Lamoille North Mathematics Curriculum
Vision/Mission
The LNSU community is committed to providing a mathematical experience that encompasses Language,
Number Sense, Problem Solving and Perseverance. As a student progresses Pre-K-12, they will discover the
interconnectedness of the world through the lens of mathematics. Students will respect the importance and value
of mathematical reasoning while becoming productive citizens and lifelong learners.
Big Idea: The Language of Mathematics
Big Idea: Mathematical Habits of Mind
Mathematics is a Universal Language that is
understood and communicated through reading
writing, speaking and listening.
Perseverance is essential for learning
mathematics. Mathematical disequilibrium builds
confidence through a celebration of errors and
risk-taking.
Big Idea: Application and Problem Solving
Big Idea: Number Sense
Number Sense is the foundation for mathematics.
Mathematical thinking requires a progression
from a concrete to an abstract understanding of
number.
Problem Solving is an effective application of
mathematics. Mathematical Problem Solving is the
synthesis and transfer of concepts and skills to
consider and evaluate all possible solutions.
Mathematics Learning Principles
The LNSU Learning Principles are an essential component of the LNSU mathematics curriculum. Research-based and closely
aligned to the Common Core’s Standards for Mathematical Practice, these Learning Principles will help guide our supervisory union
to common teaching practices inherent in excellent mathematics instruction. Following these Principles in conjunction with the
LNSU Cross-Curricular Learning Principals, teachers in the LNSU will support students in developing a strong understanding of
math concepts from Pre-K through 12th grade. Successfully implemented, these Learning Principles will help ensure that students
throughout our supervisory union meet the grade-level benchmarks in mathematics. In order to achieve the results we seek, all
teachers must have a firm understanding of mathematics content and the history of mathematics. Differentiated learning
opportunities for students must be provided by content, process and product, taking into consideration readiness, interest and
learning profile. Teachers will need to monitor student progress through the six levels of knowing (Sharma, 2008) in order to ensure
student mastery of the material. Sufficient time must be devoted to mathematics instruction in all educational settings, so that
students get the practice they need to work towards solid understanding and an ability to transfer and apply their mathematical
knowledge. With these Principles as our guide, schools across the LNSU will be able to provide consistently excellent instruction in
mathematics to meet our goals at all levels.
Lamoille North Mathematics Curriculum
Page |1
Learning Principles: Mathematics
1. Students learn mathematics best when they make sense of problems and persevere in solving them with
precision.
This principle is applied instructionally when

teachers engage students in authentic educational opportunities for practical application of
mathematics that connects to students’ daily lives.

students analyze givens, constraints, relationships and goals before attempting to solve problems.

students develop and communicate plans for solving problems.

students check their answers using a different method and continually ask themselves, “Does this
make sense?”
2. Students learn mathematics best when they reason abstractly and quantitatively.
This principle is applied instructionally when

teachers and students make sense of quantities and their relationships.

students are able to move fluently between concrete and abstract representations of a problem
solving situation.
3. Students learn mathematics best when they construct viable arguments and critique the reasoning of
others.
This principle is applied instructionally when

teachers engage students in mathematical discourse.

students justify their conclusions and communicate them to others.

students can listen to or read arguments of others, decide whether they make sense, and ask
useful questions to clarify or improve the arguments.
Lamoille North Mathematics Curriculum
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4. Students learn mathematics best when they use and/or construct models to both gain and show their
mathematical understanding.
This principle is applied instructionally when

teachers use common instructional models that are appropriate, efficient, and elegant (i.e. tens
frames, area models, graphs, tables, etc.).

students apply the mathematical models they know to solve problems arising in everyday life,
society, and the workplace.

students routinely interpret their mathematical results in the context of the situation and reflect
on whether the results make sense, possibly improving the model if it has not served its purpose.
5. Students learn mathematics best when they use appropriate tools strategically.
This principle is applied instructionally when

students consider the available tools when solving a
mathematical problem.

teachers learn about, demonstrate and model meaningful
use of technological tools for mathematical problem solving.

students use various technological tools to explore and
deepen their understanding of concepts.
6. Students learn mathematics best when they attend to precision.
This principle is applied instructionally when

teachers and students use and state the meaning of math symbols precisely, including consistent
and appropriate use of the equals sign.

teachers and students understand and use math language consistently in communicating their
thinking and reasoning with others.

students strive to calculate accurately and efficiently.

students use specific units and label quantities appropriately in context to a problem.
Lamoille North Mathematics Curriculum
Page |3
7. Students learn mathematics best when they look for and make use of patterns and structure.
This principle is applied instructionally when

students look for and express patterns in their mathematical reasoning to generalize methods,
formulas and utilize shortcuts to efficiently solve a wide variety of problems.

teachers engage students in the discovery, use and application of our place value system and the
number properties of mathematics.
Lamoille North Mathematics Curriculum
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Pre-Kindergarten
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Pre-Kindergarten Essential Questions:




How are numbers used differently within our environment and what questions can they answer?
How do we know one number is greater than another?
What are some different ways you can make the same number?
How do quantities change when objects are combined or separated?
Pre-Kindergarten Benchmarks
Kindergarten Benchmarks
(What students need to know and be able to do)
Intensive Focus: Number
Intensive Focus: Number
Apply and connect concepts to Number
(a) Know that the last word they state in counting tells how
many up to 10.
(b) Count forward and backward by ones to 10.
(c) 1:1 correspondence in counting objects to 10.
(d) Associate a number with a set of objects and order sets by
quantity.
(e) Use multiple models to identify and make patterns to 5.
(f) Comparing, combining and separating sets of objects using
words such as “more than, less than, same as”.
Lamoille North Mathematics Curriculum
August 16, 2011
Apply and connect concepts to Number
(a)




(b)
(c)
(d)
(e)
Know numbers to 10.
Symbolic form (5)
Sound of the word (“five”)
Visual cluster (subitizing, composing and decomposing) (XXXXX)
Automaticity of addition facts to 10.
Count forward by 1’s from any number within the range of 0 to 100.
Count backwards by 1’s from 30.
Skip count by 10’s to 100.
Demonstrate 1:1 correspondence with a set of objects within 10.
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Updated:
Pre-Kindergarten
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Pre-Kindergarten Essential Questions:





What shapes do we see in our environment?
How can we compare or manipulate 2-dimensional or 3-dimensional shapes?
How can simple shapes be composed to form larger shapes?
How do we measure?
How can we describe the order and position of objects and events?
Pre-Kindergarten Benchmarks
(What students need to know and be able to do)
Intensive Focus: Number
Apply and connect concepts to Number
(a) Identify and locate common shapes (circle,
square and triangle) in their environment.
(b) Describe the order and position of objects from
the child’s perspective, using language such as,
“behind, on top of, next, under, etc.”
(c) Group objects by simple attributes (shape and
size).
(d) Order, compare and describe objects according to
their size.
(e) Place events in a logical sequential order in the
immediate present.
(f) Build pictures and designs by combining two- and
three-dimensional shapes.
Lamoille North Mathematics Curriculum
August 16, 2011
Kindergarten Benchmarks
Intensive Focus: Number
Apply and connect concepts to Number
(a) Identify and name common shapes and figures (2-dimensional drawing of
a shape) regardless of their orientation or overall size (circle, square,
rectangle, hexagon and triangle, sphere, cylinder, cone and cube).
(b) Describe the relative position of objects in their environment using terms
such as above, below, beside, in front of, behind and next to.
(c) Sort shapes by simple attributes.
(d) Compose simple shapes to form larger shapes. For example: Can you join
these two triangles with full sides touching to make a rectangle?
(e) Identify uniform non-conventional units to measure length.
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Updated:
Pre-Kindergarten
Functions and Algebra Concepts
Pre-Kindergarten
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
Pre-Kindergarten Essential Questions:

Can you recognize a pattern?
Pre-Kindergarten Benchmarks
(What students eed to know and be able to do)
Intensive Focus: Number
Apply and connect concepts to Number
(a) Recognize repeating patterns.
Lamoille North Mathematics Curriculum
August 16, 2011
Kindergarten Benchmarks
Intensive Focus: Number
Apply and connect concepts to Number
(a) Recognize, create, and identify patterns with two or more elements.
(b) Compose numbers less than or equal to 10 into pairs in more than one
way. (i.e. by using objects or drawing, 5 = 2 + 3, 5 = 4 + 1)
 Apply the commutative and associative properties
Page |7
Updated:
Kindergarten
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Kindergarten Essential Questions:





How are numbers used differently in our daily lives and what questions can they answer?
How do we know one number is greater or less than another?
What are some different ways you can represent the same number?
What are some ways to efficiently count large collections?
What is equality?
Pre-Kindergarten
Benchmarks
Kindergarten Benchmarks
(What students need to know and be able to do)
First Grade Benchmarks
Intensive Focus: Number
Intensive Focus: Number
Intensive Focus: Additive Reasoning
Apply and connect concepts to Number
Apply and connect concepts to Number
(a) Know that the last word
they state in counting tells
how many up to 10.
(b) Count forward and
backward by ones to 10.
(c) 1:1 correspondence in
counting objects to 10.
(d) Associate a number with a
set of objects and order sets
by quantity.
(e) Use multiple models to
identify and make patterns
to 5.
(f) Comparing, combining and
separating sets of objects
using words such as “more
than, less than, same as”.
(a) Know numbers to 10.
 Symbolic form (5)
 Sound of the word (“five”)
 Visual cluster (subitizing,
composing and decomposing)
(XXXXX)
 Automaticity of addition facts to
10.
(b)Count forward by 1’s from any number
within the range of 0 to 100.
(c) Count backwards by 1’s from 30.
(d) Skip count by 10’s to 100.
(e) Demonstrate 1:1 correspondence
with a set of objects within 10.
Lamoille North Mathematics Curriculum
August 16, 2011
Apply and connect concepts to Addition
(a) Count forward and backward by 1’s to 120, crossing the century mark, starting
and stopping at any given number. In this range, students will read, write, order,
sequence, compare and represent written numbers with objects.
(b) Automaticity for addition facts within 20.
(c) Counting forward and backward (from a multiple of the base number)
i. by 10’s to 120,
ii. by 2’s to 20,
iii. by 5’s to 120.
(d) Count and identify one more, one less and ten more, ten less than a number
within 100.
(e) Add and subtract multiples of ten in the range of 10 to 90 (50 – 30)
(f) Show a representation of up to 3-digit numbers using manipulatives, models,
and visual or abstract representations (place value).
(g) Compare two 3-digit numbers using the symbols for greater than, less than or
equal.
(h) Use the equal sign appropriately to show equality of numbers and expressions.
(i) Writes expressions using addition and subtraction signs.
(j) Use commutative and associative properties as strategies to compose and
decompose numbers in context of addition and subtraction problems.
(k) Demonstrate understanding of part/whole (inverse relationship) of addition and
subtraction.
Page |8
Updated:
Kindergarten
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Kindergarten Essential Questions:



How can simple shapes be composed to form larger shapes?
How can we describe the order and position of objects and events?
How are geometry and measurement used in our daily lives?
Pre-Kindergarten Benchmarks
Kindergarten Benchmarks
First Grade Benchmarks
Intensive Focus: Number
(What students need to know and be able to do)
Intensive Focus: Number
Intensive Focus: Additive Reasoning
Apply and connect concepts to Number
Apply and connect concepts to Number
Apply and connect concepts to Addition
(a) Identify and locate common
shapes (circle, square and
triangle) in their environment.
(b) Describe the order and position of
objects from the child’s
perspective, using language such
as, “behind, on top of, next,
under, etc.”
(c) Group objects by simple attributes
(shape and size).
(d) Order, compare and describe
objects according to their size.
(e) Place events in a logical sequential
order in the immediate present.
(f) Build pictures and designs by
combining two- and threedimensional shapes.
(a) Identify and name common shapes and
figures (2-dimensional drawing of a
shape) regardless of their orientation
or overall size (circle, square, rectangle,
hexagon and triangle, sphere, cylinder,
cone and cube).
(b) Describe the relative position of objects
in their environment using terms such
as above, below, beside, in front of,
behind and next to.
(c) Sort shapes by simple attributes.
(d) Compose simple shapes to form larger
shapes. For example: Can you join
these two triangles with full sides
touching to make a rectangle?
(e) Identify uniform non-conventional units
to measure length.
(a) Sort or classify common shapes and figures
(circles, triangles, squares, rectangles,
rhombi, trapezoids, hexagons, sphere,
cylinder, cone and cube).
(b) Name and draw all basic two-dimensional
shapes.
(c) Compose 2-dimensional shapes and 3dimensional shapes (cube, rectangular
prism) to create a composite shape and
compose new shapes from the composite
shape.
(d) 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.
(e) Tell and write time in hours and half-hours
using analog and digital clocks.
(f) Express the length of an object using uniform
non-conventional units placed end to end
(example, paperclips).
(g) Identify coins and their values (penny, nickel,
dime, quarter).
Lamoille North Mathematics Curriculum
August 16, 2011
Page |9
Updated:
Kindergarten
Standard 7.8 Functions and Algebra Concepts
Kindergarten
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
Kindergarten Essential Questions:


Given a pattern what is coming next?
Given a number, how many different combinations of whole numbers can make a number?
Pre-Kindergarten Benchmarks
Kindergarten Benchmarks
Intensive Focus: Number
(What students need to know and be able to do)
Intensive Focus: Number
Apply and connect concepts to Number
Apply and connect concepts to Number
(a) Recognize repeating
patterns.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Recognize, create, and identify patterns
with two or more elements.
(b) Compose numbers less than or equal to
10 into pairs in more than one way. (i.e.
by using objects or drawing, 5 = 2 + 3, 5 =
4 + 1)Apply the commutative and
associative properties
First Grade Benchmarks
Intensive Focus: Additive Reasoning
Apply and connect concepts to Addition
(a) Identify and extend a variety of
patterns.
(b) Know and apply the meaning of the
equal sign when relating two
equivalent models for addition. Knows
equality/same as between two
expressions (4+1 = 2+3).
(c) Find the value of a place holder
(variable) in an addition equation
relating three whole numbers. 8 + a =
11)
P a g e | 10
Updated:
Grade 1
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
First Grade Essential Questions:






How are numbers used differently in our daily lives and what questions can they answer?
How does the position of a digit affect its quantity?
What are some different ways you can represent the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What are some ways to efficiently count large collections?
What is equality?
First Grade Benchmarks
Kindergarten Benchmarks
Second Grade Benchmarks
Intensive Focus: Number
(What students need to know and be able to do)
Intensive Focus: Additive Reasoning
Intensive Focus: Additive Reasoning
Apply and connect concepts to Number
Apply and connect concepts to Addition
Apply and connect concepts to Addition and Subtraction
Know numbers to 10.
Symbolic form (5)
Sound of the word (“five”)
Visual cluster (subitizing,
composing and decomposing)
(XXXXX)
 Automaticity of addition facts
to 10.
(b) Count forward by 1’s from any
number within the range of 0 to 100.
(c) Count backwards by 1’s from 30.
(d) Skip count by 10’s to 100.
(e) Demonstrate 1:1
correspondence
with a set of objects within 10.
(a)



(a) Count forward and backward by 1’s to 120, crossing the century
mark, starting and stopping at any given number. In this range,
students will read, write, order, sequence, compare and
represent written numbers with objects.
(b) Automaticity for addition facts within 20.
(c) Counting forward and backward (from a multiple of the base
number)
i. by 10’s to 120,
ii. by 2’s to 20,
iii. by 5’s to 120.
(d) Count and identify one more, one less and ten more, ten less
than a number within 100.
(e) Add and subtract multiples of ten in the range of 10 to 90 (50 –
30)
(f) Show a representation of up to 3-digit numbers using
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Automaticity of addition and subtraction facts
to 20.
(b) Count forward and backward from a given
number to a new century (for example: 295 to
320, 620 to 595, etc.) within 1000.
(c) Read and write numbers within 1000 using
base-ten numerals, number names and
expanded form.
(d) Compare two 3-digit numbers based on
meanings of hundreds, tens and ones using
greater than, less than and equals signs.
(e) Forward and backward skip count by 2, 5, 10,
and 100 from any given number within 1000.
(f) Forward and backward skip count by 25’s
(quarters) in the context of money.
(g) Add and subtract up to four 2-digit numbers
using strategies based on place value and
properties of operations.
(h) Add and subtract within 1000, using various
P a g e | 11
Updated:
representations of place value and properties
manipulatives, models, and visual or abstract representations
of addition and subtraction. Relate the
(place value).
strategy to a written equation. (using a
(g) Compare two 3-digit numbers using the symbols for greater
number line, partial addends, composing and
than, less than or equal.
decomposing numbers, etc.).
(h) Use the equal sign appropriately to show equality of numbers
(i) Determine whether a group of objects (up to
20) has an odd or even number of members,
and expressions.
e.g. by pairing objects or counting them by 2s;
(i) Writes expressions using addition and subtraction signs.
write an equation to express an even number
(j) Use commutative and associative properties as strategies to
as a sum of two equal addends.
compose and decompose numbers in context of addition and
(j) Connects the multiplication sign with the
subtraction problems.
concept of repeated addition and “groups of”.
(k) Demonstrate understanding of part/whole (inverse relationship) (k) Automaticity of multiplication facts of 1, 2, 5
and 10.
of addition and subtraction.
(l) 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?
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 12
Updated:
Grade 1
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
First Grade Essential Questions:




How can simple shapes be composed and decomposed to form a new shape?
What can we use to measure and how do we use it?
How are geometry and measurement used in our daily lives?
What is the importance of lines, angles and shapes within our environment?
First Grade Benchmarks
Kindergarten Benchmarks
Second Grade Benchmarks
Intensive Focus: Number
(What students need to know and be able to do)
Intensive Focus: Additive Reasoning
Intensive Focus: Additive Reasoning
Apply and connect concepts to Number
Apply and connect concepts to Addition
Apply and connect concepts to Addition and Subtraction
(a) Identify and name common
shapes and figures (2dimensional drawing of a shape)
regardless of their orientation or
overall size (circle, square,
rectangle, hexagon and triangle,
sphere, cylinder, cone and
cube).
(b) Describe the relative position of
objects in their environment
using terms such as above,
below, beside, in front of, behind
and next to.
(c) Sort shapes by simple attributes.
(d) Compose simple shapes to form
larger shapes. For example: Can
you join these two triangles with
full sides touching to make a
rectangle?
(e) Identify uniform nonconventional units to measure
length.
(a) Sort or classify common shapes and figures (circles,
triangles, squares, rectangles, rhombi, trapezoids,
hexagons, sphere, cylinder, cone and cube).
(b) Name and draw all basic two-dimensional shapes.
(c) Compose 2-dimensional shapes and 3-dimensional
shapes (cube, rectangular prism) to create a composite
shape and compose new shapes from the composite
shape.
(d) 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.
(e) Tell and write time in hours and half-hours using
analog and digital clocks.
(f) Express the length of an object using uniform nonconventional units placed end to end (example,
paperclips).
(g) Identify coins and their values (penny, nickel, dime,
quarter).
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Name and identify circles, triangles, squares, rectangles,
rhombi, trapezoids, hexagons, spheres, cylinders, cones
and cubes in the environment.
(b) Name, draw, and identify triangles, quadrilaterals and
pentagons by attributes, number of angles and/or
number of equal sides.
(c) Use models to calculate perimeter in the context of
addition and area using a standard square unit.
(d) Partition circles and rectangles into two, three or four
equal shares; name the shares using the words halves,
thirds, half of, a third of, etc.
(e) Rename a whole as two halves, three thirds, four
fourths.
(f) Show through models or drawings that equal shares of
an identical whole need not have the same shape.
(g) Use area and set models to name proper fractions:
halves, thirds, and fourths; tenths and hundredths in the
context of money.
(h) Tell and write time from analog and digital clocks to the
nearest 5 minutes, using a.m. and p.m.
(i) Measure the length of an object by selecting and using
appropriate tools and standard units of measure (inches
and centimeters).
P a g e | 13
Updated:
Grade 1
Functions and Algebra Concepts
Grade 1
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
First Grade Essential Questions:


In what ways do patterns help us to be efficient in math?
What does the equals sign mean?
Kindergarten Benchmarks
First Grade Benchmarks
Intensive Focus: Number
(What students need to know and be able to do)
Intensive Focus: Additive Reasoning
Intensive Focus: Additive Reasoning
Apply and connect concepts to Number
Apply and connect concepts to Addition
Apply and connect concepts to Addition and Subtraction
(a) Recognize, create, and identify
patterns with two or more
elements.
(b) Compose numbers less than or
equal to 10 into pairs in more than
one way. (i.e. by using objects or
drawing, 5 = 2 + 3, 5 = 4 + 1
 Apply the commutative and
associative properties
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Identify and extend a variety of patterns.
(b) Know and apply the meaning of the equal
sign when relating two equivalent models
for addition. Knows equality/same as
between two expressions (4+1 = 2+3).
(c) Find the value of a place holder (variable)
in an addition equation relating three
whole numbers. ( 8 + a = 11)
Second Grade Benchmarks
(a) Identify, extend and create a variety of
patterns (linear and non-numeric)
represented in models, tables or sequences
by extending the pattern to the next
element, or finding a missing element
(2,4,6,__ ,10).
(b) Use addition to find the total number of
objects arranged in arrays up to 5 rows
and/or 5 columns; write an equation to
express the total as a sum of equal addends.
(c) 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.
(d) Find the value of a place holder in an
addition equation and subtraction equation
relating three whole numbers. (i.e. find the
missing addend and subtrahend that makes
the equation true in an equation such as 8 +
__= 11, 8 – ___ = 5)
P a g e | 14
Updated:
Grade 2
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Second Grade Essential Questions







How are numbers used differently in our daily lives and what questions can they answer?
How does the position of a digit affect its quantity?
Why do we need zero?
What are some different ways you can represent the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What is multiplication?
What is equality?
First Grade Benchmarks
Second Grade Benchmarks
Third Grade Benchmarks
Intensive Focus: Additive Reasoning
(What students need to know and be able to do)
Intensive Focus: Additive Reasoning
Apply and connect concepts to Addition
Apply and connect concepts to Addition and Subtraction
Apply and connect concepts to multiplication
(a) Automaticity of addition and subtraction facts to
20.
(b) Count forward and backward from a given
number to a new century (for example: 295 to
320, 620 to 595, etc.) within 1000.
(c) Read and write numbers within 1000 using baseten numerals, number names and expanded
form.
(d) Compare two 3-digit numbers based on
meanings of hundreds, tens and ones using
greater than, less than and equals signs.
(e) Forward and backward skip count by 2, 5, 10,
and 100 from any given number within 1000.
(f) Forward and backward skip count by 25’s
(quarters) in the context of money.
(g) Add and subtract up to four 2-digit numbers
using strategies based on place value and
(a) Read, write, order, and compare whole
numbers to 100,000.
(b) Use place value understanding to round
whole numbers to the nearest 10 or 100.
(c) Forward and backward skip count by 3
and 4 from any given number within
1000.
(d) Use commutative, associative and
distributive properties to multiply and
divide.
(e) Fluently multiply and divide whole
numbers with products less than 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.
(f) Automaticity of multiplication facts to 10
x 10.
(g) Multiply one-digit whole numbers by
multiples of ten in the range of 10-90
(a) Count forward and backward by 1’s to
120, crossing the century mark, starting
and stopping at any given number. In
this range, students will read, write,
order, sequence, compare and
represent written numbers with
objects.
(b) Automaticity for addition facts within
20.
(c) Counting forward and backward (from
a multiple of the base number)
i. by 10’s to 120,
ii. by 2’s to 20,
iii. by 5’s to 120.
(d) Count and identify one more, one less
and ten more, ten less than a number
within 100.
(e) Add and subtract multiples of ten in the
range of 10 to 90 (50 – 30)
Lamoille North Mathematics Curriculum
August 16, 2011
Intensive Focus: Multiplicative Reasoning
P a g e | 15
Updated:
(f) Show a representation of up to 3-digit
numbers using manipulatives, models,
and visual or abstract representations
(place value).
(g) Compare two 3-digit numbers using the
symbols for greater than, less than or
equal.
(h) Use the equal sign appropriately to
show equality of numbers and
expressions.
(i) Writes expressions using addition and
subtraction signs.
(j) Use commutative and associative
properties as strategies to compose
and decompose numbers in context of
addition and subtraction problems.
(k) Demonstrate understanding of
part/whole (inverse relationship) of
addition and subtraction.
(e.g. 9 x 80, 5 x 60), using strategies based
properties of operations.
on place value or properties of
(h) Add and subtract within 1000, using various
operations.
representations of place value and properties of (h) Use multiplication and division within 100
addition and subtraction. Relate the strategy to
to solve word problems in situations
a written equation. (using a number line, partial
involving equal groups, arrays, and
measurement quantities, e.g., by using
addends, composing and decomposing numbers,
drawings and equations with a symbol for
etc.).
the unknown number to represent the
(i) Determine whether a group of objects (up to 20)
problem.
has an odd or even number of members, e.g. by (i) Recognize and relate common benchmark
pairing objects or counting them by 2s; write an
fractions (halves, thirds, fourths, sixths,
eighths and tenths) using set, linear
equation to express an even number as a sum of
and/or area models.
two equal addends.
 Write a fraction for in the
(j) Connects the multiplication sign with the
form of the numerator over
concept of repeated addition and “groups of”.
the denominator (a/b).

Show the location of a
(k) Automaticity of multiplication facts of 1, 2, 5 and
benchmark fraction on a
10.
number line. For example,
(l) Solve word problems involving dollar bills,
given ¾, a student will divide
a number line (0 to 1) into 4
quarters, dimes, nickels and pennies, using $
equal parts and label where
and ¢ symbols appropriately. Example: If you
¾ is on that line.
have 2 dimes and 3 pennies, how many cents do
 Generate and use reasoning
to
compare
simple
you have?
equivalent fractions (e.g.
1/2=2/4, 4/6=2/3, 6/6=1)
and non-equivalent fractions
(with same numerator or
the same denominator).
Record the results of the
comparisons
with
the
symbols >, =, <.
 Recognize that comparisons
of two fractions are valid
only when the two fractions
refer to the same whole ( ½
of $10.00 ≠ ½ of $20.00)
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 16
Updated:
Grade 2
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Second Grade Essential Questions:




How can simple shapes be composed and decomposed to form a new shape?
What appropriate tools and standard units can we use to measure and how do we use them?
How are geometry and measurement used in our daily lives?
What is the importance of lines, angles and shapes within our environment?
Second Grade Benchmarks
First Grade Benchmarks
Third Grade Benchmarks
Intensive Focus: Additive Reasoning
(What students need to know and be able to do)
Intensive Focus: Additive Reasoning
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to Addition
Apply and connect concepts to Addition and Subtraction
Apply and connect concepts to multiplication
(a) Sort or classify common shapes and figures
(circles, triangles, squares, rectangles,
rhombi, trapezoids, hexagons, sphere,
cylinder, cone and cube).
(b) Name and draw all basic two-dimensional
shapes.
(c) Compose 2-dimensional shapes and 3dimensional shapes (cube, rectangular
prism) to create a composite shape and
compose new shapes from the composite
shape.
(d) 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.
(e) Tell and write time in hours and half-hours
using analog and digital clocks.
(f) Express the length of an object using
uniform non-conventional units placed end
to end (example, paperclips).
(g) Identify coins and their values (penny,
nickel, dime, quarter).
(a) Name and identify circles, triangles, squares, rectangles, rhombi,
trapezoids, hexagons, spheres, cylinders, cones and cubes in the
environment.
(b) Name, draw, and identify triangles, quadrilaterals and pentagons
by attributes, number of angles and/or number of equal sides.
(c) Use models to calculate perimeter in the context of addition and
area using a standard square unit.
(d) Partition circles and rectangles into two, three or four equal
shares; name the shares using the words halves, thirds, half of, a
third of, etc.
(e) Rename a whole as two halves, three thirds, four fourths.
(f) Show through models or drawings that equal shares of an
identical whole need not have the same shape.
(g) Use area and set models to name proper fractions: halves, thirds,
and fourths; tenths and hundredths in the context of money.
(h) Tell and write time from analog and digital clocks to the nearest 5
minutes, using a.m. and p.m.
(i) Measure the length of an object by selecting and using
appropriate tools and standard units of measure (inches and
centimeters).
(a) Sort polygons (e.g. triangles,
rhombi, rectangles and others) by
their attributes (e.g., having four
sides), and define the subcategory
(e.g. quadrilaterals).
(b) Identify and create congruent
shapes.
(c) Calculate the area of a rectangle
by using the length x width
formula.
(d) Calculate the perimeter of a
rectangle by adding multiple sides
(2 lengths + 2 widths).
(e) Calculate the perimeter and area
of a rectangle through real world
problems with known sides and/or
with one unknown side.
(f) Create a rectangle given a
specified area or perimeter.
(g) Tell time to the nearest minute
using an analog or digital clock and
determine elapsed time.
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 17
Updated:
Grade 2
Functions and Algebra Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
Second Grade Essential Questions:


In what ways do patterns help us to be efficient in math?
How do numeric patterns help us to understand and use place value systems?
First Grade Benchmarks
Second Grade Benchmarks
Intensive Focus: Additive Reasoning
(What students need to know and be able to do)
Intensive Focus: Additive Reasoning
Apply and connect concepts to Addition
Apply and connect concepts to Addition and Subtraction
(a)
(b)
(c)
Identify and extend a variety of
patterns.
Know and apply the meaning of
the equal sign when relating two
equivalent models for addition.
Knows equality/same as between
two expressions (4+1 = 2+3).
Find the value of a place holder
(variable) in an addition equation
relating three whole numbers. 8
+ a = 11)
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Identify, extend and create a variety of patterns
(linear and non-numeric) represented in models,
tables or sequences by extending the pattern to
the next element, or finding a missing element
(2,4,6,__ ,10).
(b) Use addition to find the total number of objects
arranged in arrays up to 5 rows and/or 5 columns;
write an equation to express the total as a sum of
equal addends.
(c) 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.
(d) Find the value of a place holder in an addition
equation and subtraction equation relating three
whole numbers. (i.e. find the missing addend and
subtrahend that makes the equation true in an
equation such as 8 + __= 11, 8 – ___ = 5)
Third Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to multiplication
(a) Identify arithmetic patterns (including
patterns in the addition table or
multiplication table) and explain the
patterns using properties of operations. (i.e.
observe that 4 times a number is always
even, and explain why 4 times a number can
be decomposed into two equal addends.)
(b) Find the value of the variable in a
multiplication equation relating three whole
numbers. (i.e. find the value of the variable
that makes the equation true in each of the
equations 8 x a = 48, 6 x 6 = a)
P a g e | 18
Updated:
Grade 3
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Third Grade Essential Questions






How are numbers used differently in our daily lives and what questions can they answer?
What is a fraction?
What are some different ways you can represent the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What is division?
What is equality?
Second Grade Benchmarks
Third Grade Benchmarks
Fourth Grade Benchmarks
Intensive Focus: Additive Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to Addition and Subtraction
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
(a) Automaticity of addition and
subtraction facts to 20.
(b) Count forward and backward from a
given number to a new century (for
example: 295 to 320, 620 to 595, etc.)
within 1000.
(c) Read and write numbers within 1000
using base-ten numerals, number
names and expanded form.
(d) Compare two 3-digit numbers based
on meanings of hundreds, tens and
ones using greater than, less than and
equals signs.
(e) Forward and backward skip count by 2,
5, 10, and 100 from any given number
within 1000.
(f) Forward and backward skip count by
25’s (quarters) in the context of
money.
(g) Add and subtract up to four 2-digit
numbers using strategies based on
(a) Read, write, order, and compare whole numbers to
100,000.
(b) Use place value understanding to round whole
numbers to the nearest 10 or 100.
(c) Forward and backward skip count by 3 and 4 from
any given number within 1000.
(d) Use commutative, associative and distributive
properties to multiply and divide.
(e) Fluently multiply and divide whole numbers with
products less than 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.
(f) Automaticity of multiplication facts to 10 x 10.
(g) Multiply one-digit whole numbers by multiples of
ten in the range of 10-90 (e.g. 9 x 80, 5 x 60), using
strategies based on place value or properties of
operations.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Read, write, order and compare all
whole numbers through equivalency,
composition, decomposition and/or
place value.
(b) Apply the properties of numbers
(factor, multiple, remainder,
divisibility tests for 2,3,4,5,6,8, and
10).
(c) Automaticity of division facts 1 – 10.
(d) Illustrate and explain the relationship
between multiplication and division
equations using sharing, repeated
subtraction, partitioning and area (up
to four digit dividends and one digit
divisors with remainders).
(e) Multiply two-digit by two-digit and
one digit by four digit numbers
applying the commutative, associative
and distributive properties.
(f) Create a representation of any positive
fractional number as a part to whole
relationship in area, set, or linear
model.
P a g e | 19
Updated:
(h)
(i)
(j)
(k)
(l)
place value and properties of
operations.
Add and subtract within 1000, using
various representations of place value
and properties of addition and
subtraction. Relate the strategy to a
written equation. (using a number
line, partial addends, composing and
decomposing numbers, etc.).
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.
Connects the multiplication sign with
the concept of repeated addition and
“groups of”.
Automaticity of multiplication facts of
1, 2, 5 and 10.
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?
Lamoille North Mathematics Curriculum
August 16, 2011
(h) 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.
(i) Recognize and relate common benchmark
fractions (halves, thirds, fourths, sixths, eighths
and tenths) using set, linear and/or area models.
 Write a fraction for in the form of the
numerator over the denominator
(a/b).
 Show the location of a benchmark
fraction on a number line.
For
example, given ¾, a student will
divide a number line (0 to 1) into 4
equal parts and label where ¾ is on
that line.
 Generate and use reasoning to
compare simple equivalent fractions
(e.g. 1/2=2/4, 4/6=2/3, 6/6=1) and
non-equivalent fractions (with same
numerator
or
the
same
denominator). Record the results of
the comparisons with the symbols >,
=, <.
 Recognize that comparisons of two
fractions are valid only when the two
fractions refer to the same whole ( ½
of $10.00 ≠ ½ of $20.00)
(g) Compose and decompose fractions
into unit fractions by using a visual
fraction model. Example: 3/8 = 1/8
+ 1/8 + 1/8
(h) Add and subtract fractions and mixed
numbers with like denominators.
(i) Writes decimals within the context of
money and common fractions (i.e.
0.25 = ¼ = one quarter = 25 cents)
(j) Students will recognize and generate
equivalent fractions.
(k) Comparing two fractions with
different numerators and
denominators using <, > and =, and
justify their solution using a
representation.
P a g e | 20
Updated:
Grade 3
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Third Grade Essential Questions:




How are geometry and measurement used in our daily lives?
How can we sort, classify and/or compare 2-dimensional or 3-dimensional shapes?
What appropriate tools and standard units can we use to measure and how do we use them?
How are perimeter and area measured or calculated?
Second Grade Benchmarks
Third Grade Benchmarks
Fourth Grade Benchmarks
Intensive Focus: Additive Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to Addition and Subtraction
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
(a) Name and identify circles, triangles, squares,
rectangles, rhombi, trapezoids, hexagons, spheres,
cylinders, cones and cubes in the environment.
(b) Name, draw, and identify triangles, quadrilaterals
and pentagons by attributes, number of angles
and/or number of equal sides.
(c) Use models to calculate perimeter in the context of
addition and area using a standard square unit.
(d) Partition circles and rectangles into two, three or
four equal shares; name the shares using the words
halves, thirds, half of, a third of, etc.
(e) Rename a whole as two halves, three thirds, four
fourths.
(f) Show through models or drawings that equal shares
of an identical whole need not have the same shape.
(g) Use area and set models to name proper fractions:
halves, thirds, and fourths; tenths and hundredths in
the context of money.
(h) Tell and write time from analog and digital clocks to
the nearest 5 minutes, using a.m. and p.m.
(i) Measure the length of an object by selecting and
using appropriate tools and standard units of
measure (inches and centimeters).
(a) Sort polygons (e.g. triangles, rhombi,
rectangles and others) by their attributes
(e.g., having four sides), and define the
subcategory (e.g. quadrilaterals).
(b) Identify and create congruent shapes.
(c) Calculate the area of a rectangle by using
the length x width formula.
(d) Calculate the perimeter of a rectangle by
adding multiple sides (2 lengths + 2
widths).
(e) Calculate the perimeter and area of a
rectangle through real world problems
with known sides and/or with one
unknown side.
(f) Create a rectangle given a specified area
or perimeter.
(g) Tell time to the nearest minute using an
analog or digital clock and determine
elapsed time.
(a) Identify and describe the
properties of angles.
(b) Identify line-symmetric figures
and draw lines of symmetry.
(c) Express area measurements in
proper notation of square units.
(d) Match congruent figures using
reflections, translations, or
rotations.
(e) Use four operations to solve word
problems involving distances,
intervals of time, liquid volumes,
masses of objects and money.
(f) Solve problems using the
Cartesian coordinate system
(Quadrant I) to locate coordinates
and to represent data from tables.
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 21
Updated:
Grade 3
Functions and Algebra Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
Third Grade Essential Questions:


In what ways do patterns help us to be efficient in math?
How do numeric patterns help us to understand place value systems?
Second Grade Benchmarks
Third Grade Benchmarks
Fourth Grade Benchmarks
Intensive Focus: Additive Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to Addition and Subtraction
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
(a) Identify, extend and create a variety of
patterns (linear and non-numeric)
represented in models, tables or sequences
by extending the pattern to the next
element, or finding a missing element
(2,4,6,__ ,10).
(b) Use addition to find the total number of
objects arranged in arrays up to 5 rows
and/or 5 columns; write an equation to
express the total as a sum of equal
addends.
(c) 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.
(d) Find the value of a place holder in an
addition equation and subtraction equation
relating three whole numbers. (i.e. find the
missing addend and subtrahend that makes
the equation true in an equation such as 8 +
__= 11, 8 – ___ = 5)
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Identify arithmetic patterns (including
patterns in the addition table or
multiplication table) and explain the
patterns using properties of
operations. (i.e. observe that 4 times a
number is always even, and explain
why 4 times a number can be
decomposed into two equal addends.)
(b) Find the value of the variable in a
multiplication equation relating three
whole numbers. (i.e. find the value of
the variable that makes the equation
true in each of the equations 8 x a = 48,
6 x 6 = a)
(a) Identify, describe, and extend arithmetic
patterns using concrete materials and
tables to write a rule in words or symbols
to find the next term.
(b) Find the value of the variable in a
multiplication equation and division
equation relating three whole numbers.
(i.e. find the value of the variable that
makes the equation true in each of the
equations 8 x a = 48, 6 x 6 = a, 24 ÷ a =
3)
(c) Writes simple linear algebraic expressions
involving any one of the four operations
using whole numbers and variables.
P a g e | 22
Updated:
Grade 4
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Fourth Grade Essential Questions:




How are numbers used differently in our daily lives and what questions can they answer?
What are some different ways you can represent the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What is equality?
Third Grade Benchmarks
Fourth Grade Benchmarks
Fifth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
Apply and connect concepts to fractions
(a) Read, write, order, and compare whole
numbers to 100,000.
(b) Use place value understanding to round whole
numbers to the nearest 10 or 100.
(c) Forward and backward skip count by 3 and 4
from any given number within 1000.
(d) Use commutative, associative and distributive
properties to multiply and divide.
(e) Fluently multiply and divide whole numbers
with products less than 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.
(f) Automaticity of multiplication facts to 10 x 10.
(g) Multiply one-digit whole numbers by multiples
of ten in the range of 10-90 (e.g. 9 x 80, 5 x 60),
using strategies based on place value or
properties of operations.
(h) 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.
(a) Read, write, order and compare all whole numbers
through equivalency, composition, decomposition
and/or place value.
(b) Apply the properties of numbers (factor, multiple,
remainder, divisibility tests for 2,3,4,5,6,8, and 10).
(c) Automaticity of division facts 1 – 10.
(d) Illustrate and explain the relationship between
multiplication and division equations using sharing,
repeated subtraction, partitioning and area (up to
four digit dividends and one digit divisors with
remainders).
(e) Multiply two-digit by two-digit and one digit by
four digit numbers applying the commutative,
associative and distributive properties.
(f) Create a representation of any positive fractional
number as a part to whole relationship in area, set,
or linear model.
(g) Compose and decompose fractions into unit
fractions by using a visual fraction model.
Example: 3/8 = 1/8 + 1/8 + 1/8
(h) Add and subtract fractions and mixed numbers with
(a) Master the properties of
numbers (prime
factorization, composite,
divisibility tests).
(b) Apply the properties of
numbers (Greatest Common
Factor, Least Common
Multiple, Commutative,
Associative, and Distributive
Properties).
(c) Fluency with efficient
procedures for dividing
whole numbers, understand
why the procedures work (on
the basis of place value and
properties of operations), and
use them to solve problems.
(d) Efficiently and fluently add
and subtract fractions and
mixed numbers with unlike
denominators.
(e) Efficiently and fluently
multiply and divide a fraction
or a mixed number with a
whole number.
(f) Convert fractions into
decimals and percents and
make comparisons using <, >
Lamoille North Mathematics Curriculum
August 16, 2011
Intensive Focus: Fractions
P a g e | 23
Updated:
(i) Recognize and relate common benchmark
fractions (halves, thirds, fourths, sixths, eighths
and tenths) using set, linear and/or area
models.

Write a fraction for in the form of the
numerator over the denominator
(a/b).
 Show the location of a benchmark
fraction on a number line. For
example, given ¾, a student will
divide a number line (0 to 1) into 4
equal parts and label where ¾ is on
that line.
 Generate and use reasoning to
compare simple equivalent fractions
(e.g. 1/2=2/4, 4/6=2/3, 6/6=1) and
non-equivalent fractions (with same
numerator or the same
denominator). Record the results of
the comparisons with the symbols >,
=, <.
 Recognize that comparisons of two
fractions are valid only when the two
fractions refer to the same whole ( ½
of $10.00 ≠ ½ of $20.00)
Lamoille North Mathematics Curriculum
August 16, 2011
like denominators.
(i) Writes decimals within the context of money and
common fractions (i.e. 0.25 = ¼ = one quarter = 25
cents)
(j) Students will recognize and generate equivalent
fractions.
(k) Comparing two fractions with different numerators
and denominators using <, > and =, and justify their
solution using a representation.
and =.
(g) Able to read and write
numbers in standard,
expanded, exponential form
and scientific notation.
(h) Know 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 its place to its
left.
P a g e | 24
Updated:
Grade 4
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Fourth Grade Essential Questions:





How are geometry and measurement used in our daily lives?
What is the importance of lines, angles, shapes and space within our environment?
What appropriate tools and standard units can we use to measure and how do we use them?
What is volume?
What are congruent figures?
Third Grade Benchmarks
Fourth Grade Benchmarks
Fifth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
Apply and connect concepts to fractions
(a) Identify and describe the properties of angles.
(b) Identify line-symmetric figures and draw lines
of symmetry.
(c) Express area measurements in proper notation
of square units.
(d) Match congruent figures using reflections,
translations, or rotations.
(e) Use four operations to solve word problems
involving distances, intervals of time, liquid
volumes, masses of objects and money.
(f) Solve problems using the Cartesian coordinate
system (Quadrant I) to locate coordinates and
to represent data from tables.
(a) Identify and describe the properties
of angles.
(b) Identify line-symmetric figures and
draw lines of symmetry.
(c) Express area measurements in
proper notation of square units.
(d) Match congruent figures using
reflections, translations, or
rotations.
(e) Use four operations to solve word
problems involving distances,
intervals of time, liquid volumes,
masses of objects and money.
(f) Solve problems using the
Cartesian coordinate system
(Quadrant I) to locate coordinates
and to represent data from tables.
(a) Sort polygons (e.g. triangles, rhombi,
rectangles and others) by their attributes
(e.g., having four sides), and define the
subcategory (e.g. quadrilaterals).
(b) Identify and create congruent shapes.
(c) Calculate the area of a rectangle by using
the length x width formula.
(d) Calculate the perimeter of a rectangle by
adding multiple sides (2 lengths + 2
widths).
(e) Calculate the perimeter and area of a
rectangle through real world problems with
known sides and/or with one unknown
side.
(f) Create a rectangle given a specified area or
perimeter.
(g) Tell time to the nearest minute using an
analog or digital clock and determine
elapsed time.
Lamoille North Mathematics Curriculum
August 16, 2011
Intensive Focus: Fractions
P a g e | 25
Updated:
Grade 4
Functions and Algebra Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
Fourth Grade Essential Questions:


In what ways do patterns help us to be efficient in math?
How do numeric patterns help us to understand place value systems?
Third Grade Benchmarks
Fourth Grade Benchmarks
Fifth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
Apply and connect concepts to fractions
(a) Identify, describe, and extend arithmetic patterns
using concrete materials and tables to write a rule in
words or symbols to find the next term.
(b) Find the value of the variable in a multiplication
equation and division equation relating three whole
numbers. (i.e. find the value of the variable that
makes the equation true in each of the equations 8 x
a = 48, 6 x 6 = a, 24 ÷ a = 3)
(c) Writes simple linear algebraic expressions
involving any one of the four operations using
whole numbers and variables.
(a) Represent and interpret data
using line plots.
(a) Identify arithmetic patterns (including
patterns in the addition table or
multiplication table) and explain the patterns
using properties of operations. (i.e. observe
that 4 times a number is always even, and
explain why 4 times a number can be
decomposed into two equal addends.)
(b) Find the value of the variable in a
multiplication equation relating three whole
numbers. (i.e. find the value of the variable
that makes the equation true in each of the
equations 8 x a = 48, 6 x 6 = a)
Lamoille North Mathematics Curriculum
August 16, 2011
Intensive Focus: Fractions
P a g e | 26
Updated:
Grade 4
Data Statistics and Probability Concepts
PK-12 Enduring Understanding:
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Fourth Grade Essential Questions:


How is probability used in our daily lives?
In what ways can a chance event be measured?
Third Grade Benchmarks
Fourth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
(What students need to know and be able to do)
Intensive Focus: Multiplicative Reasoning
Apply and connect concepts to multiplication
Apply and connect concepts to multiplication and division
(a) Know the probability of a chance event is a
number between zero and one in the context
of fractions.
Lamoille North Mathematics Curriculum
August 16, 2011
Fifth Grade Benchmarks
Intensive Focus: Fractions
Apply and connect concepts to fractions
(a)Represent and interpret data using line plots.
P a g e | 27
Updated:
Grade 5
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Fifth Grade Essential Questions:






How are numbers used differently in our daily lives and what questions can they answer?
What are some different ways you can make the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What is equality?
How do you know if your answer is reasonable?
How can inverse relationships be used to solve problems?
Fourth Grade Benchmarks
Fifth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
(What students need to know and be able to do)
Intensive Focus: Fractions
Apply and connect concepts to multiplication and division
Apply and connect concepts to fractions
Sixth Grade Benchmarks
Intensive Focus: Proportional
Reasoning
Apply and connect concepts to ratios and
proportions.
(a) Read, write, order and compare all whole numbers through
equivalency, composition, decomposition and/or place value.
(b) Apply the properties of numbers (factor, multiple, remainder,
divisibility tests for 2,3,4,5,6,8, and 10).
(c) Automaticity of division facts 1 – 10.
(d) Illustrate and explain the relationship between multiplication
and division equations using sharing, repeated subtraction,
partitioning and area (up to four digit dividends and one digit
divisors with remainders).
(e) Multiply two-digit by two-digit and one digit by four digit
numbers applying the commutative, associative and distributive
properties.
(f) Create a representation of any positive fractional number as a
part to whole relationship in area, set, or linear model.
(g) Compose and decompose fractions into unit fractions by using
a visual fraction model. Example: 3/8 = 1/8 + 1/8 + 1/8
(h) Add and subtract fractions and mixed numbers with like
denominators.
(i) Writes decimals within the context of money and common
fractions (i.e. 0.25 = ¼ = one quarter = 25 cents)
(j) Students will recognize and generate equivalent fractions.

Comparing two fractions with different numerators and
denominators using <, > and =, and justify their solution using a
representation.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Master the properties of numbers (prime factorization,
composite, divisibility tests).
(b) Apply the properties of numbers (Greatest Common Factor,
Least Common Multiple, Commutative, Associative, and
Distributive Properties).
(c) Fluency with efficient procedures for dividing whole numbers,
understand why the procedures work (on the basis of place
value and properties of operations), and use them to solve
problems.
(d) Efficiently and fluently add and subtract fractions and mixed
numbers with unlike denominators.
(e) Efficiently and fluently multiply and divide a fraction or a
mixed number with a whole number.
(f) Convert fractions into decimals and percents and make
comparisons using <, > and =.
(g) Able to read and write numbers in standard, expanded,
exponential form and scientific notation.
(h) Know 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 its place to its left.
P a g e | 28
(a) Order and compare
rational numbers
(fractions, decimals,
percents, integers and
exponential forms with
whole number bases and
exponents).
(b) Add, subtract, multiply and
divide with integers.
(c) Demonstrate a conceptual
understanding of rational
numbers in respect to
ratios through using
models, explanations or
other representations.
(d) Compare quantities and
solve problems using ratios
(i.e. a/b, a:b, and a÷b),
rates percents (i.e. 25% of
a value) or proportions in a
variety of contexts.
Updated:
Grade 5
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Fifth Grade Essential Questions:




How are geometry and measurement used in our daily lives?
What is the importance of lines, angles, shapes and space within our environment?
What appropriate tools and standard units can we use to measure and how do we use them?
What is surface area?
Fourth Grade Benchmarks
Fifth Grade Benchmarks
Sixth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
(What students need to know and be able to do)
Intensive Focus: Fractions
Intensive Focus: Proportional Reasoning
Apply and connect concepts to multiplication and division
Apply and connect concepts to fractions
Apply and connect concepts to ratios and proportions.
(a) Identify and describe the properties of angles.
(b) Identify line-symmetric figures and draw lines
of symmetry.
(c) Express area measurements in proper notation
of square units.
(d) Match congruent figures using reflections,
translations, or rotations.
(e) Use four operations to solve word problems
involving distances, intervals of time, liquid
volumes, masses of objects and money.
(f) Solve problems using the Cartesian coordinate
system (Quadrant I) to locate coordinates and
to represent data from tables.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Recognize, name, describe and connect shapes,
figures and diagrams. (including volume and
surface area)
(b) Measure and use units of measurement and
make conversions within systems.
(c) Sort two-dimensional figures in a hierarchy
based on properties.
(d) Measure and calculate the area of right
triangles.
(e) Calculate elapsed and accrued time to the
nearest minute.
(a) Name, compare and describe
attributes and/or properties of
three-dimensional shapes.
(b) Compare and contrast the
properties of two dimensional
shapes.
(c) Measure and calculate the area
of triangles and quadrilaterals.
(d) Measure and calculate the
volume and surface area of
rectangular prisms using proper
notation.
(e) Measure and calculate radius,
diameter and circumference of a
circle.
(f) Solve problems using the
Cartesian coordinate system (all
quadrants) to locate coordinates
and to represent data from
tables.
P a g e | 29
Updated:
Grade 5
Functions and Algebra Concepts, Data Statistics and Probability Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Fifth Grade Essential Questions:

How does data help us to answer questions in our daily lives?
Fourth Grade Benchmarks
Fifth Grade Benchmarks
(What students need to know and be able to do)
Sixth Grade Benchmarks
Intensive Focus: Multiplicative Reasoning
Intensive Focus: Fractions
Intensive Focus: Proportional Reasoning
Apply and connect concepts to multiplication and
division
Apply and connect concepts to fractions
Apply and connect concepts to ratios and proportions.
Represent and interpret data
using line plots.
7.8
(a) Describe the patterns (linear and non-linear) of change using
tables, graphs, and simple symbolic rules.
(b) Write and evaluate simple algebraic expressions with a
variable on one side of the equality using substitution. (i.e.
solve for y if y=3x-6 and x = 3).
(c) Describe the slope of linear relationships as faster, slower,
bigger or smaller in a variety of problem situations.
(d) Describe how change in the value of one variable relates to
change in the value of a second variable. (y=kx, y=mx +b)
(e) Identify the difference between an independent and
dependent variable.
7.9
(a) Make and test conjectures in designing fair games.
(b) Know that a set of a data collected to answer a statistical
question has a distribution which can be described by its
center, spread, and overall shape.
(c) Know the difference between measure of central tendency
value (mean, median, and mode) and the measure of variation
value. (range, standard deviation, variance)
(a) Identify, describe, and extend
arithmetic patterns using concrete
materials and tables to write a rule in
words or symbols to find the next term.
(b) Find the value of the variable in a
multiplication equation and division
equation relating three whole numbers.
(i.e. find the value of the variable that
makes the equation true in each of the
equations 8 x a = 48, 6 x 6 = a, 24 ÷ a
= 3)
(c) Writes simple linear algebraic
expressions involving any one of the
four operations using whole numbers
and variables.
Lamoille North Mathematics Curriculum
August 16, 2011
(a)
P a g e | 30
Updated:
Grade 6
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Sixth Grade Essential Questions:





How are numbers used differently in our daily lives and what questions can they answer?
What are some different ways you can make the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What is equality?
How do you know if your answer is reasonable?



What is a ratio?
What is a proportion?
What is a rational number?
Fifth Grade Benchmarks
Intensive Focus: Fractions
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Apply and connect concepts to fractions
Master the properties of numbers (prime factorization, composite,
divisibility tests).
Apply the properties of numbers (Greatest Common Factor, Least
Common Multiple, Commutative, Associative, and Distributive
Properties).
Fluency with efficient procedures for dividing whole numbers,
understand why the procedures work (on the basis of place value
and properties of operations), and use them to solve problems.
Efficiently and fluently add and subtract fractions and mixed
numbers with unlike denominators.
Efficiently and fluently multiply and divide a fraction or a mixed
number with a whole number.
Convert fractions into decimals and percents and make
comparisons using <, > and =.
Able to read and write numbers in standard, expanded,
exponential form and scientific notation.
Know 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 its place to its left.
Lamoille North Mathematics Curriculum
August 16, 2011
Sixth Grade Benchmarks
Seventh Grade Benchmarks
(What students need to know and be able to do)
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions
Apply and connect concepts to ratios and proportions.
(a) Efficiently and fluently multiply and
divide fractions and mixed numbers.
(b) Order and compare rational numbers
(fractions, decimals, percents, integers
and exponential forms with whole
number bases and exponents).
(c) Add, subtract, multiply and divide with
integers.
(d) Demonstrate a conceptual understanding
of rational numbers in respect to ratios
through using models, explanations or
other representations.
(e) Compare quantities and solve problems
using ratios (i.e. a/b, a:b, and a÷b), rates
percents (i.e. 25% of a value) or
proportions in a variety of contexts.
(a) Use ratio and proportionality to solve a wide
variety of percent problems, including
problems involving discounts, interest, taxes,
tips and percent increase or decrease.
(b) Apply properties of operations and equalities
to addition, subtraction, multiplication and
division with rational numbers.
(c) Compare quantities and solve problems using
percent of a quantity, part of a whole as a
percentage and one quantity as a percent of
another.
(d) Apply the conventions of order of operations
including parentheses, brackets or exponents.
(e) Demonstrates understanding of the relative
magnitude of numbers by ordering, comparing
or identifying equivalent rational numbers
across number formats including integers,
absolute values and scientific notation.
P a g e | 31
Updated:
Grade 6
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Sixth Grade Essential Questions:




How are geometry and measurement used in our daily lives?
What is the importance of lines, angles, shapes and space within our environment?
What appropriate tools and standard units can we use to measure and how do we use them?
How are formulas for area derived?
Fifth Grade Benchmarks
Sixth Grade Benchmarks
Seventh Grade Benchmarks
Intensive Focus: Fractions
(What students need to know and be able to do)
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to fractions
Apply and connect concepts to ratios and proportions
Apply and connect concepts to ratios and proportions.
(a) Recognize, name, describe and connect shapes,
figures and diagrams. (including volume and
surface area)
(b) Measure and use units of measurement and
makes conversions within systems.
(c) Sort two-dimensional figures in a hierarchy
based on properties.
(d) Measure and calculate the area of right
triangles.
(e) Calculate elapsed and accrued time to the
nearest minute.
(a) Name, compare and describe attributes and/or
properties of three-dimensional shapes.
(b) Compare and contrast the properties of two
dimensional shapes.
(c) Measure and calculate the area of triangles
and quadrilaterals.
(d) Measure and calculate the volume and surface
area of rectangular prisms using proper
notation.
(e) Measure and calculate radius, diameter and
circumference of a circle.
(f) Solve problems using the Cartesian coordinate
system (all quadrants) to locate coordinates
and to represent data from tables.
(a) Construct and identify triangles
from three given measures of
angles and/or sides. (Triangle
Inequality Property)
(b) Know the formulas for the area
and circumference of a circle
and use them to solve problems.
(c) Apply the formulas for the
volumes of prisms and cylinders.
(Volume = Area of base x
Height)
(d) Use properties of angle
relationships from two and/or
three intersecting lines and/or
two parallel lines cut by a
transversal.
(e) Applies concepts of similarity
using a constant of
proportionality/scale factor.
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 32
Updated:
Grade 6
Functions and Algebra Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Sixth Grade Essential Questions:




In what ways do patterns help us to be efficient in math?
How do numeric patterns help us to understand place value systems?
What is a variable?
What effects rate?
Sixth Grade Benchmarks
Fifth Grade Benchmarks
Seventh Grade Benchmarks
Intensive Focus: Fractions
(What students need to know and be able to do)
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to fractions
Apply and connect concepts to ratios and proportions
Apply and connect concepts to ratios and proportions.
(a) Represent and interpret data
using line plots.
(a) Describe the patterns (linear and non-linear) of
change using tables, graphs, and simple
symbolic rules.
(b) Write and evaluate simple algebraic expressions
with a variable on one side of the equality using
substitution. (i.e. solve for y if y=3x-6 and x =
3).
(c) Describe the slope of linear relationships as
faster, slower, bigger or smaller in a variety of
problem situations.
(d) Describe how change in the value of one
variable relates to change in the value of a
second variable. (y=kx, y=mx +b)
(e) Identify the difference between an independent
and dependent variable.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Solve single variable multi-step linear
equations.
(b) Graph proportional relationships and
identify the unit rate as the slope of the
related line.
(c) Identify and apply direct proportional
relationships. (y/x = k, or y=kx)
(d) Identify and apply inverse proportional
relationships. (xy=k, or y=k/x)
P a g e | 33
Updated:
Grade 6
Data Statistics and Probability Concepts
PK-12 Enduring Understanding:
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Sixth Grade Essential Questions:



How is probability used in our daily lives?
How does data help us to answer questions in our daily lives?
How do the measures of central tendency and variation help us interpret data?
Sixth Grade Benchmarks
Fifth Grade Benchmarks
Seventh Grade Benchmarks
Intensive Focus: Fractions
(What students need to know and be able to do)
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to fractions
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to ratios and proportions.
(a) Represent and interpret data
using line plots.
(a) Make and test conjectures in designing fair
games.
(b) Know that a set of a data collected to answer
a statistical question has a distribution which
can be described by its center, spread, and
overall shape.
(c) Know the difference between measure of
central tendency value (mean, median, and
mode) and the measure of variation value.
(range, standard deviation, variance)
(a) Identify or create representations that best
display a given set of data or situation.
(b) Compare and contrast patterns, trends, or
distribution in data using measures of central
tendency.
(c) State the effects of outliers on measures of
central tendency.
(d) State the experimental or theoretical probability
of an event in which the sample space may or
may not contain equally likely outcomes.
(e) Use random sample techniques to draw
inferences about a population.
(f) Use counting techniques to solve problems.
(g) Construct probability models and use them to
find probabilities of events.
(h) Find and/or solve the probabilities of compound
events.
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 34
Updated:
Grade 7
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Seventh Grade Essential Questions:







How are numbers used differently in our daily lives and what questions can they answer?
What are some different ways you can make the same number?
What do you consider when choosing the most efficient strategy to solve problems?
What is equality?
How do you know if your answer is reasonable?
How is order important in mathematics?
What role does the symbolic language of mathematics have in solving problems?
Sixth Grade Benchmarks
Seventh Grade Benchmarks
(What students need to know and be able to do)
Eighth Grade Benchmarks
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to algebra
(a) Use ratio and proportionality to solve a wide variety of
percent problems, including problems involving discounts,
interest, taxes, tips and percent increase or decrease.
(b) Apply properties of operations and equalities to addition,
subtraction, multiplication and division with rational
numbers.
(c) Compare quantities and solve problems using percent of a
quantity, part of a whole as a percentage and one quantity as
a percent of another.
(d) Apply the conventions of order of operations including
parentheses, brackets or exponents.
(e) Demonstrates understanding of the relative magnitude of
numbers by ordering, comparing or identifying equivalent
rational numbers across number formats including integers,
absolute values and scientific notation.
(a) Estimate and/or calculate the
square root, cube root, square or
cube of a number, radicals and
integer exponents.
(b) Know there are numbers that
are not rational and approximate
them by rational numbers (e.g.,
Π, √2)
(c) Demonstrates understanding of
the relative magnitude of
numbers by ordering or
comparing rational numbers and
common irrational numbers
(√2).
(a) Order and compare rational numbers
(fractions, decimals, percents,
integers and exponential forms with
whole number bases and exponents).
(b) Add, subtract, multiply and divide
with integers.
(c) Demonstrate a conceptual
understanding of rational numbers in
respect to ratios through using
models, explanations or other
representations.
(d) Compare quantities and solve
problems using ratios (i.e. a/b, a:b,
and a÷b), rates percents (i.e. 25% of a
value) or proportions in a variety of
contexts.
Lamoille North Mathematics Curriculum
August 16, 2011
Intensive Focus: Algebraic Thinking
P a g e | 35
Updated:
Grade 7
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Seventh Grade Essential Questions:





How are geometry and measurement used in our daily lives?
What is the importance of lines, angles, shapes and space within our environment?
What appropriate tools and standard units can we use to measure and how do we use them?
How are formulas for area and volume derived?
What is similarity?
Sixth Grade Benchmarks
Seventh Grade Benchmarks
(What students need to know and be able to do)
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to ratios and proportions.
(a) Name, compare and describe attributes and/or
properties of three-dimensional shapes.
(b) Compare and contrast the properties of two
dimensional shapes.
(c) Measure and calculate the area of triangles and
quadrilaterals.
(d) Measure and calculate the volume and surface
area of rectangular prisms using proper
notation.
(e) Measure and calculate radius, diameter and
circumference of a circle.
(f) Solve problems using the Cartesian coordinate
system (all quadrants) to locate coordinates
and to represent data from tables.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Construct and identify triangles from
three given measures of angles and/or
sides. (Triangle Inequality Property)
(b) Know the formulas for the area and
circumference of a circle and use them to
solve problems.
(c) Apply the formulas for the volumes of
prisms and cylinders. (Volume = Area of
base x Height)
(d) Use properties of angle relationships from
two and/or three intersecting lines and/or
two parallel lines cut by a transversal.
(e) Applies concepts of similarity using a
constant of proportionality/scale factor.
Eighth Grade Benchmarks
Intensive Focus: Algebraic Thinking
Apply and connect concepts to algebra
(a) Use the Pythagorean Theorem to
find a missing side of a right triangle
and in problem solving situations.
(b) Use the Polygon Interior Angle Sum
Theorem to find the sum of the
angles in a convex polygon of any
number of sides.
(c) Apply the formulas for volumes of
cones and spheres.
(d) Apply concepts of similarity to
determine the impact of scaling on
the volume or surface area of
figures.
(e) Identify and perform transformations
of figures, including reflections,
translations, and rotations within the
Cartesian coordinate system.
P a g e | 36
Updated:
Grade 7
Functions and Algebra Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Seventh Grade Essential Questions:





In what ways do patterns help us to be efficient in math?
How do numeric patterns help us to understand place value systems?
What effects rate?
What is a proportional relationship?
What is a linear relationship?
Sixth Grade Benchmarks
Seventh Grade Benchmarks
(What students need to know and be able to do)
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to ratios and proportions.
(a)
(b)
(c)
(d)
(e)
Describe the patterns (linear and
non-linear) of change using tables,
graphs, and simple symbolic rules.
Write and evaluate simple algebraic
expressions with a variable on one
side of the equality using
substitution. (i.e. solve for y if y=3x6 and x = 3).
Describe the slope of linear
relationships as faster, slower,
bigger or smaller in a variety of
problem situations.
Describe how change in the value of
one variable relates to change in the
value of a second variable. (y=kx,
y=mx +b)
Identify the difference between an
independent and dependent variable.
Lamoille North Mathematics Curriculum
August 16, 2011
(a) Solve single variable multi-step linear
equations.
(b) Graph proportional relationships and
identify the unit rate as the slope of the
related line.
(c) Identify and apply direct proportional
relationships. (y/x = k, or y=kx)
(d) Identify and apply inverse proportional
relationships. (xy=k, or y=k/x)
Eighth Grade Benchmarks
Intensive Focus: Algebraic Thinking
Apply and connect concepts to algebra
(a) Apply the conventions of order of operations to algebraic
expressions.
(b) Use numeric, graphic, symbolic and tabular strategies to solve
problems involving linear functions.
(c) Solve one and two step linear equations with one variable (i.e.
with rational number coefficients and constants).
(d) Solve two expressions through application of commutative,
associative or distributive properties and/or order of operations or
substitution and identify its equality or inequality.
(e) Read, write, compare and interpret tables, charts, and graphs to
make comparisons and predictions of linear functions.
(f) Find the slope of a line.
(g) Graph proportional relationships interpreting the unit rate as the
slope of a graph.
(h) Show by graphing the connection between proportional
relationships, lines and linear equations. (y=mx, y=mx+b)
(i) Solve a system of linear equations.
(j) Solve and use systems of two linear equations in two variables
and relate the systems to pairs of lines that intersect, are parallel,
or are the same line, in the plane.
(k) Find the domain and range and their application in problem
solving contexts.
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Grade 7
Data Statistics and Probability Concepts
PK-12 Enduring Understanding:
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Seventh Grade Essential Questions:




How is probability used in our daily lives?
How does data help us to answer questions in our daily lives?
How do the measures of central tendency and variation help us interpret data?
How do probability models help us make inferences about a population?
Seventh Grade Benchmarks
Sixth Grade Benchmarks
Intensive Focus: Proportional Reasoning
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to ratios and proportions.
(a) Make and test conjectures in
designing fair games.
(b) Know that a set of a data
collected to answer a statistical
question has a distribution
which can be described by its
center, spread, and overall
shape.
(c) Know the difference between
measure of central tendency
value (mean, median, and
mode) and the measure of
variation value. (range, standard
deviation, variance)
Eighth Grade Benchmarks
(What students need to know and be able to do)
(a) Identify or create representations that best display
a given set of data or situation.
(b) Compare and contrast patterns, trends, or
distribution in data using measures of central
tendency.
(c) State the effects of outliers on measures of central
tendency.
(d) State the experimental or theoretical probability
of an event in which the sample space may or
may not contain equally likely outcomes.
(e) Use random sample techniques to draw inferences
about a population.
(f) Use counting techniques to solve problems.
(g) Construct probability models and use them to find
probabilities of events.
(h) Find and/or solve the probabilities of compound
events.
Lamoille North Mathematics Curriculum
August 16, 2011
Intensive Focus: Algebraic Thinking
Apply and connect concepts to algebra
(a)
(b)
(c)
(d)
Interpret representations to analyze data.
Estimate line of best fit to analyze
situations.
Identify the sample from which statistics
were developed.
Evaluate the possible association between
bivariate data as positive, negative or no
correlation.
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Updated:
Grade 8
Arithmetic, Number, and Operation Concepts
PK-12 Enduring Understandings:


Students will understand that place, digit, and quantity define a number’s value.
Students will understand and use the relationships between and among the operations to efficiently solve problems.
Eighth Grade Essential Questions:







How are numbers used differently in our daily lives and what questions can they answer?
What are irrational numbers?
What are some different ways you can make or approximate a number?
What do you consider when choosing the most efficient strategy to solve problems?
What is equality?
How do you know if your answer is reasonable?
What role does the symbolic language of mathematics have in solving problems?
Seventh Grade Benchmarks
Eighth Grade Benchmarks
Intensive Focus: Proportional Reasoning
(What students need to know and be able to do)
Intensive Focus: Algebraic Thinking
Apply and connect concepts to ratios and proportions.
Apply and connect concepts to algebra
(a) Use ratio and proportionality to solve a wide variety of
percent problems, including problems involving discounts,
interest, taxes, tips and percent increase or decrease.
(b) Apply properties of operations and equalities to addition,
subtraction, multiplication and division with rational
numbers.
(c) Compare quantities and solve problems using percent of a
quantity, part of a whole as a percentage and one quantity as
a percent of another.
(d) Apply the conventions of order of operations including
parentheses, brackets or exponents.
(e) Demonstrates understanding of the relative magnitude of
numbers by ordering, comparing or identifying equivalent
rational numbers across number formats including integers,
absolute values and scientific notation.
(a) Estimate and/or calculate the square root, cube root, square or cube of
a number, radicals and integer exponents.
(b) Know there are numbers that are not rational and approximate them
by rational numbers (e.g., Π, √2)
(c) Demonstrates understanding of the relative magnitude of numbers by
ordering or comparing rational numbers and common irrational
numbers (√2).
Lamoille North Mathematics Curriculum
August 16, 2011
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Updated:
Grade 8
Geometry and Measurement Concepts
PK-12 Enduring Understanding:
 Students will understand that objects in our world can be described and compared according to their physical and spatial
attributes and relationships.
Eighth Grade Essential Questions:






How are geometry and measurement used in our daily lives?
What is the importance of lines, angles, shapes and space within our environment?
What appropriate tools and standard units can we use to measure and how do we use them?
What formulas do you consider when choosing the most effective strategy to solve problems?
What is scaling?
What is a transformation?
Seventh Grade Benchmarks
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
(a)
(b)
(c)
(d)
(e)
Construct and identify triangles from three given
measures of angles and/or sides. (Triangle Inequality
Property)
Know the formulas for the area and circumference of
a circle and use them to solve problems.
Apply the formulas for the volumes of prisms and
cylinders. (Volume = Area of base x Height)
Use properties of angle relationships from two and/or
three intersecting lines and/or two parallel lines cut by
a transversal.
Applies concepts of similarity using a constant of
proportionality/scale factor.
Lamoille North Mathematics Curriculum
August 16, 2011
Eighth Grade Benchmarks
(What students need to know and be able to do)
Intensive Focus: Algebraic Thinking
Apply and connect concepts to algebra
(a) Use the Pythagorean Theorem to find a missing side of a right triangle
and in problem solving situations.
(b) Use the Polygon Interior Angle Sum Theorem to find the sum of the
angles in a convex polygon of any number of sides.
(c) Apply the formulas for volumes of cones and spheres.
(d) Apply concepts of similarity to determine the impact of scaling on the
volume or surface area of figures.
(e) Identify and perform transformations of figures, including reflections,
translations, and rotations within the Cartesian coordinate system.
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Updated:
Grade 8
Functions and Algebra Concepts
PK-12 Enduring Understanding:
 Students will understand that algebra is a tool used to extend and generalize patterns in mathematics
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Eighth Grade Essential Questions:





How is order important in mathematics?
How are properties important in mathematics?
What effects rate?
What is a proportional relationship versus a non-proportional relationship?
How are systems of linear equations used to solve problems?
Eighth Grade Benchmarks
Seventh Grade Benchmarks
(What students need to know and be able to do)
Intensive Focus: Algebraic Thinking
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
(a) Solve single variable multistep linear equations.
(b) Graph proportional
relationships and identify
the unit rate as the slope of
the related line.
(c) Identify and apply direct
proportional relationships.
(y/x = k, or y=kx)
(d) Identify and apply inverse
proportional relationships.
(xy=k, or y=k/x)
Apply and connect concepts to algebra
(a) Apply the conventions of order of operations to algebraic expressions.
(b) Use numeric, graphic, symbolic and tabular strategies to solve problems involving linear functions.
(c) Solve one and two step linear equations with one variable (i.e. with rational number coefficients and
constants).
(d) Read, write, compare and interpret tables, charts, and graphs to make comparisons and predictions of
linear functions.
(e) Find the slope of a line.
(f) Graph proportional relationships interpreting the unit rate as the slope of a graph.
(g) Show by graphing the connection between proportional relationships, lines and linear equations.
(y=mx, y=mx+b)
(h) Solve a system of linear equations.
(i) Solve two equations or simplify expressions through application of commutative, associative or
distributive properties and/or order of operations or substitution and identify its equality or inequality.
(j) Solve and use systems of two linear equations in two variables and relate the systems to pairs of lines
that intersect, are parallel, or are the same line, in the plane.
(k) Find the domain and range and their application in problem solving contexts.
Lamoille North Mathematics Curriculum
August 16, 2011
P a g e | 41
Updated:
Grade 8
Data Statistics and Probability Concepts
PK-12 Enduring Understanding:
 Students will understand that data is used to analyze, predict, hypothesize, conclude and make decisions.
Eighth Grade Essential Questions:




How does data help us to answer questions in our daily lives?
How do the measures of central tendency and variation help us interpret data?
How do algebraic models help us make inferences about a population?
How do you know statistics are valid and reliable?
Eighth Grade Benchmarks
Seventh Grade Benchmarks
(What students need to know and be able to do)
Intensive Focus: Algebraic Thinking
Intensive Focus: Proportional Reasoning
Apply and connect concepts to ratios and proportions.
(a) Make and test conjectures in
designing fair games.
(b) Know that a set of a data
collected to answer a statistical
question has a distribution which
can be described by its center,
spread, and overall shape.
(c) Know the difference between
measure of central tendency
value (mean, median, and mode)
and the measure of variation
value. (range, standard
deviation, variance)
Lamoille North Mathematics Curriculum
August 16, 2011
Apply and connect concepts to algebra
(a)
(b)
(c)
(d)
Construct and interpret representations to analyze data.
Estimate line of best fit to analyze situations.
Identify the sample from which statistics were developed.
Evaluate the possible association between bivariate data as positive, negative or no
correlation.
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Updated:
What does it mean to know mathematics?
Development of
Mastery and
competence
Linguistic
Conceptual
Beginning
Approaches
Masters/demonstrates competence
Exceeds
Student uses own
experience and language
to communicate
Student uses prescribed math
language to communicate
Student knows meaning of each word
and combination. Has internalized
language in math containers and uses
mathematical words and symbols
accurately and fluently to
communicate mathematical ideas.
Extends learning by using
linguistic, conceptual and
procedural mastery to go
deeper, can generalize …
Student makes
connections with
previous learning & is
working toward making
models
Student models the concept
and creates a variety of
representations (pictorial,
graphical..)
(From intuitive to
concrete)
Student generalizes, creating
symbolic work using formal language
and symbols. Can explain the
connection between the concrete
example and the abstract concept
using words, symbols & pictures.
Procedural
Student accesses a
variety of procedures to
find mathematical
solutions
Uses a variety of procedures
to achieve results, including
most efficient, accurately and
consistently.
Executes most efficient procedure
correctly & consistently. Can
articulate the connection between
the concept and procedure.
Skill (Fact
mastery)
Counts or uses
manipulatives to achieve
results. Working toward
automatization.
Beginning to automatize facts.
Uses an efficient mental or
manual procedure. Accurate
but not yet immediate,
develops strategies to
automatize facts
Demonstrates automatization of
facts(immediate and accurate).
Fluency in procedural skills allows
students flexibility in combining
methods to solve.
Make connections
between current learning
and other phenomena, in
math or other subject
areas
Solve multi-step and nonroutine application
problems.
Teach someone else
Lamoille North Mathematics Curriculum
August 16, 2011
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Updated:
Levels of Knowing Mathematics
Intuitive
Learners develop
math
understanding
and achieve
linguistic,
conceptual and
procedural
mastery, the
study of
mathematics will
include…
Making
Connections
to previous
learning;
extending
previous
schemas or
crating new
schemas to
include new
concepts,
Concrete
Representational
Models which
are exact,
efficient and
elegant. Can
transfer from
one model to
other models.
Must vary and
include
Discrete,
Continuous,
integrated,
universal
Representations
of the models with
pictures, figures,
tables, number
line, coordinate
axes, graphs,
diagrams,
graphing
calculator
Lamoille North Mathematics Curriculum
August 16, 2011
Abstract
Symbolical, abstract
representation or
generalization using
formal language &
symbols,
Making explicit
connections between
concrete and pictorial
models and abstract
formula or procedure
Application
Communication
Integrated
applications to
problems in the
form of word
problems,
interdisciplinary,
extracurricular,
projects, modeling
Explanations and
defense of ideas both
verbally and in writing
through written,
graphical,
compugraphic,
concrete, tests, exams,
peer teaching, designing
tests, oral math
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