Natrona County School District #1
Mission Statement
The Natrona County School District empowers every learner to grow, excel and be
successful contributors to the local and global community.
K-12 Mathematics Curriculum
Natrona County School District Mathematics Education Curriculum
Table of Contents
Acknowledgements........................................................................................................................ 3
Introduction ...................................................................................................................................... 4
Grade Level Pacing Guides
Kindergarten ...................................................................................................................... 6
1st Grade ............................................................................................................................ 18
2nd Grade ........................................................................................................................... 26
3rd Grade............................................................................................................................ 33
4th Grade ............................................................................................................................ 43
5th Grade ............................................................................................................................ 51
Course 1 – 6th Grade .................................................................................................... 59
Course 2 – 7th Grade .................................................................................................... 67
Accelerated Math – 7th Grade .................................................................................. 74
Course 3 – 8th Grade .................................................................................................... 84
Algebra I ............................................................................................................................ 90
Geometry........................................................................................................................... 97
Algebra II........................................................................................................................ 105
NCSD Math Vocabulary
K – 8 Critical Vocabulary ........................................................................................ 113
Kindergarten ................................................................................................................ 117
First Grade ..................................................................................................................... 122
Second Grade ............................................................................................................... 125
Third Grade ................................................................................................................... 127
Fourth Grade ................................................................................................................ 129
Fifth Grade ..................................................................................................................... 131
NCSD Common Math Assessment Information
K – 5 .................................................................................................................................. 134
6 – 8 .................................................................................................................................. 135
High School ................................................................................................................... 136
CCSS-M Resources and Visuals
Overview of Domains ............................................................................................... 138
Standards of Mathematical Practice ................................................................. 139
Instructional Shifts .................................................................................................... 143
Long Range Plan ........................................................................................................................ 144
NCSD Standards-Based Terminology .............................................................................. 145
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Natrona County School District Mathematics Education Curriculum
Acknowledgements
Thank you to the members of the Math Curriculum Committee for their hard work on this
curriculum:
Kindergarten
Course 1 (6th Grade)
Brandy Bentz
Cheryl Anderson
DeLaine Britt
Emily Quintana
Liz Harris
Sarah Willis
Carrie Patterson
Course 2 (7th Grade)
1st Grade
Accelerated Math (7th Grade)
Laurie Kilts
Teresa Bunker
Angela Ourth
Jann Keller
James Russell
2nd Grade
David Sunday
Kathy Christensen
Noelle Clark
Course 3 (8th Grade)
Angela Cavalier
3rd Grade
Kerin Dillon
Jennifer Bonnett
Jeanine Pickering
Buddy Johnson
Bud Sorenson
Ashley Ujvary
High School
4th Grade
Jessica Bratton
Kristin Fauss
Rebecca Byer
Stan Hahn
Timothy Fauss
Amy Radden
Ashlie Howell
Marla Switzer
th
5 Grade
Roger Switzer
Karen Bayert
Debra White
Cynthia Brachtenbach
Tonya Munari
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Natrona County School District Mathematics Curriculum
Introduction
Mathematics is the language that defines the blueprint of the universe. Mathematics is
woven into all parts of our lives, and is more than a list of skills to be mastered. The
essence of mathematics is the ability to employ critical thinking and reasoning to solve
problems. To be successful in mathematics, one must see mathematics as sensible, useful,
and worthwhile. The development and maturation of students’ conceptual understanding
and application of the processes and procedures of mathematics is the driving force behind
the 2011 Wyoming Mathematics Content and Performance Standards.
2011 WYOMING MATHEMATICS CONTENT AND PERFORMANCE STANDARDS DOCUMENT
The purpose of this document is to provide teachers with a horizontally and vertically
articulated progression of the math standards. The intent is to ensure a guaranteed and
viable curriculum for students as they progress through their educational journey in
Natrona County School District. This document will provide guidance to teachers as they
plan their instruction. Critical and supporting vocabulary has been identified to provide
further guidance.
2011 WYOMING MATHEMATICS CONTENT AND PERFORMANCE STANDARDS
RATIONALE:
The Common Core State Standards for Mathematics is a state-led effort to
establish a single set of clear educational standards that states can share and
voluntarily adopt. Including the Common Core State Standards into the 2011
Wyoming Mathematics Content and Performance Standards prepares
Wyoming students to be competitive on the national and world stage. These
standards are a set of specific, rigorous expectations that build students’
conceptual understanding, mathematical language, and application of
processes and procedures coherently from one grade to the next so all
students will be prepared for post-secondary experiences. The use of
technology is expected throughout all levels of the standards. The focus areas
for each grade level and conceptual category establish a depth of knowledge
as opposed to a breadth of knowledge across multiple standards in each
grade level or content area. The Standards for Mathematical Practices
describe the essential ways of thinking and habits of mind that are the
hallmark of a mathematically literate and informed citizen.
The Common Core State Standards for mathematics stress both conceptual
understanding and procedural skills to ensure students learn and can apply
the critical information needed to succeed at each level. This creates a
vertical articulation where the mathematics learned in elementary school
provides the foundation for the study of statistics, probability, ratio and
proportion, geometry, and algebra in middle school. This is, in turn, the
bedrock upon which the knowledge needed for success in colleges and
careers can be developed in the high school.
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Natrona County School District Mathematics Curriculum
Algorithmic knowledge is no longer sufficient when preparing our students
to become globally competitive. The knowledge of good practitioners goes
beyond algorithmic learning and allows them to picture the problem and the
many roads that may lead to absolution. They realize that mathematics is
applicable outside of the classroom and are confident in their ability to apply
mathematical concepts to all aspects of life. The symbiotic nature of the
Standards of Mathematical Practice allows students to deepen their
understandings of mathematical concepts and cultivates their autonomy as
mathematically literate and informed citizens. Employing mathematics as a
means of synthesizing complex concepts and making informed decisions is
paramount to success in all post-secondary endeavors.
MATH CURRICULUM COMMITTEE PROCESS
The alignment of math curriculum and shift towards a standards based approach in math
education began in NCSD during the 2013-2014 school year with the adoption of a new
primary resource. During professional development that school year teachers could
choose to work in teams to create district pacing guides for each grade level and address
district-wide concerns regarding a guaranteed and viable curriculum in math, common
assessments, and the appropriate use of instructional resources. This committee work has
continued as teacher teams have worked to revise the pacing guides, discuss critical
vocabulary, and work on common assessments.
In March 2015 the math curriculum committee surveyed teachers to gather input and were
charged with making final revisions to the district pacing guides, identifying critical
vocabulary, and revising district common assessments. Committee members were
sensitive to the concerns of teachers and worked diligently to make appropriate changes to
meet the needs of the entire district. As this work moves forward, teacher committees will
continue to engage in the alignment of standards and assessment in math.
Back to Table of Contents
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Natrona County School District Mathematics Curriculum
Kindergarten Math Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the Critical Areas of Focus defined by
Common Core State Standards for Math.
First Quarter
Second Quarter
Third Quarter
Fourth Quarter
K.CC Know number names
and the count sequence.
K.CC Know number names
and the count sequence.
K.CC.1 Count to 75 by ones
and to100 by tens.
K.CC.1 Count to 100 by
ones.
K.CC.1 Count to 20 by ones.
K.CC.1 Count to 50 by ones.
K.CC.2 Count forward
beginning from a given
number within the known
sequence (instead of having
to begin at 1). Within the
range of 0-20.
K.CC.3 Write numbers from 0
to 5.
K.CC.1 Count to100 by tens.
K.CC.2 Count forward
beginning from a given
number within the known
sequence (instead of having
to begin at 1). Within the
range of 0-75.
K.CC.3 Write numbers from 0
to 15.
K.CC.2 Count forward
beginning from a given
number within the known
sequence (instead of having
to begin at 1). Within the
range of 0-100.
K.CC.3 Write numbers from 0
to 20.
K.CC.3 Represent a number
of objects with a written
numeral 0-15 (with 0
representing a count of no
objects).
K.CC.3 Represent a number
of objects with a written
numeral 0-20 (with 0
representing a count of no
objects).
K.CC.2 Count forward
beginning from a given
number within the known
sequence (instead of having
to begin at 1). Within the
range 0-50.
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Natrona County School District Mathematics Curriculum
K.CC.3 Represent a number
of objects with a written
numeral 0-5 (with 0
representing a count of no
objects).
K.CC.3 Write numbers from 0
to 10.
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 (groups
up to ten objects).
K.G Identify and describe
shapes (squares, circles,
triangles, rectangles,
hexagons)
K.CC.3 Represent a number
of objects with a written
numeral 0-10 (with 0
representing a count of no
objects).
K.CC.7 Compare two
numbers between 1 and 10
presented as written
numerals.
Back to Table of Contents
K.CC.4a Count to tell the
number of objects:
Understand the relationship
between numbers and
quantities; connect
counting to cardinality.
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 is the same
regardless of their
arrangement or the order in
which they were counted.
K.CC.5 Count to answer
“how many?” questions
about as many as 20 things
arranged in a circle, or as
many as 10 things in a
scattered configuration.
K.CC.4c Understand that
each successive number
name refers to a quantity
that is one larger.
K.OA.5 Fluently subtract
within 5.
K.OA.2 Solve addition and
subtraction word problems
and add and subtract within
10, e.g., by using objects or
drawings to represent the
problem.
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Natrona County School District Mathematics Curriculum
K.G.1 Describe objects in the
environment using names of
shapes.
K.OA.1 Represent addition
with objects, fingers, mental
images, drawings, sounds
(e.g., claps), acting out
situations, verbal
explanations, expressions, or
equations. (Drawings need
not show details, but should
show the mathematics in
the problem. This applies
wherever drawings are
mentioned in the
Standards).
K.G.2 Correctly name twodimensional shapes
regardless of their
orientations or overall size.
K.OA.3 Decompose
numbers less than or equal
to 10 into pairs in more than
one way, e.g., by using
objects or drawings, and
record each decomposition
by a drawing or equation
(e.g., 5 = 2 + 3 and 5 = 4 + 1).
K.OA.4 For any number from
1 to 9, find the number that
makes 10 when added to
the given number, e.g., by
using objects or drawings,
and record the answer with
a drawing or equation.
K.G.4 Analyze and compare
two- and three- dimensional
shapes, in different sizes and
orientations using informal
language to describe their
similarities, differences, parts
(e.g., number of sides and
vertices/”corners”) and
other attributes (e.g. having
sides of equal length).
Back to Table of Contents
K.CC.5 Count to answer
“how many?” questions
about as many as 20 things
arranged in a line, a
rectangular array.
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 (e.g.,
18 = 10 + 8); understand that
these numbers are
composed of ten ones and
one, two, three, four, five, six,
seven, eight, or nine ones.
K.CC.5 Given a number from K.G.1 Identify and describe
1–20, count out that many
shapes (cubes, cones,
objects.
cylinders, and spheres).
Describe objects in the
environment using names of
shapes.
K.OA.1 Represent addition
and subtraction with
objects, fingers, mental
images, drawings, sounds
(e.g., claps), acting out
situations, verbal
explanations, expressions, or
equations. (Drawings need
not show details, but should
show the mathematics in
the problem. This applies
wherever drawings are
K.G.2 Correctly name threedimensional shapes
regardless of their
orientations or overall size.
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Natrona County School District Mathematics Curriculum
mentioned in the
Standards).
K.G.5 Model shapes in the
world by building shapes
from components (e.g.,
sticks and clay balls) and
drawing shapes.
K.MD.2 Directly compare
two objects with a
measurable attribute in
common, to see which
object has “more of”/“less
of” the attribute, and
describe the difference. For
example, directly compare
the heights of two children
and describe one child as
taller/shorter.
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.
K.OA.2 Solve addition word
problems and add within 10,
e.g., by using objects or
drawings to represent the
problem.
K.G.3 Identify shapes as twodimensional (lying in a
plane, “flat”) or threedimensional (“solid”).
K.OA.5 Fluently add within 5.
K.G.4 Analyze, compare,
create, and compose
shapes: Analyze and
compare three-dimensional
shapes, in different sizes and
orientations, using informal
language to describe their
similarities, differences, parts
(e.g., number of
vertices/“corners”) and
other attributes (e.g., having
sides of equal length).
K.G.1 Describe the relative
positions of objects using
terms such as above, below,
beside, in front of, behind,
and next to.
K.MD.1 Describe measurable
attributes of objects, such as
length or weight.
Back to Table of Contents
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Natrona County School District Mathematics Curriculum
K.G.6 Compose simple
shapes to form larger
shapes. For example, “Can
you join these two triangles
with full sides touching to
make a rectangle?
11
14
13
12
Pacing Guide Section 1: Counting & Cardinality, Geometry
Standards:
First Quarter
Clarification:
K.CC
K.CC. Identify Numbers
K.CC.1
Verbally count to 20 by ones.
Know number names and the count sequence.
K.CC.1
Count to 100 by ones and by tens.
K.CC.2
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
K.CC.2
Within the range of 0-20
K.CC.3
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a
count of no objects).
K.CC.6
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects
in another group, e.g., by using matching and counting strategies (groups up to ten objects).
K.CC.3
Write numbers within the
range 0-5. Represent a
number of objects with
written numeral within the
range 0-5. Reversals are
accepted, but not place value
reversals like 16/61.
K.G
Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).
K.G Identify 2-dimensional
shapes
K.G.1
Describe objects in the environment using names of shapes, and describe the relative positions of these objects
using terms such as above, below, beside, in front of, behind, and next to.
K.G.1 Naming 2-dimensional
shapes only; exclude relative
positions
K.G.2
Correctly name shapes regardless of their orientations or overall size.
K.G.2
2-dimensional shapes
Back to Table of Contents
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Natrona County School District Mathematics Curriculum
K.G.4
Analyze and compare two- and three- dimensional shapes, in different sizes and orientations using informal
language to describe their similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other
attributes (e.g. having sides of equal length).
K.G.4
2-dimensional shapes
(Students must describe # of
corners/vertices and sides
without being prompted. (If
they have not differentiated
between the square and
rectangle you may ask “how
are they are different?”).
K.G.5
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
K.G.5
2-dimensional shapes
Vocabulary:
Critical
Resource:
MyMath
Chapter 1 Lesson 2, 4-9
Chapter 2 Lesson 8
Chapter 11 Lesson 1-4, 8, 9
number, count, zero, one, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen, fifteen, sixteen, seventeen,
eighteen, nineteen, twenty, less (than/least),
greater (than/more), fewer, equal, square, circle,
triangle, rectangle, hexagon, sides, corners,
vertices, curved, 2-dimensional
Supporting
flat, alike, attributes, different, group, large,
model, objects, order, same, sequence, shape,
small, straight, digits, make, rhombus, trapezoid
Instructional Days: 30 days
Flex Days: 5 days (field trips, reteach/enrich, Discover days)
Pacing Guide Section 2: Counting & Cardinality, Geometry, Operations & Algebraic Thinking, Measurement & Data
Second Quarter
Standards:
Clarification:
K.CC
K.CC. Identify Numbers
Know number names and the count sequence.
K.CC.1
Count to 100 by ones and by tens.
K.CC.1
Verbally count to 50 by ones.
Verbally count to 100 by
tens.
K.CC.2
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
K.CC.2
Within the range of 0-50
Back to Table of Contents
Page 11 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
K.CC.3
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a
count of no objects).
K.CC.7
Compare two numbers between 1 and 10 presented as written numerals.
K.OA.1
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out
situations, verbal explanations, expressions, or equations. (Drawings need not show details, but should show the
mathematics in the problem. This applies wherever drawings are mentioned in the Standards).
K.OA.1
Addition only in the range of
0-10
K.OA.3
Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings,
and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
K.OA.3 (2+3 and 3+2 are
considered different)
K.OA.4
For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using
objects or drawings, and record the answer with a drawing or equation.
K.MD.2
Directly compare two objects with a measurable attribute in common, to see which object has “more of”/“less of”
the attribute, and describe the difference. For example, directly compare the heights of two children and describe
one child as taller/shorter.
K.MD.3
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.
K.G.1
Describe objects in the environment using names of shapes, and describe the relative positions of objects using
terms such as above, below, beside, in front of, behind, and next to.
Back to Table of Contents
K.CC.3
Write numbers within the
range 0-10. Represent a
number of objects with
written numeral within the
range 0-10. Reversals are
accepted, but not place value
reversals like 16/61.
K.CC.7
Use the vocabulary greater
than, less than, or equal
to/same- not symbols.
K.MD.2 (Must use vocabulary
longer/shorter)
K.G.1
Orally describe the relative
position of objects.
(Acceptable vocabulary:
above, on top, over, below,
under, beneath, on the
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Natrona County School District Mathematics Curriculum
bottom, next to, beside, to
the left, to the right, in front
of, behind, in back)
K.G.6
Compose simple shapes to form larger shapes. For example, “Can you join these two triangles with full sides
touching to make a rectangle?
Vocabulary:
Critical
Resource:
number, count, zero, one, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen, fifteen, sixteen, seventeen,
eighteen, nineteen, twenty, greater (than/more),
less (than/least), equal to, addition
(add/plus/combine), number sentence, length,
longer, shorter, sort, triangle, square, rectangle,
hexagon, 2-dimensional, sides, vertices, fewer
Support
color, group, order, above, on top, over, below,
under, beneath, on the bottom, next to, beside, to
the left, to the right, in front of, behind, in back,
alike, attributes, compose, decompose, different,
large, measure, objects, order, same, sequence,
shape, small, digits, equations, make, rhombus,
symbol, trapezoid, compare
MyMath
Chapter 2 Lesson 3, 6, 7
Chapter 3 Lesson 8, 10
Chapter 4 Lesson 1-9
Chapter 5 Lesson 1-7
Chapter 8 Lesson 1-4, 6
Chapter 9 Lesson 1-5
Chapter 10 Lesson 1-4
Chapter 11 Lesson 7
Instructional Days: 35 days
Flex Days: 4 days (field trips, reteach/enrich, Discover days)
Pacing Guide Section 3: Counting & Cardinality, Operations & Algebraic Thinking, Measurement & Data
Standards:
Third Quarter
Clarification:
K.CC.1
Count to 100 by ones and by tens.
K.CC.1
Verbally count to 75 by ones.
K.CC.2
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
K.CC.3
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a
count of no objects).
Back to Table of Contents
K.CC.3
Write numbers within the
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Natrona County School District Mathematics Curriculum
range 0-15. Represent a
number of objects with
written numeral within the
range 0-15. Reversals are
accepted, but not place value
reversals like 16/61. If the
objects are miscounted but
they write the correct
numeral for what they
counted, the answer is
correct.
K.CC.4
Understand the relationship between numbers and quantities; connect counting to cardinality.
a. When counting objects, say the number names in the standard order, pairing each object with one and only
one number name and each number name with one and only one object.
b. Understand that the last number name said tells the number of objects counted. The number of objects is the
same regardless of their arrangement or the order in which they were counted.
c. Understand that each successive number name refers to a quantity that is one larger.
K.CC.4a
In the range of 1-20.
K.CC.4b
In the range of 1-20.
K.CC.4c
In the range of 1-20.
K.CC.5
Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a
circle, or as many as 10 things in a scattered configuration; given a number from 1 – 20, count out that many
objects.
K.CC.5 Students must
independently count
correctly without prompting.
K.OA.1
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out
situations, verbal explanations, expressions, or equations. (Drawings need not show details, but should show the
mathematics in the problem. This applies wherever drawings are mentioned in the Standards).
K.OA.1
In the range of 0-10. If
students verbally explain
how they solved the
problems that is correct.
K.OA.2
Solve addition and subtraction word problems and add within 10, e.g., by using objects or drawings to represent
the problem.
K.OA.2 Addition word
problems only.
K.OA.5
Fluently add and subtract within 5.
K.OA.5 (Addition only) Must
answer with automaticity
(Cannot count by 1’s, count
on, or build with fingers).
Back to Table of Contents
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Natrona County School District Mathematics Curriculum
K.MD.1
Describe measurable attributes of objects, such as length or weight.
Vocabulary:
Critical
K.MD.1 Students should
suggest a measurement of
length, height, width, or
weight.
Resource:
number, count, zero, one, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen, fifteen, sixteen, seventeen,
eighteen, nineteen, twenty, addition (add, plus,
combine), height, tall(er), short(er), length,
long(er), width, wide(r), narrow(er), weight,
lighter, heavy, heavier, equal to, more, number
sentence, subtraction (take away/minus)
Support
equation, facts, group, measure, objects, order,
digits, equations, make, symbol
MyMath
Chapter 1 Lesson 1, 3, 10, 11
Chapter 2 Lesson 1, 2, 4, 5, 9
Chapter 3 Lesson 1-3
Chapter 6 Lesson 1-6
Chapter 8 Lesson 5
Instructional Days: 35 days
Flex Days: 3 days (field trips, reteach/enrich, Discover days)
Pacing Guide Section 4: Counting & Cardinality, Geometry, Operations & Algebraic Thinking,
Number & Operation in Base Ten
Standards:
Fourth Quarter
Clarification:
K.CC.1
Count to 100 by ones and by tens.
K.CC.2
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
K.CC.3
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a
count of no objects).
Back to Table of Contents
K.CC.1
Verbally count to 100 by
ones.
K.CC.3
Write numbers within the
range 0-20. Represent a
number of objects with
written numeral within the
range 0-20. Reversals are
accepted, but not place value
reversals like 16/61.
Page 15 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
K.CC.5
Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a
circle, or as many as 10 things in a scattered configuration; given a number from 1 – 20, count out that many
objects.
K.OA.2
Solve addition and subtraction word problems and add within 10, e.g., by using objects or drawings to represent
the problem.
K.OA.5
Fluently add and subtract within 5.
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 (e.g., 18 = 10 + 8);
understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or
nine ones.
K.NBT.1
Compose (put together) and
decompose (take apart).
If the student writes 10+8 or
8+10 the equation is still
correct.
K.G.1
Describe objects in the environment using names of shapes, and describe the relative positions of objects using
terms such as above, below, beside, in front of, behind, and next to.
K.G.1
3-dimensional objects
K.G.2
Correctly name shapes regardless of their orientations or overall size.
K.G.2
3-dimensional shapes
K.G.3
Identify shapes as two-dimensional (lying in a plane, “flat”) or three- dimensional (“solid”).
K.G.3
2-D and 3-D shape cards
K.G.4
Analyze and compare two- and three- dimensional shapes, in different sizes and orientations using informal
language to describe their similarities, differences, parts (e.g., number of sides and vertices/”corners”) and other
attributes (e.g. having sides of equal length).
K.G.4
3-dimensional shapes
Vocabulary:
Critical
K.CC.5 Students must
independently count
correctly without prompting.
K.OA.5 (Subtraction only)
Must answer with
automaticity (Cannot count
by 1’s, count down, or build
with fingers).
Resource:
number, count, zero, one, two, three, four, five,
six, seven, eight, nine, ten, eleven, twelve,
thirteen, fourteen, fifteen, sixteen, seventeen,
eighteen, nineteen, twenty, addition (add, plus,
MyMath
Chapter 3 Lesson 4-7, 9
Chapter 6 Lesson 7
Chapter 7 Lesson 1-5
Chapter 12 Lesson 1-5
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Page 16 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
combine), subtraction (minus/take away), cone,
sphere, cylinder, cube, 3-dimensional, 2dimensional, edges, faces, corners, vertices,
curved, equal, more, number sentence, sides
Support
flat, alike, attribute, compose, create, decompose,
different, facts, group, objects, order, roll, same,
sequence, shape, solid, stack, straight, digits,
equations, make, symbol, slide, solve
Instructional Days: 35 days
Flex Days: 4 days (field trips, reteach/enrich, Discover days)
Back to Table of Contents
Page 17 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
1st Grade Math Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math.
43 Days
Pacing Guide Section 1
Quarter 1
● Addition and Subtraction Concepts Measurement
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary in the CCSSM document)
NCSD clarification 1.OA.1 Within 10; with solving for unknowns
1.OA.3
Apply properties of operations as strategies to add and subtract. (Students need not use
formal terms for these properties.) 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.)
NCSD clarification 1.OA.3 Commutative Property
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 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.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.
Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five,
six, seven, eight, or nine ones.
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.
Vocabulary
Critical
Vocabulary
bundle, compose, decompose, difference, digit, equations, even, length, odd, ones, place
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Page 18 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
value, sum, tens, unknown
Supporting
Vocabulary
longer, longest, addition (add, plus, combine), subtraction (take away, minus)
Resource
My Math Chapter 1 Addition Concepts (17 days)
● Lesson 1 - Addition Stories
● Lesson 2 - Model Addition
● Lesson 3 - Addition Number Sentences
● Lesson 4 - Add 0
● Lesson 5 - Vertical Addition
● Lesson 6 - Problem-Solving Strategy: Write a Number Sentence
● Lesson 7 - Ways to Make 4 and 5
● Lesson 8 - Ways to make 6 and 7
● Lesson 9 - Ways to Make 8
● Lesson 10 - Ways to Make 9
● Lesson 11 - Ways to Make 10
● Lesson 12 - Find Missing Parts of 10
● Lesson 13 - True and False Statements
My Math Chapter 2 Subtraction Concepts (18 days)
● Lesson 1 - Subtraction Stories
● Lesson 2 - Model Subtraction
● Lesson 3 - Subtraction Number Sentences
● Lesson 4 - Subtract 0 and All
● Lesson 5 - Vertical Subtraction
● Lesson 6 - Problem Solving Strategy: Draw a Diagram
● Lesson 7 - Compare Groups
● Lesson 8 - Subtract from 4 and 5
● Lesson 9 - Subtraction from 6 and 7
● Lesson 10 - Subtract from 8
● Lesson 11 - Subtract from 9
● Lesson 12 - Subtract from 10
● Lesson 13 - Relate Addition and Subtraction
● Lesson 14 - True and False Statements
My Math Chapter 5 Place Value (4 days)
● Lesson 1 - Numbers 11 to 19
● Lesson 2 - Tens
My Math Chapter 8 Measurement and Time (4 days)
● Lesson 1 - Compare Lengths
● Lesson 2 - Compare and Order Lengths
● Lesson 3 - Nonstandard Units of Length
● Lesson 4 - Problem-Solving Strategy: Guess, Check, and Revise
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Page 19 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
46 Days
Pacing Guide Section 2
Quarter 2
● Addition and Subtraction Strategies
● Geometry
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary in the CCSSM document)
NCSD clarification
1.OA.1 - Within 20; with solving for unknowns.
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).
NCSD clarification
1.OA.6 - Addition within 10.
1.OA.7
Understand the meaning of the equal sign, and determine I 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.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.
Understand the following as special cases:
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five,
six, seven, eight, or nine tens (and 0 ones).
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 <.
NCSD clarification
1.NBT.3 - Compare 2 one-digit numbers.
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. (Students do not need to learn formal names such as
“right rectangular prism.”)
NCSD clarification
1.G.2 - Both 2-dimensional and 3-dimensional shapes.
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Page 20 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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.
Vocabulary
Critical
Vocabulary
2 dimensional, 3 dimensional, circle, compose, decompose, difference, equal to, equations,
fourths, greater than, half circle, halves, less than, quarter circle, quarters, rectangle,
rectangular prism, sides, sum, symbol, trapezoid, triangular prism, vertex
Supporting
Vocabulary
Resource
My Math Chapter 3 Addition Strategies to 20 (12 days)
● Lesson 1 - Count on 1, 2, o r3
● Lesson 2 - Count On Using Pennies
● Lesson 3 - Use a Number Line to Add
● Lesson 4 - Use Doubles to Add
● Lesson 5 - Use Near Doubles to Add
● Lesson 6 - Problem-Solving Strategy: Act It Out
● Lesson 7 - Make 10 to Add
● Lesson 8 - Add in Any Order
● Lesson 9 - Add Three Numbers
My Math Chapter 4 Subtraction Strategies to 20 (11 days)
● Lesson 1 - Count Back 1, 2, or 3
● Lesson 2 - Use a Number Line to Subtract
● Lesson 3 - Use Doubles to Subtract
● Lesson 4 - Problem-Solving Strategy: Write a Number Sentence
● Lesson 5 - Make 10 to Subtract
● Lesson 6 - Use Related Facts to Add and Subtract
● Lesson 7 - Fact Families
● Lesson 8 - Missing Addends
My Math Chapter 9 Two-Dimensional Shapes and Equal Shares (10 days)
● Lesson 1 - omit
● Lesson 2 - omit
● Lesson 3 - omit
● Lesson 4 - Compare Shapes
● Lesson 5 - Composite Shapes
● Lesson 6 - More Composite Shapes
● Lesson 7 - Problem-Solving Strategy: Use Logical Reasoning
● Lesson 8 - Equal Parts
● Lesson 9 - Halves
● Lesson 10 - Quarters and Fourths
My Math Chapter 10 Geometry (7 days)
● Lesson 1 - Cubes and Prisms
● Lesson 2 - Cones and Cylinders
● Lesson 3 - Problem Solving: Look for a Pattern
● Lesson 4 - Combine 3-D Shapes
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Page 21 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
42 Days Pacing Guide Section 3
Quarter 3
● Place Value
● Two-Digit Addition and Subtraction
● Time
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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.
NCSD clarification
1.OA.2 - Within 10.
1.OA.3
Apply properties of operations as strategies to add and subtract. (Students need not use
formal terms for these properties.) 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.)
NCSD clarification
1.OA.3 - Associative Property.
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).
NCSD clarification
1.OA.6 - Addition and subtraction within 20.
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating to
three whole numbers. For example, determine the unknown number that makes the
equation true in each of the equations 8 + ? = 11, 5 = � – 3, 6 + 6 = �.
NCSD clarification
1.OA.8 - Combinations to 10.
1.NBT.2
Understand that the two digits of a two-digit number represent amounts of tens and ones.
Understand the following as special cases:
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five,
six, seven, eight, or nine tens (and 0 ones).
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 <.
NCSD clarification
1.NBT.3 - Compare 2 two-digit numbers.
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 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.
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Page 22 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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
Give 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.
Vocabulary
Critical
Vocabulary
compose, decompose, difference, equations, sum
Supporting
Vocabulary
false, true
Resource
My Math Chapter 5 Place Value (21 days)
● Lesson 1 - addressed in Quarter 1
● Lesson 2 - addressed in Quarter 1
● Lesson 3 - Count by Tens Using Dimes
● Lesson 4 - Ten and Some More
● Lesson 5 - Tens and Ones
● Lesson 6 - Problem-Solving Strategy: Make a Table
● Lesson 7 - Numbers to 100
● Lesson 8 - Ten More, Ten Less
● Lesson 9 - Count by Fives Using Nickels
● Lesson 10 - Use Models to Compare Numbers
● Lesson 11 - Use Symbols to Compare Numbers
● Lesson 12 - Numbers to 120
● Lesson 13 - Count to 120
● Lesson 14 - Read and Write Numbers to 120
My Math Chapter 6 Two-Digit Addition (21 days)
● Lesson 1 - Add Tens
● Lesson 2 - Count On Tens and Ones
● Lesson 3 - Add Tens and Ones
● Lesson 4 - Problem-Solving Strategy: Guess, Check, and Revise
● Lesson 5 - Add Tens and Ones with Regrouping
● Lesson 6 - Subtract Tens
● Lesson 7 - Count Back by 10s
● Lesson 8 - Relate Addition and Subtraction of Tens
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Page 23 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
45 Days Pacing Guide Section 4
Quarter 4
● Data
● Geometry
Clarification
Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary in the CCSSM document)
NCSD clarification
1.OA.1 - Within 20. Solving questions for the unknown.
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.
NCSD clarification
1.OA.2 - Within 20.
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.
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).
NCSD clarification
1.OA.6 - Addition and subtraction within 20.
1.OA.8
Determine the unknown whole number in an addition or subtraction equation relating to
three whole numbers. For example, determine the unknown number that makes the
equation true in each of the equations 8 + ? = 11, 5 = � – 3, 6 + 6 = �.
NCSD clarification
1.OA.8 - Combinations to 20.
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 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.MD.3
Tell and write time in hours and half-hours using analog and digital clocks.
1.MD.4
Organize, represent, and interpret data with up to three categories; ask and answer questions
about the total number of data points, how many in each category, and how many more or
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Page 24 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
less are in one category than in another.
NCSD clarification
1.MD.4 - Graphing can be embedded within learning cycles throughout the year.
Vocabulary
Critical
Vocabulary
analog, compose, decompose, difference, digital, equations, fact family, half hour, hour,
sum
Supporting
Vocabulary
Resource
My Math Chapter 7 Organize and Use Graphs (12 days)
● Lesson 1 - Tally Charts
● Lesson 2 - Problem-Solving Strategy: Make a Table
● Lesson 3 - Make Picture Graphs
● Lesson 4 - Read Picture Graphs
● Lesson 5 - Make Bar Graphs
● Lesson 6 - Read Bar Graphs
My Math Chapter 8 Measurement and Time (10 days)
● Lesson 5 - Time to the Hour: Analog
● Lesson 6 - Time to the Hour: Digital
● Lesson 7 - Time to the Half Hour: Analog
● Lesson 8 - Time to the Half Hour: Digital
● Lesson 9 - Time to the Hour and Half Hour
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Page 25 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
2nd Grade Math Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math.
43 Days
Pacing Guide Section 1
● Addition and Subtraction Concepts
● Number Patterns
● Place Value to 1,000
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary in the CCSSM
document)
NCSD clarification
2.OA.1 - Within 20.
2.OA.2
Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from
memory all sums of two one-digit numbers.
2.OA.3
Determine whether a group of objects (up to 20) has an odd or even number of members,
e.g., by pairing objects or counting them by 2s; write an equation to express an even
number as a sum of two equal addends.
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.
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. Understand the following as
special cases:
a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three,
four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.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.
NCSD clarification
2.NBT.5 - Within 20.
2.NBT.9
Explain why addition and subtraction strategies work, using place value and the properties
of operations.
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Page 26 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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.
Vocabulary
Critical
Vocabulary
addend, base ten numerals, expanded form, fact family, hundreds, number line, number
name, place value
Supporting
Vocabulary
columns, digits, equations, equivalent, expression, ones, sum, symbol, tens, unknown, whole
numbers, rows
Resource
My Math Chapter 1 Applying Addition and Subtraction Concepts (18 days)
● Lesson 1 - Addition Properties
● Lesson 2 - Count on to Add
● Lesson 3 - Doubles and Near Doubles
● Lesson 4 - Make a 10
● Lesson 5 - Add Three Numbers
● Lesson 6 - Problem-Solving Strategy: Write a Number Sentence
● Lesson 7 - Count Back to Subtract
● Lesson 8 - Subtract All and Subtract Zero
● Lesson 9 - Use Doubles to Subtract
● Lesson10 - Relate Addition and Subtraction
● Lesson 11 - Missing Addends
● Lesson 12 - Fact Families
● Lesson 13 - Two-Step Word Problems
My Math Chapter 2 Number Patterns (12 days)
● Lesson 1 - Skip Count on a Hundred Chart
● Lesson 2 - Skip Count by 2s, 5s, and 10s
● Lesson 3 - Problem Solving Strategy: Find a Pattern
● Lesson 4 - Repeated Addition
● Lesson 5 - Repeated Addition with Arrays
● Lesson 6 - Even and Odd Numbers
● Lesson 7 - Sums of Equal Numbers
My Math Chapter 5 Place Value to 1,000 (13 days)
● Lesson 1 - Hundreds
● Lesson 2 - Hundreds, Tens, and Ones
● Lesson 3 - Place Value to 1,000
● Lesson 4 - Problem Solving Strategy: Use Logical Reasoning
● Lesson 5 - Read and Write Numbers to 1,000
● Lesson 6 - Count by 5s, 10s, and 100s
● Lesson 7 - Compare Numbers to 1,000
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Page 27 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
46 Days
Pacing Guide Section 2
● Two-Digit Addition and Subtraction
● Measurement
● Time
● Money
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary in the CCSSM
document)
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.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.
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?
Vocabulary
Critical
Vocabulary
analog, compose, decompose, digital, fact family, number line
Supporting
Vocabulary
a.m., dime, dollar bill, equations, equivalent, expression, hour, hour hand, minute hand,
minutes, nickel, p.m., penny, quarter, quarter past, sum, symbol, unknown, whole numbers
Resource
My Math Chapter 3 Add Two-Digit Numbers (11 days)
● Integrate 2.MD.6 Adding and Subtracting on a number line*needs to be supplemented
● Lesson 1 - Take Apart Tens to Add
● Lesson 2 - Regroup Ones as Tens
● Lesson 3 - Add a Two-Digit Number
● Lesson 4 - Add Two-Digit Numbers
● Lesson 5 - Rewrite Two-Digit Addition
● Lesson 6 - Add Three or Four Two-Digit Numbers
● Lesson 7 - Problem Solving Strategy: Make a Model
My Math Chapter 4 Subtract Two-Digit Numbers (13 days)
● Lesson 1 - Two-Digit Fact Families
● Lesson 2 - Take Apart Tens to Subtract
● Lesson 3 - Regroup a Ten as Ones
● Lesson 4 - Subtract From a Two-Digit Number
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Page 28 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
●
●
●
●
●
Lesson 5 - Subtract Two-Digit Numbers
Lesson 6 - Rewrite Two-Digit Subtraction
Lesson 7 - Check Subtraction
Lesson 8 - Problem-Solving Strategy: Write a Number Sentence
Lesson 9 - Two-Step Word Problems
My Math Chapter 8 Money (10 days)
● Lesson 1 - Pennies, Nickels, and Dimes
● Lesson 2 - Quarters
● Lesson 3 - Count Coins
● Lesson 4 - Problem-Solving Strategy: Act it Out
● Lesson 5 - Dollars
My Math Chapter 10 Time (12 days)
● Lesson 1 - Time to the Hour
● Lesson 2 - Time to the Half Hour
● Lesson 3 - Problem Solving Strategy: Find a Pattern
● Lesson 4 - Time to the Quarter Hour
● Lesson 5 - Time to Five Minute Intervals
● Lesson 6 - A.M. and P.M.
38 Days
Pacing Guide Section 3
● Add Three-Digit Numbers
● Data Analysis
● Geometry
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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 problems (See
Glossary) using information presented in a bar graph.
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.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.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.G.1
Recognize and draw shapes having specified attributes, such as a given number of angles or
a given number of equal faces. (Sizes are compared directly or visually, not compared by
measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
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,
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Natrona County School District Mathematics Curriculum
three thirds, four fourths. Recognize that equal shares of identical wholes need not have the
same shape.
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. (See Glossary in the CCSSM
document)
Vocabulary
Critical
Vocabulary
fact family, pentagon, picture graph, quadrilateral
Supporting
Vocabulary
angle, data, data table, edges, equal shares, equations, equivalent, expression, faces, fourths,
fraction, halves, horizontal, key, line plot, partition, quarters, rectangular prism, scale, sum,
survey, symbol, tally mark, thirds, unknown, vertical, whole numbers
Resource
My Math Chapter 6 Add Three-Digit Numbers (12 days)
● Lesson 1 - Make a Hundred to Add
● Lesson 2 - Add Hundreds
● Lesson 3 - Mentally Add 10 or 100
● Lesson 4 - Regroup Ones to Add
● Lesson 5 - Regroup Tens to Add
● Lesson 6 - Add Three-Digit Numbers
● Lesson 7 - Rewrite Three-Digit Addition
● Lesson 8 - Problem-Solving Strategy: Guess, Check and Revise
My Math Chapter 9 Data Analysis (12 days)
● Lesson 1 - Take a Survey
● Lesson 2 - Make Picture Graphs
● Lesson 3 - Analyze Picture Graphs
● Lesson 4 - Make Bar Graphs
● Lesson 5 - Analyze Bar Graphs
● Lesson 6 - Problem-Solving Strategy: Make a Table
● Lesson 7 - Make Line Plots
● Lesson 8 - Analyze Line Plots
My Math Chapter 12 Geometric Shapes and Equal Shares (14 days)
● Lesson 1 - Two-Dimensional Shapes
● Lesson 2 - Sides and Angles
● Lesson 3 - Problem-Solving Strategy: Draw a Diagram
● Lesson 4 - Three-Dimensional Shapes
● Lesson 5 - Faces, Edges, and Solids
● Lesson 6 - Relate Shapes and Solids
● Lesson 7 - Halves, Thirds, and Fourths
● Lesson 8 - Area
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Natrona County School District Mathematics Curriculum
35 Days
Pacing Guide Section 4
● Subtract Three-Digit Numbers
● Measurement - Customary and Metric Lengths
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary)
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.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.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.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.8
Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a
given number 100-900.
Vocabulary
Critical
Vocabulary
centimeter, customary system, fact family, feet/foot, inch, meter, metric system, yard
Supporting
Vocabulary
equations, equivalent, expression, length, measuring tape, meter stick, ruler, symbol, unit,
unknown, yard stick
Resource
My Math Chapter 7 Subtract Three-Digit Numbers (16 days)
● Lesson 1 - Take Apart Hundreds to Subtract
● Lesson 2 - Subtract Hundreds
● Lesson 3 - Mentally Subtract 10 or 100
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Page 31 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
●
●
●
●
●
●
Lesson 4 - Regroup Tens
Lesson 5 - Regroup Hundreds
Lesson 6 - Subtract Three-Digit Numbers
Lesson 7 - Rewrite Three- Digit Subtraction
Lesson 8 - Problem-Solving Strategy: Write a Number Sentence
Lesson 9 - Subtract Across Zeros
My Math Chapter 11 Customary and Metric Lengths (19 days)
● Lesson 1 - Inches
● Lesson 2 - Feet and Yards
● Lesson 3 - Select and Use Customary Tools
● Lesson 4 - Compare Customary Lengths
● Lesson 5 - Relate Inches, Feet, and Yards
● Lesson 6 - Problem-Solving Strategy: Using Logical Reasoning
● Lesson 7 - Centimeters and Meters
● Lesson 8 - Select and Use Metric Tools
● Lesson 9 - Compare Metric Lengths
● Lesson 10 - Relate Centimeters and Meters
● Lesson 11 - omit
● Lesson 12- Measurement Data
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Natrona County School District Mathematics Curriculum
3rd Grade Math Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
38-43
Days*
Pacing Guide Section 1
● Place Value, Addition and Subtraction Concepts, Geometry, Perimeter
* Five days leeway for first week routines and NWEA testing.
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
3.NBT.1
Use place value understanding to round whole numbers to the nearest 10 or 100.
3.NBT.2
Fluently add and subtract within 1000 using strategies and algorithms based on place value,
properties of operations, and/or the relationship between addition and subtraction.
3.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. *
*This standard is limited to problems posed with whole numbers and having whole
number answers; students should know how to perform operations in the
conventional order when there are no parentheses to specify a particular order
(Order of Operations).
NCSD Clarification
3.OA.8 - Addition and subtraction only
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.
NCSD Clarification
3.OA.9 - Addition only
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.
3.MD.8
Solve real world and mathematical problems involving perimeters of polygons, including
finding the perimeter given the side lengths, finding an unknown side length, and
exhibiting rectangles with the same perimeter and different areas or with the same area and
different perimeters.
Vocabulary
Critical
Vocabulary
angle(s), parallel, parallelogram, perimeter, place value, polygon, unit, whole numbers
Supporting
Vocabulary
fact family, quadrilateral, sum, difference
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Natrona County School District Mathematics Curriculum
Resource
My Math Chapter 1 Place Value (9 days)
● Lesson 1 - Place Value Through Thousands
● Lesson 2 - Compare Numbers
● Lesson 3 - Order Numbers
● Lesson 4 - Round to the Nearest Ten
● Lesson 5 - Round to the Nearest Hundred
● Lesson 6 - Problem-Solving Investigation: Use the Four-Step Plan
My Math Chapter 2 Addition (12 days)
● Lesson 1 - Addition Properties
● Lesson 2 - Patterns in the Addition Table
● Lesson 3 - Addition Patterns
● Lesson 4 - Add Mentally
● Lesson 5 - Estimate Sums
● Lesson 6 - Hands On: Use Models to Add
● Lesson 7 - Add Three-Digit Numbers
● Lesson 8 - Add Four-Digit Numbers
● Lesson 9 - Problem-Solving Investigation: Reasonable Answers
My Math Chapter 14 Geometry (5 days)
● Use manipulatives
● Lesson 1 - Hands On: Angles
● Lesson 2 - Polygons
● Lesson 4 - Quadrilaterals
● Lesson 5 - Shared Attributes of Quadrilaterals
My Math Chapter 13 Perimeter and Area (3 days)
● Use additional resources to practice with 2- and 3- digit side lengths
● Lesson 1 - Hands On: Find Perimeter
● Lesson 2 - Perimeter
My Math Chapter 3 Subtraction (9 days)
● Lesson 1 - Subtract Mentally
● Lesson 2 - Estimate Differences
● Lesson 3 - Problem-Solving Investigation: Estimate or Exact Answer
● Lesson 4 - Hands On: Subtraction with Regrouping
● Lesson 5 - Subtract Three-Digit Numbers
● Lesson 6 - Subtract Four-Digit Numbers
● Lesson 7 - Subtract Across Zeros
41 Days
Pacing Guide Section 2
● Multiplication and Division Concepts
● Patterns
● Connecting Area to Multiplication
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Students should be aware of the different types of operation symbols
(𝑖. 𝑒. ,×,∙,∗,÷, 𝑎⁄𝑏 , 𝑎𝑛𝑑 𝑙𝑜𝑛𝑔 𝑑𝑖𝑣𝑖𝑠𝑜𝑛 𝑠𝑡𝑟𝑢𝑐𝑡𝑢𝑟𝑒)
Standards
3.OA.1
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5
groups of 7 objects each. For example, describe a context in which a total number of
objects can be expressed as 5 × 7.
3.OA.2
Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of
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Page 34 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
objects in each share when 56 objects are partitioned equally into 8 shares, or as a number
of shares when 56 objects are partitioned into equal shares of 8 objects each. For example,
describe a context in which a number of shares or a number of groups can be expressed as
56 ÷ 8.
3.OA.4
Determine the unknown whole number in a multiplication or division equation relating
three whole numbers. For example, determine the unknown number that makes the
equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
NCSD Clarification
3.OA.4 - Fact fluency for 0-5, 10
3.OA.5
Apply properties of operations as strategies to multiply and divide. (Students need not use
formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is
also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 =
15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of
multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2)
= (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
NCSD Clarification
3.OA.5 - Commutative Property of Multiplication only
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.
NCSD Clarification
3.OA.6 - Fact fluency for 0-5, 10
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 from memory all products of two
one-digit numbers.
NCSD Clarification
3.OA.7 - Fact fluency for 0-5, 10
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.*
*This standard is limited to problems posed with whole numbers and having whole
number answers; students should know how to perform operations in the
conventional order when there are no parentheses to specify a particular order
(Order of Operations).
NCSD Clarification
3.OA.8 - Multiplication and division only
3.NBT.3
Multiply one-digit whole numbers by multiples of 10 in the range 10–90 (e.g., 9 × 80, 5 ×
60) using strategies based on place value and properties of operations.
3.MD.5
Recognize area as an attribute of plane figures and understand concepts of area
measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square
unit” of area, and can be used to measure area.
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Page 35 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
b. A plane figure which can be covered without gaps or overlaps by n unit squares is
said to have an area of n square units.
3.MD.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and
improvised units).
3.MD.7
Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show
that the area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in
the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning.
Vocabulary
Critical
Vocabulary
division, equal part, factor, label, multiplication, partitioned, product, quotient, unit,
variable, whole numbers
Supporting
Vocabulary
area, array, difference, fact family, interpret, multiple, sum
Resource
My Math Chapter 4 Understand Multiplication (9 days)
● Lesson 1 - Hands On: Model Multiplication
● Lesson 2 - Multiplication as Repeated Addition
● Lesson 3 - Hands On: Multiply with Arrays
● Lesson 4 - Arrays and Multiplication
● Lesson 5 - Problem-Solving Investigation: Make a Table
● Lesson 6 - Use Multiplication to Find Combinations
My Math Chapter 5 Understand Division (9 days)
● Lesson 1 - Hands On: Model Division
● Lesson 2 - Division as Equal Sharing
● Lesson 3 - Relate Division and Subtraction
● Lesson 4 - Hands On: Relate Division and Multiplication
● Lesson 5 - Inverse Operations
● Lesson 6 - Problem-Solving Investigations: Use Models
My Math Chapter 6 Multiplication and Division Problems (12 days)
● Lesson 1 - Patterns in the Multiplication Table
● Lesson 2 - Multiply by 2
● Lesson 3 - Divide by 2
● Lesson 4 - Multiply by 5
● Lesson 5 - Divide by 5
● Lesson 6 - Problem-Solving Investigation: Look for a Pattern
● Lesson 7 - Multiply by 10
● Lesson 8 - Multiples of 10
● Lesson 9 - Divide by 10
My Math Chapter 7 Multiplication and Division (11 days)
● Lesson 1 - Multiply by 3
● Lesson 2 - Divide by 3
● Lesson 3 - Hands On: Double a Known Fact
● Lesson 4 - Multiply by 4
● Lesson 5 - Divide by 4
● Lesson 6 - Problem-Solving Investigation: Extra or Missing Information
● Lesson 7 - Multiply by 0 and 1
● Lesson 8 - Divide with 0 and 1
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Page 36 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
My Math Chapter 13 Perimeter and Area (no additional days)
● Area as an array model
● Embed area lessons within multiplication lessons by connecting the array model to area of a rectangle.
● Lesson 5 - Hands On: Tile Rectangle to Find Area
● Lesson 6 - Areas of Rectangles
40 Days* Pacing Guide Section 3
3 days
● Fractions
leeway for
● Measurement Related to Fractions
PAWS
● Geometry Related to Fractions
● Data Analysis
● Multiplication and Division Concepts
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary)
3.OA.4
Determine the unknown whole number in a multiplication or division equation relating
three whole numbers. For example, determine the unknown number that makes the
equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
NCSD Clarification
3.OA.4 - Fact Fluency for 6, 7, 8, 9
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.
NCSD Clarification
3.OA.6 - Fact fluency for 6, 7, 8, 9
3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship between
multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or
properties of operations. By the end of Grade 3, know from memory all products of two
one-digit numbers.
NCSD Clarification
3.OA.7 - Fact fluency for 6, 7, 8, 9
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.
3.MD.7
Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole number side lengths by tiling it, and show
that the area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in
the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number
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Page 37 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
side lengths a and b + c is the sum of a × b and a × c. Use area models to represent
the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them
into non-overlapping rectangles and adding the areas of the non-overlapping parts,
applying this technique to solve real world problems.
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.
a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to
1 as the whole and partitioning it into b equal parts. Recognize that each part has
size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the
number line.
b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b
from 0. Recognize that the resulting interval has size a/b and that its endpoint
locates the number a/b on the number line.
3.NF.3
Explain equivalence of fractions in special cases, and compare fractions by reasoning about
their size.
a. Understand two fractions as equivalent (equal) if they are the same size, or the same
point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3).
Explain why the fractions are equivalent, e.g., by using a visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that are equivalent to
whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6;
locate 4/4 and 1 at the same point of a number line diagram.
d. Compare two fractions with the same numerator or the same denominator by
reasoning about their size. Recognize that comparisons are valid only when the two
fractions refer to the same whole. Record the results of comparisons with the
symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction
model.
3.G.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction
of the whole. For example, partition a shape into 4 parts with equal area, and describe the
area of each part as 1/4 of the area of the shape.
Vocabulary
Critical
Vocabulary
denominator, division, equal part, factor, fraction, multiplication, numerator, partitioned,
polygon, product, quotient, unit, variable, whole numbers
Supporting
Vocabulary
array, equivalent, fact family, multiple
Supporting
Strategy
number line
Resource
My Math Chapter 8 Apply Multiplication and Division (9 days)
● Lesson 1 - Multiply by 6
● Lesson 2 - Multiply by 7
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Page 38 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
●
●
●
●
Lesson 3 - Divide by 6 and 7
Lesson 4 - Multiply by 8
Lesson 5 - Multiply by 9
Lesson 6 - Divide by 8 and 9
My Math Chapter 12 Represent and Interpret Data (12 days)
● Lesson 1 - Collect and Record Data
● Lesson 2 - Draw Scaled Picture Graphs
● Lesson 3 - Draw Scaled Bar Graphs
● Lesson 4 - Relate Bar Graphs to Scaled Picture Graphs
● Lesson 5 - Draw and Analyze Line Plots
● Lesson 6 - Hands On: Measure to Halves and Fourths of a Inch
● Lesson 7 - Collect and Display Measurement Data
My Math Chapter 10 Fractions (11 days)
○ Use manipulatives
● Lesson 1 - Unit Fractions
● Lesson 2 - Part of a Whole
● Lesson 3 - Part of a Set
● Lesson 4 - Problem -Solving Investigation: Draw a Diagram
● Lesson 5 - Hands on: Fractions on a Number Line
● Lesson 6 - Equivalent Fractions
● Lesson 7 - Fractions as One Whole
● Lesson 8 - Compare Fractions
My Math Chapter 14 Geometry (3 days)
● Lesson 3 - Hands on: Triangles
● Lesson 7 - Partition Shapes
My Math Chapter 13 Perimeter and Area (5 days)
○ May need additional resources
● Lesson 3 - Hands On: Understand Area
● Lesson 4 - Measure Area
● Lesson 8 - Area of Composite Figures
● Lesson 9 - Area and Perimeter
44 Days
4 days
leeway for
NWEA and
final week.
Pacing Guide Section 4:
● Multiplication and Division Word Problems
● Measuring Time and Capacity
● Properties of Multiplication
● Strategies for Two-Digit Multiplication
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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. (See Glossary)
3.OA.4
Determine the unknown whole number in a multiplication or division equation relating
three whole numbers. For example, determine the unknown number that makes the
equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?.
NCSD Clarification
3.OA.4 - Fact Fluency for 11, 12
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Page 39 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
3.OA.5
Apply properties of operations as strategies to multiply and divide. (Students need not use
formal terms for these properties.) Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is
also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 =
15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of
multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2)
= (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)
NCSD Clarification
3.OA.5 - Associative and Distributive Properties of Multiplication
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.
NCSD Clarification
3.OA.6 - Fact fluency and strategies for 11, 12
3.OA.7
Fluently multiply and divide within 100, using strategies such as the relationship
between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 =
8) or properties of operations. By the end of Grade 3, know from memory all products
of two one-digit numbers.
NCSD Clarification
3.OA.7 - Fact fluency and strategies for 11, 12
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. (This
standard is limited to problems posed with whole numbers and having whole number
answers; students should know how to perform operations in the conventional order when
there are no parentheses to specify a particular order (Order of Operations).)
NCSD Clarification
3.OA.8 - All four operations
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.
3.MD.2
Measure and estimate liquid volumes and masses of objects using standard units of grams
(g), kilograms (kg), and liters (l). (Excludes compound units such as cm 3 and finding the
geometric volume of a container.) Add, subtract, multiply, or divide to solve one-step word
problems involving masses or volumes that are given in the same units, e.g., by using
drawings (such as a beaker with a measurement scale) to represent the problem. [Excludes
multiplicative comparison problems (problems involving notions of “times as much”; see
Glossary, Table 2])
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
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Page 40 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
marked off in appropriate units— whole numbers, halves, or quarters.
3.MD.5
Recognize area as an attribute of plane figures and understand concepts of area
measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square
unit” of area, and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is
said to have an area of n square units.
3.MD.6
Measure areas by counting unit squares (square cm, square m, square in, square ft, and
improvised units).
3.MD.7
Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show
that the area is the same as would be found by multiplying the side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in
the context of solving real world and mathematical problems, and represent wholenumber products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number
side lengths a and b + c is the sum of a × b and a × c. Use area models to represent
the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them
into non-overlapping rectangles and adding the areas of the non-overlapping parts,
applying this technique to solve real world problems.
Vocabulary
Critical
Vocabulary
division, factor, label, liquid volume, multiplication, product, quotient, scale, unit, variable,
whole numbers
Supporting
Vocabulary
area, array, fact family, feet, inches, multiple
Resource
My Math Chapter 11 Measurement (11 days)
● Lesson 1 - Hands On: Estimate and Measure Capacity
● Lesson 2 - Solve Capacity Problems
● Lesson 3 - Hands On: Estimate and Measure Mass
● Lesson 4 - Solve Mass Problems
● Lesson 5 - Tell Time to the Minute
● Lesson 6 - Time Intervals
● Lesson 7 - Problem-Solving Investigation: Work Backward
My Math Chapter 8 Apply Multiplication and Division (4 days)
● Lesson 7 - Problem-Solving Investigation: Make an Organized List
● Lesson 8 - Multiply by 11 and 12
● Lesson 9 - Divide by 11 and 12
My Math Chapter 9 Properties and Equations (12 days)
● Lesson 1 - Hands On: Take Apart to Multiply
● Lesson 2 - The Distributive Property
● Lesson 3 - Hands On: Multiply Three Factors
● Lesson 4 - The Associative Property
● Lesson 5 - Write Expressions
● Lesson 6 - Evaluate Expressions
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Natrona County School District Mathematics Curriculum
●
●
●
Lesson 7 - Write Equations
Lesson 8 - Solve Two-Step Word Problems
Lesson 9 - Problem-Solving Investigation: Use Logical Reasoning
My Math Chapter 13 Perimeter and Area (3 days)
● Lesson 7 - Hands On: Area and the Distributive Property
● Lesson 10 - Problem-Solving Investigation: Draw a Diagram
My Math Chapter 14 Geometry (2 days)
○ Need to find more similar practice problems
○ Apply and Transfer Skills to Real Life Application, need additional resources.
● Lesson 6 - Problem-Solving Investigation: Guess, Check, and Revise
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Natrona County School District Mathematics Curriculum
4th Grade Math Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
29 Days
Pacing Guide Section 1
● Place Value
● Addition and Subtraction
● Multiplication Concepts
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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.
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.
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.
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Natrona County School District Mathematics Curriculum
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.
4.G.1
Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and
parallel lines. Identify these in two-dimensional figures.
NCSD Clarification
4.G.1 – Assessed on NCSD Benchmark 4 but needs to be addressed prior to PAWS
Vocabulary
Critical
Vocabulary
algorithm, factors, inequalities, multiples, multiplicative inverse, quotient
Supporting
Vocabulary
area model, array, expanded form, expression, operation, place value, product
Resource
My Math Chapter 1 Place Value (7 days)
● Lesson 1 - Place Value
● Lesson 2 - Read and Write Multi-Digit Numbers
● Lesson 3 - Compare Numbers
● Lesson 4 - Order Numbers
● Lesson 5 - Use Place Value to Round
My Math Chapter 2 Add and Subtract Whole Numbers (9 days)
● Lesson 2 - Addition and Subtraction Patterns
● Lesson 3 - Add and Subtract Mentally
● Lesson 4 - Estimate Sums and Differences
● Lesson 5 - Add Whole Numbers
● Lesson 6 - Subtract Whole Numbers
● Lesson 7 - Subtract Across Zeros
● Lesson 9 - Solve Multi-Step Word Problems
My Math Chapter 3 Understand Multiplication and Division (10 days)
● Lesson 1 - Relate Multiplication and Division
● Lesson 3 - Multiplication as Comparison
● Lesson 4 - Compare to Solve Problems
● Lesson 5 - Multiplication Properties and Division Rules
● Lesson 6 - The Associative Property of Multiplication
● Lesson 7 - Factors and Multiples
○ Include more lessons covering Factors and Multiples
● Lesson 8 - Problem-Solving Investigation: Reasonable Answers
37 Days
Pacing Guide Section 2
● Multiplication and Division
● Area and Perimeter
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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
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Natrona County School District Mathematics Curriculum
concepts of place value and division.
4.NBT.3
Use place value understanding to round multi-digit whole numbers to any place.
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.
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.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.
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.
NCSD Clarification
4.G.2 – Assessed on NCSD Benchmark 4 but needs to be addressed prior to PAWS
Vocabulary
Critical
Vocabulary
dividend, divisor, factors, inequalities, meter, multiples, remainder, yard
Supporting
Vocabulary
area, composite, formula, perimeter, prime
Resource
My Math Chapter 4 Multiply with One-Digit Numbers (11 days)
● Lesson 1 - Multiples of 10, 100, and 1000
● Lesson 2 - Round to Estimate Products
● Lesson 3 - omit
● Lesson 4 - Hands On: Use Models to Multiply
● Lesson 5 - Multiply by a Two-Digit Number
○ Focus on partial products, area model, or distributive property (not standard algorithm)
● Lesson 6 - omit
● Lesson 7 - The Distributive Property
● Lesson 8 - Multiply with Regrouping
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Natrona County School District Mathematics Curriculum
●
●
●
Lesson 9 - Multiply by a Multi-Digit Number
Lesson 10 - Problem Solve: Estimate or Exact Answer
Lesson 11 - Multiply Across Zeros
My Math Chapter 5 Multiply with Two-Digit Numbers (7 days)
○ Focus on partial products, area model, or distributive property (not standard algorithm)
● Lesson 1 - Multiply by Tens
● Lesson 2 - Estimate Products
● Lesson 3 - Hands On: Use the Distributive Property to Multiply
● Lesson 4 - Multiply by a two-Digit Number
● Lesson 5 - Solve Multi-Step Word Problems
My Math Chapter 6 Divide by One-Digit Numbers (9 days)
○ Focus on partial products, area model, or distributive property (not standard algorithm)
● Lesson 1 - Divide Multiples of 10, 100, and 1,000
● Lesson 5 - Divide with Remainders
● Lesson 6 - Interpret Remainders
● Lesson 8 - Hands On: Distributive Property and Partial Quotients
● Lesson 9 - Divide Greater Numbers
● Lesson 10 - Quotients with Zeros
● Lesson 11 - Solve Multi-Step Word Problems
My Math Chapter 13 Perimeter and Area (7 days)
● Lesson 1 - Measure Perimeter
● Lesson 2 - Problem-Solving Investigations: Solve a Simpler Problem
● Lesson 3 - Hands On: Model Area
● Lesson 4 - Measure Area
● Lesson 5 - Relate Area and Perimeter
34 Days
Pacing Guide Section 3
● Fractions/Decimals
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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.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 1/2. Recognize that comparisons are valid only when the two fractions refer to the
same whole. Record the results of comparisons with symbols >, =, or <, and justify the
conclusions, e.g., by using a visual fraction model.
4.NF.3
Understand a fraction a/b with a > 1 as a sum of fractions 1 / b.
a. Understand addition and subtraction of fractions as joining and separating parts
referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same denominator in more
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Page 46 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
than one way, recording each decomposition by an equation. Justify
decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8
+ 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.
c. Add and subtract mixed numbers with like denominators, e.g., by replacing each
mixed number with an equivalent fraction, and/or by using properties of operations
and the relationship between addition and subtraction.
d. Solve word problems involving addition and subtraction of fractions referring to the
same whole and having like denominators, e.g., by using visual fraction models and
equations to represent the problem.
4.NF.4
Apply and extend previous understandings of multiplication to multiply a fraction by a
whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual fraction
model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the
equation 5/4 = 5 × (1/4).
b. Understand a multiple of a/b as a multiple of 1/b, and use this understanding to
multiply a fraction by a whole number. For example, use a visual fraction model to
express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n ×
(a/b) = (n × a)/b.)
c. Solve word problems involving multiplication of a fraction by a whole number, e.g.,
by using visual fraction models and equations to represent the problem. For
example, if each person at a party will eat 3/8 of a pound of roast beef, and there
will be 5 people at the party, how many pounds of roast beef will be needed?
Between what two whole numbers does your answer lie?
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. (Students
who can generate equivalent fractions can develop strategies for adding fractions with
unlike denominators in general. But addition and subtraction with unlike denominators in
general is not a requirement at this grade.) 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.
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.
NCSD Clarification
4.G.1 – Assessed on NCSD Benchmark 4 but needs to be addressed prior to PAWS
Vocabulary
Critical
Vocabulary
common denominator, decimal, decimal point, equivalent fraction, factors, multiples,
numerator
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Natrona County School District Mathematics Curriculum
Supporting
Vocabulary
denominator, hundredths, improper fractions, mixed numbers
Resource
My Math Chapter 8 Fractions (11 days)
● Lesson 1 - Factors and Multiples
● Lesson 2 - Prime and Composite Numbers
● Lesson 3 - Hands On: Model Equivalent Fractions
● Lesson 4 - Equivalent Fractions
● Lesson 5 - Simplest Form
○ Use models
● Lesson 6 - Compare and Order Fractions
● Lesson 7 - Use Benchmark Fractions to Compare and Order
● Lesson 9 - Mixed Numbers
● Lesson 10 - Mixed Numbers and Improper Fractions
My Math Chapter 9 Operations with Fractions (10 days)
● Lesson 1 - Hands On: Use Models to Add Like Fractions
● Lesson 2 - Add Like Fractions
● Lesson 3 - Hands On: Use Models to Subtract Like Fractions
● Lesson 4 - Subtract Like Fractions
● Lesson 6 - Add Mixed Numbers
● Lesson 7 - Subtract Mixed Numbers
● Lesson 8 - Hands On: Model Fractions and Multiplication
● Lesson 9 - Multiply Fractions by Whole Numbers
My Math Chapter 10 Fractions and Decimals (10 days)
● Lesson 1 - Hands On: Place Value Through Tenths and Hundredths
● Lesson 2 - Tenths
● Lesson 3 - Hundredths
● Lesson 4 - Hands On: Model Decimals and Fractions
● Lesson 5 - Decimals and Fractions
● Lesson 6 - Use Place Value and Models to Add
● Lesson 7 - Compare and Order Decimals
● Lesson 8 - Problem-Solving Investigation: Extra or Missing Information
Mini Lessons on 4.G.3 (3 days)
● Standard 4.G.3 will not be on the benchmark, but it needs to be addressed prior to PAWS.
43 Days
Pacing Guide Section 4
● Measurement
● Geometry
● Patterns and Sequences
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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.
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
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Natrona County School District Mathematics Curriculum
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.4
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8).
Solve problems involving addition and subtraction of fractions by using information
presented in line plots. For example, from a line plot find and interpret the difference in
length between the longest and shortest specimens in an insect collection.
4.MD.5
Recognize angles as geometric shapes that are formed wherever two rays share a common
endpoint, and understand concepts of angle measurement:
a. An angle is measured with reference to a circle with its center at the common
endpoint of the rays, by considering the fraction of the circular arc between the
points where the two rays intersect the circle. An angle that turns through 1/360 of a
circle is called a “one-degree angle,” and can be used to measure angles.
b. An angle that turns through n one-degree angles is said to have an angle measure of
n degrees.
4.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.
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.
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.
Vocabulary
Critical
Vocabulary
acute angle, angle, endpoint (vertex), line of symmetry, meter, obtuse angle, right angle,
yard
Supporting
Vocabulary
capacity, centimeter, cup, customary system, degree, gallon, gram, hour, intersecting lines,
interval (time), kilogram, line, line plot, line segment, liter, metric system, milliliter,
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Natrona County School District Mathematics Curriculum
minute, ounce, parallel lines, pattern, perpendicular lines, pint, point, pound, protractor,
quart, ray
Resource
My Math Chapter 11 Customary Measurement (12 days)
● Lesson 1 - Customary Units of Length
● Lesson 2 - Convert Customary Units of Length
● Lesson 3 - Customary Units of Capacity
● Lesson 4 - Convert Customary Units of Capacity
● Lesson 5 - Customary Units of Weight
● Lesson 6 - Convert Customary Units of Weight
● Lesson 7 - Convert Units of Time
● Lesson 8 - Display Measurement Data in a Line Plot
● Lesson 9 - Solve Measurement Problems (More lessons over line plot are needed.)
● Lesson 10 - Problem-Solving Investigation: Guess, Check, and Revise
My Math Chapter 12 Metric Measurement (7 days)
● Lesson 1 - Metric Units of Length
● Lesson 2 - Metric Units of Capacity
● Lesson 3 - Metric Units of Mass
● Lesson 5 - Convert Metric Units
● Lesson 6 - Solve Measurement Problems
My Math Chapter 14 Geometry (13 days)
● Lesson 1 - Draw Points, Lines, and Rays
● Lesson 2 - Draw Parallel and Perpendicular Lines
● Lesson 3 - Hands On: Model Angles
● Lesson 4 - Classify Angles
● Lesson 5 - Measure Angles
● Lesson 6 - Draw Angles
● Lesson 7 - Solve Problems with Angles
● Lesson 8 - Triangles
● Lesson 9 - Quadrilaterals
● Lesson 10 - Draw Lines of Symmetry
● Lesson 11 - Problem-Solving Investigation: Make a Model
My Math Chapter 7 Patterns and Sequences (11 days)
● Lesson 1 - Non-numeric Patterns
● Lesson 2 - Numeric Patterns
● Lesson 3 - Sequences
● Lesson 4 - Problem-Solving Investigation: Look for a Pattern
● Lesson 5 - Addition and Subtraction Rules
● Lesson 6 - Multiplication and Division Rules
● Lesson 7 - Order of Operations
● Lesson 8 - Hands On: Equations with Two Operations
● Lesson 9 - Equations with Multiple Operations
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Natrona County School District Mathematics Curriculum
5th Grade Math Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
42 Days
Pacing Guide Section 1
● Place Value
● Multiplication and Division
Clarification
Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
Understand the place value system.
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.
a. Read and write decimals to thousandths using base-ten numerals, number names,
and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 ×
(1/100) + 2 × (1/1000).
b. Compare two decimals to thousandths based on meanings of the digits in each
place, using >, =, and < symbols to record the results of comparisons.
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.
Vocabulary
Critical
Vocabulary
braces, brackets, composite, cubed, dividend, divisor, expression, factors, hundredths,
parentheses, prime factorization, product, quotient, rule, squared, standard form, tenths,
thousandths
Supporting
Vocabulary
algorithm, base, decimal point (and), expanded form, inequalities, number line, remainder
Resource
My Math Chapter 1 Place Value (9 days)
● Lesson 1 - Place Value Through Millions
● Lesson 2 - Compare and Order Whole Number Through Millions
● Lesson 3 - Hands On: Model Fractions and Decimals
● Lesson 4 - Represent Decimals
● Lesson 5 - Hands On: Understand Place Value
● Lesson 6 - Place Value Through Thousandths
● Lesson 7 - Compare Decimals
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Natrona County School District Mathematics Curriculum
●
●
Lesson 8 - Order Whole Numbers and Decimals
Lesson 9 - Problem-Solving Investigation: Use the Four-Step Plan
My Math Chapter 2 Multiply Whole Numbers (10 days)
● Lesson 1 - Prime Factorization
● Lesson 2 - Hands On: Prime Factorization Patterns
● Lesson 3 - Powers and Exponents
● Lesson 4 - Multiplication Patterns
● Lesson 5 - Problem-Solving Investigation: Make a Table
● Lesson 6 - Hands On: Use Partial Products and the Distributive Property
● Lesson 7 - The Distributive Property
● Lesson 8 - Estimate Products
● Lesson 9 - Multiply by One-Digit Numbers
● Lesson 10 - Multiply by Two-Digit Numbers
My Math Chapter 3 Divide by a One-Digit Divisor (13 days)
● Lesson 1 - Relate Division to Multiplication
● Lesson 2 - Hands On: Division Models
● Lesson 3 - Two-Digit Dividends
● Lesson 4 - Division Patterns
● Lesson 5 - Estimate Quotients
● Lesson 6 - Hands On: Division Models with Greater Numbers
● Lesson 7 - Hands On: Distributive Property and Partial Quotients
● Lesson 8 - Divide Three- and Four-Digit Dividends
● Lesson 9 - Place the First Digit
● Lesson 10 - Quotients with Zeros
● Lesson 11 - Hands On: Use Models to Interpret the Remainder
● Lesson 12 - Interpret the Remainder
● Lesson 13 - Problem-Solving Investigation: Determine Extra or Missing Information
My Math Chapter 4 Divide by a Two-Digit Divisor (6 days)
● Lesson 1 - Estimate Quotients
● Lesson 2 - Hands On: Divide Using Base-Ten Blocks
● Lesson 3 - Divide by a Two-Digit Divisor
● Lesson 4 - Adjust Quotients
● Lesson 5 - Divide Greater Numbers
● Lesson 6 - Problem-Solving Investigation: Solve a Simpler Problem
38 Days
Pacing Guide Section 2
● Algebraic Thinking, Introduction to Fractions and Decimals
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
5.OA.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions
with these symbols.
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.
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
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Page 52 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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.
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.
NCSD clarification
5.NBT.7 - Add and subtract decimals will be assessed.
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.5
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of
the other factor, without performing the indicated multiplication.
NCSD clarification
5.NF.5 – Only 5.NF.5a addressed in this section of the pacing guide
Vocabulary
Critical
Vocabulary
braces, brackets, dividend, divisor, expression, factors, greatest common factor (GCF),
hundredths, least common denominator, least common multiple, mixed numbers,
parentheses, quotient, rule, scaling, simplest form, standard form, tenths, thousandths
Supporting
Vocabulary
algorithm, decimal point (and), denominator, expanded form, inequalities, multiples,
number line, numerator
Resources
My Math Chapter 7 Expressions and Patterns (6 days)
● Lesson 1 - Hands On: Numerical Expressions
● Lesson 2 - Order of Operations
● Lesson 3 - Write Numerical Expressions
● Lesson 4 - Problem-Solving Investigation: Work Backwards
● Lesson 5 - Hands On: Generate Patterns
● Lesson 6 - Patterns
My Math Chapter 5 Add and Subtract Decimals (10 days)
● Lesson 1 - Round Decimals
● Lesson 2 - Estimate Sums and Differences
● Lesson 3 - Problem-Solving Investigation: Estimate or Exact Answer
● Lesson 4 - Hands On: Add Decimals Using Base-Ten Blocks
● Lesson 5 - Hands On: Add Decimals Using Models
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Natrona County School District Mathematics Curriculum
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Lesson 6 - Add Decimals
Lesson 7 - Addition Properties
Lesson 8 - Hands On: Subtract Decimals Using Base-Ten Blocks
Lesson 9 - Hands On: Subtract Decimals Using Models
Lesson 10 - Subtract Decimals
My Math Chapter 6 Multiply and Divide Decimals (14 days)
● Lesson 1 - Estimate Products of Whole Numbers and Decimals
● Lesson 2 - Hands On: Use Models to Multiply
● Lesson 3 - Multiply Decimals by Whole Numbers
● Lesson 4 - Hands On: Use Models to Multiply Decimals
● Lesson 5 - Multiply Decimals
● Lesson 6 - Multiply Decimals by Powers of Ten
● Lesson 7 - Problem-Solving Investigation: Look for a Pattern
● Lesson 8 - Multiplication Properties
● Lesson 9 - Estimate Quotients
● Lesson 10 - Hands On: Divide by Decimals
● Lesson 11 - Divide Decimals by Whole Numbers
● Lesson 12 - Hands On: Use Models to Divide Decimals
● Lesson 13 - Divide Decimals
● Lesson 14 - Divide Decimals by Powers of Ten
My Math Chapter 8 Fractions and Decimals (8 days)
● Lesson 1 - Fractions and Division
● Lesson 2 - Greatest Common Factor
● Lesson 3 - Simplest Form
● Lesson 4 - Problem-Solving Investigation: Guess, Check, and Revise
● Lesson 5 - Least Common Multiple
● Lesson 6 - Compare Fractions
● Lesson 7 - Hands On: Use Models to Write Fractions as Decimals
● Lesson 8 - Write Fractions as Decimals
34 Days
Pacing Guide Section 3 All fraction computation in this section (+ - * /)
● Fraction Operations
● Volume
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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
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Natrona County School District Mathematics Curriculum
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.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts;
equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a
visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this
equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares
of the appropriate unit fraction side lengths, and show that the area is the same as
would be found by multiplying the side lengths. Multiply fractional side lengths to
find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.5
Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of
the other factor, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a
product greater than the given number (recognizing multiplication by whole
numbers greater than 1 as a familiar case); explaining why multiplying a given
number by a fraction less than 1 results in a product smaller than the given number;
and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of
multiplying a/b by 1.
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. (Students able to multiply fractions in
general can develop strategies to divide fractions in general, by reasoning about the
relationship between multiplication and division. But division of a fraction by a fraction is
not a requirement at this grade.)
a. Interpret division of a unit fraction by a non-zero whole number, and compute such
quotients. For example, create a story context for (1/3) ÷ 4, and use a visual
fraction model to show the quotient. Use the relationship between multiplication and
division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients.
For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to
show the quotient. Use the relationship between multiplication and division to
explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole
numbers and division of whole numbers by unit fractions, e.g., by using visual
fraction models and equations to represent the problem. For example, how much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How
many 1/3-cup servings are in 2 cups of raisins?
5.MD.1
Convert among different-sized standard measurement units within a given measurement
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Natrona County School District Mathematics Curriculum
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.
a. A cube with side length 1 unit, called a “unit cube,” is said to have “one cubic unit”
of volume, and can be used to measure volume.
b. A solid figure which can be packed without gaps or overlaps using n unit cubes is
said to have a volume of n cubic units.
5.MD.4
Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised
units.
5.MD.5
Relate volume to the operations of multiplication and addition and solve real world and
mathematical problems involving volume.
a. Find the volume of a right rectangular prism with whole-number side lengths by
packing it with unit cubes, and show that the volume is the same as would be found
by multiplying the edge lengths, equivalently by multiplying the height by the area
of the base. Represent threefold whole-number products as volumes, e.g., to
represent the associative property of multiplication.
b. Apply the formulas V=l×w×h and V=b×h for rectangular prisms to find volumes of
right rectangular prisms with whole number edge lengths in the context of solving
real world and mathematical problems.
c. Recognize volume as additive. Find volumes of solid figures composed of two nonoverlapping right rectangular prisms by adding the volumes of the non- overlapping
parts, applying this technique to solve real world problems.
Vocabulary
Critical
Vocabulary
cubed, expression, improper fractions, mixed numbers, rule, simplest form, volume
Supporting
Vocabulary
composite figure, cubic, denominator, height, measure, number line, numerator
Resources
My Math Chapter 9 Add and Subtract Fractions (13 days)
● Lesson 1 - Round Fractions
● Lesson 2 - Add Like Fractions
● Lesson 3 - Subtract Like Fractions
● Lesson 4 - Hands On: Use Models to Add Unlike Fractions
● Lesson 5 - Add Unlike Fractions
● Lesson 6 - Hands On: Use Models to Subtract Unlike Fractions
● Lesson 7 - Subtract Unlike Fractions
● Lesson 8 - Problem-Solving Investigation: Determine Reasonable Answers
● Lesson 9 - Estimate Sums and Differences
● Lesson 10 - Hands On: Use Models to Add Mixed Numbers
● Lesson 11 - Add Mixed Numbers
● Lesson 12 - Subtract Mixed Numbers
● Lesson 13 - Subtract with Renaming
My Math Chapter 10 Multiply and Divide Fractions (12 days)
● Lesson 1 - Hands On: Part of a Number
● Lesson 2 - Estimate Products of Fractions
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Natrona County School District Mathematics Curriculum
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Lesson 3 - Hands On: Model Fraction Multiplication
Lesson 4 - Multiply Whole Numbers and Fractions
Lesson 5 - Hands On: Use Models to Multiply Fractions
Lesson 6 - Multiply Fractions
Lesson 7 - Multiply Mixed Numbers
Lesson 8 - Hands On: Multiplication as Scaling
Lesson 9 - Hands On: Division with Unit Fractions
Lesson 10 - Divide Whole Numbers by Unit Fractions
Lesson 11 - Divide Unit Fractions by Whole Numbers
Lesson 12 - Problem-Solving Investigation: Draw a Diagram
My Math Chapter 12 Geometry (5 days)
● Lesson 8 - Hands On: Use Models to Find Volume
● Lesson 9 - Volume of Prisms
● Lesson 10 - Hands On: Build Composite Figures
● Lesson 11 - Volume of Composite Figures
● Lesson 12 - Problem-Solving Investigation: Make a Model
23 Days
Pacing Guide Section 4
● Measurement
● Data
● Geometry
Clarification Based on pre and post testing, make decisions to skip lessons and / or add materials.
Standards
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.
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.
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.
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Natrona County School District Mathematics Curriculum
Vocabulary
Critical
Vocabulary
capacity, convert, coordinate plane, customary system, metric system, ordered pairs, plane,
quadrant, rule, scalene triangle, scaling, solid figure, vertex, volume, x-axis, x-coordinate, yaxis, y-coordinate
Supporting
Vocabulary
acute angle, area, axis, composite figure, cube, cubic, edges, equilateral triangle, faces,
height, intersecting lines, isosceles triangle, kilo, line plot, measure, measurement unit,
milli, obtuse angle, obtuse triangle, parallel, parallelogram, perimeter, polygon,
quadrilateral, rhombus, right angle, trapezoid
Resource
My Math Chapter 7 Expressions and Patterns (2 days)
● Lesson 8 - Ordered Pair
● Lesson 9 - Graph Patterns
My Math Chapter 11 Measurement (10 days)
● Lesson 1 - Hands On: Measure with a Ruler
● Lesson 2 - Convert Customary Units of Length
● Lesson 3 - Problem-Solving Investigation: Use Logical Reasoning
● Lesson 4 - Hands On: Estimate and Measure Weight
● Lesson 5 - Convert Customary Units of Weight
● Lesson 6 - Hands On: Estimate and Measure Capacity
● Lesson 7 - Convert Customary Units of Capacity
● Lesson 8 - Display Measurement Data on a Line Plot
● Lesson 9 - Hands On: Metric Rulers
● Lesson 10 - Convert Metric Units of Length
My Math chapter 12 Geometry (7 days)
● Lesson 1 - Polygons
● Lesson 2 - Hands On: Sides and Angles of Triangles
● Lesson 3 - Classify Triangles
● Lesson 4 - Hands On: Sides and Angles of Quadrilaterals
● Lesson 5 - Classify Quadrilaterals
● Lesson 6 - Hands On: Build Three-Dimensional Figures
● Lesson 7 - Three-Dimensional Figures
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Natrona County School District Mathematics Curriculum
Course 1 Math (6th Grade) Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
40 Days
Pacing Guide Section 1: The Number System
● Compute with Multi-Digit Numbers
● Multiply and Divide Fractions
Standards
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 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.)
How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?
How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip
of land with length 3/4 mi and area 1/2 square mi?
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.
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 two whole numbers with no common factor. For example, express 36
+ 8 as 4 (9 + 2).
6.NS.5
Understand that positive and negative numbers are used together to describe quantities
having opposite directions or values (e.g., temperature above/ below zero, elevation
above/below sea level, credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts, explaining the meaning of
0 in each situation.
6.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.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0
on the number line; recognize that the opposite of the opposite of a number is the
number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants
of the coordinate plane; recognize that when two ordered pairs differ only by signs,
the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical
number line diagram; find and position pairs of integers and other rational numbers
on a coordinate plane.
6.NS.7
Understand ordering and absolute value of rational numbers.
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Page 59 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
a. Interpret statements of inequality as statements about the relative position of two
numbers on a number line diagram. For example, interpret –3 > –7 as a statement
that –3 is located to the right of –7 on a number line oriented from left to right.
b. Write, interpret, and explain statements of order for rational numbers in real-world
contexts. For example, write –3 oC > –7 oC to express the fact that –3 oC is
warmer than –7 oC.
c. Understand the absolute value of a rational number as its distance from 0 on the
number line; interpret absolute value as magnitude for a positive or negative
quantity in a real-world situation. For example, for an account balance of –30
dollars, write |–30| = 30 to describe the size of the debt in dollars.
d. Distinguish comparisons of absolute value from statements about order. For
example, recognize that an account balance less than –30 dollars represents a debt
greater than 30 dollars.
6.NS.8
Solve real-world and mathematical problems by graphing points in all four quadrants of
the coordinate plane. Include use of coordinates and absolute value to find distances
between points with the same first coordinate or the same second coordinate.
Vocabulary
Critical
Vocabulary
bar notation, exponents
Supporting
Vocabulary
difference, estimate, factor, multiple, product, quotient, rational number, simplest form,
sum
Resource
Glencoe Math Course 1 Chapter 3 Compute with Multi-Digit Numbers (20 days)
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Lesson 1 - Add and Subtract Decimals
Lesson 2 - Estimate Products
Lesson 3 - Multiply Decimals by Whole Numbers
Lesson 4 - Multiply Decimals by Decimals
Lesson 5 - Divide Multi-Digit Numbers
Lesson 6 - Estimate Quotients
Lesson 7 - Divide Decimals by Whole Numbers
Lesson 8 - Divide Decimals by Decimals
Glencoe Math Course 1 Chapter 4 Multiply and Divide Fraction (20 days)
●
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●
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●
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Lesson 1 - Estimate Products of Fractions
Lesson 2 - Multiply Fractions and Whole Numbers
Lesson 3 - Multiply Fractions
Lesson 4 - Multiply Mixed Numbers
Lesson 5 - Convert Measurement Units
Lesson 6 - Divide Whole Numbers by Fractions
Lesson 7 - Divide Fractions
Lesson 8 - Divide Mixed Numbers
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Page 60 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
40 Days
Pacing Guide Section 2
● Ratios and Proportional Relationships
● Fractions, Decimals and Percent
Standards
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 ≠ 0, and use
rate language in the context of a ratio relationship. For example, “This recipe has a
ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of
sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”
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.
a. Make tables of equivalent ratios relating quantities with whole number
measurements, find missing values in the tables, and plot the pairs of values on the
coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed.
For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns
could be mowed in 35 hours? At what rate were lawns being mowed.
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100
times the quantity); solve problems involving finding the whole, given a part and
the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units
appropriately when multiplying or dividing quantities.
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 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.)
How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally?
How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip
of land with length 3/4 mi and area 1/2 square mi?
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 two whole numbers with no common factor. For example, express 36
+ 8 as 4 (9 + 2).
Vocabulary
Critical
Vocabulary
equivalent ratios, percent, ratio, scaling
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Page 61 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Supporting
Vocabulary
estimate, factor, multiple, ordered pair, proportion, unit rate
Resource
Glencoe Math Course 1 Chapter 1 Ratios and Rates (18 days)
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Lesson 1 - Factors and Multiples
Lesson 2 - Ratios
Lesson 3 - Rates
Lesson 4 - Ratio Tables
Lesson 5 - Graph Ratio Tables
Lesson 6 - Equivalent Ratios
Lesson 7 - Ratio and Rate Problems
Glencoe Math Course 1 Chapter 2 Fractions, Decimals, and Percents (22 days)
●
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Lesson 1 - Decimals and Fractions
Lesson 2 - Percent and Fractions
Lesson 3 - Percents and Decimals
Lesson 4 - Omit
Lesson 5 - Compare and Order Fractions, Decimals and Percents
Lesson 6 - Estimate with Percent
Lesson 7 - Percent of a Number
Lesson 8 - Solve Percent Problems
48 Days
Pacing Guide Section 3
● Expressions and Equations
● Functions and Inequalities
● Integers and the Coordinate Plane
Standards
6.NS.5
Understand that positive and negative numbers are used together to describe quantities
having opposite directions or values (e.g., temperature above/ below zero, elevation
above/below sea level, credits/debits, positive/negative electric charge); use positive and
negative numbers to represent quantities in real-world contexts, explaining the meaning of
0 in each situation.
6.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.
a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0
on the number line; recognize that the opposite of the opposite of a number is the
number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
b. Understand signs of numbers in ordered pairs as indicating locations in quadrants
of the coordinate plane; recognize that when two ordered pairs differ only by signs,
the locations of the points are related by reflections across one or both axes.
c. Find and position integers and other rational numbers on a horizontal or vertical
number line diagram; find and position pairs of integers and other rational numbers
on a coordinate plane.
6.NS.7
Understand ordering and absolute value of rational numbers.
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Page 62 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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.
a. Write expressions that record operations with numbers and with letters standing for
numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
b. Identify parts of an expression using mathematical terms (sum, term, product,
factor, quotient, coefficient); view one or more parts of an expression as a single
entity. For example, describe the expression 2 (8 + 7) as a product of two factors;
view (8 + 7) as both a single entity and a sum of two terms.
c. Evaluate expressions at specific values of their variables. Include expressions that
arise from formulas used in real-world problems. Perform arithmetic operations,
including those involving whole number exponents, in the conventional order when
there are no parentheses to specify a particular order (Order of Operations). For
example, use the formulas V = s3 and A = 6 s2 to find the volume and surface area
of a cube with sides of length s = 1/2.
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.
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 of 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 distances and times, and write the
equation d = 65t to represent the relationship between distance and time.
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Page 63 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Vocabulary
Critical
Vocabulary
algebraic expressions, coefficient, dependent variable, distributive property, independent
variable, quadrant, term
Supporting
Vocabulary
absolute value, coordinate plane, equation, evaluate, factor, function, inequality, integers,
ordered pair, simplify
Resource
Glencoe Math Course 1 Chapter 6 Expressions (15 days)
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Lesson 1 - Powers and Exponents
Lesson 2 - Numerical Expressions
Lesson 3 - Algebra: Variables and Expressions
Lesson 4 - Algebra: Write Expressions
Lesson 5 - Algebra:Properties
Lesson 6 - Distributed Property
Lesson 7 - Equivalent Expressions
Glencoe Math Course 1 Chapter 7 Equations (12 days)
●
●
●
●
●
Lesson 1 - Equations
Lesson 2 - Solve and Write Addition Equations
Lesson 3 - Solve and Write Subtraction Equations
Lesson 4 - Solve and Write Multiplication Equations
Lesson 5 - Solve and Write Division Equations
Glencoe Math Course 1 Chapter 8 Functions and Inequalities (15 days)
●
●
●
●
●
●
●
Lesson 1 - Function Tables
Lesson 2 - Function Rules
Lesson 3 - Functions and Equations
Lesson 4 - Multiple Representations of Functions
Lesson 5 - Inequalities
Lesson 6 - Write and Graph Inequalities
Lesson 7 - Solve One-Step Inequalities
Glencoe Math Course 1 Chapter 5 Integers and the Coordinate Plane (6 days)
●
●
●
●
●
●
●
Lesson 1 - Integers and Graphing
Lesson 2 - Absolute Value
Lesson 3 - Compare and Order Integers
Lesson 4 - omit
Lesson 5 - omit
Lesson 6 - Coordinate Plane
Lesson 7 - Graph on the Coordinate Plane
45 Days
Pacing Guide Section 4
● Statistics and Probability
● Area, Volume, and Surface Area
Standards
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
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Page 64 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is
the same as would be found by multiplying the edge lengths of the prism. Apply the
formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional
edge lengths in the context of solving real-world and mathematical problems.
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.
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
which 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:
a. Reporting the number of observations.
b. Describing the nature of the attribute under investigation, including how it was
measured and its units of measurement.
c. Giving quantitative measures of center (median and/or mean) and variability
(interquartile range and/or mean absolute deviation), as well as describing any
overall pattern and any striking deviations from the overall pattern with reference
to the context in which the data were gathered.
d. Relating the choice of measures of center and variability to the shape of the data
distribution and the context in which the data were gathered.
Vocabulary
Critical
composite figure, interquartile range, mean/average, mean absolute deviation, measures of
Vocabulary center, median, mode, outliers, quartiles, range, surface area, vertices/vertex, volume
Supporting trapezoid, dimensions, rectangular prism, square pyramid, slant height, square units, cubic
Vocabulary units, distribution, symmetric
Resource
Glencoe Math Course 1 Chapter 9 Area (11 days)
●
●
●
●
●
●
Lesson 1 - Area of Parallelograms
Lesson 2 - Area of Triangles
Lesson 3 - Area of Trapezoids
Lesson 4 - Changes in Dimension
Lesson 5 - Polygons on the Coordinate Plane
Lesson 6 - Area of Composite Figures
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Page 65 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Glencoe Math Course 1 Chapter 10 Volume and Surface Area (12 days)
●
●
●
●
●
●
●
Lesson 1 - Volume of Rectangular Prisms
Lesson 2 - Volume of Triangular Prisms
Lesson 3 - Surface Area of Rectangular Prisms
Inquiry Lab - Nets of Triangular Prisms
Lesson 4 - Surface Area of Triangular Prisms
Inquiry Lab - Nets of Pyramids
Lesson 5 -Surface Area of Pyramids
Glencoe Math Course 1 Chapter 11 Statistical Measure (11 days)
●
●
●
●
●
Lesson 1 - Mean
Lesson 2 - Median and Mode
Lesson 3 - Measures of Variation
Lesson 4 - Mean absolute Deviation
Lesson 5 - Appropriate Measures
Glencoe Math Course Chapter 12 Statistical Displays (11 days)
●
●
●
●
●
●
Lesson 1 - Line Plots
Lesson 2 - Histograms
Lesson 3 - Box Plots
Lesson 4 - Shape and Data Distributions
Lesson 5 - Interpret Line Graphs
Lesson 6 - Select an Appropriate Display
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Page 66 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Course 2 Math (7th Grade) Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
31 Days
Pacing Guide Section 1
● The Number System
Standards
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.
a. Describe situations in which opposite quantities combine to make 0. For example,
a hydrogen atom has 0 charge because its two constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or
negative direction depending on whether q is positive or negative. Show that a
number and its opposite have a sum of 0 (are additive inverses). Interpret sums of
rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q =
p + (–q). Show that the distance between two rational numbers on the number line
is the absolute value of their difference, and apply this principle in real-world
contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions
to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by
requiring that operations continue to satisfy the properties of operations,
particularly the distributive property, leading to products such as (–1)(–1) = 1 and
the rules for multiplying signed numbers. Interpret products of rational numbers by
describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and
every quotient of integers (with non-zero divisor) is a rational number. If p and q
are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers
by describing real world contexts.
c. Apply properties of operations as strategies to multiply and divide rational
numbers.
d. Convert a rational number to a decimal using long division; know that the decimal
form of a rational number terminates in 0s or eventually repeats.
7.NS.3
Solve real-world and mathematical problems involving the four operations with rational
numbers. (Computations with rational numbers extend the rules for manipulating fractions
to complex fractions.
Vocabulary
Critical
additive inverse, integer, opposites, rational number, repeating decimal, terminating
Vocabulary decimal
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Page 67 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Supporting bar notation
Vocabulary
Resource
Glencoe Math Course 2 Chapter 3 Integers (14 days)
●
●
●
●
●
Lesson 1 - Integers and Absolute Value
Lesson 2 - Add Integers
Lesson 3 - Subtract Integers
Lesson 4 - Multiply Integers
Lesson 5 - Divide Integers
Glencoe Math Course 2 Chapter 4 Rational Numbers (17 days)
●
●
●
●
●
●
●
●
Lesson 1 - Terminating and REpeating Decimals
Lesson 2 - Compare and Order Rational Numbers
Lesson 3 - Add and Subtract Like Fractions
Lesson 4 - Add and Subtract Unlike Fractions
Lesson 5 - Add and Subtract Mixed Numbers
Lesson 6 - Multiply Fractions
Lesson 7 - Convert Between Systems
Lesson 8 - Divide Fractions
41 Days
Pacing Guide Section 2
● Expressions and Equations
Standards
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. For example, a + 0.05a =
1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
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. For example: If a woman making $25 an
hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50,
for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center
of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from
each edge; this estimate can be used as a check on the exact computation.
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.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r,
where p, q, and r are specific rational numbers. Solve equations of these forms
fluently. Compare an algebraic solution to an arithmetic solution, identifying the
sequence of the operations used in each approach. For example, the perimeter of a
rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r,
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Page 68 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
where p, q, and r are specific rational numbers. Graph the solution set of the
inequality and interpret it in the context of the problem. For example: As a
salesperson, you are paid $50 per week plus $3 per sale. This week you want your
pay to be at least $100. Write an inequality for the number of sales you need to
make, and describe the solutions.
Vocabulary
Critical
constant, equivalent expressions, like terms
Vocabulary
Supporting additive identity, algebraic expressions, associative property, coefficient, commutative
Vocabulary property, dependent variable, distributive property, independent variable, factored form,
factoring, linear expression, monomial, simplest form (factoring), solution set
Resource
Glencoe Math Course 2 Chapter 5 Expressions (17 days)
●
●
●
●
●
●
●
●
Lesson 1 - Algebraic Expressions
Lesson 2 - Sequences
Lesson 3 - Properties of Operations
Lesson 4 - The Distributive Property
Lesson 5 - Simplify Algebraic Expressions
Lesson 6 - Add Linear Expressions
Lesson 7 - Subtract Linear Expressions
Lesson 8 - Factor Linear Expressions
Glencoe Math Course 2 Chapter 6 Equations and Inequalities (20 days)
●
●
●
●
●
●
●
●
Lesson 1 - Solve One-Step Addition and Subtraction Equations
Lesson 2 - Solve Multiplication and Division Equations
Lesson 3 - Solve Equations with Rational Coefficients
Lesson 4 - Solve Two-Step Equations
Lesson 5 - More Two-Step Equations
Lesson 6 - Solve Inequalities by Addition or Subtraction
Lesson 7 - Solve Inequalities by Multiplication or Division
Lesson 8 - Solve Two-Step Inequalities
34 Days
Pacing Guide Section 3
● Ratios and Proportional Relationships
Standards
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. For example, if a person walks 1/2
mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per
hour, equivalently 2 miles per hour.
7.RP.2
Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is
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Page 69 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
proportional to the number n of items purchased at a constant price p, the
relationship between the total cost and the number of items can be expressed as t =
pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in
terms of the situation, with special attention to the points (0, 0) and (1, r) where r is
the unit rate.
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.
Vocabulary
Critical
commission, complex fraction, cross products, discount, gratuity/tip, markdown/markup,
Vocabulary percent change, percent decrease/increase, percent equation, percent error, percent
proportion, principle, proportion, proportional, rate, repeating decimal, sales tax, selling
price, simple interest, terminating decimal, unit rate
Supporting constant of variation, constant of proportionality, constant rate of change, direct variation,
Vocabulary non-proportional, slope, equivalent ratios, origin, ratio
Resource
Glencoe Math Course 2 Chapter 1 Ratios and Proportional Reasoning (16 days)
●
●
●
●
●
●
●
●
●
Lesson 1 - Rates
Lesson 2 - Complex Fractions and Unit Rates
Lesson 3 - Convert Unit Rates
Lesson 4 - Proportional and Nonproportional Relationships
Lesson 5 - Graph Proportional Relationships
Lesson 6 - Solve Proportional Relationships
Lesson 7 - Constant Rate of Change
Lesson 8 - Slope
Lesson 9 - Direct Variation
Glencoe Math Course 2 Chapter 2 Percents (18 days)
●
●
●
●
●
●
●
●
Lesson 1 - Percent of a Number
Lesson 2 - Percent and Estimation
Lesson 3 - The Percent Proportion
Lesson 4 - The Percent Equation
Lesson 5 - Percent of Change
Lesson 6 - Sales Tax, Tips, and Markups
Lesson 7 - Discount
Lesson 8 - Financial Literacy: Compound Interest
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Page 70 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
66 Days
Pacing Guide Section 4
● Geometry
● Statistics
● Probability
Standards
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 (freehand, with ruler and protractor, and with technology) 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.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.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.
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.
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. For example, estimate the mean
word length in a book by randomly sampling words from the book; predict the winner of a
school election based on randomly sampled survey data. Gauge how far off the estimate or
prediction might be.
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. For example, the mean height of players on the
basketball team is 10 cm greater than the mean height of players on the soccer team,
about twice the variability (mean absolute deviation) on either team; on a dot plot, the
separation between the two distributions of heights is noticeable.
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. For example,
decide whether the words in a chapter of a seventh-grade science book are generally
longer than the words in a chapter of a fourth-grade science book.
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Page 71 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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. For example, when rolling a number cube 600
times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly
200 times.
7.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.
a. Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For example, if
a student is selected at random from a class, find the probability that Jane will be
selected and the probability that a girl will be selected.
b. Develop a probability model (which may not be uniform) by observing frequencies
in data generated from a chance process. For example, find the approximate
probability that a spinning penny will land heads up or that a tossed paper cup will
land open-end down. Do the outcomes for the spinning penny appear to be equally
likely based on the observed frequencies?
7.SP.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation.
a. Understand that, just as with simple events, the probability of a compound event is
the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized
lists, tables and tree diagrams. For an event described in everyday language (e.g.,
“rolling double sixes”), identify the outcomes in the sample space which compose
the event.
c. Design and use a simulation to generate frequencies for compound events. . For
example, use random digits as a simulation tool to approximate the answer to the
question: If 40% of donors have type A blood, what is the probability that it will
take at least 4 donors to find one with type A blood?
Vocabulary
Critical
adjacent angles, circumference, complementary angles, cross section, diameter, double
Vocabulary box plot, double dot plot, lateral face, lateral surface, pi, radius, semi-circle, slant height,
supplementary angles, vertical angles
Supporting absolute value, biased sample, compound event, convenience sample, dependent events,
Vocabulary experimental probability, fundamental counting principle, independent event, inferences,
invalid, likelihood, measures of variability, outcome, permutation, probability, random,
relative frequency, sample, sample space, simple event, simple random sample,
simulation, statistics, systematic random sample, theoretical probability, tree diagram,
unbiased sample, valid, voluntary response sample, measures of center, coplaner, straight
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Page 72 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
angles, acute triangle, equilateral triangle, isosceles triangle, obtuse triangles, scalene
triangle
Resource
Glencoe Math Course 2 Chapter 7 Geometric Figures (15 days)
●
●
●
●
●
●
Lesson 1 - Classify Angles
Lesson 2 - Complementary and Supplementary Angles
Lesson 3 - Triangles
Lesson 4 - Scale Drawings
Lesson 5 - Draw Three Dimensional Figures
Lesson 6 - Cross Sections
Glencoe Math Course 2 Chapter 8 Measure Figures (21 days)
●
●
●
●
●
●
●
●
Lesson 1 - Circumference
Lesson 2 - Area of Circles
Lesson 3 - Area of Composite Figures
Lesson 4 - Volume of Prisms
Lesson 5 - Volume of Pyramids
Lesson 6 - Surface Area of Prisms
Lesson 7 - Surface Area of Pyramids
Lesson 8 - Volume and Surface Area of Composite Figures
Glencoe Math Course 2 Chapter 9 Probability (15 days)
●
●
●
●
●
●
●
Lesson 1 - Probability of Simple Events
Lesson 2 - Theoretical and Experimental Probability
Lesson 3 - Probability of Compound Events
Lesson 4 - Simulations
Lesson 5 - Fundamental Counting Principle
Lesson 6 - Permutations
Lesson 7 - Independent and Dependent Events
Glencoe Math Course 2 Chapter 10 Statistics (15 days)
●
●
●
●
●
Lesson 1 - Make Predictions
Lesson 2 - Unbiased and Biased Samples
Lesson 3 - Misleading Graphs and Statistics
Lesson 4 - Compare Populations
Lesson 5 - Select an Appropriate Display
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Page 73 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Accelerated Math (7th Grade) Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
48 Days
Pacing Guide Section 1
● The Number System, Expressions, Equations
Standards
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.
a. Describe situations in which opposite quantities combine to make 0. For example,
a hydrogen atom has 0 charge because its two constituents are oppositely charged.
b. Understand p + q as the number located a distance |q| from p, in the positive or
negative direction depending on whether q is positive or negative. Show that a
number and its opposite have a sum of 0 (are additive inverses). Interpret sums of
rational numbers by describing real-world contexts.
c. Understand subtraction of rational numbers as adding the additive inverse, p – q =
p + (–q). Show that the distance between two rational numbers on the number line
is the absolute value of their difference, and apply this principle in real-world
contexts.
d. Apply properties of operations as strategies to add and subtract rational numbers.
7.NS.2
Apply and extend previous understandings of multiplication and division and of fractions
to multiply and divide rational numbers.
a. Understand that multiplication is extended from fractions to rational numbers by
requiring that operations continue to satisfy the properties of operations,
particularly the distributive property, leading to products such as (–1)(–1) = 1 and
the rules for multiplying signed numbers. Interpret products of rational numbers by
describing real-world contexts.
b. Understand that integers can be divided, provided that the divisor is not zero, and
every quotient of integers (with non-zero divisor) is a rational number. If p and q
are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers
by describing real world contexts.
c. Apply properties of operations as strategies to multiply and divide rational
numbers.
d. Convert a rational number to a decimal using long division; know that the decimal
form of a rational number terminates in 0s or eventually repeats.
7.NS.3
Solve real-world and mathematical problems involving the four operations with rational
numbers. (Computations with rational numbers extend the rules for manipulating fractions
to complex fractions.
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.
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Page 74 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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
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 an integer power of 10 to
estimate very large or very small quantities, and to express how many times as much one
is than the other. For example, estimate the population of the United States as 3 × 108 and
the population of the world as 7 × 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.
Vocabulary
Critical
additive inverse, constant, equivalent expressions, integer, like terms, opposites, rational
Vocabulary number, repeating decimal, terminating decimal
Supporting additive identity, algebraic expressions, associative property, bar notation, coefficient,
Vocabulary commutative property, dependent variable, distributive property, independent variable,
factored form, factoring, linear expression, monomial, simplest form (factoring), solution
set
Resource
Glencoe Math 7th Accelerated Chapter 1 The Language of Algebra (12 days)
●
●
●
●
●
●
●
Lesson 1 - A Plan for Problem Solving
Lesson 2 - Words and Expressions
Lesson 3 - Variables and Expressions
Lesson 4 - Properties of Numbers
Lesson 5 - Problem-Solving Strategies
Lesson 6 - Ordered Pairs and Relations
Lesson 7 - Words, Equations, Tables and Graphs
Glencoe Math 7th Accelerated Chapter 2 Operations with Integers (13 days)
●
●
●
●
●
●
Lesson 1 - Integers and Absolute Value
Lesson 2 - Adding Integers
Lesson 3 - Subtracting Integers
Lesson 4 - Multiplying Integers
Lesson 5 - Dividing Integers
Lesson 6 - Graphing in Four Quadrants
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Page 75 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Glencoe Math 7th Accelerated Chapter 3 Operations with Rational Numbers (11 days)
●
●
●
●
●
●
Lesson 1 - Fractions and Decimals
Lesson 2 - Rational Numbers
Lesson 3 - Multiplying Rational Numbers
Lesson 4 - Dividing Rational Numbers
Lesson 5 - Adding and Subtracting Like Fractions
Lesson 6 - Adding and Subtracting Unlike Fractions
Glencoe Math 7th Accelerated Chapter 4 Powers and Roots (12 days)
●
●
●
●
●
●
●
Lesson 1 - Powers and Exponents
Lesson 2 - Negative Exponents
Lesson 3 - Multiplying and Dividing Monomials
Lesson 4 - Scientific Notation
Lesson 5 - Compute with Scientific Notation
Lesson 6 - Square Roots and Cube Roots
Lesson 7 - The Real Number System
30 Days
Pacing Guide Section 2
● Ratios & Proportions
Standards
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. For example, if a person walks 1/2
mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per
hour, equivalently 2 miles per hour.
7.RP.2
Recognize and represent proportional relationships between quantities.
a. Decide whether two quantities are in a proportional relationship, e.g., by testing for
equivalent ratios in a table or graphing on a coordinate plane and observing
whether the graph is a straight line through the origin.
b. Identify the constant of proportionality (unit rate) in tables, graphs, equations,
diagrams, and verbal descriptions of proportional relationships.
c. Represent proportional relationships by equations. For example, if total cost t is
proportional to the number n of items purchased at a constant price p, the
relationship between the total cost and the number of items can be expressed as t =
pn.
d. Explain what a point (x, y) on the graph of a proportional relationship means in
terms of the situation, with special attention to the points (0, 0) and (1, r) where r is
the unit rate.
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.
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.
Vocabulary
Critical
commission, complex fraction, cross products, discount, gratuity/tip, markdown/markup,
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Page 76 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Vocabulary percent change, percent decrease/increase, percent equation, percent error, percent
proportion, principle, proportion, proportional, rate, repeating decimal, sales tax, selling
price, simple interest, terminating decimal, unit rate
Supporting constant of variation, constant of proportionality, constant rate of change, direct variation,
Vocabulary non-proportional, slope, equivalent ratios, origin, ratio
Resource
Glencoe Math 7th Accelerated Chapter 5 Ratios, Proportions, and Similar Figures (15 days)
●
●
●
●
●
●
●
●
●
●
Lesson 1 - Ratios
Lesson 2 - Unit Rates
Lesson 3 - Complex Fractions and Unit Rates
Lesson 4 - Converting Rates
Lesson 5 - Proportional and NonProportional Relationships
Lesson 6 - Graphing Proportional Relationships
Lesson 7 - Solving Proportions
Lesson 8 - Scale Drawings and Models
Lesson 9 - Similar Figures
Lesson 10 - Indirect Measurement
Glencoe Math 7th Accelerated Chapter 6 Percents (12 days)
●
●
●
●
●
●
Lesson 1 - Using the Percent Proportion
Lesson 2 - Find Percent of a Number Mentally
Lesson 3 - Using the Percent Equation
Lesson 4 - Percent of Change
Lesson 5 - Discount and Markup
Lesson 6 - Simple and Compound Interest
42 Days
Pacing Guide Section 3
● Expressions
● Equations
● Linear Functions
Standards
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. For example, a + 0.05a =
1.05a means that “increase by 5%” is the same as “multiply by 1.05.”
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. For example: If a woman making $25 an
hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50,
for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center
of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from
each edge; this estimate can be used as a check on the exact computation.
7.EE.4
Use variables to represent quantities in a real-world or mathematical problem, and
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Page 77 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
construct simple equations and inequalities to solve problems by reasoning about the
quantities.
a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r,
where p, q, and r are specific rational numbers. Solve equations of these forms
fluently. Compare an algebraic solution to an arithmetic solution, identifying the
sequence of the operations used in each approach. For example, the perimeter of a
rectangle is 54 cm. Its length is 6 cm. What is its width?
b. Solve word problems leading to inequalities of the form px + q > r or px + q < r,
where p, q, and r are specific rational numbers. Graph the solution set of the
inequality and interpret it in the context of the problem. For example: As a
salesperson, you are paid $50 per week plus $3 per sale. This week you want your
pay to be at least $100. Write an inequality for the number of sales you need to
make, and describe the solutions.
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.
a. Give examples of linear equations in one variable with one solution, infinitely
many solutions, or no solutions. Show which of these possibilities is the case by
successively transforming the given equation into simpler forms, until an
equivalent equation of the form x = a, a = a, or a = b results (where a and b are
different numbers).
b. 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.
8.EE.8
Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection
satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspection. For
example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot
simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two
variables. For example, given coordinates for two pairs of points, determine
whether the line through the first pair of points intersects the line through the
second pair.
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.
8.F.2
Compare properties 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
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Page 78 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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.
Vocabulary
Critical
constant, equivalent expressions, like terms
Vocabulary
Supporting additive identity, algebraic expressions, associative property, coefficient, commutative
Vocabulary property, dependent variable, distributive property, independent variable, factored form,
factoring, linear expression, monomial, simplest form (factoring), solution set
Resource
Glencoe Math 7th Accelerated Chapter 7 Algebraic Expressions (10 days)
●
●
●
●
●
Lesson 1 - The Distributive Property
Lesson 2 - Simplifying Algebraic Expressions
Lesson 3 - Adding Linear Expressions
Lesson 4 - Subtracting Linear Expressions
Lesson 5 - Factoring Linear Expressions
Glencoe Math 7th Accelerated Chapter 8 Equations and Inequalities (14 days)
●
●
●
●
●
●
●
●
Lesson 1 - Solving Equations with Rational Coefficients
Lesson 2 - Solving Two-Step Equations
Lesson 3 - Writing Equations
Lesson 4 - More Two-Step Equations
Lesson 5 - Solving Equations with Variables on Each Side
Lesson 6 - Inequalities
Lesson 7 - Solving Inequalities
Lesson 8 - Solving Multi-Step Equations and Inequalities
Glencoe Math 7th Accelerated Chapter 9 Linear Functions (12 days)
●
●
●
●
●
●
●
Lesson 1 - Functions
Lesson 2 - Representing Linear Functions
Lesson 3 - Constant Rate of Change and Slope
Lesson 4 - Direct Variation
Lesson 5 - Slope-Intercept Form
Lesson 6 - Solve Systems of Equations by Graphing
Lesson 7 - Solve Systems of Equations Algebraically
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Page 79 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
45 Days
Pacing Guide Section 4
● Geometry
● Pythagorean Theorem
● Statistics & Probability
Standards
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 (freehand, with ruler and protractor, and with technology) 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.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.5
Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.
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.
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. For example, estimate the mean
word length in a book by randomly sampling words from the book; predict the winner of a
school election based on randomly sampled survey data. Gauge how far off the estimate or
prediction might be.
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. For example, the mean height of players on the
basketball team is 10 cm greater than the mean height of players on the soccer team,
about twice the variability (mean absolute deviation) on either team; on a dot plot, the
separation between the two distributions of heights is noticeable.
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. For example,
decide whether the words in a chapter of a seventh-grade science book are generally
longer than the words in a chapter of a fourth-grade science book.
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Page 80 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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. For example, when rolling a number cube 600
times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly
200 times.
7.SP.8
Find probabilities of compound events using organized lists, tables, tree diagrams, and
simulation.
a. Understand that, just as with simple events, the probability of a compound event is
the fraction of outcomes in the sample space for which the compound event occurs.
b. Represent sample spaces for compound events using methods such as organized
lists, tables and tree diagrams. For an event described in everyday language (e.g.,
“rolling double sixes”), identify the outcomes in the sample space which compose
the event.
c. Design and use a simulation to generate frequencies for compound events. For
example, use random digits as a simulation tool to approximate the answer to the
question: If 40% of donors have type A blood, what is the probability that it will
take at least 4 donors to find one with type A blood?
8.G.1
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken 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.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
angle-angle criterion for similarity of triangles. For example, arrange three copies of the
same triangle so that the sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
8.G.6
Explain a proof of the Pythagorean Theorem and its converse.
NCSD Clarification – 8.G.6
The 7th Grade Accelerated McGraw-Hill resource does not address Pythagorean Theorem.
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Natrona County School District Mathematics Curriculum
Utilize other resources to address these standards.
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.
NCSD Clarification – 8.G.7
The 7th Grade Accelerated McGraw-Hill resource does not address Pythagorean Theorem.
Utilize other resources to address these standards.
8.G.8
Apply the Pythagorean Theorem to find the distance between two points in a coordinate
system.
NCSD Clarification – 8.G.8
The 7th Grade Accelerated McGraw-Hill resource does not address Pythagorean Theorem.
Utilize other resources to address these standards.
8.G.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
Vocabulary
Critical
adjacent angles, circumference, complementary angles, cross section, diameter, double
Vocabulary box plot, double dot plot, lateral face, lateral surface, pi, radius, semi-circle, slant height,
supplementary angles, vertical angles
Supporting absolute value, biased sample, compound event, convenience sample, dependent events,
Vocabulary experimental probability, fundamental counting principle, independent event, inferences,
invalid, likelihood, measures of variability, outcome, permutation, probability, random,
relative frequency, sample, sample space, simple event, simple random sample,
simulation, statistics, systematic random sample, theoretical probability, tree diagram,
unbiased sample, valid, voluntary response sample, measures of center, coplaner, straight
angles, acute triangle, equilateral triangle, isosceles triangle, obtuse triangles, scalene
triangle
Resource
Glencoe Math 7th Accelerated Chapter 10 Statistics and Probability (14 days)
●
●
●
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●
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●
Lesson 1 - Measures of Center
Lesson 2 - Measures of Variability
Lesson 3 - Mean Absolute Deviation
Lesson 4 - Compare Populations
Lesson 5 - Using Sampling to Predict
Lesson 6 - Probability of Simple Events
Lesson 7 - Theoretical and Experimental Probability
Lesson 8 - Probability of Compound Events
Glencoe Math 7th Accelerated Chapter 11 Congruence, Similarities, and Transformations (14 days)
●
●
●
●
●
●
●
●
Lesson 1 - Angle and Line Relationships
Lesson 2 - Triangles
Lesson 3 - Polygons
Lesson 4 - Translations and Reflections on the Coordinate Plane
Lesson 5 - Rotations on the Coordinate Plane
Lesson 6 - Congruence and Transformations
Lesson 7 - Dilations on the Coordinate Plane
Lesson 8 - Similarity and Transformations
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Page 82 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Glencoe Math 7th Accelerated Chapter 12 Volume and Surface Area (16 days)
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●
●
Lesson 1 - Circles and Circumference
Lesson 2 - Area of Circles
Lesson 3 - Area of Composite Figures
Lesson 4 - Three-Dimensional Figures
Lesson 5 - Volume of Prisms
Lesson 6 - Volume of Cylinders
Lesson 7 - Volume of Pyramids, Cones and Spheres
Lesson 8 - Surface Area of Prisms
Lesson 9 - Surface Area of Cylinders
Lesson 10 - Surface Area of Pyramids and Cones
Pythagorean Theorem school resources
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Page 83 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Course 3 Math (8th Grade) Pacing Guide
The Pacing Guide has been aligned to Wyoming State Standards with consideration to the
Critical Areas of Focus defined by Common Core State Standards for Math and the
Wyoming PAWS Blueprint.
45 Days
Pacing Guide Section 1
● Real Numbers
● Equations in One Variable
Standards
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.
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 x 2 =
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 an integer power of 10 to
estimate very large or very small quantities, and to express how many times as much one
is than the other. For example, estimate the population of the United States as 3 × 108 and
the population of the world as 7 × 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.
8.EE.7
Solve linear equations in one variable.
a. Give examples of linear equations in one variable with one solution, infinitely
many solutions, or no solutions. Show which of these possibilities is the case by
successively transforming the given equation into simpler forms, until an
equivalent equation of the form x = a, a = a, or a = b results (where a and b are
different numbers).
b. Solve linear equations with rational number coefficients, including equations
whose solutions require expanding expressions using the distributive property and
collecting like terms.
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Page 84 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Vocabulary
Critical
irrational, scientific notation, solve (an equation)
Vocabulary
Supporting base, coefficient, cube root, exponent, identity, multiplicative inverse, monomial, null set,
Vocabulary perfect cube, perfect square, power, radical sign, rational number, real number, repeating
decimal, square root, terminating decimal
Resource
Glencoe Math Chapter 1 Real Numbers (25 days)
●
●
●
●
●
●
●
●
●
●
Lesson 1 - Rational Numbers
Lesson 2 - Powers and Exponents
Lesson 3 - Multiply and Divide Monomials
Lesson 4 - Powers of Monomials
Lesson 5 - Negative Exponents
Lesson 6 - Scientific Notation
Lesson 7 - Compute with Scientific Notation
Lesson 8 - Roots
Lesson 9- Estimate Roots
Lesson 10- Compare Real Numbers
Glencoe Math Chapter 2 Equations in One Variable (20 days)
●
●
●
●
●
Lesson 1 - Solve Equations with Rational Coefficients
Lesson 2 - Solve Two-Step Equations
Lesson 3 - Write Two-Steps Equations
Lesson 4 - Solve Equations with Variables on Each Side
Lesson 5 - Solve Multi-Step Equations
36 Days
Pacing Guide Section 2
● Equations in Two Variables
● Functions
Standards
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.8
Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables
correspond to points of intersection of their graphs, because points of intersection
satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate
solutions by graphing the equations. Solve simple cases by inspection. For
example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot
simultaneously be 5 and 6.
c. Solve real-world and mathematical problems leading to two linear equations in two
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Page 85 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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.
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.
8.F.2
Compare properties 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.
Vocabulary
Critical
constant of proportionality, constant rate of change, direct variation, linear equations,
Vocabulary linear functions, slope, slope-intercept form, substitution, x-intercept, y-intercept
Supporting constant of variation, linear relationship, point-slope form, rise, run, standard form,
Vocabulary systems of equations, continuous data, dependent variable, discrete data, domain, function,
function table, independent variable, non-linear function, quadratic function, qualitative
graph, range, relation
Resource
Glencoe Math Chapter 3 Equations in Two Variables (18 days)
●
●
●
●
●
●
●
●
Lesson 1 - Constant Rate of Change
Lesson 2 - Slope
Lesson 3 - Equations in y=mx Form
Lesson 4 - Slope-Intercept Form
Lesson 5 - Graph a Line Using Intercepts
Lesson 6 - Write Linear Equations
Lesson 7 - Solve Systems of Equations by Graphing
Lesson 8 - Solve Systems of Equations Algebraically
Glencoe Math Chapter 4 Functions (18 days)
●
●
●
●
●
Lesson 1 - Represent Relationships
Lesson 2 - Relations
Lesson 3 - Functions
Lesson 4 - Linear Functions
Lesson 5 - Compare Properties of Functions
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Page 86 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
●
●
●
●
Lesson 6 - Construct Functions
Lesson 7 - Linear and Nonlinear Functions
Lesson 8 - Quadratic Functions
Lesson 9- Qualitative Graphs
31 Days
Pacing Guide Section 3
● Triangles and the Pythagorean Theorem
● Transformations
Standards
8.G.1
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken 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.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
angle-angle criterion for similarity of triangles. For example, arrange three copies of the
same triangle so that the sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
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.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.
Vocabulary
Critical
Vocabulary
alternate exterior angles, alternate interior angles, angle sum (interior of a figure),
corresponding angles, dilation, exterior angles, hypotenuse, image, leg (of a right
triangle), pre-image, Pythagorean Theorem, reflection, rotation, transformation,
translation, transversal
Supporting
Vocabulary
distance formula, equiangular, interior angle, parallel lines, perpendicular lines, regular
polygons, remote interior angles, angle of rotation, center of dilation, center of rotation,
congruent, line of reflection, rotational symmetry
Resource
Glencoe Math Chapter 5 Triangles and the Pythagorean Theorem (18 days)
●
Lesson 1 - Lines
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Page 87 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
●
●
●
●
●
●
Lesson 2 - omit
Lesson 3 - Angles of Triangles
Lesson 4 - Polygons and Angles
Lesson 5 - The Pythagorean Theorem
Lesson 6 - Use the Pythagorean Theorem
Lesson 7 - Distance on the Coordinate Plane
Glencoe Math Chapter 6 Transformations (13 days)
●
●
●
●
Lesson 1 - Translations
Lesson 2 - Reflections
Lesson 3 - Rotations
Lesson 4 - Dilations
49 Days
Pacing Guide Section 4
● Congruence and Similarity
● Volume
● Scatter Plots and Data Analysis
Standards
8.G.1
Verify experimentally the properties of rotations, reflections, and translations:
a. Lines are taken to lines, and line segments to line segments of the same length.
b. Angles are taken to angles of the same measure.
c. Parallel lines are taken 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.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
angle-angle criterion for similarity of triangles. For example, arrange three copies of the
same triangle so that the sum of the three angles appears to form a line, and give an
argument in terms of transversals why this is so.
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.G.9
Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve
real-world and mathematical problems.
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 by judging the closeness of the data points to the line.
Back to Table of Contents
Page 88 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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?
Vocabulary
Critical
line of best fit, linear association, negative association, nonlinear association, positive
Vocabulary association, relative frequency, scatterplot, two-way table
Supporting corresponding parts, indirect measurement, scale factor, similar polygons, composite
Vocabulary solids, hemisphere, sphere, volume, bivariate data, distribution, 5-number summary, mean
absolute deviation, qualitative data, quantitative data, standard deviation, symmetric
Resource
Glencoe Math Chapter 7 Congruency and Similarities (17 days)
●
●
●
●
●
●
●
Lesson 1 - Congruence of Transformations
Lesson 2 - Congruence
Lesson 3 - Similarity and Transformations
Lesson 4 - Properties of Similar Polygons
Lesson 5 - Similar Triangles and Indirect Measurement
Lesson 6 - Slope and Similar Triangles
Lesson 7 - Area and Perimeter of Similar Figures
Glencoe Math Chapter 8 Volume and Surface Area (17 days)
●
●
●
●
●
Lesson 1 - Volume of Cylinders
Lesson 2 - Volume of Cones
Lesson 3 - Volume of Spheres
Lesson 4 and Lesson 5 - omit
Lesson 6 - Changes in Dimensions
Glencoe Math Chapter 9 Scatter Plots and Data Analysis (15 days)
●
●
●
●
●
●
Lesson 1 - Scatter Plots
Lesson 2 - Lines of Best Fit
Lesson 3 - Two-Way Tables
Lesson 4 - Descriptive Statistics
Lesson 5 - Measures of Variation
Lesson 6 - Analyze Data Distributions
Back to Table of Contents
Page 89 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Algebra I Pacing Guide
Algebra I 1ST SEMESTER
CCSS Mathematical Content
Reason quantitatively and use units to solve problems.
N.Q.1 - Use Clusters as a way to understand problems and to guide the solution of
multi-step problems; choose and interpret Clusters 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.
Interpret the structure of expressions.
A.SSE.1 - Interpret expressions that represent a quantity in terms of its context. Limit to
linear expressions and to exponential expressions with integer exponents.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a
single entity.
Create equations that describe numbers or relationships.
A.CED.1 - Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.
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. For example, represent inequalities describing nutritional and cost
constraints on combinations of different foods.
A.CED.4 - Rearrange formulas to highlight a quantity of interest, using the same
reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight
resistance R.
Back to Table of Contents
CCSS Mathematical
Practice
Content and
Educator Notes
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and critique
the reasoning of
others.
4. Model with
mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make
use of structure.
8. Look for and express
regularity in repeated
reasoning.
2.6, 2.7, 2.8 (N.Q.1)
2.6 extended lab
(N.Q.2)
1.3 extended lab
(N.Q.3)
*Rounding
throughout the year
1.1, 1.4 (A.SSE.1a)
1.2, 1.3 (A.SSE.1b)
1.5, 2.1-2.5, 3.2
(A.CED.1)
3.1, 3.4, 3.5, 3.6,
4.1-4.6 (A.CED.2)
4.2 (A.CED.3)
2.8, 4.1 (A.CED.4)
Page 90 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Understand solving equations as a process of reasoning and explain the
reasoning.
A.REI.1 - Explain each step in solving a simple equation as following from the equality of
numbers asserted at the previous step, starting from the assumption that the original
equation has a solution. Construct a viable argument to justify a solution method. Focus on
and master linear equations and be able to extend and apply their reasoning to other types
of equations in future courses.
Solve equations and inequalities in one variable
A.REI.3 - Solve linear equations and inequalities in one variable, including equations with
coefficients represented by letters. Extend earlier work with solving linear equations to
solving linear inequalities in one variable and to solving literal equations that are linear in
the variable being solved for. Include simple exponential equations that rely only on
application of the laws of exponents, such as 5 x=125 or 2x=1/16
Solve systems of equations
A.REI.5 – Prove that, given a system of 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. Focus on justification of the methods
used. Include cases where the two equations describe the same line (yielding infinitely
many solutions) and cases where two equations describe parallel lines (yielding no
solution); connect to to GPE.5 when it is taught in Geometry, which requires students to
prove the slope criteria for parallel line.
Represent and solve equations and inequalities graphically
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).
Focus on linear and exponential equations and be able to adapt and apply that learning to
other types of equations in future courses.
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. Focus on cases where f(x)
and g(x) are linear or exponential.
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 variables as the intersection of the corresponding halfplanes.
Back to Table of Contents
1.5, 2.2-2.6
(A.REI.1)
1.5, 2.2-2.8
(A.REI.3)
6.4 (A.REI.5)
6.1-6.5 (A.REI.6)
1.6, 1.7, 3.1, 3.2,
3.4 (A.REI.10)
6.1(A.REI.11)
5.6, 6.6(A.REI.12)
Page 91 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Understand the concept of a function and use functions notation
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.3 - Recognize that sequences are functions, sometimes defined recursively, whose
domain is a subset of the integers. For example, the Fibonacci sequence is defined
recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Students should experience a variety of types of situations modeled by functions. Detailed
analysis of any particular class of functions at this stage is not advised. Students should
apply these concepts throughout their future mathematics courses.
Draw examples from linear and exponential functions. Draw connection between F.IF.3
and F.BF.2 which required students to write arithmetic and geometric sequences.
Emphasize arithmetic and geometric sequences as examples of linear and exponential
functions.
Interpret functions that arise in applications in terms of a context
F.IF.4 - For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity. Focus on linear and
exponential functions.
F.IF.6 - Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph. Focus on linear functions and exponential functions whose domain is a subset of
the integers.
Analyze functions using different representations
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. Focus on linear
and exponential functions. Include comparisons of two functions presented algebraically.
For example, compare the growth of two linear functions, or two exponential functions.
a. Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.9 - Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For example,
given a graph of one quadratic function and an algebraic expression for another, say which
has the larger maximum.
Back to Table of Contents
1.6, 1.7 (F.IF.1)
1.7, 3.6, 4.3 (F.IF.2)
3.5 (F.IF.3)
1.8, 3.1, 4.1 (F.IF.4)
3.3 (F.IF.6)
3.1,3.2,3.4,4.1
(F.IF.7a)
1.7,3.6,4.3 (F.IF.9)
Page 92 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Build a function that models a relationship between two quantities
F.BF.1 - Write a function that describes a relationship between two quantities. Limit to
F.BF.1a, 1b, and 2 to linear and exponential functions.
a. Determine an explicit expression, a recursive process, or steps for calculation from
a context.
b. Combine standard function types using arithmetic operations. For example, build a
function that models the temperature of a cooling body by adding a constant
function to a decaying exponential, and relate these functions to the model.
F.BF.2 – Write arithmetic and geometric sequences both recursively and with an explicit
formula, use them to model situations, and translate between the two form.
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 graphs using
technology. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
F.BF.4 – Find the 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.
Construct and compare linear, quadratic, and exponential models and solve
problems
F.LE.1 - Distinguish between situations that can be modeled with linear functions and with
exponential functions.
a. Prove that linear functions grow by equations 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.
F.LE.2 - Construct linear and exponential functions, including arithmetic and geometric
sequences, given a graph, a description of a relationship, or two input-output pairs (include
reading these from a table). Draw on and consolidate previous work in grade 8 on finding
equations for lines and linear functions.
F.LE.3 - Observe using graphs and tables that a quantity increasing exponentially
eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a
polynomial function. Limit to comparisons between linear and exponential models.
Interpret expressions for functions in terms of the situation they model
F.LE.5 - Interpret the parameters in a linear or exponential function in terms of a context.
Back to Table of Contents
1.7,3.1,3.4,3.6,4.14.6 (F.BF.1a)
4.2 (F.BF.1b)
3.5, 7.7, 7.8 (F.BF.
2)
9.3(F.BF.3)
4.7 (F.BF.4)
3.3, 3.5 (F.LE.1a)
3.5, 3.6 (F.LE.1b)
7.6, 7.7 (F.LE.1c)
3.5, 3.6, 4.2-4.6
(F.LE.2)
9.6 (F.LE.3)
3.4, 4.1, 4.5
(F.LE.5)
Page 93 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Summarize, represent, and interpret data on two categorical and quantitative
variables
S.ID.6 - Represent data on two quantitative variables on a scatter plot, and describe how
the variables are related. Focus on linear models, but may be used to preview quadratic
functions in unit 5 of this course.
a. Fit a function to the data; use functions fitted to data to solve problems in the
context of the data. Use given functions or choose a function suggested by the
context. Emphasize linear and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Students model the relationship between two numerical variables. In addition to
fitting a line to data, students assess how well the model fits by analyzing residuals.
4.5, 4.6 (S.ID.6a, b,
c)
Interpret linear models
S.ID.7 - Interpret the slope (rate of change) and the intercept (constant term) of a linear
model in the context of the data.
S.ID.8 - Compute (using technology) and interpret the correlation coefficient of a linear fit.
S.ID.9 – Distinguish between correlation and causation.
4.1, 4.4, 4.5 (S.ID.
7)
4.6 (S.ID.8)
4.5 (S.ID.9)
Algebra I 2ND SEMESTER
CCSS Mathematical Content
Extend the properties of exponents to rational exponents.
N.RN.1 – Explain how the definition of the meaning of rational exponents follows from
extending the properties of integer exponents to those values, allowing for a notation for
radicals in terms of rational exponents. For example, we define (51/3)3 = 5(1/3)3 must equal
5.
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 a rational number and an irrational number is irrational; and that the product of a
nonzero rational number and an irrational number is irrational
Back to Table of Contents
CCSS Mathematical
Practice
Content and
Educator Notes
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the reasoning
of others.
4. Model with
7.3 (N.RN.1)
7.3(N.RN.2)
10-2 (N.RN.3)
Page 94 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Interpret the structure of expressions
A.SSE.1 - Interpret expressions that represent a quantity in terms of its context. Focus on
quadratic and exponential expressions.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For example, see
x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems
A.SSE.3 - Choose and produce an equivalent form of an expression to reveal and explain
properties of the quantity represented by the expression. It is important to balance
conceptual understanding and procedural fluency in work with equivalent expressions. For
example, development of skill in factoring and completing the square goes hand-in-hand
with understanding what different forms of a quadratic expression reveal.
a. Factor a quadratic expression to reveal the zeros of the function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum
value of the function it defines.
c. Use the properties of exponents to transform expressions for exponential functions.
For example the expression 1.15t can be rewritten as (1.151/12)12t = 1.01212t
(approx..) to reveal the approximate equivalent monthly interest rate if the annual
rate is 15%.
Perform arithmetic operations on polynomials
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. Focus on polynomial expressions that simplify to forms
that are linear or quadratic in a positive integer power of x.
Create equations that describe numbers or relationships
A.CED.1 - Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple rational
and exponential functions.
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. For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
Solve equations and inequalities in one variable
Back to Table of Contents
mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make
use of structure.
8. Look for and express
regularity in repeated
reasoning.
8.1, 9.1 (A.SSE.1)
7.1, 7.2, 8.6, 8.8
(A.SSE.2)
8.5, 8.6, 8.8
(A.SSE.3a)
9.3, 9.3(A.SSE.3b)
7.6extention
(A.SSE.3c)
8.1-8.4 (A.APR.1)
5.1-5.5, 8.5, 8.6
(A.CED.1)
6.1-6.5, 8.6, 8.8,
9.1, 9.2, 9.5
(A.CED.2)
5.6, 6.1, 6.2
(A.CED.3)
Page 95 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Interpret functions that arise in applications in terms of a context
F.IF.5 - Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the positive integers would
be an appropriate domain for the function.
Analyze functions using different representations
F.IF.8 - Write a function defined by an expression in different but equivalent forms to reveal
and explain different properties of the function.
a. Use the process of factoring and completing the square in a quadratic function to
show zeros, extreme values, and symmetry of the graph, and interpret these in
terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions.
For example, identify percent rate of change in functions such as y = (1.02) t, y =
(0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential
growth or decay
A.REI.4 – Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x
into an equation of the form (x-p)2=q that has the same solutions. Derive the
quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2=49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the
initial form of the equation. Recognize when the quadratic formula gives complex
solutions and write them as
9.1, 9.2 (F.IF. 5)
9.2, 9.3 (F.IF.8a)
7.1, 7.2 (F.IF.8b)
9.4, 9.5, 10.2
(A.REI.4a)
8.6, 8.7, 8.8, 9.2,
9.4, 9.5 (A.REI.4b)
for real numbers a and b.
Summarize, represent, and interpret data on a single count or measurement
variable
S.ID.1 - Represent data with plots on the real number line (dot plots, histograms, and box
plots).
S.ID.2 - Use statistics appropriate to the shape of the data distribution to compare center
(median, mean) and spread (interquartile range, standard deviation) of two or more
different data sets.
S.ID.3 - Interpret differences in shape, center, and spread in the context of the data sets,
accounting for possible effects of extreme data points (outliers).
In grades 6-8 students describe center and spread in a data distribution. Here they choose
a summary statistic appropriate to the characteristics of the data distribution, such as the
shape of the distribution or the existence of extreme data points.
Back to Table of Contents
0.13, 12.3, 12.4
(S.ID.1)
12.2, 12.3, 12.4
(S.ID.2)
12.3-12.4 (S.ID.3)
Page 96 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Geometry Pacing Guide
Geometry 1st Semester
Geometry Instructional Focus 1:
Congruence, Proof, and Constructions
Suggested Quarter/Time: 1st/30 days
CCSS Mathematical Content
Experiment with transformations in the plane
G.CO.1 - Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, distance along a line, and distance
around a circular arc.
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 as 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, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a
sequence of transformations that will carry a given figure onto another.
Understand congruence in terms of rigid motions.
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.CO.7 - Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and corresponding pairs
of angles are congruent.
G.CO.8 - Explain how the criteria for triangle congruence (ASA, SAS, and, SSS) follow
from the definition of congruence in terms of rigid motions.
Back to Table of Contents
CCSS Mathematical
Practice
Content and
Educator
Content
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique
the reasoning of others.
4. Model with
mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use
of structure.
8. Look for and express
regularity in repeated
reasoning.
G.CO.1 (1.1-1.4,
3.1-3.2, 10.1)
G.CO.2 (9.4)
G.CO.3 (9.5)
G.CO.4 (9.1-9.3)
G.CO.5 (9.1-9.4)
G.CO.6 (4.7)
G.CO.7 (4.3, 4.7)
G.CO.8 (4.4, 4.5)
Page 97 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Rigid motions are at the foundation of the definition of congruence. Students reason from
the basic properties of rigid motions (that they preserve distance and angle), which are
assumed without proof. Rigid motions and their assumed properties can be used to
establish the usual triangle congruence criteria, which can then be used to prove other
theorems.
Prove Geometric Theorems
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 to 180°; base angles of isosceles triangles are congruent; the
segment joining midpoints of two sides of a triangle is parallel to the third side and half
the length; the medians of a triangle meet at a point.
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.
Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow
diagrams, in two-column format, and using diagrams without words. Students should be
encouraged to focus on the validity of the underlying reasoning while exploring a variety of
formats for expressing that reasoning. Implementation of G.CO.10 may be extended to
include concurrence of perpendicular bisectors and angle bisectors as preparation for
G.C.3 in Unit 5.
Make geometric constructions.
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.). Copying a segment; copying an angle; bisecting a segment; bisecting an
angle; constructing perpendicular lines, including the perpendicular bisector of a line
segment; and constructing a line parallel to a given line through a point not on the line.
G.CO.13 - Construct an equilateral triangle, a square, and a regular hexagon inscribed in a
circle.
Build on prior student experience with simple constructions. Emphasize the ability to
formalize and explain how these constructions result in the desired objects.
Some of these constructions are closely related to previous standards and can be
introduced in conjunction with them.
Back to Table of Contents
G.CO.9 (2.7-2.8,
3.2, 3.5)
G.CO.10 (4.2,
4.4, 4.5, 4.6, 4.8,
5.1, 5.2, 5.3, 5.5,
5.6)
G.CO.11 (6.2,
6.4, 6.5)
G.CO.12
(Throughout)
G.CO.13
(Throughout)
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Natrona County School District Mathematics Curriculum
Geometry Instructional Focus 2:
Similarity, Proof and Trigonometry
Suggested Quarter/Time: 1st – 2nd/45 days
CCSS Mathematical Content
Understand similarity in terms of similarity transformations
G.SRT.1 - Verify experimentally the properties of dilations given by a center and a
scale factor.
G.SRT.1a - 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.
G.SRT.1b - The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.
G.SRT.2 - Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity transformations the
meaning of similarity for triangles as the equality of all corresponding pairs of angles and
the proportionality of all corresponding pairs of sides.
G.SRT.3 - Use the properties of similarity transformations to establish the AA criterion for
two triangles to be similar.
Prove theorems involving similarity
G.SRT.4 - Prove theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the Pythagorean
Theorem proved using triangle similarity.
G.SRT.5 - Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Define trigonometric ratios and solve problems involving right triangles
G.SRT.6 - Understand that by similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.7 - Explain and use the relationship between the sine and cosine of
complementary angles.
G.SRT.8 - Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
Back to Table of Contents
CCSS Mathematical
Practice
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique
the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
Content and
Educator
Content
G.SRT.1 (9.6 a
and b)
G.SRT.2 (7.2,
7.6)
G.SRT.3 (7.3)
G.SRT.4 (7.3,
7.4, 7.5, 8.1)
G.SRT.5 (4.34.5) (7-3, 7-4, 75, 7-6, 8-1)
G.SRT.6 (8-3, 84)
G.SRT.7 (8.4)
G.SRT.8 (8.5)
Page 99 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Apply geometric concepts in modeling situations
G.MG.1 - Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).*
G.MG.2 - Apply concepts of density based on area and volume in modeling situations
(e.g., persons per square mile, BTUs per cubic foot).
G.MG.3 - Apply geometric methods to solve design problems (e.g., designing an object or
structure to satisfy physical constraints or minimize cost; working with typographic grid
systems based on ratios).
Focus on situations well modeled by trigonometric ratios for acute angles.
Apply trigonometry to general triangles
G.SRT.9 - (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an
auxiliary line from a vertex perpendicular to the opposite side.
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).
With respect to the general case of the Laws of sines and Cosines, the definitions of sine
and cosine must be extended to obtuse angles.
G.MG.1
(Throughout)
G.MG.2 (Ch 12)
G.MG.3
(Throughout)
Geometry 2nd Semester
Geometry Instructional Focus 3:
Extending to Three Dimensions
Suggested Quarter/Time: 3rd/20 days
CCSS Mathematical Content
Explain volume formulas and use them to solve problems.
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. Use dissection arguments,
Cavalieri’s principle, and informal limit arguments.
G.GMD.3 - Use volume formulas for cylinders, pyramids, cones, and spheres to solve
problems.
Back to Table of Contents
CCSS Mathematical
Practice
1.
2.
Make sense of problems
and persevere in solving
them.
Reason abstractly and
quantitatively.
Content and
Educator
Content
G.GMD.1 (11.3,
12.5, 12.6)
G.GMD.3 (11.3,
12.5, 12.6)
Page 100 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Informal arguments for area and volume formulas can make use of the way in which area
and volume scale under similarity transformations: when one figure in the plane results from
another by applying a similarity transformation with scale factor k, its area is k2 times the
area of the first.
Similarly, volumes of solid figures scale by k3 under a similarity transformation with scale
factor k.
Visualize the relation between two-dimensional and three-dimensional objects.
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.
Apply geometric concepts in modeling situations.
G.MG.1 - Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as a cylinder).
Focus on situations that require relating two- and three-dimensional objects, determining
and using volume, and the trigonometry of general triangles.
3.
4.
5.
6.
7.
8.
Construct viable
arguments and critique
the reasoning of others.
Model with mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
G.GMD.4 (12.1)
G.MG.1
(Throughout)
Geometry Instructional Focus 4:
Connecting Algebra and Geometry through Coordinates
Suggested Quarter/Time: 3rd/20 days
CCSS Mathematical Content
Use coordinates to prove simple geometric theorems algebraically
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.GPE.5 - Prove the slope criteria for parallel and perpendicular lines and uses 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.GPE.7 - Use coordinates to compute perimeters of polygons and areas of triangles
Back to Table of Contents
CCSS Mathematical
Practice
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique
the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
Content and
Educator
Content
G.GPE.4 (6.26.4, 6.6)
G.GPE.5 (3.33.4)
G.GPE.6 (8.7,
10.8)
G.GPE.7 (1.6,
11.1)
Page 101 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
and rectangles, e.g., using the distance formula.
This provides practice with the distance formula and its connection with the Pythagorean
theorem.
This unit has a close connection with the next unit. For example, a curriculum might
merge G.GPE.1 and the Unit 5 treatment of G.GPE.4 with the standards in this unit.
Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation
for a line through two points using similar right triangles.
Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra I
involving systems of equations having no solution or infinitely many solutions.
strategically.
6. Attend to precision.
7. Look for and make use of
structure.
8. Look for and express
regularity in repeated
reasoning.
Translate between the geometric description and the equation for a conic section.
G.GPE.2 - Derive the equation of a parabola given a focus and directrix.
The directrix should be parallel to a coordinate axis.
Geometry Instructional Focus 5:
Circles With and Without Coordinates
Suggested Quarter/Time: 4th/25 days
CCSS Mathematical Content
Understand and apply theorems about circles
G.C.1 - Prove that all circles are similar.
G.C.2 - Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles; inscribed
angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent
where the radius intersects the circle.
G.C.3 - Construct the inscribed and circumscribed circles of a triangle, and prove properties
of angles for a quadrilateral inscribed in a circle.
G.C.4 - (+) Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles
G.C.5 - Derive using similarity the fact that the length of the arc intercepted by an angle is
proportional to the radius, and define the radian measure of the angle as the constant of
proportionality; derive the formula for the area of a sector.
Back to Table of Contents
CCSS Mathematical
Practice
1.
2.
3.
4.
5.
6.
7.
Make sense of problems
and persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and critique
the reasoning of others.
Model with mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and make use
Content and
Educator
Content
G.C.1 (10.1)
G.C.2 (10.2-10.4)
G.C.3 (10.4,
10.5)
G.C.4 (Plus)
G.C.5 (10.2,
11.3)
Page 102 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Emphasize the similarity of all circles. Note that by similarity of sectors with the same
central angle, arc lengths are proportional to the radius. Use this as a basis for introducing
radian as a unit of measure. It is not intended that it be applied to the development of
circular trigonometry in this course.
8.
of structure.
Look for and express
regularity in repeated
reasoning.
Translate between the geometric description and the equation for a conic section
G.GPE.1 - Derive the equation of a circle of given center and radius using the Pythagorean
Theorem; complete the square to find the center and radius of a circle given by an equation.
G.GPE.1 (10.8)
Use coordinates to prove simple geometric theorems algebraically
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).
Include simple proofs involving circles.
G.GPE.4 (6.26.4, 6.6)
Apply geometric concepts in modeling situations
G.MG.1 - Use geometric shapes, their measures, and their properties to describe objects
(e.g., modeling a tree trunk or a human torso as acylinder).
Focus on situations in which the analysis of circles is required.
G.MG.1
(Throughout)
Geometry Instructional Focus 6:
Applications of Probability
Suggested Quarter/Time: 4th/20 days
CCSS Mathematical Content
Understand independence and conditional probability and use them to interpret
data.
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,” “and,” “not”).
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
Back to Table of Contents
CCSS Mathematical
Practice
1.
2.
3.
Make sense of problems
and persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
Content and
Educator
Content
S.CP.1 (13.6)
S.CP.2 (13.5)
Page 103 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
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 object being classified. Use the two-way table as a sample space to
decide if events are independent and to approximate conditional probabilities. For
example, collect data from a random sample of students in your school on their favorite
subject among math, science, and English. Estimate the probability that a randomly
selected student from your school will favor science given that the student is in tenth
grade. Do the same for other subjects and compare the results.
S.CP.5 - Recognize and explain the concepts of conditional probability and independence
in everyday language and everyday situations. For example, compare the chance of
having lung cancer if you are a smoker with the chance of being a smoker if you have lung
cancer.
Build on work with two-way tables from Algebra I Unit 3 (S.ID.5) to develop understanding
of conditional probability and independence.
Use the rules of probability to compute probabilities of compound events in a
uniform probability model
S.CP.6 - Find the conditional probability of A given B as the fraction of B’s outcomes that
also belong to A, and interpret the answer in terms of the model.
S.CP.7 - Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the
answer in terms of the model.
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.
4.
5.
6.
7.
8.
arguments and critique
the reasoning of others.
Model with mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and make use
of structure.
Look for and express
regularity in repeated
reasoning.
S.CP.3 (13.5)
S.CP.4 (13.5 Lab
only)
S.CP.5
(Throughout Ch
13)
S.CP.6 (13.5 lab
only)
S.CP.7 (13.6)
Use probability to evaluate outcomes of decisions
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).
This unit sets the stage for work in Algebra II, where the ideas of statistical inference are
introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e.
incomplete information) requires an understanding of probability concepts.
Back to Table of Contents
Page 104 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Algebra II Pacing Guide
Algebra II 1ST SEMESTER
CCSS Mathematical Content
Perform arithmetic operations with complex numbers
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 i 2 = –1 and the commutative, associative, and distributive
properties to add, subtract, and multiply complex numbers.
Use complex numbers in polynomial identities and equations
N.CN.7 – Solve quadratic equations with real coefficients that have complex solutions.
Interpret the structure of expressions
A.SSE.1- Interpret expressions that represent a quantity in terms of its context. Extend to
polynomial and rational expressions.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single
n
entity. For example, interpret P(1+r) as the product of P and a factor not depending
on P.
CCSS Mathematical
Practice
1.
2.
3.
4.
5.
6.
7.
8.
A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For example, see
x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
Make sense of problems
and persevere in solving
them.
Reason abstractly and
quantitatively.
Construct viable
arguments and critique
the reasoning of others.
Model with mathematics.
Use appropriate tools
strategically.
Attend to precision.
Look for and make use
of structure.
Look for and express
regularity in repeated
reasoning.
Content and
Educator
Notes
4.4 (N.CN.1)
4.4 (N.CN.2)
4.5, 4.6 (N.CN.7)
5.1, 5.2 (A.SSE.1a)
4.7(A.SSE.1b)
4.3, 5.1 (A.SSE.2)
Perform arithmetic operations on polynomials
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. Extend beyond the quadratic polynomials found in
Algebra I.
5.1 (A.APR.1)
Understand the relationship between zeros and factors of polynomials.
A.APR.2-Know and apply the Remainder Theorem: For a polynomial p(x) and a number a,
5.6 (A.APR.2)
Back to Table of Contents
Page 105 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
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 polynomial.
Use polynomial identities to solve problems
A.APR.4 - Prove polynomial identities and use them to describe numerical relationships.
For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate
Pythagorean triples.
Rewrite rational expressions
A.APR.6 - Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form
q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less
than the degree of b(x), using inspection, long division, or, for the more complicated
examples, a computer algebra system.
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.
Include combinations of linear, polynomial, rational, radical, absolute value, and
exponential functions.
Create equations that describe numbers or relationships.
A.CED.1 - Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple
rational and exponential functions.
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 nonviable options in a modeling
context. For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
A.CED.4 - Rearrange formulas to highlight a quantity of interest, using the same reasoning
as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.
For A.CED.1, use all available types of functions to create such equations, including root
functions, but constrain to simple cases. While functions used in A.CED.2, 3, and 4 will
often be linear, exponential, or quadratic the types of problems should draw from more
complex situations than those addressed in Algebra I. For example, finding the equation of
a line through a given point perpendicular to another line allows one to find the distance
Back to Table of Contents
4.3, 4.5, 4.6
(A.APR.3)
5.7 extension
(A.APR.4)
5.2 (A.APR.6)
In sections 4-2, 6-3
and 7-2 graphing
technology labextend to include
intersection with a
line (A.REI.11)
4.7, 4.8, 7.1, 7.8
(A.CED.1)
4.1, 4.2, 4.7
(A.CED.2)
4.8, 7.8(A.CED.3)
4.7 (A.CED.4)
Page 106 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
from a point to a line. Note that the example given for A.CED.4 applies to earlier instances
of this standard, not to the current course.
Interpret functions that arise in applications in terms of a context.
F.IF.4 - For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.
F.IF.5 - Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the positive integers would
be an appropriate domain for the function.
F.IF.6 - Calculate and interpret the average rate of change of a function (presented
symbolically or as a table) over a specified interval. Estimate the rate of change from a
graph. Focus on linear functions and exponential functions whose domain is a subset of
the integers.
Emphasize the selection of a model function based on behavior of data and context.
Analyze functions using different representations
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.
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.
Relate F.IF.7c to the relationship between zeros of quadratic functions and their factored
form.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and
trigonometric functions, showing 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 function.
F.IF.9 - Compare properties of two functions each represented in a different way
(algebraically, graphically, numerically in tables, or by verbal descriptions). For example,
given a graph of one quadratic function and an algebraic expression for another, say which
has the larger maximum.
Focus on applications and how key features relate to characteristics of a situation, making
selection of a particular type of function model appropriate.
Back to Table of Contents
4.1, 4.2, 5.4
(F.IF.4)
2.6, 4.1 (F.IF.5)
4.7 extension
(F.IF.6)
4.1, 5.4 (F.IF.7)
4.7 (F.IF.8)
4.1 (F.IF.9)
Page 107 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Building new functions from existing 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. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
2.7, 4.7 (F.BF.3)
Algebra II 2ND SEMESTER
CCSS Mathematical Content
Create equations that describe numbers or relationships.
A.CED.1 - Create equations and inequalities in one variable and use them to solve
problems. Include equations arising from linear and quadratic functions, and simple rational
and exponential functions.
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 nonviable options in a modeling
context. For example, represent inequalities describing nutritional and cost constraints on
combinations of different foods.
Interpret functions that arise in applications in terms of a context.
F.IF.4 - For a function that models a relationship between two quantities, interpret key
features of graphs and tables in terms of the quantities, and sketch graphs showing key
features given a verbal description of the relationship. Key features include: intercepts;
intervals where the function is increasing, decreasing, positive, or negative; relative
maximums and minimums; symmetries; end behavior; and periodicity.
F.IF.5 - Relate the domain of a function to its graph and, where applicable, to the
quantitative relationship it describes. For example, if the function h(n) gives the number of
person-hours it takes to assemble n engines in a factory, then the positive integers would
be an appropriate domain for the function.
Back to Table of Contents
CCSS Mathematical
Practice
1. Make sense of problems
and persevere in solving
them.
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique
the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools
strategically.
6. Attend to precision.
7. Look for and make use
of structure.
8. Look for and express
regularity in repeated
reasoning.
Content and
Educator
Notes
4.7, 4.8, 7.1, 7.4,
7.5, 7.8 (A.CED.1)
4.1, 4.2, 4.7, 6.3,
7.1 (A.CED.2)
4.8, 7.8(A.CED.3)
4.1, 4.2, 5.4, 6.3
(F.IF.4)
2.6, 4.1, 6.3
(F.IF.5)
Page 108 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Analyze functions using different representations
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.
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.
Relate F.IF.7c to the relationship between zeros of quadratic functions and their
factored form.
e. Graph exponential and logarithmic functions, showing intercepts and end behavior,
and trigonometric functions, showing period, midline, and amplitude.
Interpret the structure of expressions
A.SSE.1- Interpret expressions that represent a quantity in terms of its context. Extend to
polynomial and rational expressions.
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single
n
entity. For example, interpret P(1+r) as the product of P and a factor not depending
on P.
A.SSE.2 - Use the structure of an expression to identify ways to rewrite it. For example, see
x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as
(x2 – y2)(x2 + y2).
Write expressions in equivalent forms to solve problems
A.SSE.4 - Derive 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. Consider extending A.SSE.4 to infinite geometric series in curricular
implementations of this course description.
Represent and solve equations and inequalities graphically.
Understand solving equations as a process of reasoning and explain the reasoning.
A.REI.2 - Solve simple rational and radical equations in one variable, and give examples
showing how extraneous solutions may arise.
Extend to simple rational and radical equations.
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.
Back to Table of Contents
4.1, 5.4, 6.3
(F.IF.7)
5.1, 5.2 (A.SSE.1a)
4.7, 6.5, 7.1
(A.SSE.1b)
4.3, 5.1, 6.5, 7.5
(A.SSE.2)
7.2, 10.3 (A.SSE.4)
6.7, 8.6 (A.REI.2)
In sections 4-2, 6-3
and 7-2 graphing
technology labextend to include
intersection with a
line (A.REI.11)
Page 109 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Include combinations of linear, polynomial, rational, radical, absolute value, and
exponential functions.
Build a function that models a relationship between two quantities
F.BF.1 - Write a function that describes a relationship between two quantities.
b. Combine standard function types using arithmetic operations. For example, build a
function that models the temperature of a cooling body by adding a constant function to
a decaying exponential, and relate these functions to the model.
Develop models for more complex or sophisticated situations than in previous courses.
Building new functions from existing 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. Include recognizing even and odd functions from their graphs and algebraic
expressions for them.
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. For example, f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠
1.
Use transformations of functions to find models as students consider increasingly more
complex situations.
For F.BF.3, note the effect of multiple transformations on a single graph and the common
effect of each transformation across function types.
Extend F.BF.4a to simple rational, simple radical, and simple exponential functions;
connect F.BF.4a to F.LE.4.
Construct and compare linear, quadratic, and exponential models and solve
problems
F.LE.4 - For exponential models, express as a logarithm the solution to abct = d where a, c,
and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Consider extending this unit to include the relationship between properties of logarithms
and properties of exponents, such as the connection between the properties of exponents
and the basic logarithm property that log xy = log x +log y.
Summarize, represent, and interpret data on a single count or measurement
variable
S.ID.4 - Use the mean and standard deviation of a data set to fit it to a normal distribution
and to estimate population percentages. Recognize that there are data sets for which such
Back to Table of Contents
6.1 without
composition
(F.BF.1)
2.7, 4.7 (F.BF.3)
6.2 without domain
restrictions and no
composition
(F.BF.4)
7.2, 7.8 (F.LE.4)
Seek additional
resources (S.ID.4)
Page 110 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate
areas under the normal curve.
While students may have heard of the normal distribution, it is unlikely that they will have
prior experience using it to make specific estimates. Build on students’ understanding of
data distributions to help them see how the normal distribution uses area to make
estimates of frequencies (which can be expressed as probabilities). Emphasize that only
some data are well described by a normal distribution.
Understand and evaluate random processes underlying statistical experiments
S.IC.1 - Understand statistics as a process for making inferences about population
parameters based on a random sample from that population.
S.IC.2 - Decide if a specified model is consistent with results from a given data-generating
process, e.g., using simulation. For example, a model says a spinning coin falls heads up
with probability 0.5. Would a result of 5 tails in a row cause you to question the model?
For S.IC.2, include comparing theoretical and empirical results to evaluate the
effectiveness of a treatment.
Seek additional
resources (S.IC.1)
Make inferences and justify conclusions from sample surveys, experiments, and
observational studies
S.IC.3 - Recognize the purposes of and differences among sample surveys, experiments,
and observational studies; explain how randomization relates to each.
S.IC.4 - Use data from a sample survey to estimate a population mean or proportion;
develop a margin of error through the use of simulation models for random sampling.
S.IC.5 - Use data from a randomized experiment to compare two treatments; use
simulations to decide if differences between parameters are significant.
S.IC.6 - Evaluate reports based on data.
In earlier grades, students are introduced to different ways of collecting data and use
graphical displays and summary statistics to make comparisons. These ideas are revisited
with a focus on how the way in which data is collected determines the scope and nature
of the conclusions that can be drawn from that data. The concept of statistical significance
is developed informally through simulation as meaning a result that is unlikely to have
occurred solely as a result of random selection in sampling or random assignment in an
experiment.
For S.IC.4 and 5, focus on the variability of results from experiments - that is, focus on
statistics as a way of dealing with, not eliminating, inherent randomness.
Seek additional
resources (S.IC.3)
Extend the domain of trigonometric functions using the unit circle
F.TF.1 - Understand radian measure of an angle as the length of the arc on the unit circle
subtended by the angle.
Back to Table of Contents
Seek additional
resources (S.IC.2)
Seek additional
resources (S.IC.4)
Seek additional
resources (S.IC.5)
Seek additional
resources (S.IC.6)
12.2 (F.TF.1)
Page 111 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
F.TF.2 - Explain how the unit circle in the coordinate plane enables the extension of
trigonometric functions to all real numbers, interpreted as radian measures of angles
traversed counterclockwise around the unit circle.
Model periodic phenomena with trigonometric functions.
F.TF.5 - Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.
Prove and apply trigonometric identities.
F.TF.8 - Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin (θ), cos
(θ), or tan (θ), given sin (θ), cos (θ), or tan (θ), and the quadrant of the angle.
An Algebra II course with an additional focus on trigonometry could include the (+)
standard F.TF.9: Prove the addition and subtraction formulas for sine, cosine, and
tangent and use them to solve problems. This could be limited to acute angles in Algebra
II.
Back to Table of Contents
12.2 (F.TF.2)
12.7(F.TF.5)
12.2 (F.TF.8)
Page 112 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
NCSD Math Vocabulary
Research by Marzano indicates that the importance of direct vocabulary instruction cannot be overstated. Vocabulary
provides essential background knowledge and is linked to academic achievement. NCSD Math Curriculum Committee
members worked to create a vertically aligned vocabulary progression for which key terms were identified and labeled as
critical or supporting. Critical vocabulary words are those words that all students are expected to master by the end of the
grade and will appear on common assessments. Based on research about vocabulary acquisition, committee members
attempted to limit the number of Tier 3 critical math vocabulary to fewer than 30 words. Supporting vocabulary words are
those words students should be exposed to throughout the year. The purpose of this work is to focus instruction and provide
guidance to teachers. Teachers will be asked to provide feedback in order to guide to revision of these lists.
The following table includes the critical Tier 3 vocabulary words that students should master by the end of the school year.
NCSD Math Vocabulary K - 8
Kindergarten
First Grade Second Grade Third Grade Fourth Grade
Fifth Grade
Sixth Grade
Seventh Grade
Eighth Grade
2-dimensional
2dimensional
addend
angle(s)
acute angle
braces
algebraic
expressions
additive inverse
alternate
exterior angles
3-dimensional
3dimensional
analog
denominator
algorithm
brackets
bar notation
adjacent angles
alternate
interior angles
addition (add,
plus, combine)
analog
base ten
numerals
division
angle
capacity
coefficient
circumference
angle sum
(interior of a
figure)
circle
bundle
centimeter
(cm)
equal part
common
denominator
composite
composite
figure
commission
constant of
proportionality
cone
circle
compose
factor
decimal
convert
dependent
variable
complementary
angles
constant rate of
change
corners
compose
customary
system
fraction
decimal point
coordinate
plane
distributive
property
complex fraction
corresponding
angles
count
decompose
decompose
label
dividend
cubed
equivalent
ratios
constant
dilation
cube
difference
digital
liquid volume
divisor
customary
system
exponents
cross products
direct variation
curved
digit
endpoint
(vertex)
dividend
independent
variable
cross section
exterior angle
expanded form multiplication
Back to Table of Contents
Page 113 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
NCSD Math Vocabulary K - 8
Kindergarten
First Grade Second Grade Third Grade Fourth Grade
Fifth Grade
Sixth Grade
Seventh Grade
Eighth Grade
cylinder
digital
fact family
numerator
equivalent
fraction
divisor
interquartile
range
diameter
hypotenuse
edges
equal to
feet/foot (ft)
parallel
factors
expression
mean /
average
discount
image
eight
equations
hundreds
parallelogram
inequalities
factors
mean
absolute
deviation
double box plot
irrational
eighteen
even
inch (in.)
partitioned
line of
symmetry
greatest
common
factor (gcf)
measures of
center
double dot plot
leg (of a right
triangle)
eleven
fact family
meter
perimeter
meter (m)
hundredths
median
equivalent
expressions
line of best fit
equal (to)
fourths
metric system
place value
multiples
improper
fractions
mode
gratuity / tip
linear
association
faces
greater
than
number line
polygon
multiplicative least common
inverse
denominator
outliers
integer
linear equations
fewer
half circle
number name
product
numerator
least common
multiple
percent
lateral face
linear functions
fifteen
half hour
pentagon
quotient
obtuse angle
metric system
quadrant
lateral surface
negative
association
five
halves
picture graph
scale
quotient
mixed
numbers
quartiles
like terms
nonlinear
association
four
hour
place value
unit
remainder
ordered pairs
range
markdown/
markup
positive
association
fourteen
length
quadrilateral
variable
right angle
parentheses
ratio
opposites
pre-image
greater (than)
(more)
less than
yard (yd)
whole
numbers
yard (yd)
plane
scaling
percent change
Pythagorean
Theorem
heavy/heavier
odd
prime
factorization
surface area
percent
decrease/increase
reflection
height
ones
product
term (in a
sequence)
percent equation
relative
frequency
Back to Table of Contents
Page 114 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
NCSD Math Vocabulary K - 8
Kindergarten
First Grade Second Grade Third Grade Fourth Grade
Fifth Grade
Sixth Grade
Seventh Grade
Eighth Grade
hexagon
place value
quadrant
vertices /
vertex
percent error
rotation
length
quarter
circle
quotient
volume
percent proportion
scatterplot
less (than) (least)
quarters
rule
pi
scientific
notation
lighter
rectangle
scalene
triangle
principle
slope
long/longer
rectangular
prism
scaling
proportion
slope-intercept
form
narrow/narrower
sides
simplest form
proportional
solve
(an equation)
nine
sum
solid figure
radius
substitution
nineteen
symbol
squared
rate
transformation
number
tens
standard form
rational number
translation
number sentence
trapezoid
tenths
repeating decimal
transversal
one
triangular
prism
thousandths
sales tax
two-way table
rectangle
unknown
vertex
selling price
x-intercept
seven
vertex
volume
semi circle
y-intercept
seventeen
x-axis
simple interest
short/shorter
x-coordinate
slant height
y-axis
supplementary
angles
y-coordinate
terminating
decimal
sides
six
sixteen
unit rate
sort
vertical angles
sphere
square
Back to Table of Contents
Page 115 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
NCSD Math Vocabulary K - 8
Kindergarten
First Grade Second Grade Third Grade Fourth Grade
Fifth Grade
Sixth Grade
Seventh Grade
Eighth Grade
subtraction (take
away, minus)
tall/taller
ten
thirteen
three
triangle
twelve
twenty
two
vertices
weight
wide/wider
width
zero
Back to Table of Contents
Page 116 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Kindergarten Critical Math Vocabulary
Please note: These words are critical to be mastered by the end of the year. This chart aligns the critical vocabulary to our
NCSD Pacing Guides. Supporting words are listed at the bottom of each column.
Critical Vocabulary
2-dimensional
Benchmark 1
Benchmark 2
X
X
Benchmark 3
X
X
3-dimensional
X
addition (add, plus, combine)
circle
X
X
X
X
X
cone
corners
X
count
X
X
X
X
X
X
cube
curved
Benchmark 4
X
X
cylinder
X
edges
X
eight
X
X
X
X
eighteen
X
X
X
X
eleven
X
X
X
X
equal (to)
X
X
X
X
X
faces
fewer
X
Back to Table of Contents
X
Page 117 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
fifteen
X
X
X
X
five
X
X
X
X
four
X
X
X
X
fourteen
X
X
X
X
greater (than) (more)
X
X
X
X
heavier
X
height
X
hexagon
X
length
less (than) (least)
X
X
X
X
X
X
X
X
lighter
X
long/longer
X
X
narrower
nine
X
X
X
X
nineteen
X
X
X
X
number
X
X
X
X
X
X
X
X
X
number sentence
one
X
X
rectangle
X
X
seven
X
X
Back to Table of Contents
X
X
X
Page 118 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
seventeen
X
short/shorter
X
X
X
X
X
sides
X
X
X
six
X
X
X
X
sixteen
X
X
X
X
X
sort
X
sphere
square
X
X
X
subtraction (take away, minus)
X
taller
X
X
ten
X
X
X
X
thirteen
X
X
X
X
three
X
X
X
X
triangle
X
X
twelve
X
X
X
X
twenty
X
X
X
X
two
X
X
X
X
vertices
X
X
X
X
weight
X
wider
X
Back to Table of Contents
Page 119 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
X
width
X
zero
X
SUPPORTING
TIER 1 WORDS
flat
color
SUPPORTING
TIER 2 WORDS
alike
attribute
different
group
large
model
objects
order
same
sequence
shape
small
straight
above
alike
attribute
behind
below
beneath
beside
combine
compose
decompose
different
group
in back of
in front of
large
measure
next to
objects
on the bottom
on top
order
over
same
sequence
shape
small
to the left
to the right
under
Back to Table of Contents
X
X
flat
combine
facts
group
measure
objects
order
alike
attribute
combine
compose
create
decompose
different
facts
group
model
objects
order
roll
same
sequence
shape
solid
stack
straight
Page 120 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
SUPPORTING TIER 3 WORDS
digits
make
rhombus
trapezoid
digits
equations
make
rhombus
symbol
take apart
trapezoid
digits
equations
make
symbol
take apart
digits
equations
make
ones
symbol
take apart
tens
SUPPORTING CONCEPTS
how many
how many
how many
how many
SUPPORTING STRATEGIES
counting on
ten frame
counting on
ten frame
counting on
ten frame
counting on
ten frame
Back to Table of Contents
Page 121 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
1st Grade Critical Math Vocabulary
Please note: These words are critical to be mastered by the end of the year. This chart aligns the critical vocabulary to our
NCSD Pacing Guides. Supporting words are listed at the bottom of each column.
Critical Vocabulary
Benchmark 1
Benchmark 2
2-dimensional
X
3-dimensional
X
Benchmark 3
X
analog
bundle
Benchmark 4
X
X
circle
compose
X
X
X
X
decompose
X
X
X
X
difference
X
X
X
X
digit
X
X
digital
X
equal to
equations
X
even
X
X
X
X
X
fact family
fourths
X
greater than
X
half circle
X
Back to Table of Contents
Page 122 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
X
half hour
X
halves
X
hour
length
X
X
less than
odd
X
ones
X
place value
X
quarter circle
X
quarters
X
rectangle
X
rectangular prism
X
sides
X
sum
X
X
X
trapezoid
X
triangular prism
X
unknown
X
X
symbol
tens
X
X
X
vertex
Back to Table of Contents
Page 123 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
Supporting Vocabulary
addition (add, plus,
combine)
longer
longest
subtraction (take
away, minus)
Back to Table of Contents
false
true
Page 124 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
2nd Grade Critical Math Vocabulary
Please note: These words are critical to be mastered by the end of the year. This chart aligns the critical vocabulary to our
NCSD Pacing Guides. Supporting words are listed at the bottom of each column.
Critical Vocabulary
addend
Benchmark 1
Benchmark 3
X
X
X
centimeter (cm)
X
compose
X
customary system
decompose
X
digital
X
expanded form
X
fact family
X
X
X
X
X
feet/foot
hundreds
Benchmark 4
X
analog
base ten numerals
Benchmark 2
X
inch (in)
X
meter
X
metric system
X
number line
X
number name
X
Back to Table of Contents
X
Page 125 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
X
pentagon
place value
X
picture graph
X
quadrilateral
X
X
yard
Supporting Vocabulary
columns
digits
equations
equivalent
expressions
ones
rows
sum
symbol
tens
unknown
whole numbers
Back to Table of Contents
a.m.
dime
dollar bill
equations
equivalent
expression
hour
hour hand
minute hand
minutes
nickel
p.m.
penny
quarter
quarter past
sum
symbol
unknown
whole numbers
angle
data
data table
edges
equal shares
equations
equivalent
expression
faces
fourths
fraction
halves
horizontal
key
line plot
partition
quarters
rectangular prims
scale
sum
survey
symbol
tally mark
thirds
unknown
vertical
whole numbers
equations
equivalent
expression
length
measuring tape
meter stick
ruler
symbol
unit
unknown
yard stick
Page 126 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
3rd Grade Critical Math Vocabulary
Please note: These words are critical to be mastered by the end of the year. This chart aligns the critical vocabulary to our
NCSD Pacing Guides. Supporting words are listed at the bottom of each column.
Critical Vocabulary
angle(s)
Benchmark 1
Benchmark 2
Benchmark 3
Benchmark 4
X
X
denominator
division
X
X
equal part
X
X
factor
X
X
X
X
X
fraction
X
label
X
X
liquid volume
X
multiplication
X
X
X
numerator
parallel
X
parallelogram
X
X
partitioned
perimeter
X
place value
X
polygon
X
X
X
product
Back to Table of Contents
X
X
X
Page 127 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
X
quotient
X
X
X
scale
unit
X
variable
whole numbers
SUPPORTING WORDS
X
fact family
quadrilateral
sum
difference
SUPPORTING STRATEGY
area
array
difference
fact family
interpret
multiple
sum
X
X
X
X
X
X
X
X
X
array
equivalent
fact family
multiple
area
array
fact family
feet
inches
multiple
number line
Back to Table of Contents
Page 128 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
4th Grade Critical Math Vocabulary
Please note: These words are critical to be mastered by the end of the year. This chart aligns the critical vocabulary to our
NCSD Pacing Guides. Supporting words are listed at the bottom of each column.
Critical Vocabulary
Benchmark 1
Benchmark 2
Benchmark 3
X
acute angle
algorithm
Benchmark 4
X
X
angle
common denominator
X
decimal
X
decimal point
X
dividend
X
divisor
X
X
endpoint (vertex)
X
equivalent fraction
factors
X
X
inequalities
X
X
X
X
line of symmetry
X
meter (m)
multiples
X
multiplicative inverse
X
X
X
X
X
numerator
Back to Table of Contents
Page 129 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
X
obtuse angle
quotient
X
X
remainder
X
right angle
X
yard (yd)
Supporting Vocabulary
area model
array
expanded form
expression
operation
place value
product
Back to Table of Contents
area
composite
formula
perimeter
prime
X
denominator
hundredths
improper fractions
mixed numbers
capacity
centimeter
cup
customary system
degree
gallon
gram
hour
intersecting lines
interval (time)
kilogram
line
line plot
line segment
liter
metric system
milliliter
minute
ounce
parallel lines
pattern
perpendicular lines
pint
point
pound
protractor
quart
ray
Page 130 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
5th Grade Critical Math Vocabulary
Please note: These words are critical to be mastered by the end of the year. This chart aligns the critical vocabulary to our
NCSD Pacing Guides. Supporting words are listed at the bottom of each column.
Critical Vocabulary
Benchmark 1
Benchmark 2
braces
X
X
brackets
X
X
Benchmark 3
X
capacity
composite
Benchmark 4
X
convert
X
coordinate plane
X
cubed
X
X
X
customary system
dividend
X
X
divisor
X
X
expression
X
X
factors
X
X
X
greatest common factor (GCF)
hundredths
X
X
X
X
improper fractions
least common denominator
X
least common multiple
X
Back to Table of Contents
Page 131 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
X
metric system
X
mixed numbers
X
X
ordered pairs
parentheses
X
X
X
plane
prime factorization
X
product
X
X
quadrant
quotient
X
X
rule
X
X
X
X
X
scalene triangle
scaling
X
simplest form
X
X
X
X
solid figure
squared
X
standard form
X
X
tenths
X
X
thousandths
X
X
X
vertex
X
volume
Back to Table of Contents
X
Page 132 of 146, Revised by committee June 2015
Natrona County School District Mathematics Curriculum
x-axis
X
x-coordinate
X
y-axis
X
y-coordinate
X
Supporting Words
(These words are introduced or reinforced)
algorithm
base
decimal point (and)
expanded form
inequalities
number line
remainder
Back to Table of Contents
algorithm
decimal point (and)
denominator
expanded form
inequalities
multiples
number line
numerator
composite figure
cubic
denominator
height
measure
number line
numerator
acute angle
area
axis
composite figure
cube
cubic
edges
equilateral triangle
faces
height
intersecting lines
isosceles triangle
kilo
line plot
measure
measurement unit
milli
obtuse angle
obtuse triangle
parallel
parallelogram
perimeter
polygon
quadrilateral
rhombus
right angle
trapezoid
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Natrona County School District Mathematics Curriculum
NCSD Common Assessment Information
2015-2016 NCSD Math Benchmark Summary K-5
The purpose of these common benchmark assessments is to monitor progress towards mastery of standards.
IEP and 504 accommodations should be followed according to the student plan.
Benchmarks are to be administered to all students, regardless of IEP or 504, without review.
Grade
Benchmark 1 Benchmark 2
Benchmark 3
Benchmark 4
Guidelines for all students
11 questions
14 questions
13 questions
13 questions
Kindergarten
Kindergarten
Interviews
Each benchmark is given one-on-one.
Grade 1
Read Aloud
16 questions
16 questions
14 questions
15 questions
Grade 2
Independent
20 questions
22 questions
23 questions
19 questions
45-60 minutes
Grade 3
Independent
24 questions
24 questions
25 questions
22 questions
Teacher
Discretion
45-60 minutes
Grade 4
Independent
45-60 minutes
Grade 5
Independent
Grade 1
Each benchmark can be read aloud in a whole
group setting or smaller learning
stations. Students will need scratch paper, pencil,
or whiteboard.
Grades 2 - 5
Second grades must read all benchmarks on their
own.
Teachers can read a vocabulary word but NOT
give the definition or an example.
Students will need scratch paper, pencil, or
whiteboard.
22 questions
23 questions
30 questions
26 questions
No calculators are allowed on any benchmark.
24 questions
23 questions
25 questions
45-60 minutes
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18 questions
Each benchmark assessment is to be completed in
one 45-60 minute class session. This is done in
order to build fluency and use effective and
efficient strategies.
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2015-2016 NCSD Math Benchmark Summary Course 1 – Course 3 (Grades 6-8)
The purpose of these common benchmark assessments is to monitor progress towards mastery of standards.
IEP and 504 accommodations should be followed according to the student plan.
Benchmarks are to be administered to all students, regardless of IEP or 504, without review.
Grade / Course
Grade 6-Course 1
Benchmark 1
Benchmark 2
Benchmark 3
Benchmark 4
17 Questions
20 Questions
22 Questions
18 Questions
Calculator: no
Calculator: yes
Calculator: yes
Calculator: yes
Guidelines for all students
Grade 6-8
Independent
-Teachers can read a vocabulary word but NOT give the
definition or an example.
Time: 40-60 minute
Grade 7-Course 2
Independent
18 Questions
18 Questions
18 Questions
20 Questions
Calculator: no
Calculator: yes
Calculator: yes
Calculator: yes
-The benchmark assessment is designed to be completed
in one 40-60 minute class period; the reason being is to
build fluency and to use effective and efficient strategies.
Time: 40-60 minute
Grade 7
Accelerated
Independent
-Use of calculators is noted at the beginning of each
benchmark.
18 Questions
18 Questions
22 Questions
18 Questions
Calculator: yes
Calculator: yes
Calculator: yes
Calculator: yes
21 Questions
21 Questions
19 Questions
20 Questions
Calculator: no
Calculator: yes
Calculator: yes
Calculator: yes
Time: 40-60 minute
Grade 8- Course 3
Independent
Time: 40-60 minute
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Natrona County School District Mathematics Curriculum
2015-2016 NCSD Math Benchmark Summary High School
Benchmarks are to be administered to all students without review.
Course
Benchmark 1
Benchmark 2
Guidelines for all students
34 Questions
34 Questions
Independent
Calculator: yes
Calculator: yes
Time: 40-60 minute
Testing window:
Mid-point of course
Testing window:
End of Course
-IEP and 504 accommodations should be
followed according to the student plan.
22 Questions
34 Questions
●
●
●
Independent
Calculator: yes
Calculator: yes
Time: 40-60 minute
Testing window:
Mid-point of course
Testing window:
End of Course
30 Questions
26 Questions
Independent
Calculator: yes
Calculator: yes
Time: 40-60 minute
Testing window:
Mid-point of course
Testing window:
End of Course
Algebra 1
Geometry
Algebra 2
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High School
Scientific or Graphing calculator allowed
Graph paper allowed
Scratch paper allowed
Use of textbook and/or class notes NOT allowed
-The benchmark assessment is designed to be
completed in one 40-60 minute class period, the
reason being is to build fluency and to use
effective and efficient strategies.
Geometry
Provide formula sheet from back of Glencoe
geometry book or provide a copy (see next
page).
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Natrona County School District Mathematics Curriculum
Goemetry Formula Sheet
Back to Table of Contents
Page 137 of 146, Revised by committee June
2015
Natrona County School District Mathematics Curriculum
Common Core State Standards for Mathematics Resources and Visuals
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Natrona County School District Mathematics Curriculum
Standards of Mathematical Practice
Math Practice Standard 1: Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution.
They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a
solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler
forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if
necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window
on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between
equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for
regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem.
Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does
this make sense?" They can understand the approaches of others to solving complex problems and identify correspondences between
different approaches.
Math Practice Standard 2: Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary
abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent
it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—
and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols
involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved;
attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and
objects.
Math Practice Standard 3: Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing
arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to
analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate
them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into
account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two
plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it
is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments
can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine
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Natrona County School District Mathematics Curriculum
domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense,
and ask useful questions to clarify or improve the arguments.
Math Practice Standard 4: Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the
workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might
apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to
solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students
who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that
these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such
tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw
conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make
sense, possibly improving the model if it has not served its purpose.
Math Practice Standard 5: Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and
paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions
about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically
proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors
by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can
enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically
proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of
concepts.
Math Practice Standard 6: Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in
their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They
are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate
accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary
grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine
claims and make explicit use of definitions.
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Natrona County School District Mathematics Curriculum
Math Practice Standard 7: Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and
seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes
have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In
the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a
geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift
perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects.
For example, they can see 5 - 3(x -y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5
for any real numbers x and y.
Math Practice Standard 8: Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper
elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude
they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line
through (1, 2) with slope 3, middle school students might abstract the equation (y - 2)/(x - 1) = 3. Noticing the regularity in the way terms
cancel when expanding (x - 1)(x + 1), (x - 1)(x2 + x + 1), and (x - 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a
geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to
the details. They continually evaluate the reasonableness of their intermediate results.
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Natrona County School District Mathematics Curriculum
Crosswalk of Common Core Instructional Shifts: Mathematics
Both the 6 instructional shifts articulated by the NY State Department of Education and the 3 instructional shifts outlined by Student Achievement Partners help
educators understand the major changes required by the Common Core in terms of curricular materials and classroom instruction in Mathematics.
6 Shifts: EngageNY
www.engageny.org
3 Shifts: Student Achievement Partners
www.achievethecore.org
1: Focus: Teachers use the power of the eraser and significantly narrow and deepen the scope of how time
and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are
prioritized in the standards so that students reach strong foundational knowledge and deep conceptual
understanding and are able to transfer mathematical skills and understanding across concepts and
grades.
=
2: Coherence: Principals and teachers carefully connect the learning within and across grades so that,
for example, fractions or multiplication spiral across grade levels and students can build new
understanding onto foundations built in previous years. Teachers can begin to count on deep conceptual
understanding of core content and build on it. Each standard is not a new event, but an extension of
previous learning.
1: Focus strongly where the Standards focus
2: Coherence: Think across grades, and link to major topics
=
within grades
3: Fluency: Students are expected to have speed and accuracy with simple calculations; teachers structure
class time and/or homework time for students to memorize, through repetition, core functions (found in the
attached list of fluencies) such as multiplication tables so that they are more able to understand and
manipulate more complex concepts.
4: Deep Understanding: Teachers teach more than “how to get the answer” and instead support students’
ability to access concepts from a number of perspectives so that students are able to see math as more than
a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of core
math concepts by applying them to new situations as well as writing and speaking about their
understanding.
3: Rigor: Require fluency, application, and deep
understanding
5: Application: Students are expected to use math and choose the appropriate concept for application
even when they are not prompted to do so. Teachers provide opportunities at all grade levels for students to
apply math concepts in “real world” situations. Teachers in content areas outside of math, particularly
science, ensure that students are using math – at all grade levels – to make meaning of and access content.
=
6: Dual Intensity: Students are practicing and understanding. There is more than a balance between
these two things in the classroom – both are occurring with intensity. Teachers create opportunities for
students to participate in “drills” and make use of those skills through extended application of math
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concepts. The amount of time and energy spent practicing andBack
understanding
learning
environments is
driven by the specific mathematical concept and therefore, varies throughout the given school year.
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Natrona County School District Mathematics Curriculum
Long-Range Plan
SUBJECTS
13/14
14/15
15/16 16/17 17/18 18/19 19/20 20/21 21/22 22/23
ELA
C
VC
R/A
VA
C
VC
R/A
VA
Health
C
VC
R/A
VA
C
VC
R/A
VA
PE
C
VC
R/A
VA
C
VC
R/A
Social Studies
C
VC
R/A
VA
C
VC
R/A
Fine & Performing Arts
C
VC
R/A
VA
C
VC
R/A
Science
C
VC
R/A
VA
C
VC
Foreign Language
C
VC
R/A
VA
C
VC
C
VC
R/A
VA
C
C
VC
R/A
VA
C
Math
C, A
VC, VA
Career & Vocational Education
VA
KEY
C=DEVELOPMENT OF CURRICULUM
DOCUMENTS
VC=IMPLEMENTATION & VALIDATION OF
NEW CURRICULUM
R=RESOURCE SELECTION
A=DEVELOPMENT OF ASSESSMENTS
VA=IMPLEMENTATION & VALIDATION OF
ASSESSMENTS
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NCSD Standards-Based Terminology
District Guaranteed & Viable Curriculum
The District’s guaranteed & viable curriculum is identified as the learning curriculum that guarantees an equal
opportunity for the learning for all students. It also guaranteed adequate time for educators to teach content and for
students to learn it. It guarantees that the curriculum being taught is the same curriculum that will be assessed. It is
viable when adequate time is ensured to teach all determined essential content. The district’s curriculum is composed
of identified learning components that are the essential learning that students must know or do in order to perform at
the mastery level of the identified standards.
State & District Content Standards
The state & district content standards are the minimum content expectations that students must meet as defined by the
state board of education. They provide a common understanding among educators as to what students should learn at
particular grades. The standards are not the curriculum.
District Vertical Pacing Guide
A district vertical Pacing Guide is the purposeful sequencing of a subject area’s route for teaching and learning
expectations across multiple developmental stages, ages or vertical grade levels. The progressions illustrate progress
toward the mastery of content skills as students move throughout their K-12 learning experience. The exiting stage is
defined as college and career readiness for the graduate.
District Grade Level Pacing Guide
A district grade-level pacing guide is the district timeline for logical and progressive sequencing showing what a grade
level content area should master over the course of an academic school year. Pacing guides include the timeline for
interim benchmark assessments and summative grade level assessments for the district and/or the state.
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Natrona County School District Mathematics Curriculum
NCSD Standards-Based Terminology
Classroom Curriculum Map
A classroom curriculum map is the application of the District Grade Level Pacing Guide in an expanded format to guide
an educator in mapping out the essential outcomes of the grade level curriculum while differentiating to the needs of
their students. It is designed to provide an overall picture of the what, the when and the length of relevant content
outcomes to be mastered during a school year.
Classroom Unit
A classroom unit targets the learning of outcomes and their components over a cycle/chunk of a few days to a few
weeks. It contains all three stages for a learning cycle/chunk:
 Desired results/learning
 Assessment/evidence
 Learning plan
Classroom Lesson Plan
The classroom lesson plan is a detailed instructional lesson that is used to plan and guide the daily learning activities.
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