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Finding Focus for
Mathematics Instruction –
Grade 1
Huron Intermediate School District
March 25, 2010
Finding Focus for Mathematics Instruction – Grade 1
Huron Intermediate School District
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Introduction
When teachers plan instruction, they draw on many sources such as state assessment
standards, local curriculum guides, textbook materials, and supplemental assessment
resources. These documents serve as useful sources of information, and it is neither necessary
nor desirable to replace them.
Michigan’s Grade-Level Content Expectations (GLCEs) describe in detail many ways in which
students can demonstrate their mastery of the mathematics curriculum. The GLCEs do not,
however, describe the big ideas and enduring understandings that students must develop in
order to achieve these expectations. The GLCEs describe products of student learning, but
they do not describe the thinking that must take place within the minds of students as they learn.
It is the purpose of this document to focus on the fundamental mathematical ideas that form the
basis of elementary and middle school instruction. Although a variety of research materials
were used in the development of this document, several sources were relied on quite heavily.
In 2006, the National Council of Teachers of Mathematics (NCTM) released Curriculum Focal
Points for Prekindergarten through Grade 8 Mathematics. The Focal Points describe big topics,
or focus areas, for each grade level.
In May, 2009, the Michigan Department of Education published the Michigan Focal Points Core
and Extended Designations. In that document, the NCTM Focal Points were adjusted to align
with Michigan’s GLCEs. The new core and extended designations for the MEAP reflect
Michigan’s Focal Points.
This document is structured around Michigan’s Focal Points and supporting documents, with
significant content included from two other documents:
Charles, Randall I. “Big Ideas and Understandings as the Foundation for Elementary
and Middle School Mathematics.” NCSM Journal of Mathematics Education Leadership.
Spring-Summer, 2005. vol. 8, no. 1, pp. 9 – 24.
“Chapter 4: Curricular Content.” Foundations for Success: The Final Report of the
National Mathematics Advisory Panel. U.S. Department of Education, 2008. pp. 15 – 25.
Particular thanks go to Ruth Anne Hodges for her contributions to this project.
references to research are cited throughout the document.
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Focusing on Mathematics at Grade 1
Grade
Grade 1
#1
Michigan Focal Point
Related GLCE Topics
Targeted Vocabulary
Developing understanding
of addition and subtraction
and strategies for basic
addition facts and related
subtraction facts









Count, write, and order
numbers
Add and subtract whole
numbers
Solve problems
Work with money




Count by/skip count
Count backwards
Quantity/amount/value
Order
Compare (i.e., same
as, greater than, fewer
than, equal to, bigger,
smaller, smallest,
largest)
More / increase
Less / decrease
Addition-- add (+),
sum, combine/join/put
together, extend/
count on/more
Subtraction—subtract
(-), difference,
compare, take away
Fact family
Number sentence
Equals / equal to (=)
Number path / line




Grade 1
#2
Grade 1
#3
Developing an
understanding of whole
number relationships,
including grouping tens
and ones

Explore place value

Ones / tens
Developing an
understanding of linear
measurement and facility
in measuring lengths

Estimate and measure
length



Length
Measure
Shorter/shortest,
taller/tallest,
longer/longest
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National Math Panel Benchmarks
By the end of
Grade 4,
students should
be proficient
with the addition
and subtraction
of whole
numbers.
By the end of
Grade 5,
students should
By the end of
Grade 6,
students should
By the end of
Grade 7,
students should
be proficient
with comparing
fractions and
decimals and
common
percents, and
with the addition
and subtraction
of fractions and
decimals.
be proficient
with
multiplication
and division of
fractions and
decimals.
be proficient
with all
operations
involving
positive and
negative
fractions.
be able to solve
problems
involving
perimeter and
area of triangles
and all
quadrilaterals
having at least
one pair of
parallel sides
(i.e., trapezoids).
be able to
analyze the
properties of
two-dimensional
shapes and
solve problems
involving
perimeter and
area and
analyze the
properties of
threedimensional
shapes and
solve problems
involving surface
area and
volume.
be proficient
with
multiplication
and division of
whole numbers.
be able to
identify and
represent
fractions and
decimals, and
compare them
on a number line
or with other
common
representations
of fractions and
decimals.
Geometry and Measurement
Fluency with Fractions and Decimals
Fluency with
Whole Numbers
By the end of
Grade 3,
students should
be proficient
with all
operations
involving
positive and
negative
integers.
be able to solve
problems
involving
percent, ratio,
and rate and
extend this work
to
proportionality.
be familiar with
the relationship
between similar
triangles and the
concept of the
slope of a line.
Taken from The National Mathematics Advisory Panel Final Report, 2008, p.20
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Michigan Focal Points Leading to Proficiency with Addition and Subtraction
Big
Mathematical
Ideas


The same
value can be
represented
in many
equivalent
ways
You can only
add or
subtract
things that
are the same
(that have the
same unit)
Kindergarten
Grade 1
Grade 2
Grade 3
Focal Point #1:
Focal Point #2:
Focal Point #1:
Focal Point #4:
Representing,
comparing, and
ordering whole
numbers and
joining and
separating sets
Developing an
understanding of
whole number
relationships,
including grouping in
tens and ones
Developing an
understanding of
the base-ten
numeration system
and place-value
concepts
Developing an
understanding of
fractions and
fraction
equivalence
Focal Point #1:
Focal Point #2:
Developing
understandings of
addition and
subtraction and
strategies for basic
addition facts and
related subtraction
facts
Developing quick
recall of addition
facts and related
subtraction facts
and fluency with
multi-digit addition
and subtraction
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NATIONAL M ATH
PANEL
BENCHMARK:
By the end of
Grade 3, students
should be
proficient with the
addition and
subtraction of
whole numbers.
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Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies
for basic addition facts and related subtraction facts
Kindergarten
Grade 1
Grade 2
Developing understandings
of addition and subtraction
and strategies for basic
addition facts and related
subtraction facts (NCTM-1st)
Developing quick recall of
addition facts and related
subtraction facts and
fluency with multidigit
addition and subtraction
(NCTM-2nd)
Count, write, and order
numbers
Count, write, and order
numbers
Count, write, and order whole
numbers
Compose and decompose
numbers
Add and subtract whole
numbers
Add and subtract whole
numbers
Add and subtract numbers
Solve problems using addition
and subtraction of length,
money, and time (M.PS.01.08)
Measure, add, and subtract
length
Work with money
Solve measurement problems
involving length, money, and
perimeter (M.PS.02.10 ,
M.TE.02.11)
Tell time and solve time
problems
Record, add, and subtract
money
National Math Panel Benchmark:
By the end of Grade 3, students should be
proficient with the addition and subtraction
of whole numbers
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross over GLCE topics associated with another focal point
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Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies
for basic addition facts and related subtraction facts
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #1 (Numbers)
The set of real numbers is infinite, and each real number can be associated with a unique
point on the number line.

Counting tells how many items there are altogether. The number of objects in a set is the
last number named when the objects are counted.

Counting a set 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.

Quantities are measured or counted using same-sized units.

0 represents a set with no objects.

0 added to or subtracted from a number does not change the number.

All quantities have a unit. Sometimes the unit is unspoken (i.e., 23 ones).

The counting numbers can be put in a path so that numbers are in order. The position of
two numbers on the number path shows which number is larger.

The smallest counting number is one, but a number path can start at any number. There is
no largest counting number.

If a number path is used for counting on or counting back, the numbers must be counted by
the same unit all the way through.
o 7, 8, 9, 10, 11, 12 is a valid number path because each number is one more than
the number before it
o 2, 4, 6, 8, 10, 12 is a valid number path because the difference between one
number and the next is always 2
o 1, 2, 3, 6, 7, 15 is a list of numbers in order but can not be used for counting
because the difference the difference between one number and the next keeps
changing

The number line uses segments of equal length to represent number. When using a
number line to add, subtract, or count, count the “jumps” between numbers, not the tick
marks.
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
The distance between any two consecutive counting numbers on a given number line is the
same. In other words, the distance between one counting number and the next is a unit of
one.
Big Idea #2 (The Base Ten Numeration System)
The base ten numeration system is a scheme for recording numbers using digits 0-9,
groups of ten, and place value.

The same quantity can be expressed in many equivalent forms using different units:
o 3 red teddy bears is the same number of objects as 3 teddy bears
o 23 ones is equivalent to 2 tens and 3 ones
o 24 tens is equivalent to 240 ones
o 100 pennies has the same value as 1 dollar
o 2 feet has the same length as 24 inches
o 5+2=9–2
o 1/2 = 2/4
o 3x = 2x + x

Rational numbers can be written as a sum of ones, tens, hundreds, and other types
of ten-groups (powers of 10).
o 23 = 2 tens + 3 ones = 20 + 3
o 874 = 8 hundreds + 7 tens + 4 ones = 800 + 70 + 4
o 12.3 = 1 ten + 2 ones + 3 tenths = 10 + 2 + 0.3
Teacher note: 10n, . . . 103, 102, 101, 100, 10-1, 10-2, 10-3, . . . 10-n

Ones can only be added to ones, tens can only be added to tens, and so forth.
Place value makes this easy to see, so that we can add and subtract faster.

Sets of ten, one hundred and so forth must be perceived as single entities when
interpreting numbers using place value (e.g., 1 hundred is one group; it is 10 tens or
100 ones).

When there are no groups of a particular size, 0 is used as a place holder. Since
adding 0 does not change the value of a number, the 0 is not used in expanded
notation.
o

302 = 300 + 2
In Base 10 notation, each place is ten times the value of the place to the right:
o
o
o
o
o
o
5 tens = 50 ones
2 hundreds = 20 tens
400 tens = 4 thousands
3 ones = 30 tenths
20 hundredths = 2 tenths
4 tenths = 400 thousandths
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Big Idea #3 (Equivalence)
Any number, measure, numerical expression, algebraic expression, or equation can be
represented in an infinite number of ways that have the same value.

Only numbers that have the same unit can be added or subtracted. The sum or difference
also has the same unit. Sometimes, it is useful to change a quantity to a different form in
order to add or subtract quantities.
o 4 teddy bears + 5 teddy bears = 9 teddy bears
o 2 dogs + 3 cats is 2 dogs and 3 cats, but 2 pets + 3 pets = 5 pets
o 2 ones + 4 ones = 6 ones
o 2 tens + 4 ones = 2 tens and 4 ones, but 20 ones + 4 ones = 24 ones
o 5 nickels + 8 nickels = 13 nickels
o 5 dimes + 8 nickels = 5 dimes and 8 nickels, but 5 dimes + 4 dimes = 9 dimes.
OR: 50 pennies + 40 pennies = 90 pennies
o 1/2 + 2/2 = 3/2
o 1/2 + 2/3 can only be simplified by finding equivalent fractions: 3/6 + 4/6 = 7/6
Big Idea #5 (Operation Meanings & Relationships)
The same number sentence (e.g. 12 – 4 = 8) can be associated with different concrete or
real-world situations, AND different number sentences can be associated with the same
concrete or real-world situation.

Real-world situations for addition and subtraction:
o Subtraction can be used to describe a situation where some items are taken
away from a set: John had 12 pennies and gave 4 away; how many did he have
left?
o Subtraction can also be used to compare two quantities: John has 12 pennies,
and Mary has 4. How many more pennies does John have than Mary?
o Addition can be used to describe a situation where two sets of items are
combined: John has 8 pennies, and Mary has 4. How many pennies do they
have all together?
o Addition can also be used to count on from a number: John had 8 pennies, and
he found 4 more. How many does he have now?

The real-world actions for addition and subtraction of whole numbers are the same for
operations with fractions, decimals, and integers.

Only numbers that have the same unit can be added or subtracted. The sum or
difference also has the same unit. Sometimes, it is useful to change a quantity to a
different form in order to add or subtract quantities.
o 4 teddy bears + 5 teddy bears = 9 teddy bears
o 2 dogs + 3 cats is 2 dogs and 3 cats, but 2 pets + 3 pets = 5 pets
o 2 ones + 4 ones = 6 ones
o 2 tens + 4 ones = 2 tens and 4 ones, but 20 ones + 4 ones = 24 ones
o 5 nickels + 8 nickels = 13 nickels
o 5 dimes + 8 nickels = 5 dimes and 8 nickels, but 5 dimes + 4 dimes = 9 dimes.
OR: 50 pennies + 40 pennies = 90 pennies
o 1/2 + 2/2 = 3/2
o 1/2 + 2/3 can only be simplified by finding equivalent fractions: 3/6 + 4/6 = 7/6
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Big Idea #6 (Properties)
For a given set of numbers there are relationships that are always true, and these are the
rules that govern arithmetic and algebra.

Sets can be joined (added) in any order. Addition facts can be “turned around.”
o 3 + 5 = 8 and 5 + 3 = 8 (addition is commutative)
o 3 + 6 + 4 = 6 + 4 + 3 = 10 + 3 = 13 (associative property applies because
addition is commutative)

Sets are separated (subtracted) by starting with the total number of objects. Subtraction
facts do not “turn around” the same way as addition facts (subtraction is not commutative),
but each subtraction fact has a related subtraction fact.
o 8 – 5  5 – 8. However, 8 – 5 = 3 and 8 – 3 = 5.

Adding zero to a number (or subtracting zero from a number) does not change the number.
Big Idea #7 (Basic Facts & Algorithms)
Basic facts and algorithms for operations with rational numbers use notions of
equivalence to transform calculations into simpler ones.

Subtraction is the inverse of addition. Any addition problem has related subtraction
problems, and any subtraction problem has related addition problems.

Any subtraction calculation can be solved by adding up, and addition can be used to check
subtraction.

Any unknown addition or subtraction fact can be found by using known facts.
o 5 is one more than 4, so 4 + 1 = 5.
o 7 is one less than 8, so 8 – 1 is 7. That means that 8 – 7 = 1.
o 5 + 5 is 10 and 5 + 3 is 8. 3 more than 10 is 13, so 5 + 8 = 13.
[ The light gray points are related to the same Big Idea and topic, but are addressed at a later
grade level.]
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Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies
for basic addition facts and related subtraction facts
INSTRUCTIONAL IMPLICATIONS
In Chapter 6 of their book Learning and Teaching Early Math: The Learning Trajectories
Approach, Douglas H. Clements and Julie Sarama (2009) 1 address three topics of increasing
sophistication: arithmetic combinations (“facts”), place value, and multi-digit addition and
subtraction. Each of these topics builds on composition of number, a central idea in the
development of number sense. In this section, we provide a summary of the research
implications regarding arithmetic combinations.
A robust understanding of arithmetic combinations involves both counting-based strategies and
part-whole relationships. Counting-based strategies rely on an understanding of contextual
situations that can be modeled with objects or pictures, as described in Table 1 on the next
page. Both the situation (context) and the nature of the unknown value impact the difficulty of
the problem. For some situations, which value is unknown does not significantly affect the
difficulty of the problem; for other situations, different unknowns lead to problems with large
differences in difficulty. In “Add to” situations, for example2,
Result unknown problems are easy, change unknown problems are moderately
difficult, and start unknown are the most difficult. This is due in large part to the
increasing difficulty children have modeling, or “act outing,” each type (p. 62).
In other words, students find it much easier to answer the question “3 and 2 more is how
many?” (result unknown) than to determine “how many do you start with if 2 more makes 5?”
(start unknown). Therefore, in sequencing arithmetic instruction, teachers should find where a
child is in the learning trajectory and tailor instructional tasks to move the child forward.
In addition to counting strategies that can be modeled within the different problem types
described earlier, children also develop arithmetic combinations through their understanding of
part-whole relationships. To start, children separate a group of objects in various ways. For
example, a child could separate 7 objects into a pile of 5 and a pile of 2, or 6 and 1, or 0 and 7.
These separations are represented as addition problems so that, eventually, a child can
produce all of the number combinations composing a given number. For example, 6 is
represented as 5+1, as 4+2, and so forth.
Clements, Douglas H. and Sarama, Julie (2009) Learning and Teaching Early Math: The
Learning trajectories Approach. Routledge: New York, NY.
2
The authors refer to this situation as “Join” or “Change Plus.”
1
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Table 1
Add to
Take from
Put together/
Take apart
Compare
Result Unknown
Change Unknown
Start Unknown
Two bunnies sat on the
grass. Three more
bunnies hopped there.
How many bunnies are
on the grass now?
2+3=?
Two bunnies were sitting on
the grass. Some more
bunnies hopped there. Then
there were five bunnies. How
many bunnies hopped over
to the first two?
2+?=5
Some bunnies were sitting
on the grass. Three more
bunnies hopped there. Then
there were five bunnies. How
many bunnies were on the
grass before?
?+3=5
Five apples were on
the table. I ate two
apples. How many
apples are on the table
now?
5–2=?
Five apples were on the
table. I ate some apples.
Then there were three
apples. How many apples did
I eat?
5–?=3
Some apples were on the
table. I ate two apples. Then
there were three apples.
How many apples were on
the table before?
?–2=3
Total Unknown
Addend Unknown
Both Addends Unknowns
Three red apples and
two green apples are
on the table. How many
apples are on the
table?
3+2=?
Five apples are on the table.
Three are red and the rest
are green. How many apples
are green?
3 + ? = 5, 5 – 3 = ?
Grandma has five flowers.
How many can she put in her
red vase and how many in
her blue vase?
5 = 0 + 5, 5 = 5 + 0
5 = 1 + 4, 5 = 4 + 1
5 = 2 + 3, 5 = 3 + 2
Difference Unknown
Bigger Unknown
Smaller Unknown
(“How many more?”
version):
Lucy has two apples.
Julie has five apples.
How many more apples
does Julie have than
Lucy?
(“How many fewer?”
version):
Lucy has two apples.
Julie has five apples.
How many fewer
apples does Lucy have
than Julie?
2 + ? = 5, 5 – 2 = ?
(Version with “more”):
(Version with “more”):
Julie has three more apples
than Lucy. Lucy has two
apples. How many apples
does Julie have?
Julie has three more apples
than Lucy. Julie has five
apples. How many apples
does Lucy have?
(Version with “fewer”):
(Version with “fewer”):
Lucy has 3 fewer apples than
Julie. Lucy has two apples.
How many apples does Julie
have?
2 + 3 = ?, 3 + 2 = ?
Lucy has 3 fewer apples than
Julie. Julie has five apples.
How many apples does Lucy
have?
5 – 3 = ?, ? + 3 = 5
“Table 1” taken from page 65 of the 3/10/2010 draft of the Common Core State Standards for Mathematics .
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What Research Says about Basic Facts and Fluency
According to Clements and Sarama (2009),
World-wide research shows that the way most people in the U.S. think about
arithmetic combinations and children’s learning of them, and the language they
use may harm more than help . . . For example, we hear about “memorizing
facts” and “recalling your facts.” This is misleading. . . children move through a
long developmental progression to reach the point where they can compose
numbers. Further they also should learn about arithmetic properties, patterns,
and relationships as they do so, and that knowledge, along with intuitive
magnitude and other knowledge and skills, ideally is learned simultaneously and
in an integrated fashion with knowledge of arithmetic combinations. (pp. 82-83)
Research suggests that producing basic combinations is not just a simple “lookup” process. Retrieval is an important part of the process, but many brain
systems help. . . So, when children really know 8 – 3 = 5, they also know that 3 +
5 = 8, 8 – 5 = 3, and so forth, and all these “facts” are related. (p. 83)
So what does this mean? The authors highlight several implications of the research (p. 83):
1. Children need considerable practice, distributed across time.
2. Because counting strategies did not activate the same brain systems, we need to guide
children to move to more sophisticated composition strategies.
3. Practice should not be “meaningless drill” but should occur in a context of making sense
of the situation and the number relationships.
4. Children who are strong in calculations know and use multiple strategies, so teachers
should not teach “one correct procedure.”
Many rote approaches to basic fact recall, such as timed tests, flash cards, or more emphasis
on easier arithmetic problems than harder problems, were found to have no positive – and
sometimes negative – effect. Instead, teachers should use strategies that emphasis reasoning
strategies along with flexible use of number combinations. By developing the understanding
and skills described below, students can become fluent, flexible users of addition and
subtraction.
1. Understand commutativity: 3 + 2 has the same value as 2 + 3.
2. Understand associativity: 5 + 3 + 2 can be seen as 8 + 2 or 5 + 5.
3. Find all the pairs of numbers “hiding inside” other numbers (decompose numbers). For
example, 6 = 0 + 6; 6 = 1+ 5; 6 = 2 + 4; 6 = 3 + 3; 6 = 4 + 2; 6 = 5 + 1; 6 = 6 + 0.
4. Use special patterns such as “doubles” (4 + 4) and “doubles-plus-one” (7 + 8).
5. Understand that adding one to a number is the next number (the n + 1 rule). This should
be understood before “doubles-plus-one.”
6. Know that adding 0 does not change a number (the n + 0 rule).
7. Use the Break-Apart-to-Make-Tens (BAMT) strategy. Before developing this strategy,
children
a. Have a solid knowledge of numerals and counting
b. Understand the structure of the teens numbers (10 + another number)
c. Can solve addition and subtraction of numbers with totals less than 10
d. Can find “break-apart” partners of numbers less than or equal to 10
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e. Use the 10s structure to solve addition and subtraction problems involving teen
numbers (10 + 2 = 12; 18 – 8 = 10)
f. Add and subtract with three addends using 10s (4 + 6 + 3 = 10 + 3 = 13; 15 – 5 –
9 = 10 – 9 = 1)
When students have reached this point in the learning progression, they are ready to
develop understanding of the BAMT strategy. Essentially, students break apart one
number in order to make a 10 and some ones. For example, 9 + 4 becomes 9 + 1 + 3 =
10 + 3 = 13. For more information on the four phases to develop this strategy, see
Clements and Sarama (2009), pp. 86-87.
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Grade 1 Focal Point #1: Developing understandings of addition and subtraction and strategies
for basic addition facts and related subtraction facts
RELATED GLCES
Number and Operations
Count, write, and order numbers
N.ME.01.01
Count to 110 by 1’s, 2’s, 5’s, and 10’s, starting from any number in the
sequence; count to 500 by 100’s and 10’s; use ordinals to identify position
in a sequence, e.g., 1st, 2nd, 3rd.
N.ME.01.02
Read and write numbers to 110 and relate them to the quantities they
represent.
N.ME.01.03
Order numbers to 110; compare using phrases such as “same as”, “more
than”, “greater than”, “fewer than”, use = symbol. Arrange small sets of
numbers in increasing or decreasing order, e.g., write the following from
smallest to largest: 21, 16, 35, 8.
N.ME.01.04
Identify one more than, one less than, 10 more than, and 10 less than for
any number up to 100.
N.ME.01.05
Understand that a number to the right of another number on the number
line is bigger and that a number to the left is smaller.
N.ME.01.06
Count backward by 1’s starting from any number between 1 and 100.
Add and subtract whole numbers
N.ME.01.08
List number facts (partners inside of numbers) for 2 through 10, e.g., 8 =
7 + 1 = 6 + 2 = 5 + 3 = 4 + 4; 10 = 8 + 2 = 2 + 8.
N.MR.01.09
Compare two or more sets in terms of the difference in number of
elements.
N.MR.01.10
Model addition and subtraction for numbers through 30 for a given
contextual situation using objects or pictures; explain in words; record
using numbers and symbols; solve.
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N.MR.01.11
Understand the inverse relationship between addition and subtraction,
e.g., subtraction “undoes” addition: if 3 + 5 = 8, we know that 8 – 3 = 5
and 8 – 5 = 3; recognize that some problems involving combining, “taking
away”, or comparing can be solved by either operation.
N.FL.01.12
Know all the addition facts up to 10 + 10, and solve the related
subtraction problems fluently.
N.MR.01.13
Apply knowledge of fact families to solve simple open sentences for
addition and subtraction, such as:  + 2 = 7 and 10 -  = 6.
N.FL.01.014 Add three one-digit numbers.
N.FL.01.15
Calculate mentally sums and differences involving: a two-digit number
and a one-digit number without regrouping; a two-digit number and a
multiple of 10.
N.FL.01.16
Compute sums and differences through 30 using number facts and
strategies, but no formal algorithm.
Measurement
Solve problems
M.PS.01.08
Solve one-step word problems using addition and subtraction of length,
money and time, including “how much more/less”, without mixing units.
Work with money
M.UN.01.04
Identify the different denominations of coins and bills.
M.UN.01.05
Match one coin or bill of one denomination to an equivalent set of
coins/bills of other denominations, e.g., 1 quarter = 2 dimes and 1 nickel.
M.UN.01.06
Tell the amount of money: in cents up to $1, in dollars up to $100. Use
the symbols $ and ¢.
M.PS.01.07
Add and subtract money in dollars only or in cents only.
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FROM THE 3/10/2010 DRAFT OF THE COMMON CORE STANDARDS
Number—Operations and the Problems They Solve 1-NOP
Addition and subtraction
1. Understand the properties of addition.
a. Addition is commutative. For example, if 3 cups are added to a stack of 8 cups,
then the total number of cups is the same as when 8 cups are added to a stack
of 3 cups; that is, 8 + 3 = 3 + 8.
b. Addition is associative. For example, 4 + 3 + 2 can be found by first adding 4 + 3
= 7 then adding 7 + 2 = 9, or by first adding 3 + 2 = 5 then adding 4 + 5 = 9.
c. 0 is the additive identity.
2. Explain and justify properties of addition and subtraction, e.g., by using representations
such as objects, drawings, and story contexts. Explain what happens when:
a. The order of addends in a sum is changed in a sum with two addends.
b. 0 is added to a number.
c. A number is subtracted from itself.
d. One addend in a sum is increased by 1 and the other addend is decreased by 1.
Limit to two addends.
3. Understand that addition and subtraction have an inverse relationship. For example, if 8
+ 2 = 10 is known, then 10 – 2 = 8 and 10 – 8 = 2 are also known.
4. Understand that when all but one of three numbers in an addition or subtraction equation
are known, the unknown number can be found. Limit to cases where the unknown
number is a whole number.
5. Understand that addition can be recorded by an expression (e.g., 6 + 3), or by an
equation that shows the sum (e.g., 6 + 3 = 9). Likewise, subtraction can be recorded by
an expression (e.g., 9 – 5), or by an equation that shows the difference (e.g., 9 – 5 = 4).
Describing situations and solving problems with addition and subtraction
6. Understand that addition and subtraction apply to situations of adding-to, taking-from,
putting together, taking apart, and comparing. (Table- page14).
7. Solve word problems involving addition and subtraction within 20, e.g., by using objects,
drawings and equations to represent the problem. Students should work with all of the
addition and subtraction situations shown in the Table on page 14, solving problems with
unknowns in all positions, and representing these situations with equations that use a
symbol for the unknown (e.g., a question mark or a small square). Grade 1 students
need not master the more difficult problem types.
8. Solve word problems involving addition of three whole numbers whose sum is less than
or equal to 20.
Number—Base Ten 1-NBT
Adding and subtracting in base ten
5. Calculate mentally, additions and subtractions within 20.
a. Use strategies that include counting on; making ten (for example, 7 + 6 = 7 + 3 +
3 = 10 + 3 = 13); and decomposing a number (for example, 17 – 9 = 17 – 7 – 2 =
10 – 2 = 8).
6. Demonstrate fluency in addition and subtraction within 10.
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7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like
units (tens and tens, ones and ones) and sometimes it is necessary to compose or
decompose a higher value unit.
8. Given a two-digit number, mentally find 10 more or 10 less than the number, without
having to count.
9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and
two-digit numbers.
10. Explain addition of two-digit numbers using concrete models or drawings to show
composition of a ten or a hundred.
11. Add two-digit numbers to two-digit numbers using strategies based on place value,
properties of operations, and/or the inverse relationship between addition and
subtraction; explain the reasoning used.
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Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including
grouping in tens and ones
Kindergarten
Grade 1
Grade 2
Representing, comparing,
and ordering whole
numbers and joining and
separating sets
(NCTM -K)
Developing an
understanding of whole
number relationships,
including grouping in tens
and ones (NCTM-1st)
Developing an
understanding of the baseten numeration system and
place-value concepts
(NCTM-2nd)
Count, write, and order
numbers
Count, write, and order
numbers
Count, write, and order whole
numbers
Compose and decompose
numbers
Explore place value
Understand place value
Work with unit fractions
Add and subtract numbers
Explore number patterns
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross over GLCE topics associated with another focal point
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Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including
grouping in tens and ones
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #1 (Numbers)
The set of real numbers is infinite, and each real number can be associated with a unique
point on the number line.

Counting tells how many items there are altogether. The number of objects in a set is the
last number named when the objects are counted.

Counting a set 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.

Quantities are measured or counted using same-sized units.

0 represents a set with no objects.

0 added to or subtracted from a number does not change the number.

All quantities have a unit. Sometimes the unit is unspoken (i.e., 23 ones).

The counting numbers can be put in a path so that numbers are in order. The position of
two numbers on the number path shows which number is larger.

The smallest counting number is one, but a number path can start at any number. There is
no largest counting number.

If a number path is used for counting on or counting back, the numbers must be counted by
the same unit all the way through.
o 7, 8, 9, 10, 11, 12 is a valid number path because each number is one more than
the number before it
o 2, 4, 6, 8, 10, 12 is a valid number path because the difference between one
number and the next is always 2
o 1, 2, 3, 6, 7, 15 is a list of numbers in order but can not be used for counting
because the difference the difference between one number and the next keeps
changing

The number line is a linear (length) model, not a set (object) model. When using a number
line to add, subtract, or count, count the “jumps” between numbers, not the tick marks.
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
The distance between any two consecutive counting numbers on a given number line is the
same. In other words, the distance between one counting number and the next is a unit of
one.
Big Idea #2 (The Base Ten Numeration System)
The base ten numeration system is a scheme for recording numbers using digits 0-9,
groups of ten, and place value.

The same quantity can be expressed in many equivalent forms using different units:
o 3 red teddy bears is the same number of objects as 3 teddy bears
o 23 ones is equivalent to 2 tens and 3 ones
o 24 tens is equivalent to 240 ones
o 100 pennies has the same value as 1 dollar
o 2 feet has the same length as 24 inches
o 5+2=9–2
o 1/2 = 2/4
o 3x = 2x + x

Any real number can be written as a sum of ones, tens, hundreds, tenths, hundredths, and
other types of ten-groups (powers of 10). You can add the value of the digits together to get
the value of the number.
o 23 = 2 tens + 3 ones = 20 + 3
o 874 = 8 hundreds + 7 tens + 4 ones = 800 + 70 + 4
o 12.3 = 1 ten + 2 ones + 3 tenths = 10 + 2 + 0.3
o -78 = - (7 tens + 8 ones) or -7 tens + -8 ones
Teacher note: 10n, . . . 103, 102, 101, 100, 10-1, 10-2, 10-3, . . . 10-n

Ones can only be added to ones; tens can only be added to tens, and so forth. Place value
makes this easy to see, so that we can add and subtract faster.

Sets of ten, one hundred and so forth must be perceived as single entities when interpreting
numbers using place value (e.g., 1 hundred is one group; it is 10 tens or 100 ones).

When there are no groups of a particular size, 0 is used as a place holder. Since adding 0
does not change the value of a number, the 0 is not used in expanded notation.
o 302 = 300 + 2

In Base 10 notation, each place is ten times the value of the place to the right:
o 5 tens = 50 ones
o 2 hundreds = 20 tens
o 400 tens = 4 thousands
o 3 ones = 30 tenths
o 20 hundredths = 2 tenths
o 4 tenths = 400 thousandths

The value of each digit depends on its position within the number.

Decimal place value is an extension of whole number place value.
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
The base-ten numeration system extends infinitely to very large and very small numbers
(e.g., millions and millionths).
Big Idea #3 (Equivalence)
Any number, measure, numerical expression, algebraic expression, or equation can be
represented in an infinite number of ways that have the same value.

The same quantity can be expressed in many equivalent forms using different units:
o 3 red teddy bears is the same number of objects as 3 teddy bears
o 23 ones is equivalent to 2 tens and 3 ones
o 24 tens is equivalent to 240 ones
o 100 pennies has the same value as 1 dollar
o 2 feet has the same length as 24 inches
o 5+2=9–2
o 1/2 = 2/4
o 3x = 2x + x
Big Idea #4 (Comparison)
Numbers, expressions, and measures can be compared by their relative values.
[ The light gray points are related to the same Big Idea and topic, but are addressed at a later
grade level.]
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Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including
grouping in tens and ones
INSTRUCTIONAL IMPLICATIONS
Preschool children begin to understand the process of making groups with equal
numbers of objects. Such grouping, and the knowledge of the special grouping
into tens, appears not to be related to counting skill. However, experience with
additive composition does appear to contribute to knowledge of grouping and
place value.
Teachers often believe that their students understand place value because they
can, for example, put digits into “tens and one charts.” However, ask these
students what the “1” in “16” means and they are as likely to say “one” (and
mean 1 singleton) as they are to say “one ten.” (Clements and Sarama (2009),
pp. 88-893)
There is more to understanding place value than naming the place value positions or aligning
ones and tens for column addition. Children need concrete experiences to make sense of what
it means to group objects by tens and ones. However, these experiences must be connected
to, not separate from, the language and symbolism of mathematics. In their book Learning and
Teaching Early Math: The Learning Trajectories Approach, Douglas H. Clements and Julie
Sarama (2009) identify several learning experiences that help children to make sense of place
value and grouping:






Solving simple addition problems in early years lays the foundation for understanding
place value
Students should group sets of objects into groups of tens and ones while discussing the
ideas of place value
When grouping objects, make links to number words (twenty-two, that’s 2 tens and 2
ones) and to the written numerals (“22”)
It takes many experiences to fix “ten” in students’ minds as a benchmark. Any activity
where students collect or count objects sorted into tens can form this link:
o Roll two number cubes and take that many pennies. If you have 10 or more
pennies, you must trade 10 pennies for 1 dime before your turn is over. Take
turns, continuing until someone has 100 cents (10 dimes).
o The Michigan Mathematics Program Improvement project (MMPI) describes
similar Trading Games using Base 10 blocks. For more information, visit
www.michiganmathematics.org (Chapter 2, Page 4, Activities 6 and 7).
Use number language that symbolizes composing and decomposing. For example,
when reading “52” say “fifty-two, that’s 5 tens and 2 ones.”
Experiences such as the trading games described above, along with use of number
language that emphasizes place value, can help students to see “ten” as a unit in and of
itself, laying the foundation for counting by tens, and later, adding groups of ten.
Clements, Douglas H. and Sarama, Julie (2009). Learning and Teaching Early Math: The
Learning Trajectories Approach. Routledge: New York, NY.
3
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Even with the strategies above, some children still view the digits in a multidigit number as if
they were singletons. “Secret code cards” can help to make explicit that the value of each digit
depends on its position in the number. The MMPI project has templates for place value number
cards that can be used in this way (www.michiganmathematics.org, Chapter 2, Page 7, Activity
15).
It is common to use manipulatives and pictures in early elementary instruction. However, it is
essential not simply to do “hands-on activities,” but to connect those activities to the
fundamental ideas – and symbolism – of mathematics. Clements and Sarama (2009) describe
one process by which this happens:
High-quality instruction often uses manipulatives or other objects to demonstrate
and record quantities. Further, such manipulatives are used consistently enough
that they becomes tools for thinking. They are discussed to explicate the placevalue ideas. They are used to solve problems, including arithmetic problems.
Finally, they are replaced by symbols (p. 90).
This process of using concrete objects to form strong mental images connected to concepts,
connecting those images to pictures and symbols, and eventually withdrawing the objects, is an
example of the research-based Concrete-Representational-Abstract strategy4.
In addition to connecting manipulatives to concepts and symbolism, it is also important to select
robust models that apply to a variety of mathematical contexts. When done well, mathematical
modeling is one of three “powerful practices” in mathematics and science.5 When selecting
manipulatives with which to model the base 10 place value system, choose the model
appropriate to your student’s needs:
1. Bustable and proportional. Some models, like bundled straws or linked unifix
cubes, are proportional in that 10 ones is the same physical size as 1 ten.
Additionally, the model is “bustable,” meaning that 1 ten can be taken apart
into 10 ones.
2. Tradable and proportional. Base 10 blocks are an example of a tradable and
proportional model. 1 ten is the same size as 10 ones, but a ten cannot be
broken into 10 ones – it must be traded for 10 ones.
3. Tradable and non-proportional. Money, such as pennies and dimes, is an
example of a tradable and non-proportional model. One dime has the same
value as 10 pennies, but the dime appears physically smaller than the 10
pennies.
For examples of a variety of place value manipulatives, along with strategies to use them, refer
to the MMPI project (www.michiganmathematics.org, Chapter 2).
4
“Concrete-Representational-Abstract Instructional Approach.” The Access Center: Improving
Outcomes for All Students K-8. Accessed on 3-22-2010 from
http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp
Carpenter, Thomas P. and Romberg, Thomas A. (2004) Powerful Practices in Mathematics
and Science. Learning Point Associates: Naperville, IL.
5
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Grade 1 Focal Point #2: Developing an understanding of whole number relationships, including
grouping in tens and ones
RELATED GLCES
Number and Operations
Explore place value
N.ME.01.07
Compose and decompose numbers through 30, including using bundles
of tens and units, e.g., recognize 24 as 2 tens and 4 ones, 10 and 10 and
4, 20 and 4, and 24 ones.
Count, write, and order numbers – see Focal Point #1
N.ME.01.01
Count to 110 by 1’s, 2’s, 5’s, and 10’s, starting from any number in the
sequence; count to 500 by 100’s and 10’s; use ordinals to identify position
in a sequence, e.g., 1st, 2nd, 3rd.
N.ME.01.02
Read and write numbers to 110 and relate them to the quantities they
represent.
N.ME.01.03
Order numbers to 110; compare using phrases such as “same as”, “more
than”, “greater than”, “fewer than”; use = symbol. Arrange small sets of
numbers in increasing or decreasing order, e.g., write the following from
smallest to largest: 21, 16, 35, 8.
N.ME.01.04
Identify one more than, one less than, 10 more than, and 10 less than for
any number up to 100.
N.ME.01.05
Understand that a number to the right of another number on the number
line is bigger and that a number to the left is smaller.
N.ME.01.06
Count backward by 1’s starting from any number between 1 and 100.
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FROM THE 3/10/2010 DRAFT OF THE COMMON CORE STANDARDS
Number—Base Ten 1-NBT
Numbers up to 100
1. Read and write numbers to 100.
2. Starting at any number, count to 100 or beyond.
3. Understand that when comparing two-digit numbers, if one number has more tens, it is
greater; if the amount of tens is the same in each number, then the number with more ones
is greater.
4. Compare and order two-digit numbers based on meanings of the tens and ones digits, using
> and < symbols to record the results of comparisons.
Adding and subtracting in base ten
5. Calculate mentally, additions and subtractions within 20.
b. Use strategies that include counting on; making ten (for example, 7 + 6 = 7 + 3 +
3 = 10 + 3 = 13); and decomposing a number (for example, 17 – 9 = 17 – 7 – 2 =
10 – 2 = 8).
6. Demonstrate fluency in addition and subtraction within 10.
7. Understand that in adding or subtracting two-digit numbers, one adds or subtracts like units
(tens and tens, ones and ones) and sometimes it is necessary to compose or decompose a
higher value unit.
8. Given a two-digit number, mentally find 10 more or 10 less than the number, without having
to count.
9. Add one-digit numbers to two-digit numbers, and add multiples of 10 to one-digit and twodigit numbers.
10. Explain addition of two-digit numbers using concrete models or drawings to show
composition of a ten or a hundred.
11. Add two-digit numbers to two-digit numbers using strategies based on place value,
properties of operations, and/or the inverse relationship between addition and subtraction;
explain the reasoning used.
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Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in
measuring lengths
Kindergarten
Grade 1
Measurement
Measurement
Ordering objects by
measurable attributes
(NCTM-K)
Developing an
understanding of linear
measurement and facility in
measuring lengths (NCTM2nd)
Explore other measurement
attributes
Grade 2
Estimate and measure length
Solve problems using addition
and subtraction of length,
money, and time
(M.PS.01.08)
Solve measurement problems
involving length, money, and
perimeter (M.PS.01.10,
M.TE.02.11)
Geometry
Geometry
Describing shapes and
space (NCTM-K)
Composing and
decomposing geometric
shapes (NCTM-1st)
Create, explore, and describe
shapes
Identify and describe shapes
Explore geometric patterns
Work with unit fractions
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross over GLCE topics associated with another focal point
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Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in
measuring lengths
BIG MATHEMATICAL IDEAS AND UNDERSTANDINGS
Big Idea #17 (Measurement)
Some attributes of objects are measurable and can be quantified using unit amounts.

Measurement involves a selected attribute of an object (length, area, mass, volume,
capacity) and a comparison of the object being measured against a unit of the same
attribute.

The larger the unit of measure, the fewer units it takes to measure the object.

A given measurement can be expressed in many equivalent forms of different units of the
same attribute or dimension:
o 2 feet = 24 inches
o 1 cubic yard = 27 cubic feet

The magnitude of the attribute to be measured and the accuracy needed determines the
appropriate measurement unit.

A figure or object can be constructed from or decomposed into figures of the same
dimension. The measurement of a given attribute of the object is equal to the sums of the
measurements of the components of the object for that attribute:
o if a polygon is decomposed into other polygons, the area of the original polygon
is equal to the sum of the areas of the component polygons
o the perimeter of a polygon can be found by adding together the lengths of the
sides
o if an angle is composed from smaller angles, the measure of the total angle is
equal to the sums of the measures of the component angles
o if a box is composed from smaller boxes, the total volume of the box is equal to
the sum of the volumes of the component boxes
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Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in
measuring lengths
INSTRUCTIONAL IMPLICATIONS
This may seem obvious, but if children are going to compare and order objects by measurable
attributes, they must do so with actual objects. The Concrete-Representational-Abstract (CRA)
strategy, which has a strong research base, suggests that drawing pictures is a good way to
connect the concrete to the representational6. Children first compare and order real objects,
drawing pictures of what happened. Eventually, children can compare pictures of objects.
In their book Learning and Teaching Early Math: The Learning Trajectories Approach, Douglas
H. Clements and Julie Sarama (2009) suggest these instructional tasks for five- and six-yearolds exploring length:
What’s the Missing Step? (p.171)
Children see stairs made from connecting cubes from 1 to 6. They cover their
eyes and the teacher hides one step. They uncover their eyes and identify the
missing step, telling how they knew.
X-Ray Vision 1 (p. 171)
Children place Counting Cards, 1 to 6 or more, in order, face down. Then they
take turns pointing to the cards and using their “x-ray vision” to tell which card it
is.
Length Riddles (p.171)
Ask questions such as, “You write with me and I am 7 cubes long. What am I?”
As children develop (around age 7), provide fewer cues (e.g., only the length)
and only one unit per child so they have to iterate (repeatedly “lay down”) a
single unit to measure.
Measure with physical or drawn units (p.171)
Focus on long, thin units such as toothpicks cut to one inch sections. Explicit
emphasis should be given to the linear nature of the unit. That is children should
learn that, when measuring with, say, centimeter cubes, it is the length of one
edge that is the linear unit – not the area of a face or volume of the cube.
It is around age 7 that children relate size and number of units explicitly and can make
statements such as, “If you measure with centimeters instead of inches, you’ll need more of
them, because each one is smaller.”7 The authors also note that children may be able to draw a
line to a given length before they measure objects accurately, and they suggest this instructional
task:
“Concrete-Representational-Abstract Instructional Approach.” The Access Center: Improving
Outcomes for All Students K-8. Accessed on 3-22-2010 from
http://www.k8accesscenter.org/training_resources/CRA_Instructional_Approach.asp
7 Clements, Douglas H. and Sarama, Julie. (2009) Learning and Teaching early Math: The Learning
Trajectories Approach. Routledge: New York, NY. p. 172.
6
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Use line-drawing activities to emphasize how you start at the 0 (zero point) and
discuss how, to measure objects, you have to align the object to that point.
Similarly, explicitly discuss what the intervals and the number represent,
connecting these to end-to-end length measuring with physical units. (p. 172)
Children reach the “length measurer” stage around age 8. At this stage, the child sees the
length of a bent path as the sum of its parts (not the distance between the endpoints). He is
able to measure length. He knows the need for identical units and understands the relationship
between different units. He can use partitions of a unit, use the zero point on rulers,
understands accumulation of distance, and begins to estimate. Here is an example of an
instructional task appropriate for children at this level:
Children create units of units, such as a “footstrip” consisting of traces of their
feet glued to a roll of adding-machine tape. They measure in different-sized units
(e.g., 15 paces or 3 footstrips each of which has 5 paces) and accurately relate
these units. They also discuss how to deal with leftover space, to count it as a
whole unit or as part of a unit. (p. 172).
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Grade 1 Focal Point #3: Developing an understanding of linear measurement and facility in
measuring lengths
RELATED GLCES
Measurement
Estimate and measure length
M.UN.01.01
Measure the lengths of objects in non-standard units, e.g., pencil length,
shoe lengths, to the nearest whole unit.
M.UN.01.02
Compare measured lengths using the words shorter, shortest, longer,
longest, taller, tallest, etc.
FROM THE 3010-2010 DRAFT OF THE COMMON CORE STANDARDS
Measurement and Data 1-MD
Length measurement
1. Order three objects by length; compare the length of two objects indirectly by using a
third object.
2. Understand that the length of an object can be expressed numerically by using another
object as a length unit (such as a paper-clip, yardstick, or inch length on a ruler). The
object to be measured is partitioned into as many equal parts as possible with the same
length as the length unit. The length measurement of the object is the number of length
units that span it with no gaps or overlaps. For example, “I can put four paperclips end to
end along the pencil, so the pencil is four paperclips long.”
3. Measure the length of an object by using another object as a length unit.
Time measurement
4. Tell time from analog clocks in hours and half- or quarter-hours.
Representing and interpreting data
5. Organize, represent, and interpret data with several categories; ask and answer
questions about the total number of data points, how many in each category, and how
many more or less are in one category than in another.
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Grade 1 GLCEs not related to a focal point
Key:
Builds on previous grade(s)
Related to topics within or beyond mathematics
Later grade at which topic relates to a focal point
Kindergarten
Explore concepts of time
Grade 2
Grade 1
Tell time
Grade 2
Grade 2
Understand meaning of
multiplication and division
Grades 3, 4, 5
Create and describe shapes
Grade 2
Understand the concept of
area Grade 3
Create and describe patterns
involving geometric objects
Read thermometers
Use pictographs
Use coordinate systems
Create, interpret, and solve
problems involving
pictographs
Key:
bold, non-italic = Michigan Curriculum Focal Points
non-bold, non-italic = GLCE topics associated with that focal point
non-bold, italic = Cross over GLCE topics associated with another focal point
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Grade 1 GLCEs not related to a focal point
Approximately 70% - 80% of Tier 1 instruction should relate to the grade-level Focal Points identified
previously. No more than 20% - 30% of Tier 1 instruction should be devoted to the following GLCEs,
which are not related to a focal point.
Measurement
Tell time
M.UN.01.03
Tell time on a twelve-hour clock face to the hour and half-hour.
Geometry
Create and describe shapes
G.GS.01.01
Create common two-dimensional and three-dimensional shapes, and
describe their physical and geometric attributes, such as color and shape.
G.LO.01.02
Describe relative position of objects on a plane and in space, using words
such as above, below, behind, in front of.
Create and describe patterns involving geometric objects
G.SR.01.03
Create and describe patterns, such as repeating patterns and growing
patterns using number, shape, and size.
G.SR.01.04
Distinguish between repeating and growing patterns.
G.SR.01.05
Predict the next element in a simple repeating pattern.
G.SR.01.06
Describe ways to get to the next element in simple repeating patterns.
Data and Probability
Use pictographs
D.RE.01.01
Collect and organize data to use in pictographs.
D.RE.01.02
Read and interpret pictographs.
D.RE.01.03
Make pictographs of given data using both horizontal and vertical forms of
graphs; scale should be in units of one and include symbolic
representations, e.g.,  represents one child.
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FROM THE 3/10/2010 DRAFT OF THE COMMON CORE STANDARDS
Geometry 1-G
Shapes, their attributes, and spatial reasoning
1. Distinguish between defining attributes (e.g., triangles are closed and three-sided)
versus non-defining attributes (e.g., color, orientation, overall size) for a wide variety of
shapes.
2. Understand that shapes can be joined together (composed) to form a larger shape or
taken apart (decomposed) into a collection of smaller shapes. Composing multiple
copies of some shapes creates tilings. In this grade, “circles,” “rectangles,” and other
shapes include their interiors as well as their boundaries.
3. Compose two-dimensional shapes to create a unit, using cutouts of rectangles, squares,
triangles, half-circles, and quarter-circles. Form new shapes by repeating the unit.
4. Compose three-dimensional shapes to create a unit, using concrete models of cubes,
right rectangular prisms, right circular cones, and right circular cylinders. Form new
shapes by repeating the unit. Students do not need to learn formal names such as “right
rectangular prism.”
5. Decompose circles and rectangles into two and four equal parts. Describe the parts
using the words halves, fourths, and quarters, and using the phrases half of, fourth of,
and quarter of. Describe the whole as two of, or four of the parts. Understand that
decomposing into more equal shares creates smaller shares.
6. Decompose two-dimensional shapes into rectangles, squares, triangles, half-circles, and
quarter-circles, including decompositions into equal shares.
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Suggested First Grade Vocabulary
Taken from Huron County Mathematics Curriculum Framework
January 3, 2006
Number and Operations
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add
addend
addition
altogether
backward
base ten block
between
combination
compare*
counting back
counting on
counting up
difference
digit
doubles
doubles plus one
eight
equal (=)
equation
estimate
even*
fact family
facts
fair share
fifth
first
five
forward*
four
fourth
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fraction
greater than (>)
half
hundreds
left
less
less than (<)
mental math
minus
more
most
nine
number line
number sentences
numbers
odd*
one
one less
one more
one-fourth
one-half
ones
one-third
operation
order
ordinal numbers*
pattern
place
place value (hundreds,
ten, ones)
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plus
quantity
record
same
same as
second
sequence*
sets
seven
six
sixth
skip count
strategy
subtract
subtraction
sum
taking away
ten
tens
third
three
turn-arounds
two
unit
whole
zero
* Instructional term on which student might not be assessed
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Measurement
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add
addition
afternoon
analog clock*
between
calendar
capacity
Celsius
cent sign (¢)
centimeter
cents
circumference
clock
coin(s)
colder
cooler
cup
date
degree
difference
digital clock*
dime
distance
dollar bills
dollar sign ($)
equals
estimate
evening
gallon
gram
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half-hour
height
hotter
hour
hour hand
inch
increase
last month
last week
last year
length
liter
longer
longest
measure
measurement
meter
metric
minute
minute hand
money
month
morning
next month
next week
next year
nickel
night
non-standard unit*
penny
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pint
pound
quart
quarter
ruler
scale
second hand
shorter
shortest
solution
standard unit
subtraction
sum
taller
tallest
temperature
thermometer*
time
today
tomorrow
unit
volume
warmer
week
weight
width
year
yesterday
* Instructional term on which student might not be assessed
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Geometry
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above/below
angle
balance
box
category
change
circle
closed
color
cone
construct
corner
cube
curve
cylinder
decrease*
dot
edge
enlarge*
face
flip
folding
front/back
geoboard
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group
growing
hexagon
inside/outside
large
left
line
next to
objects
on
open
out
oval
overlap
parallel
part
pattern
perpendicular
rectangle
reduce*
repeating
right
same shape
same size
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segment
shape
shrinking*
side
size
slide
small
solid
sort
sphere
square
symmetry
tangrams
texture
thick/thin
tile
top
trapezoid
triangle
turn
under/over
up
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record
represent
results
same
sort
survey
table
tally
tally marks
traits
Data and Probability
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bar graph
chance
chart
collect
compare
estimate
experiment
fewer
graph
growth
hypothesis /guess
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interpret
least
less
line graph
more
most
organize
pictograph
pictograph graph
picture graph
predict
* Instructional term on which student might not be assessed
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