptual categories portray a coherent view of high school mathematics; a student’s
work with functions, for example, crosses a number of traditional course boundaries,
potentially up through and including calculus.
Modeling is best interpreted not as a collection of isolated topics but in relation to other
standards. Making mathematical models is a Standard for Mathematical Practice, and
specific modeling standards appear throughout the high school standards indicated by a
star symbol (★). The star symbol sometimes appears on the heading for a group of
standards; in that case, it should be understood to apply to all standards in that group.
New Jersey Student Learning Standards for Mathematics
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Mathematics | High School—Number and
Quantity
Numbers and Number Systems. During the years from kindergarten to eighth grade, students
must repeatedly extend their conception of number. At first, “number” means “counting
number”: 1, 2, 3... Soon after that, 0 is used to represent “none” and the whole numbers are
formed by the counting numbers together with zero. The next extension is fractions. At first,
fractions are barely numbers and tied strongly to pictorial representations. Yet by the time
students understand division of fractions, they have a strong concept of fractions as numbers and
have connected them, via their decimal representations, with the base-ten system used to
represent the whole numbers. During middle school, fractions are augmented by negative
fractions to form the rational numbers. In Grade 8, students extend this system once more,
augmenting the rational numbers with the irrational numbers to form the real numbers. In high
school, students will be exposed to yet another extension of number, when the real numbers are
augmented by the imaginary numbers to form the complex numbers.
With each extension of number, the meanings of addition, subtraction, multiplication, and
division are extended. In each new number system—integers, rational numbers, real numbers,
and complex numbers—the four operations stay the same in two important ways: They have the
commutative, associative, and distributive properties and their new meanings are consistent with
their previous meanings.
Extending the properties of whole-number exponents leads to new and productive notation. For
example, properties of whole-number exponents suggest that (5
1/3
)
3
should be 5
(1/3)3
= 5
1
= 5 and
that 5
1/3
should be the cube root of 5.
Calculators, spreadsheets, and computer algebra systems can provide ways for students to
become better acquainted with these new number systems and their notation. They can be used
to generate data for numerical experiments, to help understand the workings of matrix, vector,
and complex number algebra, and to experiment with non-integer exponents.
Quantities. In real world problems, the answers are usually not numbers but quantities: numbers
with units, which involves measurement. In their work in measurement up through Grade 8,
students primarily measure commonly used attributes such as length, area, and volume. In high
school, students encounter a wider variety of units in modeling, e.g., acceleration, currency
conversions, derived quantities such as person-hours and heating degree days, social science
rates such as per-capita income, and rates in everyday life such as points scored per game or
batting averages. They also encounter novel situations in which they themselves must conceive
the attributes of interest. For example, to find a good measure of overall highway safety, they
might propose measures such as fatalities per year, fatalities per year per driver, or fatalities per
vehicle-mile traveled. Such a conceptual process is sometimes called quantification.
Quantification is important for science, as when surface area suddenly “stands out” as an
important variable in evaporation. Quantification is also important for companies, which must
conceptualize relevant attributes and create or choose suitable measures for them.
New Jersey Student Learning Standards for Mathematics
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Number and Quantity Overview
The Real Number System
• Extend the properties of exponents to
rational exponents
• Use properties of rational and irrational
numbers.
Quantities
• Reason quantitatively and use units to solve
problems
The Complex Number System
• Perform arithmetic operations with complex
numbers
• Represent complex numbers and their
operations on the complex plane
• Use complex numbers in polynomial identities and equations
Vector and Matrix Quantities
• Represent and model with vector quantities.
• Perform operations on vectors.
• Perform operations on matrices and use matrices in applications.
New Jersey Student Learning Standards for Mathematics
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The Real Number System N -RN
A. Extend the properties of exponents to rational exponents.
1. Explain how the definition of the meaning of rational exponents follows from extending the
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents. For example, we define 5
1/3
to be the cube root of 5 because we want
(5
1/3
)
3
= 5(
1/3
)
3
to hold, so (5
1/3
)
3
must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
B. Use properties of rational and irrational numbers.
3. Explain why the sum or product of two rational numbers is rational; that the sum of a
rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
Quantities
★
N -Q
A. Reason quantitatively and use units to solve problems.
1. Use units as a way to understand problems and to guide the solution of multi-step problems;
choose and interpret units consistently in formulas; choose and interpret the scale and the
origin in graphs and data displays.
2. Define appropriate quantities for the purpose of descriptive modeling.
3. Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
The Complex Number System N -CN
A. Perform arithmetic operations with complex numbers.
1. Know there is a complex number i such that i
2
= –1, and every complex number has the form
a + bi with a and b real.
2. Use the relation i
2
= –1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
3. (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of
complex numbers.
B. Represent complex numbers and their operations on the complex plane.
4. (+) Represent complex numbers on the complex plane in rectangular and polar form
(including real and imaginary numbers), and explain why the rectangular and polar forms of a
given complex number represent the same number.
5. (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation.
For example, (-1 + √3i)
3
= 8 because (-1+ √3i) has modulus 2 and argument 120°.
6. (+) Calculate the distance between numbers in the complex plane as the modulus of the
difference, and the midpoint of a segment as the average of the numbers at its endpoints.
C. Use complex numbers in polynomial identities and equations.
7. Solve quadratic equations with real coefficients that have complex solutions.
8. (+) Extend polynomial identities to the complex numbers. For example, rewrite x
2
+ 4 as (x +
2i)(x – 2i).
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63
9. (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.
Vector and Matrix Quantities N -VM
A. Represent and model with vector quantities.
1. (+) Recognize vector quantities as having both magnitude and direction. Represent vector
quantities by directed line segments, and use appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v).
2. (+) Find the components of a vector by subtracting the coordinates of an initial point from
the coordinates of a terminal point.
3. (+) Solve problems involving velocity and other quantities that can be represented by
vectors.
B. Perform operations on vectors.
4. (+) Add and subtract vectors.
a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that
the magnitude of a sum of two vectors is typically not the sum of the magnitudes.
b. Given two vectors in magnitude and direction form, determine the magnitude and
direction of their sum.
c. Understand vector subtraction v – w as v + (–w), where –w is the additive inverse of w,
with the same magnitude as w and pointing in the opposite direction. Represent vector
subtraction graphically by connecting the tips in the appropriate order, and perform vector
subtraction component-wise.
5. (+) Multiply a vector by a scalar.
a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their
direction; perform scalar multiplication component-wise, e.g., as c(v
x
, v
y
) = (cv
x
, cv
y
).
b. Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction
of cv knowing that when |c|v ≠0, the direction of cv is either along v (for c > 0) or against v
(for c < 0).
C. Perform operations on matrices and use matrices in applications.
6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a
game are doubled.
8. (+) Add, subtract, and multiply matrices of appropriate dimensions.
9. (+) Understand that, unlike multiplication of numbers, matrix multiplication for square
matrices is not a commutative operation, but still satisfies the associative and distributive
properties.
10. (+) Understand that the zero and identity matrices play a role in matrix addition and
multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square
matrix is nonzero if and only if the matrix has a multiplicative inverse.
11. (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable
dimensions to produce another vector. Work with matrices as transformations of vectors.
12. (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute
value of the determinant in terms of area.
New Jersey Student Learning Standards for Mathematics
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Mathematics | High School—Algebra
Expressions. An expression is a record of a computation with numbers, symbols that represent
numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of
evaluating a function. Conventions about the use of parentheses and the order of operations
assure that each expression is unambiguous. Creating an expression that describes a
computation involving a general quantity requires the ability to express the computation in
general terms, abstracting from specific instances.
Reading an expression with comprehension involves analysis of its underlying structure. This may
suggest a different but equivalent way of writing the expression that exhibits some different
aspect of its meaning. For example, p + 0.05p can be interpreted as the addition of a 5% tax to a
price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price
by a constant factor.
Algebraic manipulations are governed by the properties of operations and exponents, and the
conventions of algebraic notation. At times, an expression is the result of applying operations to
simpler expressions. For example, p + 0.05p is the sum of the simpler expressions p and 0.05p.
Viewing an expression as the result of operation on simpler expressions can sometimes clarify its
underlying structure.
A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic
expressions, perform complicated algebraic manipulations, and understand how algebraic
manipulations behave.
Equations and inequalities. An equation is a statement of equality between two expressions,
often viewed as a question asking for which values of the variables the expressions on either side
are in fact equal. These values are the solutions to the equation. An identity, in contrast, is true
for all values of the variables; identities are often developed by rewriting an expression in an
equivalent form.
The solutions of an equation in one variable form a set of numbers; the solutions of an equation
in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate
plane. Two or more equations and/or inequalities form a system. A solution for such a system
must satisfy every equation and inequality in the system.
An equation can often be solved by successively deducing from it one or more simpler equations.
For example, one can add the same constant to both sides without changing the solutions, but
squaring both sides might lead to extraneous solutions. Strategic competence in solving includes
looking ahead for productive manipulations and anticipating the nature and number of solutions.
Some equations have no solutions in a given number system, but have a solution in a larger
system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of
2x + 1 = 0 is a rational number, not an integer; the solutions of x
2
– 2 = 0 are real numbers, not
rational numbers; and the solutions of x
2
+ 2 = 0 are complex numbers, not real numbers.
The same solution techniques used to solve equations can be used to rearrange formulas. For
example, the formula for the area of a trapezoid, A = ((b
1
+b
2
)/2)h, can be solved for h using the
same deductive process.
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Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of
the properties of equality continue to hold for inequalities and can be useful in solving them.
Connections to Functions and Modeling. Expressions can define functions, and equivalent
expressions define the same function. Asking when two functions have the same value for the
same input leads to an equation; graphing the two functions allows for finding approximate
solutions of the equation. Converting a verbal description to an equation, inequality, or system of
these is an essential skill in modeling.
New Jersey Student Learning Standards for Mathematics
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Algebra Overview
Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solve
problems
Arithmetic with Polynomials and Rational
Functions
• Perform arithmetic operations on polynomials
• Understand the relationship between zeros
and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational expressions
Creating Equations
• Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities
• Understand solving equations as a process of reasoning and explain the
reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically
New Jersey Student Learning Standards for Mathematics
67
Seeing Structure in Expressions A-SSE
A. Interpret the structure of expressions
1. Interpret expressions that represent a quantity in terms of its context.
★
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
For example, interpret P(1+r)
n
as the product of P and a factor not depending on P
2. Use the structure of an expression to identify ways to rewrite it. For example, see x
4
– y
4
as
(x
2
)
2
– (y
2
)
2
, thus recognizing it as a difference of squares that can be factored as (x
2
–
y
2
)(x
2
+ y
2
).
B. Write expressions in equivalent forms to solve problems
3. Choose and produce an equivalent form of an expression to reveal and explain properties of
the quantity represented by the expression.
★
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value
of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions. For
example the expression 1.15
t
can be rewritten as (1.15
1/12
)
12t
≈1.012
12t
to reveal the
approximate equivalent monthly interest rate if the annual rate is 15%.
4. Derive and/or explain the formula for the sum of a finite geometric series (when the
common ratio is not 1), and use the formula to solve problems. For example, calculate
mortgage payments.
★
Arithmetic with Polynomials and Rational Expressions A -APR
A. Perform arithmetic operations on polynomials
1. Understand that polynomials form a system analogous to the integers, namely, they are
closed under the operations of addition, subtraction, and multiplication; add, subtract, and
multiply polynomials.
B. Understand the relationship between zeros and factors of polynomials
2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomial.
C. Use polynomial identities to solve problems
4. Prove polynomial identities and use them to describe numerical relationships. For example,
the difference of two squares; the sum and difference of two cubes; the polynomial identity
(x
2
+ y
2
)
2
= (x
2
– y
2
)
2
+ (2xy)
2
can be used to generate Pythagorean triples.
5. (+) Know and apply the Binomial Theorem for the expansion of (x + y)
n
in powers of x and y
for a positive integer n, where x and y are any numbers, with coefficients determined for
example by Pascal’s Triangle.
1
1
The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.
New Jersey Student Learning Standards for Mathematics
68
D. Rewrite rational expressions
6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) +
r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the
degree of b(x), using inspection, long division, or, for the more complicated examples, a
computer algebra system.
7. (+) Understand that rational expressions form a system analogous to the rational numbers,
closed under addition, subtraction, multiplication, and division by a nonzero rational
expression; add, subtract, multiply, and divide rational expressions.
Creating Equations
★
A -CED
A. Create equations that describe numbers or relationships
1. Create equations and inequalities in one variable and use them to solve problems. Include
equations arising from linear and quadratic functions, and simple rational and exponential
functions.
2. Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
3. Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or nonviable options in a modeling context. For
example, represent inequalities describing nutritional and cost constraints on combinations of
different foods.
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving
equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
Reasoning with Equations and Inequalities A -REI
A. Understand solving equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers
asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
2. Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
B. Solve equations and inequalities in one variable
3. Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
4. Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an
equation of the form (x – p)
2
= q that has the same solutions. Derive the quadratic formula
from this form.
b. Solve quadratic equations by inspection (e.g., for x
2
= 49), taking square roots, completing
the square, the quadratic formula and factoring, as appropriate to the initial form of the
equation. Recognize when the quadratic formula gives complex solutions and write them
as a ± bi for real numbers a and b.
C. Solve systems of equations
5. Prove that, given a system of two equations in two variables, replacing one equation by the
sum of that equation and a multiple of the other produces a system with the same solutions.
New Jersey Student Learning Standards for Mathematics
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6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
7. Solve a simple system consisting of a linear equation and a quadratic equation in two
variables algebraically and graphically. For example, find the points of intersection between
the line y = –3x and the circle x
2
+ y
2
= 3.
8. (+) Represent a system of linear equations as a single matrix equation in a vector variable.
9. (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations
(using technology for matrices of dimension 3 × 3 or greater).
D. Represent and solve equations and inequalities graphically
10. Understand that the graph of an equation in two variables is the set of all its solutions
plotted in the coordinate plane, often forming a curve (which could be a line).
11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y
= g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately,
e.g., using technology to graph the functions, make tables of values, or find successive
approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational,
absolute value, exponential, and logarithmic functions.
★
12. Graph the solutions to a linear inequality in two variables as a half plane (excluding the
boundary in the case of a strict inequality), and graph the solution set to a system of linear
inequalities in two variables as the intersection of the corresponding half-planes.
New Jersey Student Learning Standards for Mathematics
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Mathematics | High School—Functions
Functions describe situations where one quantity determines another. For example, the return
on $10,000 invested at an annualized percentage rate of 4.25% is a function of the length of time
the money is invested. Because we continually make theories about dependencies between
quantities in nature and society, functions are important tools in the construction of
mathematical models.
In school mathematics, functions usually have numerical inputs and outputs and are often
defined by an algebraic expression. For example, the time in hours it takes for a car to drive 100
miles is a function of the car’s speed in miles per hour, v; the rule T(v) = 100/v expresses this
relationship algebraically and defines a function whose name is T.
The set of inputs to a function is called its domain. We often infer the domain to be all inputs for
which the expression defining a function has a value, or for which the function makes sense in a
given context.
A function can be described in various ways, such as by a graph (e.g., the trace of a seismograph);
by a verbal rule, as in, “I’ll give you a state, you give me the capital city;” by an algebraic
expression like f(x) = a + bx; or by a recursive rule. The graph of a function is often a useful way of
visualizing the relationship of the function models, and manipulating a mathematical expression
for a function can throw light on the function’s properties.
Functions presented as expressions can model many important phenomena. Two important
families of functions characterized by laws of growth are linear functions, which grow at a
constant rate, and exponential functions, which grow at a constant percent rate. Linear functions
with a constant term of zero describe proportional relationships.
A graphing utility or a computer algebra system can be used to experiment with properties of
these functions and their graphs and to build computational models of functions, including
recursively defined functions.
Connections to Expressions, Equations, Modeling, and Coordinates.
Determining an output value for a particular input involves evaluating an expression; finding
inputs that yield a given output involves solving an equation. Questions about when two
functions have the same value for the same input lead to equations, whose solutions can be
visualized from the intersection of their graphs. Because functions describe relationships
between quantities, they are frequently used in modeling. Sometimes functions are defined by a
recursive process, which can be displayed effectively using a spreadsheet or other technology.
New Jersey Student Learning Standards for Mathematics
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Functions Overview
Interpreting Functions
• Understand the concept of a function and use
function notation
• Interpret functions that arise in applications
in terms of the context
• Analyze functions using different
representations
Building Functions
• Build a function that models a relationship
between two quantities
• Build new functions from existing functions
Linear, Quadratic, and Exponential Models
• Construct and compare linear and exponential models and solve problems
• Interpret expressions for functions in terms of the situation they model
Trigonometric Functions
• Extend the domain of trigonometric functions using the unit circle
• Model periodic phenomena with trigonometric functions
• Prove and apply trigonometric identities
New Jersey Student Learning Standards for Mathematics
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Interpreting Functions F-IF
A. Understand the concept of a function and use function notation
1. Understand that a function from one set (called the domain) to another set (called the
range) assigns to each element of the domain exactly one element of the range. If f is a
function and x is an element of its domain, then f(x) denotes the output of f corresponding to
the input x. The graph of f is the graph of the equation y = f(x).
2. Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1)
= 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
B. Interpret functions that arise in applications in terms of the context
4. For a function that models a relationship between two quantities, interpret key features of
graphs and tables in terms of the quantities, and sketch graphs showing key features given a
verbal description of the relationship. Key features include: intercepts; intervals where the
function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.
★
5. Relate the domain of a function to its graph and, where applicable, to the quantitative
relationship it describes. For example, if the function h(n) gives the number of person-hours it
takes to assemble n engines in a factory, then the positive integers would be an appropriate
domain for the function.
★
6. Calculate and interpret the average rate of change of a function (presented symbolically or
as a table) over a specified interval. Estimate the rate of change from a graph.
★
C. Analyze functions using different representations
7. Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
★
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
b. Graph square root, cube root, and piecewise-defined functions, including step functions
and absolute value functions.
c. Graph polynomial functions, identifying zeros when suitable factorizations are available,
and showing end behavior.
d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations
are available, and showing end behavior.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing period, midline, and amplitude.
8. Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to show
zeros, extreme values, and symmetry of the graph, and interpret these in terms of a
context.
b. Use the properties of exponents to interpret expressions for exponential functions. For
example, identify percent rate of change in functions such as y = (1.02)
t
, y = (0.97)
t
, y =
(1.01)
12t
, y = (1.2)
t/10
, and classify them as representing exponential growth or decay.
New Jersey Student Learning Standards for Mathematics
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9. Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of
one quadratic function and an algebraic expression for another, say which has the larger
maximum.
Building Functions F-BF
A. Build a function that models a relationship between two quantities
1. Write a function that describes a relationship between two quantities.
★
a. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
b. Combine standard function types using arithmetic operations. For example, build a
function that models the temperature of a cooling body by adding a constant function to a
decaying exponential, and relate these functions to the model.
c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a
function of height, and h(t) is the height of a weather balloon as a function of time, then
T(h(t)) is the temperature at the location of the weather balloon as a function of time.
2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between the two forms.
★
B. Build new functions from existing functions
3. Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for
specific values of k (both positive and negative); find the value of k given the graphs.
Experiment with cases and illustrate an explanation of the effects on the graph using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
4. Find inverse functions.
a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write
an expression for the inverse. For example, f(x) =2 x
3
or f(x) = (x+1)/(x–1) for x ≠1.
b. (+) Verify by composition that one function is the inverse of another.
c. (+) Read values of an inverse function from a graph or a table, given that the function has
an inverse.
d. (+) Produce an invertible function from a non-invertible function by restricting the domain.
5. (+) Use the inverse relationship between exponents and logarithms to solve problems
involving logarithms and exponents
.
Linear and Exponential Models
★
F-LE
A. Construct and compare linear and exponential models and solve problems
1. Distinguish between situations that can be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval
relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per
unit interval relative to another.
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2. Construct linear and exponential functions, including arithmetic and geometric sequences,
given a graph, a description of a relationship, or two input-output pairs (include reading
these from a table).
3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds
a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
4. Understand the inverse relationship between exponents and logarithms. For exponential
models, express as a logarithm the solution to ab
ct
= d where a, c, and d are numbers and the
base b is 2, 10, or e; evaluate the logarithm using technology.
B. Interpret expressions for functions in terms of the situation they model
5. Interpret the parameters in a linear or exponential function in terms of a context.
Trigonometric Functions F-TF
A. Extend the domain of trigonometric functions using the unit circle
1. Understand radian measure of an angle as the length of the arc on the unit circle subtended
by the angle.
2. Explain how the unit circle in the coordinate plane enables the extension of trigonometric
functions to all real numbers, interpreted as radian measures of angles traversed
counterclockwise around the unit circle.
3. (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for
π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent
for πx, π+x, and 2π–x in terms of their values for x, where x is any real number.
4. (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
B. Model periodic phenomena with trigonometric functions
5. Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
★
6. (+) Understand that restricting a trigonometric function to a domain on which it is always
increasing or always decreasing allows its inverse to be constructed.
7. (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts;
evaluate the solutions using technology, and interpret them in terms of the context.
★
C. Prove and apply trigonometric identities
8. Prove the Pythagorean identity sin
2
(θ) + cos
2
(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ)
given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
9. (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them
to solve problems.
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Mathematics | High School—Modeling
Modeling links classroom mathematics and statistics to everyday life, work, and decision-making.
Modeling is the process of choosing and using appropriate mathematics and statistics to
analyze empirical situations, to understand them better, and to improve decisions. Quantities
and their relationships in physical, economic, public policy, social, and everyday situations can be
modeled using mathematical and statistical methods. When making mathematical models,
technology is valuable for varying assumptions, exploring consequences, and comparing
predictions with data.
A model can be very simple, such as writing total cost as a product of unit price and number
bought, or using a geometric shape to describe a physical object like a coin. Even such simple
models involve making choices. It is up to us whether to model a coin as a three-dimensional
cylinder, or whether a two-dimensional disk works well enough for our purposes. Other
situations—modeling a delivery route, a production schedule, or a comparison of loan
amortizations—need more elaborate models that use other tools from the mathematical
sciences. Real-world situations are not organized and labeled for analysis; formulating tractable
models, representing such models, and analyzing them is appropriately a creative process. Like
every such process, this depends on acquired expertise as well as creativity.
Some examples of such situations might include:
• Estimating how much water and food is needed for emergency relief in a devastated
city of 3 million people, and how it might be distributed.
• Planning a table tennis tournament for 7 players at a club with 4 tables, where each
player plays against each other player.
• Designing the layout of the stalls in a school fair so as to raise as much money as
possible.
• Analyzing stopping distance for a car.
• Modeling savings account balance, bacterial colony growth, or investment growth.
• Engaging in critical path analysis, e.g., applied to turnaround of an aircraft at an airport.
• Analyzing risk in situations such as extreme sports, pandemics, and terrorism.
• Relating population statistics to individual predictions.
In situations like these, the models devised depend on a number of factors: How precise an
answer do we want or need? What aspects of the situation do we most need to understand,
control, or optimize? What resources of time and tools do we have? The range of models that we
can create and analyze is also constrained by the limitations of our mathematical, statistical, and
technical skills, and our ability to recognize significant variables and relationships among them.
Diagrams of various kinds, spreadsheets and other technology, and algebra are powerful tools for
understanding and solving problems drawn from different types of real-world situations.
One of the insights provided by mathematical modeling is that essentially the same mathematical
or statistical structure can sometimes model seemingly different situations. Models can also shed
light on the mathematical structures themselves, for example, as when a model of bacterial
growth makes more vivid the explosive growth of the exponential function.
The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the
situation and selecting those that represent essential features, (2) formulating a model by
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creating and selecting geometric, graphical, tabular, algebraic, or statistical representations
that describe relationships between the variables, (3) analyzing and performing operations on
these relationships to draw conclusions, (4) interpreting the results of the mathematics in
terms of the original situation, (5) validating the conclusions by comparing them with the
situation, and then either improving the model or, if it
is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices,
assumptions, and approximations are present throughout this cycle.
In descriptive modeling, a model simply describes the phenomena or summarizes them in a
compact form. Graphs of observations are a familiar descriptive model— for example, graphs of
global temperature and atmospheric CO
2
over time.
Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with
parameters that are empirically based; for example, exponential growth of bacterial colonies
(until cut-off mechanisms such as pollution or starvation intervene) follows from a constant
reproduction rate. Functions are an important tool for analyzing such problems.
Graphing utilities, spreadsheets, computer algebra systems, and dynamic geometry software are
powerful tools that can be used to model purely mathematical phenomena (e.g., the behavior of
polynomials) as well as physical phenomena.
Modeling Standards: Modeling is best interpreted not as a collection of isolated topics but rather
in relation to other standards. Making mathematical models is a Standard for Mathematical
Practice, and specific modeling standards appear throughout the high school standards indicated
by a star symbol (
★
).
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Mathematics | High School—Geometry
An understanding of the attributes and relationships of geometric objects can be applied in
diverse contexts—interpreting a schematic drawing, estimating the amount of wood needed to
frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most
efficient use of material.
Although there are many types of geometry, school mathematics is devoted primarily to plane
Euclidean geometry, studied both synthetically (without coordinates) and analytically (with
coordinates). Euclidean geometry is characterized most importantly by the Parallel Postulate,
that through a point not on a given line there is exactly one parallel line. (Spherical geometry, in
contrast, has no parallel lines.)
During high school, students begin to formalize their geometry experiences from elementary and
middle school, using more precise definitions and developing careful proofs. Later in college
some students develop Euclidean and other geometries carefully from a small set of axioms.
The concepts of congruence, similarity, and symmetry can be understood from the perspective of
geometric transformation. Fundamental are the rigid motions: translations, rotations, reflections,
and combinations of these, all of which are here assumed to preserve distance and angles (and
therefore shapes generally). Reflections and rotations each explain a particular type of symmetry,
and the symmetries of an object offer insight into its attributes—as when the reflective
symmetry of an isosceles triangle assures that its base angles are congruent.
In the approach taken here, two geometric figures are defined to be congruent if there is a
sequence of rigid motions that carries one onto the other. This is the principle of superposition.
For triangles, congruence means the equality of all corresponding pairs of sides and all
corresponding pairs of angles. During the middle grades, through experiences drawing triangles
from given conditions, students notice ways to specify enough measures in a triangle to ensure
that all triangles drawn with those measures are congruent. Once these triangle congruence
criteria (ASA, SAS, and SSS) are established using rigid motions, they can be used to prove
theorems about triangles, quadrilaterals, and other geometric figures.
Similarity transformations (rigid motions followed by dilations) define similarity in the same way
that rigid motions define congruence, thereby formalizing the similarity ideas of "same shape"
and "scale factor" developed in the middle grades. These transformations lead to the criterion for
triangle similarity that two pairs of corresponding angles are congruent.
The definitions of sine, cosine, and tangent for acute angles are founded on right triangles and
similarity, and, with the Pythagorean Theorem, are fundamental in many real-world and
theoretical situations. The Pythagorean Theorem is generalized to non-right triangles by the Law
of Cosines. Together, the Laws of Sines and Cosines embody the triangle congruence criteria for
the cases where three pieces of information suffice to completely solve a triangle. Furthermore,
these laws yield two possible solutions in the ambiguous case, illustrating that Side-Side-Angle is
not a congruence criterion.
Analytic geometry connects algebra and geometry, resulting in powerful methods of analysis and
problem solving. Just as the number line associates numbers with locations in one dimension, a
pair of perpendicular axes associates pairs of numbers with locations in two dimensions. This
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correspondence between numerical coordinates and geometric points allows methods from
algebra to be applied to geometry and vice versa. The solution set of an equation becomes a
geometric curve, making visualization a tool for doing and understanding algebra. Geometric
shapes can be described by equations, making algebraic manipulation into a tool for geometric
understanding, modeling, and proof. Geometric transformations of the graphs of equations
correspond to algebraic changes in their equations.
Dynamic geometry environments provide students with experimental and modeling tools that
allow them to investigate geometric phenomena in much the same way as computer algebra
systems allow them to experiment with algebraic phenomena.
Connections to Equations. The correspondence between numerical coordinates and geometric
points allows methods from algebra to be applied to geometry and vice versa. The solution set of
an equation becomes a geometric curve, making visualization a tool for doing and understanding
algebra. Geometric shapes can be described by equations, making algebraic manipulation into a
tool for geometric understanding, modeling, and proof.
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Geometry Overview
Congruence
• Experiment with transformations in the plane
• Understand congruence in terms of rigid
motions
• Prove geometric theorems
• Make geometric constructions
Similarity, Right Triangles, and Trigonometry
• Understand similarity in terms of similarity
transformations
• Prove theorems involving similarity
• Define trigonometric ratios and solve
problems involving right triangles
• Apply trigonometry to general triangles
Circles
• Understand and apply theorems about circles
• Find arc lengths and areas of sectors of circles
Expressing Geometric Properties with Equations
• Translate between the geometric description and the equation for a conic
section
• Use coordinates to prove simple geometric theorems algebraically
Geometric Measurement and Dimension
• Explain volume formulas and use them to solve problems
• Visualize relationships between two-dimensional and three-dimensional
objects
Modeling with Geometry
• Apply geometric concepts in modeling situations
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Congruence G-CO
A. Experiment with transformations in the plane
1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment,
based on the undefined notions of point, line, distance along a line, and distance around a
circular arc.
2. Represent transformations in the plane using, e.g., transparencies and geometry software;
describe transformations as functions that take points in the plane as inputs and give other
points as outputs. Compare transformations that preserve distance and angle to those that
do not (e.g., translation versus horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed
figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of
transformations that will carry a given figure onto another.
B. Understand congruence in terms of rigid motions
6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of
a given rigid motion on a given figure; given two figures, use the definition of congruence in
terms of rigid motions to decide if they are congruent.
7. Use the definition of congruence in terms of rigid motions to show that two triangles are
congruent if and only if corresponding pairs of sides and corresponding pairs of angles are
congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
C. Prove geometric theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles are congruent;
when a transversal crosses parallel lines, alternate interior angles are congruent and
corresponding angles are congruent; points on a perpendicular bisector of a line segment are
exactly those equidistant from the segment’s endpoints.
10. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints
of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are congruent,
opposite angles are congruent, the diagonals of a parallelogram bisect each other, and
conversely, rectangles are parallelograms with congruent diagonals.
D. Make geometric constructions
12. Make formal geometric constructions with a variety of tools and methods (compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.).
Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing
perpendicular lines, including the perpendicular bisector of a line segment; and constructing a
line parallel to a given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
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Similarity, Right Triangles, and Trigonometry G-SRT
A. Understand similarity in terms of similarity transformations
1. Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and
leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
2. Given two figures, use the definition of similarity in terms of similarity transformations to
decide if they are similar; explain using similarity transformations the meaning of similarity
for triangles as the equality of all corresponding pairs of angles and the proportionality of all
corresponding pairs of sides.
3. Use the properties of similarity transformations to establish the AA criterion for two triangles
to be similar.
B. Prove theorems involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle
divides the other two proportionally, and conversely; the Pythagorean Theorem proved using
triangle similarity.
5. Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
C. Define trigonometric ratios and solve problems involving right triangles
6. Understand that by similarity, side ratios in right triangles are properties of the angles in the
triangle, leading to definitions of trigonometric ratios for acute angles.
7. Explain and use the relationship between the sine and cosine of complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
D. Apply trigonometry to general triangles
9. (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite side.
10. (+) Prove the Laws of Sines and Cosines and use them to solve problems.
11. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown
measurements in right and non-right triangles (e.g., surveying problems, resultant forces).
Circles G-C
A. Understand and apply theorems about circles
1. Prove that all circles are similar.
2. Identify and describe relationships among inscribed angles, radii, and chords. Include the
relationship between central, inscribed, and circumscribed angles; inscribed angles on a
diameter are right angles; the radius of a circle is perpendicular to the tangent where the
radius intersects the circle.
3. Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
4. (+) Construct a tangent line from a point outside a given circle to the circle.
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B. Find arc lengths and areas of sectors of circles
5. Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
Expressing Geometric Properties with Equations G-GPE
A. Translate between the geometric description and the equation for a conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem;
complete the square to find the center and radius of a circle given by an equation.
2. Derive the equation of a parabola given a focus and directrix.
3. (+)Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum
or difference of distances from the foci is constant.
B. Use coordinates to prove simple geometric theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or
disprove that a figure defined by four given points in the coordinate plane is a rectangle;
prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing
the point (0, 2).
5. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric
problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes
through a given point).
6. Find the point on a directed line segment between two given points that partitions the
segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,
e.g., using the distance formula.
★
Geometric Measurement and Dimension G-GMD
A. Explain volume formulas and use them to solve problems
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and
informal limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
★
B. Visualize relationships between two-dimensional and three-dimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and
identify three-dimensional objects generated by rotations of two-dimensional objects.
Modeling with Geometry G-MG
A. Apply geometric concepts in modeling situations
1. Use geometric shapes, their measures, and their properties to describe objects (e.g.,
modeling a tree trunk or a human torso as a cylinder).
★
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons
per square mile, BTUs per cubic foot).
★
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3. Apply geometric methods to solve design problems (e.g., designing an object or structure to
satisfy physical constraints or minimize cost; working with typographic grid systems based on
ratios).
★
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Mathematics | High School—Statistics and Probability
★
Decisions or predictions are often based on data—numbers in context. These decisions or
predictions would be easy if the data always sent a clear message, but the message is often
obscured by variability. Statistics provides tools for describing variability in data and for making
informed decisions that take it into account.
Data are gathered, displayed, summarized, examined, and interpreted to discover patterns and
deviations from patterns. Quantitative data can be described in terms of key characteristics:
measures of shape, center, and spread. The shape of a data distribution might be described as
symmetric, skewed, flat, or bell shaped, and it might be summarized by a statistic measuring
center (such as mean or median) and a statistic measuring spread (such as standard deviation or
interquartile range). Different distributions can be compared numerically using these statistics or
compared visually using plots. Knowledge of center and spread are not enough to describe a
distribution. Which statistics to compare, which plots to use, and what the results of a
comparison might mean, depend on the question to be investigated and the real-life actions to
be taken.
Randomization has two important uses in drawing statistical conclusions. First, collecting data
from a random sample of a population makes it possible to draw valid conclusions about the
whole population, taking variability into account. Second, randomly assigning individuals to
different treatments allows a fair comparison of the effectiveness of those treatments. A
statistically significant outcome is one that is unlikely to be due to chance alone, and this can be
evaluated only under the condition of randomness. The conditions under which data are
collected are important in drawing conclusions from the data; in critically reviewing uses of
statistics in public media and other reports, it is important to consider the study design, how the
data were gathered, and the analyses employed as well as the data summaries and the
conclusions drawn
.
Random processes can be described mathematically by using a probability model: a list or
description of the possible outcomes (the sample space), each of which is assigned a probability.
In situations such as flipping a coin, rolling a number cube, or drawing a card, it might be
reasonable to assume various outcomes are equally likely. In a probability model, sample points
represent outcomes and combine to make up events; probabilities of events can be computed by
applying the Addition and Multiplication Rules. Interpreting these probabilities relies on an
understanding of independence and conditional probability, which can be approached through
the analysis of two-way tables.
Technology plays an important role in statistics and probability by making it possible to generate
plots, regression functions, and correlation coefficients, and to simulate many possible outcomes
in a short amount of time
.
Connections to Functions and Modeling. Functions may be used to describe data; if the data
suggest a linear relationship, the relationship can be modeled with a regression line, and its
strength and direction can be expressed through a correlation coefficient.
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Mathematical Practices
1. Make sense of problems and persevere in
solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the
reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated
reasoning
Statistics and Probability Overview
Interpreting Categorical and Quantitative
Data
• Summarize, represent, and interpret data on
a single count or measurement variable
• Summarize, represent, and interpret data on
two categorical and quantitative variables
• Interpret linear models
Making Inferences and Justifying Conclusions
• Understand and evaluate random processes
underlying statistical experiments
• Make inferences and justify conclusions
from sample surveys, experiments and
observational studies
Conditional Probability and the Rules of
Probability
• Understand independence and conditional probability and use them to
interpret data
• Use the rules of probability to compute probabilities of compound events
in a uniform probability model
Using Probability to Make Decisions
• Calculate expected values and use them to solve problems
• Use probability to evaluate outcomes of decisions
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Interpreting Categorical and Quantitative Data S-ID
A. Summarize, represent, and interpret data on a single count or measurement variable
1. Represent data with plots on the real number line (dot plots, histograms, and box plots).
2. Use statistics appropriate to the shape of the data distribution to compare center (median,
mean) and spread (interquartile range, standard deviation) of two or more different data
sets.
3. Interpret differences in shape, center, and spread in the context of the data sets, accounting
for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal distribution and to
estimate population percentages. Recognize that there are data sets for which such a
procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas
under the normal curve.
B. Summarize, represent, and interpret data on two categorical and quantitative variables
5. Summarize categorical data for two categories in two-way frequency tables. Interpret
relative frequencies in the context of the data (including joint, marginal, and conditional
relative frequencies). Recognize possible associations and trends in the data.
6. Represent data on two quantitative variables on a scatter plot, and describe how the
variables are related.
a. Fit a function to the data (including with the use of technology); use functions fitted to
data to solve problems in the context of the data. Use given functions or choose a function
suggested by the context. Emphasize linear and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals, including with the
use of technology.
c. Fit a linear function for a scatter plot that suggests a linear association.
C. Interpret linear models
7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the
context of the data.
8. Compute (using technology) and interpret the correlation coefficient of a linear fit.
9. Distinguish between correlation and causation.
Making Inferences and Justifying Conclusions S-IC
A. Understand and evaluate random processes underlying statistical experiments
1. Understand statistics as a process for making inferences about population parameters based
on a random sample from that population.
2. Decide if a specified model is consistent with results from a given data-generating process,
e.g., using simulation. For example, a model says a spinning coin falls heads up with
probability 0.5. Would a result of 5 tails in a row cause you to question the model?
B. Make inferences and justify conclusions from sample surveys, experiments, and
observational studies
3. Recognize the purposes of and differences among sample surveys, experiments, and
observational studies; explain how randomization relates to each.
4. Use data from a sample survey to estimate a population mean or proportion; develop a
margin of error through the use of simulation models for random sampling.
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5. Use data from a randomized experiment to compare two treatments; use simulations to
decide if differences between parameters are significant.
6. Evaluate reports based on data.
Conditional Probability and the Rules of Probability S-CP
A. Understand independence and conditional probability and use them to interpret data
1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or
categories) of the outcomes, or as unions, intersections, or complements of other events
(“or,” “and,” “not”).
2. Understand that two events A and B are independent if the probability of A and B occurring
together is the product of their probabilities, and use this characterization to determine if
they are independent.
3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret
independence of A and B as saying that the conditional probability of A given B is the same as
the probability of A, and the conditional probability of B given A is the same as the probability
of B.
4. Construct and interpret two-way frequency tables of data when two categories are
associated with each object being classified. Use the two-way table as a sample space to
decide if events are independent and to approximate conditional probabilities. For example,
collect data from a random sample of students in your school on their favorite subject among
math, science, and English. Estimate the probability that a randomly selected student from
your school will favor science given that the student is in tenth grade. Do the same for other
subjects and compare the results.
5. Recognize and explain the concepts of conditional probability and independence in everyday
language and everyday situations. For example, compare the chance of having lung cancer if
you are a smoker with the chance of being a smoker if you have lung cancer.
B. Use the rules of probability to compute probabilities of compound events in a uniform
probability model
6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong
to A, and interpret the answer in terms of the model.
7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in
terms of the model.
8. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) =
P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
9. (+) Use permutations and combinations to compute probabilities of compound events and
solve problems
.
Using Probability to Make Decisions S-MD
A. Calculate expected values and use them to solve problems
1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each
event in a sample space; graph the corresponding probability distribution using the same
graphical displays as for data distributions.
2. (+) Calculate the expected value of a random variable; interpret it as the mean of the
probability distribution.
3. (+) Develop a probability distribution for a random variable defined for a sample space in
which theoretical probabilities can be calculated; find the expected value. For example, find
the theoretical probability distribution for the number of correct answers obtained by
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88
guessing on all five questions of a multiple-choice test where each question has four choices,
and find the expected grade under various grading schemes.
4. (+) Develop a probability distribution for a random variable defined for a sample space in
which probabilities are assigned empirically; find the expected value. For example, find a
current data distribution on the number of TV sets per household in the United States, and
calculate the expected number of sets per household. How many TV sets would you expect to
find in 100 randomly selected households?
B. Use probability to evaluate outcomes of decisions
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and
finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings
from a state lottery ticket or a game at a fast food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a
high-deductible versus a low-deductible automobile insurance policy using various, but
reasonable, chances of having a minor or a major accident.
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
7. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game).
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Note on courses and transitions
The high school portion of the Standards for Mathematical Content specifies the mathematics all students
should study for college and career readiness. These standards do not mandate the sequence of high
school courses. However, the organization of high school courses is a critical component to
implementation of the standards.
The standards themselves do not dictate curriculum, pedagogy, or delivery of content. In particular, states
may handle the transition to high school in different ways. For example, many students in the U.S. today
take Algebra I in the 8
th
grade, and in some states this is a requirement. The K-7 standards contain the
prerequisites to prepare students for Algebra I by 8th grade, and the standards are designed to permit
states to continue existing policies concerning Algebra I in 8
th
grade.
A second major transition is the transition from high school to post-secondary education for college and
careers. The evidence concerning college and career readiness shows clearly that the knowledge, skills, and
practices important for readiness include a great deal of mathematics prior to the boundary defined by (+)
symbols in these standards. Indeed, some of the highest priority content for college and career readiness
comes from Grades 6-8. This body of material includes powerfully useful proficiencies such as applying
ratio reasoning in real-world and mathematical problems, computing fluently with positive and negative
fractions and decimals, and solving real-world and mathematical problems involving angle measure, area,
surface area, and volume. Because important standards for college and career readiness are distributed
across grades and courses, systems for evaluating college and career readiness should reach as far back in
the standards as Grades 6-8. It is important to note as well that cut scores or other information generated
by assessment systems for college and career readiness should be developed in collaboration with
representatives from higher education and workforce development programs, and should be validated by
subsequent performance of students in college and the workforce.
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Glossary
Addition and subtraction within 5, 10, 20, 100, or 1000. Addition or subtraction of two whole numbers
with whole number answers, and with sum or minuend in the range 0-5, 0-10, 0-20, or 0-100, respectively.
Example: 8 + 2 = 10 is an addition within 10, 14 – 5 = 9 is a subtraction within 20, and 55 – 18 = 37 is a
subtraction within 100.
Additive inverses. Two numbers whose sum is 0 are additive inverses of one another. Example: 3/4 and –
3/4 are additive inverses of one another because 3/4 + (– 3/4) = (– 3/4) + 3/4 = 0.
Associative property of addition. See Table 3 in this Glossary.
Associative property of multiplication. See Table 3 in this Glossary.
Bivariate data. Pairs of linked numerical observations. Example: a list of heights and weights for each
player on a football team.
Box plot. A method of visually displaying a distribution of data values by using the median, quartiles, and
extremes of the data set. A box shows the middle
50% of the data.
1
Commutative property. See Table 3 in this Glossary.
Complex fraction. A fraction A/B where A and/or B are fractions (B nonzero).
Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct
result in every case when the steps are carried out correctly. See also: computation strategy.
Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a
fixed order, and may be aimed at converting one problem into another. See also: computation algorithm.
Congruent. Two plane or solid figures are congruent if one can be obtained from the other by rigid motion
(a sequence of rotations, reflections, and translations).
Counting on. A strategy for finding the number of objects in a group without having to count every
member of the group. For example, if a stack of books is known to have 8 books and 3 more books are
added to the top, it is not necessary to count the stack all over again. One can find the total by counting
on—pointing to the top book and saying “eight,” following this with “nine, ten, eleven. There are eleven
books now.”
Dot plot. See: line plot.
Dilation. A transformation that moves each point along the ray through the point emanating from a fixed
center, and multiplies distances from the center by a common scale factor.
Expanded form. A multi-digit number is expressed in expanded form when it is written as a sum of single-
digit multiples of powers of ten. For example, 643 = 600 + 40 + 3.
Expected value. For a random variable, the weighted average of its possible values, with weights given by
their respective probabilities.
First quartile. For a data set with median M, the first quartile is the median of the data values less than M.
Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the first quartile is 6.
2
See also: median, third
quartile, interquartile range.
Fraction. A number expressible in the form a/b where a is a whole number and b is a positive whole
number. (The word fraction in these standards always refers to a non-negative number.) See also: rational
number.
Identity property of 0. See Table 3 in this Glossary.
Independently combined probability models. Two probability models are said to be combined
independently if the probability of each ordered pair in the combined model equals the product of the
original probabilities of the two individual outcomes in the ordered pair.
1
Adapted from Wisconsin Department of Public Instruction, http://dpi.wi.gov/standards/mathglos.html , accessed March 2, 2010.
2
Many different methods for computing quartiles are in use. The method defined here is sometimes called the Moore and McCabe
method. See Langford, E., “Quartiles in Elementary Statistics,” Journal of Statistics Education Volume 14, Number 3 (2006).
New Jersey Student Learning Standards for Mathematics
91
Integer. A number expressible in the form a or –a for some whole number a.
Interquartile Range. A measure of variation in a set of numerical data, the interquartile range is the
distance between the first and third quartiles of the data set. Example: For the data set {1, 3, 6, 7, 10, 12,
14, 15, 22, 120}, the interquartile range is 15 – 6 = 9. See also: first quartile, third quartile.
Line plot. A method of visually displaying a distribution of data values where each data value is shown as a
dot or mark above a number line. Also known as a dot plot.
3
Mean. A measure of center in a set of numerical data, computed by adding the values in a list and then
dividing by the number of values in the list.
4
Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120},
the mean is 21.
Mean absolute deviation. A measure of variation in a set of numerical data, computed by adding the
distances between each data value and the mean, then dividing by the number of data values. Example:
For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the mean absolute deviation is 20.
Median. A measure of center in a set of numerical data. The median of a list of values is the value
appearing at the center of a sorted version of the list—or the mean of the two central values, if the list
contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median
is 11.
Midline. In the graph of a trigonometric function, the horizontal line halfway between its maximum and
minimum values.
Multiplication and division within 100. Multiplication or division of two whole numbers with whole
number answers, and with product or dividend in the range 0-100. Example: 72 ÷ 8 = 9.
Multiplicative inverses. Two numbers whose product is 1 are multiplicative inverses of one another.
Example: 3/4 and 4/3 are multiplicative inverses of one another because 3/4 × 4/3 = 4/3 × 3/4 = 1.
Number line diagram. A diagram of the number line used to represent numbers and support reasoning
about them. In a number line diagram for measurement quantities, the interval from 0 to 1 on the diagram
represents the unit of measure for the quantity.
Percent rate of change. A rate of change expressed as a percent. Example: if a population grows from 50 to
55 in a year, it grows by 5/50 = 10% per year.
Probability distribution. The set of possible values of a random variable with a probability assigned to
each.
Properties of operations. See Table 3 in this Glossary.
Properties of equality. See Table 4 in this Glossary.
Properties of inequality. See Table 5 in this Glossary.
Properties of operations. See Table 3 in this Glossary.
Probability. A number between 0 and 1 used to quantify likelihood for processes that have uncertain
outcomes (such as tossing a coin, selecting a person at random from a group of people, tossing a ball at a
target, or testing for a medical condition).
Probability model. A probability model is used to assign probabilities to outcomes of a chance process by
examining the nature of the process. The set of all outcomes is called the sample space, and their
probabilities sum to 1. See also: uniform probability model.
Random variable. An assignment of a numerical value to each outcome in a sample space.
Rational expression. A quotient of two polynomials with a non-zero denominator.
Rational number. A number expressible in the form a/b or – a/b for some fraction a/b. The rational
numbers include the integers.
Rectilinear figure. A polygon all angles of which are right angles.
Rigid motion. A transformation of points in space consisting of a sequence of one or more translations,
reflections, and/or rotations. Rigid motions are here assumed to preserve distances and angle measures.
3
Adapted from Wisconsin Department of Public Instruction, op. cit.
4
To be more precise, this defines the arithmetic mean.
New Jersey Student Learning Standards for Mathematics
92
Repeating decimal. The decimal form of a rational number. See also: terminating decimal.
Sample space. In a probability model for a random process, a list of the individual outcomes that are to be
considered.
Scatter plot. A graph in the coordinate plane representing a set of bivariate data. For example, the heights
and weights of a group of people could be displayed on a scatter plot.
5
Similarity transformation. A rigid motion followed by a dilation
Tape diagram. A drawing that looks like a segment of tape, used to illustrate number relationships. Also
known as a strip diagram, bar model, fraction strip, or length model.
Terminating decimal. A decimal is called terminating if its repeating digit is 0.
Third quartile. For a data set with median M, the third quartile is the median of the data values greater
than M. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the third quartile is 15. See also:
median, first quartile, interquartile range.
Transitivity principle for indirect measurement. If the length of object A is greater than the length of
object B, and the length of object B is greater than the length of object C, then the length of object A is
greater than the length of object C. This principle applies to measurement of other quantities as well.
Uniform probability model. A probability model which assigns equal probability to all outcomes. See also:
probability model.
Vector. A quantity with magnitude and direction in the plane or in space, defined by an ordered pair or
triple of real numbers.
Visual fraction model. A tape diagram, number line diagram, or area model.
Whole numbers. The numbers 0, 1, 2, 3, ….
5
Adapted from Wisconsin Department of Public Instruction, op. cit.
New Jersey Student Learning Standards for Mathematics
93
Table 1. Common addition and subtraction situations.
6
Result Unknown Change Unknown Start Unknown
Add to
Two bunnies sat on the grass.
Three more bunnies hopped
there. How many bunnies are
on the grass now?
2 + 3 = ?
Two bunnies were sitting on
the grass. Some more bunnies
hopped there. Then there
were five bunnies. How many
bunnies hopped over to the
first two?
2 + ? = 5
Some bunnies were sitting on
the grass. Three more bunnies
hopped there. Then there
were five bunnies. How many
bunnies were on the grass
before?
? + 3 = 5
Take from
Five apples were on the table. I
ate two apples. How many
apples are on the table now?
5 – 2 = ?
Five apples were on the table.
I ate some apples. Then there
were three apples. How many
apples did I eat?
5 – ? = 3
Some apples were on the
table. I ate two apples. Then
there were three apples. How
many apples were on the
table before?
? – 2 = 3
Total Unknown Addend Unknown Both Addends Unknown
1
Put Together/
Take Apart
2
Three red apples and two green
apples are on the table. How
many apples are on the table?
3 + 2 = ?
Five apples are on the table.
Three are red and the rest are
green. How many apples are
green?
3 + ? = 5, 5 – 3 = ?
Grandma has five flowers.
How many can she put in her
red vase and how many in her
blue vase?
5 = 0 + 5, 5 = 5 + 0
5 = 1 + 4, 5 = 4 + 1
5 = 2 + 3, 5 = 3 + 2
Difference Unknown Bigger Unknown Smaller Unknown
Compare
3
(“How many more?” version):
Lucy has two apples. Julie has
five apples. How many more
apples does Julie have than
Lucy?
(“How many fewer?” version):
Lucy has two apples. Julie has
five apples. How many fewer
apples does Lucy have than
Julie?
2 + ? = 5, 5 – 2 = ?
(Version with “more”):
Julie has three more apples
than Lucy. Lucy has two
apples. How many apples
does Julie have?
(Version with “fewer”):
Lucy has 3 fewer apples than
Julie. Lucy has two apples.
How many apples does Julie
have?
2 + 3 = ?, 3 + 2 = ?
(Version with “more”):
Julie has three more apples
than Lucy. Julie has five
apples. How many apples
does Lucy have?
(Version with “fewer”):
Lucy has 3 fewer apples than
Julie. Julie has five apples.
How many apples does Lucy
have?
5 – 3 = ?, ? + 3 = 5
1
These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the
equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as.
2
Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic
situation, especially for small numbers less than or equal to 10.
3
For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using
less for the smaller unknown). The other versions are more difficult.
6
Adapted from Box 2-4 of National Research Council (2009, op. cit., pp. 32, 33).
New Jersey Student Learning Standards for Mathematics
94
Table 2. Common multiplication and division situations.
7
Unknown Product
Group Size Unknown
(“How many in each group?”
Division)
Number of Groups Unknown
(“How many groups?”
Division)
3 x 6 = ?
3 x ? = 18, and 18 ÷ 3 = ? ? x 6 = 18, and 18 ÷ 6 = ?
Equal Groups
There are 3 bags with 6 plums
in each bag. How many plums
are there in all?
Measurement example. You
need 3 lengths of string, each
6 inches long. How much string
will you need altogether?
If 18 plums are shared equally
into 3 bags, then how many
plums will be in each bag?
Measurement example. You
have 18 inches of string, which
you will cut into 3 equal pieces.
How long will each piece of
string be?
If 18 plums are to be packed 6
to a bag, then how many bags
are needed?
Measurement example. You
have 18 inches of string, which
you will cut into pieces that are
6 inches long. How many pieces
of string will you have?
Arrays,
4
Area
5
There are 3 rows of apples
with 6 apples in each row. How
many apples are there?
Area example. What is the area
of a 3 cm by 6 cm rectangle?
If 18 apples are arranged into 3
equal rows, how many apples
will be in each row?
Area example. A rectangle has
area 18 square centimeters. If
one side is 3 cm long, how long
is a side next to it?
If 18 apples are arranged into
equal rows of 6 apples, how
many rows will there be?
Area example. A rectangle has
area 18 square centimeters. If
one side is 6 cm long, how long
is a side next to it?
Compare
A blue hat costs $6. A red hat costs
3 times as much as the
blue hat. How much does the
red hat cost?
Measurement example. A
rubber band is 6 cm long. How
long will the rubber band be
when it is stretched to be 3
times as long?
A red hat costs $18 and that is
3 times as much as a blue hat
costs. How much does a blue
hat cost?
Measurement example. A
rubber band is stretched to be
18 cm long and that is 3 times
as long as it was at first. How
long was the rubber band at
first?
A red hat costs $18 and a blue
hat costs $6. How many times
as much does the red hat cost
as the blue hat?
Measurement example. A
rubber band was 6 cm long at
first. Now it is stretched to be
18 cm long. How many times as
long is the rubber band now as
it was at first?
General
a × b = ? a × ? = p, and p ÷ a = ? ? × b = p, and p ÷ b = ?
4
The language in the array examples shows the easiest form of array problems. A harder form is to use the terms rows and columns: The apples in
the grocery window are in 3 rows and 6 columns. How many apples are in there? Both forms are valuable.
5
Area involves arrays of squares that have been pushed together so that there are no gaps or overlaps, so array problems include these especially
important measurement situations.
7
The first examples in each cell are examples of discrete things. These are easier for students and should be given
before the measurement examples.
New Jersey Student Learning Standards for Mathematics
95
Table 3. The properties of operations. Here a, b and c stand for arbitrary numbers in a given number
system. The properties of operations apply to the rational number system, the real number system, and the
complex number system.
Associative property of addition (a + b) + c = a + (b + c)
Commutative property of addition a + b = b + a
Additive identity property of 0 a + 0 = 0 + a = a
Existence of additive inverses For every a there exists –a so that a + (–a) = (–a) + a = 0.
Associative property of multiplication (a b) c = a (b c)
Commutative property of multiplication a b = b a
Multiplicative identity property of 1 a 1 = 1 a = a
Existence of multiplicative inverses For every a ≠ 0 there exists 1/a so that a 1/a = 1/a a =
1.
Distributive property of multiplication over addition a (b + c) = a b + a c
Table 4. The properties of equality. Here a, b and c stand for arbitrary numbers in the rational, real, or
complex number systems.
Reflexive property of equality
a = a
Symmetric property of equality
If a = b, then b = a.
Transitive property of equality
If a = b and b = c, then a = c.
Addition property of equality
If a = b, then a + c = b + c.
Subtraction property of equality
If a = b, then a – c = b – c.
Multiplication property of equality
If a = b, then a x c = b x c.
Division property of equality
If a = b and c ≠, then a ÷c = b
÷
c.
Substitution property of equality
If a = b, then b may be substituted for a
in any expression containing a.
Table 5. The properties of inequality. Here a, b and c stand for arbitrary numbers in the rational or real
number systems.
Exactly one of the following is true: a < b, a = b, a > b.
If a > b and b > c then a > c.
If a > b, then b < a.
If a > b, then –a < –b.
If a > b, then a ± c > b ± c.
If a > b and c > 0, then a x c > b xc.
If a > b and c < 0, then a c < b xc.
If a > b and c > 0, then a ÷c > b ÷c.
If a > b and c < 0, then a ÷c < b ÷c.
New Jersey Student Learning Standards for Mathematics
96
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