Download THE NUTLEY PUBLIC SCHOOLS

Nutley Public Schools
Mathematics
Curriculum
November 2016
Draft Adopted by the Nutley Board of Education on November, 2016
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Nutley Public Schools
Mathematics Curriculum
Introduction
Mathematics instruction provides the foundation to help students achieve in all curricular areas.
Through this curriculum, students will learn basic mathematical skills including number sense,
operations, cardinality, measurement and the use of data. Students will utilize these skills to move
into higher levels of learning including the areas of fractions, decimals, geometry, and algebra
skills. Students will learn ways in which these skills can be transferred to all other subject areas.
The mathematics curriculum, written to the New Jersey Student Learning Standards, addresses
various components of mathematics instruction that build from year to year. Each curricular unit
includes interdisciplinary and technology connections to bridge learning in various content areas.
The units also include an identification of the writing activities that will occur with each
mathematics unit. While some of the themes are repeated at various grade levels, the content spirals
and addresses the students’ needs at each level.
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Table of Contents
Kindergarten ................................................................................................................................... 4
Grade 1
................................................................................................................................... 65
Grade 2
................................................................................................................................... 95
Grade 3 .............................................................................................................................................. 125
Grade 4
................................................................................................................................... 155
Grade 5
....................................................................................................................................... 184
Grade 6
................................................................................................................................... 218
Pre-Algebra
................................................................................................................................... 240
Foundations of Algebra ..................................................................................................................... 266
Algebra 1 ........................................................................................................................................... 295
Geometry ........................................................................................................................................... 330
Algebra 2 ........................................................................................................................................... 361
Pre-Calculus ...................................................................................................................................... 402
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Nutley Public Schools
Kindergarten
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Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 1:
Numbers to 5
Summary and Rationale
Many children come to school with a basic understanding of counting and numbers. Counting is a
fundamental skill in the development of number sense and understanding numbers is the beginning of
math literacy.
In this unit, children learn to read and write numerals 1 to 5 and investigate how to sort objects using one
attribute. They look for sameness and differences with such attributes as size, number, and color. Sorting
and classifying skills are necessary as children work with patterns, geometric shapes, and data. These
skills help children understand the need for order and organization in daily life. The sorting activities are
directly connected to the numerals and quantities 1 through 5.
Counting up to 5 objects is the most basic form of numerical capability. Being able to distinguish between
same and different characteristics of objects and pictures leads to being able to sort and classify.
Pacing
Three (3) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
Measurement and Data
K.MD.1
Describe and compare measurable attributes. Describe measurable attributes of objects, such
as length or weight. Describe several measurable attributes of a single object.
K.MD.2
Directly and compare two measurable attributes. Directly compare two objects with a
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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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be used to count, label, order, identify, measure, and describe things.
Numbers can be represented in many ways.
Numbers can be ordered and compared.
When counting, the last number named in the sequence is the total of the group of objects.
Essential Questions
How do we use numbers?
Why are numbers important?
How can you show various numbers?
What is the order of numbers 1-5?
Why do we count?
When do we count?
Can everything be counted?
Evidence of Learning
Guided Practice
Formative Assessments
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Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Number Names: one, two, three, four, five
 The Count Sequence
 Colors: blue, green, red, yellow, white, black
 Comparative words: same, not the same, different
 Descriptive words: big, small, long, short, tall
 To count up to 5 objects by saying number names in the standard order and saying just one number
name for each for each number counted.
 To recognize the relationship between the number of objects and their respective numerals.
 To identify same and different attributes of objects such as color, size, and shape.
Students will be able to:
 Count from 1 to 5.
 Count groups of 1,2,3,4, and 5.
 Read and write the numerals 1 to 5.
 Match and sort.
 Look for, understand, and identify sameness, not the same, and different.
 Spot and understand differences.
 Spot differences between two pictures.
 Make subtle differences in two pictures.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Numbers to 5
www.morestarfall.com
www.numbernuts.com
www.primarygames.com
www.youtube.com/harrykindergarten
Language Arts Integration
Math Journal
Ten Black Dots by Donald Crews
Suggested Resources
Math In Focus Resources Chapter 1: Numbers to 5
Student Activity Cards
Teacher Activity Cards
Numeral Cards
Connecting Tubes
Counters
Attribute Blocks
www.hmhlearning.com
www.hmheducation.com/mathinfocus
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Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 2:
Numbers to 10
Summary and Rationale
Counting and an understanding of one-to-one correspondence is a child’s entrance into the world of
mathematics. Through repetition of counting, children develop a visual sense of small quantities and
relate those quantities to number words.
This unit includes a variety of matching activities in which children will find two groups that have the
same number of objects, which will provide practice and reinforcement with counting while developing a
visual meaning of number.
All children will eventually be able to match groups of equal objects without counting and be able to look
at a larger group of objects and estimate the number of objects in the group.
Counting from 0 to 9 is the next step from being able to count from 1 to 5. Being able to deduce one more
or one less of a set of objects forms the basis of simple addition and subtraction.
Pacing
Four (4) weeks
Standards
Counting and Cardinality
K.CC.2
K.CC.3
K.CC.4a
K.CC.4b
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
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.
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
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counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
K.CC.6
Compare numbers. 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. (Include groups with up to 10 objects.)
K.CC.7
Compare numbers. Compare two numbers between 1 and 10 presented as written numerals.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be used to count, label, order, identify, measure, and describe things.
Numbers can be represented in many ways.
Numbers can be ordered and compared.
Counting from 0 to 9 is the next step from being able to count from 1 to 5.
When counting, the last number named in the sequence is the total of the group of objects.
Essential Questions
How do we use numbers?
Why are numbers important?
How can you show various numbers?
What is the order of numbers 1-10?
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Why do we count?
When do we count?
Can everything be counted?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Number Names: one, two, three, four, five, six, seven, eight, nine, zero
 The Count Sequence
 Comparative words: one more, one less, the same number
 To count up to 5 objects by saying number names in the standard order and saying just one number
name for each for each number counted.
 To count 0 to 9 objects by saying one number name for each object and realizing the last number
named tells how many.
 To recognize the relationship between the number of objects and their respective numerals.
 To compare two sets of objects, or a set of objects and a numeral, to determine if there is a difference
of one more, one less, or the same number of objects.
Students will be able to:
 Count from 1 to 9.
 Use 0 to 9 to tell the number of objects.
 Read and write the numerals 1 to 9.
 Pair number names with numerals.
 Use 0 to 9 to tell the number of objects.
 Pair up sets of objects with other sets of the same quantity.
 Use one more, one less, and the same number to describe differences between groups of objects.
 Pair up sets of objects one-to-one with other sets of the same quantity.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 2: Numbers to 10
www.numbernuts.com
www.primarygames.com
www.morestarfall.com
www.youtube.com/harrykindergarten
Language Arts Integration
Math Journal
Ten Black Dots by Donald Crews
Suggested Resources
Math In Focus Resources Chapter 2: Numbers to 10
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Student Activity Cards
Numeral Cards
Dot Cards
Connecting Tubes
Number Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
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Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 3:
Order by Length or Weight
Summary and Rationale
Measurement applies mathematics in a way that young children can easily understand and relate to realworld experiences. Children begin by touching, examining, and comparing objects to develop awareness
of attributes, such as length, size, and weight. Children can see and feel these differences, which leads to
comparing and ordering objects based on their attributes.
Using tools to measure length and weight connects the geometry of physical objects to numbers. In this
unit, children begin to measure by comparing visually and by feel, laying the foundation for using nonstandard units in later grades.
Pacing
Two (2) weeks
Standards
Measurement and Data
K.MD.1
Describe and compare measurable attributes. Describe measurable attributes of objects, such
as length or weight. Describe several measurable attributes of a single object.
K.MD.2
Directly and compare two measurable attributes. 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.
K.MD.3
Classify objects and count the number of objects in each category. Classify objects into
given categories; count the numbers of objects in each category and sort the categories by
count. (Limit category counts to be less than or equal to 10.)
Mathematical Practices
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
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Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Groups of objects can be compared and ordered by length, size, and weight.
Ordering and comparing objects form the basics of measurement.
Essential Questions
How do we order and compare numbers?
How do we order and compare objects?
What do we look for when comparing and ordering objects?
How are objects alike, different, and the same?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Comparative words: same size, different size, bigger than, taller than, smaller than, shorter than,
heavier, lighter
 Ordering words: biggest, middle-sized, smallest, longest, shortest, heaviest, lightest, heavier, lighter
 To order objects from smallest to biggest, shortest to longest, and lightest to heaviest.
 To estimate measureable attributes such as weight through visual aids.
 To use comparative vocabulary to express differences in size, length, and weight.
Students will be able to:
 Pair up sets of objects.
 Order objects by size.
 Use comparing words.
 Order objects according to length.
 Order objects according to weight.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 3: Order by Length or Weight
Writing Integration
Math Journal
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Suggested Resources
Math In Focus Resources Chapter 3: Order by Length and Weight
Student Activity Cards
Connecting Tubes
www.hmhlearning.com
www.hmheducation.com/mathinfocus
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Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 4:
Counting Up and Down Between 0 and 10
Summary and Rationale
As children learn and practice counting skills, they become aware of connections to other math topics
such as comparing and ordering of numbers and quantities. Counting is also a key strategy that children
use to find the total of combined groups.
In this unit, children count up to 10 and down from 10. Children use their fingers, as well as
manipulatives and pictures, to determine the number that is one more or one less than a given number.
Children combine and take away objects, and then count to find the result. These activities help children
develop and understanding of the meaning of addition and subtraction, which are introduces in this unit.
Composing and decomposing provide a strong foundation for mental addition and subtraction. Counting
and comparing numbers build on sequencing and numerical order.
Pacing
Four (4) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.2
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
K.CC.3
Know number names and the count sequence. 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).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
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apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.3
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. 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 drawing
or equation (e.g., 5=2+3 and 5=4+1).
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP. 8 Look for an express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be used to count, label, order, identify, measure, and describe things.
Numbers can be represented in many ways.
Numbers can be ordered and compared.
When counting, the last number named in the sequence is the total of the group of objects.
Manipulatives and pictures can be used to determine the number that is one more or one less than a given
number.
When counting, the next number in the sequence is one more.
Essential Question
How do we use numbers?
Why are numbers important?
How can you show various numbers?
What is the order of numbers 1-10?
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Why do we count?
When do we count?
Can everything be counted?
What can we use to help us count up to 10?
What can we use to help us count down from 10?
How do you determine one more or one less than a number?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 To compose and decompose numbers through 5 to build a strong foundation in number facts
 To count up to 10 and compare numerals and sets using terms more, less, fewer, and same number.
 To count how many in all, which is the most basic form of addition.
 The concept of one more.
 Fingers can represent a set of objects up to 10.
 Fingers and toes can represent a set of objects up to 20.
 The meaning of same, more, and how many more.
 The meaning of less and fewer.
Students will be able to:
 Compose and decompose numbers through 5.
 Pair number names with numerals.
 Order numbers 0 to 10.
 Determine one more.
 Show the meaning of same and more.
 Show the meaning of less and fewer.
 Use more and less to compare number values.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 4: Counting and Numbers 0 through 10
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 4: Counting and Numbers 0 through 10
Teacher Activity Cards
Numeral Cards
Dot Cards
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Connecting Tubes
Number Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
18
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 5:
Size and Position
Summary and Rationale
In order to find their way around the home, neighborhood, and school, children must acquire spatial skills.
In this unit, children begin to describe objects and their relative position to one another by identifying
objects that are on top of, under, next to, behind, in front of, and inside other objects.
Children will eventually learn to distinguish between left and right and give more precise descriptions of
location.
Size and fit forms the basis to understanding capacity and positional words allow for acquisition of spatial
skills. Ordering events allows for the development of understanding order and time.
Pacing
Two (2) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.3
Know number names and the count sequence. 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).
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
Measurement and Data
K.MD.1
Describe and compare measurable attributes. Describe measurable attributes of objects, such
as length or weight. Describe several measurable attributes of a single object.
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K.MD.2
K.MD.3
Directly and compare two measurable attributes. 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.
Classify objects and count the number of objects in each category. Classify objects into
given categories; count the numbers of objects in each category and sort the categories by
count. (Limit category counts to be less than or equal to 10.)
Geometry
K.G.1
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). 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.
Mathematical Practices
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Objects can be described and classified by size.
Descriptive words can be used to describe positions and movement of objects.
Essential Question
How are objects alike, different, and the same?
What words can we use to describe size?
Why do we use positional words?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:

Descriptive words: big, small, long
20

Comparative words: bigger, smaller, same size


To estimate whether objects can fit into containers of various sizes.
Positional vocabulary which will allow them to describe the location of an object in a spatial
arrangement: on top of, under, next to, behind, between, beside, in front of, in back of, inside, outside
To order events using the terms before and after.

Students will be able to:
 Use size comparisons such as big or small.
 Explore the idea that only a few big objects fit into small spaces, however many small objects fir into
big spaces.
 Identify positions of objects in space.
 Use appropriate positional language to describe and compare.
 Use language such as before or after to describe relative position in a sequence of events.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 5: Size and Position
www.apples4theteacher.com
www.gamequarium.com
www.funbrain.com
www.primarygames.com
Language Arts Integration
Math Journal
A Pig is Big by Douglas Florian
Big and Little by Steve Jenkins
Suggested Resources
Math In Focus Resources Chapter 5: Size and Position
Student Activity Cards
Connecting Tubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
21
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 6:
Numbers to 20
Summary and Rationale
Children learn to count in increments, first to 5 or 10, and then to 20. The understanding of one-to-one
correspondence is developed by asking children to point to each object and say the number word. It is also
crucial for children to understand that each number that they say is one more than the number before it,
which leads to the understanding of one more and one less.
It is also important for children to realize that when counting, the last number named in the sequence is
the total of the group of objects, which will help later when learning addition.
Counting up to 20 serves as an introduction to counting two-digit numbers. Mastering knowledge of
numerical sense up to 20, as well as its sequence, is the stepping stone to counting numbers up to 100.
Pacing
Three (3) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.2
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
K.CC.6
Compare numbers. 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. (Include groups with up to 10 objects.)
K.CC.7
Compare numbers. Compare two numbers between 1 and 10 presented as written numerals.
Operations and Algebraic Thinking
22
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.4
Understanding addition as putting together and adding to, and understand subtraction as
taking apart and taking from. 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.
Numbers – Base Ten
K.NBT.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 (such as 18=10+8); understand that these numbers are composed of ten ones and
one, two, three, four, five, six, seven, eight, or nine ones.
Geometry
K.G.1
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). 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.
Mathematical Practices
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be used to count, label, order, identify, measure, and describe things.
Numbers can be represented in many ways.
Numbers can be ordered and compared.
When counting, the last number named in the sequence is the total of the group of objects.
Manipulatives and pictures can be used to determine the number that is one more or one less than a given
number.
When counting, the next number in the sequence is one more and the previous number in thesequence is
one less.
23
Essential Questions
How do we use numbers?
Why are numbers important?
How can you show various numbers?
What is the order of numbers 11-20?
Why do we count?
When do we count?
Can everything be counted?
How do you determine one more or one less than a number?
How can we compare numbers?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Number Names: ten, eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen,
twenty
 Comparative Words: more, fewer, greater than, less than
 To count up to 20 objects by using one-to-one correspondence.
 To compare and sequence numbers to 20.
Students will be able to:
 Count to 10.
 Read and write the numeral 10.
 Count from 10 to 20.
 Use ten-frames to count on.
 Read and write the numerals form 10 to 20.
 Compare and order groups of up to 20 objects.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter6: Numbers 0 to 20
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 6: Numbers 0 to 20
24
Numeral Cards
Connecting Tubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
25
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 7:
Geometry
Summary and Rationale
Geometry helps children describe the world around them. In kindergarten, geometry instruction focuses
on expanding and enhancing children’s prior knowledge and understanding of shapes that they have
acquired by observing that the world that surrounds them. Children learn more precise names for shapes
and how to describe them, and begin to compare and contrast them.
It is helpful to provide many concrete examples of flat and solid shapes to help children make real-world
connections and it is crucial that they are able to identify examples and non-examples of different shapes.
Knowing the basic properties of two and three-dimensional shapes is the first step to understanding
geometry. Creating patterns using shapes will enable children to later better understand more complex
patterns such as number patterns.
Pacing
Two (2) weeks
Standards
Geometry
K.G.1
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). 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.
K.G.2
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). Correctly name shapes regardless of their orientations or overall
size.
K.G.3
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). Identify shapes as two-dimensional (lying in a plane, “flat”).
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
26
Interdisciplinary Connections
Standard
Integration of Technology
Standard x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Shapes can be described, identified, compared, and classified.
Some shapes have flat faces, edges, and corners, and some do not.
Essential Questions
What are the names of some shapes?
Where do we find flat shapes around us?
In what ways can geometric solids be matched to real-life objects?
How can I put shapes together and take them apart to form other shapes?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Parts of a Shape: face, edge, corner
 Solid Shape Names: cube, cone, cylinder, sphere, pyramid
 Flat Shape Names: circle, triangle, square, rectangle, hexagon
 Descriptive Words: big, small
 Shape patterns
 Some shapes have flat faces, edges, and corners, and some do not.
Students will be able to:
 Recognize and name basic solid and flat shapes.
 Describe basic solid and flat shapes.
 Recognize the relationship between solid and flat shapes.
 Draw flat shapes.
 Identify basic flat shapes within a scene.
 Make a picture using basic flat shapes.
 Identify and extend a shape pattern.
27
Integration
Art: make shape collages and sculptures
Social Studies: identify shapes around us
Music: shape songs
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 7: Solid and Flat Shapes
www.brainpopjr.com
www.morestarfall.com
www.primarygames.com
www.gamequarium.com
www.fun4thebrain.com
www.theteachersguide.com
www.math-play.com/Geometry-Math-Games.html
www.apples4theteacher.com
Language Arts Integration
The Wings on a Flea: A Book about Shapes by Ed Emberly
What is a Square? By Rebecca Kai Dotlich
Cubes, Cones, Cylinders, and Spheres by Tana Hoban
The Shape of Things by Dayle Ann Dodds
Math Journal
Suggested Resources
Math In Focus Resources Chapter 7: Solid and Flat Shapes
Student Activity Cards
Solid Shapes
Attribute Blocks
www.hmhlearning.com
www.hmheducation.com/mathinfocus
28
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 8:
Skip Counting to 100
Summary and Rationale
Skip-counting is a skill that can help children connect many mathematics topics. It provides a shorter way
to count objects when objects are grouped in twos, fives, or tens. Skip-counting can lead to addition of
two, five, or ten, and is also related to future topics such as multiplication, number patterns, and functions.
In this unit, children also explore the basics of place value, which is the foundation of the number system.
As they use concrete models to explore numbers through 100, they will learn to use groups of tens and
ones to represent and name these greater numbers. This leads to the beginning of an understanding of twodigit numbers and their magnitudes, as well as to the discovery of number patterns in the hundred-chart
and useful counting skills.
Counting by 2s, 5s, and 10s lends to the foundation to simple multiplication. Mastering two-digit numbers
up to 100 allows for development of place value knowledge.
Pacing
Three (3) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
Mathematical Practices
K-12.MP.4 Model with mathematics.
Interdisciplinary Connections
29
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be used to count, label, order, identify, measure, and describe things.
Numbers can be represented in many ways.
Numbers can be ordered and compared.
When counting, the last number named in the sequence is the total of the group of objects.
When counting by ones, students need to understand that the next number in the sequence is one more.
When counting by tens, the next number in the sequence is “ten more” (or one more group of ten).
Essential Questions
Why are numbers important?
How can you show various numbers?
Why do we count?
When do we count?
Can everything be counted?
How do you determine one more or one less than a number?
What is skip-counting?
Why do we skip-count?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Number Names: ten, twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety, hundred
 Counting words: pairs, twos, fives, tens, tally
30
Students will be able to:
 Recognize and use pairs for counting.
 Count by 2s, 5s, and 10s up to 20.
 Use the counting by 2s sequence to count up to 20 objects.
 Keep count of numbers using tallies.
 Count to 49, to 79, to 100.
 Count by 2s, 5s, and 10s up to 100.
 Count from any given number to 49, to 79, to 100.
Integration
Science: thermometers- temperature reading
Art: 100 chart color pictures
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 8: Skip Counting to 100
Interactive hundreds chart
www.abcya.com
www.softschools.com
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 8: Skip Counting to 100
Connecting Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
31
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 9:
Comparing Sets
Summary and Rationale
As students continue to develop counting skills, their numbers sense become stronger and they are soon
ready to compare quantities. Through one-to-one correspondence activities, children are able to identify a
number that is greater or less than another and further learn how much more or less one number is than
another. These concepts pave the way to an understanding of addition and subtraction.
Comparing sets is the most basic form of subtraction. Combining sets lays the foundations for addition.
Pacing
Four (4) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.2
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4a
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.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.2
Solve subtraction word problems and subtract within 10,e.g.,by using objects or drawing to
represent the problem.
K.OA.5
Demonstrate fluency for addition and subtraction within 5.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
32
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for an express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Counting is a strategy for finding the total of combined groups.
When counting, the last number named in the sequence is the total number of objects in the group(s) .
Essential Questions
How can we compare groups of objects?
What happens when we combine groups or sets?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Comparative words: fewer, less, more, most, fewest.
 To compare sets of up to 20 to find the difference between the two sets.
 To compare countable sets using the terms fewer and more, and uncountable sets using the terms less
and more.
 To combine sets to find how many in all.
 Number Trains
Students will be able to:
 Compare sets of up to 20 objects.
 Use and understand fewer, less, more, most, and fewest.
 Recognize and understand number trains.
 Count the difference through comparing sets in one-to-one correspondence.
Integration
Technology Integration
33
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 9: Comparing Sets
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 9: Comparing Sets
Dot Cards
Connecting Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
34
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 10:
Ordinal Numbers
Summary and Rationale
Sequence and order are important concepts in all school subjects and in daily life. When children learn to
count, they learn that numbers follow sequential order.
Numbers can be used for different purposes and they can also be represented in different ways. One
example is the relationship between cardinal and ordinal numbers.
Learning to order things, be it events, physical position, or preferences, sets the pace for basic
understanding of sequence and patterns.
Pacing
Two (2) weeks
Standards
Measurement and Data
K.MD.2
Directly and compare two measurable attributes. 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.
Mathematical Practices
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
35
Ordinal words can be used to sequence events. Examples are first, next, last, first, second, third, before
and after.
Ordinal words can be used to describe physical position and ranks of preference.
Essential Questions
What words can be used to describe physical position?
What words can be used to sequence events?
Why is sequencing events important?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 To order 3- and 4-step events using the terms first, next, last, second, and third.
 To order physical position as well as relate order to the terms before and after.
 To order their preferences, make picture graphs, and make deductions based on the picture graphs.
Students will be able to:
 Sequence events.
 Use and understand first, next, and last to sequence events.
 Use and understand first, second, and third to sequence events, in terms of physical position, and to
rank personal preferences.
 Understand before and after in terms of physical position.
 Make picture graphs based on preferences.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 10: Ordinal Numbers
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 10: Ordinal Numbers
Student Activity Cards
www.hmhlearning.com
www.hmheducation.com/mathinfocus
36
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 11:
Calendar Patterns
Summary and Rationale
The unit presents another application of sequence- the days of the week and the months of the year.
Everything children do, from attending school, to playing sports, to watching television, is related to the
concept of time. Children should recognize the names of the days of the week and the months of the year
and understand their relationship. The dates in a month also help strengthen numerical sense.
Knowing the days of the week and the months of the year is the most basic form of time awareness.
Pacing
One (1) week
Standards
Measurement and Data
K.MD.2
Directly and compare two measurable attributes. 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.
K.MD.3
Classify objects and count the number of objects in each category. Classify objects into
given categories; count the numbers of objects in each category and sort the categories by
count. (Limit category counts to be less than or equal to 10.)
Mathematical Practices
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
37
Enduring Understandings
Calendars are used to show days, weeks, and months of a year.
Essential Questions
What are the days of the week?
What are the months of the year?
How can the months of the year be compared?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 To name and order the days of the week
 To name and order the months of the year
 Other terms related to time: day, week, year, today, tomorrow, yesterday
 Comparative words: warmer, cooler
Students will be able to:
 Name and know the days of the week and the months of the year and how many there are.
 Use and understand today, tomorrow, and yesterday.
 Read and understand a weekly calendar.
 Order the days of the week and the months of a year.
 Make and interpret pictographs.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 11: Calendar Patterns
www.brainpopjr.com
Language Arts Integration
The Grouchy Ladybug by Eric Carle
What Time is it Mr. Crocodile? By Judy Sierra
Math Journal
Suggested Resources
Math In Focus Resources Chapter 11: Calendar Patterns
Student Activity Cards
www.hmhlearning.com
www.hmheducation.com/mathinfocus
38
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 12:
Counting on and Counting Back
Summary and Rationale
In this unit, children learn the counting on and counting back strategies for problem solving. Within the
context of these activities, children build an understanding of, and familiarity with, number pairs that
make ten. The activities in this unit also increase awareness of addition and subtraction.
Counting on and counting back helps in the familiarity of number facts. This in turn helps with mental
math and simple addition and subtraction.
Pacing
Two (2) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.2
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
K.CC.6
Compare numbers. 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. (Include groups with up to 10 objects.)
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
39
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.3
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. 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 drawing
or equation (e.g., 5=2+3 and 5=4+1).
K.OA.4
Understanding addition as putting together and adding to, and understand subtraction as
taking apart and taking from. 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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be used to count, label, order, identify, measure, and describe things.
Numbers can be represented in many ways.
Numbers can be ordered and compared.
When counting, the last number named in the sequence is the total of the group of objects.
Manipulatives and pictures can be used to determine the number that is one more or one less than a given
number.
When counting, the next number in the sequence is one more and the previous number in thesequence is
one less.
Essential Questions
Why do we count?
When do we count?
40
Can everything be counted?
How do you determine one more or one less than a number?
How can we compare numbers?
When do we count on?
When do we count back
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 To count on and count back.
 To find the difference between two sets using several strategies such as finger counting and one-toone correspondence.
Students will be able to:
 Count back using fingers and other representations.
 Count up and back to fins the difference between two sets.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 12: Counting On and Counting Back
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 12: Counting On and Counting Back
Teacher Activity Cards
Numeral Cards
Counters
Number Cubes
Connecting Cubes
www.hmhlearning.com
www.hmheducation.com/mathinfocus
41
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 13:
Patterns
Summary and Rationale
Kindergarten children should be encourages to look for patterns in their environment, such as tile patterns
or the patterns in routine events. Most kindergarteners already recognize simple patterns and many find
them appealing because of their need for organization and structure. The simple patterns children use and
create in kindergarten provide a basis for increasingly complex patterns.
Extending and creating patterns not only reinforces the properties of shapes, but also sets the foundation
for more complex patterns such as number patterns.
Pacing
One (1) week
Standards
Geometry
K.G.1
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). 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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
42
Instructional Focus
Enduring Understandings
Patterns are used everywhere Examples are word patterns, art patterns, math patterns, and behavior
patterns.
Patterns can be described, reproduced, extended, and created.
Essential Questions
What is a pattern?
How do we make patterns?
How do we determine what comes next in a given pattern?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 Pattern Unit
 Repeating Patterns
 To learn to create and extend repeating patterns by identifying the pattern unit and duplicating it.
Students will be able to:
 Recognize, extend, and create a repeating pattern.
 Identify a missing portion of a repeating pattern.
 Create ABABAB, AABAAB, ABBABB, and ABCABC repeating patterns.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 13: Patterns
www.brainpopjr.com
Language Arts Integration
Pattern Fish by Trudy Harris
Math Journal
Suggested Resources
Math In Focus Resources Chapter 13: Patterns
Student Activity Cards
Attribute Blocks
www.hmhlearning.com
www.hmheducation.com/mathinfocus
43
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 14:
Number Facts
Summary and Rationale
In this unit, children extend their counting abilities by counting and combining groups of objects and
counting on to find differences, which are both important readiness skills for addition and subtraction.
Activities that require children to count groups of objects and then find out how many more are needed to
make 10 not only provides a foundation for addition and subtraction, but can also lead an understanding
that 10 is the basis of our numeration system. Counting and grouping tens is a basic place-value concept.
Number bonds are introduced in this unit to create a mental picture of the relationship between a number
and the parts that combine to make it, and also provide a visualization of the composing and decomposing
of a number.
Number bonds lay the foundation for basic number facts, and especially for solving problems that involve
missing addends.
Combining sets, composing and decomposing, as well as counting on all play a part in familiarizing the
child with number facts. They also form the background for addition and subtraction.
Pacing
Two (2) weeks
Standards
Counting and Cardinality
K.CC.2
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
44
K.CC.6
Compare numbers. 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. (Include groups with up to 10 objects.)
Numbers- Base Ten
K.NBT.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 (such as 18=10+8); understand that these numbers are composed of ten ones and
one, two, three, four, five, six, seven, eight, or nine ones.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.3
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. 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 drawing
or equation (e.g., 5=2+3 and 5=4+1).
K.OA.4
Understanding addition as putting together and adding to, and understand subtraction as
taking apart and taking from. 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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
CPI #
Instructional Focus
Enduring Understandings
One number can be represented by combining two smaller numbers.
Numbers can be composed and decomposed using five and ten-frames.
A number line can be helpful when counting on.
A difference between two numbers can be found by counting on.
Essential Questions
45
In what ways can a number be represented using two other numbers?
What is a strategy for finding the difference between two numbers?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 Number facts to 10.
 To combine sets and to see how many more are needed for values up to 15.
 To compose and decompose numbers up to 20 by counting on and other strategies.
Students will be able to:
 Compose and decompose numbers through 10.
 Combine sets to make 5, 6, 7, 8, 9, and 10.
 Compose and decompose numbers to 20 with five-frames and ten-frames.
 Count on using a number line.
 Count on to find the difference.
 Combine two sets to find how many more for sums through 15.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 14: Number Facts
Language Arts Integration
Math Journal
Suggested Resources
Math In Focus Resources Chapter 14: Number Facts
Teacher Activity Cards
Numeral Cards
Connecting Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
46
Nutley Public Schools
Mathematics
Kindergarten
Unit 15:
Length and Height
Summary and Rationale
Measuring length has a variety of applications in the real world. Measurement also connects ideas in the
number strand with geometry concepts. In this unit, children are introduced to measuring lengths and
heights using nonstandard units instead of rulers to get at the idea that any length can be measured with
any same-sized unit.
Measuring lengths and heights using nonstandard units paves the way not only for geometry, bust also
exposes the child to the skill of estimation.
Pacing
Two (2) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4a
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.
K.CC.4b
Understand that the last number name said tells the number of objects counted. The number
of objects id the same regardless of their arrangement or the order in which they are
counted.
K.CC.4c
Understand that each successive number name refers to a quantity that is one larger.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.2
Solve subtraction word problems and subtract within 10,e.g.,by using objects or drawing to
represent the problem.
Measurement and Data
K.MD.1
Describe and compare measurable attributes. Describe measurable attributes of objects, such
as length or weight. Describe several measurable attributes of a single object.
K.MD.2
Directly and compare two measurable attributes. Directly compare two objects with a
47
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Measuring length has a variety of applications in the real world.
Any length can be measured with any same-sized unit.
It takes a number and a unit to express a measurement.
More units are needed to measure a longer or taller object than a shorter object.
Measurements change depending on the size of the unit.
When comparing two lengths, one end of each length must match.
When measuring, length or height start at the beginning of the object and finish measuring at the end of
the object.
Essential Questions
How do we tell which object is longer?
How do we tell which object is taller?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
48



To compare lengths of objects using the terms long, short, longer, shorter, longest, tallest and
shortest.
To compare lengths and heights of objects using nonstandard units of measurement, such as
connecting cubes and paperclips.
To find the difference in lengths in terms of nonstandard units.
Students will be able to:
 Compare lengths.
 Use nonstandard units to measure and compare lengths and heights.
 Find differences in lengths using nonstandard units.
 Use the terms tallest and shortest in terms of height.
Integration
Art: draw objects to compare sizes (tall flower and short flower)
Science: measure ingredients for experiments-measure objects for theme of the month (ex. Insectsflowers)
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 15: Length and Height
www.apples4theteacher.com
www.gamequarium.com
www.funbrain.com
www.primarygames.com
Language Arts Integration
Inch by Inch by Leo Lionni
Biggest, Strongest, Fastest by Steve Jenkins
Math Journal
Suggested Resources
Math In Focus Resources Chapter 15: Length and Height
Numeral Cards
Connecting Cubes
www.hmhlearning.com
www.hmheducation.com/mathinfocus
49
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 16:
Classifying and Sorting
Summary and Rationale
Sorting and classifying skills help children to identify patterns, describe geometric objects, and analyze
data. When children classify and sort, they develop skills that will help them with other mathematical
strands.
Classifying and sorting objects by one, two, and three attributes exposes children to a wide variety of
color, shape, size, and pattern-related vocabulary. It also lays the foundation to learning about graphs in
the future.
Pacing
(1) week
Standards
Measurement and Data
K.MD.1
Describe and compare measurable attributes. Describe measurable attributes of objects, such
as length or weight. Describe several measurable attributes of a single object.
K.MD.2
Directly and compare two measurable attributes. 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.
K.MD.3
Classify objects and count the number of objects in each category. Classify objects into
given categories; count the numbers of objects in each category and sort the categories by
count. (Limit category counts to be less than or equal to 10.)
Geometry
K.G.2
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones,
cylinders, and spheres). Correctly name shapes regardless of their orientations or overall
size.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
50
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Objects can be classified using one, two, or three attributes such as color, size, shape, and other special
features.
Objects can be sorted using one or two attributes such as color, size, shape, and other special features.
Essential Questions
Why do we classify objects?
How do we classify objects?
Why and how do we sort objects?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 To sort.
 To identify attributes and pick out the ‘odd one out’ in a set of objects.
 To sort and classify objects according to up to three attributes.
Students will be able to:
 Classify objects using one attribute, two attributes, and three attributes. (color, size, shape, special
features)
 Identify objects that do not belong to a set.
 Sort objects using one attribute or two attributes. (color, size, shape, special features)
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 16: Classifying and Sorting
Language Arts Integration
Math Journal
51
Suggested Resources
Math In Focus Resources Chapter 16: Classifying and Sorting
Connecting Cubes
Counters
Attribute Blocks
www.hmhlearning.com
www.hmheducation.com/mathinfocus
52
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 17:
Addition Stories
Summary and Rationale
In this unit, children learn to show addition using objects, pictures, models, number, and words. Using
symbols to represent mathematical situations is one of the beginning skills of algebra.
Addition is one of four number operations in mathematics. Simple addition and interpreting number
sentences from numbers stories forms the basis to more complex addition situations that include three or
more addends.
Pacing
One (1) week
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4
Count to tell the number of objects. Understand the relationship between numbers and
quantities; connect counting to cardinality.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.2
Solve subtraction word problems and subtract within 10,e.g.,by using objects or drawing to
represent the problem.
K.OA.3
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. 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 drawing
or equation (e.g., 5=2+3 and 5=4+1).
K.OA.5
Demonstrate fluency for addition and subtraction within 5.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
53
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for an express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Addition is the joining of two sets.
Pictures and manipulatives can be used to model addition.
Essential Question
What happens when we combine groups or sets?
Why do I need to know how to add?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 To deduce addition sentences from addition stories and write them using the symbols + and =.
 Mathematical vocabulary: plus, is equal to, numbers sentence
 Fluency with addition facts to 5.
Students will be able to:
 Use symbols and numerals to write number sentences.
 Represent addition stories with addition sentences.
Integration
Art- draw number stories
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 17: Addition Stories
www.gamequarium.com
www.fun4thebrain.com
www.theteachersguide.com
54
www.gamequarium.com
www.fun4thebrain.com
www.theteachersguide.com
Language Arts Integration
The Napping House by Audrey Wood
One Monday Morning by Uri Shulevitz
Quack and Count by Keith Baker
12 Ways to Get to 11 by Eve Merriam
Math Journal
Suggested Resources
Math In Focus Resources Chapter 17: Addition Stories
Numeral Cards
Symbol Cards
Connecting Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
55
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 18:
Subtraction Stories
Summary and Rationale
Story problems are a common context for applying subtraction ideas. Subtraction stories provide children
with opportunities to demonstrate understanding of simple separating and comparison subtraction
problems. The activities in this unit connect and continue to develop basic concepts in number, algebra,
and problem solving strands.
Subtraction is one of the four number operations in mathematics. Simple subtraction and interpreting
number sentences form number stories forms the basis to more complex subtraction situations.
Pacing
Two (2) weeks
Standards
Counting and Cardinality
K.CC.1
Know number games and the count sequence. Count to 10 by ones and by tens.
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.4
Count to tell the number of objects. Understand the relationship between numbers and
quantities; connect counting to cardinality.
K.CC.6
Compare numbers. 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. (Include groups with up to 10 objects.)
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.2
Solve subtraction word problems and subtract within 10,e.g.,by using objects or drawing to
represent the problem.
K.OA.3
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. 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 drawing
or equation (e.g., 5=2+3 and 5=4+1).
Mathematical Practices
56
K-12.MP.1
K-12.MP.2
K-12.MP.4
K-12.MP.6
K-12.MP.8
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Model with mathematics.
Attend to precision.
Look for an express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Subtraction describes the process of separating from a whole.
Pictures and manipulatives can be used to model subtraction.
Essential Questions
What happens when we take items away from a group?
Why do I need to know how to subtract?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 To form subtraction sentences from subtraction stories and write them using the symbols - and =.
 Mathematical vocabulary: minus, left, how many more
 Compare sets by one-to-one correspondence, and then write subtraction sentences to represent the
subtraction situation.
 Fluency with subtraction facts to 5.
Students will be able to:
 Use symbols and numerals to write number sentences.
 Represent subtraction stories with subtraction sentences.
 Compare two sets and show the number sentence to answer how many more.
Integration
ELA- Dr. Seuss- use stories for subtraction manipulatives or to write number sentences (ex. Fox in Sox)
Art- draw number stories
57
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 18: Subtraction Stories
www.primarygames.com
www.funbrain.com
www.gamequarium.com
Language Arts Integration
Ten In the Bed by Penny Dale
Five Little Monkeys jumping on the Bed by Eileen Christelow
Math Journal
Suggested Resources
Math In Focus Resources Chapter 18: Subtraction Stories
Numeral Cards
Symbol Cards
Connecting Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
58
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 19:
Measurement
Summary and Rationale
Prior to this unit, the basis of measurement instruction is on comparing objects and situations. During this
unit, children engage using actual measurement tools during physical activities, such as comparing the
weights of two objects on a balance. Children continue the uses of nonstandard units to allow them to
think about the effect of the size of the units on the final measure.
Measurement involving nonstandard units reinforces the skill of estimation. Introducing capacities and
time also forms the basis of volume and more complex duration situations in the future.
Pacing
Two (2) weeks
Standards
Counting and Cardinality
K.CC.3
Know number names and the count sequence. Write numbers from 0 to 10. Represent a
number of objects with a written numeral 0-10 (with 0 representing a count of no objects).
K.CC.6
Compare numbers. 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. (Include groups with up to 10 objects.)
Measurement and Data
K.MD.1
Describe and compare measurable attributes. Describe measurable attributes of objects, such
as length or weight. Describe several measurable attributes of a single object.
K.MD.2
Directly and compare two measurable attributes. 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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
59
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Weight cannot always be judged by size.
Larger does not always mean heavier.
Essential Questions
How do we tell which object is heavier?
Does a larger size always lead to a heavier object?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 To compare weights of objects by using a balance scale.
 To measure and compare weights using nonstandard units.
 To compare capacities of containers using the terms holds more, holds less, and hold the same
amount.
 Compare events and decide which takes more or less time.
Students will be able to:
 Compare weights using nonstandard units.
 Compare containers according to capacity.
 Use the terms holds more, holds less, and hold the same amount.
 Compare events according to duration.
Integration
Science: measure ingredients for experiments-measure objects for theme of the month (ex. Insectsflowers)
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 19: Measurement
www.apples4theteacher.com
www.gamequarium.com
www.funbrain.com
www.primarygames.com
Language Arts Integration
60
Biggest, Strongest, Fastest by Steve Jenkins
Math Journal
Suggested Resources
Math In Focus Resources Chapter 19: Measurement
Balance Scale
Connecting Cubes
www.hmhlearning.com
www.hmheducation.com/mathinfocus
61
Nutley Public Schools
Mathematics
Grade Kindergarten
Unit 20:
Money
Summary and Rationale
Using money is a real-world application of concepts taught in the numbers and operations strand of
mathematics. In this unit, children apply counting strategies when counting coins such as pennies, nickels,
and dimes. Learning to count coins also complements what children already know about sorting, likeness,
and differences, counting, adding, subtracting, and using number sentences.
Pacing
One (1) week
Standards
Counting and Cardinality
K.CC.2
Know number names and the count sequence. Count forward beginning from a given
number within the known sequence (instead of having to begin at 1).
K.CC.4
Count to tell the number of objects. Understand the relationship between numbers and
quantities; connect counting to cardinality.
K.CC.5
Count to tell the number of objects. Count to answer “how many?” questions with as many
as 20 things arranged in a line, a rectangular array, or a circle, or 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.
Operations and Algebraic Thinking
K.OA.1
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. Represent addition and subtraction up to 10 with objects,
fingers, mental images, drawings (drawings need not show details, but should show the
mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations,
expressions or equations.
K.OA.2
Solve subtraction word problems and subtract within 10,e.g.,by using objects or drawing to
represent the problem.
K.OA.3
Understand addition as putting together and adding to, and understand subtraction as taking
apart and taking apart from. 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 drawing
or equation (e.g., 5=2+3 and 5=4+1).
Mathematical Practices
K-12.MP.4 Model with mathematics.
62
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Each coin represents a specific money value.
Coins have values and are combined to create larger amounts of money.
Essential Questions
Why is it important to identify the coins?
Why do I need to know how to count the value of coins?
Why do we have money?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn
 To identify coins by appearance and value: penny, nickel, dime, quarter
 Other vocabulary: cent, change
 To add coins to pay for objects with a combined value of up to 10 cents.
 The concept of receiving change for a payment.
Students will be able to:
 Identify penny, nickel, dime, and quarter and the value of each coin.
 Add coins up to 10 cents.
 Use pennies to buy up to three objects (up to 10 cents).
 Identify and recognize different combinations of coins that make up 10 cents.
Integration
Social Studies: Presidents on the coins and symbols on the back of the coins
Science: what are coins made of
Dramatic Play: set up a store or post office to practice with coins
Art: coin rubbings
63
Music: coin songs
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 20: Money
www.moneyinstructor.com
www.usmint.gov
Language Arts Integration
The Coin Counting Book by Rozanne Williams
Benny’s Pennies by Pat Brisson
Henry’s Pennies by Louise Greep McNamara
Math Journal
Suggested Resources
Math In Focus Resources Chapter 20: Money
Teacher Activity Cards
Plastic Coins
www.hmhlearning.com
www.hmheducation.com/mathinfocus
64
Nutley Public Schools
Grade 1
65
Nutley Public Schools
Mathematics
Grade 1
Unit 1:
Numbers to 10
Summary and Rationale
In this unit, children use countable objects to develop the association between the physical representation
of the number, the number symbol, and the number word. Besides counting objects in a set, and creating a
set within a given number of objects, children also differentiate between numbers of objects in sets, a skill
that forms a basis for number comparison. They learn to recognize relationships between numbers, such
as 1 more than and 1 less than.
Using countable objects and a math balance, children are led to see how a given number can be made
from two smaller numbers. The part-whole analysis through number bonds forms the basis for the basis of
the concept of adding two numbers to give another number.
Children add by counting on and by using number bonds. They learn to construct addition stories from
pictures and solve real-world problems by writing addition sentences.
Children use strategies such as the take-away concept, number bonds, counting on, and counting back to
identify and learn subtraction facts. They write subtraction sentences to represent familiar situations, and
begin to see the inverse relationship between addition and subtraction by using number bonds.
Pacing
Seven (7) weeks
Standards
Number and Operations in Base Ten
1.NBT.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.
1.NBT.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.
Operations and Algebraic Thinking
1.OA.1
Use addition and subtraction within 20 to solve word problems involving situations of
66
1.OA.3
1.OA.4
1.OA.5
1.OA.6
1.OA.7
1.OA.8
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.1
Apply properties of operations as strategies to add and subtract.2 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.)
Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8. Add and subtract within 20.
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
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).
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.
Determine the unknown whole number in an addition or subtraction equation relating 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 = _.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for an express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers to 10 can be counted and compared.
Number bonds can be used to show parts and whole.
Addition can be used to find how many in all.
67
Subtraction can be used to find how may are left.
Subtraction is the opposite of addition.
Essential Questions
What are some strategies for addition?
What are some strategies for subtraction?
Why and when do we add and subtract?
How are addition and subtraction related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will learn:
 Numbers to 10
 Comparative words: same, more, fewer, greater than, less than, more than
 Pattern
 Number bonds, part, whole
 Add, plus (+), equal to (=)
 Addition sentence, addition story
 Subtract, take away, minus (-)
 Subtraction sentence, subtraction story
 Fact family
Students will be able to:
 Count from 0 to 10 objects.
 Read and write 0 to 10 in numbers and words.
 Compare two sets of objects by using one-to-one correspondence.
 Identify the set that has more, fewer, or the same number of objects.
 Identify the number that is greater than or less than another number.
 Make number patterns.
 Use connecting cubes or a math balance to find number bonds.
 Find different number bonds for numbers to 10.
 Count on to add and to subtract.
 Take away to subtract.
 Count back to subtract.
 Use number bonds to add in any order and to subtract.
 Write and solve addition and subtraction sentences.
 Tell addition and subtraction stories about pictures.
 Recognize related addition and subtraction sentences.
 Write fact families and use them to solve real-world problems.
68
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Numbers to 10
Math In Focus On-line Resources Chapter 2: Number Bonds
Math In Focus On-line Resources Chapter 3: Addition Facts to 10
Math In Focus On- line Resources Chapter 4: Subtraction Facts to 10
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 1: Numbers to 10
Math In Focus Resources Chapter 2: Number Bonds
Math In Focus Resources Chapter 3: Addition Facts to 10
Math In Focus Resources Chapter 4: Subtraction Facts to 10
Connecting Cubes
Counters
Counting Tape
Math balance
www.hmhlearning.com
www.hmheducation.com/mathinfocus
69
Nutley Public Schools
Mathematics
Grade 1
Unit 2:
Shapes and Patterns
Summary and Rationale
Children have learned in Kindergarten to identify, name, and describe a variety of plane and solid shapes.
In this unit, children classify and compare plane and solid shapes based on the geometric properties, using
the appropriate vocabulary for describing shapes. They make composite shapes, models, and patterns
with these shapes.
Mathematical concepts in geometry can be related to objects in the real word, so children are encourages
to use basic shapes and spatial reasoning to model objects in their environment.
Pacing
Two (2) weeks
Standards
Geometry
1.G.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.
1.G.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.1
1.G.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
70
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Plane and solid shapes are found in the real-world and can be compared by their geometric attributes and
properties..
Patterns can be identified and compared by looking at the plane and solid shapes that are involved.
Essential Questions
What is a pattern?
Where are flat shapes found in the real-world?
Where are solid shapes found in the real-world?
Where are patterns found in the real-world?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Flat and solid shape names
 Parts: side, corner
 Descriptive words: size, shape, color
 Comparative words: alike, different
 Sort
 Stack, slide, roll
 Repeating pattern
Students will be able to:
 Identify, classify, and describe plane and solid shapes.
 Make same and different shapes.
 Combine and separate plane and solid shapes.
 Identify plane and solid shapes in real life.
 Use plane and solid shapes to identify, extend and create patterns.
Integration
71
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 5: Shapes and Patterns
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 5: Shapes and Patterns
Teacher Activity Cards
Attribute Blocks
Geometric Solids
www.hmhlearning.com
www.hmheducation.com/mathinfocus
72
Nutley Public Schools
Mathematics
Grade 1
Unit 3:
Numbers and Facts to 20
Summary and Rationale
Ordering numbers and number positions with ordinal numbers are key number concepts. In this unit,
children use ordinal numbers from first to tenth to describe order and position of objects and persons.
They enhance their vocabulary with position words including in front of, before, and after to describe
position of something relative to another.
As an introduction to the concept of place value, children count to 20 using pictorial representations of
concrete objects. They recognize numbers 11 to 20 as one group of ten and a particular number of ones.
This is a key stage and sets the foundation for developing the idea of tens and ones and being able to
make sense of two-digit numbers.
Children compare numbers and establish number relationships such as greater than and less than. They
identify patterns from these number relationships and extend the patterns.
Children learn more strategies for addition and subtraction as they solve problems that include numbers
between 10 and 20. These strategies include grouping into tens and ones, number bonds, and using double
facts to add and subtract. Children use addition and subtraction sentences to solve real-world problems.
Pacing
Six (6) weeks
Standards
Numbers and Operations in Base 10
1.NBT.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.
1.NBT.2
Understand that the two digits of a two-digit number represent amounts of tens and ones.
1.NBT.2a
Understand that 10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.2b Understand that the numbers from 11 to 19 are composed of a ten and one, two, three, four,
five, six, seven, eight, or nine ones.
1.NBT.3
Compare two two-digit numbers based on meanings of the tens and ones digits, recording
the results of comparisons with the symbols >, =, and <.
1.NBT.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
73
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.
Operations and Algebraic Thinking
1.OA.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.1
1.OA.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.
1.OA.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8. Add and subtract within 20.
1.OA.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).
1.OA.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.
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating 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 = _.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers and words can be used to describe order and position.
Numbers to 20 can be counted, ordered, and compared.
74
Different strategies can be used to add and subtract.
Essential Questions
What numbers can be used to describe order?
What numbers can be used to describe position?
Why and when do we add?
Why and when do we subtract?
How are addition and subtraction related?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Ordinal Numbers
 Position Words
 Counting to 20
 Place Value
 Comparing Numbers
 Making Patterns
 Ordering Numbers
 Addition and Subtraction Strategies
 Addition and Subtraction Facts
Students will be able to:
 Use ordinal numbers.
 Use position words to name relative positions.
 Count on from 10 to 20.
 Read and write 11 to 20 in numbers and words.
 Use a place value chart to show numbers up to 20.
 Show objects up to 20 as tens and ones.
 Compare numbers to 20.
 Order numbers by making number patterns.
 Use different strategies to add one and two-digit numbers.
 Subtract a one-digit from a two-digit number with and without regrouping.
 Solve real world problems.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 6: Ordinal Numbers and Position
75
Math In Focus On-line Resources Chapter 7: Numbers to 20
Math In Focus On-line Resources Chapter 8: Addition and Subtraction Facts to 20
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 6: Ordinal Numbers and Position
Math In Focus Resources Chapter 7: Numbers to 20
Math In Focus Resources Chapter 8: Addition and Subtraction Facts to 20
Teacher Activity Cards
Connecting cubes
Number cubes
Unit cubes
Ten frames
Tens rods
Place-value chart
www.hmhlearning.com
www.hmheducation.com/mathinfocus
76
Nutley Public Schools
Mathematics
Grade 1
Unit 4:
Measurement and Data
Summary and Rationale
As an introduction to measuring length, children compare the lengths of two objects both directly (by
comparing them with each other) and indirectly (by comparing both with a third object), and they order
several objects according to length. Children use common objects as non-standard units to measure and
compare length. Their spatial awareness is exercised by having tem recognize vertical length as height.
Children integrate their understanding of numbers and measurement through an introduction to weight.
They compare weight and learn to measure weight suing a balance and use common objects as nonstandard units to measure and compare weight.
Children’s counting skills are utilized in the collection of data. They are led to see how the data collected
can be compiled into picture graphs or bar graphs. The strategy of using tally marks is a way to organize
data better. Children interpret and make sense of the data from the diagrams.
Pacing
Six (6) weeks
Standards
Measurement and Data
1.MD.1
Order three objects by length; compare the lengths of two objects indirectly by using a third
object.
1.MD.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.
Operations and Algebraic Thinking
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating 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 = _.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.5 Use appropriate tools strategically.
77
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The weight of things can be compared and measured with nonstandard units.
Using different nonstandard units may give different measurements for the same item.
Most measurements have some degree of uncertainty.
Picture graphs, tally charts, and bar graphs can be used to display data.
Essential Question
Why do we measure?
Why do we need standardized units of measurement?
How does what we measure influence how we measure?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 start line
 about
 unit
 as heavy as
Students will be able to:
 Compare two lengths using the terms tall/taller, long/longer, and short/shorter.
 Compare two lengths by comparing each with a third length.
 Compare more than two lengths using the terms tallest, longest, and shortest.
 Use a common starting point when comparing lengths.
 Measure lengths using nonstandard units.
 Compare the weight of two things using the terms heavy, heavier, light, lighter, and as heavy as.
78












Compare the weight of more than two objects using the terms lightest and heaviest.
Use a nonstandard object to find the weight of things (such as a balance).
Compare weight using a nonstandard object as a unit of measurement.
Use the term “unit” to describe length and when writing the weight of things.
Explain why there is a difference in a measurement when using different nonstandard units.
Count measurement units in a group of ten and ones.
Collect and organize data.
Show data as a picture graph.
Draw picture graphs.
Make a tally chart.
Show data in a bar graph.
Interpret data shown in a picture graph and a bar graph.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 9: Length
Math In Focus On-line Resources Chapter 10: Weight
Math In Focus On-line Resources Chapter 11: Picture Graphs and Bar Graphs
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 9: Length
Math In Focus Resources Chapter 10: Weight
Math In Focus Resources Chapter 11: Picture Graphs and Bar Graphs
Teacher Activity Cards
Balance Scale
Connecting Cubes
Number Cubes
Calculator
www.hmhlearning.com
www.hmheducation.com/mathinfocus
79
Nutley Public Schools
Mathematics
Grade 1
Unit 5:
Place Value
Summary and Rationale
Counting on to 40 is a smooth progression from where children stopped, at 20, in previous units. The
general form in the numbers in words from 20 to 40 gives children a sense of how the numbers beyond 40
may be written.
Children us place-value charts to show numbers to 40. The place-value chart enables children to make
comparisons between two or more numbers, when tens are different or when tens are equal. In being able
to compare two numbers, children apply this knowledge to ordering numbers in ascending and descending
order. With children familiar with the counting, comparing, and ordering of numbers to 40, they are then
able to identify number patterns that occur through addition and subtraction. All of these activities build
the foundation that children will rely on when they learn about numbers to 100.
Pacing
Two (2) weeks
Standards
Numbers and Operations in Base Ten
1.NBT.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.
1.NBT.2
Understand that the two digits of a two-digit number represent amounts of tens and ones.
1.NBT.2a
Understand that 10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.2c
Understand that 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).
1.NBT.3
Compare two two-digit numbers based on meanings of the tens and ones digits, recording
the results of comparisons with the symbols >, =, and <.
Operations and Algebraic Thinking
1.OA.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating 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 = _.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.5 Use appropriate tools strategically.
80
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Place-value charts can be used to show numbers to 40.
Missing numbers in a number pattern can sometimes be identified by adding or subtracting.
Essential Questions
How does finding patterns help in counting?
How are some patterns created?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Use of Place-Value Chart
 Use of counting tape.
Students will be able to:
 Count on from 21 to 40.
 Read and write 21 to 40 in numbers and words.
 Use a place-value chart to show numbers up to 40.
 Show objects up to 40 as tens and ones.
 Use a strategy to compare numbers to 40.
 Order numbers to 40.
 Find the missing numbers in a number pattern.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 12: Numbers to 40
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 12: Numbers to 40
81
Connecting cubes
Place-value chart
www.hmhlearning.com
www.hmheducation.com/mathinfocus
82
Nutley Public Schools
Mathematics
Grade 1
Unit 6:
Addition and Subtracting Regrouping
Summary and Rationale
In this unit, children progress to the standard vertical form of addition and subtraction of numbers based
on place value. In teaching children to regroup, they are encouraged to use place-value charts to correctly
align the digits and to record the regrouping process. The frequent use of place-value charts leads children
away from a dependence on concrete representations which are not feasible when later dealing with larger
numbers. Children are also reminded that addition can be used to check subtraction. Children also solve
real-world problems involving addition and subtraction.
Children use number bonds to add and subtract mentally. They add and subtract mentally by also using
double facts, and using the strategies of add the ones, add the tens, subtract the ones, and subtract the
tens.
Pacing
Four (4) weeks
Standards
Number and Operations in Base Ten
1.NBT.2
Understand that the two digits of a two-digit number represent amounts of tens and ones.
1.NBT.2a
Understand that 10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.2c
Understand that 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).
1.NBT.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.
1.NBT.5
Given a two-digit number, mentally find 10 more or 10 less than the number, without having
to count; explain the reasoning used.
1.NBT.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.
Operations and Algebraic Thinking
1.OA.1
Use addition and subtraction within 20 to solve word problems involving situations of
83
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.1
1.OA.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.
1.OA.3
Apply properties of operations as strategies to add and subtract.2 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.)
1.OA.4
Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8. Add and subtract within 20.
1.OA.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.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).
1.OA.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.
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating 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 = _.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Whole numbers can be added and subtracted with or without regrouping.
Number bonds can help you add and subtract mentally.
84
Essential Questions
What are some strategies for adding mentally?
What are some strategies for subtracting mentally?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Regroup
 Mental math
 Doubles fact
Students will be able to:
 Add a two-digit number and a one-digit number without and with regrouping.
 Add two-digit numbers without and with regrouping.
 Subtract a one-digit number from a two-digit number without and with regrouping.
 Subtract a two-digit number from a two-digit number without and with regrouping.
 Add three one-digit numbers.
 Use addition and subtraction facts to solve real-world problems.
 Mentally add and subtract one-digit numbers.
 Mentally add a one-digit number to a two-digit number.
 Mentally add a two-digit number to tens.
 Mentally subtract a one-digit number from a two-digit number.
 Mentally subtract tens from a two-digit number.
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 13: Addition and Subtraction
Math In Focus On-line Resources Chapter 14: Mental Math Strategies
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 13: Addition and Subtraction
Math In Focus Resources Chapter 14: Mental Math Strategies
Base ten blocks
Counters
Number cards
Number cubes
Number bonds
Place-value chart
Counting tape
Math balance and weights
www.hmhlearning.com
www.hmheducation.com/mathinfocus
85
Nutley Public Schools
Mathematics
Grade 1
Unit 7:
Calendar and Time
Summary and Rationale
A mathematical concept that is associated with time is the ability to arrange events in order using a
calendar or a clock. In this unit, children learn to read a calendar in terms of the days of the week and the
months of a year and to write the date. They also learn to read and show time to the hour and to the half
hour on a clock.
With the ability to read both the calendar and the clock, children are able to relate the notion of time, day,
month, and year to their everyday lives.
Pacing
Two (2) weeks
Standards
Measurement and Data
1.MD.3
Tell and write time in hours and half-hours using analog and digital clocks.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Calendars are used to show days, weeks, and months of a year.
Clocks are used to read time of the day.
86
Essential Question
What are the characteristics of morning, afternoon, and evening?
Why do we need to know what time it is?
How do we tell time to the hour?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 calendar
 seasons
 terms: o’clock, minute hand, hour hand, half past, half hour
Students will be able to:
 Read a calendar.
 Name the days of the week, months of the year, and seasons.
 Write the date.
 Use the term o’clock to tell the time to the hour.
 Read and show time to the hour on a clock.
 Read time to the half hour.
 Use the term half past
 Relate time to daily activities.
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 15: Calendar and Time
Language Arts Integration
Suggested Resources
Math In Focus On-line Resources Chapter 15: Calendar and Time
Calendar
Demonstration Clock
www.hmhlearning.com
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87
Nutley Public Schools
Mathematics
Grade 1
Unit 8:
Numbers to 100
Summary and Rationale
In this unit, children learn to count on from 40 to 100. In knowing that a two-digit number is made up of
tens and ones, children count in tens before counting the remaining ones when identifying a two-digit
number. Children learn to represent numbers to 100 using place-value charts and strategies to compare
and order them. Once children can order numbers, they observe number patterns and identify missing
numbers patterns with numbers to 100.
Children extend the vertical form of addition and subtraction to numbers to 100 through two methods:
counting on/back and using place-value charts. Through these methods, children add and subtract with
and without regrouping using numbers to 100.
Pacing
Four (4) weeks
Standards
Number and Operations in Base Ten
1.NBT.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.
1.NBT.2
Understand that the two digits of a two-digit number represent amounts of tens and ones.
1.NBT.2a
Understand that 10 can be thought of as a bundle of ten ones — called a “ten.”
1.NBT.2c
Understand that 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).
1.NBT.3
Compare two two-digit numbers based on meanings of the tens and ones digits, recording
the results of comparisons with the symbols >, =, and <.
1.NBT.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.
1.NBT.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.
Operations and Algebraic Thinking
88
Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by
finding the number that makes 10 when added to 8. Add and subtract within 20.
1.OA.5
Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).
1.OA.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.
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating 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 = _.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
1.OA.4
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers to 100 can be added and subtracted with and without regrouping.
Regrouping is needed when the addition of ones exceeds nine, and when the subtraction of ones cannot be
carried out because of insufficient ones.
Essential Questions
What are some strategies for addition and subtraction using numbers to 100?
When is regrouping necessary in addition?
When is regrouping necessary in subtraction?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
89
Students will know or learn:
Students will be able to:
 Count on from 41 to 100.
 Read and write 41 to 100 in numbers and in words.
 Use a place-value chart to show numbers up to 100.
 Show objects up to 100 as tens and ones.
 Use a strategy to compare numbers to 100.
 Order numbers to 100.
 Find the missing numbers in a number pattern.
 Add a two-digit number and a one-digit number without and with regrouping.
 Add two-digit numbers without and with regrouping.
 Subtract a one-digit number from a two-digit number without and with regrouping.
 Subtract two-digit numbers without and with regrouping.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 16: Numbers to 100
Math In Focus On-line Resources Chapter 17: Addition and Subtraction to 100.
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 16: Numbers to 100
Math In Focus Resources Chapter 17: Addition and Subtraction to 100.Base ten blocks
Number lines
Connecting Cubes
Hundred chart
Number cubes
Place-value chart
Unit cubes
Base ten blocks
Ten rods
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
90
Nutley Public Schools
Mathematics
Grade 1
Unit 9:
Multiplication and Division
Summary and Rationale
In this unit, multiplication (but not the word) is linked to the part-whole meaning of addition. Joining
groups (parts) to find a total (whole) and the use of double facts and addition properties together form an
important basis for understanding multiplication as repeated addition. Children ass the same numbers to
understand the concept of multiplication.
Division is the opposite of multiplication. Numbers sense concepts such as counting and comparing
numbers form the groundwork of division. Children explore both meanings of division: finding the
number of equal groups of a given size and finding the size of a given number of groups. The use of
models is important at this grade level because children may need to rely heavily on manipulatives to
comprehend the two meanings of division. Children distribute items equally to understand the concept of
sharing equally and distribute items into equal groups to understand the concepts of dividing into equal
groups.
Pacing
Two (2) weeks
Standards
Geometry
1.G.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
91
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Multiplication is the same as adding equal groups.
Dividing is the same as sharing things equally or putting things in equal groups.
Division is the opposite of multiplication.
Essential Questions
How can multiplication be modeled?
How can division be modeled?
How are multiplication and division related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Terminology: same, groups, each, share, equally
Students will be able to:
 Use objects or pictures to find the total number of items in groups of the same size.
 Relate repeated addition to the concept of multiplication.
 Use objects and pictures to find the number of items in each group when sharing equally.
 Relate sharing equally to the concept of division.
 Use objects and pictures to show the concept of division as finding the number of equal groups.
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 18: Multiplication and Division
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 18: Multiplication and Division
Connecting Cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
92
Nutley Public Schools
Mathematics
Grade 1
Unit 10:
Money
Summary and Rationale
In this unit, children recall their knowledge of the penny, nickel, dime, and quarter. Children count the
value of different coins by applying the strategies of counting on and skip-counting from the coin of
greatest value by first arranging the coins in order. Children use addition and subtraction in real-world
situations that involve money. They first interpret the question, form addition and subtraction sentences
accordingly, and then apply the strategies of mental calculation and place-value to find the solution.
Pacing
Two (2) weeks
Standards
Operations and Algebraic Thinking
1.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Penny, nickel, dime, and quarter are coins that can be counted and exchanged.
93
To count the value of different coins, arrange the coins in order, begin with the coin of the greatest value,
and then count on or skip count from that coin to find the total value.
Money values can be added and subtracted.
Essential Questions
When and why do we use money?
How do we find the total value of a group of different coins?
How do you calculate change?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 New money terms: value, exchange
Students will be able to:
 Recognize and name penny, nickel, dime, quarter, and the cents symbol.
 Skip-count to find the value of a collection of coins.
 Exchange one coin for a set of coins of equal value.
 Use different combinations of coins less than 25 cents to buy things.
 Count money in cents up to $1 using the “count on” strategy.
 Choose the value of coins when buying items.
 Use different combinations of coins to show the same value.
 Add to find the cost of items.
 Subtract to find the change.
 Add and subtract money in cents (up to $1).
 Solve real world problems involving addition and subtraction of money.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 19: Money
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 19: Money
Plastic coins
Hundred chart
www.hmhlearning.com
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94
Nutley Public Schools
Grade 2
95
Nutley Public Schools
Mathematics
Grade 2
Unit 1:
Numbers to 1,000
Summary and Rationale
In this unit, children extend their concept of numbers, and learn how to count, read, and write up to 1,000.
Base- ten blocks, place-value charts, and number lines are used to develop the association between the
physical representation of the number, the number symbol, and the number word. The concept of placevalue is extended to the hundreds place value.
Children apply addition concepts to three-digit numbers. They use multiple regroupings by using base-ten
blocks and place-value charts as concrete representations, which allow them to visualize addition with
regrouping in the ones and tens place.
Children perform multi-digit subtraction with and without regrouping. They use base-ten blocks and
place-value charts as concrete representations, which aid them in visualizing the regrouping of tens as
ones, hundreds as tens, and hundreds as tens and ones. Another method of subtraction introduced in this
unit is subtraction across zeros, through which regrouping is done in the hundreds first, followed by tens
and ones.
Bar models provide a useful pictorial representation of sets as parts making up a whole. Children learn
strategies such as adding on and taking away sets represented by bar models to solve addition and
subtraction real-world problems. Children label the bars with words as well as numbers, so they can use
bar models to illustrate a problem, indicating in the model the known and the unknown parts of the whole.
Comparing sets using bar models helps children to see clearly whether to add or subtract to solve a given
problem.
Pacing
Eight (8) weeks
Standards
Operations and Algebraic Thinking
2.OA.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
2.OA.3
Work with equal groups of objects to gain foundations for multiplication
96
Number and Operations in Base Ten
2.NBT.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.
2.NBT.1a
Understand that 100 can be thought of as a bundle of ten tens — called a “hundred.”
2.NBT.1b Understand that 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.NBT.2
Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.3
Read and write numbers to 1000 using base-ten numerals, number names, and expanded
form.
2.NBT.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.
2.NBT.5
Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
2.NBT.6
Add up to four two-digit numbers using strategies based on place value and properties of
operations.
2.NBT.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.
2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties
of operations.1
Measurement and Data
2.MD.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.
2.MD.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Number concepts include demonstrating numbers in different ways.
97
Fluency in adding and subtracting basic facts is gained by discovering patterns and using strategies in
finding sums and differences.
Relationships are developed within addition and subtraction combinations.
Three digit numbers can be added with and without grouping.
Three digit numbers can be subtracted with and without regrouping.
Addition and subtraction can be shown with bar models.
Exploring place value allows students to see how patterns continue.
Three digit numbers are written in standard and expanded form.
Essential Questions
How do you use place value to describe numbers in different ways?
How can you use place value to write 3-digit numbers?
How can you compare 3-digit numbers?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Hundred
 Hundreds
 Thousand
 Standard form
 Expanded form
 Word form
 Greater than (>)
 Less than (<)
 Join
 Set
 Take away
 Compare
Students will be able to:
 Use base ten blocks and a place-value chart to recognize, read, write, and represent numbers to 1,000.
 Read and write numbers to 1,000 in standard form, expanded form, and word form.
 Use base ten blocks to compare numbers.
 Compare numbers using the terms greater than and less than.
 Compare numbers using symbols > and <.
98














Order three digit numbers.
Identify the greatest number and the least number.
Identify number patterns.
Use base-ten blocks to add numbers without and with regrouping.
Add up to three-digit numbers without and with regrouping.
Solve real world addition and subtraction problems.
Use base-ten blocks to add numbers without and with regrouping.
Subtract from three-digit numbers without and with regrouping.
Apply the inverse operations of addition and subtraction.
Use bar models to solve addition and subtraction problems.
Model addition as joining sets.
Model subtraction as taking away.
Model addition and subtraction as comparing sets.
Use bar models to solve two-step addition and subtraction problems.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Numbers to 1,000
Math In Focus On-line Resources Chapter 2: Addition up to 1,000
Math In Focus On-line Resources Chapter 3: Subtraction up to 1,000
Math In Focus On-line Resources Chapter 4: Using Bar Models: addition and Subtraction
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 1: Numbers to 1,000
Math In Focus Resources Chapter 2: Addition up to 1,000
Math In Focus Resources Chapter 3: Subtraction up to 1,000
Math In Focus Resources Chapter 4: Using Bar Models: Addition and Subtraction
Base-ten blocks
Place-value chart
Place-value mat
Numbered dice
Ten-sided dice
Number lines
Number cards
Connecting cubes
Counters
www.hmhlearning.com
www.hmheducation.com/mathinfocus
99
Nutley Public Schools
Mathematics
Grade 2
Unit 2:
Multiplication and Division
Summary and Rationale
In this unit, children move to the pictorial and symbolic phases of multiplication and division through the
emphasis on equal groups. Multiplication is used to find the number of items in a number of equal groups.
Division is the process of sharing a number of items among a number of groups either by finding the
number of items in each group or by finding the number of equal groups that can be formed. The
strategies of repeated addition and repeated subtraction are reviewed in this unit.
Children learn the multiplication facts of 2, 5, and 10 using skip-counting and dot-paper strategies.
Pictures and fingers illustrate the skip-counting strategy related to computation in multiplication. Using
the skip-counting strategy, each finger is used to represent a specific value. Using dot paper for
multiplication, each column represents the number of groups while each row represents the number of
items in each group.
Children also learn to use related multiplication facts to divide. Division is conceptualized as the inverse
of multiplication and as the equal sharing of items.
Pacing
Four (4) weeks
Standards
Operations and Algebraic Thinking
2.OA.3
Work with equal groups of objects to gain foundations for multiplication
2.OA.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.
Numbers and Operations in Base Ten
2.NBT.2
Count within 1000; skip-count by 5s, 10s, and 100s.
Mathematical Practices
K-12.MP.1
Make sense of problems and persevere in solving them.
K-12.MP.2
Reason abstractly and quantitatively.
K-12.MP.3
Construct viable arguments and critique the reasoning of others.
K-12.MP.4
Model with mathematics.
K-12.MP.5
Use appropriate tools strategically.
K-12.MP.6
Attend to precision.
100
K-12.MP.8
Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Multiplication and division use equal groups.
Any even number can be represented by two equal numbers.
Multiplication is used to find the number of items in a number of equal groups.
Division is the process of sharing a number of items among a number of groups either by finding the
number of items in each group or by finding the number of equal groups that can be formed.
Pictures and fingers illustrate the skip-counting strategy related to multiplication. Using the skip-counting
strategy, each finger is used to represent a specific value.
Using dot paper for multiplication, each column represents the number of groups while each row
represents the number of items in each group.
Related multiplication facts can be used to divide.
Division, the equal sharing of items, is the opposite of multiplication.
Essential Questions
What is multiplication?
What are some strategies for multiplication?
What is division?
What are some strategies for division?
How are multiplication and division related?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
101
Objectives
Students will know or learn:
 Times
 Equal groups
 Multiply
 Repeated addition and subtraction.
 Share
 Divide
 Multiplication and division sentence
 Multiplication and division story
 Odd and even numbers
Students will be able to:
 Used equal groups and repeated addition to multiply.
 Make multiplication stories about pictures.
 Make multiplication sentences.
 Divide to share equally.
 Divide by repeated subtraction of equal groups.
 Make groups of 2 to find odd and even numbers
 Solve multiplication and division word problems.
 Use base-ten blocks and a place-value chart to recognize, read, write, and represent numbers to 1,000.
 Count by 1s, 10s, and 100s to 1,000.
 Read and write numbers to 1,000 in standard form, expanded form, and word form.
 Skip count and use dot paper to multiply by 5 and 10.
 Use known multiplication facts to find new multiplication facts.
 Identify related multiplication facts.
 Use related multiplication facts to find related division facts.
 Write a multiplication sentence and a related division sentence.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 5: Multiplication and Division
Math In Focus On-line Resources Chapter 6: Multiplication Tables of 2, 5, and 10.
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 5: Multiplication and Division
Math In Focus Resources Chapter 6: Multiplication Tables of 2, 5, and 10
Connecting cubes
Counters
Base-ten blocks
Place-value chart
Place-value mat
Dot paper
Hundreds chart
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102
Nutley Public Schools
Mathematics
Grade 2
Unit 3:
Metric Measurement
Summary and Rationale
There are two basic systems of measurement in the United States- customary and metric. The basic units
of length in the metric system are meters and centimeters. In this unit, children estimate and measure
medium and short lengths using the standard metric units meters (m) and centimeters (cm). The meter
stick and centimeter ruler are used to illustrate length as a concept of measure to determine how long or
short an object is. The lengths of curved lines are measured with the help of a piece of string which is
paced along the curved line and then measured with a ruler. Children also draw lines of specific lengths.
Children learn that mass is a concept of measure to describe how heavy an object is. They estimate and
measure the mass of objects using the standard metric units of kilograms (kg) and grams (g). Children
also read the masses of objects from measuring scales and use a balance to determine, compare, add and
subtract the masses of objects.
Children learn that volume is the amount of liquid in a container. In Singapore Math, there is a distinction
made between capacity of container (amount of space in a container) and volume of liquid (amount of
liquid in a container). This distinction is not made in this unit because the emphasis here is on the amount
of volume of liquids, and not containers. Children compare volumes of liquids in identical and nonidentical containers and learn that the metric unit of measure for volume is liters (L). They measure the
volume of liquid in a container by using one or more measurement cups. The liquid is poured into the
measuring cup(s) to determine its volume, regardless of the capacity of the original container. The volume
of liquid in different containers is compared by comparing the number of measuring cups needed to
contain all the liquid.
Pacing
Six (6) weeks
Standards
Measurement and Data
2.MD.1
Measure the length of an object by selecting and using appropriate tools such as rulers,
yardsticks, meter sticks, and measuring tapes.
2.MD.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.
2.MD.3
Estimate lengths using units of inches, feet, centimeters, and meters.
103
2.MD.4
Measure to determine how much longer one object is than another, expressing the length
difference in terms of a standard length unit.
2.MD.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.
2.MD.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.
Numbers and Operations in Base-Ten
2.NBT.5
Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
2.NBT.6
Add up to four two-digit numbers using strategies based on place value and properties of
operations.
2.NBT.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.
Mathematical Connections
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Tools can be used to estimate and measure the lengths of objects to the nearest centimeter and meter.
Centimeter rulers and metric sticks can be used to measure and compare how long and how tall things are.
Mass is a concept of measure to describe how heavy an object is.
A scale can be used to measure and compare masses in kilograms and grams.
Volume is the amount of liquid in a container.
Liters can be used to measure volume.
Standard and non-standard units of measure are used to find measurements.
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Units of measurement used often depend of the size of an object.
Essential Question
What are some of the tools that can be used to estimate and measure the length of an object?
What are some of the tools that can be used to estimate and measure mass?
What are some of the tools that can be used to estimate and measure volume?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Meter (m)
 Width
 Length
 Unit
 Height
 Meter stick
 Centimeter (cm)
 Kilogram (kg)
 Mass
 Measuring scale
 Gram (g)
 Liter (L)
Students will be able to:
 Use a metric stick to estimate and measure length.
 Compare lengths.
 Find the difference in length of objects.
 Use a centimeter ruler to compare lengths.
 Draw a line of a given length.
 Use a centimeter ruler to measure and compare lengths of objects.
 Find the difference in centimeters in lengths of objects.
 Solve one-step and two-step problems involving length.
 Draw models to solve real-world problems.
 Use a measuring scale to measure mass in kilograms and grams.
 Compare and order masses.
 Use a measuring scale to measure mass in grams.
 Compare and order masses in grams.
 Use bar models to solve problems about mass.
 Explore and compare volume.
 Use liters to estimate, measure, and compare volume.
105

Use bar models, addition, and subtraction to solve real-world problems about volume.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 7: Metric Measurement of Length
Math In Focus On-line Resources Chapter 8: Mass
Math In Focus On-line Resources Chapter 9: Volume
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 7: Metric Measurement of Length
Math In Focus Resources Chapter 8: Mass
Math In Focus Resources Chapter 9: Volume Connecting cubes
Meter stick or measuring tape
Centimeter ruler
Measuring chart
Measuring scale
Balance scale
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106
Nutley Public Schools
Mathematics
Grade 2
Unit 4:
Mental Math and Money
Summary and Rationale
In this unit, children use place-value and number bond strategies such as adding 10 and subtracting extra
ones and adding 100 and subtracting extra tens to help them with mental addition and subtraction. The
number line is used as a visual representation to illustrate the rounding concept, which is an important
application of place value and number sense. Children round numbers to the nearest 10 and estimate sum
and difference to check reasonableness of answers.
The topic of money provides a natural application of place-value and introduction to decimal notation.
Children recognize bills and coins, and their respective values. They use the dot to separate dollars from
cents when writing money amounts in dollars and to exchange dollars as cents and vice versa. Just as in
place value, children compare money from left to right by first comparing the dollars, and then move on
to the cents.
Pacing
Five (5) weeks
Standards
Operations and Algebraic Thinking
2.OA.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
2.OA.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.
Numbers and Operations in Base-Ten
2.NBT.5
Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
2.NBT.6
Add up to four two-digit numbers using strategies based on place value and properties of
operations.
2.NBT.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
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ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2.NBT.8
Mentally, add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a
given number 100-900.
2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties
of operations.1
Measurement and Data
2.MD.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.
2.MD.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?
Mathematical Connections
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Mental math can be used when an exact answer is needed. Estimation can be used when and exact answer
is not needed.
When numbers are added, the result is called the sum. When numbers are subtracted, the result is called
the difference.
Money amounts can be shown and counted using bills and coins.
Values of money can be represented in different ways.
Essential Questions
How do you use the value of coins and bills to find the total amount of a group of money?
How is money compared?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
108
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Sum
 Difference
 Mental addition and subtraction
 Number line
 Round
 Nearest ten
 Estimate
 Reasonable answers
 $1, $5, $10, $20 bills
 Cent sign
 Dollar sign ($)
 Decimal point
Students will be able to:
 Add numbers up to 3-digits mentally with and without regrouping.
 Subtract up to 3-digit numbers mentally with and without regrouping.
 Use a number line to round numbers to the nearest ten.
 Use rounding to estimate sums and differences.
 Estimate to check reasonableness of answers.
 Recognize $1, $5, $10, and $20 bills.
 Show and count money using coins and bills up to $20.
 Write money amounts using $ and the cents symbol.
 Write dollars as cents and cents as dollars.
 Compare amounts of money using tables.
 Use bar models to solve real-world problems involving addition and subtraction of money.
 Solve word problems using $ and cents symbol.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 10: Mental Math and Estimation
Math In Focus On-line Resources Chapter 11: Money
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 10: Mental Math and Estimation
Math In Focus Resources Chapter 11: Money
Number cubes
Number cards
Paper bills
Plastic coins
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109
Nutley Public Schools
Mathematics
Grade 2
Unit 5:
Fractions
Summary and Rationale
In this unit, children use fractions to describe equal parts of a whole. They identify shapes divided into
equal fractional parts, as well as, model and name unit fractions for halves, thirds, and fourths, based on
the number of equal parts a whole is divided into. Bar model drawings are used to show fractional parts
in different ways. Visual models are further used to compare fractional parts and to add and subtract like
fractions.
Pacing
Two (2) weeks
Standards
Geometry
2.G.2
Partition a rectangle into rows and columns of same-size squares and count to find the total
number of them.
2.G.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 describe the whole as two halves,
three thirds, four fourths. Recognize that equal shares of identical wholes need not have the
same shape.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
110
Fractions can be used to describe how equal parts are related to a whole.
Bar models can be used to represent, compare, add, and subtract fractions.
Adding and subtracting like fractions is just like adding and subtracting whole numbers. (Just add or
subtract the top numbers.)
Essential Questions
What is a fraction?
How can bar model drawings be used to represent and compare fractions?
How do you add and subtract like fractions?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Equal and unequal
 Whole
 Fraction
 One-half, one-third, one-fourth
 Unit fraction
 Like fractions
Students will be able to:
 Identify whether a shape is divided into equal fractional parts.
 Read, write, and identify unit fractions for halves, thirds, and fourths.
 Show fractions and a whole using model drawings.
 Compare two or more unit fractions using models of the same size.
 Order two or more unit fractions with or without the use of models of the same size.
 Identify fractions that name more than one equal part of a whole.
 Use models to add and subtract fractions.
 Add and subtract like fractions.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 12: Fractions
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 12: Fractions
111
Connecting cubes
Triangle cards
Paper shapes
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112
Nutley Public Schools
Mathematics
Grade 2
Unit 6:
Customary Measurement and Time
Summary and Rationale
There are two basic systems of measurement in the United States- customary and metric. The basic units
of length in the customary system are feet and inches. In this unit, children learn to estimate and measure
lengths of objects using a foot ruler. To further reinforce children’s understanding of length, they draw
lines of specific lengths and apply these strategies along with addition and subtraction skills to solve realworld one and two-step word problems involving length.
Children learn to read time based on the position of the minute hand on the clock and use skip-counting
strategies to tell how many minutes have passed and to read and write time in hours and minute using
numerals and words. In addition, children will order events by time and determine how much time has
elapsed.
Pacing
Four (4) weeks
Standards
Operations and Algebraic Thinking
2.OA.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
Number and Operations in Base Ten
2.NBT.5
Fluently add and subtract within 100 using strategies based on place value, properties of
operations, and/or the relationship between addition and subtraction.
2.NBT.6
Add up to four two-digit numbers using strategies based on place value and properties of
operations.
2.NBT.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.
113
2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties
of operations.1
Measurement and Data
2.MD.1
Measure the length of an object by selecting and using appropriate tools such as rulers,
yardsticks, meter sticks, and measuring tapes.
2.MD.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.
2.MD.3
Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.4
Measure to determine how much longer one object is than another, expressing the length
difference in terms of a standard length unit.
2.MD.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.
2.MD.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.
2.MD.7
Tell and write time from analog and digital clocks to the nearest five minutes, using a.m.
and p.m.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Standard and non-standard units of measure are used to find measurements. Units of measurement used
often depend of the size of an object.
Every measurement is an estimate. The precision of measurement depends on the size of the unit used to
measure the object. The smaller the unit used, the more precise the measurement.
Tools can be used to estimate and measure the lengths of objects to the nearest inch and foot.
Inch rulers and yardsticks can be used to measure and compare how long and how tall things are.
Time if the day can be shown in different ways.
Duration of an event is determined by reading a clock and comparing the beginning time to the ending
114
time.
Essential Question
What are some of the tools that can be used to estimate and measure the length of an object?
How do you read times on a digital and analog clock?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Foot/feet (ft)
 Ruler
 Yardstick
 Length, width, height
 Inch (in.)
 A.M.
 P.M.
Students will be able to:
 Use a ruler to estimate and measure length.
 Compare lengths.
 Find differences in lengths of objects.
 Use a ruler to measure length to the nearest inch.
 Draw parts of lines of given lengths.
 Use an inch ruler to measure and compare lengths.
 Find the difference in lengths of objects in inches.
 Measure the same objects in inches and feet.
 Solve one and two-step problems involving length.
 Draw bar models to solve real-world problems.
 Use the minute hand to show and tell the number for every five minutes after the hour.
 Show and tell time in hours and minutes.
 Use A.M. and P.M. to show morning, afternoon, or night.
 Order events by time.
 Determine how much time has passed.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 13: Customary Measurement of Length
Math In Focus On-line Resources Chapter 14: Time
Language Arts Integration
Suggested Resources
115
Math In Focus On-line Resources Chapter 13: Customary Measurement of Length
Math In Focus On-line Resources Chapter 14: Time
Foot-long rulers
Measuring tape
Measurement chart
Analog clock
Paper clocks
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116
Nutley Public Schools
Mathematics
Grade 2
Unit 7:
Multiplication Facts and Bar Models
Summary and Rationale
In this unit, children learn the multiplication facts of 3 and 4 using skip-counting and dot-paper strategies.
Since division is the inverse of multiplication, children use related multiplication facts to divide and apply
this inverse relationship to write family facts and division sentences from related multiplication sentences.
Children use bar models to solve real-world multiplication and division problems, particularly those that
involve measurement and money. Multiplication is conceptualized as finding the total number of items,
given the number of groups. Division is conceptualized as sharing or dividing a set of items into equal
groups, so that each group has the same number of items.
Pacing
Four (4) weeks
Standards
Operations and Algebraic Thinking
2.OA.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.
Measurement and Data
2.MD.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.
2.MD.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
117
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Multiplication is finding the total number of items, given the number of groups.
In the skip-counting strategy for multiplication, each skip is used to represent a specific value.
In the dot-paper strategy, each column of dots represents the number of groups while each row of dots
represents the number of items in each group.
Division is sharing or dividing a set of items into equal groups, so that each group has the same number of
items.
Known multiplication facts can be used to find other multiplication and division facts.
Bar models can be used to solve multiplication and division problems.
Essential Questions
In your own words, what is multiplication? Division?
How does the skip-counting strategy for multiplication work?
How does the dot-paper strategy for multiplication work?
How are multiplication and division related and how does this help us with number facts?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Related multiplication facts.
Students will be able to:
 Skip count by 3s and 4s.
 Solve multiplication and division word problems.
 Use dot paper to multiply by 3 and 4.
 Use known multiplication facts to find new multiplication facts.
 Identify related multiplication facts.
 Find division facts using related multiplication facts.
 Write a multiplication sentence and a related division sentence.
118



Use bar models to solve real-world multiplication and division problems.
Write multiplication and division sentences to solve real-world problems.
Use bar models to solve real-world problems on measurement and money.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 15: Multiplication Tables of 3 and 4
Math In Focus On-line Resources Chapter 16: Bar Models
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 15: Multiplication Tables of 3 and 4
Math In Focus Resources Chapter 16: Bar Models
Dot paper
Multiplication chart
Number wheel
Number cards
Number stickers
Number cubes
Counters
Multiplication cards
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119
Nutley Public Schools
Mathematics
Grade 2
Unit 8:
Data
Summary and Rationale
Collecting, organizing, reporting, and interpreting data are important activities related to children’s
everyday experiences. In this unit, children analyze more complex picture graphs in which the reading,
analysis, and interpretation of the graphs involves symbols that may represent more than one item. This
allows symbols to stand for multiples of a number so that larger numbers can be represented. These type
of picture graphs turn out to be more presentable, easier to read and make, and assist children in solving
real-world problems.
Pacing
Two (2) weeks
Standards
Measurement and Data
2.MD.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.
2.MD.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 problems1 using
information presented in a bar graph.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
120
Enduring Understandings
Data can be collected and organized in different charts to display information that can be used to solve
problems.
Picture graphs use pictures to show data about things you can count.
Essential Questions
How do picture graphs represent multiplication?
How can you collect data and organize that data to show information?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Symbol
 Key
 Line plot
Students will be able to:
 Read, analyze, and interpret picture graphs.
 Complete picture graphs.
 Make picture graphs.
 Make a line plot to show data.
 Solve real-world problems using picture graphs.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 17: Picture Graphs
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 17: Picture Graphs
Counters
Picture graph
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121
Nutley Public Schools
Mathematics
Grade 2
Unit 9:
Geometry
Summary and Rationale
In this unit, children recognize, identify, and describe parts of lines and curves that make up plane and
solid shapes. They learn to combine parts of lines and curves to draw plane shapes and to identify,
classify, and count flat and curved surfaces of solid shapes by using their senses of sight and touch.
Children also discover which properties of shapes allow them to slide, stack, or roll.
Two new plane shapes are introduced, the trapezoid and the hexagon. Children combine smaller plane
shapes to make larger ones, and separate larger shapes to make smaller ones. They do the same using
solid shapes to make and deconstruct models. Activities that involve drawing and copying shapes onto dot
and square grid paper act as preparation for learning symmetry and congruence in later grades.
Pacing
Four (4) weeks
Standards
Geometry
2.G.1
Recognize and draw shapes having specified attributes, such as a given number of angles or
a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and
cubes
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Properties of parts of lines, curves, and surfaces can be seen and felt.
Objects with flat surfaces can slide.
122
Objects that have more than one flat surface can be stacked.
Objects that have curved surfaces can be rolled.
Planes and solid shapes can be identified and classified. They can be separated and combined to make
other shapes.
Essential Questions
Which objects can slide? Stack? Roll?
Where do we see plane shapes in the real-world?
Where so we see solid shapes in the real-world?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Part of a line
 Curve
 Flat surface
 Curved surface
 Slide
 Stack
 Roll
 Plane shape
 Hexagon
 Trapezoid
 Figure
 Quadrilateral
 Pentagon
 Angle
Students will be able to:
 Recognize, identify, and describe parts of lines and curves.
 Draw parts of lines and curves.
 Identify, classify, and count flat and curved surfaces.
 Identify solids that can stack, slide, and/or roll.
 Recognize and identify plane shapes. Combine smaller plane shapes to make larger plane shapes.
 Separate larger plane shapes into smaller plane shapes.
 Combine and separate plane shapes in figures.
 Draw plane shapes and figures on dot paper and square grid paper.
 Identify quadrilaterals and pentagons.
123

Recognize and draw shapes having a given number of angles.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 18: Lines and Surfaces
Math In Focus On-line Resources Chapter 19: Shapes and Patterns
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 18: Lines and Surfaces
Math In Focus Resources Chapter 19: Shapes and Patterns
Solid shapes
Attribute blocks
Paper shapes
Dot and grid paper
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124
Nutley Public Schools
Grade 3
125
Nutley Public Schools
Mathematics
Grade 3
Unit 1:
Numbers to 10,000
Summary and Rationale
In this unit, students use daily examples of objects seen around them to count and compare numbers to
10,000. They differentiate between numbers by comparing them and learn to order numbers in ascending
or descending order.
The mental math strategies in this unit rely on composing and decomposing numbers through number
bonds. Students use number bonds and estimation strategies in mental calculation to find and check sums
and differences with and without regrouping.
Students apply previously learned addition strategies such as using base-ten blocks, place-value charts,
vertical form, and the part-whole concept to add and subtract greater numbers with and without
regrouping.
Drawing bar models help students to visualize and construct concrete examples to help them make sense
of the relationship between the values given in real-world problems. It requires students to understand the
mathematical concepts underlying the word problems and equips them with a strong conceptual
foundation in mathematics to solve real-word problems and even the most challenging problems.
Recommended Pacing
Eight (8) weeks
Standards
Number and Operations in Base Ten
3.NBT.1
Use place value understanding to round to the nearest 10 or 100.
3.NBT.2
Fluently add and subtract within 1,000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction.
Operations and Algebraic Thinking
3.OA.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.OA.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.
Mathematical Practices
126
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers to 10,000 can be counted and compared.
Number bonds and estimation strategies can be used to find and check sums and differences.
Greater numbers can be added the same way two-digit numbers are added, with or without regrouping.
Greater numbers can be subtracted with or without regrouping.
Addition and subtraction bar models can be used to solve two-step real-world problems.
Essential Questions
How do we add greater numbers?
How do we subtract greater numbers?
Ho can bar models help solve real-world addition and subtraction problems?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Student will know or learn:
 Word form, standard form, expanded form
 Digit, place-value chart, place-value strips
 Greater than, less than, least, greatest
 Rule
 Number line
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Rounding
Reasonable estimate, overestimate
Leading digit
Front-end estimation
Regroup
Sum, difference
Student will be able to:
 Use base-ten blocks and a place-value chart to count, read, write, and represent numbers to 10,000.
 Count by 1s, 10s, 100s, and 1,000s to 10,000.
 Read and write numbers to 10,000 in standard form, expanded form, and word form.
 Use base-ten blocks and place value to compare and order numbers.
 Add two-digit numbers mentally with and without regrouping.
 Subtract two-digit numbers mentally with and without regrouping.
 Use different strategies to add two-digit numbers close to 100 mentally.
 Round numbers to estimate sums and differences.
 Use front-end estimation to estimate sums and differences.
 Add greater numbers with and without regrouping.
 Use base-ten blocks to subtract with and without regrouping.
 Use base-ten blocks to subtract across zeros.
 Write subtraction number sentences.
 Solve subtraction word problems.
 Use bar models to solve two-step real-world problems involving addition and subtraction.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Numbers to 10,000.
Math In Focus On-line Resources Chapter 2: Mental Math and Estimation
Math In Focus On-line Resources Chapter 3: Addition Up to 10,000
Math In Focus On-line Resources Chapter 4: Subtraction Up to 10,000
Math In Focus On-line Resources Chapter 5: Using Bar Models: Addition and Subtraction
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 1: Numbers to 10,000.
Math In Focus Resources Chapter 2: Mental Math and Estimation
Math In Focus Resources Chapter 3: Addition Up to 10,000
Math In Focus Resources Chapter 4: Subtraction Up to 10,000
Math In Focus Resources Chapter 5: Using Bar Models: Addition and Subtraction
Base-ten blocks
Place-value chart
Place-value strips
Place-value mat
Number cards
Number cubes
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128
Nutley Public Schools
Mathematics
Grade 3
Unit 2:
Multiplication and Division
Summary and Rationale
Both multiplication and division are associated with the part-whole concept. In this unit, students multiply
using different models such as number lines, dot paper, and area models. These use skip-counting to
multiply mentally and base-ten blocks and place-value charts to multiply greater numbers with and
without regrouping.
Division concepts are extended to division situations where there may be remainders. Students are made
aware that the dividend does not always divide exactly into equal groups, but sometimes leaves a
remainder. Students learn the steps of vertical division (long division) to divide with or without
regrouping or a remainder.
The conceptual skills between multiplication and division are strengthened. Students use related
multiplication facts to divide. They apply the inverse relationship of multiplication and division to write
division statements from multiplication sentences.
Bar models are used to solve different kinds of multiplication and division word problems. Drawing bar
models provides students with a systematic means of organizing information and determining the
calculations needed to solve the problem. Bar models simplify the problems by showing clearly hwat
steps need to be taken to answer the question.
Recommended Pacing
Nine (9) weeks
Standards
Number Operations in Base Ten
3.NBT.3
Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 x 80, 5 x
60) using strategies based on place value and properties of operations.
Operations and Algebraic Thinking
3.OA.1
Interpret products of whole numbers, e.g. interpret 5x7 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 x 7.
3.OA.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 equals 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
129
be expressed as 56 ÷ 8.
3.OA.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.
3.OA.4
Determine the unknown number in a division equation elating three whole numbers. For
example, determine the unknown number that makes the equation true in each of the
equations 8 x ? = 48, 5 = ? ÷ 3, 6 x 6 = ?
3.OA.5
Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24
is known, then 4 x 6 = 24is also known. (Commutative property of multiplication.) 3 x 5 x 2
can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30.
(Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can
find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)
3.OA.6
Understand division as an unknown- factor problem. For example, find 32 ÷ 8 by finding the
number that makes 32 when multiplied by 8.
3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or
properties of operations. By the end of Grade 3, know form memory all products of two
one-digit numbers.
3.OA.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.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Many models can be used to multiply.
Mental math can be used to multiply.
Numbers up to three digits can be multiplied with or without regrouping.
130
There can be remainders when dividing to make equal groups or when sharing equally. The dividend does
not always divide exactly into equal groups, but sometimes leaves a remainder.
Bar models can be used to solve different kinds of multiplication and division word problems.
Essential Questions
What strategies can be used to multiply and divide?
How do multiplication facts help you divide?
How are multiplication and division related?
What is a remainder and when does a division problem result in a remainder?
How do bar models help solve multiplications and division word problems?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Skip
 Dot paper
 Number line
 Commutative Property
 Associative Property
 Multiplication Property of One
 Multiplication Property of Zero
 Array model
 Area model
 Equal groups
 Product
 Quotient, remainder
 Even number, odd number
 Twice, double
Students will be able to:
 Use multiplication properties.
 Understand multiplication by using array models.
 Practice multiplication facts of 6.
 Understand multiplication by using area models.
 Practice multiplication facts of 7.
 Understand multiplication by using number lines.
 Practice multiplication facts of 8.
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Practice multiplication facts of 9.
Divide to find the number of items in each group.
Identify related multiplication and division facts.
Write and express division sentences for real-world problems.
Divide to find the number of groups.
Multiply ones, tens, and hundreds mentally.
Multiply ones, tens, and hundreds with and without regrouping.
Use related multiplication facts to divide.
Use patterns to divide multiples of 10 and 100.
Divide a one-digit number or a two-digit number by a one-digit number with or without remainder.
Use different strategies to identify odd and even numbers.
Use base-ten blocks and place-value to divide two-digit numbers without regrouping and remainders.
Use base-ten blocks and place-value to divide two-digit numbers by a one-digit number with
regrouping, with or without remainders.
Use bar models to solve one and two -step multiplication and division word problems.
Choose the correct operations in two-step word problems.
Recognize number relationships.
Solve two-step problems using the four operations.
Represent the unknown quantities with letters.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 6: Multiplication of 6,7,8, and 9
Math In Focus On-line Resources Chapter 7: Multiplication
Math In Focus On-line Resources Chapter 8: Division
Math In Focus On-line Resources Chapter 9: Using Bar Models: Multiplication and Division
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 6: Multiplication of 6,7,8, and 9
Math In Focus Resources Chapter 7: Multiplication
Math In Focus Resources Chapter 8: Division
Math In Focus Resources Chapter 9: Using Bar Models: Multiplication and Division
Number lines
Dot paper
Counters
Connecting cubes
Number cubes
Number train
Number cards
Area models
Multiplier cards
Number board
Question cards
Base-ten blocks
Place-value mat
Game cards
132
Paper strips
Multiplication/division flashcards
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133
Nutley Public Schools
Mathematics
Grade 3
Unit 3:
Money
Summary and Rationale
Students come into contact with money almost every day of their lives. As students deal with daily
purchases, especially on their own, it is imperative for them to be able to mentally work the basic
operations with money.
In Grade 2, students learned how to count, read, and write money, convert cents to dollars and vice versa,
and to make change. In this Grade 3 unit, these concepts are reviewed and extended to amounts of money
up to $100.
Since the concept of decimals is not introduced until Grade 4, the decimal point in money is presented as
a dot separating dollars and cents.
Recommended Pacing
Two (2) weeks
Standards
Numbers and Operations in Base Ten
3.NBT.2
Fluently add and subtract within 1,000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
134
Enduring Understandings
We come into contact with money almost every day of our lives.
You can add and subtract money the same way you add and subtract whole numbers.
Essential Questions
When do we come into contact with money?
How do you add and subtract money?
When do we add and subtract money?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Bills up to $100
Students will be able to:
 Add money in different ways with and without regrouping.
 Subtract money in different ways with and without regrouping.
 Solve up to two-step real-world problems involving addition and subtraction of money.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 10: Money
www.cashout.com
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 10: Money
Bills and coin manipulatives
Bills and coin cut-outs
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Nutley Public Schools
Math
Grade 3
Unit 4:
Metric Measurement
Summary and Rationale
In this unit, students use metric units of measurement to measure length, mass, and volume. They read
tools measuring length, mass, and volume in metric units and also convert between metric units of
measurement.
Students use bar models to solve one and two-step real-world problems involving the addition,
subtraction, multiplication, division, and metric measurements.
Recommended Pacing
Three (3) weeks
Standards
Numbers and Operations in Base Ten
3.NBT.2
Fluently add and subtract within 1,000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction.
Operations and Algebraic Thinking
3.OA.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.
3.OA.4
Determine the unknown number in a division equation elating three whole numbers. For
example, determine the unknown number that makes the equation true in each of the
equations 8 x ? = 48, 5 = ? ÷ 3, 6 x 6 = ?
3.OA.5
Apply properties of operations as strategies to multiply and divide. Examples: If 6 x 4 = 24
is known, then 4 x 6 = 24is also known. (Commutative property of multiplication.) 3 x 5 x 2
can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30.
(Associative property of multiplication.) Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can
find 8 x 7 as 8 x (5 + 2) = (8 x 5) + (8 x 2) = 40 + 16 = 56. (Distributive property.)
3.OA.6
Understand division as an unknown- factor problem. For example, find 32 ÷ 8 by finding the
number that makes 32 when multiplied by 8.
3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 x 5 = 40, one knows 40 ÷ 5 = 8) or
properties of operations. By the end of Grade 3, know form memory all products of two
one-digit numbers.
Measurement and Data
3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams
136
(g), kilograms (kg), and liters (L). Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same unit, e.g., by using
drawings (such as a beaker with a measurement scale) to represent the problem.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Length, mass, and volume can be measured using metric units of measurement.
Bar models can be used to solve one and two-step problems on measurements.
Essential Question
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Meter (m), centimeter (cm), kilometer (km)
 distance
 Gram (g), kilogram (kg)
 Volume
 Capacity
 Liter (L), milliliter (mL)
Students will be able to:
 Use meters, centimeters, and kilometers as units of measurement of length.
 Estimate and measure length.
 Convert units of measurement.
 Read scales in kilograms and grams.
 Estimate and find actual masses of objects using different scales.
 Estimate and find volume of liquid in liters and milliliters.
 Find the volume and capacity of a container.
137
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Draw bar models to solve one and two-step measurement problems.
Choose the operation to solve one and two-step problems.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 11: Metric Length, Mass, and Volume
Math In Focus On-line Resources Chapter 12: Real-World Problems: Measurement
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 11: Metric Length, Mass, and Volume
Math In Focus Resources Chapter 12: Real-World Problems: Measurement
Measuring tape or meter stick
Centimeter square grid
Weighing scale
Metric measuring cups
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138
Nutley Public Schools
Mathematics
Grade 3
Unit 5:
Data
Summary and Rationale
In this unit, students use bar graphs and line plots to organize data. Bar graphs are used to compare data.
Line plots show how data is spread out. Students begin to work with bar graphs that contain scales in
skips of two or greater and read and interpret bard graphs to solve real-world problems. Line plots are also
introduced for the first time. Students use these line plots to organize data and show frequency of an
event. They relate both graphs and plots to real-life problem situations.
Recommended Pacing
Two (2) weeks
Standards
Measurement and Data
3.MD.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.
3.MD.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, and quarters.
Number and Operations- Fractions
3.NF.3
Explain equivalence of fractions in special cases and compare fractions by reasoning about
their size.
3.NF.3c
Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 3/1; recognize 6/1 = 6; locate 4/4 and 1 at
the same point of a number line diagram.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
139
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Bar graphs and line plots help to organize data.
Bar graphs are used to compare data.
Line plots show how data is spread out.
Essential Question
Why do we put data in bar graphs and line plots?
How do bar graphs and line plots help us better understand information?
What kinds of questions can be answered using line plots and bar graphs?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Vertical, horizontal
 Axis, scale
 Line plot
 Survey
Students will be able to:
 Make bar graphs with scales using data in picture graphs and tally charts.
 Read and interpret data from bar graphs.
 Solve problems using bar graphs.
 Make a line plot to represent and interpret data.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 13: Bar Graphs and Line Plots
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 13: Bar Graphs and Line Plots
140
Bar charts and graphs
Centimeter square grids
Tally charts
Number lines
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141
Nutley Public Schools
Mathematics
Grade 3
Unit 6:
Fractions
Summary and Rationale
In this unit, student s begin to work with fractions of wholes that are divided into more than four equal
parts. Students learn concepts such as equivalent fractions and identifying fractions of a set. Through this
unit, students develop an understanding of the meanings and uses of fractions, such as representing parts
of a whole, parts of a set, and points or distances on a number line.
Working with wholes that are divided into more than four equal parts, students use a variety of skills to
find equivalent fractions, compare fractions, and order fractions. They use models to add and subtract like
fractions.
Recommended Pacing
Three (3) weeks
Standards
Measurement and Data
3.MD.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, and quarters.
3.MD.7
Relate area to the operations of multiplication and addition.
3.MD.7a
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.
3.MD.7b
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.
3.MD.7c
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 x b and a x c. Use area models to represent the
distributive property in mathematical reasoning.
Number and Operations- Fractions
3.NF.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.
3.NF.2
Understand a fraction as a number on the number line; represent fractions on a number line
diagram.
3.NF.2a
Represent a fraction 1/b on a number line diagram by defining the interval 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.
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3.NF.2b
3.NF.3
3.NF.3a
3.NF.3b
3.NF.3c
3.NF.3d
Represent a fraction a/b on a number line diagram by marking off a lengths of 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.
Explain equivalence of fractions in special cases and compare fractions by reasoning about
their size.
Understand two fractions as equivalent (equal) if they are the same size, or the same point
on a number line.
Recognize and generate simple equivalent fractions, e.g., ½ = 2/4, 4/6 = 2/3). Explain why
the fractions are equivalent, e.g., by using a visual fraction model.
Express whole numbers as fractions, and recognize fractions that are equivalent to whole
numbers. Examples: Express 3 in the form 3 = 3/1; recognize 6/1 = 6; locate 4/4 and 1 at
the same point of a number line diagram.
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.
Geometry
3.G.1
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 ¼ of the area of the shape.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Fractions can be used to describe parts of a region.
Fractions can be used to describe parts of a set.
Essential Questions
How can fractions be modeled?
How are fractional parts compared?
143
What are equivalent fractions?
Where are fractions found in the real world?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Student will know or learn:
 Whole
 Equal parts
 Numerator, denominator
 Equivalent fractions
 Number line
 Simplest form
 Benchmark
 Like fractions, unlike fractions
Students will be able to:
 Read, write, and identify fractions from wholes with more than four parts.
 Identify numerator and denominator.
 Use models and a number line to identify equivalent fractions.
 Use multiplications and division to find equivalent fractions.
 Write fractions in simplest form.
 Compare and order fractions using benchmark fractions.
 Show fractions as points or distances on a number line.
 Add two or three fractions with sums to one.
 Subtract a like fraction from another like fraction or one-whole.
 Read, write, and identify fractions of a set.
 Find the number of items in a fraction of a set.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 14: Fractions
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 14: Fractions
Connecting cubes
Fraction circle cut-outs
Centimeter square grid paper
Fraction strips
Fraction number lines
Fraction manipulatives
144
Fraction Bingo
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Nutley Public Schools
Mathematics
Grade 3
Unit 7:
Customary Measurement
Summary and Rationale
In this unit, students measure length, weight, and capacity using customary units. With the exposure to
real-world problems, students are able to make sense of what they learn in parallel context situations
encountered in everyday life.
Recommended Pacing
Two (2) weeks
Standards
Measurement and Data
3.MD.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, and quarters.
3.MD.7
Relate area to the operations of multiplication and addition.
3.MD.7a
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.
3.MD.7b
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.
3.MD.7c
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 x b and a x c. Use area models to represent the
distributive property in mathematical reasoning.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
146
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Length, weight, and capacity can be measured using customary units.
Length can be measured in inches, feet, yards, and miles.
Weight can be measured in ounces, pounds, and tons.
Capacity can be measured in cups, pints, quarts, and gallons.
Essential Questions
When and how do we measure length?
When and how do we measure weight?
Where do we find units of capacity such as cups, pints, quarts, or gallons?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Student will know or learn:
 Inch (in.), half-inch, foot (ft), yard (yd), mile (mi)
 Ounce (oz), pound (lb), ton (T)
 Cup (c), pint (pt), quart (qt), gallon (gal)
Students will be able to:
 Use inch, foot, yard, and mile as units of measurement for lengths.
 Estimate and measure given lengths.
 Use referents to measure lengths.
 Use ounce, pound, and ton as units of measurement for weight.
 Read scales in ounce (oz) and pound (lb).
 Estimate and find actual weights of objects by using different scales.
 Use referents to measure weight.
 Measure capacity with cup (c), pint (pt), quart, (qt), and gallon (gal).
 Estimate and find the actual capacity of a container.
 Relate units of capacity to one another.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 15: Customary Length, Weight, and Capacity
147
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 15: Customary Length, Weight, and Capacity
www.hmhlearning.com
www.hmheducation.com/mathinfocus
148
Nutley Public Schools
Mathematics
Grade 3
Unit 8:
Time and Temperature
Summary and Rationale
In this unit, students learn about reading and telling time to the minute. They convert time units in hours
and minutes, add and subtract time, and use time to find when activities start and end, or how long an
activity will last.
Students are introduced to the concept of temperature, in conjunction with the reading of the Fahrenheit
thermometer. They use temperature to describe weather conditions and solve real-world problems
involving temperature.
Recommended Pacing
Two (2) weeks
Standards
Measurement and Data
3.MD.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.
Mathematical Practices
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Time is a measurement concept that can be used to tell when activities start and end, or how long an
149
activity will last.
Temperature is a measurement concept that can be used to understand what the weather will be like.
Essential Questions
Why is it important to tell time?
When is it appropriate to estimate telling time?
Why is it important to find elapsed time?
How is temperature useful to us?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Student will know or learn:
 Hours (h), minutes (min)
 Hour past, minute to
Students will be able to:
 Tell time to the minute.
 Read time on a digital clock.
 Change minutes to hours or hours to minutes.
 Add and subtract time with and without regrouping.
 Find elapsed time.
 Read a Fahrenheit thermometer.
 Choose the appropriate tool and unit to measure temperature.
 Use a referent to measure temperature.
 Solve up to two-step word problem on time.
 Solve word problems involving temperature.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 16: Time and Temperature
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 16: Time and Temperature
www.hmhlearning.com
www.hmheducation.com/mathinfocus
150
Nutley Public Schools
Mathematics
Grade 3
Unit 9:
Geometry
Summary and Rationale
In this unit, an angle is defined as two line segments that share the same endpoint. This is done for
simplicity, as the term ray is not introduced until Grade 4. Students identify and relate angles,
perpendicular lines, and parallel lines to real-life objects and are encouraged to see angles and lines in
planes shapes and three-dimensional objects.
Students identify the sides and angles of closed polygons. They classify polygons and learn the names of
special polygons and quadrilaterals. They learn the concepts of congruency symmetry and how to check
for congruency, determine symmetric figures, and draw a line of symmetry to produce congruent halves.
Students combine geometry and measurement by learning the concepts of area and perimeter. In finding
area and perimeter, students use appropriate units for figures of different sizes and explore the relationship
between them.
Recommended Pacing
Five (5) weeks
Standards
Measurement and Data
3.MD.5
Recognize area as an attribute of plane figures and understand concepts of area
measurement.
3.MD.5a
Recognize that 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.
3.MD.5b
Recognize that a plane figure which can be covered without gaps or overlaps by n unit
square is said to have an area of n square units.
3.MD.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and nonstandard units.
3.MD.7
Relate area to the operations of multiplication and addition.
3.MD.7a
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.
3.MD.7b
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.
3.MD.7c
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 x b and a x c. Use area models to represent the
151
distributive property in mathematical reasoning.
3.MD.7d
Recognize area as additive. Find areas of rectilinear figures by decomposing them into nonoverlapping rectangles and adding the areas of the non-overlapping parts, applying this
technique to solve real world problems.
3.MD.8
Solve real world and mathematical problems involving perimeters of polygons, including
finding 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.
Numbers and Operations in Base Ten
3.NBT.2
Fluently add and subtract within 1,000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction.
Geometry
3.G.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Angles and lines can be found all around us and can be described with special names.
An angle is defined as two line segments that share the same endpoint.
Polygons can be classified by the number of sides, corners, and angles.
Figures can be congruent, or symmetrical, or both.
The perimeter is the distance around a polygon or the sum of the lengths of its sides.
The area is the amount of space which a polygon takes up.
Essential Questions
152
How is an angle formed?
How can angles be compared?
What are congruent figures?
How are area and perimeter different? Are they related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Student will know or learn:
 Point
 Angle
 Line, line segment
 Endpoint
 Right angle
 Perpendicular lines, is perpendicular to
 Parallel lines, is parallel to
 Plane figure, open figure, closed figure
 Polygon
 Vertex
 Quadrilateral
 Parallel
 Rhombus
 Parallelogram
 Pentagon
 Octagon
 Tangram
 Flip, slide, turn, rotate
 Congruent
 Symmetry, line of symmetry
 Area
 Square units: square centimeter (cm2), square inch (in2), square meter (m2), square foot (ft2)
 Perimeter
Students will be able to:
 Find angles in plane shapes and real-world objects.
 Compare the number of sides and angles of plane shapes.
 Make a right angle.
 Compare right angles to a right angle.
 Identify right angles in plane shapes.
 Define and identify perpendicular lines.
 Define and identify parallel lines.
153

















Identify open and closed figures.
Identify special polygons and quadrilaterals.
Classify polygons by the number of sides, vertices, and angles.
Classify quadrilaterals by parallel sides, length of sides, and angles.
Combine and separate polygons to make other polygons.
Identify a slide, flip, and turn.
Slide, flip, and turn shapes to make congruent figures.
Identify congruent figures.
Identify symmetric figures.
Use folding to find a line of symmetry.
Use square units to find the area of plane figures made of squares and half squares.
Compare areas of plane figures and make plane figures of the same area.
Use square centimeter and square inch to find and compare the area of figures.
Use square meters and square feet to find and compare the area of plane figures.
Estimate the area of small and large surfaces.
Find the perimeter of figures formed using small squares.
Compare the area and perimeter of two figures.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 17: Angles and Lines
Math In Focus On-line Resources Chapter 18: Two-Dimensional Shapes
Math In Focus On-line Resources Chapter 19: Area and Perimeter
www.mathematicsgames.com
Language Arts Integration
An Illusionary Tale by Arline Baum and Joseph Baum
Math Journal
Suggested Resources
Math In Focus Resources Chapter 17: Angles and Lines
Math In Focus Resources Chapter 18: Two-Dimensional Shapes
Math In Focus Resources Chapter 19: Area and Perimeter
Centimeter square grid paper
Paper strips
Square grid paper
Geoboards
Tangram cut-outs
Congruent shape cards
Square tiles
Half-square tiles
Measuring tape
www.hmhlearning.com
www.hmheducation.com/mathinfocus
154
Nutley Public Schools
Grade 4
155
Nutley Public Schools
Mathematics
Grade 4
Unit 1:
Whole Numbers
Summary and Rationale
Number theory, the study of whole numbers and their properties, has a long history an is still an active
field of inquiry. In this unit, place value concepts are reviewed and extended to the ten thousands place.
Students represent numbers to 100,000 in various ways and apply what they know about comparing
numbers to larger numbers.
Students build on their knowledge of rounding numbers toe estimate sums, differences, products, and
quotients and use estimation skills to determine if an answer is reasonable. They determine if estimates or
exact answers are needed and apply estimation skills in real-world situations.
Students are introduced to factors, multiples, least common multiples, and greatest common factors in this
unit. They use basic multiplication and division facts to find factors and multiples, break down whole
numbers into factors, and multiply them to get multiples.
Pacing
Five (5) weeks
Standards
Number and Operations in Base Ten
4.NBT.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.
4.NBT.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.
4.NBT.3
Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Operations and Algebraic Thinking
4.OA.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.
4.OA.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.
4.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Whole numbers can be compared and ordered according to the place value of their digits.
When two factors are multiplied, the product is a multiple of both numbers.
Knowing factors and multiples of numbers can help in estimating products and quantities.
Essential Questions
What is a factor?
What is a multiple? How are factors and multiples related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Ten thousand
 Hundred thousand










Standard form, word form, expanded form
Reasonable estimate
Front-end estimation
Rounding
Product, quotient
Factor, common factor
Greatest common factor (GCF)
Prime number, composite number
Multiple, common multiple
Least common multiple (LCM)
Students will be able to:
 Write numbers to 100,000 in standard form, word form, and expanded form.
 Compare and order numbers to 100,000.
 Identify how much more or less one number is than another number.
 Find the rule in a number pattern.
 Add multi-digit numbers with and without regrouping.
 Subtract multi-digit numbers with and without regrouping.
 Round numbers to estimate sums, differences, products, and quotients.
 Estimate to check that an answer is reasonable.
 Decide whether an exact answer or an estimate is needed.
 Find the common factors and greatest common factor of two whole numbers.
 Identify prime numbers and composite numbers.
 Find multiples of whole numbers.
 Find common multiples and the least common multiple of two or more numbers.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Place Value of Whole Numbers
Math In Focus On-line Resources Chapter 2: Estimation and Number Theory
Language Arts Integration
Sea Squares by Joy N. Hulme
Each Orange had 8 Slice: A Counting Book by Paul Giganti
Suggested Resources
Math In Focus Resources Chapter 1: Place Value of Whole Numbers
Math In Focus Resources Chapter 2: Estimation and Number Theory
Place-value chart
Place-value chips
Number cards
Prime numbers table
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 4
Unit 2:
Multiplication and Division
Summary and Rationale
In this unit, students advance to multiplying and dividing mutli-digit numbers. The place-value concept,
which students are familiar with, is used in multiplication and division. Students discover that division is
the inverse of multiplication and use estimation to check the reasonableness of answers.
Pacing
Four (4) weeks
Standards
Number and Operations in Base Ten
4.NBT.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.
4.NBT.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.
4.NBT.3
Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
4.NBT.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.
4.NBT.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.
Operations and Algebraic Thinking
4.OA.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.
4.OA.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
4.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Place value is used to multiply and divide multi-digit numbers.
Estimation can be used to check the reasonableness of an answer.
Division is the inverse of multiplication.
Essential Question
When would it be useful to multiply than to use repeated addition?
When will you use division in your daily life?
How are multiplication and division related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Round, estimate
 Product
 Regroup

Quotient, remainder
Students will be able to:
 Multiply multi-digit numbers by o one-digit number using an array model.
 Use different methods to multiply up to four-digit numbers by one-digit numbers, with or without
regrouping.
 Multiply two two-digit numbers using an area model.
 Multiply by two-digit numbers with and without regrouping.
 Estimate products.
 Model regrouping in division.
 Divide a three-digit number by a one-digit number with regrouping.
 Divide up to a four-digit number by a one-digit number with regrouping, and with and without
remainders.
 Estimate quotients.
 Solve real-world problems.
 Solve multi-step word problems using the four operations.
 Represent the problems with a letter standing for the unknown quantity.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 3: Whole Number Multiplication and Division
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 3: Whole Number Multiplication and Division
Number cubes
Base-ten blocks
Place-value chart
Place-value chips
Calendar
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 4
Unit 3:
Data and Probability
Summary and Rationale
Data and probability is a recurring phenomenon in everyday life.
Tables and graphs are the visual tools for showing and analyzing data. Data that is tabulated or plotted on
graphs can be reviewed easily, and visually elicit patterns and trends. Comparing, analyzing, and
classifying are just some of the thinking skills students apply as they look for these patterns and trends.
Students are introduced to line graphs, which are graphs with two numerical axes and shows data
continuously from left to right. Students use the four operations of whole numbers when they analyze
data presented in graphs and tables to solve problems.
Students learn how to use different tools to analyze data such as average, mean, median, and probability.
They apply their understanding of place value and graphs to develop and use stem-and-leaf plots.
The ability to predict the likelihood of an outcome is a practical skill. Students learn to describe the
possibility of the occurrence of different outcomes and express the probability of an outcome as a
fraction. They also have opportunities to solve real-world problems and demonstrate the ability to make
predictions based on given data.
Pacing
Five (5) weeks
Standards
Number and Operations – Fractions
4.NF.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.
4.NF.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.3c
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Graphs and tables are visual tools for showing and analyzing data.
Information can be analyzed to find a typical value for a data set.
Line graphs have two numerical axes and show data continuously from left to right.
Data can be analyzed to predict the likelihood of an event happening.
Probability is the likelihood that a desired outcome will occur.
Probability is a value between 0 and 1 and can be represented as a fraction.
Essential Questions
How does organizing data make it easier to understand?
How is data on a line plot interpreted?
How is data on a stem-and-leaf interpreted?
What does probability tell us?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Data
 Table















Tally chart
Row, column, intersection,
Line graph
Horizontal axis, vertical axis
Average, mean
Median
Mode
Range
Line plot
Stem-and-leaf plot
Outlier
Outcome
Certain, more likely, equally likely, less likely, impossible
Favorable outcome
Probability
Students will be able to:
 Collect, organize, and interpret data in a table.
 Create a table form data in a tally chart and a bar graph.
 Read and interpret data in a table, using rows, columns, and intersections.
 Make, read, and interpret line graphs.
 Choose an appropriate graph to display a given data set.
 Describe a data set using the average or mean.
 Find the mean, median, mode, and range of a set of data.
 Make an interpret line plots.
 Organize and represent data in a stem-and-leaf plot.
 Use a stem-and-leaf plot to find median, mode, and range.
 Decide whether an outcome is certain, more likely, equally likely, less likely, or impossible.
 Determine the probability of an event.
 Express probability as a fraction.
 Solve real-world problems involving probability and measures of central tendency.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 4: Tables and Line Graphs
Math In Focus On-line Resources Chapter 5: Data and Probability
www.gamequarium.com/data.html (Landmarks games: Range, mode, median)
www.shodor.org/interactivate/activities/StemAndLeafPlotter/ (Stem-and-leaf)
Language Arts Integration
How Much is a Million? by David Schwartz
If You Made a Million by David Schwartz
Suggested Resources
Math In Focus Resources Chapter 4: Tables and Line Graphs
Math In Focus Resources Chapter 5: Data and Probability
Tally charts
Survey tables
Grid paper
Connecting cubes
Counters
Line plots
Number cards
Number cubes
Spinners
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 4
Unit 4:
Fractions and Mixed Numbers
Summary and Rationale
In this unit, students learn how to add and subtract like and unlike fractions with and without renaming.
They are introduced to the concept of fractions of a set, and will apply this knowledge to solve real-world
problems. Terms such as numerator and denominator are used throughout this unit. Fraction circles and
bar models are used to illustrate the concepts. Students apply their knowledge if finding common factors
and multiples to add and subtract unlike but related fractions
Pacing
Four (4) weeks
Standards
Number and Operations-Fractions
4.NF.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.
4.NF.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 ½. 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.
4.NF.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.3a
Understand addition and subtraction of fractions as joining and separating parts referring to
the same whole.
4.NF.3b
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.
4.NF.3c
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.
4.NF.3d
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.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction by a
whole number.
4.NF.4a
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).
4.NF.4b
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.)
4.NF.4c
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?
Measurement and Data
4.MD.1
Know relative sizes of measurement units within one system of units including km, m, cm;
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), ...
4.MD.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.
Operations and Algebraic Thinking
4.OA.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
4.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Fractions and mixed numbers are used to name whole and parts of a whole.
Fractions and mixed numbers can be added and subtracted.
To add or subtract unlike fractions, find a common multiple of both denominators and use that as the
denominator of both fractions.
Essential Questions
How is an improper fraction converted into a mixed number?
How do you add or subtract unlike fractions and mixed numbers?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Numerator, denominator
 Equivalent fraction
 Unlike fraction
 Mixed number
 Simplest form
 Improper fraction
 Fraction bar
 Division rule
 Multiplication rule
Students will be able to:
 Compare unlike fractions using the symbols >, =, or < .
 Find equivalent fractions.
 Add and subtract unlike fractions.
 Write a mixed number for a model.
 Draw models to represent mixed numbers.
 Write an improper fraction for a model.
 Express mixed numbers as improper fractions.
 Use multiplication and division to rename improper fractions and mixed numbers.
 Add fractions to get mixed-number sums.
 Subtract fractions from whole numbers.
 Use a bar model to represent a fraction of a set.
 Find a fractional part of a number.
 Multiply a fraction and a whole number.



Solve real-world problems involving fractions.
Show measurements in a line plot with a scale of fractions of a unit.
Solve problems by adding and subtracting fractions from data in a line plot.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 6: Fractions and Mixed Numbers
Language Arts Integration
Top 10 of Everything: 200 by R. Ash
Scholastic Kid’s Almanac for the 21st Century by E. Pascoe and D. Kops
Suggested Resources
Math In Focus Resources Chapter 6: Fractions and Mixed Numbers
Fraction strips
Fraction circles
Fraction bar models
Number cubes
Connecting cubes
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 4
Unit 5:
Decimals
Summary and Rationale
Decimals are an extension of the base-ten system of writing whole numbers. Decimals can represent
amounts that are parts of a whole and are useful for representing numbers less than one and numbers
between consecutive whole numbers. In this unit, students learn to recognize, compare, and round
decimals in tenths and hundredths. Number lines are used to represent, compare, and round decimals.
Students learn that the period used to separate dollars and cents in money is called a decimal point, which
is used to separate the whole number part and the fractional part. Students make the connection between
equivalent fractions and decimals through models and number lines.
Students add and subtract decimals by using the same algorithms as whole numbers.
Pacing
Five (5) weeks
Standards
Measurement and Data
4.MD.1
Know relative sizes of measurement units within one system of units including km, m, cm;
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), ...
4.MD.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.
Numbers and Operations in Base Ten
4.NBT.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.
4.NBT.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
170
place, using >, =, and < symbols to record the results of comparisons.
4.NBT.3
Use place value understanding to round multi-digit whole numbers to any place.
4.NBT.4
Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Numbers and Operations- Fractions
4.NF.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.
4.NF.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 ½. 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.
4.NF.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
4.NF.3a
Understand addition and subtraction of fractions as joining and separating parts referring to
the same whole.
4.NF.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 100.2 For
example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
4.NF.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.
4.NF.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.
Operations and Algebraic Thinking
4.OA.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.
4.OA.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” to
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
171
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Decimals are another way to show amounts that are parts of a whole.
Decimals are useful for representing numbers less than one and numbers between consecutive whole
numbers.
A decimal has a decimal point to the right of the ones place and digits to the right of the decimal point.
The decimal point is used to separate the whole number part and the fractional part.
Decimals can be added and subtracted in the same ways as whole numbers.
Essential Questions
How are decimals and factions related?
What is the significance of the decimal point?
How are decimals added and subtracted?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Tenth, hundredth
 Decimal form
 Decimal point
 Expanded form
 Placeholder zero
 Equivalent fraction and decimal
Students will be able to:
 Read and write tenths in decimal and fractional forms
 Represent and interpret tenths models.
 Read and write hundredths in decimal and fractional forms
 Represent and interpret hundredths models.
 Compare and order decimals.
 Complete number patterns.
 Round decimals to the nearest whole number or tenth.
 Express a fraction as a decimal and a decimal as a fraction.
172


Add and subtract decimals up to two decimal places.
Solve real-world problems involving addition and subtraction of decimals.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 7: Decimals
Math In Focus On-line Resources Chapter 8: Adding and Subtracting Decimals
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 7: Decimals
Math In Focus Resources Chapter 8: Adding and Subtracting Decimals
Decimal place-value chart
Place-value chips
Centimeter ruler
Ten-sided dice
Measuring tape
Decimal cards
Fraction cards
Decimal bars
Unit cubes
Decimal squares in tenths and hundredths
www.hmhlearning.com
www.hmheducation.com/mathinfocus
173
Nutley Public Schools
Mathematics
Grade 4
Unit 6:
Geometry
Summary and Rationale
In this unit, students learn that angles can be seen everywhere around them. Angles are formed when two
rays or sides of a figure meet. Students estimate angle measures, measure angles with a protractor, and are
introduced to the degree symbol. They also learn to draw angles to 180° using a protractor and make
connections between angles and turns.
Students extend their knowledge of lines to line segments and continue to explore parallel and
perpendicular lines. They learn to use a drawing triangle to draw perpendicular, parallel, horizontal, and
vertical line segments when a grid is not provided.
Students learn the properties of squares and rectangles. They apply their knowledge of angles and parallel
and perpendicular line segments to identify and define squares and rectangles. Students also decompose
shapes that are made up of squares and rectangles. These use the properties of squares and rectangles to
find unknown angles measures and side lengths of figures.
Pacing
Six (6) weeks
Standards
Geometry
4.G.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Identify these in two-dimensional figures.
4.G.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.
Measurement and Data
4.MD.1
Know relative sizes of measurement units within one system of units including km, m, cm;
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), ...
4.MD.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.
4.MD.5
Recognize angles as geometric shapes that are formed wherever two rays share a common
endpoint, and understand concepts of angle measurement:
4.MD.5a
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.
4.MD.5b
An angle that turns through n one-degree angles is said to have an angle measure of n
degrees.
4.MD.6
Measure angles in whole-number degrees using a protractor. Sketch angles of specified
measure.
4.MD.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.
Operations and Algebraic Thinking
4.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Angles can be seen and measured when two rays or sides of a shape meet.
Line segments can go up and down, from side to side, and in every direction.
Parallel lines never intersect.
Perpendicular lines intersect at right angles.
A horizontal line is one that is parallel to the level around. A vertical line is one that is perpendicular to a
horizontal line.
Squares and rectangles are four- sided figures with special properties.
A square has four sides of equal length and four right angles. The opposite sides are parallel.
A rectangle has four sides and four right angles. The opposite sides are parallel and have the same length.
A square is a subset of a rectangle. All squares are rectangles, but not all rectangles are squares.
Essential Questions
How is an angle formed?
How are angles measured?
How are parallel and perpendicular line segments different?
How are vertical and horizontal lines related?
Are all squares also classified as rectangles?
Area all rectangles also classified as squares?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Ray
 Vertex
 Protractor
 Degrees
 Inner scale, outer scale
 Acute angle
 Obtuse angle
 Right angle
 Straight angle
 Turn
 Additive
 Perpendicular line segments
 Drawing triangle





Parallel line segments
Base
Horizontal lines, vertical lines
Square
Rectangle
Students will be able to:
 Estimate and measure angles with a protractor.
 Estimate whether the measure of an angle is less than or greater than a right angle (90°).
 Use a protractor to draw acute and obtuse angles.
 Relate ¼, ½, ¾, and full turns to the number of right angles (90°).
 Understand that an angle that turns through 1/360 of a circle is called a “one-degree angle.”
 Find unknown angles using addition or subtraction.
 Solve addition and subtraction problems to find unknown angles on a diagram in real-world problems.
 Draw perpendicular line segments.
 Draw parallel line segments.
 Identify vertical and horizontal lines.
 Understand and apply properties of squares and rectangles.
 Find unknown angle measures and side lengths of squares and rectangles.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 9: Angles
Math In Focus On-line Resources Chapter 10: Perpendicular and Parallel Lines
Math In Focus On-line Resources Chapter 11: Perpendicular and Parallel Lines
www.mathleague.com/help/geometry/angles.htm (Angle notes)
www.mathplayground.com/measuringangles.html (Using a protractor)
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 9: Angles
Math In Focus Resources Chapter 10: Perpendicular and Parallel Lines
Math In Focus Resources Chapter 11: Squares and Rectangles
Protractor
Table for Measuring angles
Angle strips
Straightedge
Drawing triangle
Centimeter ruler
Centimeter grid paper
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 4
Unit 7:
Measurement
Summary and Rationale
In this unit, students learn to find the area and perimeter of figures using formulas. They find the
perimeter of composite figures. Students apply the properties of squares and rectangles to find one side of
a square or rectangle given its perimeter or area. They also solve real-world problems involving area and
perimeter of figures.
Pacing
Five (5) weeks
Standards
Measurement and Data
4.MD.1
Know relative sizes of measurement units within one system of units including km, m, cm;
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), ...
4.MD.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.
4.MD.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.
Operations and Algebraic Thinking
4.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Area and perimeter of a square, rectangle, or composite figure can be found by counting squares or using
a formula. The area formula for any rectangle is length x width.
Area is the amount of surface covered by a figure and is measured in square units.
Area can be measured by counting the number of same-sized units of area that cover the shape without
gaps overlaps.
Perimeter is the distance around a figure.
Essential Questions
What is area?
What is perimeter?
How are perimeter and area related? How are they different?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Length
 Width
 Composite figure
Students will be able to:














Understand the relative sizes of measurement units.
Convert metric units of length.
Convert customary units of length.
Convert metric units of mass.
Convert customary units of weight.
Convert units of time.
Use the four operations to solve word problems involving distance, time, volume, mass, and money.
Represent measurement quantities using line diagrams.
Estimate the area of a rectangle using grid squares.
Find the area of a rectangle using a formula.
Solve problems involving the area and perimeter of squares and rectangles.
Find the perimeter and area of a composite figure.
Solve word problems involving estimating areas of figures.
Solve word problems involving area and perimeter of composite figures.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 12: Area and Perimeter
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 12: Area and Perimeter
Geoboards
Dot paper
Square grid paper
Centimeter square grid paper
straightedge
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 4
Unit 8:
Symmetry and Tessellations
Summary and Rationale
In this unit, students learn to identify lines of symmetry of figures and to make symmetric shapes and
patterns. They also learn to identify figures with rotational symmetry. The hands-one activities in this unit
involve folding and cutting patterns to understand line symmetry and rotational symmetry. Students also
complete and create symmetric patterns on grid paper.
Students learn to recognize tessellations, identify the repeated shape used in a tessellation, and recognize
shapes that can tessellate. Students also build on their knowledge of symmetry (both line and rotational)
and congruence to design and analyze simple tessellations.
Pacing
Four (4) weeks
Standards
Geometry
4.G.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.
Operations and Algebraic Thinking
4.OA.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” to
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Figures can have line and rotational symmetry. Figures with rotational symmetry may or may not have
line symmetry.
A tessellation is made when a shape (or shapes) is repeated, covering a plane (or surface) without gaps or
overlaps to form patterns.
Essential Questions
Do all figures with rotational symmetry also have line symmetry?
Do all figures with line symmetry also have rotational symmetry?
Which shapes tessellate? Which do not tessellate and why?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Line of symmetry
 Symmetric figure
 Rotation
 Rotational symmetry
 Center of rotation
 Clockwise, counter-clockwise
 Tessellation
 Repeated shape
 Slide, rotate, flip, modify
Students will be able to:
 Identify a line of symmetry of a figure.
 Relate rotational symmetry to turns.
 Trace a figure to determine whether it has rotational symmetry.
 Draw a shape or pattern about a line of symmetry and check for rotational symmetry.
 Complete a symmetric shape or pattern.
 Create symmetric patterns on grid paper.
 Recognize and make tessellations.
 Identify the unit shape used in a tessellation
 Tessellate shapes in different ways.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 13: Symmetry
Math In Focus On-line Resources Chapter 14: Tessellations
www.math-play.com/Polygon-Game.html (Polygons)
www.mrnussbaum.com/shapeinvaders.htm (Polygon and shapes games)
www.coolmath4kids.com/tesspag1.html (Tessellations notes)
Language Arts Integration
The Greedy Triangle by Marilyn Burns
Math Curse by Jon Scieska
Suggested Resources
Math In Focus Resources Chapter 13: Symmetry
Math In Focus Resources Chapter 14: Tessellations
Shape cut-outs
Drawing triangle
Grid paper
Straightedge
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Grade 5
Nutley Public Schools
Mathematics
Grade 5
Unit 1:
Whole Numbers
Summary and Rationale
In this unit, students represent six-digit and seven-digit numbers in word, standard, and expanded forms.
They extend place-vale to larger numbers as they compare and order. The concept of negative numbers is
very briefly explored by using number lines and real-world situations. Students are asked to identify to
identify rules for number patterns and then complete the patterns. They also estimate sums, differences,
products, and quotients through several methods: rounding, using compatible numbers, and front-end
estimation with adjustment.
Students learn to use the basic functions of a calculator, multiply and divide using patterns and
conventional algorithms, simplify numerical expressions using order of operations, and solve real-world
problems involving multiplication and division using bar models and other strategies.
Pacing
Six (6) weeks
Standards
Number and Operations in Base Ten
5.NBT.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.
5.NBT.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.
5.NBT.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.6
Find whole-number quotients of whole numbers with up to four-digit dividends and twodigit 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.
Operations and Algebraic Thinking
5.OA.1
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.
5.OA.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
5.OA.3
as 18932 + 921, without having to calculate the indicated sum or product.
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.
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Whole numbers can be written in different ways.
Numbers can be compared and rounded, according to their place value.
Patterns can be used to help you multiply and divide numbers.
Numeric expressions can be simplified using the order of operations.
Multiplication and division can be used to solve real-world problems.
Essential Questions
What is the order of operations and why is it necessary?
What kind of number patterns can help when multiplying and dividing by multiples of 10?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Hundred, thousand
 Periods
 Million
 Place-value
 Greater than (>), less than (<)
 Front-end estimation, with adjustment
 Compatible numbers
 Product, factor
 Exponent, base
 Square, cube
 Quotient, dividend, divisor, remainder
 Numeric expression
 Order of operations
Students will be able to:
 Count by ten thousands and hundred thousands to 10,000,000.
 Read and write numbers to 10,000,000 in standard form, in word form, and in expanded form.
 Identify the place value of any digit in numbers to 10,000,000.
 Identify and complete a number pattern.
 Find a rule for a number pattern.
 Round numbers to the nearest thousand.
 Locate numbers on a number line.
 Use rounding to estimate or check sums, difference, and products.
 Used related multiplication facts to estimate quotients.
 Use a calculator to add, subtract, multiply, and divide whole numbers.
 Multiply and divide numbers by 10, 100, or 1,000 using patterns.
 Multiply and divide numbers up to 4 digits by multiples of 10, 100, or 1,000.
 Use rounding to estimate products.
 Multiply whole numbers by 10 squared or 10 cubed.
 Multiply and divide a 2, 3, or 4-digit number by a 2-digit number.
 Use rounding and related multiplication facts to estimate quotients.
 Use order of operations to simplify a numeric expression.
 Evaluate numerical expressions with parentheses, brackets, and braces.
 Use efficient strategies to solve multi-step problems involving multiplications and division.
 Express and interpret the product or quotient appropriately.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Whole Numbers
Math In Focus On-line Resources Chapter 2: Whole Numbers Multiplication and Division
www.funbrain.com/tens/index.html (Place value)
www.gamequarium.com/estimation.html (Estimation)
www.mathplayground.com/division02.html (Division with remainders)
www.kidsnumbers.com/long-division.php (Division practice games)
www.mathplayground.com/WordProblemsWithKatie2.html (Multiplication and division number stories)
www.math-play.com/Order-of-Operations-Millionaire/order-of-operations-millionaire.html (Order of
operations)
Language Arts Integration
Counting on Frank by Rod Clement
How Much is a Million? by David M. Schwartz
Math Talk by Theoni Pappas
Speed Mathematics by Bill Handley
A Remainder of One by Elinor J. Pinczes
One Hundred Hungry Ants by Elinor J. Pinczes
Suggested Resources
Math In Focus Resources Chapter 1: Whole Numbers
Math In Focus Resources Chapter 2: Whole Numbers Multiplication and Division
Place-value chips
Place-value charts
Number lines
Calculators
Multiplication tables
Division tables
Number cards
Symbol cards
Numerical expressions table
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 2:
Fractions and Mixed Numbers
Summary and Rationale
In this unit, students learn to add and subtract unlike fractions and mixed numbers by rewriting the
fractions as like fractions using the concept of least common denominator and equivalent fractions. They
are encouraged to recognize the relationships between fractions, mixed numbers, division expressions,
and decimals. Learning to represent the same number in different ways is a necessary skill for the study of
algebra.
Students learn how to multiply and divide whole numbers, proper fractions, improper fractions, and
mixed numbers in any combinations. Using manipulatives such as fraction bars and circles or using the
pictorial approach by drawing bar models are ideal ways of demonstrating the concepts.
Pacing
Six (6) weeks
Standards
Number and Operations- Fractions
5.NF.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.)
5.NF.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.
5.NF.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?
5.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction or whole
number by a fraction.
5.NF.4a
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.)
5.NF.4b
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.NF.5
Interpret multiplication as scaling (resizing), by
5.NF.5a
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.
5.NF.5b
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.
5.NF.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.
5.NF.7
Apply and extend previous understandings of division to divide unit fractions by whole
numbers and whole numbers by unit fractions.1
5.NF.7a
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.
5.NF.7b
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.
5.NF.7c
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?
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Unlike fractions and mixed numbers can be added and subtracted by first rewriting them with like
denominators.
Whole numbers, fractions, and mixed numbers can be multiplied or divided in any combination.
Dividing a fraction by a whole number is equivalent to multiplying it by the reciprocal of the whole
number.
Essential Questions
What is the first step in adding or subtracting unlike fractions or mixed numbers?
Does the order in which you multiply fractions matter?
How do you divide a fraction by a whole number?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know and learn:
 Multiple, least common multiple (LCM)
 Least common denominator
 Equivalent fractions
 Benchmarks
 Division expression
 Mixed number
 Product, common factor
 Proper fraction, improper fraction
 Reciprocal
Students will be able to:
 Add and subtract two unlike fractions where one denominator is not a multiple of the other.
 Estimate sums of fractions and differences between fractions.
 Understand and apply the relationships between fractions, mixed numbers, and division expressions.
 Express fractions, division expressions, and mixed numbers as decimals.
 Add and subtract mixed numbers with or without renaming.
 Estimate sums of mixed numbers and differences between mixed numbers.
 Solve real-world problems involving fractions and missed numbers.
 Compare the size of a product to the size of its factors.
 Multiply proper fractions.
 Multiply improper fractions by proper or improper fractions.




Multiply a mixed number by a whole number.
Divide a fraction by a whole number.
Divide a whole number by a unit fraction.
Solve real-world problems involving multiplication and division of whole numbers, proper fractions,
improper fractions, and mixed numbers.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 3: Adding and Subtracting Fractions and Mixed Numbers
Math In Focus On-line Resources Chapter 4: Multiplying and Dividing Fractions and Mixed Numbers
www.aaamath.com/fra43ax2.htm (Comparing fractions with like denominators)
www.aaamath.com/fra43bx2.htm (Comparing fractions with unlike denominators)
www.funbrain.com/fract/index.html (Equivalent fraction game)
www.math-play.com/adding-and-subtracting-fractions-game.html (Adding and subtracting fraction game)
www.everydaymathonline.com
www.aaamath.com/fra66dx2.htm (Adding mixed numbers)
www.aaamath.com/fra66ex2.htm (Subtracting mixed numbers)
www.mathplayground.com/fractions_mult.html (Multiplying fractions)
Language Arts Integration
Gator Pie by Louise Matthews
Eating Fractions by Bruce McMillian
Fourscore and 7: Investigating in American History by Betsy Franco
Tiger Math by Ann Whitehead Nagda
Jim and the Beanstalk by Jean Cushman
Fraction Action by Loreen Leedy
Suggested Resources
Math In Focus Resources Chapter 3: Adding and Subtracting Fractions and Mixed Numbers
Math In Focus Resources Chapter 4: Multiplying and Dividing Fractions and Mixed Numbers
Fraction circles
Decimal models
Table of Measures
Grid paper
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 3:
Algebra
Summary and Rationale
Algebra is a language that is used to create mathematical models of real-world situations and handle
problems that we cannot solve using just arithmetic. Rather than using words, algebra uses symbols to
make statements about things. In this unit, students will earn to write both numerical and algebraic
expressions and equations that correspond to given situations. They also learn to simplify and evaluate
expressions, and use expressions, inequalities, and equations to solve real-world problems.
Students learn that variables represent numbers whose exact values are not yet specified. They also learn
that expressions in different forms can be equivalent, as they rewrite expressions to represent a quantity in
different way by simplifying it. Students will solve simple equations by using number sense, properties
of operations, and the idea of maintaining equality on both sides of an equation.
Pacing
Two (2) weeks
Standards
Operations and Algebraic Thinking
5.OA.1
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.
5.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Algebra is a language that is used to create mathematical models of real-world situations and handle
problems that we cannot solve using just arithmetic. Rather than using words, algebra uses symbols to
make statements about things.
Variables represent numbers whose exact values are not yet specified.
Numerical expressions can involve addition, subtraction, multiplication, and division. Numerical
expressions that involve an unknown value are called algebraic expressions.
To solve an equation is to find the exact value of the unknown variable. To do so, inverse operations are
used along with the properties of operations and equality are used on both sides of the equation.
Essential Questions
What does algebra allow us to do?
What is a variable?
Why are variables helpful?
What are the guidelines for solving an equation?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Numerical expression
 Variable
 Algebraic expression
 Evaluate
 Simplify
 Like terms
 Inequality
 Equation
 Equality properties
 Solve
Students will be able to:
 Recognize, write, and evaluate simple algebraic expressions in one variable.
 Simplify algebraic expressions in one variable.
 Write an evaluate inequalities.
 Solve simple equations.
 Solve real-world problems involving algebraic expressions.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 5: Algebra
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 5: Algebra
Recording sheets
Number cards
Letter cards
Balance
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 4:
Geometry: Triangles and Area
Summary and Rationale
In this unit, students learn to find the area of a triangle by correctly identifying and using its base and
height. Students compare the area of a triangle with the area of its corresponding rectangle, that is, a
rectangle sharing the same base and height as that triangle.
Pacing
Two (2) weeks
Standards
Number and Operations – Fractions
5.NF.4b
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.
Mathematical Practices
K-12.MP.4 Model with mathematics.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Base and height are measurements that are used to find the area of a triangle.
The base of a rectangle always perpendicular to its height.
The base is a side of the triangle. However, depending on the triangle, the height may or not be another
side of the triangle. The height must ne perpendicular to the base.
In right triangles, the base and the height are both sides of the triangle. They are the sides that are
perpendicular to each other, forming the right angle. In other triangles, the sides are not perpendicular, so
a dotted line is drawn to represent the height.
Essential Questions
How can you calculate the area of a triangle?
How are the base and height of a rectangle or triangle related?
Are the base and the height always both sides of the triangle?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Vertex
 Side
 Angle
 Base
 Height
 Perpendicular
 Area
 Right triangle
 Acute triangle
 Obtuse triangle
Students will be able to:
 Find the area of a rectangle with fractional side lengths by counting square units, and by using a
formula.
 Identify the base given the height of a triangle.
 Identify the height given the base of the triangle.
 Find the area of a triangle given its base and height.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 6: Area of a Triangle
www.aaamath.com/geo78_x3.htm (Area of a rectangle)
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 6: Area of a Triangle
Grid paper
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 5:
Ratio
Summary and Rationale
In this unit, students learn to compare two numbers using division and express this comparison as a ratio.
They apply the concepts of equivalent ratios, part-whole, part-part, and whole-part comparisons to solve
one and two-step real-world problems involving ratios. The study of ratios in this unit is extended to
involve three quantities.
Pacing
Two (2) weeks
Standards
Number and Operations – Fractions
5.NF.5
Interpret multiplication as scaling (resizing), by
5.NF.5a
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Two numbers can be compared by subtraction.
Two or more numbers or quantities can also be compared by division and the comparison expressed as a
ratio.
A ratio expresses the relationship between two numbers or quantities that have the same units.
A ratio can be written as a fraction or using a colon.
Essential Questions
In what different ways can numbers be compared?
What is a ratio?
In what different ways can a ratio be written?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Ratio
 Term
 Equivalent ratios
 Simplest form
 Greatest common factor (GCF)
Students will be able to:
 Read and write ratios.
 Find equivalent ratios.
 Interpret ratios given in fraction form.
 Write ratios in fraction form to find how many times as large as one number another number is.
 Read and write ratios with three quantities.
 Express equivalent ratios with three quantities.
 Solve real-world problems involving ratios and fractions.
 Solve real-world problems involving ratios with three quantities.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 7: Ratio
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 7: Ratio
Counters
Connecting cubes
Ten frames
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 6:
Decimals
Summary and Rationale
In this unit, students are introduced to the place value of decimals through thousandths. In the process,
they learn to how to read and write decimals through thousandths, identify the relationship between
fractions and decimals, compare and order decimals, and round decimals to the nearest hundredth.
Students use patterns to help them multiply and divide decimals by one-digit whole numbers. They also
learn conventional algorithms for multiplying and dividing decimals by whole numbers, make reasonable
estimates of decimal sums, differences, products, and quotients. Students solve problems involving
decimals, including multi-step problems and problems involving measurement.
Pacing
Five (5) weeks
Standards
Number and Operations in Base Ten
5.NBT.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.
5.NBT.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.
5.NBT.3
Read, write, and compare decimals to thousandths.
5.NBT.3a
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).
5.NBT.3b Compare two decimals to thousandths based on meanings of the digits in each place, using
>, =, and < symbols to record the results of comparisons.
5.NBT.4
Use place value understanding to round decimals to any place.
5.NBT.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.
Number and Operations – Fractions
5.NF.5
Interpret multiplication as scaling (resizing), by
5.NF.5b
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Decimals are another way of writing fractions or mixed numbers?
Thousandths can be represented with three decimal places or as fractions.
Decimals can be multiplied and divided in the same way as whole numbers.
Essential Questions
How are fractions and decimals related?
How is multiplying and dividing different when working with decimals compared to when working with
whole numbers?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Thousandth
 Equivalent
 Estimate
 Divisor
Students will be able to:
 Read and write thousandths in decimal and fraction forms.
 Represent and interpret thousandths in models or in place-value charts.
 Write a fraction with denominator 1,000 as a decimal.
 Compare and order decimals to 3 decimal places.
 Round decimals to the nearest hundredth.
 Rewrite decimals as fractions and mixed numbers in simplest form.
 Multiply and divide tenths and hundredths by a one-digit whole number.
 Multiply and divide tenths and hundredths by 10, 100, and 1,000.
 Multiply and divide tenths and hundredths by multiples of 10, 100, and 1,000.
 Multiply decimals by 10 squared or 10 cubed.
 Round quotients to the nearest tenth or hundredth.
 Estimate decimal sums, differences, products, and quotients.
 Solve real-world problems involving decimals.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 8: Decimals
Math In Focus On-line Resources Chapter 9: Multiplying and Dividing Decimals
www.funbrain.com/football/ (Decimal division game)
www.mrnussbaum.com/deathdeciamials.htm (Converting fractions to decimals game)
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 8: Decimals
Math In Focus Resources Chapter 9: Multiplying and Dividing Decimals
Base-ten blocks
Place-value chips
Place-value charts
Number lines
Connecting cubes
Multiplication tables
Division tables
Tables of Measures
Measuring tape
Rulers
Bill and coin cut-outs
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 7:
Percent
Summary and Rationale
In this unit, students are introduced to the concept of percent. They learn that percent can be expressed as
a fraction with a denominator of 100 and study the relationship between fractions, decimals, and percents.
Students find percent of a number and solve real-world problems involving percent, including concepts
such as sales tax, discount, and interest.
Pacing
Two (2) weeks
Standards
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Percent is another way of expressing a part of a whole.
A percent can be expressed as a fraction with a denominator of 100.
Percents are used everywhere in the real world. When fractions are expressed as percents, they already
have a common denominator, which makes for much easier computation.
Essential Question
What does percent mean?
Where and when are percents used?
Why is percent helpful?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Percent
 Sales tax
 Meals tax
 Discount
 Interest
Students will be able to:
 Relate and compare percents.
 Express fractions as percents.
 Use different ways to find the number represented by a percent.
 Solve real-world problems involving percents.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 10: Percent
www.aaamath.com/rat61cx2.htm (Fractions to percents)
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 10: Percent
Percent models
Number lines
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 8:
Data and Probability
Summary and Rationale
In this unit, students learn to make and interpret double bar graphs as well as graph linear equations on
coordinate grids. They apply their understanding of whole numbers, fractions, and decimals as they
construct and analyze double bar graphs.
Students learn to read and plot points on a coordinate grid, and to graph an equation that represents a
functional relationship between two quantities.
Students learn to find and compare experimental and theoretical probabilities. They learn to list all
possible combinations for compound events. These lists are then used to find probabilities of compound
events.
Pacing
Three (3) weeks
Standards
Measurement and Data
5.MD. 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.
5.MD.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.
Geometry
5.G.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).
5.G.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.
Operations and Algebraic Thinking
5.OA.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Displaying data in a graph highlights some features of the data.
In a double bar graph, each item on the horizontal axis is represented by two bars, one for each set of data.
The graph of an equation on a coordinate grid often represents a functional relationship between two
quantities.
Probability measures the likelihood of an event occurring.
Experimental probability is used mainly for events that are not equally likely, such as weather predictions.
Its calculation is based on how frequently a desired outcome happens compared to the total number of
trials or experiments.
Essential Questions
How is a double bar graph useful in displaying and analyzing data?
What are the differences between theoretical and experimental probability?
How does a systematic list help determine probabilities of compound events?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Double bar graph
 Key
 Coordinate grid, coordinate plane
 X-axis, y-axis
 Ordered pair
 X-coordinate, y-coordinate
 Origin
 Straight line graph
 Equation
 Favorable outcome
 Theoretical probability
 Experimental probability
Students will be able to:
 Make and interpret a double bar graph.
 Make a line plot to represent data given in fractions of a unit.
 Use operations on fractions to solve problems on the information presented.
 Read and plot points on a coordinate grid.
 Graph an equation.
 List and count all possible combinations.
 Draw a tree diagram to show all possible combinations.
 Use multiplication to find the number of possible combinations.
 Find the experimental probability of an outcome.
 Compare the results of an experiment with the theoretical probability.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 11: Graphs and Probability
http://exchange.smartteck.com/#tab=0 (Interactive SmartBoard lessons)
www.mrnussbaum.com/cardsharks.htm (Probability)
www.mrnussbaum.com/coolgraphing.htm (Several graphs to practice organizing data)
www.mathplayground.com/locate_aliens.html (Coordinates)
www.mrnussbaum.com/stockshelves.htm (Coordinates)
www.emgames.com
www.gamequarium.com/data.html (Rules, tables, and graphs)
www.everydaymathonline.com
Language Arts Integration
Jumanji by Chris Van Allsburg
About Probability by Martha Wetson
Suggested Resources
Math In Focus Resources Chapter 11: Graphs and Probability
Number cubes
Graphs
Rulers
Measuring tape
Coordinate grids
Grid paper
Connecting cubes
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 9:
Two-Dimensional Geometry & Measurement
Summary and Rationale
In this chapter, students are introduced to properties of angles on a line, angles at a point, and vertical
angles. Students verify thee properties through hands-on activities and then apply them to finding
unknown angle measures without the use of a protractor.
Students learn the properties of triangles and four-sided figures. They learn to identify special triangles
such as right, isosceles, and equilateral triangles, categorized by angle measures and/or side lengths.
Students learn and apply the Triangle Sum Property to solve problems. Triangle inequalities are also
taught in this unit. As students carry out hands-on activities, they discover for themselves that the sum of
the lengths of any two sides of a triangle is always greater than the length of the third side. Students also
extend their knowledge of the properties of four-sided figures to parallelogram, rhombus, and trapezoid.
Pacing
Five (5) weeks
Standards
Geometry
5.G.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.
5.G.4
Classify two-dimensional figures in a hierarchy based on properties.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The sum of angle measures on a line is 180°.
The sum of angle measures at a point is 360°.
Vertical angles have equal measures.
Properties of geometric figures state relationships among angles or sides of the figures.
Triangles and four-sided figures have their own special properties.
Triangles can be classified by their side lengths and by their angle measures.
A scalene triangle is a triangle with no equal angles or sides.
An isosceles triangle is a triangle with two equal sides (legs) and two equal angles (base angles).
An equilateral triangle is a triangle with three equal sides and three equal angles.
Triangle Sum Property: The sum of the angle measures of a triangle is 180°.
The sum of the lengths of any two sides of a triangle is always greater than the length of the third side.
A parallelogram is a four-sided figure (a quadrilateral) with two pairs of opposite sides that are parallel
and equal. Opposite angles of a parallelogram are equal.
A rectangle is a parallelogram with four right angles.
A rhombus is a parallelogram with four equal sides.
A square is a parallelogram with four right angles and four equal sides.
A trapezoid is a four-sided figure (a quadrilateral) with one pair of opposite sides. (Therefore, a trapezoid
is not a parallelogram.
Essential Questions
What is the sum of the measures of angles on a line? At a point?
What is true about vertical angles?
In what ways can triangles be classified?
Can line segments of any lengths form a triangle?
Why isn’t a trapezoid classified as a parallelogram?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Angles on a line
 Angles at a point
 Intersecting lines
 Vertical angles
 Equilateral triangle
 Isosceles triangle
 Scalene triangle
 Right triangle
 Acute triangle
 Obtuse triangle
 Parallelogram
 Rhombus
 Trapezoid
Students will be able to:
 Understand and apply the property that the sum of the angle measures on a line is 180°.
 Understand and apply the property that the sum of the angle measures at a point is 360°.
 Understand and apply the property that vertical angles have equal measures.
 Classify triangles by the lengths of their side lengths and angle measures.
 Understand and apply the property that the sum of the angle measures of a triangle is 180°.
 Understand and apply the properties of right, isosceles, and equilateral triangles.
 Understand that the sum of the lengths of any two sides of a triangle is greater than the length of the
third side.
 Understand and apply the properties of parallelogram, rhombus, and trapezoid.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 12: Angles
Math In Focus On-line Resources Chapter 13: Properties of Triangles and Four-Sided Figures
Language Arts Integration
The Greedy Triangle by Marilyn Burns
Suggested Resources
Math In Focus Resources Chapter 12: Angles
Math In Focus Resources Chapter 13: Properties of Triangles and Four-Sided Figures
Protractors
Inch and centimeter rulers
Tracing paper
Drawing triangles
Grid paper
Shape cut-outs
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 5
Unit 10:
Three-Dimensional Geometry and Measurement
Summary and Rationale
In this unit, students learn to recognize three-dimensional solid shapes and identify nets that can form
some of these solids. The solids emphasized in this unit are prisms, pyramids, cylinders, and cones.
Students learn the fundamental concepts and vocabulary of solid shapes, such as vertex/vertices, edges,
faces, and bases. Models play an important role in the study of solids. Students see, touch, and manipulate
models of solids to consolidate their understanding.
Students build solids using unit cubes, draw cubes and rectangular prisms on dot paper, find the surface
areas of cubes and prisms, and find the volumes of cubes, rectangular prisms, and liquids in rectangular
containers.
Students are expected to recognize area as an attribute of two-dimensional shapes and volume as an
attribute of three-dimensional shapes. Because a net is a two-dimensional representation of a threedimensional solid, it can be used to find the surface area of a solid.
There are to different contexts to understand volume- the capacity of containers, and the amount of space
taken up by objects. Both of these contexts are explored in this unit. Volume is also measured using both
solid and liquid measures.
Pacing
Five (5) weeks
Standards
Measurement and Data
5.MD. 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.
5.MD.3
Recognize volume as an attribute of solid figures and understand concepts of volume
measurement.
5.MD.3a
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.
5.MD.3b
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.
5.MD.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and non-
standard units.
Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
5.MD.5a
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.
5.MD.5b
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.
5.MD.5c
Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non-overlapping parts,
applying this technique to solve real world problems.
Number and Operations in Base Ten
5.NBT.5
Fluently multiply multi-digit whole numbers using the standard algorithm.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
5.MD.5
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Solid figures can be identified and classified by the number of faces, edges, and vertices.
The volume of cubes and rectangular prisms can be expressed and the number of cubic units they contain.
The surface area of a solid is the sum of the areas of its faces.
Area is as an attribute of two-dimensional shapes and volume is as an attribute of three-dimensional
shapes. Because a net is a two-dimensional representation of a three-dimensional solid, it can be used to
find the surface area of a solid.
Volume is the capacity of containers, and is also the amount of space taken up by objects.
Volume is also measured using both solid and liquid measures.
Essential Questions
How is the volume of any prism calculated?
How is volume different from surface area?
When is volume calculated in daily life?
When is surface area calculated in daily life?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Face, base, edge, vertex
 Prism
 Rectangular prism, triangular prism
 Pyramid
 Square pyramid, triangular pyramid
 Net
 Cylinder
 Sphere
 Cone
 Unit cube
 Surface area
 Right triangle
Students will be able to:
 Identify and classify prisms and pyramids.
 Identify the solid figure that can be formed from a net.
 Identify and classify cylinders, spheres, and cones.
 Build solids using unit cubes.
 Determine the number of unit cubes in an irregular solid.
 Draw a cube and a rectangular prism on dot paper.
 Complete a partially drawn cube and rectangular prism on dot paper.
 Find the surface area of a prism by adding the area of each face.
 Find the volumes of cubes and rectangular prisms.
 Find the volume of a solid constructed by unit cubes.
 Compare volumes of cubes, rectangular prisms, and other objects.
 Use a formula to find the volume of a rectangular prism.
 Find the capacity of a rectangular container.
 Find the volume of a figure composed of two rectangular prisms.
 Solve word problems involving volumes of rectangular prisms, liquids, and of figures composed of
two rectangular prisms.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 14: Three-Dimensional Shapes
Math In Focus On-line Resources Chapter 15: Volume of Cube and Rectangular Prism
www.mathsisfun.com/geometry/prisms.html (Volume of right prisms –Notes and practice)
Language Arts Integration
Room for Ripley by Stuart J. Murphy
G is for Googoi by David M. Schwartz
Suggested Resources
Math In Focus Resources Chapter 14: Three-Dimensional Shapes
Math In Focus Resources Chapter 15: Volume of Cube and Rectangular Prism
Solid shapes
Number cubes
Net of solid shapes
Connecting cubes
Dot paper
Containers
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Grade 6
Nutley Public Schools
Mathematics
Grade 6
Unit 1:
Positive and Negative Numbers
Summary and Rationale
In this unit, students learn that a single number can be represented in many ways. A number line can help
students compare and order positive and negative whole numbers by visualizing the relationship among
the numbers’ positions. Students apply their knowledge of prime factorization to find the greatest
common factor and the least common multiple of a set of numbers. They also apply their knowledge of
squaring and cubing to evaluate numerical expressions and extend the order of operations to include
exponents. Students solve problems involving absolute value and see how it can be used to interpret realworld situations involving positive and negative numbers.
Pacing
Four (4) weeks
Standards
The Number System
6.NS.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 the two whole numbers with no common factor. For example, express 36 + 8 as
4 (9 + 2).
6.NS.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.
6.NS.6a
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.
6.NS.7
Understand ordering and absolute value of rational numbers.
6.NS.7a
Interpret statements of inequality as statements about 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 orientated from left to right.
6.NS.7b
Write, interpret, and explain statements of order for rational numbers in real-world contexts.
For example, write -3 °C > -7°C to express the fact that -3°C is warmer than -7 °C.
6.NS.7c
Understand that 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, doe an account balance of -30 dollars, write │-30│= 30 to describe
the size of the debt on dollars.
Distinguish comparisons of absolute value from statements about order. For example,
recognize that an account balance lee than -30 dollars represents a debt greater than 30
dollars.
Expressions and Equations
6.EE.1
Write and evaluate numerical expressions involving whole number exponents.
6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.2c
Evaluate expressions at specific values of their variables. Include expressions that arise from
formulas used in real-world problems. Perform arithmetic operations, including those
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 formula V = s3 and
A = 6s2 to find the volume and surface area of a cube with side lengths s= ½ .
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
6.NS.7d
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Whole numbers, fractions, decimals, and positive and negative numbers can be represented in several
ways.
Negative numbers are the opposites of positive numbers. For every positive number, there is a
corresponding negative number.
On a horizontal number line, numbers increase in value from left to right. Positive numbers are to the
right of 0 and negative numbers are to the left of zero. The number 0 is neither positive or negative.
On a vertical number line, numbers increase from the bottom to the top.
Prime numbers are numbers that have exactly two factors, the number itself and one. The first prime
number is 2.
Composite numbers are numbers greater than one that have more than two factors.
The Fundamental Theorem of Arithmetic states that every integer greater than one can be expressed as a
product of prime factors in only one way (except for the order of the factors).
The squares and cubes of whole numbers are called perfect squares and perfect cubes.
The order of operations is a set of rules for evaluating expressions to ensure that the solution is always the
same.
The absolute value of a number is the distance from 0 on a number line. Because it represents distance,
the absolute value of a number is always positive. Two opposites have the same absolute value. The
greater the absolute vale of a negative number, the smaller the number.
Essential Questions
How can we compare and order numbers?
How can positive and negative numbers represent real-world quantities?
Why is the order of operations necessary?
What is absolute value?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Number line
 Positive number, negative number, opposite
 Composite number
 Prime factor
 Common factor, greatest common factor (GCF)
 Common multiple, least common multiple (LCM)
 Square (of a number)
 Exponent, base (of an exponent)
 Perfect square
 Square root
 Cube (of a number)
 Perfect cube
 Cube root
 Absolute value
Students will be able to:
 Represent whole numbers, fractions, and decimals on a number line.
 Interpret and write statements of inequality for two given positive numbers using the symbols > or
<.












Express a whole number as a product of its prime factors.
Find the common factors and the greatest common factor (GCF) of two whole numbers.
Find the common multiples and the least common multiple (LCM) of two whole numbers.
Find the square of a number.
Find the square root of a perfect square.
Find the cube of a number.
Find the cube root of a perfect cube.
Evaluate numerical expressions involving whole number exponents.
Use negative numbers to represent real-world quantities.
Represent, compare, and order positive and negative numbers on a number line.
Understand that the absolute value of a number is the distance from 0 on a number line.
Interpret absolute value as the magnitude for a positive or negative quantity in a real-world situation.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: Positive Numbers and the Number Line
Math In Focus On-line Resources Chapter 2: Negative Numbers and the Number Line
www.studyisland.com
www.ixl.com
www.brainpop.com
Language Arts Integration
Suggested Resources
Math In Focus On-line Resources Chapter 1: Positive Numbers and the Number Line
Math In Focus On-line Resources Chapter 2: Negative Numbers and the Number Line
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 6
Unit 2:
Multiplying and Dividing Fractions and Decimals
Summary and Rationale
In this unit, students learn how to divide fractions and to multiply and divide decimals. They apply
multiplication skills to real-world problems involving fractions and decimals. Students draw and revise
bar models to aid in solving multi-step real-world problems.
Pacing
Three (3) weeks
Standards
The Number System
6.NS.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 ¾ 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 ½ lb of chocolate equally? How
many ¾-cup servings are in a 2/3 of a cup of yogurt? How wide is a rectangular strip of
land with length ¾ mi and area ½ square mi?
6.NS.2
Fluently divide multi-digit numbers using the standard algorithm.
6.NS.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard
algorithm for each operation.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understanding
Whole number concepts can be extended to fractions and decimals when more precise calculations are
needed.
Estimating a product or quotient can help determine where to place the decimal point.
Essential Question
Why are fractions and decimals necessary?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Reciprocals
Students will be able to:
 Divide a fraction, whole number, or mixed number by a fraction or a mixed number
 Multiply a decimal by a decimal.
 Divide a whole number or a decimal by a decimal.
 Solve problems involving fractions and decimals.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 3: Multiplying and Dividing Fractions and Decimals
www.studyisland.com
www.ixl.com
www.brainpop.com
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 3: Multiplying and Dividing Fractions and Decimals
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 6
Unit 3:
Ratio, Rates, and Percent
Summary and Rationale
In this unit, students extend concepts learned with fractions to ratios. Along with writing equivalent ratios
and writing ratios in simplest form, they use comparison models and the unitary method to solve many
types of ratio problems involving two or three quantities, two related sets, and ratios that change.
Students then extend their knowledge of ratios to the concept of rate They use the unitary method and bar
models to find rates and unit rates, and solve real-world rate problems.
Students use bar models to visualize percents, and solve problems using both the unitary method and
traditional methods. Students write equivalent fractions, decimals, and percents. They use bar models to
visualize and solve problems that involve finding a percent given a part and a whole, finding a part given
its percent and the whole, finding the whole given a part and its percent, and finding percent increase,
percent decrease, or amount of increase or decrease.
Pacing
Eight (8) weeks
Standards
Ratios and Proportional Relationships
6.RP.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.”
6.RP.2
Understand the concept of a unit rate a/b associated with a ratio a:b with b not equal to zero,
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 ¾ cup of flour for each cup of sugar.”
“We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
6.RP.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.
6.RP.3a
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.
6.RP.3b
Solve unit rate problems including those involving unit pricing and constant speed. For
example, if it took 7 hours to row 4 lawns, then at that rate, how many lawns could be
mowed in 35 hours? At what rate were lawns being mowed?
6.RP.3c
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.
6.RP.3d
Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
The Number System
6.NS.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 ¾ 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 ½ lb of chocolate equally? How
many ¾-cup servings are in a 2/3 of a cup of yogurt? How wide is a rectangular strip of
land with length ¾ mi and area ½ square mi?
6.NS.2
Fluently divide multi-digit numbers using the standard algorithm.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A ratio can be used to compare two quantities, and can be used to solve problems.
A rate can be used to compare one quantity to another quantity, and can be used to solve problems.
Percent is a concept used to compare quantities expressed per hundred.
Percent means “per hundred” or “out of 100.” A given percent is a numerator on a fraction with a
denominator of 100.
Essential Questions
How are the concepts of ratio, rate, and percent used to solve real world problems?
Where when are ratios, rate, and percents used?
How are fractions, decimals, and percents related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Ratio
 Equivalent ratios
 Simplest form
 Rate, unit rate
 Speed, average speed
 Percent
 Base
 Sales tax
 Commission
 Interest, interest rate
 Markup
 Discount
Students will be able to:
 Write ratios to compare two quantities.
 Interpret ratios given in fraction form.
 Use a ratio to find what fraction one quantity is of another or how many times as great one is as the
other.
 Write equivalent rations.
 Write ratios in simplest form
 Compare ratios.
 Solve real-world problems involving ratios.
 Solve unit rate problems including unit pricing and constant speed.
 Solve problems involving unit rates and rates.
 Understand percent notation.
 Write equivalent fractions, decimals, and percents.
 Find the percent of a number.
 Solve problems involving percent, percent increase, and percent decrease.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 4: Ratio
Math In Focus On-line Resources Chapter 5: Rates
Math In Focus On-line Resources Chapter 6: Percent
www.studyisland.com
www.ixl.com
www.brainpop.com
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 4: Ratio
Math In Focus Resources Chapter 5: Rates
Math In Focus Resources Chapter 6: Percent
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 6
Unit 4:
Algebra
Expressions, Equations, and Inequalities
Summary and Rationale
In this unit, students write algebraic expressions to represent situations in the world around them. They
learn to use variables to represent unknown quantities and to correctly identify the terms in algebraic
expressions. Students expand and factor algebraic expressions and use substitution to evaluate algebraic
expressions for given values.
Students relate the use of bar models and number properties to algebraic expressions in order to solve
real-word problems, expand algebraic expressions, and recognize equivalent algebraic expressions.
Students learn to think of the = symbol as meaning that two expressions have the same value. This leap in
abstraction should be accompanied by as much work with wither a balanced scale as possible. Students
use inverse operations to “get the variable alone” on one side of an equal sign to solve an equation. This
reliance on the properties if equality establishes a strong base for future work in algebra. Students can use
substitution to check the accuracy of a solution to an equation. The solution is the value that makes the
equation a true statement.
Students learn to think of the symbols > and < as meaning that two expressions have different values. This
concept can be visualized using an unbalanced scale. Students are also introduced to the symbols ≥ and ≤,
which expands their conception of how two quantities or expressions compare. They extend their use of
substitution to determine whether a given number is a solution to an inequality and use number lines to
represent a visual solution to one-variable inequalities.
The term linear equation is introduced in this unit. Students learn that the graphed solutions of simple
two-variable equations are lines that contain an infinite number of solutions, including not just whole
numbers, but also fractions, mixed numbers, and decimals.
Pacing
Six (6) weeks
Standards
Expressions and Equations
6.EE.2
Write, read, and evaluate expressions in which letters stand for numbers.
6.EE.2a
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.
6.EE.2b
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.
6.EE.2c
Evaluate expressions at specific values of their variables. Include expressions that arise from
formulas used in real-world problems. Perform arithmetic operations, including those
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 formula V = s3 and
A = 6s2 to find the volume and surface area of a cube with side lengths s= ½ .
6.EE.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.
6.EE.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. Reason about and solve one-variable equations and inequalities.
6.EE.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.EE.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.
6.EE.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
6.EE.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 in the form x > c or x < c have
infinitely many solutions; represent solutions of such inequalities on number line diagrams.
6.EE.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 distance and times, and write the equation d=65t to
represent the relationship between distance and time.
The Number System
6.NS.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.
6.NS.6b
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.
6.NS.6c
6.NS.8
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.
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Variables are used to represent unknown quantities.
Algebraic expressions can be used to describe situations and solve real-world problems.
Algebraic expressions are sometimes called variable expressions because they contain one or more
variables.
To evaluate an algebraic expression for a given value, substitute the value in for the variable and simplify
the remaining numerical expression.
When simplifying numerical or algebraic expressions, the expression obtained after simplifying is
equivalent to the original expression.
The = symbol as meaning that two expressions have the same value.
To solve a one-variable equation, use inverse operations to “get the variable alone” on one side of an
equal sign to solve an equation. Substitution to check the accuracy of a solution to an equation. The
solution is the value that makes the equation a true statement.
The symbols > and < as meaning that two expressions have different values.
The solutions to a one-variable inequality can be represented using a number line.
The graphed solutions of simple two-variable linear equations are lines that contain an infinite number of
solutions, including not just whole numbers, but also fractions, mixed numbers, and decimals.
Equations and inequalities can be used to describe situations and solve real-world problems.
The coordinate plane is formed by the intersection of two number lines. The plane is divided into four
regions, called quadrants. The origin is the place where the two number lines intersect.. An ordered pair is
a pair of numbers that describes the location of a point in a coordinate plane.
Any point on a coordinate plane can be named by an ordered pair of numbers, and you can graph any
ordered pair of real numbers as a point on the plane.
An ordered pair (x, y) is ordered because the horizontal coordinate is named first. If the order is reversed
the location of the point is changed.
Essential Questions
What is the difference between an algebraic expression and an equation?
How are equations different from inequalities?
How do you evaluate an algebraic expression given a specific value for the variable?
How do you solve a simple equation?
What is the best way to show the solutions to a one-variable inequality? Why?
What do the solutions to a linear equation look like? Can the solutions be counted?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Variable
 Algebraic expression
 Terms
 Evaluate
 Substitute
 Simplify
 Coefficient
 Like terms
 Equivalent expressions
 Expand
 Factor
 Equation
 Solution
 Linear equation
 Independent variable, dependent variable
 Inequality
 Coordinate plane



Origin
Quadrants
Ordered pair, x-coordinate, y-coordinate
Students will be able to
 Use variables to write algebraic expressions.
 Evaluate algebraic expressions for given values of the variable.
 Simplify algebraic expressions in one variable.
 Recognize the expression obtained after simplifying is equivalent to the original expression.
 Expand and factor algebraic expressions.
 Solve real-world problems involving algebraic expressions.
 Solve equations in one variable
 Express the relationship between two quantities as a linear equation.
 Use a table or graph to represent a linear equation.
 Use substitution to determine whether a given number is a solution of an inequality.
 Represent the solutions of an inequality on a number line.
 Solve real-world problems by writing equations and inequalities.
 Name and graph points on a coordinate plane.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 7: Algebraic Expressions
Math In Focus On-line Resources Chapter 8: Equations and Inequalities
Math In Focus On-line Resources Chapter 9: The Coordinate Plane
www.studyisland.com
www.ixl.com
www.brainpop.com
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 7: Algebraic Expressions
Math In Focus Resources Chapter 8: Equations and Inequalities
Math In Focus Resources Chapter 9: The Coordinate Plane
Rulers
Yardsticks
Algebra tiles
Balance scale
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 6
Unit 5:
Geometry
Summary and Rationale
Students will explore and identify the differences between a two dimensional figure and a threedimensional figure. Formulas for calculating perimeter, area, and circumference for two-dimensional
figures as well as formulas for calculating surface area and volume of three-dimensional figures will be
modeled.
Calculating measurements for two and three-dimensional figures is often used by homeowners when
working on household projects. Many contractors, such as, painters, carpenters, masons, and landscapers
will use these formulas when problem solving to calculate cost and materials.
Pacing
Five (5) weeks
Standards
Geometry
6.G.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.
6.G.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
= lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in
the context of solving real world and mathematical problems.
6.G.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.
6.G.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Perimeter and circumference are used to calculate the distance around a two-dimensional figure.
Area is used to calculate the covering of a two-dimensional figure and is measured in square units.
The area of irregular polygons can often be found by separating it into familiar figures, finding the area of
each smaller figure, and adding the areas together.
A three-dimensional figure has the three dimensions of length, width, and height.
Prisms and pyramids are named by the shapes of their bases.
Surface area is used to calculate the covering of a three-dimensional figure. A net of a three-dimensional
figure can be used to calculate the surface area. The surface area is the total area of all faces.
Volume is used to calculate the capacity of a three-dimensional container.
Essential Question
Why is it important to be able to calculate the perimeter, circumference, and area, of two-dimensional
figures?
How is surface area different from volume?
Why is it important to be able to calculate the surface area and volume of three-dimensional figures?
Evidence of Learning (Assessments)
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Two-dimensional geometric figures have an area, and a perimeter or circumference

Three-dimensional geometric figures have a surface area and volume
Students will be able to:
 Calculate the area and perimeter of a triangle, and quadrilateral, and area and circumference of a circle
 Calculate the area of composite shapes
 Calculate the volume and surface area of a right rectangular prism
 Draw polygons on a coordinate plane given the coordinates of vertices
 Represent three-dimensional figures using nets to calculate surface areas
 Use knowledge of area, perimeter, circumference, volume, and surface area to solve real world
problems
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 10: Area of Polygons
Math In Focus On-line Resources Chapter 11: Circumference and Area of a Circle
Math In Focus On-line Resources Chapter 12: Surface Area and Volume of Solids
www.studyisland.com
www.ixl.com
www.brainpop.com
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 10: Area of Polygons
Math In Focus Resources Chapter 11: Circumference and Area of a Circle
Math In Focus Resources Chapter 12: Surface Area and Volume of Solids
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Grade 6
Unit 6:
Statistics
Summary and Rationale
Students will discover how to collect, organize, and display data. Choosing the appropriate graph is
situational, yet specific graphs are used for displaying different types of data. Students will calculate the
probability of an event occurring, based on comparing the possible outcomes to the total outcomes. These
ratios are used to make predictions on the chance the event will occur.
Many companies in the business world will use statistical analysis to make very important decisions.
Advertising and marketing companies collect, organize, and display data, then analyze the information.
Pacing
Four (4) weeks
Standards
Statistics and Probability
6.SP.1
Recognize a statistical question as one that anticipates variability in the data related to the
question and accounts for it in the answers.
6.SP.2
Understand that a set of data collected to answer a statistical question has a distribution that
can be described by its center, spread, and overall shape.
6.SP.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.
6.SP.4
Display numerical data in plots on a number line, including dot plots, histograms, and box
plots.
6.SP.5
Summarize numerical data sets in relation to their context such as by: reporting the number
of observations, and giving quantitative measures of center and variability.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Statistical data identifies central tendency, or patterns used to make inference that influence the decision
making process.
There are different statistics for describing the “the center” of a numerical data set. A measure of center
describes how the data within a set is centered.
The mean, median, and mode are common measures for central tendency of a data set.
A frequency table shows how often each data value occurs.
A box-and-whisker plot displays and analyzes how a data set is distributed by emphasizing five key
values and dividing the data into four equal parts.
A bar graph uses horizontal or vertical bars to display numerical information. When the numerical
information is grouped into equal intervals, the bar graph is called a histogram.
Data can be described through observations of peeks, gaps, clusters, and symmetry of lack of symmetry.
Survey questions are statistical questions that usually have more than one answer when asked of a group
of people. The questions should contain language that is neutral and does not lead people to answer in a
particular way.
Essential Questions
How can statistical data and probability be used to influence decisions and make predictions?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Statistical variability
 Display of data
 Mean
 Median
 Mode






Range
Frequency tables, dot plots
Bar graph, histogram
Box-and-whisker plot
Statistical question
Shapes of distribution
Students will be able to:
 Recognize a statistical question as one that anticipates a range in the collected data
 Collect, organize, and display data on appropriate graph; including bar graph, line graph, stem and
leaf plot, and histogram.
 Calculate measures of central tendency; mean, median, and mode, for a set of data
 Identify and predict trends and deviations of data given measures of central tendency and variability
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 13: Introduction to Statistics
Math In Focus On-line Resources Chapter 14: Measures of Central Tendency
www.studyisland.com
www.ixl.com
www.brainpop.com
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 13: Introduction to Statistics
Math In Focus Resources Chapter 14: Measures of Central Tendency
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Pre-Algebra
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 1:
Real Numbers
Summary and Rationale
In this unit, students extend their knowledge of numbers (whole numbers, integers, fractions, and
decimals) to irrational numbers. They identify the numbers that make up the set of rational numbers and
those that make up the set of real numbers. They locate numbers from both sets on the number line.
Students learn about significant digits and precision and their importance in measurement.
Students add and subtract integers with the same sign and with different signs. They learn how to add
integers to their opposites and subtract integers by adding their opposites. Students find the distance
between two integers on a number line.
Students multiply and divide integers and then evaluate expressions that include any combination of
operations.
Students extend their operations skills to rational numbers, including decimals and percents, and they use
their new skills to solve real-world problems.
Pacing
Seven (7) weeks
Standards
The Number System
7.NS.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.
7.NS.1a
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?
7.NS.1c
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.
7.NS.1d
Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions to
multiply and divide rational numbers.
7.NS.2a
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.
7.NS.2c
Apply properties of operations as strategies to multiply and divide rational numbers.
7.NS.2d
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.
7.NS.3
Solve real-world and mathematical problems involving the four operations with rational
numbers.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Real numbers are represented as points on an infinite number line and are used to count, measure,
estimate, or approximates quantities.
A rational number is a number that can be written as a fraction, in the form a/b. Between every pair of
rational numbers, there is another rational number.
Between any two real numbers, there is another real number. The real numbers, which contain both
rational and irrational numbers, complete the real number line.
The square root of a number increases as the number increases.
The results of a calculation with measurements should not have more significant digits than the data used
for the calculation. A result should never be more precise than the measures from which it was
calculated.
The operations of addition, subtraction, multiplication, and division can be applied to rational numbers,
including negative numbers.
Subtraction is the same as adding the opposite.
The product of two integers with the same sign is positive, and the product of two integers with different
signs is negative. The same generalizations hold for division.
Essential Questions
What is a rational number? How is it different form an irrational number?
What happens when you multiply or divide two negative numbers?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Opposites
 Set of Integers, Positive integers, negative integers
 Rational numbers, positive fractions, negative fractions
 Terminating decimal
 Repeating decimal
 Irrational numbers
 Approximate
 Real number
 Real number line
 Significant digits
 Precision
 Additive inverse
 Zero pair
 Complex fraction
 Least common denominator (LCM)
Students will be able to:
 Find the absolute values of rational numbers.
 Express numbers in m/n form.
 Locate rational numbers on the number line.
 Write rational numbers as terminating or repeating decimals using long division.
 Compare rational numbers on the number line.
 Understand irrational numbers and how they fill in the number line.
 Use rational numbers to locate irrational numbers approximately on the number line.
 Show that irrational numbers are characterized by non-terminating and non-repeating decimal
representation.
 Introduce the real number system and the real number line.
 Introduce rules to identify significant digits in a given number.
 Determine if trailing zeros of an integer are significant.



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





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Round integers and decimals to a specified number of significant digits.
Add integers with the same sign.
Add integers to their opposites.
Add integers with different signs.
Subtract integers by adding their opposites.
Find the distance between two integers on a number line.
Multiply and divide integers.
Use addition, subtraction, multiplication, and division with integers.
Add and subtract rational numbers.
Multiply and divide rational numbers.
Add and subtract decimals.
Multiply or divide numbers in decimal or percent form.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 1: The Real Number System
Math In Focus On-line Resources Chapter 2: Rational Number Operations
Brain Pop "Adding and Subtracting Integers"
Brain Pop "Square Roots" "Order of Operations", "Fibonacci Sequence"
Brain Pop "Rational and Irrational Numbers", "Converting Fractions to Decimals",
"Adding and Subtracting Fractions", "Multiplying and Dividing Fractions"
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 1: The Real Number System
Math In Focus Resources Chapter 2: Rational Number Operations
rulers
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 2:
Algebra
Summary and Rationale
In Grade 6, students took concepts and skills they had with numerical expressions and applied them to
basic algebraic expressions. In this unit, students extend them to more complex expressions. Students
simplify, expand, and factor increasingly complex algebraic expressions. They create bar models and
diagrams to help them visualize algebraic situations and use them to solve real-world problems.
Students learn to identify equivalent equations. They solve multi-step equations with variables on both
sides, including equations with parentheses, and they learn to solve real-world problems algebraically.
After solving equations, students learn how to solve inequalities, graph the solution set of an inequality,
and use inequalities to solve real-word problems.
Pacing
Seven (7) weeks
Standards
Expressions and Equations
7.EE.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear
expressions with rational coefficients.
7.EE.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.
7.EE.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.
7.EE.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.
7.EE.4a
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.
7.EE.4b
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Algebraic expressions containing rational numbers and several variables can be simplified, expanded, or
factored to write equivalent expressions.
Equivalent algebraic expressions are expressions that have the same value for any given value of the
variable.
Fractions and decimals function the same as integers in algebraic expressions: They can be numerical
terms and coefficients. The different types of numbers do not determine whether terms are like terms.
Subtraction is the same as adding the opposite.
When solving equations by performing inverse operations on both sides, the revised equation is
equivalent to the original equation. (Properties of Equality)
When solving equations, it is advisable to perform the order of operations “in reverse.” However, the
inverse operations can be performed in any order and the solution remains the same.
Equivalent equations are equations that have the same solution.
An inequality has a set of solutions that make it a true statement. The solution to an inequality can be
graphed as a ray on a number line.
To solve inequalities, employ the same methods used to solve equations with one exception: The
inequality symbol reverses every time both sides of the inequality are multiplied or divided by a negative
number.
The solutions to equations and inequalities can be checked by substituting a solution back into the
original.
Algebraic equations and inequalities can be used to model mathematical or real-world situations and to
find values of variables.
Essential Questions
What is the difference between solving an equation and solving an inequality?
How can equations and inequalities be modeled?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Equivalent Equations
 Solution set
 Equivalent inequalities
Students will be able to:
 Represent algebraic expressions using bar models.
 Simplify algebraic expressions with decimal and fractional coefficients by adding and subtracting like
terms.
 Simplify algebraic expressions with more than two terms.
 Simplify algebraic expressions by using the commutative property of addition.
 Simplify algebraic expressions with two variables.
 Expand algebraic expressions involving fractions, decimals, and negative factors.
 Factor algebraic expressions with two variables.
 Factor algebraic expressions with negative terms.
 Translate verbal descriptions into algebraic expressions with one or more variables.
 Translate verbal descriptions into algebraic expressions with parentheses
 Solve real-world problems using algebraic reasoning.
 Identify equivalent equations.
 Solve algebraic equations with variables on the same side of the equation.
 Solve algebraic equations with variables on both sides of the equation.
 Solve algebraic equations in factored form.
 Solve algebraic inequalities.
 Graph the solution set of an inequality on a number line.
 Solve multi-step algebraic inequalities.
 Solve real-word problems using algebraic equations and inequalities.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 3: Algebraic Expressions
Math In Focus On-line Resources Chapter 4: Algebraic Equations and Inequalities
Brain Pop "Inequalties", "Graphing and Solving Inequalities", "Equations with Variables", "Two-Step
Equations"
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 3: Algebraic Expressions
Math In Focus Resources Chapter 4: Algebraic Equations and Inequalities
Algebra tiles
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 3:
Proportional Relationships
Summary and Rationale
In this unit, students extend their knowledge of ratios and rates to the concepts of direct and inverse
proportion. They identify both direct and inverse proportion, recognize that a constant of proportionality
can be a constant rate, and solve real-world proportional-relationship problems.
Students use cross products to solve proportions. They use bar models to visualize, interpret, and solve
direct and inverse proportion problems.
Pacing
Three (3) weeks
Standards
Ratios and Proportional Relationships
7.RP.1
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and
other quantities measured in like or different units.
7.RP.2
Recognize and represent proportional relationships between quantities.
7.RP.2a
Decide whether two quantities are in a proportional relationship.
7.RP.2b
Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and
verbal descriptions of proportional relationships.
7.RP.2c
Represent proportional relationships by equations.
7.RP.2d
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.
7.RP.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Two quantities that are in a proportional relationship can be used to solve real-world and mathematical
problems.
A proportion is an equation that says two ratios are equivalent. They key to writing a proportion is making
sure that all ratios in the proportion compare quantities in the same order.
Cross products can be used to solve proportions.
A direct proportion is a relationship between two quantities in which both quantities increase or decrease
by the same factor. This factor is called the constant of proportionality. The graph of a direct proportion
relationship is a straight line that passes through the origin, but does not lie along the x or y-axis.
An inverse proportion is a relationship between two quantities in which one quantity increases as the
other decreases or vice versa.The product of the two quantities remains constant. In an inverse proportion,
on quantity The graph of an inverse proportion relationship is a curve that never crosses the x or y-axis.
Essential Questions
What is a proportion?
How is proportion solved?
How is a direct proportion different from an inverse proportion?
What are some real-life examples of direct proportions? Inverse proportions?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Proportion
 Direct proportion
 Constant of proportionality
 Cross products
 Inverse proportion
Students will be able to:
 Identify direct and inverse proportion.
 Recognize that a constant of proportionality can be a constant rate.
 Use a graph to interpret direct and inverse proportion.
 Solve real-world direct and inverse proportion problems.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 5: Direct and Inverse Proportions
Language Arts Integration
Suggested Resources
Math In Focus On-line Resources Chapter 5: Direct and Inverse Proportions
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 4:
Geometry
Summary and Rationale
In this unit, students explore and apply the properties of complementary angles, supplementary angles,
adjacent angles, angles on a line, angles at a point, vertical angles, pairs of angles formed by parallel lines
and a transversal, as well as interior and exterior angles of a triangle.
Students use algebra throughout the unit to solve geometric problems involving angle measures. As they
apply and use angle sum properties, students write algebraic equations and solve them in order to identify
unknown angle measures. When an angle measures are related to a ratio, students use bar models and the
unitary method to identify angle measures.
Students learn to construct angle bisectors and perpendicular bisectors, and also explore conditions that
determine whether a triangle with a given set of dimensions is unique or not. The formal constructions
they perform serve as an introduction to deductive reasoning skills they will further develop in later
geometry courses.
Students also study scale drawings, learn to identify scale factors, and solve scale problems.
Pacing
Six (6) weeks
Standards
Expressions and Equations
7.G.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.
7.G.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.
7.G.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The sum of the measures of angles on a line is 180°.
The sum of the measures of angles at a point is 360°.
Adjacent angels are two angles that share a common vertex and a side, but have no common interior
points.
Complementary angles are two angles whose angle measures total 90°. Supplementary angles are two
angles whose angle measures total 180°.
Two or more angles that have the same measure are congruent angles.
Vertical angles are either pair of non-adjacent angles formed when two lines intersect. Vertical angles
have equal measures.
When a transversal intersect two lines, several angles are formed. Alternate interior angles are on opposite
sides of the transversal, but inside the two lines. Alternate exterior angles are on opposite sides of the
transversal, but outside the two lines. Corresponding angles are pairs of angles that are on the same side
of the transversal and on the same side of the given lines. When a transversal intersects two parallel lines,
many of these angle pairs formed are congruent.
An interior angle of a triangle is an angle inside the triangle. An exterior angle of a triangle is formed by
one side of a triangle and the extension of an adjacent side.
A unique triangle refers to a triangle with a specific size and shape. It refers not a to a single triangle, but
to all triangles with the same size and shape.
Similar figures are figures that are identical in shape, but not the same size. The side lengths of similar
figures are proportional.
Essential Questions
What are adjacent angles?
How is an exterior angle of a triangle formed?
When a transversal intersects two parallel lines, how many and what kind of angles are formed?
What are congruent figures?
How are congruent figures different from similar figures?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Complementary angles, supplementary angles
 Adjacent angles
 Vertical angles
 Congruent angles
 Transversal
 Alternate interior angles, alternate exterior angles
 Corresponding angles
 Interior angles, exterior angles
 Bisect, bisector,
 Angle bisector
 Perpendicular bisector
 Equidistant
 Midpoint
 Included side, included angle
 Scale, scale factor
Students will be able to:
 Explore the properties of complementary angles and supplementary angles.
 Explore the properties of adjacent angles.
 Explore and apply the properties of angles at a point.
 Explore and apply the properties of vertical angles.
 Identify the types of angles formed by parallel lines and a transversal.
 Write and solve equations to find unknown angle measures in figures.
 Explore and apply the properties of the interior angles of a triangle.
 Explore and apply the properties of the exterior angles of a triangle.
 Understand the meaning of and construct an angle bisector.
 Understand the meaning of and construct a perpendicular bisector.
 Construct a triangle with given measures.
 Determine whether a unique triangle, more than one triangle, or no triangle can be drawn from
given side lengths.
 Construct a rectangle, square, rhombus, or parallelogram.


Identify the scale factor in diagrams.
Solve problems involving scale drawings of geometric figures.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 6: Angle Properties and Straight Lines
Math In Focus On-line Resources Chapter 7: Geometric Construction
Brain Pop "Proportions", "Similar Figures", "Scale Drawing"
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 6: Angle Properties and Straight Lines
Math In Focus Resources Chapter 7: Geometric Construction
Protractor
Compass
Straightedge
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 5:
Measurement
Summary and Rationale
In this unit, students identify cylinders, cones and pyramids, both as solids and from their nets. They also
identify the shapes of certain cross sections of these solids. Students explore the concepts of surface area
and volume of three-dimensional shapes including prisms, cylinders, pyramids, and cones. They discover
relationships between the volumes of prisms and pyramids and cylinders and cones to discover, justify,
and apply surface area and volume formulas. Students also use the formulas to find volume and surface
areas of three-dimensional composite shapes and solve real-world problems.
Pacing
Three (3) weeks
Standards
Geometry
7.G.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.
7.G.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.
7.G.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Solids such as pyramids, cylinders, and spheres are all around. Surface areas and volumes of these figures
can be calculated to solve real-world problems.
Volume is a measure of the space enclosed within a solid figure.
Surface area is the sum of the areas of the faces and lateral surfaces of a solid figure.
Right solids have a central axis that is perpendicular to the base. Oblique solids are slanted.
A cylinder is a solid with a curved surface and two parallel bases that are congruent circles.
A cone is a solid with a circular base, a curved surface, and a vertex.
A lateral surface is the curved surface of a cone or cylinder.
The slant height of a cone is the distance from the vertex to any point on the edge of the base.
A plane is a flat surface that extended infinitely in two dimensions.
A plane intersecting a solid creates a cross section only if the plane passes through the interior of the
solid.
A sphere is a solid whose every point is the same distance from its center. Every cross section of a sphere
is a circle. The cross section that contains the center of the sphere is the cross section with the greatest
possible area, and is called the great circle.
The slant height of a pyramid is the distance from the vertex to the midpoint of any edge of the base.
Essential Questions
Where do we find solids around us?
How is volume different form surface area?
How is the net of a cone or a cylinder different form that of a pyramid or prism?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Cylinder
 Cone
 Lateral surface
 Slant height
 Sphere
 Plane
 Cross section
 Volume
 Surface area
Students will be able to:
 Recognize cylinders, cones, and spheres.
 Identify cross sections of solids.
 Find the volume and surface area of cylinders, cones and spheres.
 Find the volume of pyramids.
 Solve real- world problems involving cylinders, cones, pyramids, spheres, and composite figures.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 8: Volume of Surface Area and Solids
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 8: Volume of Surface Area and Solids
Rulers
Cylindrical objects
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 6:
Statistics
Summary and Rationale
In this unit, students learn to identify measures of variation. They divide a data set into quartiles and
identify interquartile range. Students draw and interpret stem-and-leaf plots and box-and-whisker plots,
and learn to find mean absolute deviation.
Students learn about population and samples. The apply different random sampling methods, use statistics
from a sample to make inferences about a population, and use an inference to estimate a population mean.
Students also make comparative inferences about two populations using two sets of sample statistics.
Pacing
Three (3) weeks
Standards
Statistics and Probability
7.SP.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.
7.SP.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.
7.SP.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.
7.SP.4
Use measures of center and measures of variability for numerical data from random samples
to draw informal comparative inferences about two populations.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Measures of central tendency and measures of variation are used to draw conclusions about populations.
The median divides a data set into two halves. The medians of the lower half and the upper half divide the
set into equal fourths. The median of the lower half is the first quartile. The median of the set if the
second quartile. The median of the upper half is the third quartile.
The interquartile range gives a good idea of the values that are typical of the data set. It describes that
central 50% of the values in a data set and is the difference between the third quartile and the first
quartile.
A stem-and-leaf plot displays data in a way that emphasizes the range of the data set. A box-and-whicker
plot emphasizes the three quartiles, as well as the lower and upper extremes of the data.
The mean absolute deviation (MAD) is another useful measure of variation. It is the average of the
distances of all the values in a set from the mean. This value gives a good sense of how tightly data in a
set is clustered around the mean. The greater the MAD of a data set, the more spread out its values are
from the mean.
When it is impossible or impractical to study an entire population, a sample population can be used to
obtain data and draw conclusions. Such conclusions are called inferences, which are approximations, and
not facts. The more representative of a population a sample is, the more likely it is that the sample data
will be useful and valid.
To select a random and unbiased sample, every member of the population must have an equal chance of
being selected and the selection of members is independent of each other. Three different random
sampling methods are simple random sampling, stratified random sampling, and systematic random
sampling.
Essential Questions
What are quartiles?
What do stem-and-leaf plots emphasize about a data set? Box-and-whisker plots?
Why are random sampling methods used?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Measure of variation
 Range
 First/second/third quartile
 Lower/upper quartile
 Interquartile range
 Stem-and-leaf plot
 Box plot
 Box-and-whisker plot
 5-point summary
 Mean absolute deviation
 Population
 Sample, sample size, random sample
 Unbiased, biased sample
 Simple random/stratified/systematic random sampling
 Inference
Students will be able to:
 Introduce the concept of measures of variation.
 Understand and solve problem involving quartiles and interquartile range.
 Represent data in a tem-and-leaf plot.
 Make conclusions and solve word problems involving stem-and-leaf plots.
 Draw and interpret box plots.
 Understand mean absolute deviation.
 Solve problems involving box plots and mean absolute deviation.
 Understand the concept of a population and samples.
 Understand and apply different random sampling methods.
 Simulate random sampling.
 Make and use inferences about a population to estimate its population mean.
 Make comparative inferences about two populations.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 9: Statistics
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 9: Statistics
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Pre-Algebra
Unit 7:
Probability
Summary and Rationale
In this unit, students learn about chance processes, and measuring the likelihood of events. They learn to
distinguish between theoretical and experimental probability and begin to recognize that as the number of
trials increases in a n experiment with chance process, the experimental probability measures tend to
approach the values of theoretical probability measures.
Pacing
Three (3) weeks
Standards
Statistics and Probability
7.SP.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.
7.SP.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.
7.SP.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.
7.SP.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation.
7.SP.8a
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.
7.SP.8b
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.
7.SP.8c
Design and use a simulation to generate frequencies for compound events.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Some events are more likely than others. You can use probability to describe how likely an event is to
occur.
The probability of an event is represented by a rational number from 0 to 1.
The theoretical probability of an event is equal to the number of favorable outcomes to the event divided
by the number of equally likely outcomes.
Mutually exclusive events are events that cannot both occur simultaneously.
The complement of an event consists of all the outcomes in a sample space that are not favorable to the
event.
Experimental probability is based on the observed frequency of an event during a number of trials, or an
experiment. As the number of trials increases, experimental probability tends to approach theoretical
probability measures.
Probability distributions can be displayed in a table, bar graph, or histogram.
Essential Questions
What is probability?
What are the differences between theoretical and experimental probability? How are they related?
Evidence of Learning
Guided Practice
Formative Assessments
Pre and Post Chapter Assessments
Benchmark Assessments
Objectives
Students will know or learn:
 Outcome,
 Sample space
 Event
 Probability











Fair, biased
Venn diagram
Mutually exclusive
Complementary events, complement
Relative frequency
Observed frequency
Experimental probability
Theoretical probability
Probability model
Probability distribution
Uniform probability model, non-uniform probability model
Students will be able to:
 Understand the concepts of outcomes, events, and sample space and apply them to everyday life.
 Find the probability of events.
 Use Venn diagrams to illustrate events and their relationships.
 Solve real-world problems involving probability.
 Find relative frequencies, interpret them as probabilities and use them to make predictions.
 Compare relative frequencies to theoretical probabilities.
 Understand and apply uniform probability models and non-uniform probability models
 Compare experimental probability with theoretical probability.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
Math In Focus On-line Resources Chapter 10: Probability
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 10: Probability
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Foundations of Algebra
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 0:
Operations and Equations with Rational Numbers
Summary and Rationale
In this unit students will develop an understanding of rational numbers and the different forms that
numbers can take. Students will use appropriate operations to solve problems involving fractions and
decimals. They will learn how to manipulate positive and negative fractions, and mixed numbers using
addition, subtraction, multiplication, and division to solve problems in everyday contexts. They will find
solutions to application problems using equations. Lastly, students will study solving two-step equations.
Pacing
Three (3) weeks
Standards
The Number System
8.NS.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.
Expressions and Equations
8.EE.7
Solve linear equations in one variable.
 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).
 Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and
collecting like terms.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A rational number is any number than can be written as a fraction a/b, where a and b are integers and b ≠
0.
Fractions can be expressed as terminating or repeating decimals.
Non-repeating numbers such as π and
2 cannot be written as fractions so they are not rational.
The rules to manipulate rational numbers using addition, subtraction, multiplication, and division are the
same as those for integers and fractions.

To solve a one-step equation, use the Properties of Equality to isolate the variable on one side of the
equation.
To solve two-step equations, use inverse operations to undo each operation in reverse order of the order of
operations.
Essential Questions
How do different forms of rational numbers help solve problems?
What does it mean to have unknowns in an equation?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Rational numbers.
 Operations with rational numbers.
 Equations with rational numbers.
 Positive and negative fractions.
 Two-step equations.
Students will be able to:
 Write rational numbers in equivalent forms.
 Add, subtract, multiply, and divide fractions, mixed numbers, and decimals.
 Solve equations involving all types of rational numbers.
 Solve two-step equations.
Integration
Technology Integration
Math In Focus Virtual Manipulatives
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Teacher-made materials, red and yellow counters, algebra tiles, magnetic fraction pieces (bars & circles),
communicators
FACEing Math activity book
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 1:
Exponents and Roots
Summary and Rationale
In this unit students will learn the basic skills involving exponents and powers. The students will know
the first 20 perfect squares and the first 10 perfect cubes by the end of the unit. They will simplify real
number expressions using integer exponents and the laws of exponents. They also will learn how to
operate with numbers in scientific notation. Students will learn to approximate the values of irrational
numbers by estimating square and cube roots. They will learn the relationship between the subsets of the
real number system.
Pacing
Five (5) weeks
Standards
The Number System
8.NS.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.
8.NS.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., π2). 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.1
Know and apply the properties of integer exponents to generate equivalent numerical
expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27.
8.EE.2
Use square root and cube root symbols to represent solutions to equations of the form x2 = p
and x3 = 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.
8.EE.3
Use numbers expressed in the form of a single digit times a whole-number 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 times 108 and
the population of the world as 7 times 109, and determine that the world population is more
than 20 times larger.
8.EE.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Science
Discuss how scientific notation is used to describe distances in the solar system.
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
There are rules for operating with numerical expressions involving exponents.
Exponents are used to express very large and very small numbers in scientific notation.
There are rules for operating with square and cube roots.
There is an inverse relationship between squares and square roots.
There is a difference between estimation and actual value.
There is a relationship between subsets of the real number system.
If a positive integer is not a perfect square, then its square root is irrational.
Irrational numbers are non-terminating, non-repeating decimals.
Essential Question
How are the rules of multiplying and dividing powers helpful?
How can we tell when quantities are equal? What about irrational quantities?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Exponents
 Roots
 Laws of exponents
 Scientific notation
Students will be able to:
 Add, subtract, multiply, and divide expressions with integer exponents.
 Express large and small numbers in scientific notation.
 Apply scientific notation to real-world situations.
 Recite the first 20 perfect squares and the first 10 perfect cubes.
 Evaluate square and cube roots.
 Estimate square and cube roots.
 Identify the set or sets any real number belongs to.
 Order all real numbers, including irrationals, on a real number line.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 1: Exponents
Math In Focus On-line Resources Chapter 2: Scientific Notation
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 1: Exponents
Math In Focus Resources Chapter 2: Scientific Notation
Teacher-made materials (such as square root flash cards, scientific notation matching game etc.),
geoboards, communicators, graphic organizer for the Real Number System.
FACEing Math activity book
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 2:
Linear Equations and Relationships
Summary and Rationale
This unit focuses on linear equations and linear functions. Students will identify constant rates of change
to distinguish proportional and non-proportional relationships. They will solve multi-step problems
involving direct variation. Students will interpret the slope and x- and y-intercepts when graphing a linear
equation. They will learn to symbolically represent and solve real-world situations that involve linear
equations. Students will be able to write a linear equation and make a prediction if given a table, graph,
or verbal description. Through the use of tables and graphs, students will represent, analyze, and solve
real-world problems related to linear equations and systems of linear equations. Students will translate
among verbal, tabular, graphical, and algebraic representations of linear functions.
Pacing
Five (5) weeks
Standards
Expressions and Equations
8.EE.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.
8.EE.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.
8.EE.7
Solve linear equations in one variable.
 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).
 Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and
collecting like terms.
8.EE.8
Analyze and solve pairs of simultaneous linear equations.
 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.
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.
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.l
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
8.F.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.
8.F.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 = s2 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.
8.F.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.
8.F.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
If an equation is linear, then a constant change in the x-value corresponds to a constant change in the yvalue.
The slope of a linear equation represents the rate of change.
The slope of a line may be calculated by finding the ratio of the rise to the run for any two points on the
line.
Horizontal lines have a slope of zero since the rise is zero for any run, and zero divided by a nonzero
number is zero.
Vertical lines have a slope that is undefined because any two points on the line will have a run of zero,
and division by zero is undefined.
The x-intercept of a line is the value of x where the line crosses the x-axis (where y = 0) and the yintercept of a line is the value of y where the line crosses the y-axis (where x = 0).
The slope-intercept form for a linear equation is y = mx + b where m is the slope and b is the y-intercept.
Parallel lines have the same slope and perpendicular lines have slopes that are the negative reciprocals of
each other.
There are zero, one, or infinitely many solutions to a system of equations, depending on whether the lines
are parallel, intersecting, or coincident.
If a function has a constant rate of change then it is a linear function.
A direct variation is a special type of linear function that can be written in the form y = kx where k is a
nonzero constant.
A linear relationship can be represented as verbal descriptions, functions, graphs, and tables.
Essential Questions
Why does graphing a linear equation result in a straight line?
Where in the real world can you find and what are the important attributes of linear patterns and linear
relationships?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know and be able to:
 Identify and graph linear equations.








Find the slope of a line and use slope to understand and draw graphs.
Use slope and intercepts to graph linear equations.
Use the slope-intercept form to write an equation of a line.
Solve a system of linear equations by graphing.
Identify and write linear functions.
Recognize direct variation by graphing tables of data and checking for constant ratios.
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.
Compare linear functions represented in different ways.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 3: Algebraic Linear Equations
Math In Focus On-line Resources Chapter 4: Lines and Linear Equations
Math In Focus On-line Resources Chapter 5: Systems of Linear Equations
www.brainpop.com,
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 3: Algebraic Linear Equations
Math In Focus Resources Chapter 4: Lines and Linear Equations
Math In Focus Resources Chapter 5: Systems of Linear Equations
Math In Focus Resources Chapter 6: Functions
Teacher-made materials, communicators, graphing “line” transparencies
FACEing Math activity book
Technology: www.brainpop.com, http://www.math-play.com/math-jeopardy.html
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 3:
Multi-Step Equations & Systems
Summary and Rationale
In this unit, students will learn how to use the properties of real numbers to rewrite complex expressions
involving parentheses and like terms. They will find solutions to application problems using algebraic
equations. They will use these properties to solve multi-step equations, including those with variables on
both sides. Finally, students will determine if an ordered pair is a solution to a system of equations and be
able to solve a system of equations.
Pacing
Three (3) weeks
Standards
Expressions and Equations
8.EE.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.
8.EE.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.
8.EE.7
Solve linear equations in one variable.
 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).
 Solve linear equations with rational number coefficients, including equations whose
solutions require expanding expressions using the distributive property and
collecting like terms.
8.EE.8
Analyze and solve pairs of simultaneous linear equations.
 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.
 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.
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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard

Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The strategy for solving multi-step equations and equations with variables on both sides is overall the
same for solving equations.
The properties of real numbers can be used to rewrite complex expressions involving parentheses and like
terms and to solve multi-step equations.
A system of equations is a set of two or more equations that each involves the same set of two or more
variables.
A solution of a system of equations is a set of values that are solutions of all of the equations in the
system.
Essential Questions
How are mathematical properties helpful in simplifying expressions and equations?
Why is using equations to solve problems useful?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Multi-step equations
 Complex expressions involving parenthesis
 System of equations
Students will be able to:
 Apply the distributive property to rewrite algebraic expressions.
 Combine like terms in an expression.
 Solve multi-step equations.
 Use the properties of Equality to solve multi-step equations.
 Solve equations with variables on both sides of the equal sign.
 Solve systems of equations algebraically.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 3: Algebraic Linear Equations
Math In Focus On-line Resources Chapter 4: Lines and Linear Equations
Math In Focus On-line Resources Chapter 5: Systems of Linear Equations
Math In Focus On-line Resources Chapter 6: Functions
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 3: Algebraic Linear Equations
Math In Focus Resources Chapter 4: Lines and Linear Equations
Math In Focus Resources Chapter 5: Systems of Linear Equations
Math In Focus Resources Chapter 6: Functions
Teacher-made materials, communicators
FACEing Math activity book
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 4:
Graphs and Functions
Summary and Rationale
In this unit students will locate ordered pairs of rational numbers on a coordinate plane. They will be
introduced to expressions, relations, and functions. They will learn to generate different representations of
data using tables, graphs, and equations. Lastly, students will use functions to describe relationships
among data.
Pacing
Three (3) weeks
Standards
Functions
8.F.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
8.F.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.
8.F.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 = s2 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.
8.F.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.
8.F.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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Ordered pairs of rational numbers can be located on a coordinate plane.
Functions can be used to describe relationships among data.
Relations can be used to assign members of one set to members of another set. A function, a specific type
of relation, assigns each member of one set to a unique number of another set.
Relationships can be described and generalizations made for mathematical situations that repeat in a
predictable way. These generalizations can then be represented in a variety of forms including tables,
graphs, equations, and in words.
Essential Questions
Why do we need to study the relationship between two numbers?
How are ordered pairs, graphs, and tables used to represent relationships between two quantities? What
could be understood from a function by examining its multiple representations?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Functions
 Graphing lines
 Relationships between numbers
 Depiction of relationships between quantities
Students will be able to:
 Write solutions of equations in two variables as ordered pairs.
 Graph points on the coordinate plane.
 Interpret information given in a graph and make a graph to model the situation.
 Represent functions with tables, graphs, and equations.
 Generate different representations of the same data.
 Identify and graph linear equations.
 Find the slope of a line and use slope to understand and draw graphs.
 Use slope and intercepts to graph linear equations.
 Use the slope-intercept form to write an equation of a line.
 Solve a system of linear equations by graphing.
 Identify and write linear functions.
 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.
 Compare two linear functions represented in different ways.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 3: Algebraic Linear Equations
Math In Focus On-line Resources Chapter 4: Lines and Linear Equations
Math In Focus On-line Resources Chapter 5: Systems of Linear Equations
Math In Focus On-line Resources Chapter 6: Functions
www.brainpop.com
www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 3: Algebraic Linear Equations
Math In Focus Resources Chapter 4: Lines and Linear Equations
Math In Focus Resources Chapter 5: Systems of Linear Equations
Math In Focus Resources Chapter 6: Functions
Teacher-made materials, communicators, Lucky Coordinates game
FACEing Math activity book
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 5:
The Pythagorean Theorem and Measurement
Summary and Rationale
In this unit students will use the Pythagorean Theorem to solve problems with right triangles. Finally,
students will use the Pythagorean Theorem to find the distance between points on a coordinate plane and
solve real-life problems
Students will also focus on using formulas to solve problems involving geometric figures. They will use
these formulas to find the volume of prisms, cylinders, pyramids, cones, and spheres. They will build
upon their experience with polygons to develop these formulas.
Pacing
Two (2) weeks
Standards
Expressions and Equations
8.EE.2
Use square root and cube root symbols to represent solutions to equations of the form x2 = p
and x3 = 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.
Geometry
8.G.6
Explain a proof of the Pythagorean Theorem and its converse.
8.G.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.G.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate
system.
8.G.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.6 Attend to precision.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
History
Discuss the mathematician Pythagoras.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The Pythagorean Theorem is an historically and practically important application of squares and square
roots.
The Pythagorean Theorem and its converse can be used to calculate lengths of line segments in 2- and 3dimensional geometric objects.
The Pythagorean Theorem can be used to find the distances between points on the coordinate plane and
able to find real-world distances.
Prior experience with polygons can be used to develop formulas for the volume of prisms, cylinders,
pyramids, and cones.
Pyramids and cones are 1/3 the volume of prisms and cylinders of the same base and height.
An understanding of cylinder and cone volumes gives the opportunity to explore Archimedes’ formula for
the volume of a sphere.
Finding the volume of three-dimensional objects with or without curved surfaces is useful in solving
problems.
Essential Questions
In what situations will we need to solve problems dealing with measurements of right triangles?
What attributes of three-dimensional objects are important to be able to measure and quantify and why?
Why must you know the area of the base to find the volume of prisms, cylinders, pyramids, and cones?
Evidence of Learning (Assessments)
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Integration
Technology Integration
Math In Focus On-line Resources Chapter 7: The Pythagorean Theorem
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
What’s Your Angle, Pythagoras? By Julie Ellis
Suggested Resources
Math In Focus On-line Resources Chapter 7: The Pythagorean Theorem
geoboards, communicators
Teacher-made materials, 3-d figures (fill with rice or water)
FACEing Math activity book
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
dot paper
graph paper
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 6:
Geometric Relationships and Transformations
Summary and Rationale
In this unit students will solve problems involving angles created by parallel lines cut by transversals:
vertical, alternate interior, alternate exterior, and corresponding angles. They will demonstrate that the
sum of the angles in a triangle is 180 degrees. Students will apply transformations (translations,
reflections, and rotations) to plane figures in the coordinate plane. They will learn how the transformation
of a figure affects the location on the coordinate plane.
Pacing
Three (3) weeks
Standards
Geometry
8.G.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.
8.G.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.
8.G.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional
figures using coordinates.
8.G4
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.
8.G.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 angleangle 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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
When parallel lines are cut by a transversal congruent angles are formed.
The sum of the angles in a triangle always equals 180° and it can be proven by the Triangle Sum
Theorem.
Students will learn how the transformation of a figure affects the location on the coordinate plane.
Transformational geometry can be used to describe motions, patterns, designs, and properties of shapes in
the real world.
When a figure (pre-image) is rotated, the vertex and its image are the same distance from the center of
rotation and all the angles formed by the vertex, the center of rotation, and the image of that vertex are
congruent.
When a figure is reflected, each vertex and its image are an equal, perpendicular distance from the line of
reflection.
When a figure is translated, each point in the pre-image moves the same distance and in the same
direction.
Essential Questions
Describe the relationship between two parallel lines cut by a transversal and the angles that are formed.
How can we copy shapes and make precise drawings without measuring tools?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Geometric relationships
 Transformations
 Angles
 Parallel lines
 Transversals
 Translation
 Map
 Image
 Invariant
 Reflection, line of reflection
 Rotation, center of rotation, angle of rotation, clockwise, counter-clockwise
Students will be able to:
 Classify angles and find their measures.
 Identify parallel and perpendicular lines and the angles formed by the transversal.
 Explore the relationship among the angles of a triangle.
 Find unknown angles and identify possible side lengths in triangles.
 Perform transformation of figures, including reflections, translations and rotations.
 Give precise mathematical directions for performing reflections, rotations, and translations.
 Draw compositions of translations, reflections, and rotations.
 Identify and create dilations of plane figures.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 8: Geometric Transformations
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 8: Geometric Transformations
Teacher-made materials, Anglegs, communicators
FACEing Math activity book
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 7:
Congruence and Similarity
Summary and Rationale
In this unit students will study congruence and similarity. They will use proportional relationships in
similar figures to find missing measurements. They will use similar triangles to solve problems that
include height and distance. Students will generate similar figures using dilations. Students will use
critical attributes to define congruency. They will learn how the transformation of a figure affects its
congruency.
Pacing
Three (3) weeks
Standards
Geometry
8.G.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.
8.G.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.
8.G.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 angleangle 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.
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
If two figures are similar one is an enlargement or reduction of the other.
Two polygons are similar if corresponding angles are congruent and corresponding sides are proportional.
Given two similar figures, it is often possible to find an unknown side length in one of the figures by
setting up and solving a proportion.
After a dilation, the image is similar to the original figure.
The scale factor in a dilation determines precisely how much the size changes.
Figures with the same shape and size are congruent.
That if we know that corresponding sides and angles are congruent, we can conclude that the polygons are
congruent.
Two figures are congruent if one figure can be transformed into the other through a series of translations,
reflections, and rotations.
Essential Question
How can proportional reasoning be applied to problem-solving situations involving similar figures?
How are the results of a transformation different than the original figure? How are they similar?
Evidence of Learning (Assessments)
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Congruence
 Similarity
 Proportional relationships
 Similar triangles
 Corresponding angles, corresponding sides
 Statement of congruence
 Dilation
Students will be able to
 Determine whether figures are similar and find missing dimensions in similar figures.
 Identify and create dilations of plane figures using the scale factor.
 Use properties of congruent figures to solve problems.
 Identify transformations as similarity or congruence transformations.
 Identify the image of a figure after a combined transformation is performed, and determine whether
the final image is similar or congruent to the original.
 Use similar triangles in a coordinate plane to explore slope.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 9: Congruence and Similarity
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Cut Down to Size at High Noon By Scott Sundby
Suggested Resources
Math In Focus Resources Chapter 9: Congruence and Similarity
Teacher-made materials, communicators
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
protractors
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Mathematics
Course: Foundations of Algebra
Unit 8:
Statistics and Probability
Summary and Rationale
In this unit students will organize and construct scatter plots. They will construct lines of best fit. They
will select and construct appropriate displays to convey information and make conjectures about possible
relationships amongst two different variables.
Students will also learn to identify whether compound events are independent or dependent. They will
apply the multiplication and addition probability rules to compute probabilities of compound events, both
independent and dependent.
Pacing
Six (6) weeks
Standards
Statistics and Probability
8.SP.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.
8.SP.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.
8.SP.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.
8.SP.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?
Mathematical Practices
K-12.MP.1 Make sense of problems and persevere in solving them.
K-12.MP.2 Reason abstractly and quantitatively.
K-12.MP.3 Construct viable arguments and critique the reasoning of others.
K-12.MP.4 Model with mathematics.
K-12.MP.5 Use appropriate tools strategically.
K-12.MP.7 Look for and make use of structure.
K-12.MP.8 Look for and express regularity in repeated reasoning.
Interdisciplinary Connections
Standard
Integration of Technology
Standard
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A scatter plot is a graph that shows bivariate data; that is, data for which there are two variables for each
observation, such as height and weight.
If a correlation exists within data plotted on a scatter plot, a line of best fit (trend line) can be drawn and a
linear equation formulated.
A two-way table displays two-variable data by collecting it into rows and columns.
Collecting and analyzing data can answer some questions, and the question to be answered determines the
data that needs to be collected, how best to collect it, and how to visually represent it.
Probability of simple events can be used to compute the probability of compound events, either
independent or dependent.
Essential Questions
How do scatter plots and lines of best fit enable you to make predictions about data?
What are the ways in which data can be collected, analyzed, and represented to answer questions that are
important to us?
What is the difference between independent events and dependent events in terms of probability?
Evidence of Learning
Ongoing observation
Class work
Problem of the Day/Week
Homework
Quizzes/Tests
Projects
Objectives
Students will know or learn:
 Scatter plots
 Data analysis
 Making predictions using data
 Lines of best fit
 Simple event
 Compound event
 Possibility diagram
 Tree diagram
Students will be able to:
 Create a scatter plot of the data between two variables to see if there is a relationship.
 Estimate the patterns in scatter plots.
 Estimate the line of best fit in scatter plots and use the line of best fit to solve problems and make
predictions.
 Construct and interpret two-way tables.
 Understand and represent compound events.
 Use possibility diagrams to find probability of compound events.
 Understand independent and dependent events.
 Use multiplication rule and addition rule of probability to solve problems with independent events.
 Use rules of probability to solve problems with dependent events.
Integration
Technology Integration
Math In Focus On-line Resources Chapter 10: Statistics
Math In Focus On-line Resources Chapter 11: Probability
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
Language Arts Integration
Suggested Resources
Math In Focus Resources Chapter 10: Statistics
Math In Focus Resources Chapter 11: Probability
Teacher-made materials
www.brainpop.com
http://www.math-play.com/math-jeopardy.html
www.hmhlearning.com
www.hmheducation.com/mathinfocus
Nutley Public Schools
Algebra 1
Nutley Public Schools
Mathematics
Course: Algebra 1
Unit 1:
Properties of Real Numbers
Summary and Rationale
This unit revisits and further explores the characteristics of real numbers and operations of real numbers.
Pacing
Two (2) weeks
Standards
Number and Quantity
N-RN-3
Explain why the sum or product of two rational numbers is rational; that the sum of the
rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
N-Q-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.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be classified by their characteristics. All real numbers can be represented on the graph of a
real number line. The location of some numbers must be approximated (irrational numbers).
The definition of a square root can be used to find the exact square root of some nonnegative numbers,
called perfect squares. The square roots of other non-negative numbers that are not perfect squares can be
approximated.
Relationships that are always true for real numbers are called properties. The Properties of Real Numbers
are rules used to rewrite and compare expressions.
All real numbers can be added or subtracted using a number line model or using rules involving absolute
value. Subtracting a real number is equivalent to adding its opposite: a – b = a + (-b)
The rules for multiplying and dividing real numbers are related to the properties of real numbers and the
definitions of operations. The product or quotient of two real numbers with different signs is negative.
The product or quotient of two real numbers with the same sign is positive.
Essential Questions
How can we compare and contrast numbers?
How do operations affect numbers?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Properties of real numbers
 Square roots
 Operations of real numbers
Students will be able to:
 Graph, compare, and order real numbers.
 Classify numbers within the different sets: (natural numbers, whole numbers, integers, rational
numbers, and irrational numbers.)
 Find the opposite and the absolute value of a number.
 Add, subtract, multiply, and divide real numbers.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 1
Unit 2:
Connections to Algebra
Summary and Rationale
In the transition from arithmetic to algebra, attention shifts from arithmetic operations (addition,
subtraction, multiplication, and division) to the use of properties of these operations. All arithmetic facts
and algebra follow from these properties. Unit 2 introduces students to variables and expressions.
Students will learn to write and evaluate expressions with unknown values and will use the properties of
real numbers to simplify expressions.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-RN-1
Explain how the definition of the meaning of rational exponents follows from extending
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents.
N-RN-3
Explain why the sum or product of two rational numbers is rational; that the sum of the
rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficents.
b. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-SSE-3
Choose and produce and 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 expression for exponential functions.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Quantities are used to form expressions, equations, and inequalities. An expression refers to a quantity,
but does not make a statement about it.
A single quantity may be represented by many different expressions. Expressions that represent the same
quantity are equivalent expressions.
Algebra uses symbols, called variables, to represent quantities that are unknown or that vary.
Mathematical phrases and real-world relationships can be represented using symbols and operations.
These mathematical phrases are called algebraic expressions.
When simplifying an expression, operations must be performed in correct order, known as the Order of
Operations.
Powers are used to shorten the representation of repeated multiplication.
The Distributive Property can be used to simplify the product of a number and a sum or difference.
An algebraic expression can also be simplified by combining the parts of the expression that are alike.
Essential Questions
How are properties related to algebra?
How can you use algebra to represent quantities?
Can expressions that appear to be different be equivalent?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Properties of operations
 Variables
 Distributive property
Students will be able to:
 Evaluate variable expressions.
 Evaluate expressions involving powers.
 Use the Order of Operations to simplify numerical and variable expressions.
 Translate words into mathematical and algebraic symbols.
 Model and solve real-life problems using algebra.
 Use the distributive property.

Identify and combine like terms.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 1
Unit 3:
Solving Linear Equations – An Extension
Summary and Rationale
Unit 3 connects and extends the ideas introduced in Unit 2 to solving one-variable equations. Students
will learn to rewrite an equation in order to make the statement about its variable as simple as possible.
Through the use of the properties of real numbers and equality, inverse operations, and other algebraic
properties, students will learn to transform an equation into equivalent and more simple equations in order
to isolate the variable and determine a solution(s). Students will use this process to solve one-step, twostep, and multi-step linear equations, equations with variables on both sides, absolute value equations, as
well as proportions.
Pacing
Three (3) weeks
Standards
Algebra
A-CED-1
A-REI-1
Create equations and inequalities in one variable and use them to solve problems.
Explain each step in solving a simple equations as following from the equality of numbers
asserted in the previous step, starting from the assumption that the original equations has a
solution. Construct a viable argument to justify a solution method.
A-REI-3
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Quantities are used to form expressions, equations, and inequalities. An expression refers to a quantity,
but does not make a statement about it. An equation (or an inequality) is a statement about the quantities it
mentions. Using variables in place of numbers in equations (or inequalities) allows the statement of
relationships among numbers that are unknown or unspecified.
A single quantity may be represented by many different expressions. The facts about a quantity may be
represented by many different equations (or inequalities.)
Equations are used to represent the relationship between two quantities that have the same value.
Equations can describe, explain, and predict various aspects of the real world.
Equivalent equations have the same solution(s). An algebraic equation can be represented using the
symbols in an infinite number of ways, where each representation has the same solution(s).
Properties of real numbers and equality, along with the use of inverse operations, can transform an
equation into one or a series of equivalent simpler equations. The properties of real numbers and equality
can be used repeatedly to isolate the variable. This process is used to find solutions to one-variable
equations. The process is also used to isolate a particular variable in a formula that contains two or more
variables.
Absolute value equations can be solved by first isolating the absolute value expression (if necessary), then
writing a pair of linear equations, and then solving each equation separately, which will yield two
solutions.
Ratio and rates can be used to compare quantities and make conversions in order to solve problems.
A proportion is an equation that states two ratios are equal. Proportionality involves a relationship in
which the ratio of two quantities remains constant as the corresponding values of the quantities change. In
a proportional relationship, there are an infinite number of ratios equal to this constant ratio. If two ratios
are equal, and a quantity in one of the ratios is unknown, the unknown quantity can be found by writing
and solving a proportion.
Percents represent another application of proportions. The percent proportion can be used to solve for any
one of the missing components and to solve percent increase and percent decrease problems. Percent
problems can also be solved using the percent equation.
Essential Questions
Can equations that appear to be different be equivalent?
How can you solve equations?
What kinds of relationships can proportions represent?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Expressions
 Equations
 Inequalities
Students will be able to:
 Solve linear equations using addition, subtraction, multiplication, and division. (One Step)
 Use two or more steps to solve a linear equation.
 Solve equations that have variables on both sides.
 Solve a formula for one of its variables.
 Use ratios and rates to solve real-life problems.
 Solve percent problems.
 Solve absolute-value equations.
 Solve proportions.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 1
Unit 4:
Solving Linear Equations (One-Variable)
Summary and Rationale
Unit 4 connects and extends the ideas introduced in Unit 3 to solving one-variable inequalities. Students
will learn to rewrite an inequality to make the statement about its variable as simple as possible. Through
the use of the properties of real numbers and inequality, inverse operations, and other algebraic properties,
students will learn to transform an inequality into equivalent, simpler inequalities in order to isolate the
variable and determine a solution(s). Students will use this process to solve one-step, two-step, and multistep, compound, and absolute value inequalities. Useful information about one- variable inequalities,
including solutions, can be found by analyzing their number line graphs. The types of solutions vary
predictably, based on the type of inequality.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-CED-1
Create equations and inequalities in one variable and use them to solve problems.
A-REI-3
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Quantities are used to form expressions, equations, and inequalities. An expression refers to a quantity,
but does not make a statement about it. An equation (or an inequality) is a statement about the quantities it
mentions. Using variables in place of numbers in equations (or inequalities) allows the statement of
relationships among numbers that are unknown or unspecified.
A single quantity may be represented by many different expressions. The facts about a quantity may be
represented by many different equations (or inequalities.)
Inequalities are used to state that two quantities that DO NOT have the same value. Inequalities can
describe, explain, and predict various aspects of the real world.
An inequality is a mathematical sentence that uses an inequality symbol to compare the values of two
expressions. Inequalities can be represented with symbols. Their solutions can be represented on a
number line.
Properties of real numbers and inequality, along with the use of inverse operations, can transform an
inequality into one or a series of equivalent simpler inequalities. The properties of real numbers and
inequality can be used repeatedly to isolate the variable. This process is used to find solutions to onevariable inequalities (including multi-step and compound inequalities.)
Many properties of equality hold true for inequalities. One major difference is as follows: When
multiplying or dividing both sides of an inequality by a negative number, it is necessary to reverse the
inequality sign.
The solutions of a compound inequality are either the overlap or combination of the solution sets of
distinct inequalities. The graph of a compound inequality with the word and contains the overlap of the
graphs of the two inequalities. The graph of a compound inequality with the word or contains each graph
of the two inequalities.
Absolute value inequalities can be solved by first isolating the absolute value expression (if necessary),
then writing the appropriate compound inequality that satisfies the condition, and then solving
accordingly, which will yield a solution set to be graphed.
Essential Questions
How can you represent relationships between quantities that are not equal?
Can inequalities that appear to be different be equivalent?
How can you solve inequalities?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 One-variable linear equations
 Multi-step inequalities
 Compound inequalities
Students will be able to:




Solve and graph one-step inequalities in one variable using addition, subtraction, multiplication, and
division.
Solve and graph multi-step inequalities in one variable.
Solve and graph compound inequalities.
Solve absolute-value inequalities in one variable.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 1
Unit 5:
An Introduction to Functions
Summary and Rationale
Unit 5 introduces the topic of functions, with a specific focus on linear functions and their graphs.
Throughout Units 5 and 6, students will have the opportunity to see that useful information about onevariable equations, including solutions, can be found by analyzing the graphs of their related two-variable
functions.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
c. Interpret parts of an expression, such as terms, factors, and coefficients.
d. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-CED-1
Create equations and inequalities in one variable and use them to solve problems.
A-CED-2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A-CED-3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Functions
F-IF-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).
F-IF-2
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F-IF-4
For a function that models a relationships 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.
F-IF-5
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F-IF-6
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
F-IF-8
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
F-IF-9
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
F-BF-1
Write a functions 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 functions types using arithmetic operations.
c. Compose functions
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Given a two-variable equation, the value of one quantity can be found if the value of the other is known.
A table can be used to display the relationship between the quantities, which would also represent a set of
solutions of the equation. The set of all solutions of the equation forms its graph on a coordinate plane.
The graph will show solutions that are in the table, will visually represent the relationship between the
two variable quantities that are changing, and can also show solutions to the equation that are not in the
table.
A function is a relationship between variables in which each value of the input variable (value in the
domain) is associated with a unique value of the output variable (value in the range.) In order to determine
if an equation or a set of ordered pairs represents a function, the solutions of the equation or the ordered
pairs can be organized in a table or plotted on a graph. If the table of values shows that each value in one
set is paired with exactly one value in the other set, the relation is a function. The vertical line test uses
the graph to determine whether a relation is a function.
A linear function is a function whose graph is a line. A nonlinear function is a function whose graph is not
part of a line. Both linear and nonlinear functions can be represented in a variety of ways, such as words,
tables, two-variable equations or rules, sets of ordered pairs, and graphs. Each representation is
particularly useful in certain situations.
Many real world mathematical problems can be modeled and represented algebraically and graphically by
functions. A function that models a real world situation can be represented using an equation or graph that
can be used to make estimates or predictions about future occurrences. A real-world graph of a function
should only show points that make sense in the given situation.
Essential Questions
How can you represent quantities and relationships on a graph?
How can you represent and describe functions?
Can functions model real-world situations?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Functions
 Graphs
Students will be able to:
 Plot points in a coordinate plane.
 Graph linear equations on a coordinate by using a table of solution values.
 Represent relations using sets of ordered pairs, tables, mappings, and graphs.
 Determine whether an equation or relation is a function using a table of solution values or a graph.
 Identify the domain and range of a function.
 Find the value of a function for a given element of the domain.
 Use function notation.
Integration
Technology Integration
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Mathematics
Course: Algebra 1
Unit 6:
Linear Equations and Functions (Graphing and Writing)
Summary and Rationale
Unit 6 connects and extends the ideas introduced in Unit 5 by allowing students to take a closer look at
the characteristics and properties of linear functions, and their equations and graphs. Students will learn
how the slope of a line affects its graph, different graphing methods, and how to write and obtain
important information from linear equations in different forms.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-CED-2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A-CED-3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A-REI-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).
Functions
F-IF-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).
F-IF-2
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F-IF-4
For a function that models a relationships 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.
F-IF-5
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F-IF-6
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
F-IF-7
F-IF-8
F-IF-9
F-LE-1
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. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
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.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship between variables in which each value of the input variable (value in the
domain) is associated with a unique value of the output variable (value in the range.) Functions can be
represented in a variety of ways, such as words, tables, two-variable equations or rules, sets of ordered
pairs, and graphs. Each representation is particularly useful in certain situations. Some important families
of functions are developed through transformations of the simplest form of the function.
Two ratios are proportional if they have the same ratio in each instance where they are measured. With
linear functions, the slope of the line is the ratio of the vertical change to the horizontal change. The slope
of a line can be positive, negative, zero, or undefined. The ratio of slope remains the same when measured
between any two points on a line, so it is proportional.
If the ratio of two variables is constant, then the variables have a special linear relationship, called a direct
variation. The equation of a direct variation is y = kx. Its graph is a line that passes through the origin and
has a slope of k.
A linear equation can be represented using three equation forms: slope-intercept form, point-slope form,
and standard form. All forms are useful in writing the equations of linear functions given a graph or
certain characteristics and also provide effective and efficient graphing methods.
The particular form of a linear equation often suggests a particular graphing method. The standard form
makes it easy to find x and y intercepts and draw graphs quickly using two points. The slope-intercept
form makes it possible to graph the line easily starting with one point and obtaining several others by
moving according to the slope.
The relationship between two lines can be determined by comparing their slopes and y-intercepts obtained
from graphs or equations.
Scatter plots provide a way to graph ordered pairs to determine whether two sets of real world numerical
data are related. If two sets of data are related, it may be possible to use the graph of an estimated line to
model the data and use it to make estimates or predictions about values.
Absolute value equations can be graphed using a table of values or more quickly by shifting the graph of
y = x.
Essential Questions
What does the slope of a line indicate about the line?
What information does the equation of a line give you?
How can you make predictions based on a scatter plot?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Linear equations
 Functions
 Graphing
 Slope
Students will be able to:
 Determine the slope of a line given a graph or by using the slope formula given two points.
 Identify and write the equation of a direct variation.
 Find the x and y intercepts of a linear equation and use them to draw the graph
 Use slope and a point to graph a line.
 Graph horizontal and vertical lines.
 Use slope-intercept form, point-slope form, and standard form to write an equation of a line.
 Write an equation of a line given two points on the line.
 Write and use a linear equation to solve a real-life problem.





Write the equations of perpendicular and parallel lines
Graph equations of lines in slope-intercept form, point-slope form, and standard form.
Graph ordered pairs of real life data sets in a scatter plot to determine whether a relationship exists
between the two data sets.
Write and use a linear equation or function to solve a real-life problem.
Graph absolute value functions.
Integration
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Writing Integration
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Mathematics
Course: Algebra 1
Unit 7:
Linear Systems of Equations and Inequalities
Summary and Rationale
Unit 7 connects and extends the concepts from Units 2 and 3 about finding solutions to equations and
inequalities to solving systems of equations and inequalities. Students will learn to solve systems of
equations and inequalities by graphing and through algebraic methods such as substitution and
elimination. Students will have the opportunity to see that useful information about equations and
inequalities (including solutions) can be found by analyzing their graphs. Furthermore, the numbers and
types of solutions vary predictably, based on the types of equations and graphs in the system.
Pacing
Three (3) weeks
Standards
Algebra
A-CED-2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A-CED-3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A-REI-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.
A-REI-6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
A-REI-7
Solve a simple system consisting of a linear equations and a quadratic equation in two
variables algebraically and graphically.
A-REI-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 variable as the intersections of the corresponding half-planes.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Solving an equation (or inequality) is the process of rewriting the equation (or inequality) to make what it
says about its variable(s) as simple as possible. Properties of numbers, equality, and inequality can be
used to transform an equation (or inequality) into equivalent, simpler equations (or inequalities) in order
to find solutions.
The solution of a system of equations is the set of ordered pairs that satisfy both equations in the system.
When solving a system of linear equations, there are possible types of solutions: one solution (the point of
intersection of the two lines), no solution (The lines do not intersect.), or an infinite number of solutions
(The equations in the system represent the exact same line.)
Systems of equations can be solved in more than one way. Three methods are graphing, substitution, and
elimination. The best method to use depends on the forms of the given equations and how precise the
solution should be. The graphing method involves graphing each equation and finding the intersection
point, if one exists. When a system has at least one equation that can be solved for a variable, the system
can be efficiently solved using substitution. Some equations of a system are written in a way that makes
eliminating a variable the best method to use.
A linear inequality in two variables has an infinite number of solutions. Solutions to a linear inequality in
two variables can be represented in a coordinate plane as the set of all points on one side of a boundary
line.
Solutions of a system of linear inequalities can be graphed in the coordinate plane. The graph of the
solution of a system of linear inequalities is the region where the graphs of the individual inequalities
overlap.
Real world problems can be modeled and solved using linear inequalities and systems of linear equations
and inequalities.
Essential Questions
How can you solve a system of equations or inequalities?
Can systems of equations model real-world situations?
When and how do you know when the use of one method is more efficient than another for a particular
problem?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Linear systems of equations
 Inequalities
 Graphing
Students will be able to:
 Solve a linear system by graphing, substitution, and elimination methods.
 Identify the number of solutions of a linear system.
 Determine if the lines in a system are parallel, perpendicular, or neither.
 Use linear systems to solve real-life problems.
 Solve and graph a two-variable linear inequality.
 Graph a system of linear inequalities.
Integration
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Writing Integration
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Mathematics
Course: Algebra 1
Unit 8:
Exponents and Radicals
Summary and Rationale
Unit 8 expands on students’ understandings and skills related to exponential expressions. The unit also
introduces concepts related to the square root operation. Students will extend the use of exponents to
include zero and negative exponents and will use the Properties of Exponents and Radicals to simplify
and perform operations on expressions containing exponents and radicals.
Pacing
Four (4) weeks
Standards
Number and Quantity
N-RN-1
Explain how the definition of the meaning of rational exponents follows from extending
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents.
N-RN-2
Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
N-RN-3
Explain why the sum or product of two rational numbers is rational; that the sum of the
rational number and an irrational number is irrational; and that the product of a nonzero
rational number and an irrational number is irrational.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
e. Interpret parts of an expression, such as terms, factors, and coefficients.
f. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-REI-2
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A single quantity may be represented by many different expressions. The facts about a quantity may be
expressed by many different equations (or inequalities).
Powers are used to shorten the representation of repeated multiplication. The Multiplication Properties of
Exponents must be used when simplifying and performing operations on numerical or algebraic
expressions containing powers
The idea of exponents can be extended to include zero and negative exponents. A nonzero number to the
zero power is equal to 1. a-n is the reciprocal of an.
The Properties of Exponents make it easier to simplify products or quotients of powers with the same base
or powers raised to a power or products raised to a power.
To multiply powers with the same base, add the exponents. To raise a power to a power, multiply the
exponents. To raise a product to a power, raise each factor to the power and multiply.
To divide powers with the same base, subtract the exponents. To raise a quotient to a power, raise the
numerator and the denominator to the power and simplify.
The opposite of squaring a number is taking the square root of a number. Square roots are written with a
radical symbol √. The number or expression inside the radical symbol is called the radicand.
Positive real numbers have two square roots. Zero has only one square root: zero. Negative numbers do
not have real square roots because the square of every real number is either positive or zero.
The square of an integer is a perfect square. Therefore the square root of a perfect square is in integer. The
square root of a nonnegative number that is not a perfect square is an irrational number. The exact value
of an irrational number can be represented using a radical. The decimal representation of an irrational
number is an approximation that can either be estimated or determined by a calculator.
An algebraic expression that contains a radical is called a radical expression. The simplest form of a
radical expression is an expression that has no perfect square factors other than 1 in the radicand, no
fractions in the radicand, and no radicals in the denominator of a fraction. Properties of Real Numbers and
Radicals (or Square Roots) can be used to simplify expressions that contain radicals, as well as to
multiply and divide radicals.
Rationalizing the denominator of a radical expression removes the radical from the denominator of the
expression. The denominators of some radical expressions can be rationalized by multiplying by
conjugates.
Essential Questions
How can you simplify expressions involving exponents?
How do you know when an expression with exponents is completely simplified?
How do you simplify a radical (or square root) expression?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Exponents
 Radicals
 Square roots
 Radical expression
 Distributive property
Students will be able to:
 Use properties of exponents to evaluate and simplify expressions.
 Evaluate a square root.
 Simplify square roots and radical expressions.
 Simplify radical expressions involving addition, subtraction, and multiplication.
Integration
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Mathematics
Course: Algebra 1
Unit 9:
Polynomials
Summary and Rationale
Unit 9 connects and extends the big ideas introduced in Unit 8 to polynomials. In this unit, students will
have more opportunities to see that two algebraic expressions that appear to be different can be equivalent
as they apply the properties of real numbers to the addition, subtraction, and multiplication of polynomial
expressions, and as they factor polynomials. They will use the Properties of Real Numbers and
Exponents, particularly the Commutative and Associative Properties and the Distributive Property to
manipulate polynomial expressions, and multiply and factor polynomials.
Pacing
Four (4) weeks
Standards
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
g. Interpret parts of an expression, such as terms, factors, and coefficients.
h. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-APR-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.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A single quantity may be represented by many different expressions. The facts about a quantity may be
expressed by many different equations (or inequalities).
A monomial is a number, a variable, or the product of a number and one or more variables with whole
number exponents. The degree of a monomial is the sum of the exponents of the variables in the
monomial.
Monomials can be used to form larger expressions called polynomials. A polynomial is a monomial or a
sum of monomials. A polynomial of two terms is a binomial. A polynomial of three terms is a trinomial.
A polynomial is usually written in standard form, which means that the terms are arranged in decreasing
order, from largest exponent to smallest exponent. The degree of a polynomial in one variable is the
largest exponent of that variable.
Polynomials can be added and subtracted. To add or subtract polynomials, add or subtract like terms.
The Properties of Real Numbers can be used to multiply a monomial by a polynomial or to simplify the
product of binomials. To multiply polynomials, use The Distributive Property or FOIL pattern. Simplify
by using the Multiplication Properties of Exponents and then combining like terms.
Special product patterns occur when multiplying polynomials, particularly when multiplying two
binomials. These patterns include the Sum and Difference Patterns and the Square of a Binomial Pattern.
One important fact is that with exception to the Sum and Difference Patterns, when you multiply two
binomials, the result is a trinomial.
Some trinomials and some polynomials of a degree greater than two can be factored to equivalent forms
which are the product of two binomials. Factoring a polynomial reverses the multiplication process. To
factor a polynomial means to use the Properties of Real Numbers to rewrite it as a product of factors.
Completely factoring a polynomials can involve one or more of the following methods and strategies:
factoring out the GCF (greatest common factor), factoring by grouping (commonly used when factoring
polynomials with four terms), the “unfoil” or “sum/product” method (commonly used for factoring
trinomials), and special product or sum and difference of cube patterns (used for special binomials).
If a polynomial has four or more terms, it may be possible to group the terms and factor binomials from
the groups. This method is called “Factor by Grouping.”
The signs and factors of the coefficients of a trinomial can be used indicate how the trinomial can be
factored.
To factor a trinomial means to “undistributed” or “unfoil” so that it is written as a product of two
binomials (factored form). The sum/product method is the most efficient method when factoring the
simplest trinomials of the form: ax2 + bx + c and a = 1. When a does not equal one, the sum/product
method can still be used in combination with the factoring by grouping method.
Some polynomials, such as trinomials that are the squares of binomials, or binomials that are the
differences of two squares, can be factored by reversing the rules for multiplying binomials that contain
special product patterns.
The following is a step-by-step factoring strategy that can be used for factoring all polynomials:
1) Factor out the greatest common factor, if one exists. (GCF).
2) Does it contain four terms? Try factor by grouping method.
3) Trinomial? (three terms) Does a = 1? If so, “Unfoil” using sum/product method.
If a ≠ 1, Use sum/product method with factor by grouping method. (Break up
the middle term so that there are four terms.)
4) Binomial? Look for Difference of Squares, Sum or Difference of Cubes
Reminder: A sum of squares cannot be factored. (prime)
5) Repeat steps until all factors are prime.
Essential Questions
How are the properties of real numbers related to polynomials?
How do you add, subtract, and multiply polynomials?
How do you factor polynomials?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Monomials
 Polynomials
 Trinomials
 Distributive property
 FOIL pattern
Students will be able to:
 Identify polynomials by their number of terms.
 State the degree of a monomial and of a polynomial.
 Add, subtract, and multiply polynomials.
 Factor trinomials.
 Factor binomials that contain differences of squares, sum of cubes, difference of cubes.
 Factor using the distributive property.
 Factor using grouping techniques.
Integration
Technology Integration
Writing Integration
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Mathematics
Course: Algebra 1
Unit 10:
Quadratics
Summary and Rationale
In Unit 10 students will solve quadratic equations using a variety of methods. Students will learn the
characteristics of quadratic functions as they graph them on a coordinate plane and use the graph to
determine exact solutions or types of solutions. They will also use quadratic functions to model and
represent real- world situations.
Pacing
Four (4) weeks
Standards
Number and Quantity
N-Q-1
Use units as a way to understand problems and to guide the solution of multi-step problems;
choose and interpret units consistently in formula; choose and interpret the scale and the
origin in graphs and data displays.
N-CN-1
Know there is a complex number i such that i2 = - 1, and every complex number has the
form a +bi with a and b real.
N-CN-7
Solve quadratic equations with real coefficients that have complex solutions.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
i. Interpret parts of an expression, such as terms, factors, and coefficents.
j. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-SSE-3
Choose and produce and equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
d. Factor a quadratic expression to reveal the zeros of the function it defines.
e. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
f. Use the properties of exponents to transform expression for exponential functions.
A-CED-2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A-CED-3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A-REI-4
Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equations in x
A-REI-10
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 x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the
initial form of the equations. Recognize when the quadratic formula gives complex
solutions and write them as a + bi and a – bi for real numbers a and b.
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).
Functions
F-IF-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. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
All equations, (or functions) of degree one or higher are defined as polynomial equations. Linear
equations are degree one, so most yield at most one real solution. Quadratics equations are degree two
equations therefore yield at most two real solutions.
Unlike linear functions, the family of quadratic functions models certain situations where the rate of
change is not constant.
Quadratic equations contain an x2 term. Since the opposite of squaring a number is taking the square root,
this is the simplest method for solving a quadratic equation. However, not all quadratic equations can be
solved by taking the square root of both sides.
Quadratic functions are graphed by a symmetric u-shaped graph called a parabola. The equation of a
parabola written in standard form is y = ax2 + bx + c. The equation can be used to find the coordinates
of the vertex. The value of b translates the position of the axis of symmetry and the vertex of the parabola.
A table of values can be used to find points to the left and right of the vertex to form its u-shape. Graphing
a quadratic function provides another method for solving quadratic equations. The x-intercepts of the
function are the solutions to the related quadratic equation. A parabola will cross the x-axis at most two
times.
Two other methods for solving quadratic equations are The Zero-Product Property (used with factoring
methods) and The Quadratic Formula. Any quadratic equation can be solved using The Quadratic
Formula.
When solving quadratic equations, one particular method for solving may be more appropriate or
necessary over another. The best method to use depends on the forms of the given equations, the types of
solutions that exist, and how precise the solutions should be. The value of the discriminant, b2 – 4ac, of a
quadratic equation can be used to determine the number and type of solutions and can also help predict
the best solving method.
A quadratic inequality in two variables has an infinite number of solutions. The graph of a quadratic
inequality consists of the graph of all ordered pairs (x, y) (or points) that are solutions of the inequality.
Since taking the square root of a number is the inverse of squaring a number and vice versa, some radical
equations can be solved by isolating the radical expression (or square root term), and then squaring both
sides of the equation. However, squaring both sides of an equation can yield a solution that does not
satisfy the original equation. Such a solution is called an extraneous solution. When an equation is solved
by squaring both sides, all solutions must be checked in the original equation.
Essential Questions
What are the characteristics of quadratic functions? (or equations)?
How can you solve a quadratic equation?
How can you use quadratic equations and functions to model real-world situations?
How can you solve a radical equation?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Quadratics
 Polynomial equations
 Graphing
Students will be able to:
 Graph quadratic functions.
 Solve quadratic equations by graphing.
 Solve quadratic equations by factoring.
 Solve quadratic equations by using The Quadratic Formula.
 Graph quadratic inequalities.
 Solve radical equations.
Integration
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Mathematics
Course: Algebra 1
Unit 11:
Rational Expressions and Equations
Summary and Rationale
Unit 11 connects and extends the big ideas introduced in previous units to rational expressions. In this
unit, students will learn the characteristics of rational expressions as they apply and extend previously
learned concepts to add, subtract, multiply, and divide rational expressions, as well as solve rational
equations.
Pacing
Three (3) weeks
Standards
Algebra
A-SSE-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 complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-APR-6
Rewrite simple ration 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.
A-APR-7
Understand that rational expressions form a system of analogous to the rational numbers,
closed under addition, subtraction, multiplication, and division by a nonzero ration
expression; add, subtract, multiply, and divide rational expressions.
A-REI-2
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A rational number is a number than can written as the quotient (or ratio) of two integers. A rational
expression is a fraction whose numerator and denominator are nonzero polynomials.
Simplifying rational expressions is similar to reducing fractions. To simplify a rational expression, factor
the numerator and denominator, then cancel out common factors. A rational expression is in simplest
form if its numerator and denominator have no factors in common other than ± 1.
Rational expressions can be added, subtracted, multiplied, or divided using the same properties used to
perform these operations on numerical fractions. The rules for multiplying and dividing rational
expressions are the same as the previously learned rules for multiplying and dividing fractions. As with
fractions, to add or subtract rational expressions with like, or the same, denominators, combine their
numerators and write the result over the common denominator. To add or subtract rational expressions
with unlike denominators, the expressions must first be rewritten so that they have like denominators. The
like denominator must be a common multiple of the original denominator, preferably the least common
multiple, called the least common denominator (or LCD.)
Complex fractions contain one or more fractions in their numerator, in their denominator, or in both.
A rational equation is an equation that contains rational expressions. Some rational equations that are
written as a proportion can be solved using cross multiplication. (Each side of the equation must be a
single rational equation.) Other rational equations must be solved by multiplying both sides of the
equation by the least common denominator. The process of solving rational equations may produce
extraneous solutions. When extraneous solutions are testes, they do not solve the original equation.
Any value or solution that makes the denominator of a rational expression equal to zero is excluded from
the domain of the expression or equation. (These values often end up to be extraneous solutions when
solving rational equations.)
Essential Questions
What are the characteristics of a rational expression? (or equation)?
How can you solve a rational equation?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Rational expressions
 Equations
Students will be able to:
 Simplify rational expressions.
 Multiply and divide rational expressions.
 Add and subtract rational expressions with like denominators.
 Add and subtract rational expressions with unlike denominators.
 Identify values excluded from the domain.

Solve rational equations.
Integration
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Geometry
Nutley Public Schools
Mathematics
Course: Geometry
Unit 1:
Basics of Geometry
Summary and Rationale
Unit 1 introduces students to many of the basic ideas and terms in geometry. It begins with a study of
patterns and inductive reasoning which is the foundation of fundamental reasoning skills. Next it
introduces students to the basic undefined terms and defined terms of geometry and explores their
relationships. These terms will be the basis of future definitions, theorems, and postulates. Furthermore,
this unit stresses the importance of accurate notation and correctly naming geometric figures.
Pacing
Two (2) weeks
Standards
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.CO.12
Make formal geometric constructions with a variety of tools and methods (Compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc).
Copy 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 ha point not on the line.
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The stages of inductive reasoning are the foundation of basic reasoning skills and will be useful
throughout geometry to help problem solve, prove theorems true, and apply skills.
Recognizing and describing patterns can be used to problem solve and make predictions.
The fundamental terms, (points, line, plane, segment, and angle) are the building blocks used to define
geometric figures, intersections, and explain postulates and theorems to justify the geometry of the world
around you.
Correct notation and using appropriate symbols are important when naming geometric figures,
intersections, and writing proofs. This will ensure accurate solutions and help avoid confusing.
Conditional statements are logical statements used to clearly write definitions and conjectures.
Proving a statement wrong is often more efficient and effective then trying to prove a statement is true for
all cases.
Essential Question
What are the stages of inductive reasoning and how can they be used when problem solving?
Why is it easier to prove a conjecture false than true?
What are the similarities and differences of the following: AB, AB, AB, and AB ?
Are collinear points also coplanar? Are coplanar points also collinear?

What is the difference between equality and congruence?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Inductive reasoning
 Conjectures
 Collinear points
 Coplanar points
 Equality
 Congruence
Students will be able to:
 Identify and correctly name points, lines, planes, segments, and rays
 Categorize points and lines as collinear and coplanar
 Use inductive reasoning to continue patterns and make conjectures
 Name the intersection of lines and planes
 Sketch simple figures and their intersections
 Measure segments and angles
 Apply laws of logic
 Use properties of equality and congruence
Integration
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Mathematics
Course: Geometry
Unit 2:
Angles and Segments
Summary and Rationale
In Unit 2 students will use the undefined and defined terms studied in unit 1 to explore additional
properties, postulates, and theorems on the subject of angles and segments. They will you these
properties, that include the segment addition postulate, angle addition postulate, properties of bisectors,
and theorems concerning angles formed by intersecting lines, to find missing angle measurements and
segment lengths. Students will develop solid reasoning and justification skills by analyzing geometric
relationships. Also, students will have opportunities to review algebra 1 skills by setting up equations and
solving for and unknown value.
Pacing
Two (2) Weeks
Standards
Number and Quantity
N.Q.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.
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Algebra
A.CED.1
Create equations and inequalities in one variable and use them to solve problems.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.CO.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.
G.CO.12
Make formal geometric constructions with a variety of tools and methods (Compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc).
Copy 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 ha point not on the line.
G.GPE.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).
G.GPE.6
Find the point on a directed line segment between two given points that partitions the
segment in a given ratio.
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A variety of techniques of indirect measurements, the angle addition postulate, and the segment addition
postulate can be used to find unknown values and solve problem.
Properties of bisectors are the same for segments and angles and can be used to find unknown
measurements.
Complementary and Supplementary angles describe a relationship between two angles
Two intersecting lines and two parallel lines cut by a transversal form angles with specific relationships.
These relationships can be used to help find unknown measurements, help classify polygons, prove
polygons are congruent or similar, and help find area.
Two lines intersected by a transversal form angels with specific relationships that can also be used to
prove lines are parallel or perpendicular.
Measurements can be used to describe and compare real-life objects.
Analyzing geometric relationships develops reasoning and justification skills.
When two measurements are equal or equal to a known value, an equation can be written to solve for
unknown values.
Essential Question
What is the relationship between the measures of the angles formed by intersecting lines?
What are the relationships among the angels formed by two parallel lines and a transversal?
How are the angle addition postulate and segment addition postulate the same?
How can you use equations to help you find missing angle measurements and side lengths?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Angles
 Segments
 Parallel lines
 Intersecting lines
Students will be able to:
 Use the segment and angle addition postulates to find missing measurements
 Classify angles
 Find the measures of complementary and supplementary angles
 Bisect a segment and find the coordinates of the midpoint of a segment
 Bisect an angle
 Use the properties of bisectors to find missing measurements
 Identify relationships between lines (parallel, perpendicular, skew, …)
 Identify angles formed by intersecting lines as vertical angles or linear pair and use their properties of
find angle measurements
 Identify adjacent angles
 Indentify special angle relationships formed by two lines and a transversal (alternate interior, alternate
exterior, corresponding, and same side interior angles)
 Find the congruent angles formed when a transversal cuts parallel lines
 Prove lines are parallel and perpendicular using special angle relationships
Integration
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Mathematics
Course: Geometry
Unit 3:
Triangles
Summary and Rationale
In unit 3 students will study the various properties of triangles. They will first explore properties true for
all triangles and use them to find missing measurements and classify angles by their sides and angles.
Next they will explore properties true for specific classifications and use their properties to find missing
measurements. Throughout this unit students will use several skills and concepts introduced in the
previous units, including properties of segments and angles, correctly naming geometric figures and
intersections, and analyzing geometric relationships.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Algebra
A.CED.1
Create equations and inequalities in one variable and use them to solve problems.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.CO.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.
G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum of 180 degrees; 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.
G.CO.12
Make formal geometric constructions with a variety of tools and methods (Compass and
straightedge, string, reflective devices, paper folding, dynamic geometric software, etc).
Copy 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 ha point not on the line.
G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle
divided the other two proportionally, and conversely; the Pythagorean Theorem proved
using triangle similarity.
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
All triangles have explicit properties that can be proven using angle relationships, undefined and defined
terms, postulates, and theorems. The following properties are true for all triangles: interior angles add up
to 180 degrees, the measure of an exterior angle is the sum of the two non-adjacent angles, the sum of any
two sides of a triangle must be larger than the third, the shortest side of a triangle is across from the
smallest angle, and properties of medians, angle bisectors, and altitudes.
Triangles can be classified by both their sides and their angles and specific properties apply for each
classification. These properties can be used to find missing angle measurements, missing side lengths,
and to solve problems.
You can use the Pythagorean Theorem, distance formula, midpoint formula, and theorems and postulates
about angle and segment relationships to classify triangles.
Essential Question
What are some relationships among the interior angles of triangle and exterior angles of a triangle?
What do you know about the two acute interior angles in a right triangle?
How can you use interior angles to classify triangle by their sides and how can you use side lengths to
classify triangles by their angles?
Can any three lengths define a triangle?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Triangles
 Angles
 Pythagorean Theorem
Students will be able to:
 Classify triangles by their sides and by their angles
 Find angle measures in triangles
 Use exterior angles to find the measure of interior angles and vice-versa
 Use properties of isosceles and equilateral triangles to find angle and segment measurements
 Use the Pythagorean Theorem to find missing side lengths of a right triangle
 Use the converse of the Pythagorean Theorem to classify triangles by their angles
 Apply the Triangle Inequality Theorem to determine whether three sides make a triangle
 Identify the shortest and longest sides of a triangle given angle measurements
 Identify the smallest and biggest angles given side measurements
 Identify and apply properties of medians, angle bisectors, perpendicular bisectors, and altitudes
Integration
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Mathematics
Course: Geometry
Unit 4:
Congruence and Similarity
Summary and Rationale
In unit 4 students will explore the properties, similarities, and differences of congruent and similar
polygons and use them to find missing measurements and problem solve. Next, students will specifically
study congruent and similar triangles. By using previous skills including, classifying angles, solving
linear equations, finding midpoints, and using angle relationships students will prove triangles similar and
congruent. In addition, students will strengthen their reasoning and justification skills by using visual
recognition and representation to prove congruency and similarity.
Pacing
Three (3) weeks
Standards
Algebra
A.CED.1
Geometry
G.CO.1
G.CO.7
G.CO.8
G.CO. 9
G.CO.10
G.SRT.2
Create equations and inequalities in one variable and use them to solve problems.
Know precise definitions of angles, 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.
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.
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the
definition of congruence in terms of rigid motions.
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.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum of 180 degrees; 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.
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.
G.SRT.3
Use the properties of similarity transformations to establish the AA criterion for two
triangles to be similar.
G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle
divided the other two proportionally, and conversely; the Pythagorean Theorem proved
using triangle similarity.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Equality is used to state two quantizes are the same, congruency is used to state figures are the same
shape and same size, and similarity is used to state figures are the same shape but different sizes.
Previous skills such as classifying angles, solving linear equations, finding midpoints, and using angle
relationships can be used to help prove polygons similar or congruent.
SSS, SAS, ASA, AAS, and HL are five ways to prove triangles are congruent. Once two triangles are
proven congruent then you know all six of their corresponding parts are congruent.
AA, SSS similarity, and SAS similarity are three ways to prove triangles are similar. Once two triangles
are proven similar than all properties of similarity are true.
A constant ratio exists between corresponding lengths of the sides of similar figures. The ratio can be
used find unknown side lengths.
Essential Question
What is the difference between equality, congruency, and similarity?
Why is HL the only side-side-angle combination that works to prove triangles are congruent?
How is the ratio of corresponding sides of similar polygons related to the ratio of their perimeters?
Why do you only need two pairs of congruent angles to prove triangles are similar and not three?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Congruence
 Similarity
 Equality
Students will be able to:
 Identify congruent and similar polygons and their corresponding parts
 Use congruence properties to find missing angle and segment measures
 Show triangles are congruent using SSS, SAS, ASA, AAS, and HL
 Show triangles are similar using AA, SSS similarity, and SAS similarity
 Determine the ratio of similarity and use it to set up a proportion to find missing segment lengths
 Use the ratio of similarity to find perimeters
Integration
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Mathematics
Course: Geometry
Unit 5:
Quadrilaterals
Summary and Rationale
In unit 5 students will study the various properties of quadrilaterals. Using properties, postulates, and
theorems about undefined and defined terms, angles, segments, and triangles studied in previous units
students will be able to classify special quadrilaterals. In addition, students will use the properties of
special quadrilaterals to find missing angle measurements and unknown segment lengths. By analyzing
angle and segment relationships within quadrilaterals students will continue develop reasoning and
justification skills.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Algebra
A.CED.1
Create equations and inequalities in one variable and use them to solve problems.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
G.CO.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.
G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum of 180 degrees; 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.
G.CO.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.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.GPE.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)
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Previous skills including identifying angle relationships, proving lines are parallel or perpendicular, and
solving linear equations can be used to classify quadrilaterals.
Parallelograms, rectangles, rhombi, square, trapezoids, and isosceles trapezoids each have distinct
properties and these properties can be used to find missing angle measurements and side lengths, prove
triangles are congruent or similar, classify triangles, and problem solve.
Rectangles, rhombi, and squares are all special types of parallelograms and therefore have all the
properties of parallelograms as well as their own specific characteristics. (EX: A square is a rectangle but
a rectangle is not a square.)
Essential Question
What properties are true for all quadrilaterals/ parallelograms/ special parallelograms/ trapezoids?
How can you use angle relationships to classify quadrilaterals?
What are the similarities between isosceles triangles and isosceles trapezoids?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Quadrilaterals
 Parallelograms
 Trapezoids
 Triangles
 Angle relationship
Students will be able to:
 Find angle measures of a quadrilateral
 Use properties of parallelograms to find angle and segment measures
 Show that a quadrilateral is a parallelogram using parallelogram properties
 Identify and use properties of special parallelograms (rhombi, rectangles, and squares)
 Apply properties of trapezoids to find angle and segment measures
 Identify special quadrilaterals in a coordinate plane
 Identify special quadrilaterals based on limited information
Integration
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Mathematics
Course: Geometry
Unit 6:
Polygons and Area
Summary and Rationale
In unit 6 students will explore, classify, compare, and apply properties of polygons. Students will first
classify polygons by their number of sides, whether they are convex or concave, and whether they are
regular, equilateral, or equiangular. Next, students will use those classifications to find the measures of
interior and exterior angles of polygons followed by calculating the perimeter, circumference, and area of
polygons and circles. Lastly, students will review similarity and compare perimeters and areas of similar
polygons.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.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.
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.GPE.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles,
e.g., using the distance formula.
G.GMD.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.
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons
per square mile, BTU’s per cubic feet)
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Two dimensional figures can be described, classified, and analyzed by their attributes including their
number of sides, relationship between angles and side measurements, and whether it is convex or
concave.
The sum of the interior angles of a convex polygon is dependent on the number of sides the polygon has.
This can be proven using the interior angles of a triangle. The sum of the exterior angles of a convex
polygon is always 360 degrees.
Spatial sense offers ways to visualize, to interpret, and to reflect on our physical environment.
Perimeter is the distance around the figure and is measured in units. Area is the amount of surface covered
by a figure and is measured in units squared.
A change in one dimension of an object results in predictable changes in area.
Geometric figures can be represented in the coordinate plane.
Area and segment length can be used to determine the probability of hitting a particular point.
Essential Question
What is the connection between the constant ratio between corresponding lengths of the sides of similar
figures and the ratio of their perimeters and areas?
What are some real life situations where you would use perimeter and area?
What does regular mean? What does it tell you about a polygon?
How can you use the measure of and interior angle of a regular polygon to find an exterior and viceversa?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Polygons
 Area

Perimeter
Students will be able to:
 Identify and classify polygons by their number of sides
 Find the measure of the sum of interior and exterior angles of polygons
 Find the measure of an interior and exterior angle of a regular polygon
 Classify polygons as convex, concave, equilateral, equiangular, and/or regular
 Find the area and perimeter of squares, rectangles, triangles, parallelograms, trapezoids, regular
polygons, and figures made up of a combination of those figures
 Find the circumference and area of circles
 Use segments and area to find the probability of an event
Integration
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Mathematics
Course: Geometry
Unit 7:
Surface Area and Volume
Summary and Rationale
In unit 7 students will investigate the surface area and volume of solids. Using tools from previous units
students will distinguish solids by their characteristics and use those characteristics to calculate surface
area, lateral area, and volume. In addition, students will investigate similar solids and compare their
surface area and volumes.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.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.
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.GMD.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.
G.GMD.2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a
sphere and other sold figures.
G.GMD.3 Use volume formulas for cylinders, pyramids, cones and spheres to solve problems.
G.GMD.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.
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.2
Apply concepts of density based on area and volume in modeling situations (e.g., persons
per square mile, BTU’s per cubic feet)
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Three dimensional figures can be described, classified, and analyzed by their attributes including their
bases, lateral faces, and relationship between angles and side measurements.
A change in one dimension of an object results in predictable changes in surface area and volume.
Polyhedrons are solids made up of polygons.
The surface area of a solid is the sum of the areas of all their faces and is measured in units squared. The
volume of a solid is the number cubic units contained in its interior and is measured in cubic units.
Essential Question
What solids can be made using congruent regular polygons?
How do you calculate the surface area of a polyhedron?
What is the difference between height and slant height of cones and pyramids? Which one is used when
finding surface area and which one is used when finding volume? Why?
How is the volume of a pyramid related to the volume of prism with the same base and height?
How are the surface areas and volumes of similar solids related?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Surface area
 Volume
 Solid figures
Students will be able to:
 Identify and name solid figures
 Find the surface area and volume of prisms, cylinders, cones, pyramids, and spheres
 Use properties of similar solids
Integration
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Mathematics
Course: Geometry
Unit 8:
Right Triangles
Summary and Rationale
In unit 8 students will continue their exploration with triangles with exclusively studying the
characteristics and attributes of right triangles. Previous objectives will be revisited including the
Pythagorean Theorem and the converse of Pythagorean Theorem as well as new ideas such as geometric
mean and trigonometric ratios. Also, students will examine special right triangles and discover ways to
find side lengths using constant ratios. Students will continue to apply properties of angles, segments, and
triangles, reinforce their reasoning and justification skills, and review algebra 1 skills.
Pacing
Four (4) weeks
Standards
Number and Quantity
N.RN.2
Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
N.Q.2
Define appropriate quantities for the purpose of descriptive modeling
N.Q.3
Choose a level of accuracy appropriate to limitations on measurement when reporting
quantities.
Algebra
A.CED.1
Create equations and inequalities in one variable and use them to solve problems.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle
divided the other two proportionally, and conversely; the Pythagorean Theorem proved
using triangle similarity.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.SRT.6
Understand that by similarity, side ratios in right triangles are properties of angles in the
triangle, leading to the definitions of trigonometric rations for acute angles.
G.SRT.7
Explain and use the relationship between the side and cosine complementary angles.
G.SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems
G.SRT.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).
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The Pythagorean Theorem can be used to find the lengths of the sides of a right triangle and the converse
of the Pythagorean Theorem can be used to classify triangles by their angles.
Any right triangle can be split into two similar triangles when you draw the altitude from the right angle
to its opposite side. Furthermore, when an altitude is drawn in a right triangle from the right angle to its
opposite side, the altitude is the geometric mean of the two segments of the hypotenuse and the leg is the
geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
Using the Pythagorean Theorem, you can prove that the extended ratio of the side lengths of a 45-45-90
triangle is 1:1: 2 and the extended ratio of the side lengths for 30-60-90 triangles is 1: 3 : 2. The ratios
can be used to find missing side lengths.
A trigonometric ratio is a ratio of the lengths of two sides in a right triangle. Sine, Cosine, and tangent
ratios are constant for a given angle measure. These ratios can be used to find the measure of a side or an
acute angle in a right triangle.
Essential Question
Are all right triangles similar? Why or why not?
Are all 45-45-90 triangles isosceles? Why or why not?
What relationship exists among the sides of a right triangle?
How can you use the side lengths in a triangle to classify the triangle by its angle measures?
How are geometric means related to the altitude of a right triangle?
What does it mean to solve a right triangle?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:


Right triangles
Isosceles triangles
Students will be able to:
 Use The Pythagorean Theorem to find the side length of a right triangle
 Apply the properties of 30-60-90 and 45-45-90 triangles to find side measures
 Find the sine, cosine, and tangent of an acute angle
 Use basic trigonometry ratios and inverse ratios to solve right triangles
 Calculate the geometric mean given two numbers
 Apply geometric mean properties given a right triangle and an altitude drawn from the right angle to
its opposite side
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Mathematics
Course: Geometry
Unit 9:
Circles
Summary and Rationale
In unit 9 students will study the properties and characteristics of circles. First, students will identify and
name segments and lines related to circles. In addition to properties, theorems, and postulates previous
learned, students will use properties of tangents, secants, chords, arcs, central angles, and inscribed angles
to find angle measurements, arc measurements, and unknown segment lengths. Finally, students will
write equations of circles and graph them in a coordinate plane.
Pacing
Three (3) weeks
Standards
Algebra
A.CED.1
Geometry
G.C.1
G.C.2
G.C.3
G.C.4
G.C.5
G.CO.1
G.CO. 9
G.CO.10
Create equations and inequalities in one variable and use them to solve problems.
Prove that all circles are similar
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.
Construct the inscribed and circumscribed circles of a triangle, and prove properties of
angles for a quadrilateral inscribed in a circle.
Construct a tangent line from a point outside a given circle to the circle.
Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measures of the angle as the constant of
proportionality derive the formula for the area of a sector.
Know precise definitions of angles, 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.
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.
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle
sum of 180 degrees; 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.
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.
G.CO.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.GPE.1
Derive the equations of a circle given center and radius using the Pythagorean Theorem;
complete the square to find the center and the radius of a circle given by an example.
G.GPE.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)
G.MG.1
Use geometric shapes, their measures, and their properties to describe objects.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
G.CO.11
Instructional Focus
Enduring Understandings
There are several relationships between tangents, secants, and chords. These relationships can help
determine that two chords or tangents are congruent, find the length of a secant, chord, or radius, and
determine how far a chord is from the center of the circle.
Tangents, secants, and chords can be used to find the measures of angles formed inside, outside, and on
circles. Also angles insides, outside, and on the circle can be used to find the measure and lengths of arcs.
Circles in the coordinate plane can be written using a standard equation.
Circles have many connections with other geometric figures. When a polygon in inscribed in a
circle or vice-versa you can use the properties of circles to find missing angle measures and side lengths.
For example a quadrilateral can be inscribed in a circle if and only if their opposite angles are
supplementary.
Essential Question
How are the lengths of tangent segments related?
How are inscribed angles related to central angles?
How are central angles, inscribed angles, angles inside the circle, and angles outside the circle related to
their intercepted arcs?
What is the relationship between the lengths of segments in a circle formed by two intersecting chords?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Circles
 Tangents
 Secants
 Chords
 Angles
Students will be able to:
 Identify segments and lines related to circles (chord, diameter, radius, secant, and tangent)
 Use properties of tangents, chords, and secants to find segment and angle measurements
 Classify arcs by their measurements (semi, major, and minor)
 Determine the measure of central and inscribed angles using their intercepted arcs and vice-versa
 Write and graph the equation of a circle
Integration
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Mathematics
Course: Geometry
Unit 10:
Transformation
Summary and Rationale
In unit 10 students will study the motion of geometric figures in the form of transformations. They will
review and use properties of angles, segments, polygons, congruency, and similarity to identify
reflections, rotations, translations, dilations, and compositions of transformation.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.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.
Geometry
G.CO.1
Know precise definitions of angles, 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.
G.CO.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 at outputs. Compare transformations that preserve distance and angle to those that do
not (e.g., translation versus horizontal stretch).
G.CO.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and
reflections that carry it onto itself.
G.CO.4
Develop definitions of rotations, reflections, and translations in terms of angles, circles,
perpendicular lines, parallel lines, and line segments.
G.CO.5
Given a geometric figure and a rotation, reflections, 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.
G.CO.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.
G.SRT.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.
Use congruence and similarity criteria for triangles to solve problems and to prove
relationships in geometric figures.
G.MG.3
Apply geometric method 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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
G.SRT.5
Instructional Focus
Enduring Understandings
A transformation is an operation that maps a pre-image onto the image.
Translating, reflecting and rotating polygons yield congruent polygons while dilating polygons yield
similar polygons.
There are multiple ways to describe transformations including coordinate notation and verbal notation.
Both representations are important and useful.
When you perform combinations of two or more transformations it can be equivalent to performing only
one transformation. For example, the composition of two reflections results in either a translation or
rotation.
Essential Question
What transformations maintain the congruence of a figure?
What is the relationship between the line of reflections and the segment connecting a point and its image?
What happens when you reflect a figure about the x axis and then the y axis?
How can you use the value of the scale factor of a dilation to determine if it is an enlargement or
reduction?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Transformation
 Notations
 Congruence
Students will be able to:
 Identify and use properties of translations, rotations, reflections, and dilations



Describe transformations using words and coordinate notation given a diagram
Determine the number of lines of symmetry a plan figure contains
Use coordinate notation to sketch a diagram of a transformation
Integration
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Algebra 2
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 1:
Linear Equations and Inequalities (One-variable)
Summary and Rationale
Unit 1 revisits and further explores the concepts and essential topics studied in Algebra 1. The skills,
strategies, and one-variable solving techniques previously learned in Algebra 1 are crucial in the
development of a foundation for continuing the study of algebra and its applications.
Pacing
Two (2) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-CED-1
Create equations and inequalities in one variable and use them to solve problems.
A-REI-1
Explain each step in solving a simple equations as following from the equality of numbers
asserted in the previous step, starting from the assumption that the original equations has a
solution. Construct a viable argument to justify a solution method.
A-REI-3
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Numbers can be classified by their characteristics. All real numbers can be represented on the graph of a
real number line. The location of some numbers must be approximated (irrational numbers).
The definition of a square root can be used to find the exact square root of some nonnegative numbers,
called perfect squares. The square roots of other non-negative numbers that are not perfect squares can be
approximated.
Quantities are used to form expressions, equations, and inequalities. An expression refers to a quantity,
but does not make a statement about it. An equation (or an inequality) is a statement about the quantities it
mentions. Using variables in place of numbers in equations (or inequalities) allows the statement of
relationships among numbers that are unknown or unspecified.
Algebra uses symbols, called variables, to represent quantities that are unknown or that vary. A single
quantity may be represented by many different expressions. The facts about a quantity may be represented
by many different equations (or inequalities.)
Mathematical phrases and real-world relationships can be represented through algebra using symbols and
operations. These mathematical phrases are called algebraic expressions. When simplifying numerical and
algebraic expressions, operations must be performed in correct order, known as the Order of Operations.
An algebraic expression can also be simplified by combining the parts of the expression that are alike.
Equations are used to represent the relationship between two quantities that have the same value.
Equations can describe, explain, and predict various aspects of the real world.
Equivalent equations have the same solution(s). An algebraic equation can be represented using the
symbols in an infinite number of ways, where each representation has the same solution(s).
Inequalities are used to state that two quantities that DO NOT have the same value. They are
mathematical sentences that use inequality symbols to compare the values of two expressions. Inequalities
can be represented with symbols. Their solutions can be represented on a number line. Inequalities can
describe, explain, and predict various aspects of the real world.
Properties of real numbers, equality, and inequalities, along with the use of inverse operations, can
transform an equation (or inequality) into one or a series of equivalent simpler equations (or inequalities.).
These properties can be used repeatedly to isolate the variable. This process is used to find solutions to
one-variable equations (or inequalities). The process is also used to isolate a particular variable in a
formula that contains two or more variables.
Many properties of equality hold true for inequalities. One major difference is as follows: When
multiplying or dividing both sides of an inequality by a negative number, it is necessary to reverse the
inequality sign.
The solutions of a compound inequality are either the overlap or combination of the solution sets of
distinct inequalities. The graph of a compound inequality with the word and contains the overlap of the
graphs of the two inequalities. The graph of a compound inequality with the word or contains each graph
of the two inequalities.
Absolute value inequalities can be solved by first isolating the absolute value expression (if necessary),
the writing the appropriate compound inequality that satisfies the condition, and then solving accordingly,
which will yield a solution set to be graphed.
Essential Questions
How are real numbers classified, ordered, and compared?
Why is the order of operations necessary for simplifying numerical expressions?
Can expressions (or equations or inequalities) that appear to be different be equivalent?
How can you solve equations (or inequalities)?
What are the connections between the algebraic and graphical representations of one-variable linear
inequalities?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 One-variable linear equations
 Inequalities
 Irrational numbers
 Square roots
 Graphing
Students will be able to:
 Use a number line to graph, order, and compare real numbers.
 Identify the properties of real numbers and use them to perform operations.
 Evaluate algebraic expressions.
 Simplify algebraic expressions by combining like terms.
 Solve linear equations (all types).
 Rewrite equations and formulas that contain more than one variable. (Solve for one variable.)
 Solve simple and compound inequalities.
 Solve absolute equations and inequalities.
 Solve real-life problems using algebraic models, equations, and inequalities.
Integration
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Mathematics
Course: Algebra 2
Unit 2:
Linear Functions (Two-variable)
Summary and Rationale
Unit 2 enters the algebraic world of two-variables as students revisit and further explore functions,
beginning with those that are linear. Students will review and thoroughly analyze the characteristics and
properties of linear functions, their equations and graphs. Students will also review and further explore
how the slope of a line affects its graph, different graphing methods, and how to write and obtain
important information from linear equations in different forms.
Pacing
Two (2) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-CED-2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A-CED-3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A-REI-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).
A-REI-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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
A-REI-12 Graph the solutions to a linear inequality in two variable 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 variable as the intersections of the corresponding half-planes.
Functions
F-IF-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).
F-IF-2
F-IF-1
F-IF-2
F-IF-4
F-IF-5
F-IF-6
F-IF-7
F-IF-8
F-IF-9
F-LE-1
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
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).
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
For a function that models a relationships 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.
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
f. Graph linear and quadratic functions and show intercepts, maxima, and minima.
g. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
h. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
i. Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
j. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
c. 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.
d. Use the properties of exponents to interpret expressions for exponential functions.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Distinguish between situations that can be modeled with linear functions and with
exponential functions.
d. Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
e. Recognize situations in which one quantity changes at a constant rate per unit
interval relative to another.
f. Recognize situations in which a quantity grows or decays by a constant percent rate
per unit interval relative to another.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship in which one set of values defines another. A function is a relationship
between variables in which each value of the input variable (value in the domain) is associated with a
unique value of the output variable (value in the range.) In order to determine if an equation or a set of
ordered pairs represents a function, the solutions of the equation or the ordered pairs can be organized in a
table or plotted on a graph. If the table of values shows that each value in one set is paired with exactly
one value in the other set, the relation is a function. The vertical line test uses the graph to determine
whether a relation is a function. All functions can be used to model many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function.
Functions can be represented using an equation, or through a graph of the ordered pairs on a coordinate
plane that satisfy the equation. The graph of a function is a useful way of visualizing the relationship of
the function, as well as its complete domain and range.
A relationship between two variables or two sets of data is a linear function if the two variables increase
or decrease by the same amount over equal periods of time. This constant rate of change is the slope of
the linear function.
The graph of a linear function in a coordinate plane is a line. Using algebraic methods to manipulate
and/or solve the equation of a linear function can throw light on the function’s properties such as its slope
and intercepts, which can both help visualize the behavior of its graph. These strategies can result in the
use of more effective and efficient graphing methods.
The slope of a linear function is the ratio of the vertical change to the horizontal change. The slope of a
line can be positive, negative, zero, or undefined. The ratio of slope remains the same when measured
between any two points on a line, so it is proportional.
A linear function can be represented using three equation forms: slope-intercept form, point-slope form,
and standard form. All forms are useful in writing the equations of linear functions given a graph or
certain characteristics. The particular form of a linear equation often suggests a particular graphing
method. The standard form makes it easy to find x and y intercepts and draw graphs quickly using two
points. The slope-intercept form makes it possible to graph the line easily starting with one point and
obtaining several others by moving according to the slope.
The relationship between two lines can be determined by comparing their slopes and y-intercepts obtained
from graphs or equations.
Scatter plots provide a way to graph ordered pairs to determine whether two sets of real world numerical
data are related. If two sets of data are related, it may be possible to use the graph of an estimated line to
model the data and use it to make estimates or predictions about values.
A linear inequality in two variables has an infinite number of solutions. Solutions to a linear inequality in
two variables can be represented in a coordinate plane as the set of all points on one side of a boundary
line.
The graph of an absolute value function in a coordinate plane is a v-shaped graph that can be created
using a table of values or more quickly by shifting the graph of y = x.Using algebraic methods to
manipulate and/or solve the equation of a absolute value function can throw light on the function’s
properties such as its vertex and direction of opening, which can both help visualize the behavior of its
graph. These strategies can result in the use of more effective and efficient graphing methods.
Essential Questions
What are the different ways to determine if a relation is also a function?
When and why is using one method for graphing a linear equation sometimes more appropriate than
another?
How is slope used to describe a line and compare two or more lines?
What are the connections between the algebraic and graphical representations of two variable
inequalities?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
y = a (x – h) + k
y=ax–h +k
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Two-variable linear functions
 Inequalities
 Slope
 Graphing
Students will be able to:
 Represent relations and functions.
 Identify, graph, and evaluate linear functions.
 Determine slopes of lines and classify parallel and perpendicular lines.
 Graph linear equations using slope-intercept form and standard form.
 Write linear equations.
 Identify and write linear equations for direct variations.
 Use a scatter plot to identify a correlation between two sets of data.
 Approximate the equation of a best-fit line for a set of data.
 Graph linear inequalities in two variables.
 Represent and graph piecewise functions.
 Represent and graph absolute value functions.
 Use linear equations, inequalities, and functions to model and solve real-life problems.
Integration
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Writing Integration
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Mathematics
Course: Algebra 2
Unit 3:
Linear Systems
Summary and Rationale
Unit 3 connects and extends the graphing and solving concepts from previous units to finding solutions of
systems of equations and inequalities. Students will review and extend graphing and algebraic methods
for solving systems of equations and inequalities by graphing and through algebraic methods. Students
will have more opportunities to make meaningful connections between equations and inequalities, their
graphs, and the number and types of solutions that surface.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-CED-2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A-CED-3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A-REI-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.
A-REI-6
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
A-REI-7
Solve a simple system consisting of a linear equations and a quadratic equation in two
variables algebraically and graphically.
A-REI-12 Graph the solutions to a linear inequality in two variable 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 variable as the intersections of the corresponding half-planes.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Solving an equation (or inequality) is the process of rewriting the equation (or inequality) to make what it
says about its variable(s) as simple as possible. Properties of numbers, equality, and inequality can be
used to transform an equation (or inequality) into equivalent, simpler equations (or inequalities) in order
to find solutions.
Two or more linear equations/inequalities form a linear system. The solution of a system is represented by
the set of ordered pairs that satisfy every equation/inequality in the system.
When solving a system of linear equations, there are three possible types of solutions: one solution (the
point of intersection of the two lines), no solution (The lines do not intersect.), or an infinite number of
solutions (The equations in the system represent the exact same line.)
Systems of equations can be solved in more than one way. Three methods are graphing, substitution, and
elimination. The best method to use depends on the forms of the given equations and how precise the
solution should be. The graphing method involves graphing each equation and finding the intersection
point, if one exists. When a system has at least one equation that can be solved for a variable, the system
can be efficiently solved using substitution. Some equations of a system are written in a way that makes
eliminating a variable the best method to use.
A linear inequality in two variables has an infinite number of solutions. Solutions to a linear inequality in
two variables can be represented in a coordinate plane as the set of all points on one side of a boundary
line.
Solutions of a system of linear inequalities can be graphed in the coordinate plane. The graph of the
solution of a system of linear inequalities is the region where the graphs of the individual inequalities
overlap.
Real world mathematical problems can be modeled, represented algebraically and graphically, and solved
by using equations, inequalities, and systems. An equation or graph that models a real world situation can
be used to make estimates, make predictions about future occurrences, and optimize through linear
programming. Real-world graphs should only show points that make sense in the given situation.
Essential Questions
How can you use the graph and/or the equations of a linear system to determine the number of solutions,
as well as the actual solution?
When and why is using one method for solving a linear system sometimes more appropriate than another?
What are the connections between the algebraic and graphical representations of a system of linear
inequalities?
What does a three-variable system look like algebraically? Graphically? How do methods of solving and
actual solutions compare to those of two variables?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Linear systems
 Solving equations with different numbers of variables
 Graphing
Students will be able to:
 Graph and solve systems of linear equations in two-variables.
 Solve linear systems using algebraic methods.
 Graph a system of linear inequalities to determine and represent the solutions of the system.
 Solve linear programming problems.
 Solve systems of linear equations in three variables.
 Use linear systems and linear programming to model and solve real-life problems.
Integration
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Mathematics
Course: Algebra 2
Unit 4:
Matrices and Determinants
Summary and Rationale
Unit 4 involves the study of matrices, which are used throughout linear algebra to solve linear systems. This unit
will provide an introduction to matrices, their uses, and the properties of matrix operations.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-VM-6
Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
N-VM-7
Multiply matrices by scalars to produce new matrices, e.g., as when all the payoffs in a
game are doubled.
N-VM-8
Add, subtract, and multiply matrices of appropriate dimensions.
N-VM-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.
N-VM-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.
N-VM-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.
N-VM-12 Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of
the determinant in terms of area.
Algebra
A-REI-8
Represent a system of linear equations as a single matrix equation in a vector variable.
A-REI-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 x 3 or greater).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A matrix is a rectangular arrangement of numbers in rows and columns that can be used to organize and
perform operations on numerical data. The numbers in a matrix are called its entries, (or elements.)
To add or subtract matrices, add or subtract corresponding entries. Matrices can be added or subtracted
only if they have the same dimensions.
The entries in a matrix can also be multiplied by a real number. This process is called scalar
multiplication.
The product of two matrices A and B is defined if the number of columns in A is equal to the number of
rows in B.
Some of the properties of operations of real numbers differ from the properties of matrix operations. One
major difference is as follows: Matrix multiplication, in general is not commutative.
Square matrices (matrices with the same number of rows and columns) have multiplicative inverses, as
long as their determinants do not equal zero. There are formulas and procedures for determining the
inverse of a square matrix that involve the determinant of the matrix.
Matrices, their determinants and inverses, and matrix operations can be used to solve linear systems
through the application of Cramer’s Rule or a matrix equation.
Essential Questions
How do you add, subtract, or multiply matrices?
Which properties of real numbers do not hold true for matrices?
Can two different matrices have the same determinant?
How can matrices and their inverses be used to solve linear systems?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Matrices
 Determinants
 Cramer’s Rule
Students will be able to:
 Add and subtract matrices and multiply a matrix by a scalar.






Solve matrix equations.
Multiply two matrices.
Evaluate determinants of 2x2 and 3x3 matrices.
Find the inverse of a matrix.
Solve linear systems using Cramer’s Rule.
Solve linear systems using the inverse of a matrix.
Integration
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Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 5:
Quadratics
Summary and Rationale
Unit 5 extends the study of functions to quadratic functions. Students will thoroughly analyze the
characteristics and properties of quadratic functions, their equations and graphs. Students will learn
different solving and graphing methods, as well as how to write and obtain important information from
quadratic equations in different forms. During this unit, it is crucial for students to see the relationship between
the solutions of a quadratic equation, the zeros of a quadratic function, and the x-intercept’s of this function’s
graph. This connection will follow through with other types of functions that will be studied in future units. The
graph of a quadratic function is called a parabola, which is one of the four conic sections that will be studies in Unit
10. Students will also solve and graph one and two variable quadratic inequalities.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-Q-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.
N-CN-1
Know there is a complex number i such that i2 = - 1, and every complex number has the
form a +bi with a and b real.
N-CN-2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
N-CN-3
Find the conjugate of a complex number; use conjugates to find moduli and quotients of
complex numbers.
N-CN-5
Represent addition, subtraction, multiplication, and conjugation of complex numbers
geometrically on the complex plane; use properties of this representation for computation.
N-CN-7
Solve quadratic equations with real coefficients that have complex solutions.
N-CN-8
Extend polynomial identities to the complex numbers.
N-CN-9
Know the fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
k. Interpret parts of an expression, such as terms, factors, and coefficents.
l. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-SSE-3
Choose and produce and equivalent form of an expression to reveal and explain properties
A-CED-2
A-CED-3
A-REI-4
A-REI-10
A-REI-11
Functions
F-IF-1
F-IF-2
F-IF-4
F-IF-5
F-IF-6
F-IF-7
of the quantity represented by the expression.
g. Factor a quadratic expression to reveal the zeros of the function it defines.
h. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
i. Use the properties of exponents to transform expression for exponential functions.
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
Solve quadratic equations in one variable.
c. Use the method of completing the square to transform any quadratic equations in x
into an equation of the form (x – p)2 = q that has the same solutions. Derive the
quadratic formula from this form.
d. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the
initial form of the equations. Recognize when the quadratic formula gives complex
solutions and write them as a + bi and a – bi for real numbers a and b.
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).
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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
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).
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
For a function that models a relationships 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.
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
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. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
F-IF-8
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
F-IF-9
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship in which one set of values defines another. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function.
Functions can be represented using an equation, or through a graph of the ordered pairs on a coordinate
plane that satisfy the equation. The graph of a function is a useful way of visualizing the relationship of
the function, as well as its complete domain and range.
Quadratic equations are of degree two, therefore they have two solutions that can be determined from a
graph (The solutions are the x-intercepts.), or by using different algebraic methods. Four algebraic
methods are factoring, taking the square root of both sides, completing the square, or using the Quadratic
Formula. The best method to use depends on the forms and characteristics of the given equations, the
nature of the solutions, and how precise the solutions should be.
Quadratics can have real number solutions, but can also have solutions in a larger system, called the
complex numbers. There are differences between the results of operations on complex numbers from
those obtained within the real number system.
The graph of a quadratic function in a coordinate plane is a u-shaped graph, called a parabola. Using
algebraic methods to manipulate and/or solve the equation of a quadratic function can throw light on the
function’s properties and help visualize the behavior of its graph, which can result in the use of more
effective and efficient graphing methods.
The solutions of a quadratic equation ax2 + bx + c = 0 are equal to the real zeros of the related
quadratic function y = ax2 + bx + c, and the x-intercepts of this function’s graph. Zeros that are not real
are not visible on the graph of a function in a coordinate plane of real numbers.
Quadratic inequalities can be one variable or two, and both types can be solved and represented
algebraically and graphically.
Essential Questions
What does the equation and graph of a quadratic function look like?
What is the reason for having the different methods for solving quadratics and when can using one
method be necessary or more appropriate than another?
How does using the different forms of a quadratic function help us graph?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
y = a (x – h) + k
y=ax–h +k
y = a (x – h) 2 + k
How do the results of operations on real numbers differ when applied to the complex numbers?
What are the connections between the algebraic and graphical representations of both one-variable and
two variable quadratic inequalities?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Quadratics
 Quadric equations
 Inequalities
 Graphing
Students will be able to:
 Graph quadratic functions.
 Factor quadratic expressions and solve quadratic equations by factoring.
 Find the zeros of a quadratic function by factoring.
 Solve quadratic equations by taking the square root of both sides.
 Perform operations on complex numbers.
 Solve quadratic equations by completing the square.
 Use completing the square to write quadratic functions in vertex form y = a (x – h)2 + k.
 Solve quadratic equations using the quadratic formula.
 Find the discriminant of a quadratic equation and use the value to describe the nature of the solutions
and to choose the best method for solving.
 Write and graph quadratic functions in standard form (y = ax2 + bx + c), vertex form (y = a(x – h)2
+ k), and intercept form (y = a(x – p)(x – q)).
 Solve quadratic inequalities in one variable by graphing and algebraically.
 Graph quadratic inequalities in two variables to show all solutions.
 Use quadratic equations, inequalities, and functions to model and solved real-life problems.
Integration
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Mathematics
Course: Algebra 2
Unit 6:
Polynomials
Summary and Rationale
Unit 6 extends the study of functions to polynomial functions. Students will first use the Properties of Exponents to
simplify and perform operations involving powers, monomials and polynomials. Students will then thoroughly
analyze the characteristics and properties of polynomial functions, their equations and graphs. The
quadratic functions studied in Unit 5 are polynomial functions of degree two. Students will learn different
solving and graphing methods, as well as how to write and obtain important information from polynomial
equations of degree greater than two in different forms. During this unit, it is crucial for students to revisit the
relationship between the solutions of a polynomial equation, the zeros of a polynomial function, and the xintercept’s of this function’s graph.
Pacing
Four (4) weeks
Standards
Number and Quantity
N-CN-8
Extend polynomial identities to the complex numbers.
N-CN-9
Know the fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
m. Interpret parts of an expression, such as terms, factors, and coefficients.
n. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-APR-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.
A-APR-2
Know and apply the remainder Theorem: For a polynomial p(x) and a number a, the
remainder of division by x – a is p(a) = 0 if an only if (x – a) is a factor of p (x)
A-APR-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 polynomials.
A-APR-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
(x2 + y2)2 = (x2 - y2) + (2xy)2
A-APR-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
A-REI-10
A-REI-11
Functions
F-IF-1
F-IF-2
F-IF-4
F-IF-5
F-IF-6
F-IF-7
F-IF-8
F-IF-9
F-IF-8
F-IF-9
example by Pascal’s Triangle.
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).
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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
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).
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
For a function that models a relationships 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.
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
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. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
c. 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.
d. Use the properties of exponents to interpret expressions for exponential functions.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
F-BF-1
Write a functions that describes a relationship between two quantities.
d. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
e. Combine standard functions types using arithmetic operations.
f. Compose functions
F-BF-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.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Performing operations on polynomial expressions requires the accurate application of the properties of
exponents and the order of operations, as well as the ability to distinguish between like and unlike terms.
The requirements for and the results of addition and subtraction of polynomials differ greatly from those
involving multiplication, division, and powers.
A function is a relationship in which one set of values defines another. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function.
Functions can be represented using an equation, or through a graph of the ordered pairs on a coordinate
plane that satisfy the equation. The graph of a function is a useful way of visualizing the relationship of
the function, as well as its complete domain and range.
All equations of degree one or higher are defined as polynomial equations. The Fundamental Theorem of
Algebra states that the number of solutions to a one-variable polynomial equation is equal to the degree of
the polynomial. These solutions can be determined through graphing (The solutions are the x-intercepts.)
and using one more previously learned and new algebraic methods and theorems. The best method(s) to
use depends on the forms and characteristics of the given equations, the nature of the solutions, and how
precise the solutions should be.
Polynomials can have real number solutions, but can also have solutions in a larger system, called the
complex numbers. There are differences between the results of operations on complex numbers from
those obtained within the real number system.
The graphs of polynomial functions in a coordinate plane vary, yet yield various patterns. Using algebraic
methods to manipulate and/or solve the equation of a polynomial function can throw light on the
function’s properties and help visualize the behavior of its graph, which can result in the use of more
effective and efficient graphing methods.
The solutions of a polynomial equation are equal to the real zeros of the related polynomial function and
the x-intercepts of this function’s graph. Zeros that are not real are not visible on the graph of a function
in a coordinate plane of real numbers.
Essential Questions
What are the requirements for the addition and subtraction of monomials and polynomials? Are these the
same for multiplication and division? How do the results of the operations differ?
What steps should be taken when factoring a polynomial expression or equation? What specifics do you
look for and how do you know when it is completely factored?
What characteristics, other than its solutions, of a polynomial equation can be used to describe the
behavior and sketch the graph of its related function?
What theorems and methods are used to identify the total number of roots and determine the real zeros of
a polynomial function?
Which types of roots are seen on the graph of a polynomial function? Which are not visible? What
happens in the graph if there is a double root?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Polynomials
 Fundamental Theorem of Algebra
 Rational Zero Theorem
 Graphing
Students will be able to:
 Apply the properties of exponents to simplify and evaluate expressions involving powers.
 Evaluate a polynomial function.
 Graph a polynomial function.
 Add, subtract, and multiply polynomials.
 Factor polynomial expressions.
 Use factoring methods to solve polynomial equations.
 Divide polynomials using long and synthetic division.
 Relate the results of division to the solutions of a polynomial equation and to The Remainder and





Factor Theorems.
Use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function.
Use graphing technology to identify the number of and approximate the real zeros of a polynomial
function.
Use The Rational Zero Theorem to identify possible rational zeros of a polynomial function.
Use all of the above to identify all zeros of a polynomial function, real and imaginary, and to sketch a
graph.
Use polynomial equations and functions to model and solve real-life problems.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 7:
Powers, Roots, and Radicals
Summary and Rationale
Unit 7 connects and extends the familiar ideas of squares and square roots from Algebra 1 to other
exponents and roots, including those that are not whole numbers in expressions and equations. The unit
also extends the study of functions to radical and root functions.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-RN-1
Explain how the definition of the meaning of rational exponents follows from extending
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents.
N-RN-2
Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
o. Interpret parts of an expression, such as terms, factors, and coefficents.
p. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-SSE-3
Choose and produce and equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
j. Factor a quadratic expression to reveal the zeros of the function it defines.
k. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
l. Use the properties of exponents to transform expression for exponential functions.
A-REI-2
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
Functions
F-BF-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.
F-BF-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.
b. Verify by composition that one function is the inverse of another.
c. Read values of an inverse function from a graph or table, given that function has an
inverse.
d. Produce an invertible function from a non-invertible function by restricting the
domain.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Radical expressions can be rewritten using rational exponents and vice versa.
The Properties of Exponents apply to all rational exponents.
A function is a relationship in which one set of values defines another. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function.
Functions can be represented using an equation, or through a graph of the ordered pairs on a coordinate
plane that satisfy the equation. The graph of a function is a useful way of visualizing the relationship of
the function, as well as its complete domain and range.
The graphs of radical functions in a coordinate plane vary, yet yield various patterns. Using algebraic
methods to manipulate and/or solve the equation of a radical function can throw light on the function’s
properties and help visualize the behavior of its graph, which can result in the use of more effective and
efficient graphing methods.
Functions can be added, subtracted, multiplied, or divided to form a new function. They can also be
combined or repeated to form a new function, called a composite function. The new functions created
may have different domains and ranges than their parts.
Every function has an inverse however the inverse is not always another function. There is an algebraic
procedure for finding the inverse of a function. The domain of the function is the range of its inverse. The
range of a function is the domain of its inverse. (Input and output values switch.) Graphs of inverse
functions are reflections about the diagonal line y = x.
The inverse of a power function of nth degree is an nth root function. The nth degree functions that are
classified as even functions require domain restrictions.
The use of inverse operations and the properties of equality can be applied to the solving of radical
equations and equations with rational exponents. However, certain procedures may lead to invalid
solutions called extraneous solutions (or roots).
Essential Questions
How are the graphs of a function and its inverse related?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
 y = n(x – h) + k
When and how do you check for extraneous solutions when solving equations containing radicals and
rational exponents?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Powers
 Roots
 Radicals
 Exponents
Students will be able to:
 Evaluate nth roots of real numbers using both radical and rational exponent notation.
 Apply the properties of exponents to simplifying and evaluating expressions containing rational
exponents.
 Perform operations with functions, including those that contain powers.
 Find the inverse of both linear and non-linear functions.
 Graph square root and cube root functions.
 Solve equations that contain radicals and rational exponents.
 Use radical and nth root equations and functions to model and solve real-life problems.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 8:
Exponents and Logarithms
Summary and Rationale
Unit 8 extends the study of functions to exponential and logarithmic functions, which are two important families of
functions because they model many real-life situations. Students will analyze the characteristics and properties
these functions, their equations and graphs. Students will explore graphs that contain asymptotes as they
draw and recognize patterns within the behavior of exponential and logarithmic functions. Previously
learned Properties of Exponents and new Properties of Logarithms will be used to simplify expressions
and solve equations.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-RN-1
Explain how the definition of the meaning of rational exponents follows from extending
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents.
Algebra
A-SSE-1
Interpret expressions that represent a quantity in terms of its context.
q. Interpret parts of an expression, such as terms, factors, and coefficients.
r. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-SSE-3
Choose and produce and equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
m. Factor a quadratic expression to reveal the zeros of the function it defines.
n. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
o. Use the properties of exponents to transform expression for exponential functions.
A-CED-4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations.
A-REI-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).
A-REI-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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
Functions
F-IF-1
F-IF-2
F-IF-4
F-IF-5
F-IF-6
F-IF-7
F-IF-8
F-IF-9
F-BF-5
F-LE-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).
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
For a function that models a relationships 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.
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
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.
k. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
l. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
m. Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
n. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Use the inverse relationship between exponents and logarithms to solve problems involving
logarithms and exponents.
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.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship in which one set of values defines another. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function.
Functions can be represented using an equation, or through a graph of the ordered pairs on a coordinate
plane that satisfy the equation. The graph of a function is a useful way of visualizing the relationship of
the function, as well as its complete domain and range.
A relationship between two variables or two sets of data is an exponential function if the two variables
increase (grow) or decrease (decay) by the same percent over equal periods of time.
The inverse of an exponential function of a logarithmic function of the same base. Natural exponential
and logarithmic functions are of the natural base, e. Exponential functions have restrictions on the range,
therefore logarithmic functions contain domain restrictions. The graphs of these functions contain
asymptotes, which are arbitrary lines that a graph approaches as you move away from the origin.
Every function has an inverse, however the inverse is not always another function. There is an algebraic
procedure for finding the inverse of a function. The domain of the function is the range of its inverse. The
range of a function is the domain of its inverse. (Input and output values switch.) Graphs of inverse
functions are reflections about the diagonal line y = x.
The graphs of exponential and logarithmic functions in a coordinate plane vary, yet yield various patterns.
Using algebraic methods to manipulate and/or solve the equation of a exponential or logarithmic function
can throw light on the function’s properties such as its zeroes and asymptotes, and can also help visualize
the behavior of its graph. These strategies can result in the use of more effective and efficient graphing
methods.
The use of inverse operations with the properties of equality, exponents, and logarithms can be applied to
the solving of exponential and logarithmic equations. However, since the domains of logarithmic
functions contain restrictions, obtaining extraneous solutions is a possibility.
Essential Questions
What values of b does y = bx represent exponential growth? Decay?
How do you determine the asymptotes of an exponential or logarithmic graph?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k




y = a (x – h)n + k
y = n(x – h) + k
y = ab (x-h) + k
y = logb (x – h) + k
How can you use inverse functions to graph logarithmic functions?
How is solving a logarithmic equation similar to solving an exponential equation? How is it different?
Why do logarithmic equations sometimes contain extraneous solutions?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Exponents
 Logarithms
 Graphing
Students will be able to:
 Graph exponential functions, including those that model growth and decay.
 Graph exponential functions that contain the natural base e.
 Evaluate and simplify exponential expressions with base e.
 Evaluate and simplify logarithmic expressions.
 Graph logarithmic functions.
 Use the properties of logarithms.
 Solve exponential and logarithmic equations.
 Use exponential and logarithmic functions to model and solve real-life problems.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 9:
Rational Equations and Functions
Summary and Rationale
Unit 9 extends the study of functions to rational functions, which is the ratio of two polynomial functions.
Students will analyze the characteristics and properties of these functions, their equations and graphs.
Students will explore functions that contain domain and range restrictions as they draw graphs that
contain asymptotes and other forms of discontinuity. Familiar concepts involving simplifying and
performing operations on rational numbers will be extended to operations of rational expressions.
Strategies for solving equations will be extended and applied to solving rational equations.
Pacing
Three (3) weeks
Standards
Algebra
A-SSE-1
A-SSE-2
A-APR-6
A-APR-7
A-REI-2
A-REI-10
A-REI-11
Functions
Interpret expressions that represent a quantity in terms of its context.
s. Interpret parts of an expression, such as terms, factors, and coefficents.
t. Interpret complication expressions by viewing one or more of their parts as a single
entity.
Use the structure of an expression to identify ways to rewrite it.
Rewrite simple ration 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.
Understand that rational expressions form a system of analogous to the rational numbers,
closed under addition, subtraction, multiplication, and division by a nonzero ration
expression; add, subtract, multiply, and divide rational expressions.
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
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).
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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
F-IF-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).
F-IF-2
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F-IF-4
For a function that models a relationships 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.
F-IF-5
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F-IF-6
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
F-IF-7
Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
f. Graph linear and quadratic functions and show intercepts, maxima, and minima.
g. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
h. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
i. Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
j. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
F-IF-8
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
F-IF-9
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship in which one set of values defines another. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function.
Functions can be represented using an equation, or through a graph of the ordered pairs on a coordinate
plane that satisfy the equation. The graph of a function is a useful way of visualizing the relationship of
the function, as well as its complete domain and range.
A rational function is the ratio of two polynomial functions. These functions contain restrictions on their
domains and/or ranges. Therefore, their graphs contain asymptotes, holes, and/or discontinuity.
The graphs of rational functions in a coordinate plane vary, yet yield various patterns. Using algebraic
methods to manipulate and/or solve the equation of a rational function can throw light on the function’s
properties such as its zeroes, asymptotes, domain, range, and discontinuity, which all can also help
visualize the behavior of its graph. These strategies can result in the use of more effective and efficient
graphing methods.
The previously learned procedures for multiplying, dividing, simplifying, adding, and subtracting
numerical fractions are extended to performing the same operations on algebraic rational expressions.
The use of inverse operations, the properties of equalities, and cross multiplication can be applied to the
solving of rational equations. Any solutions obtained that are not within the domain of the related function
are extraneous.
Essential Questions
How can you tell whether a set of data pairs (x, y) shows inverse variation?
How do you determine the vertical and horizontal asymptotes for the graph of a rational function?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
 y = n(x – h) + k
 y = ab (x-h) + k
 y = logb (x – h) + k
 y=
a
+ k
(x - h)
How is adding (or subtracting, multiplying, dividing simplifying) rational expressions similar to adding
(or subtracting, multiplying, dividing, simplifying) numerical fractions?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Rational equations
 Functions
 Inverse variations
Students will be able to:
 Graph rational functions.
 Write and use equations for inverse and joint variations.
 Multiply and divide rational expressions.
 Add and subtract rational expressions.
 Simplify complex fractions.
 Solve rational equations.
 Use rational equations and functions to model and solve real-life problems.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 10:
Conic Sections
Summary and Rationale
Unit 10 involves the study of the four conic sections, parabolas, circles, ellipses, and hyperbolas. Conic
sections are an important part of the study of algebra and geometry because they have many different reallife applications.
Pacing
Three (3) weeks
Standards
Number and Quantity
N-Q-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.
Algebra
A-SSE-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 complication expressions by viewing one or more of their parts as a single
entity.
A-SSE-2
Use the structure of an expression to identify ways to rewrite it.
A-SSE-3
Choose and produce and 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 expression for exponential functions.
A-REI-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).
Geometry
G-GPE-1
Derive the equations of a circle given center and radius using the Pythagorean Theorem;
complete the square to find the center and the radius of a circle given by an example.
G-GPE-2
Derive the equation of a parabola given a focus and directrix.
G-GPE-3
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or
difference of the distances from the foci is constant.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The Distance Formula is an application of the previously learned Pythagorean Theorem.
A conic section is a graph of an equation of the form
Ax2 + Bxy + Cy2 +Dx +Ey + F = 0
These relations are studied in algebra because they have various real-life applications.
The only conic sections that are functions are parabolas that open upward or downward, previously
learned as quadratic functions and hyperbolas that are written in the form of a rational function.
Using algebra to manipulate the equation of a conic section, particularly the method of “completing the
square” can be used to determine the parts and properties of its graph, and can result in the use of more
effective and efficient graphing methods.
Two quadratic equations form a quadratic system. The solutions of this type of system are represented by
the set of ordered pairs that satisfy both equations in the system.
The solutions of these systems can be determined and represented both algebraically and graphically,
using the same algebraic methods previously used to solve linear systems. However, solving quadratic
systems requires methods learned for solving quadratic equations, and may yield more solutions.
Essential Questions
Given the equation of a conic section in the following form
Ax2 + Bxy + Cy2 +Dx +Ey + F = 0 how do you classify its graph? How do you rewrite the equation so
that you can obtain the information needed to draw its graph? Which conic sections are functions? Which
are relations, but not functions?
How many solutions are possible for a system containing a conic section and a line? For a system of two
conic sections? What would each possible solution case look like graphically?
What algebraic methods can be used to solve a system of equations that contains one or more quadratic
relation?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Conic sections
 Graphing
Students will be able to:
 Determine the distance between two points.
 Determine the midpoint of the line segment joining two points.
 Graph and write equations of parabolas.
 Graph and write equations of circles.
 Graph and write equations of ellipses.
 Graph and write equations of hyperbolas.
 Classify a conic section using its equation.
 Solve systems of quadratic systems algebraically and sketch the graph.
 Use conics to model and solve real-life problems.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Algebra 2
Unit 11:
Sequences and Series
Summary and Rationale
Students have studied number patterns (or sequences) since elementary school. Unit 11 will connect this
familiar exploration of number patterns to algebra as students write and use rules for sequences and series.
An arithmetic sequence has a common difference, so it is similar to the linear functions in Unit 2. A
geometric sequence has a common ratio, so it is similar to the exponential functions from Unit 8.
Pacing
Three (3) weeks
Standards
Algebra
A-SSE-4
Derive and/or explain the derivation of 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
Functions
F-IF-3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers.
F-BF-2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between two forms.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Algebra can be used to write and use rules for number patterns that exist in sequences and series. A
sequence is a list of terms that demonstrate a number pattern, while a series is an expression formed by
adding the terms of the sequence. Both can be either finite or infinite.
An arithmetic sequence has a common difference, so it is similar to a linear function. A geometric
sequence has a common ratio, so it is similar to an exponential function.
A recursive sequence is one that uses one or more pervious term(s) to obtain the next term.
Essential Questions
What is the difference between a sequence and a series?
What makes a sequence or series arithmetic? geometric?
What information do you need to find a term of an arithmetic (or geometric) series?
How do you know if a sum of an infinite geometric series exists? And if it does, what information is
needed to determine the sum?
How can summation (or sigma) notation be used to represent a series?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Sequences
 Series
Students will be able to:
 Write rules for arithmetic sequences and find sums of arithmetic series.
 Write rules for geometric sequences and find sums of geometric series.
 Find sums of infinite geometric series.
 Use summation notation to write a series.
 Evaluate and write recursive rules for sequences.
 Use sequences and series to solve real-life problems.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Pre-Calculus
Nutley Public Schools
Mathematics
Course: Pre-Calculus
Unit 1:
Linear Equations, Inequalities, and Systems
Summary and Rationale
Unit 1 revisits and further explores the concepts and essential topics studied in Algebra 2. The skills,
strategies, and one-variable solving techniques previously learned in Algebra 2 are crucial in the
development of a foundation for continuing the study of algebra and its applications.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.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.
N.VM.6
Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
N.VM.7
Multiply matrices by scalars to produce new matrices, e.g., as when all the payoffs in a
game are doubled.
N.VM.8
Add, subtract, and multiply matrices of appropriate dimensions.
N.VM.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.
N.VM.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.
N.VM.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.
N.VM.12
Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of
the determinant in terms of area.
N.VM.6
Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence
relationships in a network.
Algebra
A.CED.1
Create equations and inequalities in one variable and use them to solve problems.
A.CED.2
Create equations in two or more variables to represent relationships between quantities;
graph equations on coordinate axes with labels and scales.
A.CED.3
Represent constraints by equations or inequalities, and by systems of equations and/or
inequalities, and interpret solutions as viable or non-viable options in a modeling context.
A.REI.1
A.REI.3
A.REI.5
A.REI.6
A.REI.7
A.REI.8
A.REI.9
A.REI.10
A.REI.11
Functions
F.BF.3
F.BF.4
F.LE.1
Explain each step in solving a simple equations as following from the equality of numbers
asserted in the previous step, starting from the assumption that the original equations has a
solution. Construct a viable argument to justify a solution method.
Solve linear equations and inequalities in one variable, including equations with coefficients
represented by letters.
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.
Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on
pairs of linear equations in two variables.
Solve a simple system consisting of a linear equations and a quadratic equation in two
variables algebraically and graphically.
Represent a system of linear equations as a single matrix equation in a vector variable.
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 x 3 or greater).
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).
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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
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.
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.
b. Verify by composition that one function is the inverse of another.
c. Read values of an inverse function from a graph or table, given that function has an
inverse.
d. Produce an invertible function from a non-invertible function by restricting the
domain.
Distinguish between situations that can be modeled with linear functions and with
exponential functions.
g. Prove that linear functions grow by equal differences over equal intervals, and that
exponential functions grow by equal factors over equal intervals.
h. Recognize situations in which one quantity changes at a constant rate per unit
interval relative to another.
i. Recognize situations in which a quantity grows or decays by a constant percent rate
per unit interval relative to another.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Properties of real numbers, equality, and inequality, along with the use of inverse operations, can
transform an equation (or inequality) into one or a series of equivalent simpler equations (or inequalities.).
These properties can be used repeatedly to isolate the variable. This process is used to find solutions to
one-variable equations (or inequalities) including linear, polynomial, radical, rational, and absolute value
equations and inequalities. Solutions to these inequalities can be represented visually using a number line
and algebraically using interval notation.
Given a two-variable equation, the value of one quantity can be found if the value of the other is known.
A table can be used to display the relationship between the quantities, which would also represent a set of
solutions of the equation. The set of all solutions of the equation forms its graph on a coordinate plane.
The graph will show solutions that are in the table, will visually represent the relationship between the
two variable quantities that are changing, and can also show solutions to the equation that are not in the
table.
A function is a relationship in which one set of values defines another. Each value of the input variable
(value in the domain) is associated with a unique value of the output variable (value in the range.) In order
to determine if an equation or a set of ordered pairs represents a function, the solutions of the equation or
the ordered pairs can be organized in a table or plotted on a graph. If the table of values shows that each
value in one set is paired with exactly one value in the other set, the relation is a function. The vertical
line test uses the graph to determine whether a relation is a function. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function. Functions can be represented using an equation, or
through a graph of the ordered pairs on a coordinate plane that satisfy the equation. The graph of a
function is a useful way of visualizing the relationship of the function, as well as its complete domain and
range.
A relationship between two variables or two sets of data is a linear function if the two variables increase
or decrease by the same amount over equal periods of time. This constant rate of change is the slope of
the linear function.
The graph of a linear equation (function) in a coordinate plane is a line. The slope of the line is the ratio of
the vertical change to the horizontal change. The slope of a line can be positive, negative, zero, or
undefined. The ratio of slope remains the same when measured between any two points on a line, so it is
proportional. The relationship between two lines can be determined by comparing their slopes and yintercepts obtained from graphs or equations.
Using algebraic methods to manipulate and/or solve the equation of a linear function can throw light on
the function’s properties such as its slope and intercepts, which can both help visualize the behavior of its
graph. A linear equation can be represented using three equation forms: slope-intercept form, point-slope
form, and standard form. All forms are useful in writing the equations of linear functions given a graph or
certain characteristics and also provide effective and efficient graphing methods.
The particular form of a linear equation often suggests a particular graphing method. The standard form
makes it easy to find x and y intercepts and draw graphs quickly using two points. The slope-intercept
form makes it possible to graph the line easily starting with one point and obtaining several others by
moving according to the slope.
The Distance and Midpoint formulas are used to determine the length and midpoint of a line.
If the ratio of two variables is constant, then the variables have a special linear relationship, called a direct
variation. The equation of a direct variation is y = kx. Its graph is a line that passes through the origin and
has a slope of k.
A linear inequality in two variables has an infinite number of solutions. Solutions to a linear inequality in
two variables can be represented in a coordinate plane as the set of all points on one side of a boundary
line. Two variable linear inequalities can also be solved algebraically and the solutions can be represented
using interval notation.
Two or more linear equations/inequalities form a linear system. The solution of a system is represented by
the set of ordered pairs that satisfy every equation/inequality in the system. When solving a system of
linear equations, there are three possible types of solutions: one solution (the point of intersection of the
two lines), no solution (The lines do not intersect.), or an infinite number of solutions (The equations in
the system represent the exact same line.)
Systems of equations can be solved in more than one way. Three methods are graphing, substitution, and
elimination. The best method to use depends on the forms of the given equations and how precise the
solution should be. The graphing method involves graphing each equation and finding the intersection
point, if one exists. When a system has at least one equation that can be solved for a variable, the system
can be efficiently solved using substitution. Some equations of a system are written in a way that makes
eliminating a variable the best method to use.
The solution to a three-variable system is an ordered triple. Three-variable systems are solved
algebraically using substitution, elimination, or a combination of both methods.
Solutions of a system of linear inequalities can be graphed in the coordinate plane. The graph of the
solution of a system of linear inequalities is the region where the graphs of the individual inequalities
overlap.
Many real world mathematical problems can be modeled and represented algebraically and graphically by
functions. A function that models a real world situation can be represented using an equation or graph that
can be used to make estimates or predictions about future occurrences. A real-world graph of a function
should only show points that make sense in the given situation.
A matrix is a rectangular arrangement of numbers in rows and columns that can be used to organize and
perform operations on numerical data. The numbers in a matrix are called its entries, (or elements.) To
add or subtract matrices, add or subtract corresponding entries. Matrices can be added or subtracted only
if they have the same dimensions.
The entries in a matrix can also be multiplied by a real number. This process is called scalar
multiplication. The product of two matrices A and B is defined if the number of columns in A is equal to
the number of rows in B.
Square matrices (matrices with the same number of rows and columns) have multiplicative inverses, as
long as their determinants do not equal zero. There are formulas and procedures for determining the
inverse of a square matrix that involve the determinant of the matrix.
Matrices, their determinants and inverses, and matrix operations can be used to solve linear systems
through the application of Cramer’s Rule or the use of a matrix equation.
Essential Questions
What are the connections between the algebraic and graphical representations of one-variable linear
inequalities? Two variable inequalities?
What are the different ways to determine if a relation is also a function?
When and why is using one method for graphing a linear equation sometimes more appropriate than
another?
When and why is using one method for solving a linear system sometimes more appropriate than another?
Which properties of real numbers do not hold true for matrices?
Can two different matrices have the same determinant?
How can matrices and their inverses be used to solve linear systems?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Linear equations
 Inequalities
 Systems
 Distance Formula
 Midpoint Formula
Students will be able to:
 Use The Distance Formula to calculate the distance between two points.
 Use The Midpoint Formula to find the midpoint of a line segment.
 Calculate the slope of a line.
 Write and graph linear equations (functions) in slope-intercept and standard forms.
 Identify slopes and equations of parallel and perpendicular lines.
 Solve linear, polynomial, radical, rational, and absolute value equations.
 Solve linear, polynomial, radical and rational inequalities using graphing and algebraic methods.




Solve systems of equations (two variable and three variable systems).
Solve systems of inequalities and linear programming problems.
Add, subtract, multiply, and find determinants and inverses of matrices.
Use matrices to solve systems of equations.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Pre-Calculus
Unit 2:
Families of Functions and their Graphs
Summary and Rationale
Unit 2 is a review and thorough extension of all families of functions and the characteristics of their
graphs. Students will graph and analyze the behaviors of each family of functions, in order to develop a
deeper understanding of how altering the equations of these functions affects all families in the same or
very similar ways.
Pacing
Four (4) weeks
Standards
Number and Quantity
N.RN.2
Rewrite expressions involving radicals and rational exponents using the properties of
exponents.
N.Q.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.
Algebra
A.REI.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).
A.REI.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.
Functions
F.IF.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).
F.IF.2
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F.IF.4
For a function that models a relationships 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.
F.IF.5
Relate domain of a function to its graph and, where applicable, to the quantitative
F.IF.6
F.IF.7
F.IF.8
F.IF.9
F.BF.1
F.BF.3
F.BF.4
relationship it describes.
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
k. Graph linear and quadratic functions and show intercepts, maxima, and minima.
l. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
m. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
n. Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
o. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
e. 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.
f. Use the properties of exponents to interpret expressions for exponential functions.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Write a functions that describes a relationship between two quantities.
g. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
h. Combine standard functions types using arithmetic operations.
i. Compose functions
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.
Find inverse functions.
e. 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.
f. Verify by composition that one function is the inverse of another.
g. Read values of an inverse function from a graph or table, given that function has an
inverse.
h. Produce an invertible function from a non-invertible function by restricting the
domain.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Given a two-variable equation, the value of one quantity can be found if the value of the other is known.
A table can be used to display the relationship between the quantities, which would also represent a set of
solutions of the equation. The set of all solutions of the equation forms its graph on a coordinate plane.
The graph will show solutions that are in the table, will visually represent the relationship between the
two variable quantities that are changing, and can also show solutions to the equation that are not in the
table.
A function is a relationship in which one set of values defines another. Each value of the input variable
(value in the domain) is associated with a unique value of the output variable (value in the range.) In order
to determine if an equation or a set of ordered pairs represents a function, the solutions of the equation or
the ordered pairs can be organized in a table or plotted on a graph. If the table of values shows that each
value in one set is paired with exactly one value in the other set, the relation is a function. The vertical
line test uses the graph to determine whether a relation is a function. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function. Functions can be represented using an equation, or
through a graph of the ordered pairs on a coordinate plane that satisfy the equation. The graph of a
function is a useful way of visualizing the relationship of the function, as well as its complete domain and
range.
Functions and their graphs can be classified into families of functions: linear (lines), quadratic (u-shaped),
higher degree polynomial (also classified as cubic, even, or odd, shapes vary), square root (or radical),
absolute value (v-shaped), greatest integer (or step), and rational.
The graph of these functions in a coordinate plane can be created using a table of values or more quickly
by shifting the following parent graphs:
y = x (linear)
y = x2 (quadratic)
y = x3, y = x4, y = x5 (polynomial)
y = √x (square root or radical),
y = x, (absolute value)
y = [x] (greatest integer or step)
y = 1/x (rational)
The graphs of functions in a coordinate plane vary, yet yield various patterns within the families of
functions. Using algebraic methods to manipulate and/or solve the equation of a function can throw light
on the function’s properties such as zeros, intercepts, a vertex, direction of opening, end behavior, and
type of symmetry, if it exists. These characteristics can help visualize the sketch of its graph and also lead
to the use of more effective and efficient graphing methods.
Families of functions are all transformed (translated (or shifted), reflected, and/or dilated) in the same
ways, as the values of a, h, and k similarly affect their graphs:






y = a (x – h) + k
y=ax–h +k
y = a (x – h) 2 + k
y = a (x – h)n + k
y = n(x – h) + k
y=
a
+ k
(x - h)
Every function has an inverse however the inverse is not always another function. There is an algebraic
procedure for finding the inverse of a function. The domain of the function is the range of its inverse. The
range of a function is the domain of its inverse. (Input and output values switch.) Graphs of inverse
functions are reflections about the diagonal line y = x.
Functions can be added, subtracted, multiplied, or divided to form a new function. They can also be
combined or repeated to form a new function, called a composite function. The new functions created
may have different domains and ranges than their parts.
Essential Questions
What are the types and the differences between the following functions and how are the values of a, h,
and k used to describe their graphs?






y = a (x – h) + k
y=ax–h +k
y = a (x – h) 2 + k
y = a (x – h)n + k
y = n(x – h) + k
y=
a
+ k
(x - h)
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Families of functions
 Graphing
Students will be able to:
 Represent relations using mappings, ordered pairs, tables, and graphs.
 Determine whether a given relation is a function.
 Identify the domain and range of any relation or function.
 Use function notation and evaluate functions.
 Identify and sketch the common families of functions: simple polynomial, square root, absolute value,
step, piecewise, and rational.
 Identify odd vs. even functions, as well as the related symmetry that exists in the graphs.
 Determine and recognize inverse functions.
 Use symmetry, vertical and horizontal stretches and shifts, and reflections to graph functions.
 Perform operations with functions.
 Find composites of functions.
Integration
Technology Integration
Writing Integration
Suggested Resources
Nutley Public Schools
Mathematics
Course: Pre-Calculus
Unit 3:
Polynomial and Rational Functions
Summary and Rationale
The majority of Unit 3 is devoted to the study of polynomial functions, with a special focus on quadratic
functions, which are degree two polynomials. Students will thoroughly analyze the characteristics and
properties of these functions, their equations and graphs. Students will learn different solving and
graphing methods, as well as how to write and obtain important information from function equations in
different forms. During this unit, it is crucial for students to recognize that the solutions of an equation are
equal to both the zeros of a function and the x-intercept’s of this function’s graph. This connection will
follow through with other types of functions that will be studied in future units. The graph of a quadratic
function is called a parabola, which is also one of the four conic sections that will be further explored in
Unit 8.
The second part of the unit involves the further study of the rational function, which is the ratio of two
polynomial functions. Students will analyze the characteristics of these functions, and their equations and
graphs. Through this analysis, students will explore a family of functions that most often has domain
and/or range restrictions as they draw graphs that contain asymptotes and other forms of discontinuity.
Familiar concepts involving properties and operations of rational numbers will be extended to the process
of decomposing rational expressions into partial fractions.
Pacing
Four (4) weeks
Standards
Number and Quantity
N.CN.1
Know there is a complex number i such that i2 = - 1, and every complex number has the
form a +bi with a and b real.
N.CN.2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
N.CN.3
Find the conjugate of a complex number; use conjugates to find moduli and quotients of
complex numbers.
N.CN.7
Solve quadratic equations with real coefficients that have complex solutions.
N.CN.8
Extend polynomial identities to the complex numbers.
N.CN.9
Know the fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Algebra
A.SSE.1
Interpret expressions that represent a quantity in terms of its context.
c. Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.2
A.SSE.3
A.APR.1
A.APR.2
A.APR.3
A.APR.4
A.APR.5
A.APR.6
A.APR.7
A.REI.2
A.REI.4
A.REI.10
A.REI.11
Functions
F.IF.1
d. Interpret complication expressions by viewing one or more of their parts as a single
entity.
Use the structure of an expression to identify ways to rewrite it.
Choose and produce and equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
d. Factor a quadratic expression to reveal the zeros of the function it defines.
e. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
f. Use the properties of exponents to transform expression for exponential functions.
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.
Know and apply the remainder Theorem: For a polynomial p(x) and a number a, the
remainder of division by x – a is p(a) = 0 if an only if (x – a) is a factor of p (x)
Identify zeros of polynomials when suitable factorizations are available, and use the zeros to
construct a rough graph of the function defined by the polynomials.
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
(x2 + y2)2 = (x2 - y2) + (2xy)2
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.
Rewrite simple ration 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.
Understand that rational expressions form a system of analogous to the rational numbers,
closed under addition, subtraction, multiplication, and division by a nonzero ration
expression; add, subtract, multiply, and divide rational expressions.
Solve simple rational and radical equations in one variable, and give examples showing how
extraneous solutions may arise.
Solve quadratic equations in one variable.
e. Use the method of completing the square to transform any quadratic equations in x
into an equation of the form (x – p)2 = q that has the same solutions. Derive the
quadratic formula from this form.
f. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the
initial form of the equations. Recognize when the quadratic formula gives complex
solutions and write them as a + bi and a – bi for real numbers a and b.
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).
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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
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).
F.IF.2
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F.IF.4
For a function that models a relationships 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.
F.IF.5
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F.IF.6
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
F.IF.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. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
F.IF.9
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
F.BF.1
Write a functions 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 functions types using arithmetic operations.
c. Compose functions
F.BF.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.
F.BF.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.
b. Verify by composition that one function is the inverse of another.
c. Read values of an inverse function from a graph or table, given that function has an
inverse.
d. Produce an invertible function from a non-invertible function by restricting the
domain.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
All equations of degree one or higher are defined as polynomial equations. Familiar equations such as
linear equations (degree 1) and quadratic equations (degree 2) are both examples of polynomial equations.
The Fundamental Theorem of Algebra states that the number of solutions to a one-variable polynomial
equation is equal to the degree of the polynomial.
Quadratic equations are of degree two, therefore they have two solutions that can be determined from a
graph (The solutions are the x-intercepts.), or by using different algebraic methods. Four previously
learned algebraic methods are factoring, taking the square root of both sides, completing the square, or
using the Quadratic Formula. The best method to use depends on the forms and characteristics of the
given equations, the nature of the solutions, and how precise the solutions should be.
Solutions to polynomial equations of degree greater than two may require the use of more algebraic
methods and theorems in addition to those already mentioned such as synthetic or long division, The
Remainder Theorem, Factor Theorem, Rational Root Theorem, Des Carte’s Rule, and/or The Location
Principle.
Polynomial equations can have real number solutions, but can also have solutions in a larger system,
called the complex numbers. There are differences between the results of operations on complex numbers
from those obtained within the real number system.
A function is a relationship in which one set of values defines another. Each value of the input variable
(value in the domain) is associated with a unique value of the output variable (value in the range.) In order
to determine if an equation or a set of ordered pairs represents a function, the solutions of the equation or
the ordered pairs can be organized in a table or plotted on a graph. If the table of values shows that each
value in one set is paired with exactly one value in the other set, the relation is a function. The vertical
line test uses the graph to determine whether a relation is a function. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function. Functions can be represented using an equation, or
through a graph of the ordered pairs on a coordinate plane that satisfy the equation. The graph of a
function is a useful way of visualizing the relationship of the function, as well as its complete domain and
range.
The solutions of a polynomial equation are equal to the real zeros of the related polynomial function, and
the x-intercepts of this function’s graph. Zeros that are not real are not visible on the graph of a function
in a coordinate plane of real numbers.
The graphs of polynomial functions in a coordinate plane vary, yet yield various patterns. Using algebraic
methods to manipulate and/or solve the equation of a function can throw light on the function’s properties
such as: zeros, intercepts, a vertex, end behavior, direction of opening, domain, range, vertex, width,
intervals of increasing or decreasing, relative maxima and/or minima, identification of double or triple
roots, and other critical points. These characteristics can help visualize the sketch of its graph and can
result in the use of more effective and efficient graphing methods.
Polynomial inequalities can be one variable or two, and both types can be solved and represented
algebraically and graphically.
A rational function is the ratio of two polynomial functions. These functions contain restrictions on their
domains and/or ranges. Therefore, their graphs contain asymptotes, holes, and/or discontinuity. The
graphs of rational functions also vary, yet yield various patterns. Using algebraic methods to manipulate
and/or solve the equation of a rational function can help determine important properties such as its zeroes,
intercepts, asymptotes, domain, range, types of discontinuity, and end behavior. These characteristics can
help visualize the sketch of its graph and can lead the use of more effective and efficient graphing
methods.
Polynomial and rational functions are all transformed (translated (or shifted), reflected, and/or dilated) in
the same ways, as the values of a, h, and k similarly affect their graphs:
 y = a (x – h) + k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
 y=
a
+ k
(x - h)
Essential Questions
What is the reason for having the different methods for solving quadratics and when can the use of one
method be necessary or more appropriate than another?
What are the connections between the algebraic and graphical representations of both one-variable and
two variable quadratic inequalities?
What steps should be taken when factoring a polynomial expression or equation? What specifics do you
look for and how do you know when it is completely factored?
What characteristics, other than its solutions, of a polynomial equation can be used to describe the
behavior and sketch the graph of its related function?
What theorems and methods are used to identify the total number of roots and determine the real zeros of
a polynomial function?
Which types of roots are seen on the graph of a polynomial function? Which are not visible? What
happens in the graph if there is a double root?
How do you determine the vertical, horizontal, and/or slant asymptotes for the graph of a rational
function?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k

y=
a
(x - h)
+ k
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Polynomials
 Rational functions
 Quadratic functions
 Rational Zero Theorem
 Graphing
Students will be able to:
 Graph and analyze quadratic functions using equations in both quadratic and standard forms.
 Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions.
 Use the Fundamental Theorem of Algebra to determine the number of zeros of a polynomial function,
as well as the number of critical points such as maxima, minima, points of inflection, and turning
points.
 Use graphing technology to identify the number of and approximate the real zeros of a polynomial
function.
 Use factoring methods, The Rational Zero Theorem, long division, synthetic division, and The
Remainder and Factor Theorems to identify rational zeros of a polynomial function.
 Use all of the above, in addition to the Quadratic Formula and operations with complex numbers, to
determine all zeros of a polynomial function, (real and complex), and to sketch its graph.
 Determine horizontal, vertical, and slant asymptotes of rational functions, and use these to sketch the
graphs, identify domains and ranges, and end behaviors.
 Decompose a rational expression into partial fractions.
Integration
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Mathematics
Course: Pre-Calculus
Unit 4:
Exponential and Logarithmic Functions
Summary and Rationale
Unit 4 extends the study of functions to exponential and logarithmic functions, which are two important
families of functions because they model many real-life situations. Students will analyze the
characteristics and properties of these functions, their equations and graphs. Students will explore two
more families of functions that contain asymptotes as they draw and recognize patterns within the
behavior of exponential and logarithmic functions. Previously learned Properties of Exponents and new
Properties of Logarithms will be used to simplify expressions and solve equations.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.RN.1
Explain how the definition of the meaning of rational exponents follows from extending
properties of integer exponents to those values, allowing for a notation for radicals in terms
of rational exponents.
Algebra
A.SSE.1
Interpret expressions that represent a quantity in terms of its context.
e. Interpret parts of an expression, such as terms, factors, and coefficients.
f. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A.SSE.2
Use the structure of an expression to identify ways to rewrite it.
A.SSE.3
Choose and produce and equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
g. Factor a quadratic expression to reveal the zeros of the function it defines.
h. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
i. Use the properties of exponents to transform expression for exponential functions.
A.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in
solving equations.
A.REI.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).
A.REI.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 aprroximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.
Functions
F.IF.1
F.IF.2
F.IF.4
F.IF.5
F.IF.6
F.IF.7
F.IF.8
F.IF.9
F.BF.5
F.LE.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).
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
For a function that models a relationships 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.
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
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. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
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.
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
Understand that inverse relationships between exponents and logarithms and use this
relationship to solve problems involving logarithms and exponents.
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.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship in which one set of values defines another. Each value of the input variable
(value in the domain) is associated with a unique value of the output variable (value in the range.) In order
to determine if an equation or a set of ordered pairs represents a function, the solutions of the equation or
the ordered pairs can be organized in a table or plotted on a graph. If the table of values shows that each
value in one set is paired with exactly one value in the other set, the relation is a function. The vertical
line test uses the graph to determine whether a relation is a function. All functions can be used to model
many important phenomena.
Determining an output value of a function, given an input value, requires evaluating the algebraic
expression that is being used to represent the function. Functions can be represented using an equation, or
through a graph of the ordered pairs on a coordinate plane that satisfy the equation. The graph of a
function is a useful way of visualizing the relationship of the function, as well as its complete domain and
range.
A relationship between two variables or two sets of data is an exponential function if the two variables
increase (grow) or decrease (decay) by the same percent over equal periods of time.
The inverse of an exponential function of a logarithmic function of the same base. Natural exponential
and logarithmic functions are of the natural base, e. Exponential functions have restrictions on the range
therefore logarithmic functions contain domain restrictions. The graphs of these functions contain
asymptotes, which are arbitrary lines that a graph approaches as you move away from the origin.
The graphs of exponential and logarithmic functions in a coordinate plane vary, yet yield various patterns.
Using algebraic methods to manipulate and/or solve the equation of a exponential or logarithmic function
can throw light on the function’s properties such as its zeroes and asymptotes, and can also help visualize
the behavior of its graph. These strategies can result in the use of more effective and efficient graphing
methods.
Families of functions are all transformed (translated (or shifted), reflected, and/or dilated) in the same
ways, as the values of a, h, and k similarly affect their graphs:
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
 y=
a
+ k
(x - h)
 y = ab (x-h) + k
 y = logb (x – h) + k
Every function has an inverse however the inverse is not always another function. There is an algebraic
procedure for finding the inverse of a function. The domain of the function is the range of its inverse. The
range of a function is the domain of its inverse. (Input and output values switch.) Graphs of inverse
functions are reflections about the diagonal line y = x.
The use of inverse operations with the properties of equality, exponents, and logarithms can be applied to
the solving of exponential and logarithmic equations. In both cases, it is useful to isolate the exponential
or logarithmic first, and then use inverse operations to solve for the variable. Since the domains of
logarithmic functions contain restrictions, obtaining extraneous solutions is a possibility.
Essential Questions
What values of b does y = bx represent exponential growth? Decay?
How do you determine the asymptotes of an exponential or logarithmic graph?
What are the types and the differences between the two functions and how are the values of a, h, and k
used to describe their graphs?
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
 y = n(x – h) + k
 y=
a
+ k
(x - h)
 y = ab (x-h) + k
 y = logb (x – h) + k
How can you use inverse functions to graph logarithmic functions?
How is solving a logarithmic equation similar to solving an exponential equation? How is it different?
Why do logarithmic equations sometimes contain extraneous solutions?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Exponential functions
 Inverse functions
 Logarithmic equations
Students will be able to:
 Use the properties of exponents evaluate and simplify expressions containing rational and irrational
exponents, and those than contain the natural base e.
 Graph exponential functions and inequalities
 Graph exponential functions and inequalities, including those that involve growth, decay, and the
natural base e.
 Evaluate and simplify logarithmic expressions.
 Graph logarithmic functions.
 Evaluate and graph natural logarithmic functions.



Use the properties of logarithms to evaluate, rewrite, expand, and condense logarithmic expressions.
Solve exponential and logarithmic equations.
Use exponential and logarithmic functions to model and solve real-life problems.
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Mathematics
Course: Pre-Calculus
Unit 5:
Trigonometric Functions of Angles
Summary and Rationale
Unit 5 continues the study of trigonometry, with a specific focus on the study of angles and triangles.
Trigonometry is closely tied to both algebra and geometry. Students began their study of trigonometry in
their geometry courses with right triangles. This unit will provide opportunities for students to further
explore the more complex and comprehensive ideas behind the use of trigonometry in triangle
measurement. Students will find trigonometric functions of angles, solve triangles, and use trigonometry
to calculate the area of triangles.
Pacing
Three (3) weeks
Standards
Functions
F.TF.1
F.TF.2
F.TF.3
I.TF.4
Geometry
G.SRT.7
G.SRT.8
Understand radian measure of an angle as the length of the arc on the unit circle subtended
by the angle.
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.
Use special triangles to determine geometrically the values of sine, cosine, tangent for /3,
/4, and /6 and use the unit circle to express the values of sine, cosine, and tangent for  x,  + x and 2 - x in terms of their values for x, where x is any real number.
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric
functions.
Explain and use the relationship between the side and cosine complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
G.SRT.9
Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite side.
G.SRT.10 Prove the Laws of Sines and Cosines and use them to solve problems
G.SRT.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).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Trigonometric functions of acute angles are defined by ratios of side lengths of right triangles. These
ratios are: sine, cosine, tangent, cosecant, secant, and cotangent.
The angles 30°, 45°, and 60° occur frequently in trigonometry. These are the angles of the two special
right triangles previously learned in geometry (30°-60°-90° and 45°-45°-90°) The values of the six
trigonometric functions of these angles do not need to be memorized since they can quickly be generated
by drawing the two special right triangles and labeling the sides of each.
Trigonometric functions and their inverses can be used to find a missing side length or angle measure in a
right triangle. Finding all missing lengths and angle measures is called solving a right triangle.
An angle is formed by two rays that have a common endpoint, called the vertex. You can generate an
angle by fixing one ray, called the initial side, and rotating the other ray, called the terminal side, about
the vertex. In a coordinate plane, an angle whose vertex is at the origin and whose initial side is the
positive x-axis on in standard position. The phrase “the terminal side of θ lies in quadrant…” is simply
saying that θ lies in that quadrant. Two angles in standard position are co-terminal if their terminal sides
coincide. An angle co-terminal with a given angle can be found by adding or subtracting multiples of
360°.
Angles can be measured in degrees and in radians. The measure of an angle is determined by the amount
and direction of a rotation from the initial side to the terminal side. The angles measure is positive is the
rotation is counterclockwise and negative if the rotation is clockwise. The terminal side of an angle can
make more than one complete rotation.
To define a radian, consider a circle with radius r centered at the origin. One radian is the measure of an
angle in standard position whose terminal side intercepts the arc of length r.
Since the circumference of a circle is equal to 2πr, there are 2π radians in a circle. Therefore 2π radians is
equivalent to 360°. (π radians = 180°, and so on…)
The length of an arc and the area of a sector of a circle can be calculated using the central angle of the
circle in degrees (previously learned), and in radians.
Reference angles are used to evaluate trigonometric functions of any angle.
The Law of Sines and The Law of Cosines can be used to solve any triangle. (Determine all missing side
lengths of angle measurements.) The use of one law rather than the other depends on the known
information about he triangle. Some problems may show cases in which a triangle cannot be made under
the given conditions (No solution). Other cases yield one or two possible triangle solutions.
Trigonometric ratios can also be used to calculate the area of any triangle.
Essential Questions
How do you convert between degrees and radians?
How can you use the trigonometric ratios to find the measurements of sides or angles of a right triangle?
How can you use The Law of Sines and The Law of Cosines to solve any triangle?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Trigonometric functions
 Law of Sines
 Law of Cosines
 Heron’s Area Formula
Students will be able to:
 Describe and measure angles using radian and degree measure.
 Convert degree measure to radian measure and vice versa.
 Evaluate trigonometric functions of any angle, given a point on its terminal side.
 Evaluate trigonometric functions of any angle, with or without using a reference angle.
 Identify trigonometric functions of any angle using the period of the function, (or by identifying and
using co-terminal angles.)
 Use the Fundamental Trigonometric Identities to determine the values of all six trigonometric
functions.
 Identify trigonometric functions of special angles with or without the use of a unit circle.
 Identify and construct a unit circle and recognize its relationship to real numbers.
 Use a calculator to approximately evaluate a trigonometric function of any angle.
 Use the right triangle trigonometric ratios and inverse trigonometric functions and Pythagorean
Theorem to solve right triangles.
 Use the Law of Sines to solve oblique triangles.
 Use the Law of Cosines to solve oblique triangles.
 Use the Law of Sines and/or Heron’s Area Formula to calculate the area of an oblique triangle.
Integration
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Mathematics
Course: Pre-Calculus
Unit 6:
Trigonometric Functions of Real Numbers
Summary and Rationale
The focus of Unit 6 is the study of trigonometric functions of real numbers, used to model periodic
behavior. Students will have opportunities to see more connections between trigonometry and algebra as
they graph trigonometric functions and their inverses in a coordinate plane. Just as they have done with
other types of functions in previous units, students will analyze the characteristics of their equations and
graphs. Students will also use advanced algebra skills to evaluate trigonometric functions, verify
trigonometric identities, and solve trigonometric equations.
Pacing
Four (4) weeks
Standards
Number and Quantity
N.Q.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.
Algebra
A.REI.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).
Functions
F.IF.4
For a function that models a relationships 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.
F.IF.5
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F.IF.6
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
f. Graph linear and quadratic functions and show intercepts, maxima, and minima.
g. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
h. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
i. Graph rational functions, identifying zeros and asymptotes when suitable
F.TF.5
F.TF.6
F.TF.7
F.TF.8
F.TF.9
factorizations are available, and showing end behavior.
j. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
Choose trigonometric functions to model periodic phenomena with specified amplitude,
frequency, and midline.
Understand that restricting a trigonometric function to a domain on which it is always
increasing or always deceasing allows its inverse to be constructed.
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.
Prove the Pythagorean identity sin2 () + cos2 () = 1 and use it to find sin(), cos(), or
tan() given sin(), cos(), or tan() and the quadrant of the angle.
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to
solve problems.
Geometry
G.SRT.7
G.SRT.8
Explain and use the relationship between the side and cosine complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied
problems.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship in which one set of values defines another. Each value of the input variable
(value in the domain) is associated with a unique value of the output variable (value in the range.) In order
to determine if an equation or a set of ordered pairs represents a function, the solutions of the equation or
the ordered pairs can be organized in a table or plotted on a graph. If the table of values shows that each
value in one set is paired with exactly one value in the other set, the relation is a function. The vertical
line test uses the graph to determine whether a relation is a function. All functions can be used to model
many important phenomena.
Periodic functions have graphs that have repeating patterns that continue indefinitely. The shortest
repeating portion is called a cycle. The horizontal length of each cycle is called the period. Trigonometric
functions are periodic functions.
Determining an output value of a trigonometric function, given an input value, requires evaluating the
trigonometric expression that is being used to represent the function. Functions can be represented using
an equation, or through a graph of the ordered pairs on a coordinate plane that satisfy the equation. The
graph of a function is a useful way of visualizing the relationship of the function, as well as its complete
domain and range.
The graphs of the sine and cosine are periodic functions that both have the following characteristics: The
domain of each function is all real numbers and the range of each function is -1 ≤ y ≤ 1. Each graph has a
period of 2π (or 360°.)
The graph of tangent is a periodic function with a domain of all real numbers except odd multiples of π/2.
At odd multiples of π/2, the graph has vertical asymptotes. The range is all real numbers. The graph has a
period of π (or 180°.)
Families of functions are all transformed (translated (or shifted), reflected, and/or dilated) in the same
ways, as the values of a, h, and k similarly affect their graphs:
 y = a (x – h) + k
 y=ax–h +k
 y = a (x – h) 2 + k
 y = a (x – h)n + k
 y=
a
+ k
(x - h)
 y = ab (x-h) + k
 y = logb (x – h) + k
 y = a sin b(x – h) + k
 y = a cos b(x – h) + k
 y = a tan b(x – h) + k
Every function has an inverse however the inverse is not always another function. There is an algebraic
procedure for finding the inverse of a function. The domain of the function is the range of its inverse. The
range of a function is the domain of its inverse. (Input and output values switch.) Graphs of inverse
functions are reflections about the diagonal line y = x.
Trigonometric and inverse trigonometric functions and their graphs can be used to model and solve realworld problems, particularly those that involve periodic behavior.
Trigonometric identities are equations that are true for all values of x in their domain. Examples of
Fundamental Trigonometric Identities are: Reciprocal Identities, Tangent and Cotangent Identities,
Pythagorean Identities, Co-function Identities, and Negative Angle Identities. These trigonometric
identities can be used to evaluate trigonometric functions, simplify trigonometric expressions, and verify
other identities.
A verification of a trigonometric identity is a chain of equivalent expressions showing that one side of the
identity is equal to the other side. When verifying an identity a common and useful strategy is to begin
with the expression from one side and manipulate it algebraically until it is identical to the other side.
Properties of real numbers and equality, along with the use of inverse operations and trigonometric
identities, can transform a trigonometric equation into one or a series of equivalent simpler equations.
These properties can be used repeatedly to isolate the variable. Algebraic techniques such as factoring or
The Quadratic Formula may also be necessary during the process of finding solutions to trigonometric
equations in quadratic form. Extraneous solutions can result if an equation is being solved within a
restricted domain.
Sum, Difference, Double, and Half-Angle Formulas make it possible to determine exact trigonometric
values for less common angles using the more common angles such as 30°, 45°, 60°, etc.
Essential Questions
Why are trigonometric functions called periodic functions?
What is a trigonometric identity?
How can trigonometric identities be used to determine exact trigonometric values of uncommon angles?
How can you solve trigonometric equations?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Trigonometric functions of real numbers
 Formulas
 Graphing
Students will be able to:
 Sketch the graphs, analyze, compare, and identify domains and ranges of the basic trigonometric
functions: sine, cosine, tangent, cotangent, secant, and cosecant.
 Find the amplitude and period of a trigonometric function and use these characteristics to sketch its
graph.
 Identify and sketch translations of trigonometric graphs, (vertical shifts and phase shifts).
 Evaluate, graph and identify the domains and ranges of inverse trigonometric functions.
 Write equations for inverse trigonometric functions.
 Evaluate compositions of trigonometric functions.
 Use trigonometric functions and their inverses to model and solve real-life problems.
 Use the Fundamental Trigonometric Identities to evaluate trigonometric functions, simplify and/or
rewrite trigonometric expressions.
 Use the Fundamental Trigonometric Identities to verify other trigonometric identities.
 Use the Sum and Difference Formulas (or identities) to evaluate exact values for trigonometric
functions.
 Use the Double and Half-Angle Formulas (or identities) to evaluate exact values for trigonometric
functions.
 Use standard algebraic techniques, in addition to the objectives above, to solve trigonometric
equations.
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Mathematics
Course: Pre-Calculus
Unit 7:
Vectors
Summary and Rationale
Unit 7 discusses vectors. Vectors are an important topic in mathematics because of their application in
calculus and physics.
Pacing
Two (2) weeks
Standards
Number and Quantity
N.VM.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.
N.VM.2
Find the components of a vector by subtracting the coordinates of an initial point from the
coordinates of a terminal point.
N.VM.3
Solve problems involving velocity and other quantities that can be represented by vectors.
N.VM.4
Add and subtract vectors.
a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand
that 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.
N.VM.5
Multiply 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(vx, vy) =
(cvx, cvy).
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 or
against v.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Vectors are used to mathematically represent velocities. A vector is a quantity, or directed distance, that
ahs both magnitude and direction.
A vector is represented geometrically by a directed line segment. The length of the directed line segment
is the vector’s magnitude.
If a vector has an initial point at the origin, it is in standard position. The amplitude of the vector is the
directed angle between the positive x-axis and the vector.
The sum of two or more vectors is called the resultant of the vectors. The resultant of vectors can be
found through two methods: The Parallelogram Method and The Triangle Method (also called The TipTo Tail Method.)
Two vectors are opposites of they have the same magnitude and opposite directions.
Two vectors are parallel if and only if they have the same or opposite directions.
Two or more vectors whose sum is a given vector are called components of the given vector. Components
can have any direction.
Vectors are used in physics to represent motion or forces acting upon objects.
Vectors can be represented algebraically using ordered pairs of real numbers. Since vectors with the same
magnitude and amplitude are equal, many vectors can be represented by an ordered pair.
Essential Questions
What is a vector and why are vectors used?
How can vectors be represented?
How do you add vectors? Subtract?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Vectors
 Parametric equations
Students will be able to:










Identify equal, opposite, and parallel vectors.
Add and subtract vectors geometrically.
Find ordered pairs that represent vectors.
Add, subtract, multiply, and calculate the magnitude of vectors algebraically.
Add and subtract and calculate the magnitude of vectors in three-dimensional space.
Calculate the inner and cross products of two vectors.
Determine whether two vectors are perpendicular.
Write vector and parametric equations of lines.
Graph parametric equations.
Use vectors and right triangle trigonometry to model and solve real-life problems.
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Mathematics
Course: Pre-Calculus
Unit 8:
Polar Coordinates and Conic Sections
Summary and Rationale
This unit will introduce students to graphing on a new set of axes, the polar axes. Unit 8 involves the
study of the four conic sections, parabolas, circles, ellipses, and hyperbolas. Conic sections are an
important part of the study of algebra and geometry because they have many different real-life
applications. This unit also introduces the polar coordinate system which provides us with another method
for describing locations, in addition to the well-known coordinate plane system.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.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.
N.Q.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.
N.Q.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.
Algebra
A.SSE.1
Interpret expressions that represent a quantity in terms of its context.
g. Interpret parts of an expression, such as terms, factors, and coefficents.
h. Interpret complication expressions by viewing one or more of their parts as a single
entity.
A.SSE.2
Use the structure of an expression to identify ways to rewrite it.
A.SSE.3
Choose and produce and equivalent form of an expression to reveal and explain properties
of the quantity represented by the expression.
j. Factor a quadratic expression to reveal the zeros of the function it defines.
k. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
l. Use the properties of exponents to transform expression for exponential functions.
A.REI.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).
Geometry
G.GPE.1
Derive the equations of a circle given center and radius using the Pythagorean Theorem;
complete the square to find the center and the radius of a circle given by an example.
G.GPE.2
Derive the equation of a parabola given a focus and directrix.
G.GPE.3
Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or
difference of the distances from the foci is constant.
Statistics and Probability
S.CP.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”, “not”, “and”)
S.CP.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.
S.CP.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.
S.CP.4
Construct and interpret two-way frequency tables of data when two categories are associated
with each other being classified. Use the two-way table as a sample space to decide if events
are independent and to approximate conditional probabilities.
S.CP.5
Recognize and explain the concepts of conditional probability and independence in
everyday language and everyday situations.
S.CP.6
Find the conditional probability of A give B as the fraction of B’s outcomes that also belong
to A, and interpret the answer in terms of the model.
S.CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A – B), and interpret the answer in
terms of the model.
S.CP.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.
S.CP.9
Use permutations and combinations to compute probabilities of compound events and solve
problems.
S.MD.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.
S.MD.2
Calculate the expected value of a random variable; interpret it as the mean of the probability
distribution.
S.MD.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.
S.MD.4
Develop a probability distribution for a random variable defined for a sample space in which
probabilities are assigned empirically; find the expected value.
S.MD.5
Weigh the possible outcomes of a decision by assigning probabilities to payoff values and
finded expected values.
a. Find the expected payoff for a game of chance.
b. Evaluate and compare strategies on the basis of expected values.
S.MD.6
Use probabilities to make fair decisions (e.g., drawing by lots, using a random number
generator).
S.MD.7
Analyze decisions and strategies using probability concepts (e.g., product testing, medical
testing, pulling a hockey goalie at the end of a game).
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
The Distance Formula is an application of the previously learned Pythagorean Theorem.
A conic section is a graph of an equation of the form
Ax2 + Bxy + Cy2 +Dx +Ey + F = 0
These relations are studied in algebra because they have various real-life applications.
The only conic sections that are functions are parabolas that open upward or downward, previously
learned as quadratic functions and hyperbolas that are written in the form of a rational function.
Using algebra to manipulate the equation of a conic section, particularly the method of “completing the
square” can be used to determine the parts and properties of its graph, and can result in the use of more
effective and efficient graphing methods.
Two quadratic equations form a quadratic system. The solutions of this type of system are represented by
the set of ordered pairs that satisfy both equations in the system.
The solutions of these systems can be determined and represented both algebraically and graphically,
using the same algebraic methods previously used to solve linear systems. However, solving quadratic
systems requires methods learned for solving quadratic equations, and may yield more solutions.
Essential Questions
Given the equation of a conic section in the following form
Ax2 + Bxy + Cy2 +Dx +Ey + F = 0 how do you classify its graph? How do you rewrite the
equation so that you can obtain the information needed to draw its graph?
Which conic sections are functions? Which are relations, but not functions?
How many solutions are possible for a system containing a conic section and a line? For a system of two
conic sections? What would each possible solution case look like graphically?
What algebraic methods can be used to solve a system of equations that contains one or more quadratic
relation?
How is the polar coordinate system different from the four-quadrant coordinate plane system?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Polar coordinates


Conic sections
Graphing
Students will be able to:
 Graph and write equations of parabolas.
 Graph and write equations of circles.
 Graph and write equations of ellipses.
 Calculate eccentricities of ellipses.
 Graph and write equations of hyperbolas.
 Classify a conic section using its general equation and/or its discriminant.
 Graph and write equations and for transformed conic sections
 Solve systems of quadratic systems algebraically and sketch the graph.
 Use conics to model and solve real-life problems.
 Plot points and find multiple representations of points in the polar coordinate system.
 Convert points and equations from rectangular to polar form and vice versa.
 Graph polar equations by point plotting, and also use symmetry, zeros, maximum r-values as graphing
aides.
 Recognize and sketch special polar graphs.
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Mathematics
Course: Pre-Calculus
Unit 9:
Sequences, Series and Probability
Summary and Rationale
Students have studied number patterns (or sequences) since elementary school, and again in Algebra.
Unit 9 will connect and extend this familiar exploration of sequences and series to calculation
probabilities and binomial expansion.
Pacing
Three (3) weeks
Standards
Algebra
A.SSE.4
Derive and/or explain the derivation of the formula for the sum of a finite geometric series
(when the common ratio is not 1), and use the formula to solve problems.
Functions
F.IF.3
Recognize that sequences are functions, sometimes defined recursively, whose domain is a
subset of the integers.
F.BF.2
Write arithmetic and geometric sequences both recursively and with an explicit formula, use
them to model situations, and translate between two forms.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
Algebra can be used to write and use rules for number patterns that exist in sequences and series. A
sequence is a list of terms that demonstrate a number pattern, while a series is an expression formed by
adding the terms of the sequence. Both can be either finite or infinite.
An arithmetic sequence has a common difference, so it is similar to a linear function. A geometric
sequence has a common ratio, so it is similar to an exponential function.
A recursive sequence is one that uses one or more pervious term(s) to obtain the next term.
Probability expresses the likelihood that a particular event will occur. Data can be used to calculate
experimental probability, and mathematical properties can be used to calculate theoretical probability.
Either experimental or theoretical probability can be used to make predictions or decisions about future
events.
The probability of an event, or P(event), tells how likely it is that the event will occur. The probability of
a compound event can sometimes be found from expressions of the probabilities of simpler events.
Different methods must be used for finding the probability of two dependent events compared to finding
the probability of two independent events.
Various counting methods can be used to find the number of possible ways to choose objects with and
without order, which can be further used to develop theoretical probabilities.
Permutation and combination notation can be used to represent real-world situations.
The Binomial Theorem can be used to calculate the probability of a real-world event and can also be used
to identify the terms of a binomial expansion.
Essential Questions
What is the difference between a sequence and a series?
What makes a sequence or series arithmetic? geometric?
What information do you need to find a term of an arithmetic (or geometric) series?
How do you know if a sum of an infinite geometric series exists? And if it does, what information is
needed to determine the sum?
How can summation (or sigma) notation be used to represent a series?
How is probability related to real-world events?
How can you use the binomial theorem to calculate the probability of a real-world event?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Sequence
 Series
 Probability
Students will be able to:
 Use sequence notation, factorial notation, and summation notation to represent sequences and series.
 Write rules for arithmetic sequences that can be used to find any term in the sequence.
 Find nth partial sums of an arithmetic series.








Write rules for geometric sequences and find sums of geometric series.
Find sums of infinite geometric series.
Use sequences and series to solve real-life problems.
Use The Binomial Theorem to expand powers of binomials.
Use The Fundamental Counting Principle to solve counting problems.
Use permutations and combinations to solve counting problems.
Calculate probabilities of events: mutually exclusive and independent events, as well as complements
of events.
Calculate the probability of an event using The Binomial Theorem
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Course: Pre-Calculus
Unit 10:
Introduction to Calculus
Summary and Rationale
Unit 10 is an introduction to the most important concepts and ideas involved in the study of Calculus. The
introduction of differential calculus will be explored as students learn how to find the limit and derivative
of a function.
Pacing
Three (3) weeks
Standards
Number and Quantity
N.Q.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.
Algebra
A.REI.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).
A.REI.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.
Functions
F.IF.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).
F.IF.2
Use function notation, evaluate functions for inputs in their domains, and interpret
statements that use function notation in terms of a context.
F.IF.4
For a function that models a relationships 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.
F.IF.5
Relate domain of a function to its graph and, where applicable, to the quantitative
relationship it describes.
F.IF.6
Calculate and interpret the average rate of change of a function over a specified interval.
Estimate the rate of change from a graph.
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in
simple cases and using technology for more complicated cases.
p. Graph linear and quadratic functions and show intercepts, maxima, and minima.
q. Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.
r. Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.
s. Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.
t. Write exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showering period, midline, and amplitude.
F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and
explain different properties of the functions.
g. 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.
h. Use the properties of exponents to interpret expressions for exponential functions.
F.IF.9
Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions).
F.BF.1
Write a functions that describes a relationship between two quantities.
d. Determine an explicit expression, a recursive process, or steps for calculation from a
context.
e. Combine standard functions types using arithmetic operations.
f. Compose functions
F.BF.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.
F.BF.4
Find inverse functions.
e. 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.
f. Verify by composition that one function is the inverse of another.
g. Read values of an inverse function from a graph or table, given that function has an
inverse.
h. Produce an invertible function from a non-invertible function by restricting the
domain.
Integration of Technology
Standard x.x
CPI #
Cumulative Progress Indicator (CPI)
Instructional Focus
Enduring Understandings
A function is a relationship between variables in which each value of the input variable (domain) is
associated with a unique value of the output variable (range). Functions can be represented in a variety of
ways, such as graph, tables, equations, or words. Each representation is particularly useful in certain
situations. Some important families of functions are developed through transformations of the simplest
form of the function.
Limits provide us with language for describing how the outputs of a function behave as the inputs
approach some particular value.
Limits can also be determined as the inputs approach positive or negative infinity. In these cases, limits
can be used to describe the end behavior of a function and its graph.
The Properties of Limits can be used to calculate unfamiliar limits by using limits that we already know.
Sometimes the values of a function tend to different limits as x approaches a number c from opposite
sides. In this case, these are the right-hand limit and a left-hand limit.
The slope of a tangent to a curve can be found by calculating the limit of the slopes of secant lines. (using
the difference quotient).
The slope of a tangent to a curve can also be calculated by finding the derivative of the function through
its definition or using one or more of the Rules for Differentiation.
Essential Questions
What does the derivative of a function represent?
How can you find the derivative of a function using the definition?
How can you find the derivative of a function using the Product Rule? Quotient Rule? Chain Rule?
Evidence of Learning (Assessments)
Tests
Quizzes
Homework
Class participation
Objectives
Students will know or learn:
 Limits
 Derivatives
 Slope of a Curve
Students will be able to:
 Use limit theorems to evaluate the limit of a polynomial function.
 Explore techniques for evaluating limits: (direct substitution, dividing out, rationalizing, graphing)
 Evaluate the limits of difference quotients.
 Use a tangent line to approximate the slope of a graph at a point.
 Use the definition of slope to find exact slopes of graph.
 Find derivatives of polynomial functions.
 Use derivatives to find slopes of graphs
 Calculate derivatives using the Sum, Difference, Product, and Quotient Rules.
 Calculate derivatives using The Chain Rule.
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