Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
NUMBER SENSE & OPERATIONS
K.N.1 Count by ones to at least
20.
2.N.1 Name and write (in
numerals) whole numbers to
1000, identify the place values of
the digits, and order the numbers.
2.N.1 Name and write (in
numerals) whole numbers to
1000, identify the place values of
the digits, and order the numbers.
3.N.1 Exhibit an understanding
of the values of the digits in the
base ten number system by
reading, modeling, writing,
comparing, and ordering whole
numbers through 9,999.
Students will KNOW:
When you count, the last
number word you say tells the
number of items in the set.
Counting a set of objects in a
different order does not change
the total.
There is a number word and a
matching symbol that tell exactly
how many items are in a group.
Adding 1-10 counters to a filled
ten-frame results in the teen
numbers or the numbers 11-20.
Zero is a number that tells how
many objects there are when
there are none.
Students will KNOW:
There is a specific order to the
set of whole numbers.
Numbers 11 through 19 can be
shown as a group of 10 with
fewer than 10 left over.
The counting sequence to 100 is
built on the repetition of 0 to 9 in
the ones place of each new
decade.
In a standard numeral, the tens
are written to the left of the ones.
You can add the values of the
digits of a number together to get
the actual number.
If a number is represented with
10 or more ones, 10 ones may be
grouped to form another 10 and
the value remains the same.
Except for decade or century
changes, a number is increased or
decreased by 1 when the ones
digit is changed by 1 and a
number is increased or decreased
by 10 when the tens digit is
changed by 1.
A number ending with 5 ones is
halfway between two multiples
of 10 and numbers ending in 1, 2,
8, or 9 are closest to the tens.
Students will KNOW:
The decade numbers to 100 are
built on groups of ten with oral
names that are similar to, but not
the same as, the number of tens
counted.
In a two-digit number, the tens
digit tells how many groups of
ten and the ones digit tells the
number of ones.
Numbers 21 through 99 are
each written by joining two
number words with a hyphen.
Numbers 1 through 20 are each
represented by a unique number
word.
For any two-digit numbers, the
one with more tens is the greater
number; if the 2 numbers have
the same number of tens, then the
number with more ones is
greater.
Except for decade changes, a
number is increased or decreased
by 1 when the ones digit is
changed by 1.
10 tens make 100, and 10
hundreds make 1,000
Each digit in a three-digit
number tell how many hundreds,
Students will KNOW:
Our number system is based on
groups of ten. Whenever we get
10 in one place value, we move
to the next greater place value.
Place value can be used to write
numbers in different, but
equivalent forms.
You can regroup whole
numbers by breaking numbers
apart using place value.
Students will be able to DO:
Use objects to represent and
count the quantities 1 through 20.
Recognize and write the
numeral that describe the
quantity 0.
Students will be able to DO:
Read and write numbers up to
9,999.
Generate equivalent
representation for a number by
composing and decomposing
numbers.
Regroup a two- or three-digit
number in preparation for
subtraction.
GRADE 4
4.N.1 Exhibit an understanding
of the base ten number system by
reading, modeling, writing, and
interpreting whole numbers to at
least 100,000; demonstrating an
understanding of the values of
the digits; and comparing and
ordering the numbers.
Students will KNOW:
Place value can be used to write
numbers in different but
equivalent forms.
We use place-value periods to
help us understand, read, and
write larger numbers.
Place value can help us
compare and order numbers.
Place value relationships can
help estimate how much.
Students will be able to DO:
Use place value ideas to write
multiples of 100, 1,000, and
10,000 in different ways.
Read, write, compare, and order
numbers through 999,999.
Estimate totals made up of large
numbers.
Page 1 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
Ordering three numbers is
similar to comparing two
numbers because each number
must be compared to both of the
others.
10 tens make 1 hundred.
The number that comes before
or after a given number will
change the ones place by 1
except at the decade changes.
Students will be able to DO:
Write the numbers just before,
after, or between two given
numbers.
Order numbers through 12.
Read and write the teen
numbers as a group of 10 and
some left over.
Count and write numbers to 100
on the hundred chart.
Given a quantity shown with
tens and ones, tell how many tens
and ones there are and write the
number.
Exchange a ten for 10 ones or
10 ones for a ten and write the
new representation in expanded
form.
Given a two-digit number, write
the numbers that are 10 more/10
less and 1 more/1 less.
Estimate the positions of
numbers on a number line
marked only in multiples of 10.
Given 3 two-digit numbers,
order them from least to greatest
or from greatest to least.
GRADE 2
GRADE 3
GRADE 4
tens, or ones are in that number.
Adding together the values of
the three-digit number produces
the total value of the number.
Students will be able to DO:
Count groups of ten, up to 10
tens, and write how many.
Use groups of tens and ones to
show a given two-digit number.
Read and write number words
for given numbers.
Use place value to ompare
numbers using the greater-than,
less-than.
Identify and write numbers that
are one before, one after, or
between given numbers.
Count by hundreds to 1,000.
Count sets grouped in hundreds,
tens, and ones.
Read and write three-digit
numbers using expanded form,
standard form, and number
words.
Page 2 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
K.N.2 Match quantities up to at
least 10 with numerals and
words.
Students will KNOW:
There is a number word and a
matching symbol that tell exactly
how many items are in a group.
Numbers can be compared by
counting or by matching
corresponding sets.
There is a specific order to the
set of whole numbers.
A graph is a tool that can be
used to organize information in
order to solve problems.
When the objects in two sets are
matched one-to-one, the set
which leftover objects at the end
has more.
Students will be able to DO:
Recognize and write the
numerals that describe the
quantities 0 to 20.
Recognize number words from
0 to 10.
Use objects to order numbers
from 0 to 5 in sequence.
Solve problems by making and
reading a real graph and a picture
graph.
GRADE 1
GRADE 2
GRADE 3
2.N.2 Identify and distinguish
among multiple uses of numbers,
including cardinal (to tell how
many) and ordinal (to tell which
one in an ordered list), and
numbers as labels and as
measurements.
Students will KNOW:
Ordinal positions in a row or
list can be determined by
counting; the ordinal words are
similar to the counting words.
2.N.2 Identify and distinguish
among multiple uses of numbers,
including cardinal (to tell how
many) and ordinal (to tell which
one in an ordered list), and
numbers as labels and as
measurements.
Students will KNOW:
Ordinal positions in a row or
list can be determined by
counting and the ordinal words
are similar to the counting words.
Students will be able to DO:
Use ordinals through twentieth
to identify position.
Students will be able to DO:
Use ordinals through twentieth
to identify position.
3.N.2 Represent, order, and
compare numbers through 9,999.
Represent numbers using
expanded notation (e.g., 853 = 8
x 100 + 5 x 10 + 3), and written
out in words (e.g., eight hundred
fifty-three).
Students will KNOW:
Our number system is based on
groups of ten. Whenever we get
10 in one place value, we move
to the next greater place value.
Place value can be used to write
numbers in different, but
equivalent forms.
Place value can help us
compare and order numbers.
Students will be able to DO:
Read and write numbers
through 9,999.
Generate equivalent
representation for a number by
composing and decomposing
numbers.
Regroup a two- or three-digit
number in preparation for
subtraction.
Compare whole numbers
through 10,000.
GRADE 4
4.N.2 Represent, order, and
compare large numbers (to at
least 100,000) using various
forms, including expanded
notation, e.g., 853 = 8 x 100 + 5
x 10 + 3.
Students will KNOW:
Place value can be used to
write numbers in different but
equivalent forms.
We use place-value periods to
help us understand, read, and
write larger numbers.
Students will be able to DO:
Use place value ideas to write
multiples of 100, 1,000, and
10,000 in different ways.
Read, write, compare, and
order numbers through 999,999.
Page 3 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
GRADE 1
GRADE 2
GRADE 3
GRADE 4
K.N.3 Identify positions of
objects in sequences (e.g., first,
second) up to fifth.
KINDERGARTEN
2.N.3 Identify and represent
common fractions (1/2, 1/3, 1/4)
as parts of wholes, parts of
groups, and numbers on the
number line.
2.N.3 Identify and represent
common fractions (1/2, 1/3, 1/4)
as parts of wholes, parts of
groups, and numbers on the
number line.
4.N.4 Select, use, and explain
models to relate common
fractions and mixed numbers
(1/2, 1/3, 1/4, 1/5, 1/6, 1/8, 1/10,
1/12, and 11/2), find equivalent
fractions, mixed numbers, and
decimals, and order fractions.
Students will KNOW:
Ordinal position in a row or list
can be determined by counting
and the ordinal words are similar
to the counting words.
Students will KNOW:
A shape can be divided into any
number of equal parts in a variety
of ways.
A half of a region is one of two
equally sized parts.
A unit fraction of a region
names one of a number of
equally sized parts.
Groups can be divided into
equal parts in the same way that
shapes can be divided into equal
parts.
Students will KNOW:
”Equal parts” means that each
part is the same.
A unit fraction names one of a
number of equal parts into which
a shape or region has been
divided.
The bottom number of a
fraction tells the number of equal
parts. The top number tells how
many equal parts are being
named.
Fractions can be used to name
part of a set as well as part of a
region.
3.N.3 Identify and represent
fractions (between 0 and 1 with
denominators through 10) as
parts of unit wholes and parts of
groups. Model and represent a
mixed number (with denominator
2, 3, or 4) as a whole number and
a fraction, e.g., 1 2/3, 3 1/2.
Students will KNOW:
A region can be divided into
equal parts in different ways and
parts that are equal in size can
have different shapes. A fraction
is relative to the size of the
whole.
The denominator of a fraction
gives the number of equal parts
in all, and the numerator tells
how many equal parts are
described. A fraction is relative
to the size of the whole.
Different fractions used to name
the same amount are equivalent.
Finding the number of objects
in a fractional part of a set
involves division.
Fractions in which the
numerator is greater than the
denominator may be expressed as
mixed numbers or as improper
fractions.
Students will be able to DO:
Use the words first through fifth
to identify ordinal positions.
Students will be able to DO:
Determine whether a shape has
been divided into equal or
unequal parts and count the
number of equal parts into which
it has been divided.
Identify and show half of a
region.
Identify and show half of a
group objects.
Students will be able to DO:
Determine whether a shape has
been divided into equal or
unequal parts; identify halves,
thirds, and fourths.
Identify and show a unit
fraction of a region.
Identify and show any fraction
of a region.
Identify and show fractions of a
set of objects.
Students will be able to DO:
Identify regions that have been
divided into equal-sized parts and
divide regions into equal-sized
parts.
Identify and draw fractional
parts of regions.
Students will KNOW:
The same fractional part can
have different names that are
equivalent. Equivalent
fractions are found by
multiplying or dividing the
numerator and denominator of a
fraction by the same non-zero
number.
Fractions can be expressed in
their simplest form by dividing
the numerator and denominator
by their greatest common factor
When two fractions have the
same denominator, the greater
fraction has the greater
numerator, and when two
fractions have the same
numerator, the fraction with the
greater denominator is less.
Fractions with a common
denominator or a common
numerator are easy to compare
and order.
Students will be able to DO:
Identify fractions that are
equivalent and find fractions
equivalent to a given fraction
using models and/or a
computational procedure.
Page 4 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
K.N.4 Compare sets of up to at
least 10 concrete objects using
appropriate language (e.g., none,
more than, fewer than, same
number of, one more than) and
order numbers.
Students will KNOW:
When the objects in two sets are
matched one-to-one, the set with
leftover objects at the end has
more.
Numbers can be compared by
counting or by matching
corresponding sets.
There is a specific order to the
set of whole numbers.
A graph is a tool that can be
used to organize information in
order to solve problems.
You can use 5 and 10 as
benchmarks and compare other
numbers to these benchmarks.
On a calendar, the number to
the right or left of another
number is either one more or one
less than the other.
One-to-one correspondence can
be used to compare groups.
GRADE 1
GRADE 2
2.N.4 Compare whole numbers
using terms and symbols, e.g.,
less than, equal to, greater than
(<, =, >).
2.N.4 Compare whole numbers
using terms and symbols, e.g.,
less than, equal to, greater than
(<, =, >).
Students will KNOW:
10 is a foundational number in
our number system and can be
made up to two 5s; 5 and 10
provide points of reference to
which other numbers can be
related.
There is a specific order to the
set of whole numbers.
For any 2 two-digit numbers,
the one with more tens is the
greater number; if they have the
same number of tens, the one
with more ones is greater.
Ordering three numbers is
similar to comparing two
numbers because each number
must be compared to both of the
others.
Students will KNOW:
For any two-digit numbers, the
one with more tens is the greater
number; if the 2 numbers have
the same number of tens, then the
number with more ones is
greater. Numbers are compared
by beginning with the place of
greatest value, the place farthest
to the left, and then moving to
the right as far as is needed.
Students will be able to DO:
Compare a given number to 5
and 10.
Students will be able to DO:
Compare numbers using the
greater-than, less-than, and equal
to symbols.
Compare three-digit numbers
using the symbols <, >, and =.
GRADE 3
Find equivalent fractions using
models such as fraction strips.
Compare and order fractions.
Find the number of objects in a
fractional part of a set where the
numerator is 1.
Read and write mixed numbers
and use objects or pictures to
show mixed numbers.
Write fractions.
3.N.4 Locate on the number line
and compare fractions (between
0 and 1 with denominators 2, 3,
or 4, e.g., 2/3).
GRADE 4
Express fractions in simplest
form.
Determine which of two
fractions is greater (or less) .
Compare fractions using >, <,
and =, and order fractions.
4.N.3 Demonstrate an
understanding of fractions as
parts of unit wholes, as parts of a
collection, and as locations on
the number line.
Students will KNOW:
Different fractions used to name
the same amount are equivalent.
Fractions with a common
denominator or a common
numerator can be compared and
ordered.
A set can be considered a
whole, and fractional parts are
parts of the set. The denominator
of the fraction tells the total
number of things in the set and
the numerator tells the number of
parts being described.
Students will KNOW:
The denominator of a fraction,
set, or group gives the number of
equal parts in all, and the
numerator tells how many equal
parts are described. When the
numerator and denominator are
equal, the fraction is equal to 1 or
the entire region.
The distance between 0 and 1
on a number line can be divided
into fractional parts, and the
points can be named with
fractions.
Students will be able to DO:
Find equivalent fractions using
models such as fraction strips.
Compare and order fractions.
Identify fractional parts of sets
or groups and divide sets to show
fractional parts.
Students will be able to DO:
Identify and draw fractional
parts of a region.
Identify fractional parts of sets
or groups and divide sets to show
fractional parts.
Locate and name fractions on a
number line.
Page 5 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
Students will be able to DO:
Use one-to-one correspondence
and counting to compare groups
and determine which has more,
which has fewer, or whether the
groups are the same.
Use objects to order numbers
from 0 to 5 in sequence.
Solve problems by making and
reading a real graph and a picture
graph.
Given a number from 1 to 10,
tell whether it is more or less
than 10.
Use a number line to order
numbers from 0 through 10.
Compare two numbers to
decide which number is greater
and which is less.
Find, identify, and record
numbers through 31 on a
calendar.
K.N.5 Understand the concepts
of whole and half
Order numbers through 12.
Given 2 two-digit numbers,
determine if the first is greater
than, less than, or equal to the
second.
Given 3 two-digit numbers,
order them from least to greatest
or from greatest to least.
2.N.5 Identify odd and even
numbers and determine whether
a set of objects has an odd or
even number of elements.
2.N.5 Identify odd and even
numbers and determine whether
a set of objects has an odd or
even number of elements.
Students will KNOW:
Parts of a whole are equal when
they are the same size.
When objects or shapes are cut
into 2 equal parts, the parts are
called halves.
Students will KNOW:
Even numbers can be broken
into two equal parts; odd
numbers cannot.
Students will KNOW:
Even numbers can be broken
into two equal parts; odd
numbers cannot.
Any number that ends in 2, 4, 6,
8, or 0 is even.
Any number that ends in 1, 3, 5,
7, 9 is odd.
Students will be able to DO:
Identify equal parts of a whole
Students will be able to DO:
Given a number less than 60,
determine if it is odd or even.
GRADE 2
GRADE 3
GRADE 4
3.N.5 Recognize classes to
which a number may belong (odd
numbers, even numbers, and
multiples of numbers through
10). Identify the numbers in
those classes, e.g., the class of
multiples of 7 between 1 and 29
consists of 7, 14, 21, 28.
Students will KNOW:
Continue number patterns and
use place-value patterns to find
sums and differences.
Addition doubles facts and
multiplying by 2 give the same
result.
Patterns and properties can help
you remember multiplication
4.N.7 Recognize classes (in
particular, odds, evens; factors or
multiples of a given number; and
squares) to which a number may
belong, and identify the numbers
in those classes. Use these in the
solution of problems.
Students will KNOW:
Fractions can be expressed in
their simplest form by dividing
the numerator and denominator
by their greatest common factor.
Students will be able to DO:
Express fractions in simplest
form.
Page 6 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
and identify halves.
GRADE 2
Students will be able to DO:
Identify any number as odd or
even.
GRADE 3
GRADE 4
facts.
Multiplication facts help you
find the products for other facts.
Students will be able to DO:
Count on or count back easily
using place values,
Find products of one-digit
numbers times 0 through 10.
K.N.6 Identify U.S. coins by
name.
Students will KNOW:
The name of a penny, nickel,
dime, quarter, and dollar bill.
Students will be able to DO:
Identify a penny, nickel, dime,
quarter, and dollar bill.
2.N.6 Identify the value of all
U.S. coins, and $1, $5, $10, and
$20 bills. Find the value of a
collection of coins and dollar
bills and different ways to
represent an amount of money up
to $5. Use appropriate notation,
e.g., 69¢, $1.35.
Students will KNOW:
A penny has a value of 1 cent,
nickel has a value of 5 cents, and
a dime has a value of 10 cents;
pennies can be counted by 1s,
nickels can be counted by 5s, and
dimes can be counted by 10s.
To count a mixed group of
coins, skip count by the largest
coin present and then shift to skip
count by the next largest coin
present and keep going until no
coins remain.
A quarter is worth 25 cents.
A half-dollar has a value of 50
cents and a dollar has a value of
100 cents.
Students will be able to DO:
Identify the value of a group of
nickels and pennies through 25
2.N.6 Identify the value of all
U.S. coins, and $1, $5, $10, and
$20 bills. Find the value of a
collection of coins and dollar
bills and different ways to
represent an amount of money up
to $5. Use appropriate notation,
e.g., 69¢, $1.35.
Students will KNOW:
To count a mixed group of
coins, skip count by the greatest
coin value present then shift to
skip counting by the next greatest
coin value present and then keep
going until no coins remain.
When counting a set of coins,
begin with the coin or coins that
have the greatest value and then
count on to the coin or coin of
least value.
The number of coins in a set
does not necessarily indicate
which of two sets has the greater
value.
There are many combinations
of coins that show the same
amount of money.
One dollar is worth 100 cents.
Both a dollar bill and a dollar
3.N.8 Select and use appropriate
operations (addition, subtraction,
division, and multiplication) to
solve problems, including those
involving money.
4.N.10 Select and use
appropriate operations (addition,
subtraction, multiplication, and
division) to solve problems,
including those involving money.
Students will KNOW:
When counting money it is
often easiest to start with the bills
or coins that have the greatest
value.
Counting up rather than
subtraction is a common way to
make change. Making change
can be thought of as part (price)
plus part (change) equals whole
(amount paid)
Writing a number sentence is
one way of representing what we
know and what we need to find
out in a word problem.
The algorithm for adding and
subtracting whole numbers can
be extended to adding and
subtracting money.
Students will KNOW:
Word problems tell what is
known and what needs to be
figured out.
When counting money, it is
often easiest to start with the bills
or coins that have the greatest
value.
The kinds of numbers in a
calculation and the ease with
which one can apply different
calculation methods together
determine an appropriate
computation method.
Writing a number expression is
one way of representing what we
know in a word problem.
Word phrases that express
mathematical situations can be
translated into specific
expressions using numbers and
Students will be able to DO:
Page 7 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
cents.
Identify the value of a group of
dimes and pennies through 99
cents.
Identify the value of a group of
dimes and nickels through 95
cents.
Identify the value of a group of
dimes, nickels, and pennies
through 99 cents.
Identify a quarter and find
groups of coins that have the
same value as a quarter.
Count a collection of coins that
includes half-dollars, quarters,
dimes, nickels, and pennies.
Identify a dollar bill, a dollar
coin, a half-dollar coin, and
combinations of coins worth
amounts up to $1.00.
GRADE 2
coin stand for 100 cents.
Students will be able to DO:
Identify the value of a group of
dimes, nickels, and pennies
through 99 cents.
Count a collection of coins that
includes half-dollars, quarters,
dimes, nickels, and pennies.
Compare the values of two sets
of coins.
Show the same amount of
money using different sets of
coins.
Identify the value of a dollar
bill and a dollar coin.
GRADE 3
GRADE 4
Make change by counting on.
Find the value of money ($5
and $1 bills, half-dollars,
quarters, dimes, nickels, and
pennies)
Write number sentences for
word problems and use complete
sentences to write answers to
word problems.
Add and subtract money up to
$100.
Use multiplication facts along
with addition and subtraction to
solve problems.
operations.
Solving problems using the Try,
Check, and Revise strategy is
based on initial estimates. After
the difference between the
estimate and the desired total is
known it is easier to determine
the exact answer.
The algorithm for multiplying
whole numbers can be extended
to multiplying money.
To find the product of three
factors you can start by finding
the product of any two.
Choosing certain pairs of factors
may enable you to compute the
product of the three factors using
mental math. Properties of
whole numbers explain why you
can choose which numbers to
multiply first.
Remainders expressed as whole
numbers give a specific type of
information that is sometimes
more useful than expressing
remainders as fractions or as the
decimal part of the quotient.
Algorithms for dividing whole
numbers can be extended to
dividing money.
Writing a number sentence is
one way you can represent what
you know and what you need to
find out in a word problem.
Students will be able to DO:
Tell in words what is known
and what needs to be determined
in given word problems.
Page 8 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
K.N. 8 Estimate the number of
objects in a group and verify
results
GRADE 1
2.N.7 Demonstrate an
understanding of various
meanings of addition and
subtraction, e.g., addition as
combination (plus, combined
GRADE 2
2.N.7 Demonstrate an
understanding of various
meanings of addition and
subtraction, e.g., addition as
combination (plus, combined
GRADE 3
3.N.10 Add and subtract (up to
four-digit numbers) and multiply
(up to two-digit numbers by a
one-digit number) accurately and
efficiently.
GRADE 4
Find the value of a given
assortment of bills and coins and
tell how to make a given money
amount with the fewest bills
and/or coins.
For a variety of problems, state
the computation method to be
used and add or subtract using
that method.
Write number expressions for
phrases.
Choose and evaluate the
number expression that matches
a word phrase.
Solve problems using the Try,
Check and Revise strategy.
Reading for the main idea in a
problem helps in identifying the
operation or operations needed to
solve it. Phrases like “how
many”, “how many more”, and
“how many times as many” are
clues to the correct operation, but
these phrases also must be read
in context.
Decide how to use the quotient
and remainder to answer the
question in a division problem.
Compute and estimate quotients
involving money amounts.
Write number sentences for
word problems and use complete
sentences to write answers to
word problems.
4.N.12 Add and subtract (up to
five-digit numbers) and multiply
(up to three digits by two digits)
accurately and efficiently.
Page 9 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
Students will KNOW:
Objects can be used to
determine the approximate
quantities of similar groups.
Students will be able to DO:
Use objects to estimate the
quantities of groups.
GRADE 1
GRADE 2
with, more); subtraction as
comparison (how much less, how
much more), equalizing (how
many more are needed to make
these equal), and separation (how
much remaining).
Students will KNOW:
Most numbers can be described
in terms of two parts in a variety
of ways.
1 more than expresses the
relationship between two
numbers
1 less than expresses the
relationship between two
numbers
Addition can be used to
represent joining situations.
Given two parts, addition can
be used to name the whole.
Joining groups can be shown in
an addition sentence that uses the
plus symbol and the equals sign.
The sum of zero and a number,
or a number and zero, is that
number.
Subtraction can be used to
represent separating situations.
Given the whole and one part,
subtraction can be used to name
the other part.
Separating groups can be shown
in a subtraction sentence that
uses the minus symbol and the
equals sign.
The difference between a
number and zero is that number;
the difference between a number
and itself is zero.
with, more); subtraction as
comparison (how much less, how
much more), equalizing (how
many more are needed to make
these equal), and separation (how
much remaining).
Students will KNOW:
Two groups can be combined
and then counted to see how
many objects there are in all.
When one part is removed from
a whole, the remaining part can
be counted to see how many
remain.
Two groups can be compared
by counting the leftovers after
doing one-to-one matching.
Changing the order of the
addends does not change the
sum.
Counting on is a strategy that
can be used for solving missingaddend problems.
GRADE 3
Students will KNOW:
The algorithm for adding,
subtracting, and multiplying
whole numbers.
Students will be able to DO:
Add and subtract numbers to
four digits.
Multiply up to a two-digit
number by a one-digit number.
Students will be able to DO:
Join two groups together and
write an addition sentence to tell
how many in all.
Take away a number of objects
from a group and count to find
how many are left.
Compare two groups to find out
how many more or how many
fewer.
Use the commutative property
to find sums.
Given a quantity and one of its
parts, find the missing part by
counting on or counting back.
GRADE 4
Students will KNOW:
There are different ways to
calculate mentally. Most involve
breaking numbers apart or
replacing them with numbers that
are easy to compute with.
Three or more numbers can be
added in any order. The process
for adding two whole numbers is
just repeated when adding more
than two numbers.
The algorithm for adding and
subtracting whole numbers can
be extended to adding and
subtracting money.
The kinds of numbers in a
calculation and the ease with
which one can apply different
calculation methods together
determine an appropriate
computation method.
Making an array with placevalue blocks enables you to
model and visualize the partial
products used in the expanded
algorithm for multiplying. The
expanded algorithm involves
multiplying the parts of numbers
based on their place value.
Both the expanded and the
traditional or standard
multiplication algorithms involve
breaking the overall calculation
Page 10 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
One-to-one correspondence can
be used to compare groups.
Subtraction can be used to
represent both separating and
comparing situations.
Use counting on to find missing
parts of 1,000.
GRADE 3
Students will be able to DO:
Show ways the numbers 6, 7, 8,
9, and 10 can be divided into two
parts.
Find the numbers that are 1 and
2 more than a given number.
Tell and act out joining stories
to find how many in all.
Find the sum of two addends.
Write an addition sentence to
find the sum in a joining
situation.
Write an addition sentence
using zero.
Tell and act out separating
stories to find out how many are
left.
Find the difference between two
numbers.
Write a subtraction sentence to
find the difference in a separating
situation.
Write subtraction sentences
using zero.
Compare two groups to find out
how nany more or how many
fewer.
Write subtraction sentences to
compare and tell how many more
or how many fewer.
No comparable standard for this
2.N.8 Understand and use the
GRADE 4
into simpler calculations. The
traditional or standard algorithm
is a shortcut for the expanded
algorithm.
Both the expanded and the
traditional or standard
multiplication algorithms can be
extended to multiply greater
numbers.
Students will be able to DO:
Compute sums of numbers
mentally.
Compute differences of
numbers mentally.
Find the sums of three or more
whole numbers or money
amounts.
Use the standard algorithm to
find differences using whole
number amounts and money
amounts.
For a variety of problems, state
the computation method to be
used and add or subtract using
that method.
Make arrays with place-value
blocks to find products.
Use the standard algorithm to
multiply two-digit numbers
and/or three-digit numbers by
one-digit numbers.
Use arrays to find products
involving two-digit factors.
2.N.8 Understand and use the
3.N.6 Select, use, and explain
4.N.8 Select, use, and explain
Page 11 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
grade
inverse relationship between
addition and subtraction (e.g., 8 +
6 = 14 is equivalent to 14 – 6 =
8 and is also equivalent to 14 – 8
= 6) to solve problems and check
solutions.
inverse relationship between
addition and subtraction (e.g., 8 +
6 = 14 is equivalent to 14 – 6 =
8 and is also equivalent to 14 – 8
= 6) to solve problems and check
solutions.
various meanings and models of
multiplication and division of
whole numbers. Understand and
use the inverse relationship
between the two operations.
Students will KNOW:
Doubles facts can be used to
find differences for their related
subtraction facts.
Every addition fact has at least
one related subtraction fact.
Fact families use the same three
numbers and can be used to show
how addition and subtraction are
related.
An addition fact can be used to
find the difference in a related
subtraction fact.
Students will KNOW:
Fact families use the same three
numbers and can be used to show
how addition and subtraction are
related.
Doubles facts can be used to
find differences for their related
subtraction facts.
An addition fact can be used to
find the difference in a related
subtraction fact.
Pictures frequently contain
important information that helps
to solve problems.
Subtraction and addition are
inversely related so one can
“undo” the other.
various meanings and models of
multiplication (through 10 x
10). Relate multiplication
problems to corresponding
division problems, e.g., draw a
model to represent 5 x 6 and 30 ÷
6.
Students will KNOW:
Combining equal groups is one
meaning of multiplication.
Arrays are a special kind of
arrangement of equal groups and
multiplication can be used to find
the total.
Two ways of thinking of
division are sharing equally and
repeated subtraction.
Multiplication and division are
inverse operations.
Students will be able to DO:
Find differences by using
doubles facts.
Write the addition and
subtraction sentences that make
up a fact family.
Find differences by using
known addition facts.
Write related addition and
subtraction facts with sums
through 18.
Students will be able to DO:
Write the addition and
subtraction sentences that make
up a fact family.
Find differences by using
doubles facts.
Find differences by using
addition facts.
Use pictures to help find
missing addends in number
sentences.
Relate addition to subtraction
by using one operation to check
the other.
Students will KNOW:
Use patterns to find products
with factors of 0, 1, 2, 5, and 9.
Fractions can be expressed in
their simplest form by dividing
the numerator and denominator
by their greatest common factor.
Students will be able to DO:
Patterns can help you remember
multiplication facts.
Express fractions in simplest
form.
Students will be able to DO:
Write multiplication number
sentences for given situations
using the X symbol.
Write multiplication sentences
for arrays and use arrays to find
multiplication facts.
Use repeated subtraction to find
answers.
Give all the facts in a
multiplication/division fact
family.
Page 12 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard for this
grade
GRADE 1
GRADE 2
GRADE 3
GRADE 4
2.N.9 Know addition facts
(addends to ten) and related
subtraction facts, and use them to
solve problems.
2.N.9 Know addition facts
(addends to ten) and related
subtraction facts, and use them to
solve problems.
3.N.9 Know multiplication facts
through 10 x 10 and related
division facts, e.g., 9 x 8 = 72
and 72 ÷ 9 = 8. Use these facts to
solve related problems, e.g., 3 x 5
is related to 3 x 50.
Students will KNOW:
Given two parts, addition can
be used to name the whole.
Joining groups can be shown in
an addition sentence that uses the
plus (+) symbol and the equals
(=) sign.
The sum of zero and a number
or a number and zero is that
number.
Writing a number sentence is
one strategy that can be used to
solve a problem.
Given the whole and one part,
subtraction can be used to name
the part.
Separating groups can be shown
in a subtraction sentence that
uses the minus (-) symbol and the
equals (=) sign.
The difference between a
number and zero is that number;
the difference between a number
and itself is zero.
Subtraction can be written in a
horizontal or vertical form, both
representing the same situation.
Addition and subtraction can
both be useful in answering
joining and separating questions.
When counting on, the last
number word said tells how
Students will KNOW:
Sums to twenty and related
subtraction facts.
The difference between
combining and separating
problems.
Students will KNOW:
Addition doubles facts and
multiplying by 2 give the same
result.
Patterns and properties can help
you remember multiplication
facts.
Word problems tell what is
known and what needs to be
figured out.
Multiplication and division are
inverse operations.
You can use the inverse
relationship between
multiplication and division to
find division facts.
Patterns can help you when
dividing with 0 and 1.
Patterns can help you divide
with divisors of 10.
Place value, multiplication, and
division facts and patterns can
help you multiply by multiples of
10 and 100.
You can use multiplication facts
you know to help you find the
products for other facts.
You can use multiplication to
compare the size of two groups.
(For example, twice as many)
4.N.11 Know multiplication
facts through 12 x 12 and related
division facts. Use these facts to
solve related multiplication
problems and compute related
problems, e.g., 3 x 5 is related to
30 x 50, 300 x 5, and 30 x 500.
Students will KNOW:
Basic facts and place-value
patterns can help you multiply a
one-digit number by multiples of
10, 100, and 1,000.
Basic facts and place-value
patterns can help you multiply a
two-digit number by multiples of
10, 100, and 1,000.
Students will be able to DO:
Write an addition sentence to
find the sum of two addends up
to a sum of twenty in a joining
situation.
Tell and act out separating
stories to find out how many are
left.
Write a subtraction sentence to
find the difference in a separating
situation with numbers of twenty
or less.
After reading a story problem,
identify combining or separating
situations and choose the
appropriate strategy for solving
the problem.
Students will be able to DO:
Multiply any number by 10,
100, or 1,000.
Mentally multiply any number
by 10, 100, or 1,000.
Page 13 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
many there are all together.
Changing the order of the
addends does not change the
sum.
It saves time when you count on
if you begin with the greater
number.
Moving to the right on a
number line is one way to
represent addition.
Some of the information given
in a problem may not be needed
in order to solve the problem.
If you know a doubles fact, it
can help you figure out a
doubles-plus 1 fact.
10 is an important benchmark in
our numeration system so other
numbers can be thought of in
relation to what we know about
the number 10.
Knowing how to draw pictures
to solve problems is helpful in
checking to see that answers
make sense.
Moving to the left on a number
line is one way to show
subtraction.
When counting back, the last
number said tells how many are
left.
Doubles facts can be used to
find differences for their related
subtraction facts.
An addition fact can be used to
find the difference in a related
subtraction fact.
If you know a doubles fact, it
can help you figure out a
GRADE 2
GRADE 3
GRADE 4
Students will be able to DO:
Find products of one-digit
numbers from 0 to 10.
Give all the facts in a
multiplication/division fact
family.
Give quotients for division facts
with divisors from 0 to 10.
Recognize which numbers are
divisible by 10.
Mentally multiply any number
by 10 and 100.
Memorize multiplication facts.
Use multiplication and
comparison to find the size of a
group.
Recognize patterns on a
multiplication fact table.
Page 14 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
doubles-plus 1 fact or a doublesminus –1 fact.
There is a place-value pattern
that can be used to add 10 to a
single-digit number mentally.
Making a group of 10 can
change a difficult addition fact to
one that is easier to add mentally.
There are a variety of strategies
to use to find addition fact sums;
which one is best depends on the
addends.
Addition fact strategies can be
applied to finding sums of three
numbers.
Every addition fact has at least
one related subtraction fact.
Fact families use the same three
numbers and can be used to show
how addition and subtraction are
related.
An addition fact can be used to
find the difference in a related
subtraction fact.
Making a 10 before subtracting
can change a difficult subtraction
fact to one that is easier to
subtract mentally.
There are a variety of strategies
to use to find subtraction fact
differences; which one is best
depends on the numbers
involved.
The answer to one problem can
be used as information needed to
solve another problem.
Students will be able to DO:
Find the sum of two addends.
Page 15 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
Write an addition sentence to
find the sum in a joining
situation.
Write an addition sentence
using zero.
Solve problems by writing
addition sentences.
Find the difference between two
numbers.
Write a subtraction sentence to
find the difference in a separating
situation.
Write subtraction sentences
using zero.
Write the differences for
horizontal and vertical forms of
subtraction.
Solve problems by choosing
addition or subtraction.
Find sums by counting on 1, 2,
or 3 using counters.
Use turn around facts ( the
commutative property) to find
sums.
Count on 1, 2, or 3 to add
starting with the greater number.
Use a number line to count on
1, 2, or 3.
Solve problems by identifying
unnecessary information and
writing number sentences.
Recognize doubles as a strategy
for remembering sums.
Use doubles facts to learn
doubles-plus 1 facts.
Recognize facts that have sums
of 10.
Solve problems by drawing
pictures.
Page 16 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
Solve problems by writing
subtraction sentences.
Use a number line to count back
1 or 2.
Find differences by counting
back 1 or 2.
Find differences by using
doubles facts.
Write related addition and
subtraction facts and sentences
that make up a fact family.
Find differences by using
known addition facts.
Recognize doubles as a
strategy for remembering sums to
18.
Use doubles facts to learn
doubles-plus-1 facts and doublesminus-1 facts.
Use a pattern to add numbers 1
to 8 to the number 10.
Find sums by making a 10
when adding to 8 or 9.
Select and apply addition fact
strategies.
Use the associative property to
find sums of three numbers.
Write related addition and
subtraction facts with sums
through 18.
Find differences using a tenframe.
Select and apply subtraction
strategies to use to find
subtraction fact differences;
which one is best depends on the
numbers involved.
Solve multiple-step problems
by using the answer to the first
Page 17 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard for this
grade
GRADE 1
questions to answer the second
question.
2.N.10 Demonstrate the ability
to add and subtract three-digit
numbers accurately and
efficiently.
Standard is not
developmentally appropriate for
grade 1.
GRADE 2
GRADE 3
GRADE 4
2.N.10 Demonstrate the ability
to add and subtract three-digit
numbers accurately and
efficiently.
3.N.7 Use the commutative
(order) and identity properties of
addition and multiplication on
whole numbers in computations
and problem situations, e.g., 3 +
4 + 7 = 3 + 7 + 4 = 10 + 4.
Students will KNOW:
In the standard algorithm, ones,
tens, and hundreds are added
separately beginning with the
ones. Whenever there are 10 or
more in a column the regrouping
is recorded in the next column to
the left.
Difference between two
numbers can be estimated by
subtracting close-but-easier
numbers.
Students will KNOW:
There are certain relationships
for whole numbers and addition
that always hold true.
Arrays are a special kind of
arrangement of equal groups and
multiplication can be used to find
the total.
Patterns and properties can help
you remember multiplication
facts.
When multiplying three
numbers you can multiply the
product of any two of the
numbers by the third number.
4.N.9 Select, use, and explain
the commutative, associative,
and identity properties of
operations on whole numbers in
problem situations, e.g., 37 x 46
= 46 x 37, (5 x 7) x 2 =
5 x (7
x 2).
Students will KNOW:
Use patterns to find products
with factors of 0, 1, 2, 5, and 9.
Dollars, dimes, and pennies
represent whole numbers, tenths,
and hundredths in our decimal
number system.
When counting money, it is
often easiest to start with the bills
or coins that have the greatest
value.
There are different ways to
calculate mentally. Most involve
breaking numbers apart or
replacing them with numbers that
are easy to compute with.
Making a table can help to
represent what you know in
solving a problem.
Students will be able to DO:
Use paper and pencil to add 2
three-digit numbers with one
regrouping.
Use estimation to select two
numbers that have a given
difference.
Students will be able to DO:
Apply commutative and identity
properties in addition and
multiplication.
Use mental math to add
numbers by breaking them apart
using place value.
Add mentally by rounding with
multiples of ten.
Write multiplication sentences
for arrays and use arrays to find
multiplication facts.
Multiply three numbers (for
example, 3 X 4 X 8)
Students will be able to DO:
Give money amounts in dollars,
dimes, and pennies and in ones,
tenths, and hundredths.
Find the value of a given
assortment of bills and coins and
tell how to make a given money
amount with the fewest bills
and/or coins.
Compute sums of money
Page 18 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
mentally.
Make tables and use them to
solve word problems.
Patterns can help you remember
multiplication facts.
K.N.7 Use objects and drawings
to model and solve related
addition and subtraction
problems to ten.
Students will KNOW:
1 more than expresses the
relationship between two
numbers: 4 is more than 3.
1 less than expresses the
relationship between two
numbers: 3 is less than 4.
Addition can be used to
represent joining numbers and/or
objects.
Subtraction can be used to
represent separating numbers
and/or objects.
Given two parts, addition can
be used to name the whole.
Knowing how to draw pictures
to solve problems is helpful in
checking to see that answers
make sense.
Joining groups can be shown in
an addition phrase that use the
plus (+) sign and the equal (=)
sign.
2.N.11 Demonstrate in the
classroom an understanding of
and the ability to use the
conventional algorithms for
addition (two 3-digit numbers
and three 2-digit numbers) and
subtraction (two 3-digit
numbers).
Standard is not
developmentally appropriate for
grade 1.
2.N.11 Demonstrate in the
classroom an understanding of
and the ability to use the
conventional algorithms for
addition (two 3-digit numbers
and three 2-digit numbers) and
subtraction (two 3-digit
numbers).
Students will KNOW:
Adding or subtracting hundreds
or tens is similar to adding or
subtracting single-digit numbers.
Numbers can be mentally added
by combining any parts in any
order, but it is easiest to add the
greatest place values first.
When adding like place values
are combined. If there are 10 or
more of one value, then 10 of
them can be exchanged for 1 of
the next-higher value.
In the standard algorithm, ones,
tens, and hundreds are added
separately beginning with the
ones. Whenever there are 10 or
more in a column the regrouping
is recorded in the next column to
the left.
Horizontal addition or
subtraction problems can be
written vertically so that the
No comparable standard in this
grade.
4.N.14 Demonstrate in the
classroom an understanding of
and the ability to use the
conventional algorithms for
addition and subtraction (up to
five-digit numbers), and
multiplication (up to three digits
by two digits).
Students will KNOW:
Three or more numbers can be
added in any order. The process
for adding two whole numbers is
just repeated when adding more
than two numbers.
The algorithm for adding and
subtracting whole numbers can
be extended to adding and
subtracting money.
Both the expanded and the
traditional or standard
multiplication algorithms involve
breaking the overall calculation
into simpler calculations. The
traditional or standard algorithm
is a shortcut for the expanded
algorithm.
Both the expanded and the
traditional or standard
multiplication algorithms can be
extended to multiply greater
numbers.
Page 19 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
Separating groups can be shown
with a subtraction phrase the uses
the minus (-) symbol.
Students will be able to DO:
Use counters to show 2 to 10 in
two parts.
Use a ten-frame to show 10 in
different ways.
Act out number stories that
involve joining or separating two
groups.
Interpret illustrations that show
joining groups and write the
corresponding numbers.
Solve problems by drawing
pictures about joining two
groups.
Use the plus (+) sign to
represent joining groups when
recording addition.
Use the minus sign (-) to
represent “take way” situations
when recording subtraction.
Identify and use the equal sign;
add and write the sum or subtract
and write the difference.
GRADE 1
GRADE 2
standard algorithm can be used.
Missing parts can be found by
counting on to the known part or
by counting back from the whole.
A hundred can be regrouped for
10 tens when you need to
subtract more tens than are
present in the tens place of the
top number.
When regrouping, the numbers
written above the columns do not
change the original amount; they
simply indicate a change in the
way that amount is being
represented.
When adding 2 two-digit
numbers, you may or may not
have enough ones altogether to
make another ten.
Adding or subtracting hundreds
or tens is similar to adding or
subtracting single-digit numbers.
GRADE 3
GRADE 4
Students will be able to DO:
Add and subtract whole
numbers and money amounts (to
five digits).
Find the sums of three or more
whole numbers or money
amounts.
Use the standard algorithm to
find differences using whole
number amounts and money
amounts.
Use the standard algorithm to
multiply two-digit numbers
and/or three-digit numbers by
one-digit numbers.
Students will be able to DO:
Add and subtract multiples of
10 or 100 to and from a threedigit number without regrouping.
Add three-digit numbers
mentally without regrouping.
Use place-value models to add
2 three-digit numbers without
regrouping.
Use paper and pencil to add 2
three-digit numbers with one
regrouping.
Add 2 three-digit numbers in
vertical form when they are
given in horizontal form.
Use models to subtract three-
Page 20 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard for this
grade
GRADE 1
2.N.12 Estimate, calculate, and
solve problems involving
addition and subtraction of twodigit numbers. Describe
differences between estimates
and actual calculations.
Standard is not developmentally
appropriate for grade 1.
GRADE 2
digit numbers with regrouping.
Use the standard algorithm to
subtract three-digit numbers with
regrouping.
Subtract three-digit numbers
written in horizontal form.
Add a two-digit number to a
two-digit number using models
or mental math.
2.N.12 Estimate, calculate, and
solve problems involving
addition and subtraction of twodigit numbers. Describe
differences between estimates
and actual calculations.
Students will KNOW:
When adding 2 two-digit
numbers, you may or may not
have enough ones altogether to
make another ten.
An estimate of a two-digit sum
can be made by adding the tens
and then deciding whether the
ones will increase the sum
beyond the next multiple of ten.
An estimate of a two-digit
difference can be made by
subtracting tens then considering
the ones.
GRADE 3
GRADE 4
3.N.12 Understand and use the
strategies of rounding and
regrouping to estimate quantities,
measures, and the results of
whole-number computations
(addition, subtraction, and
multiplication) up to two-digit
whole numbers and amounts of
money to $100, and to judge the
reasonableness of the answer.
4.N.17 Select and use a variety
of strategies (e.g., front-end,
rounding, and regrouping) to
estimate quantities, measures,
and the results of whole-number
computations up to three-digit
whole numbers and amounts of
money to $1000, and to judge the
reasonableness of the answer.
Students will KNOW:
Strategies for rounding and
regrouping.
Students will KNOW:
There are different ways to
estimate sums and differences.
Most involve replacing numbers
with other numbers that are close
and easy to compute. The
numbers used determine whether
an estimate is reasonable.
The specific numbers used to
make an estimate determine
whether an estimate is
reasonable.
There are different ways to
estimate products and quotients.
Most involve replacing numbers
with other numbers that are close
and easy to compute.
Students will be able to DO:
Use rounding and regrouping
strategies to estimate quantities,
measures, and whole-number
computations.
Students will be able to DO:
Add a two-digit number to a
two-digit number using models
or mental math.
Students will be able to DO:
Page 21 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
Estimate the sum of two twodigit numbers.
Estimate the difference between
two two-digit numbers.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
3.N.11 Round whole numbers
through 1,000 to the nearest 10,
100, and 1,000.
No comparable standard in this
grade.
4.N.16 Round whole numbers
through 100,000 to the nearest
10, 100, 1000, 10,000, and
100,000.
Students will KNOW:
Strategies for rounding whole
numbers.
Students will KNOW:
Rounding is a process for
finding the multiple of 10, 100,
etc, closest to a given number.
Students will be able to DO:
Round whole numbers through
1,000 to the nearest 10, 100, and
1,000.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
GRADE 4
Use rounding and front-end
estimation to estimate sums and
differences.
Indicate whether an estimate is
reasonable.
Use rounding and compatible
numbers to estimate products.
Use rounding and place value to
estimate products of larger
numbers.
Estimate quotients
No comparable standard in this
grade.
Students will be able to DO:
Round whole numbers through
one hundred thousand.
4.N.13 Divide up to a three-digit
whole number with a single-digit
divisor (with or without
remainders) accurately and
efficiently. Interpret any
remainders.
Students will KNOW:
When you divide whole
numbers, sometimes there is a
remainder. The remainder must
be less than the divisor.
Page 22 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
Division algorithms involve
breaking the calculation into
smaller, simpler calculations
using place value.
Algorithms for dividing can be
extended to greater numbers.
Remainders expressed as whole
numbers give a specific type of
information.
The steps for dividing do not
change when there are zeros in
the quotient.
Estimation is an effective
strategy for checking the
reasonableness of a quotient.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
Students will be able to DO:
Use models to find quotients
and remainders.
Use models and the standard
algorithm to divide 2-digit and 3digit numbers by 1-digit
numbers.
Divide with zeros in the
quotient.
4.N.15 Demonstrate in the
classroom an understanding of
and the ability to use the
conventional algorithm for
division of up to a three-digit
whole number with a single-digit
divisor (with or without
remainders).
Students will KNOW:
Division algorithms involve
breaking the calculation into
smaller, simpler calculations
using place value.
Algorithms for dividing can be
Page 23 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
extended to greater numbers.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
3.N.13 Use concrete objects and
visual models to add and subtract
(only when the answer is greater
than or equal to zero) common
fractions (halves, thirds, fourths,
sixths, and eighths) with like
denominators.
Students will KNOW:
To add or subtract fractions
with like denominators add or
subtract the numerators and use
the same denominator.
Students will be able to DO:
Add and subtract fractions with
like denominators.
Students will be able to DO:
Use a standard algorithm to
divide a two-digit number by a
one-digit number.
Use a standard algorithm to
divide a three-digit number by a
one-digit number.
4.N.5 Identify and generate
equivalent forms of common
decimals and fractions less than
one whole (halves, quarters,
fifths, and tenths).
Students will KNOW:
Decimals show fractional parts
of a whole.
Students will be able to DO:
Relate decimals to common
fraction benchmarks in 10ths and
100ths.
4.N.18 Use concrete objects and
visual models to add and subtract
common fractions.
Students will KNOW:
When adding or subtracting
fractions with like denominators,
you are adding or subtracting
portions of the same size, so you
can add the numerators – the
numbers of pieces or portions –
without changing the
denominator.
To add or subtract fractions or
Page 24 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
mixed numbers with unlike
denominators, change the
number sentence to a simpler one
with like denominators. For
example, eights, fourths, and
halves or thirds and sixths where
at least one denominator stays
the same - not thirds and fourths
to equal twelfths.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
Students will be able to DO:
Add fractions with like
denominators, using models and
paper and pencil.
Add fractions with unlike
denominators, using models and
paper and pencil.
4.N.6 Exhibit an understanding
of the base ten number system by
reading, naming, and writing
decimals between 0 and 1 up to
the hundredths
Students will KNOW:
Dollars, dimes, and pennies
represent whole numbers, tenths,
and hundredths in our decimal
number system.
When counting money, it is
often easiest to start with the bills
or coins that have the greatest
value.
Decimals show fractional parts
of a whole.
Decimal place value is an
extension of a whole number
place value based on groups of
ten.
Students will be able to DO:
Page 25 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
Give money amounts in dollars,
dimes, and pennies and in ones,
tenths, and hundredths
Find the value of a given
assortment of bills and coins, and
tell how to make a given money
amount with the fewest bills
and/or coins
Read, write, and shade grids to
show tenths and hundredths
expressed as decimals.
Write decimals in tenths and
hundredths.
PATTERNS, RELATIONS, & ALGEBRA
K.P.1 Identify the attributes of
objects as a foundation for
sorting and classifying, e.g., a red
truck, a red block, and a red ball
share the attribute of being red; a
square block, a square cracker,
and a square book share the
attribute of being square shaped.
Students will KNOW:
Logical reasoning can be used
to decide what attributes a group
of objects share.
Students will be able to DO:
Solve problems by determining
the sorting rule for groups of
sorted objects.
2.P.1 Identify, reproduce,
describe, extend, and create
simple rhythmic, shape, size,
number, color, and letter
repeating patterns.
2.P.1 Identify, reproduce,
describe, extend, and create
simple rhythmic, shape, size,
number, color, and letter
repeating patterns.
Students will KNOW:
A repeating pattern is formed
when a certain set of elements is
repeated over and over. The part
that repeats is called a pattern
unit.
Different symbols and objects
can be used to show the same
pattern.
The regular repetitions of a
pattern can be used to find a
missing term in the pattern.
Students will KNOW:
Rhythmic, shape, size, number,
color, and letter repeating
patterns are mastered in grade 2.
Repeatedly adding and
subtracting the same number
results in a repeating pattern in
both the ones and the tens digit.
No comparable standard for this
grade
No comparable standard for this
grade
Students will be able to DO:
Discover a numeric pattern
made by repeatedly adding or
subtracting the same number.
Students will be able to DO:
Identify the pattern unit in a
repeating pattern.
Translate shape patterns into
Page 26 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
K.P.2 Sort and classify objects
by color, shape, size, number,
and other properties.
Students will KNOW:
Attributes can be used to
compare objects.
Attributes such as color, shape,
or size can be used to sort the
same set of objects in different
ways.
Students will be able to DO:
Identify same and different by
the attributes of color, shape,
size, and kind.
Sort objects by one or more
attribute such as color, shape,
size, or kind.
Solve problems by determining
the sorting rule for groups of
sorted objects.
K.P.3 Identify, reproduce,
describe, extend, and create
color, rhythmic, shape, number,
and letter repeating patterns with
simple attributes, e.g., ABABAB
Students will KNOW:
A repeating pattern is formed
when a certain set of elements is
repeated over and over.
In a color or shape pattern, the
colors or shapes are repeated in a
predictable way.
A pattern has a predictable,
repeating, part, that can be
GRADE 1
letters.
Solve problems by using data
from a picture.
2.P.2 Identify different patterns
on the hundreds chart.
Students will KNOW:
The counting sequence to 100
is built on the repetition of 0 to 9
in the ones place of each new
decade.
Skip counting on a hundred
chart makes visual patterns as
well as numerical patterns.
Students will be able to DO:
Count and write numbers to
100 on the hundred chart.
Use a hundred chart to skip
count by 2s, 5s, and 10s and then
find patterns.
GRADE 2
2.P.2 Identify different patterns
on the hundreds chart.
GRADE 3
GRADE 4
No comparable standard for this
grade
No comparable standard for this
grade
Students will KNOW:
Except for decade changes, a
number is increased or decreased
by 1 when the ones digit is
changed by 1.
Students will be able to DO:
Identify and write numbers that
are one before, one after, or
between given numbers.
2.P.3 Describe and create
addition and subtraction number
patterns, e.g., 1, 4, 7, 10…; or 25,
23, 21
2.P.3 Describe and create
addition and subtraction number
patterns, e.g., 1, 4, 7, 10…; or 25,
23, 21
3.P.1 Create, describe, extend,
and explain symbolic (geometric)
patterns and addition and
subtraction patterns, e.g., 2, 6,
10, …; and 50, 45, 40.…
4.P.1 Create, describe, extend,
and explain symbolic (geometric)
and numeric patterns, including
multiplication patterns like 3, 30,
300, 3000,
Students will KNOW:
The counting sequence to 100 is
built on the repetition of 0 to 9 in
the ones place of each new
decade.
When the objects are
conveniently grouped, skip
counting can be used to find out
how many there are in all.
Students will KNOW:
Repeatedly adding and
subtracting the same number
results in a repeating pattern in
both the ones digit and the tens
digit.
Number patterns can often be
discovered by adding or
subtracting the same amount
Students will KNOW:
We can count on or count back
using place values.
Applying an addition or
subtraction rule to quantities
results in a certain type of
numerical pattern.
Addition doubles facts and
multiplying by 2 give the same
Students will KNOW:
Place value can be used to write
numbers in different but
equivalent forms.
Patterns can help you remember
multiplication facts.
Patterns and breaking apart can
help you find products with
factors of 10, 11, and 12.
Page 27 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
extended.
Students will be able to DO:
Copy and extend sound,
movement, color, and shape
patterns.
Create and extend patterns.
GRADE 1
Skip counting can be used to
solve problems involving equal
groups.
Students will be able to DO:
Count and write numbers to 100
on the hundred chart.
Skip count to find the total
number of items arranged in sets
of 10s, 5s, and 2s.
Solve problems by finding
patterns in a table of related
number pairs.
GRADE 2
repeatedly.
Students will be able to DO:
Discover and write the rule for
a numeric pattern made by
repeatedly adding or subtracting
the same number.
Continue number patterns and
skip count by different amounts.
GRADE 3
GRADE 4
result.
Patterns can help you remember
multiplication facts.
Patterns are recognized when
numerical quantities are
organized in a table. The pattern
or rule that relates two quantities
can be represented using words,
tables, and symbols.
An explanation of the solution
to a problem includes
information that is known and
how you have used this
information.
Place value and multiplication
facts and patterns can help you
multiply by multiples of 10 and
100.
The pattern or rule that relates
two quantities can be represented
using words, tables, and symbols.
Recognizing the pattern that
relates two quantities helps to
extend the table.
Basic facts and place-value
patterns can help you multiply a
one-digit number by multiples of
10, 100, and 1,000.
Basic facts and place-value
patterns can help you multiply a
two-digit number by multiples of
10, 100, and 1,000.
Students will be able to DO:
Continue number patterns and
use place-value patterns to find
sums and differences.
Complete tables representing
patterns and give the rules for the
patterns.
Find products of one-digit
numbers from 0 to 10
Write to explain a pattern.
Use mental math to multiply by
multiples of 10, and 100.
K.P.4 Count by fives and tens at
least up to 50.
Students will KNOW:
Counting by 5s and 10s in a
hundred chart makes visual
patterns as well as numerical
patterns.
2.P.4 Skip count by twos, fives,
and tens up to at least 50, starting
at any number.
Students will KNOW:
Numbers 11 through 19 can be
shown as a group of 10 with
fewer than 10 left over.
Skip counting can be used to
2.P.4 Skip count by twos, fives,
and tens up to at least 50, starting
at any number.
Students will KNOW:
It is possible to skip count on
from any number by continuing
the skip counting pattern of 10s,
5s, or 2s.
No comparable standard for this
grade
Students will be able to DO:
Use place value ideas to write
multiples of 100, 1,000, and
10,000 in different ways.
Use patterns to find products
with factors of 0, 1, 2, 5, 9,
10,11, and 12.
Find the rule for a pattern
presented in a table and use the
rule to add inputs and outputs to
the table.
Multiply any number by 10,
100, or 1,000.
Mentally multiply any number
by 10, 100, or 1,000.
No comparable standard for this
grade
Page 28 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
When the objects are
conveniently grouped, counting
by 5s and 10s can be used to find
out how many there are in all up
to 50s.
solve problems involving equal
groups.
The decade numbers to 100 are
built on groups of ten with oral
names that are similar to, but not
the same as, the number of tens
counted.
Students will be able to DO:
Use objects and a number line
to count by 5s and 10s to 50.
Use a hundred chart to count by
5s and 10s to 50.
Count groups by 5s and 10s to
find the total number up to 50.
Identify and extend number
patterns based on counting by 5s
and 10s.
No comparable standard for this
grade
Students will be able to DO:
Skip count to find the total
number of items arranged in sets
of 10s, 5s, and 2s.
Count groups of ten, up to 10
tens, and write how many.
Count sets that are grouped in
10s and leftover ones.
2.P.5 Construct and solve open
sentences that have variables,
e.g., + 7 = 10.
Students will KNOW:
Fact families use the same three
numbers and can be used to show
how addition and subtraction are
related.
Students will be able to DO:
Use known fact families to find
the missing addend.
GRADE 2
GRADE 3
GRADE 4
Students will be able to DO:
Skip count on when starting
with any number.
2.P.5 Construct and solve open
sentences that have variables,
e.g., + 7 = 10.
Students will KNOW:
10 is an important benchmark in
our numeration system so other
numbers can be thought of in
relations to what we know about
the number 10.
If you know any two of the
three numbers in an addition
sentence you can find the
missing number.
Counting on is a strategy that
can be used for solving missingaddend problems.
Missing parts can be found by
counting on to the known part or
by counting back from the whole.
3.P.3 Determine the value of a
variable (through 10) in simple
equations involving addition,
subtraction, or
multiplication,(e.g., 2 + = 9)
Students will KNOW:
Relationships between numbers,
between numbers and
expressions, and between
expressions can be expressed
using variables.
Students will be able to DO:
Compare numbers and
expressions using numbers and
supply numbers that make given
inequalities/equalities true.
4.P.3 Determine values of
variables in simple equations,
e.g., 4106 – ∇ = 37, 5 = ❍ + 3,
and – ❍ = 3.
Students will KNOW:
Equations can be solved by
trying out specific values,
checking the result, and trying
other values until a true
statement results.
Students will be able to DO:
Find the solution to an equation
by testing a set of values for the
variable.
Find the solution to an equation
informally by substituting values
for the variable.
Students will be able to DO:
Page 29 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard for this
grade
GRADE 1
GRADE 2
2.P.6 Write number sentences
using +, –, <, = , and/or > to
represent
mathematical
relationships
in
everyday
situations.
Recognize facts that have sums
of 10.
Use counters to find the missing
addend in an addition sentence.
Given a quantity and one of its
parts, find the missing part by
counting on or counting back.
Use counting on to find missing
parts of 1,000.
2.P.6 Write number sentences
using +, –, <, = , and/or > to
represent
mathematical
relationships
in
everyday
situations.
Students will KNOW:
Writing a number sentence is
one strategy that can be used to
solve a problem.
10 is an important benchmark in
our numeration system so other
numbers can be thought of in
relations to what we know about
the number 10.
Knowing how to draw pictures
to solve problems is helpful in
checking to see that answers
make sense.
Writing a number sentence is
one strategy that can be used to
solve a problem.
For any 2 two-digit numbers,
the one with more tens is the
greater number; if they have the
same number of tens, the one
with more ones is greater.
You can use what you learn
form your first attempt at a
solution to make a better guess
Students will KNOW:
Meaning of symbols for +, –, <,
= , and/or >
Students will be able to DO:
Apply +, –, <, = , and/or > in
number sentences.
GRADE 3
GRADE 4
3.P.2 Determine which symbol
(<, >, or =) is appropriate for a
given number sentence, e.g., 7 x
8 .?. 49 + 6.
4.P.2 Use symbol and letter
variables (e.g., ∆, x) to represent
unknowns or quantities that vary
in expressions and in equations
or inequalities (mathematical
sentences that use =, <, >).
Students will KNOW:
Real situations can be
represented by writing variable
expressions, and these
expressions can be evaluated by
substituting values for the
variable
Word sentences that express
mathematical situations can be
translated into equations and
equations can help you solve
problems.
Students will KNOW:
Relationships between numbers,
between numbers and
expressions, and between
expressions can be expressed
using the relational symbols =
(equals), < (less than), and
>(greater than).
Students will be able to DO:
Compare numbers and
expressions using relational
symbols that make given
inequalities/equalities true.
Students will be able to DO:
Write and evaluate variable
expressions that involve a single
operation.
Write equations for word
sentences.
Page 30 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
on your next attempt to solve the
same problem.
No comparable standard for this
grade
Students will be able to DO:
Solve problems by using the
try, check, an revise strategy.
Solve problems by writing
addition sentences.
Recognize doubles as a strategy
for remembering sums.
Solve problems by drawing
pictures.
Solve problems by writing
subtraction sentences.
Solve problems by choosing
addition or subtraction.
write related addition and
subtraction facts with sums
through 18.
Solve multiple step problems by
using the answer to the first
question to answer the second
question.
2.P.7 Describe functions related
to trading, including coin trades
and measurement trades, e.g.,
five pennies make one nickel or
four cups make one quart.
Students will KNOW:
Value of pennies, dimes,
nickels, and quarters.
Students will be able to DO:
Identify a quarter and find
groups of coins that have the
2.P.7 Describe functions related
to trading, including coin trades
and measurement trades, e.g.,
five pennies make one nickel or
four cups make one quart.
No comparable standard in this
grade.
Students will KNOW:
One dollar is worth 100 cents.
Both a dollar bill and a dollar
coin stand for 100 cents.
Cups, pints, and quarts are
standard units used to measure
capacity
4.P.5 Solve problems involving
proportional relationships,
including unit pricing (e.g., four
apples cost 80¢, so one apple
costs 20¢) and map interpretation
(e.g., one inch represents five
miles, so two inches represent ten
miles).
Students will KNOW:
Making a table can help to
represent what you know in
solving a problem.
Students will be able to DO:
Make tables and use them to
Page 31 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
same value as a quarter.
GRADE 4
solve word problems.
Students will be able to DO:
Identify the value of a dollar
bill and a dollar coin.
Compare the capacities of cups,
pints, and quarts.
No comparable standard in this
grade.
4.P.4 Use pictures, models,
tables, charts, graphs, words,
number sentences, and
mathematical notations to
interpret mathematical
relationships.
Students will KNOW:
Find the rule for a pattern
presented in a table and use the
rule to add inputs and outputs to
the table.
Making a table can help to
represent what you know in
solving a problem.
Word sentences that express
mathematical situations can be
translated into equations and
equations can help you solve
problems.
Writing a number expression is
one way of representing what we
know in a word problem.
Expressions can be evaluated
by substituting given values for
the variable.
Different kinds of real-world
problems can be represented by
and solved using different
operations.
An explanation of the solution
to a problem includes
information that is known and
Page 32 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
how you have used this
information.
The pattern or rule that relates
two quantities can be represented
using words, tables, and symbols.
Recognizing the pattern that
relates two quantities helps to
extend the table.
No comparable standard in this
grade.
3.P.4 Write number sentences
using +, –, x, ÷, <, =, and/or > to
represent mathematical
relationships in everyday
situations.
Students will KNOW:
Apply number operations and
symbols to represent
mathematical relationships in
everyday situations.
Students will be able to DO:
Write number expressions for
phrases.
Students will be able to DO:
Write equations for word
sentences.
Write number expressions for
phrases.
Evaluate variable expressions
that involve a single operation.
Make tables and use them to
solve word problems.
Write stories using given
multiplication and division facts.
Write to explain a pattern.
No comparable standard in this
grade.
4.P.6 Determine how change in
one variable relates to a change
in a second variable, e.g., inputoutput tables
Students will KNOW:
Find the rule for a pattern
presented in a table and use the
rule to add inputs and outputs to
the table.
The pattern or rule that relates
two quantities can be represented
using words, tables, and symbols.
Recognizing the pattern that
relates two quantities helps to
extend the table.
Page 33 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
Students will be able to DO:
Find the rule for a pattern
presented in a table and use the
rule to add inputs and outputs to
the table.
GEOMETRY
K.G.1 Name, describe, sort, and
draw simple two-dimensional
shapes.
No comparable standard in this
grade.
Students will KNOW:
A set of objects can be sorted
by one or more attributes.
A square has four equal sides, a
rectangle has two long and two
short sides, and a triangle has
three sides.
Students will be able to DO:
Sort the same set in different
ways.
Use more than one attribute to
sort a set of objects.
Solve problems by determining
the sorting rule for groups of
sorted objects.
Identify and describe squares
and other rectangles, circles, and
triangles.
No comparable standard in this
grade.
3.G.1 Compare and analyze
attributes and other features (e.g.,
number of sides, corners,
diagonals, and lines of
symmetry) of two-dimensional
geometric shapes.
4.G.1 Compare and analyze
attributes and other features (e.g.,
number of sides, faces, corners,
right angles, diagonals, and
symmetry) of two- and threedimensional geometric shapes.
Students will KNOW:
Polygons can be described by
specific properties and named
based on the number of sides and
corners.
Triangles and quadrilaterals can
be described and named based on
the relative lengths of their sides
and the sizes of their angles.
Students will KNOW:
Both plane figures and solid
figures have attributes you can
use to classify, identify, and
show similarities and differences
among the figures.
Geometric figures can be
described by specific properties
and named based on the number
of sides, faces, corners, or
vertices.
A plane figure has two
dimensions – length and width
and a solid figure has three
dimensions – length, width, and
height.
Students will be able to DO:
Identify and classify polygons.
Identify quadrilaterals that are
squares, rectangles, and
parallelograms.
Students will be able to DO:
Identify and classify attributes
of polygons and two- and threedimensional geometric shapes.
Page 34 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
K.G.2 Describe attributes of
two-dimensional shapes, e.g.,
number of sides, number of
corners.
Students will KNOW:
A square has four equal sides, a
rectangle has two long and two
short sides, a triangle has three
sides, and a circle is round.
Students will be able to DO:
Identify and describe squares
and other rectangles, circles, and
triangles.
GRADE 1
GRADE 2
2.G.1 Describe attributes and
parts of two- and threedimensional shapes, e.g., length
of sides, and number of corners,
edges, faces, and sides.
Students will KNOW:
Many everyday objects closely
approximate standard geometric
solids.
Many solids are comprised of
flat surfaces and vertices; the flat
surfaces on prisms are called
faces.
The flat surfaces of solid figures
are plane shapes.
2.G.1 Describe attributes and
parts of two- and threedimensional shapes, e.g., length
of sides, and number of corners,
edges, faces, and sides.
Students will KNOW:
Solids that comprise flat
surfaces, vertices, and edges.
The flat surfaces of solid figures
are plane shapes.
A large shape can be made from
more than one smaller shape.
Any given polygon has the same
number of sides as angles.
Students will be able to DO:
Identify and name standard
geometric solids and recognize
them in the environment.
Count the number of flat
surfaces and vertices on
geometric solids.
Match a geometric solid to an
outline of one of its flat surfaces.
Students will be able to DO:
Identify solid figures (cone,
cube, cylinder, pyramid,
rectangular prism, sphere) and
count their flat surfaces, vertices,
and edges.
Match a geometric solid to an
outline of one of its flat surfaces
and match that flat surface to a
plane shape.
K.G.3 Name and compare threedimensional shapes.
2.G.2 Identify, describe, draw,
and compare two-dimensional
shapes, including both polygonal
(up to six sides) and curved
figures such as circles.
2.G.2 Identify, describe, draw,
and compare two-dimensional
shapes, including both polygonal
(up to six sides) and curved
figures such as circles.
Students will KNOW:
Many everyday objects closely
approximate standard geometric
solids.
Solid figures have many
properties that make them
Students will KNOW:
Many everyday objects closely
approximate standard plane
shapes.
Plane shapes have many
properties that make them
Students will KNOW:
Atributes of two-dimensional
shapes as per standard 2.G.1.
Students will be able to DO:
Recognize and name
GRADE 3
GRADE 4
3.G.4 Identify and draw parallel
lines, perpendicular lines, and
other intersecting lines.
4.G.5 Describe and draw
intersecting, parallel, and
perpendicular lines.
Students will KNOW:
Definition of parallel and
perpendicular.
Students will KNOW:
Two lines can be intersecting,
parallel or perpendicular and
angles can be classified as right,
acute, obtuse, or straight.
Students will be able to DO:
Draw parallel, perpendicular,
and other intersecting lines
3.G.2 Describe, model, draw,
compare, and classify twodimensional shapes, e.g., circles,
triangles, and quadrilaterals.
Identify and describe simple
three-dimensional shapes, e.g.,
cubes, spheres, and pyramids.
Students will KNOW:
Space figures can be grouped
by their attributes and certain
space figures can be described
and classified by the number of
faces, edges, and corners they
Students will be able to DO:
Identify important geometric
terms relating to lines, parts of a
line, and angles.
4.G.2 Describe, model, draw,
compare, and classify two- and
three-dimensional shapes, e.g.,
circles, polygons—especially
triangles and quadrilaterals—
cubes, spheres, and pyramids.
Students will KNOW:
Both plane figures and solid
figures have attributes you can
use to classify, identify, and
show similarities and differences
among the figures.
Page 35 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
different from each other.
different from one another.
Students will be able to DO:
Identify and compare spheres,
cubes, cones, and cylinders.
Students will be able to DO:
Identify and name standard
plane shapes and recognize them
in the environment.
Sort plane shapes and identify
their properties.
GRADE 2
trapezoids, parallelograms, and
hexagons and identify the
number of sides and angles in a
polygon.
GRADE 3
GRADE 4
have.
Polygons can be described by
specific properties and named
based on the number of sides and
corners.
Quadrilaterals can be described
and named based on the relative
lengths of their sides and the size
of their angles.
Written descriptions can convey
understanding of the
characteristics of geometric
figures.
Geometric figures can be
described by specific properties
and named based on the number
of sides, faces, corners, or
vertices.
Written descriptions can convey
understanding of the
characteristics of geometric
figures.
Students will be able to DO:
Identify and describe space and
geometric figures (solids) by
name and identify their
similarities and differences.
Identify quadrilaterals that are
squares, rectangles, and
parallelograms.
2.G.3 Recognize congruent
shapes.
2.G.3 Recognize congruent
shapes.
Students will KNOW:
Two shapes can have the same
shape and the same size even if
they do not share the same
orientation.
Students will KNOW:
Shapes are congruent if they are
the same shape and the same
size. Two shapes can be
congruent even if they do not
share the same orientation.
Students will be able to DO:
Identify and create figures that
are the same size and the same
shape.
Students will be able to DO:
Identify and classify (sort) twoand three-dimensional shapes.
Describe similarities and
differences in geometric figures.
Draw two-dimensional shapes.
4.G.3 Recognize similar figures.
No comparable standard in this
grade.
Students will be able to DO:
Identify and create congruent
figures.
Students will KNOW:
Congruent figures have
matching sides with the same
length and matching angles with
the same measure. Similar
figures have matching angles
with the same measure and
proportional matching sides.
Students will be able to DO:
Identify and construct
congruent and similar figures.
Page 36 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
K.G.4 Identify positions of
objects in space, and use
appropriate language (e.g.,
beside, inside, next to, close to,
above, below, apart) to describe
and compare their relative
positions.
2.G.4 Identify shapes that have
been rotated (turned), reflected
(flipped), translated (slid), and
enlarged. Describe direction of
translations, e.g., left, right, up,
down.
2.G.4 Identify shapes that have
been rotated (turned), reflected
(flipped), translated (slid), and
enlarged. Describe direction of
translations, e.g., left, right, up,
down.
Students will KNOW:
The position of objects can be
determined in relation to
surrounding objects and
described using words.
The position of objects can be
determined in terms of whether
they are to the right or to the left
of other objects.
Students will KNOW:
Slides, flips, and turns can be
used to change an object’s
position in space.
Students will KNOW:
Slides, flips, and turns can be
used to change an object’s
orientation and position in space.
Students will be able to DO:
Perform a slide, flip, or turn on
an object and identify the
resulting position.
Students will be able to DO:
Perform slide, flip, or turn on an
object and identify the resulting
orientation and position.
GRADE 3
No comparable standard in this
grade.
Students will KNOW:
Congruent figures remain
congruent through translations,
reflections, and rotations.
Congruent figures have
matching sides with the same
length and matching angles with
the same measure. Similar
figures have matching angles
with the same measure and
proportional matching sides.
Students will be able to DO:
Use the words inside, outside,
over, under, and on to describe
the position of objects.
Use the words top, middle, and
bottom to describe position.
No comparable standard for this
grade
No comparable standard for this
grade
2.G.5 Identify symmetry in twodimensional shapes.
Students will KNOW:
Symmetrical figures have two
congruent halves.
2.G.5 Identify symmetry in twodimensional shapes.
Students will KNOW:
Symmetrical figures have two
congruent halves.
Students will be able to DO:
Identify objects that show
symmetry and draw lines of
symmetry.
2.G.6 Predict the results of
putting shapes together and
taking them apart.
Students will be able to DO:
Identify and create symmetrical
shapes.
2.G.6 Predict the results of
putting shapes together and
taking them apart.
GRADE 4
4.G.7 Describe and apply
techniques such as reflections
(flips), rotations (turns), and
translations (slides) for
determining if two shapes are
congruent.
No comparable standard for this
grade
3.G.7 Predict and explain the
results of taking apart and
combining two-dimensional
shapes.
Students will be able to DO:
Identify congruent and similar
figures and determine the slide
(translation), flip (reflection), or
turn (rotation) image of a figure.
No comparable standard for this
grade
4.G.9 Predict and validate the
results of partitioning, folding,
and combining two- and threedimensional shapes.
Page 37 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard for this
grade
GRADE 1
No comparable standard in this
grade.
GRADE 3
Students will KNOW:
A new shape can be made from
by putting two or more shapes
together.
Students will KNOW:
New shapes can be made by
combining other shapes through
slides, flips, and turns.
Students will be able to DO:
Predict which pattern blocks
will fill in an outline of a shape.
Students will be able to DO:
Predict what new shape would
be formed by putting two or
more shapes together.
Students will be able to DO:
Combine and take apart twodimensional shapes to make new
shapes using slides, flips, and
turns.
2.G.7 Relate geometric ideas to
numbers, e.g., seeing rows in an
array as a model of repeated
addition.
2.G.7 Relate geometric ideas to
numbers, e.g., seeing rows in an
array as a model of repeated
addition.
Standard is not developmentally
appropriate for grade 2 because it
is a beginning multiplication
skill. Application of arrays in
multiplication is addressed in
standard 3.N.6
Standard is not
developmentally appropriate for
grade 1.
No comparable standard in this
grade.
GRADE 2
Students will KNOW:
Familiar shapes can be
combined to create unfamiliar
ones.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
No comparable standard in this
grade.
GRADE 4
Students will KNOW:
Both plane figures and solid
figures have attributes you can
use to classify, identify, and
show similarities and differences
among the figures.
Students will be able to DO:
Predict and validate the results
of partitioning, folding, and
combining two- and threedimensional shapes.
No comparable standard in this
grade.
No comparable standard in this
grade.
3.G.5 Using ordered pairs of
whole numbers and/or letters,
locate and identify points on a
grid.
Students will KNOW:
Points on a coordinate grid can
be named using ordered pairs of
numbers.
4.G.6 Using ordered pairs of
numbers and/or letters, graph,
locate, identify points, and
describe paths (first quadrant).
Students will KNOW:
Points on a coordinate grid can
be named using ordered pairs of
numbers.
Students will be able to DO:
Locate and graph ordered pairs
on a coordinate grid.
Students will be able to DO:
Locate and graph ordered pairs
on a coordinate grid and describe
paths.
4.G.4 Identify angles as acute,
right, or obtuse.
3.G.3 Identify angles as right
angles, less than a right angle,
and greater than a right angle.
Students will KNOW:
Students will KNOW:
Page 38 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard in this
grade.
GRADE 1
No comparable standard in this
grade.
GRADE 2
No comparable standard in this
grade.
GRADE 3
GRADE 4
How to identify a right angle, a
less than a right angle, and
greater than a right angle.
Lines intersect at angles that
can be identified as acute, right,
or obtuse.
Students will be able to DO:
Identify angles as right angles,
less than a right angle, and
greater than a right angle.
Students will be able to DO:
Identify important geometric
terms relating to lines, parts of a
line, and angles.
3.G.6 Identify and draw lines of
symmetry in two-dimensional
shapes.
Students will KNOW:
A line of symmetry divides a
figure into two congruent parts.
4.G.8 Identify and describe line
symmetry in two-dimensional
shapes.
Students will KNOW:
A line of symmetry divides a
figure into two congruent parts.
Students will be able to DO:
Identify and make symmetrical
figures and draw a line or lines of
symmetry.
Students will be able to DO:
Identify and make symmetrical
figures and draw a line or lines of
symmetry.
MEASUREMENT
K.M.1 Recognize and compare
the attributes of length,
volume/capacity, weight, area,
and time using appropriate
language, e.g., longer, taller,
shorter, same length; heavier,
lighter, same weight; holds more,
holds less, holds the same
amount.
Students will KNOW:
Objects can be compared and
ordered by size, volume/capacity,
weight, and length.
The size, volume/capacity,
weight, and length of an object
can be measured.
2.M.1 Identify parts of the day
(e.g., morning, afternoon,
evening), days of the week, and
months of the year. Identify dates
using a calendar.
2.M.1 Identify parts of the day
(e.g., morning, afternoon,
evening), days of the week, and
months of the year. Identify dates
using a calendar.
3.M.1 Demonstrate an
understanding of the attributes
length, area, and weight, and
select the appropriate type of unit
for measuring each attribute
using both the U.S. Customary
(English) and metric systems.
4.M.1 Demonstrate an
understanding of such attributes
as length, area, weight, and
volume, and select the
appropriate type of unit for
measuring each attribute.
Students will KNOW:
The time of an event can be
estimated and labeled with the
terms morning, afternoon, or
night.
A calendar is a chart listing the
days of the month in order,
Students will KNOW:
There are 24 hours in one day.
The hour hand goes around the
clock face two times each day.
The 12 hours from midnight to
noon are called the A.M. hours.
The 12 hours from noon to
Students will KNOW:
Customary units of length are
inches, feet, yards, and miles.
Metric units are centimeters,
meters, and kilometers.
Customary units of weight are
ounces, pounds, and tons. Metric
Students will KNOW:
The number of square units
needed to cover the region inside
a figure is its area. To find the
area of a rectangle or a square
you can multiply the number of
rows of squares by the number of
Page 39 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
The duration of two events can
be compared.
grouped in weeks.
The months of the year, like the
days of the week, have a specific
order.
midnight are called t he P.M.
hours.
A calendar is a chart that lists
the days of the month in order.
These days are groups in 7-day
weeks. A calendar helps us keep
track of the days, weeks, months,
and years.
Students will be able to DO:
Compare and order sets of
objects by size, volume/capacity,
weight, and length.
Solve problems involving the
area of shapes.
Identify the activity or event
that takes more time or less time.
No comparable standard for this
grade
Students will be able to DO:
Determine whether an event
takes place in the morning,
afternoon, or night.
Read and use a calendar to
name the days of the week.
Identify and order the months of
the year.
2.M.2 Tell time at quarter-hour
intervals on analog and digital
clocks using a.m. and p.m
Students will KNOW:
The hour hand tells the hour
while the minute hand tells the
number of minutes after the hour.
Time to the hour and half hour
can be shown on an analog clock
or on a digital clock and can be
written in two ways: ___ o’clock
or __:00.
A half an hour is 30 minutes
long.
Students will be able to DO:
Identify the hour hand and the
minute hand on a clock and tell
time to the hour.
Tell and write time to the hour
and half hour on an analog and
Students will be able to DO:
Determine whether events occur
in the A.M. or P.M. hours.
Complete, read, and use a
calendar.
2.M.2 Tell time at quarter-hour
intervals on analog and digital
clocks using a.m. and p.m
Students will KNOW:
The hour hand tells the hour
while the minute hand tells the
number of minutes after the hour.
The number of minutes before
the next hour can be determined
by looking at the position of the
minute hand.
There are 24 hours in one day.
The hour hand goes around the
clock face two times each day.
The 12 hours from midnight to
noon are called the A.M. hours.
The 12 hours from noon to
midnight are called t he P.M.
hours.
Students will be able to DO:
GRADE 3
units are grams and kilograms.
Customary units of area are
square inches, feet, and yards.
Metric units are square
centimeters, meters, and
kilometers.
Students will be able to DO:
Choose the most appropriate
unit for length, area, and weight
3.M.3 Identify time to the minute
on analog and digital clocks
using a.m. and p.m. Compute
elapsed time, using a clock for
times less than one hour (i.e.,
minutes since), and using a
calendar (e.g., days since).
Students will KNOW:
Counting by 5s is a valuable
skill when telling time on an
analog clock.
The amount of time that passes
between a starting and an ending
time is elapsed time.
Vocabulary for calendars ordinal numbers, weeks, months,
years, leap years, decades,
centuries
Students will be able to DO:
Tell time to the nearest minute
using analog and digital clocks
and using AM and PM.
Compute elapsed time of less
than one hour and days since on
GRADE 4
squares in each row.
What you know about area can
help you find the volume of solid
figures.
Common customary units of
weight and measurement.
Students will be able to DO:
Choose the most appropriate
unit of weight and measurement.
Estimate using the most
appropriate unit of weight and
measurement..
4.M.3 Identify time to the
minute on analog and digital
clocks using a.m. and p.m.
Compute elapsed time using a
clock (e.g., hours and minutes
since…) and using a calendar
(e.g., days since…).
Students will KNOW:
Telling time on an analog clock
is based on an understanding of
the relationships between
minutes, hours, and days.
The amount of time that passes
between a starting and an ending
time is elapsed time.
A calendar organizes days,
weeks, months, and years.
Students will be able to DO:
Tell time to the nearest 1
minute and 5 minutes using
analog and digital clocks and
identify times as a.m. or p.m.
Find elapsed time, starting time,
or ending time, given two of
Page 40 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 2
GRADE 3
on a digital clock.
Tell time after the hour.
Tell time before the hour.
Determine whether events occur
in the A.M. or P.M. hours.
K.M.3 Use nonstandard units to
measure length, area, weight, and
capacity.
2.M.3 Compare the length,
weight, area, and volume of two
or more objects by using direct
comparison.
2.M.3 Compare the length,
weight, area, and volume of two
or more objects by using direct
comparison.
Students will KNOW:
The length of an object can be
described by comparing it to a
defined unit of length.
The length of an object can be
estimated and verified by
measuring.
The area of a shape can be
estimated and measured by
counting how many square units
it takes to cover the shape.
Students will KNOW:
Definition of longer, shorter,
heavier, and lighter.
Students will KNOW:
The capacity of a container can
be described by comparing it to
that of a defined unit of capacity.
You can tell how heavy or light
an object is by using a balance
scale to measure its weight.
a calendar.
Apply elements (ordinal
numbers, weeks, months, years,
leap years, decades, centuries)
and relationships (organization of
days, weeks, months, and years)
in a calendar.
3.M.2 Carry out simple unit
conversions within a system of
measurement, e.g., hours to
minutes, cents to dollars, yards to
feet or inches, etc.
Students will KNOW:
There are relationships that
make it possible to convert
between any two units of time.
Measurements of the same
property such as length or weight
can be converted to equivalent
units.
Metric units can be converted to
other metric units by multiplying
or dividing by a power of 10.
Students will be able to DO:
Measure the length of objects
using nonstandard units.
Estimate the length and width
of objects and verify by
measuring in nonstandard units.
Estimate and measure length
size, weight, volume/capacity
units.
GRADE 1
Students will be able to DO:
Compare the lengths, weight,
and volume of non-standard
units.
Students will be able to DO:
Estimate, measure, and order
objects by their capacities using
nonstandard units.
Estimate and measure the
weights of different objects by
comparing each of them to a
nonstandard unit of weight.
Students will be able to DO:
Convert among units of time
and compare measurements of
time.
Change units of length and
weight to equivalent units and
compare measures.
GRADE 4
these.
Find dates on a calendar that
are certain numbers of weeks
before and after given dates.
4.M.2 Carry out simple unit
conversions within a system of
measurement, e.g., hours to
minutes, cents to dollars, yards to
feet or inches, etc.
Students will KNOW:
There are relationships that
make it possible to convert
between any two units of time.
Measurements of the same
property such as length, weight
or mass, or capacity can be
converted to equivalent units.
The magnitude and accuracy of
the measurement determine the
units used.
The metric system of
measurement is based on the
decimal system of numeration.
Metric units can be converted to
other metric units by multiplying
or dividing by a power of 10.
Students will be able to DO:
Convert among units of time
and compare measurements of
time.
Change units of length, capacity
(mass), and weight to equivalent
units and compare measures.
Page 41 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
No comparable standard for this
grade
GRADE 1
GRADE 2
GRADE 3
GRADE 4
2.M.4 Measure and compare
common objects using metric and
English units of length
measurement, e.g., centimeter,
inch.
2.M.4 Measure and compare
common objects using metric and
English units of length
measurement, e.g., centimeter,
inch.
3.M.5 Identify and use
appropriate metric and U.S.
Customary (English) units and
tools (e.g., ruler, scale,
thermometer, clock) to estimate,
measure, and solve problems
involving length, area, weight,
temperature, and time.
4.M.5 Identify and use
appropriate metric and English
units and tools (e.g., ruler, angle
ruler, graduated cylinder,
thermometer) to estimate,
measure, and solve problems
involving length, area, volume,
weight, time, angle size, and
temperature.
Students will KNOW:
An inch is a standard unit used
to measure length.
A foot is a 12-inch standard unit
used to measure length.
Students will KNOW:
Inches and feet are standard
units used to measure length.
Inches, feet, and yards are
standard units used to measure
length.
Centimeters and meters are
standard units used to measure
length.
Students will KNOW:
Length, area, and weight as per
standard 3.M.1.
Calendar as per 3.M.3
The volume of a solid figure
can be found by counting the
cubes contained in the figure.
Fractions of an inch give
measurements that are closer to
the actual lengths of objects.
Freezing and boiling
temperatures on the Celsius scale
are 00 and 1000 while freezing
and boiling temperatures on the
Fahrenheit scale are 320 and
2120.
Students will KNOW:
Customary units of length are
the inch, foot, yard, and mile.
Using smaller units of measure
results in greater accuracy when
measuring length.
Common customary units of
capacity are teaspoon,
tablespoon, cup, pint, quart,
gallon, fluid ounce.
Millimeters, centimeters,
decimeters, meters, and
kilometers are basic metric units
of length.
The liter and milliliter are two
basic metric units of capacity and
1 liter is equivalent to 1,000
milliliters.
The gram and kilogram are two
basic metric units of mass and 1
kilogram is equivalent to 1,000
grams.
Students will be able to DO:
Estimate and measure the
lengths of objects to the nearest
inch using a ruler.
Estimate and measure the
length or height of objects to the
nearest foot using a 12-inch ruler.
Students will be able to DO:
Estimate and measure the
lengths or heights of objects in
inches, feet, or yards using a
ruler.
Estimate and measure lengths in
centimeters and meters using a
centimeter ruler or meterstick.
Students will be able to DO:
DO for length, area, and weight
as per standard 3.M.1.
DO for calendar as per 3.M.3
Measure lengths to the nearest
1/2 inch and to the nearest 1/4
inch.
Read temperatures above and
below zero on Fahrenheit and
Celsius thermometers and
determine appropriate
temperatures for given activities.
Students will be able to DO:
Estimate and measure length to
the nearest inch and choose the
most appropriate customary unit
of length for a given object of
distance.
Page 42 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
Measure and draw lengths to
the nearest half, quarter, or eighth
of an inch.
Choose the most appropriate
customary unit of capacity for a
given container and estimate and
measure capacity using
customary units.
Estimate and measure length in
metric units and choose the most
appropriate metric unit of length
for an object or distance.
Estimate and measure capacity
in milliliters and liters and
choose the most appropriate
metric unit of capacity of a
container.
Estimate and measure mass in
grams and kilograms and choose
the most appropriate metric unit
of mass for an object.
No comparable standard for this
grade
2.M.5 Select and correctly use
the appropriate measurement
tools, e.g., ruler, balance scale,
thermometer.
Students will KNOW:
An inch is a standard unit used
to measure length.
A foot is a 12-inch standard unit
used to measure length.
There are many ways to
measure an object and each uses
a different tool.
The longer the unit of measure,
the fewer units it takes to
measure the length of an object.
Students will be able to DO:
Estimate and measure the
2.M.5 Select and correctly use
the appropriate measurement
tools, e.g., ruler, balance scale,
thermometer.
Students will KNOW:
Inches, feet, and yards are
standard units used to measure
length.
Centimeters and meters are
standard units used to measure
length.
Temperature is a measure of
how hot or cold something is.
Some thermometers use the
Fahrenheit scale and others use
the Centigrade
You can tell how heavy or light
an object is by using a balance
No comparable standard for this
grade
No comparable standard for this
grade
Page 43 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
lengths of objects to the nearest
inch using a ruler.
Estimate and measure the
length or height of objects to the
nearest foot using a 12-inch ruler.
Identify appropriate tools for
measuring length, weight,
capacity, and temperature.
K.M.2 Make and use estimates
of measurements from everyday
experiences.
2.M.6 Make and use estimates of
measurement, including time,
volume, weight, and area.
Students will KNOW:
Some real-world problems can
be solved using known concepts,
skills, and strategies.
Students will KNOW:
Strategies for estimating.
Students will be able to DO:
Review and apply concepts,
skills, and strategies dealing with
measurement in K.M.1.
The area of a shape can be
estimated and measured by
counting how many square units
it takes to cover the shape.
Estimate size, volume/capacity,
weight, and length.
Students will be able to DO:
Estimate weight, length, time,
and capacity.
GRADE 2
GRADE 3
GRADE 4
3.M.4 Estimate and find area and
perimeter of a rectangle, using
diagrams and grids, or by
measuring.
4.M.4 Estimate and find area
and perimeter of a rectangle,
triangle, or irregular shape using
diagrams, models, and grids or
by measuring.
Students will KNOW:
The perimeter of a figure can be
measured using units of length.
To find the perimeter of a
polygon you can add the lengths
of the sides.
The number of square units
needed to cover the region inside
a figure is its area. To find the
area of a rectangle or a square
you can multiply the number of
rows of squares by the number of
squares in each row.
Using objects can help to
represent what you know in
solving a problem.
scale to measure its weight.
Students will be able to DO:
Estimate and measure the
lengths or heights of objects in
inches, feet, or yards using a
ruler.
Estimate and measure lengths in
centimeters and meters using a
centimeter ruler or a meterstick.
Show, read, and write
temperatures shown on
Fahrenheit and Centigrade
thermometers.
Estimate and measure the
weights of different objects by
comparing each of them to a non
–standard unit of weight.
2.M.6 Make and use estimates
of measurement, including time,
volume, weight, and area.
Students will KNOW:
Some activities take minutes to
complete while others take hours
or days.
Inches, feet, and yards are
standard units used to measure
length.
Centimeters and meters are
standard units used to measure
length.
The capacity of a container can
be described by comparing it to
that of a defined unit of capacity.
A liter is a standard metric unit
used to measure capacity.
You can tell how heavy or light
Students will KNOW:
The distance around a figure
(perimeter) can be measured
using units of length.
The number of square units
needed to cover the region inside
a figure is its area.
Students will be able to DO:
Find the perimeter of rectangle
using non-standard and standard
units of length using diagrams
and grids or measuring.
Estimate or find areas of figures
in square units of rectangles.
Page 44 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
an object is by using a balance
scale to measure its weight.
Ounces and pounds are standard
units used to measure weight.
Grams and kilograms are
standard metric units used to
measure mass.
Students will be able to DO:
Estimate whether an activity
takes minutes, hours, or days to
complete.
Estimate and measure the
lengths or heights of objects in
inches, feet, or yards using a
ruler.
Estimate and measure lengths in
centimeters and meters using a
centimeter ruler or a meter stick.
Estimate, measure, and order
objects by their capacities using
nonstandard units.
Estimate and measure capacity
in liters.
Estimate and measure the
weights of different objects by
comparing each of them to a
nonstandard unit of weight.
Compare and estimate the
weights of objects in ounces and
pounds.
Compare and estimate the
masses of objects in grams and
kilograms.
GRADE 3
GRADE 4
Students will be able to DO:
Find the perimeter of a polygon
by adding the lengths of the sides
or by using a formula.
Find the area of rectangles and
irregular rectangular shapes by
counting square units or by using
a formula.
Some problems can be solved
by showing the action with
objects.
Page 45 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
K.D.1 Collect, sort, organize,
and draw conclusions about data
using concrete objects, pictures,
numbers, and graphs.
GRADE 1
GRADE 2
GRADE 3
DATA ANALYSIS, STATISTICS, & PROBABILITY
No comparable standard for this
grade
No comparable standard for this
grade
Students will KNOW:
A graph is a way to show
information in an organized,
pictorial way.
The data in a graph can be used
to answer a question.
A graph is a tool that can be
used to organize information in
order to solve problems.
3.D.1 Collect and organize data
using observations,
measurements, surveys, or
experiments, and identify
appropriate ways to display the
data.
Students will KNOW:
Tally charts are useful in
recording and organizing survey
data.
Students will be able to DO:
Use tally charts to record and
organize data.
GRADE 4
4.D.1 Collect and organize data
using observations,
measurements, surveys, or
experiments, and identify
appropriate ways to display the
data.
Students will KNOW:
A survey is a way of gathering
data.
Students will be able to DO:
Interpret survey results and
make predictions based on those
results.
Students will be able to DO:
Complete and read a real,
picture, or bar,graph.
Collect and organize data in a
reading a real, picture, or
bar,graph.
Solve problems by reading a
real, picture, or bar,graph.
No comparable standard for this
grade
2.D.1 Use interviews, surveys,
and observations to gather data
about themselves and their
surroundings.
Students will KNOW:
Interviews, surveys, and
observations are ways of
gathering data.
Students will be able to DO:
Conduct interviews, surveys,
and observations to gather data.
2.D.1 Use interviews, surveys,
and observations to gather data
about themselves and their
surroundings.
Students will KNOW:
A survey is one way to collect
data. Organized data can be used
to answer questions and solve
problems.
A pictograph uses pictures or
symbols to show data.
A bar graph shows information
in rows or columns that look like
bars.
No comparable standard for this
grade
No comparable standard for this
grade
Page 46 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
2.D.2 Organize, classify,
represent, and interpret data
using tallies, charts, tables, bar
graphs, pictographs, and Venn
diagrams; interpret the
representations.
Students will KNOW:
Writing down all the possible
ways of doing something can be
a good way to solve a problem.
A time schedule is a table that is
useful for solving problems.
Elements of a graph can be
shown in groups of 10 to tell how
many there are in all.
A bar graph can show the same
information as a picture graph,
but uses bars instead of pictures.
Tally marks provide an easily
countable way to record how
many times something happens.
Tables are tools used for
organizing information.
Students will be able to DO:
Solve problems by making
organized lists.
Solve problems by reading and
using the information in a
schedule.
Solve problems by using data
from a graph.
Collect data and organize it into
a bar graph.
Experiment and record data
GRADE 2
Students will be able to DO:
Collect data using a survey,
pictograph, bar graphs.
2.D.2 Organize, classify,
represent, and interpret data
using tallies, charts, tables, bar
graphs, pictographs, and Venn
diagrams; interpret the
representations.
Students will KNOW:
Writing down all of the possible
ways of doing something is
sometimes a good way to solve a
problem.
Data from a chart or table are
frequently helpful in solving
problems.
Show the same amount of
money using different sets of
coins.
Recording tally marks is a
useful way to organize data in a
table.
A survey is one way to collect
data. Organized data can be used
to answer questions and solve
problems.
Venn diagrams are useful for
comparing, contrasting, and
sorting data that have
overlapping attributes.
A pictograph uses pictures or
symbols to show data.
A bar graph shows information
in rows or columns that look like
bars.
Data in a number form can be
organized on charts for easy
GRADE 3
GRADE 4
3.D.2 Match representations of a
data set in the forms of tables,
line plots, pictographs, tallies, or
bar graphs with the actual data
set.
4.D.2 Match a representation of
a data set such as lists, tables, or
graphs (including circle graphs)
with the actual set of data.
Students will KNOW:
Various types of graphs are
useful in solving problems with
some types being more
appropriate than others in given
situations.
Students will KNOW:
Various types of graphs are
useful in solving problems with
some types being more
appropriate than others in given
situations.
A survey is a way of gathering
data.
Students will be able to DO:
Match a line graph, pictograph,
bar graph, and line plot to given
data to solve problems.
Students will be able to DO:
Use data in tables and tally
charts to make line graphs, bar
graphs, and pictographs to solve
problems.
Interpret survey results and
make predictions based on those
results.
Page 47 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
using tally marks.
Solve problems by making
tables.
No comparable standard for this
grade
GRADE 2
GRADE 3
GRADE 4
3.D.3 Construct and draw
conclusions from representations
of data sets in the forms of tables,
line plots, pictographs, tallies,
and bar graphs.
4.D.3 Construct, draw
conclusions, and make
predictions from various
representations of data sets,
including tables, bar graphs,
pictographs, line graphs, line
plots, and tallies.
Students will KNOW:
Reading a pictograph involves
knowing the value of each
symbol given by the key and then
multiplying and adding to find
the total for a row.
A line plot provides an effective
illustration of data and makes it
relatively easy to identify the
range.
In a bar graph, quantities can be
interpretation.
2.D.3 Formulate inferences
(draw conclusions) and make
educated guesses (conjectures)
about a situation based on
information gained from data.
Students will be able to DO:
Solve a problem by making an
organized list.
Solve a problem by using clues
and data from a chart.
There are many combinations
of coins that show the same
amount of money.
Solve problems involving
addition by using data from a
table.
Solve a problem by making a
table.
Collect and analyze data that
have been gathered using a
survey, Venn diagram,
pictograph, and bar graph.
Predict the outcomes of simple
experiments.
2.D.3 Formulate inferences
(draw conclusions) and make
educated guesses (conjectures)
about a situation based on
information gained from data.
Students will KNOW:
The regular repetitions of a
pattern can be used to find a
missing term in the pattern.
Students will KNOW:
The outcome of a situation can
be predicted based on the data
given.
Students will be able to DO:
Solve problems by using data
from a picture.
Students will be able to DO:
Predict the outcomes of simple
experiments.
Students will KNOW:
Tally charts are useful in
recording and organizing survey
data.
A line plot provides an effective
illustration of data for a small
range of data.
Pictographs and bar graphs are
used to compare data.
Comparison statements can
demonstrate understanding of the
Page 48 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
data shown in graphs.
In a line graph, points are
connected to show how data
values change over time.
The key for a pictograph
determines the number of
pictures needed to represent a
given set of data.
In a bar graph, quantities can be
compared by comparing the
lengths of the bars.
Various types of graphs are
useful in solving problems with
some types being more
appropriate than others in given
situations.
Students will be able to DO:
Use tally charts to record and
organize data.
Make, interpret, and use a line
graph, pictograph, bar graph, and
line plot to solve problems.
GRADE 4
compared by comparing the
lengths of the bars.
In a line graph, points are
connected to show how data
values change over time.
Various types of graphs are
useful in solving problems with
some types being more
appropriate than others in given
situations.
Graph scales can be chosen so
as to exaggerate or minimize
changes or differences in data.
An explanation of the solution
to a problem includes
information that is known and
how you have used this
information.
Students will be able to DO:
Read, interpret, and make
pictographs.
Read, interpret, and make line
plots,
Read, interpret, and make bar
graphs.
Interpret and make line graphs.
Use data in tables and tally
charts to make line graphs, bar
graphs, an pictographs to solve
problems.
Interpret survey results and
make predictions based on those
results.
Recognize misleading graphs
and identify the misleading
characteristics.
Write to explain a prediction.
Page 49 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
2.D.4 Decide which outcomes of
experiments are most likely.
2.D.4 Decide which outcomes of
experiments are most likely.
No comparable standard for this
grade
No comparable standard for this
grade
GRADE 3
No comparable standard in this
grade.
Students will KNOW:
An event that is more likely will
occur more often than an event
that is less likely.
Students will KNOW:
An event that is more probable
will occur more often than an
event that is less probable.
Students will be able to DO:
Describe the likelihood of an
event as more likely or less
likely.
Students will be able to DO:
Predict the outcomes of simple
experiments.
No comparable standard for this
grade
No comparable standard for this
grade
3.D.4 List and count the number
of possible combinations of
objects from two sets, e.g., how
many different outfits can one
make from a set of two sweaters
and a set of three skirts?
Students will KNOW:
Make a tree diagram or an
organized list showing possible
combinations that can be made
from two sets.
Students will be able to DO:
List all possible results for a
situation.
GRADE 4
4.D.4 Represent the possible
outcomes for a simple probability
situation, e.g., the probability of
drawing a red marble from a bag
containing three red marbles and
four green marbles.
Students will KNOW:
Words like impossible,
unlikely, likely, certain, and fair
can describe probabilities.
The probability of an event is
the ratio of favorable outcomes
to the total number of outcomes,
which can range from zero for
and impossible event to one for a
certain event.
Students will be able to DO:
Use a fraction to express the
probability of an event.
4.D.5 List and count the number
of possible combinations of
objects from three sets, e.g., how
many different outfits can one
make from a set of three shirts, a
set of two skirts, and a set of two
hats?
Students will KNOW:
Making an organized list can
help to represent what you know
and aid in solving a problem.
A tree diagram is a visual
representation of all possible
outcomes.
Students will be able to DO:
Make an organized list to
represent information given in a
problem
Page 50 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment
Gateway Regional School District
VERTICAL ARTICULATION OF MATHEMATICS STANDARDS
Grades K-4
KINDERGARTEN
GRADE 1
GRADE 2
GRADE 3
GRADE 4
List all possible results for a
situation.
No comparable standard for this
grade
No comparable standard for this
grade
No comparable standard for this
grade
No comparable standard for this
grade
4.D.6 Classify outcomes as
certain, likely, unlikely, or
impossible by designing and
conducting experiments using
concrete objects such as
counters, number cubes,
spinners, or coins.
Students will KNOW:
Words like impossible,
unlikely, likely, certain, and fair
can describe probabilities.
Students will be able to DO:
Describe a probability as likely,
unlikely, impossible, or certain.
Page 51 of 51
Aligned with November 2000 Massachusetts Mathematics Curriculum Framework and the May 2007 Supplement
January 2009 & December 2009 - minor revisions of alignment