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In this pdf students further, build a mathematical foundation that is critical to learning higher mathematics.
1.
Grade-Two Chapter
of the
Mathematics Framework
for California Public Schools:
Kindergarten Through Grade Twelve
Adopted by the California State Board of Education, November 2013
Published by the California Department of Education
Sacramento, 2015
2.
Grade Two
8
7
I
n grade two, students further build a mathematical
foundation that is critical to learning higher math-
6 ematics. In previous grades, students developed
a foundation for understanding place value, including
grouping in tens and ones. They built understanding of
5 whole numbers to 120 and developed strategies to add,
subtract, and compare numbers. They solved addition
and subtraction word problems within 20 and developed
fluency with these operations within 10. Students also
4 worked with non-standard measurement and reasoned
about attributes of geometric shapes (adapted from
Charles A. Dana Center 2012).
3
Critical Areas of Instruction
In grade two, instructional time should focus on four crit-
2 ical areas: (1) extending understanding of base-ten nota-
tion; (2) building fluency with addition and subtraction;
(3) using standard units of measure; and (4) describing
and analyzing shapes (National Governors Association
1 Center for Best Practices, Council of Chief State School
Officers [NGA/CCSSO] 2010i). Students also work
toward fluency with addition and subtraction within 20
K using mental strategies and within 100 using strategies
based on place value, properties of operations, and the
relationship between addition and subtraction. They
know from memory all sums of two one-digit numbers.
California Mathematics Framework Grade Two 119
3.
Standards for Mathematical Content
The Standards for Mathematical Content emphasize key content, skills, and practices at each
grade level and support three major principles:
l Focus—Instruction is focused on grade-level standards.
l Coherence—Instruction should be attentive to learning across grades and to linking major
topics within grades.
l Rigor—Instruction should develop conceptual understanding, procedural skill and fluency,
and application.
Grade-level examples of focus, coherence, and rigor are indicated throughout the chapter.
The standards do not give equal emphasis to all content for a particular grade level. Cluster
headings can be viewed as the most effective way to communicate the focus and coherence
of the standards. Some clusters of standards require a greater instructional emphasis than
others based on the depth of the ideas, the time needed to master those clusters, and their
importance to future mathematics or the later demands of preparing for college and careers.
Table 2-1 highlights the content emphases at the cluster level for the grade-two standards.
The bulk of instructional time should be given to “Major” clusters and the standards within
them, which are indicated throughout the text by a triangle symbol ( ). However, stan-
dards in the “Additional/Supporting” clusters should not be neglected; to do so would result
in gaps in students’ learning, including skills and understandings they may need in later
grades. Instruction should reinforce topics in major clusters by using topics in the addition-
al/supporting clusters and including problems and activities that support natural connec-
tions between clusters.
Teachers and administrators alike should note that the standards are not topics to be
checked off after being covered in isolated units of instruction; rather, they provide content
to be developed throughout the school year through rich instructional experiences present-
ed in a coherent manner (adapted from Partnership for Assessment of Readiness for College
and Careers [PARCC] 2012).
120 Grade Two California Mathematics Framework
4.
Table 2-1. Grade Two Cluster-Level Emphases
Operations and Algebraic Thinking 2.OA
Major Clusters
• Represent and solve problems involving addition and subtraction. (2.OA.1 )
• Add and subtract within 20. (2.OA.2 )
Additional/Supporting Clusters
• Work with equal groups of objects to gain foundations for multiplication. (2.OA.3–4)
Number and Operations in Base Ten 2.NBT
Major Clusters
• Understand place value. (2.NBT.1–4 )
• Use place-value understanding and properties of operations to add and subtract.
(2.NBT.5–9 )
Measurement and Data 2.MD
Major Clusters
• Measure and estimate lengths in standard units. (2.MD.1–4 )
• Relate addition and subtraction to length. (2.MD.5–6 )
Additional/Supporting Clusters
• Work with time and money. (2.MD.7–8)
• Represent and interpret data. (2.MD.9–10)
Geometry 2.G
Additional/Supporting Clusters
• Reason with shapes and their attributes. (2.G.1–3)
Explanations of Major and Additional/Supporting Cluster-Level Emphases
Major Clusters ( ) — Areas of intensive focus where students need fluent understanding and application of the core
concepts. These clusters require greater emphasis than others based on the depth of the ideas, the time needed to
master them, and their importance to future mathematics or the demands of college and career readiness.
Additional Clusters — Expose students to other subjects; may not connect tightly or explicitly to the major work of the
Supporting Clusters — Designed to support and strengthen areas of major emphasis.
Note of caution: Neglecting material, whether it is found in the major or additional/supporting clusters, will leave gaps
in students’ skills and understanding and will leave students unprepared for the challenges they face in later grades.
Adapted from Achieve the Core 2012.
California Mathematics Framework Grade Two 121
5.
Connecting Mathematical Practices and Content
The Standards for Mathematical Practice (MP) are developed throughout each grade and, together with
the content standards, prescribe that students experience mathematics as a rigorous, coherent, useful,
and logical subject. The MP standards represent a picture of what it looks like for students to under-
stand and do mathematics in the classroom and should be integrated into every mathematics lesson
for all students.
Although the description of the MP standards remains the same at all grade levels, the way these
standards look as students engage with and master new and more advanced mathematical ideas does
change. Table 2-2 presents examples of how the MP standards may be integrated into tasks appropriate
for students in grade two. (Refer to the Overview of the Standards Chapters for a description of the MP
Table 2-2. Standards for Mathematical Practice—Explanation and Examples for Grade Two
Standards for
Mathematical Explanation and Examples
Practice
MP.1 In grade two, students realize that doing mathematics involves reasoning about and solving
problems. Students explain to themselves the meaning of a problem and look for ways to
Make sense of solve it. They may use concrete objects or pictures to help them conceptualize and solve
problems and problems. They may check their thinking by asking themselves, “Does this make sense?”
persevere in They make conjectures about the solution and plan out a problem-solving approach.
solving them.
MP.2 Younger students recognize that a number represents a specific quantity. They connect the
quantity to written symbols. Quantitative reasoning entails creating a representation of a
Reason problem while attending to the meanings of the quantities.
abstractly and
quantitatively. Students represent situations by decontextualizing tasks into numbers and symbols. For
example, a task may be presented as follows: “There are 25 children in the cafeteria, and
they are joined by 17 more children. How many students are in the cafeteria?” Students
translate the situation into an equation (such as 25 + 17 = — ) and then solve the problem.
Students also contextualize situations during the problem-solving process. To reinforce stu-
dents’ reasoning and understanding, teachers might ask, “How do you know?” or “What is
the relationship of the quantities?”
MP.3 Grade-two students may construct arguments using concrete referents, such as objects,
pictures, math drawings, and actions. They practice their mathematical communication
Construct via- skills as they participate in mathematical discussions involving questions such as “How did
ble arguments you get that?”, “Explain your thinking,” and “Why is that true?” They not only explain their
and critique own thinking, but also listen to others’ explanations. They decide if the explanations make
the reasoning sense and ask appropriate questions.
of others.
Students critique the strategies and reasoning of their classmates. For example, to solve
74 – 18, students might use a variety of strategies and discuss and critique each other’s
reasoning and strategies.
MP.4 In early grades, students experiment with representing problem situations in multiple ways,
including writing numbers, using words (mathematical language), drawing pictures, using
Model with objects, acting out, making a chart or list, or creating equations. Students need opportuni-
mathematics. ties to connect the different representations and explain the connections.
122 Grade Two California Mathematics Framework
6.
Table 2-2 (continued)
Standards for
Mathematical Explanation and Examples
Practice
Students model real-life mathematical situations with an equation and check to make sure
that their equation accurately matches the problem context. They use concrete manipula-
tives or math drawings (or both) to explain the equation. They create an appropriate
problem situation from an equation. For example, students create a story problem for the
equation 43 +£ = 82, such as “There were 43 mini-balls in the machine. Tom poured in
some more mini-balls. There are 82 mini-balls in the machine now. How many balls did
Tom pour in?” Students should be encouraged to answer questions, such as “What math
drawing or diagram could you make and label to represent the problem?” or “What are
some ways to represent the quantities?”
MP.5 In second grade, students consider the available tools (including estimation) when solving
a mathematical problem and decide when certain tools might be better suited than others.
Use appro- For instance, grade-two students may decide to solve a problem by making a math drawing
priate tools rather than writing an equation.
strategically.
Students may use tools such as snap cubes, place-value (base-ten) blocks, hundreds number
boards, number lines, rulers, virtual manipulatives, diagrams, and concrete geometric shapes
(e.g., pattern blocks, three-dimensional solids). Students understand which tools are the
most appropriate to use. For example, while measuring the length of the hallway, students
are able to explain why a yardstick is more appropriate to use than a ruler. Students should
be encouraged to answer questions such as, “Why was it helpful to use ?”
MP.6 As children begin to develop their mathematical communication skills, they try to use clear
and precise language in their discussions with others and when they explain their own
Attend to reasoning.
precision.
Students communicate clearly, using grade-level-appropriate vocabulary accurately and
precise explanations and reasoning to explain their process and solutions. For example,
when measuring an object, students carefully line up the tool correctly to get an accurate
measurement. During tasks involving number sense, students consider if their answers are
reasonable and check their work to ensure the accuracy of solutions.
MP.7 Grade-two students look for patterns and structures in the number system. For example,
students notice number patterns within the tens place as they connect counting by tens to
Look for and corresponding numbers on a hundreds chart. Students see structure in the base-ten number
make use of system as they understand that 10 ones equal a ten, and 10 tens equal a hundred. Teachers
structure. might ask, “What do you notice when ?” or “How do you know if something is a
pattern?”
Students adopt mental math strategies based on patterns (making ten, fact families,
doubles). They use structure to understand subtraction as an unknown addend problem
(e.g., 50 – 33 = — can be written as 33 + — = 50 and can be thought of as “How much
more do I need to add to 33 to get to 50?”).
MP.8 Second-grade students notice repetitive actions in counting and computation (e.g., number
patterns to count by tens or hundreds). Students continually check for the reasonableness of
Look for their solutions during and after completion of a task by asking themselves, “Does this make
and express sense?” Students should be encouraged to answer questions—such as “What is happening in
regularity in this situation?” or “What predictions or generalizations can this pattern support?”
repeated
reasoning.
Adapted from Arizona Department of Education (ADE) 2010 and North Carolina Department of Public Instruction (NCDPI) 2013b.
California Mathematics Framework Grade Two 123
7.
Standards-Based Learning at Grade Two
The following narrative is organized by the domains in the Standards for Mathematical Content and
highlights some necessary foundational skills from previous grade levels. It also provides exemplars to
explain the content standards, highlight connections to Standards for Mathematical Practice (MP), and
demonstrate the importance of developing conceptual understanding, procedural skill and fluency, and
application. A triangle symbol ( ) indicates standards in the major clusters (see table 2-1).
Domain: Operations and Algebraic Thinking
In grade one, students solved addition and subtraction word problems within 20 and developed fluency
with these operations within 10. A critical area of instruction in grade two is building fluency with ad-
dition and subtraction. Second-grade students fluently add and subtract within 20 and solve addition
and subtraction word problems involving unknown quantities in all positions within 100. Grade-two
students also work with equal groups of objects to gain the foundations for multiplication.
Operations and Algebraic Thinking 2.OA
Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations
of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions,
e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1
In grade two, students add and subtract numbers within 100 in the context of one- and two-step word
problems (2.OA.1 ). By second grade, students have worked with various problem situations (add to,
take from, put together, take apart, and compare) with unknowns in all positions (result unknown,
change unknown, and start unknown). Grade-two students extend their work with addition and sub-
traction word problems in two significant ways:
l They represent and solve problems of all types involving addition and subtraction within 100,
building upon their previous work within 20.
l They represent and solve two-step word problems of all types, extending their work with one-step
word problems (adapted from ADE 2010; NCDPI 2013b; Georgia Department of Education [GaDOE]
2011; and Kansas Association of Teachers of Mathematics [KATM] 2012, 2nd Grade Flipbook).
Different types of addition and subtraction problems are presented in table 2-3.
124 Grade Two California Mathematics Framework
8.
Table 2-3. Types of Addition and Subtraction Problems (Grade Two)
Result Unknown Change Unknown Start Unknown
There are 22 marbles in a Bill had 25 baseball cards. His Some children were playing on
bag. Thomas placed 23 more mom gave him some more. the playground, and 5 more
marbles in the bag. How many Now he has 73 baseball cards. children joined them. Then
marbles are in the bag now? How many baseball cards did there were 22 children. How
22 + 23 = £ his mom give him? many children were playing
In this problem, the starting before?
quantity is provided (25 base- This problem can be repre-
ball cards), a second quantity sented by £ + 5 = 22. The
is added to that amount (some “start unknown” problems are
baseball cards), and the result difficult for students to model
quantity is given (73 baseball because the initial quantity is
cards). This question type is unknown, and therefore some
Add to more algebraic and challeng- students do not know how to
ing than a “result unknown” start a solution strategy. They
problem and can be modeled can make a drawing, where it
by a situational equation is crucial that they realize that
(25 +£= 73) that does not the 5 is part of the 22 total
immediately lead to the children. This leads to more
answer. Students can write a general solutions by subtract-
related equation ing the known addend or
(73 – 25 = £) —called a solu- counting/adding on from the
tion equation—to solve the known addend to the total.
problem.
There were 45 apples on the Andrea had 51 stickers. She Some children were lining
table. I took 12 of those apples gave away some stickers. Now up for lunch. After 4 children
and placed them in the refrig- she has 22 stickers. How many left, there were 26 children
erator. How many apples are stickers did she give away? still waiting in line. How many
on the table now? This question may be mod- children were there before?
45 – 12 = £ eled by a situational equation This problem can be modeled
(51 – £ = 22) or a solution by £ – 4 = 26. Similar to the
equation (51 – 22 = £ ). Both previous “add to (with start
Take the “take from” and “add to” unknown)” problem, “take
from questions involve actions. from (with start unknown)”
problems require a high level
of conceptual understanding.
Students need to understand
that the total is first in a sub-
traction equation and that this
total is broken apart into the 4
and the 26.
California Mathematics Framework Grade Two 125
9.
Table 2-3 (continued)
Both Addends
Total Unknown Addend Unknown
Unknown
There are 30 red apples and Roger puts 24 apples in a Grandma has 5 flowers. How
20 green apples on the table. fruit basket. Nine (9) are red many can she put in her red
How many apples are on the and the rest are green. How vase and how many in her
table? many are green?” blue vase?
30 + 20 = ? There is no direct or implied 5 = 0 + 5, 5 = 5 + 0
action. The problem involves 5 = 1 + 4, 5 = 4 + 1
Put a set and its subsets. It may
together/ be modeled by 24 – 9 = £ 5 = 2 + 3, 5 = 3 + 2
Take apart or 9 + £ = 24. This type of
problem provides students
with opportunities to
understand subtraction
as an unknown-addend
problem.
Difference Unknown Bigger Unknown Smaller Unknown
Pat has 19 peaches. Lynda (“More” version): Theo has (“More” version): David
has 14 peaches. How many 23 action figures. Rosa has has 27 more bunnies than
more peaches does Pat have 2 more action figures than Keisha. David has 28 bun-
than Lynda? Theo. How many action nies. How many bunnies
“Compare” problems involve figures does Rosa have? does Keisha have?
relationships between quan- This problem can be mod- This problem can be mod-
tities. Although most adults eled by 23 + 2 = £. eled by 28 – 27 = £. The
might use subtraction to misleading language form
(“Fewer” version): Lucy has
solve this type of problem “more” may lead students to
28 apples. She has 2 fewer
(19 – 14 = £), students will choose the wrong operation.
apples than Marcus. How
Compare often solve this problem as
many apples does Marcus (“Fewer” version): Bill has
an unknown-addend prob-
have? 24 stamps. Lisa has 2 fewer
lem (14 + £ = 19) by using
stamps than Bill. How many
a counting-up or matching This problem can be mod-
stamps does Lisa have?
strategy. In all mathemat- eled as 28 + 2 =£. The
ical problem solving, what misleading language form This problem can be
matters is the explanation “fewer” may lead students to modeled as 24 – 2 = £.
a student gives to relate a choose the wrong operation.
representation to a con-
text—not the representation
separated from its context.
Note: Further examples are provided in table GL-4 of the glossary.
126 Grade Two California Mathematics Framework
10.
For these more complex grade-two problems, it is important for students to represent the problem
situations with drawings and equations (2.OA.1 ). Drawings can be shown more easily to the whole
class during explanations and can be related to equations. Students can also use manipulatives (e.g.,
snap cubes, place-value blocks), but drawing quantities is an exercise that can be used anywhere to
solve problems and support students in describing
their strategies. Second-grade students represent Figure 2-1. Comparison Bars
problems with equations and use boxes, blanks, or
Josh has 10 markers, and Ani has 4 markers. How
pictures for the unknown amount. For example, many more markers does Josh have than Ani?
students can represent “compare” problems using
comparison bars (see figure 2-1). Students can draw
these bars, fill in numbers from the problem, and
label the bars.
One-step word problems use one operation.
Two-step word problems (2.OA.1 ) are new for
second-graders and require students to complete
two operations, which may include the same
operation or different operations.
Initially, two-step problems should not involve the most difficult subtypes of problems (e.g., “compare”
and “start unknown” problems) and should be limited to single-digit addends. There are many
problem-situation subtypes and various ways to combine such subtypes to devise two-step problems.
Introducing easier problems first will provide support for second-grade students who are still develop-
ing proficiency with “compare” and “start unknown” problems (adapted from the University of Arizona
[UA] Progressions Documents for the Common Core Math Standards 2011a).
The following table presents examples of easy and moderately difficult two-step word problems that
would be appropriate for grade-two students.
One-Step Word Problem Two-Step Word Problem Two-Step Word Problem
One Operation Two Operations, Same Two Operations, Opposite
There are 15 stickers on the page. There are 9 blue marbles and 6 red There are 39 peas on the plate.
Brittany put some more stickers marbles in the bag. Maria put in 8 Carlos ate 25 peas. Mother put 7
on the page and now there are 22. more marbles. How many marbles more peas on the plate. How many
How many stickers did Brittany put are in the bag now? peas are on the plate now?
on the page?
9 + 6 + 8 = £ or 39 – 25 + 7 = £ or
15 + £ = 22 or (9 + 6) + 8 = £ (39 – 25) + 7 = £
22 – 15 = £
Adapted from NCDPI 2013b.
Grade-two students use a range of methods, often mastering more complex strategies such as making
tens and doubles and near doubles that were introduced in grade one for problems involving single-
digit addition and subtraction. Second-grade students also begin to apply their understanding of place
value to solve problems, as shown in the following example.
California Mathematics Framework Grade Two 127
11.
One-Step Problem: Some students are in the cafeteria. Twenty-four (24) more students came in. Now there
are 60 students in the cafeteria. How many students were in the cafeteria to start with? Use drawings and
equations to show your thinking.
Student A: I read the problem and thought about how to write
it with numbers. I thought, “What and 24 makes 60?” I used
a math drawing to solve it. I started with 24. Then I added tens
until I got close to 60; I added 3 tens. I stopped at 54. Then
I added 6 more ones to get to 60. So, 10 + 10 + 10 + 6 = 36.
So, there were 36 students in the cafeteria to start with.
My equation for the problem is £ + 24 = 60. (MP.2, MP.7, MP.8)
Student B: I read the problem and thought
about how to write it with numbers. I thought,
“There are 60 total. I know about the 24. So,
what is 60 – 24?” I used place-value blocks
to solve it. I started with 60 and took 2 tens
away. I needed to take 4 more away. So,
I broke up a ten into 10 ones. Then I took
4 away. That left me with 36. So, 36 students
were in the cafeteria at the beginning.
60 – 24 = 36. My equation for the problem
is 60 – 24 = £. (MP.2, MP.4, MP.5, MP.6)
Adapted from ADE 2010, NCDPI 2013b, GaDOE 2011, and KATM 2012 (2nd Grade Flipbook).
As students solve addition and subtraction word problems, they use concrete manipulatives, pictorial
representations, and mental mathematics to make sense of a problem (MP.1); they reason abstractly
and quantitatively as they translate word problem situations into equations (MP.2); and they model
with mathematics (MP.4).
Table 2-4 presents a sample classroom activity that connects the Standards for Mathematical Content
and Standards for Mathematical Practice.
128 Grade Two California Mathematics Framework
12.
Table 2-4. Connecting to the Standards for Mathematical Practice—Grade Two
Standards Addressed Explanation and Examples
Connections to Standards for Task: Base-Ten Block Activities. This is a two-tiered approach to
Mathematical Practice problem solving with basic operations within 100. The first task
involves students seeing various strategies for adding two-digit
MP.1. Students are challenged to think
numbers using base-ten blocks. The second is an extension that
through how they would solve a poten-
builds facility in adding and subtracting such numbers.
tially unfamiliar problem situation and
to devise a strategy. The teacher can 1. The teacher should present several problem situations that
assess each student’s starting point and involve addition and subtraction in which students can use
move him or her forward from there. base-ten blocks to model their solution strategies. Such solu-
tions are made public through an overhead display or by the
MP.3. When students are asked to teacher rephrasing and demonstrating student solutions. Four
explain to their peers how they solved sample problems are provided below:
the problems, they are essentially con-
structing a mathematical argument that l Micah had 24 marbles. Sheila had 15. Micah and Sheila de-
justifies that they have performed the cided to put all of their marbles in a box. How many marbles
addition or subtraction correctly. were there altogether? (This is an addition problem that
does not require bundling ones into a ten.)
MP.7. When students begin exchanging l There were 28 boys and 35 girls on the playground at recess.
sticks and units to represent grouping How many children were there on the playground at recess?
and breaking apart tens and ones, they (This is an addition problem that requires bundling.)
are making use of the structure of the
base-ten number system to understand l There were 48 cows on a pasture. Seventeen (17) of the cows
addition and subtraction. went into the barn. How many cows are left on the pasture?
(This is a subtraction problem that does not require ex-
Standards for Mathematical Content changing a ten for ones.)
l There were 54 erasers in a basket. Twenty-six (26) students
2.OA.1. Use addition and subtraction
were allowed to take one eraser each. How many erasers
within 100 to solve one- and two-step
are left over after the children have taken theirs? (This is a
word problems involving situations of
subtraction problem involving the exchange of a ten for 10
adding to, taking from, putting togeth-
ones.)
er, taking apart, and comparing, with
unknowns in all positions, e.g. by using 2. Next, the teacher can play a game that reinforces understand-
drawings and equations with a symbol ing of addition, subtraction, and skill in doing addition and
for the unknown number to represent subtraction. Each student takes out base-ten blocks to rep-
the problem. resent a given number—for example, 45. The teacher then
asks students how many more blocks are needed to make 80.
2.NBT.5. Fluently add and subtract Students represent the difference with base-ten blocks and
within 100 using strategies based on justify how they know their answers are correct. The teacher
place value, properties of operations, can ask several variations of this same basic question; the task
and/or the relationship between addi- can be used repeatedly throughout the school year to reinforce
tion and subtraction. concepts of operations.
Classroom Connections. When students are given the opportunity
to construct their own strategies for adding and subtracting num-
bers, they reinforce their understanding of place value and the
base-ten number system. Activities such as those presented here
help build this foundation in context and through modeling num-
bers with objects (e.g., with base-ten blocks).
California Mathematics Framework Grade Two 129
13.
To solve word problems, students learn to apply various computational methods. Kindergarten students
generally use Level 1 methods, and students in first and second grade use Level 2 and Level 3 methods.
The three levels are summarized in table 2-5 and explained more thoroughly in appendix C.
Table 2-5. Methods Used for Solving Single-Digit Addition and
Subtraction Problems
Level 1: Direct Modeling by Counting All or Taking Away
Represent the situation or numerical problem with groups of objects, a drawing, or
fingers. Model the situation by composing two addend groups or decomposing a total
group. Count the resulting total or addend.
Level 2: Counting On
Embed an addend within the total (the addend is perceived simultaneously as an ad-
dend and as part of the total). Count this total, but abbreviate the counting by omitting
the count of this addend; instead, begin with the number word of this addend. The
count is tracked and monitored in some way (e.g., with fingers, objects, mental images
of objects, body motions, or other count words).
For addition, the count is stopped when the amount of the remaining addend has been
counted. The last number word is the total. For subtraction, the count is stopped when
the total occurs in the count. The tracking method indicates the difference (seen as the
unknown addend).
Level 3: Converting to an Easier Equivalent Problem
Decompose an addend and compose a part with another addend.
Adapted from UA Progressions Documents 2011a.
In grade two, students extend their fluency with addition and subtraction from within 10 to within 20
(2.OA.2 ). The experiences students have had with addition and subtraction in kindergarten (within 5)
and grade one (within 10) culminate in grade-two students becoming fluent in single-digit additions
and related subtractions, using Level 2 and Level 3 methods and strategies as needed.
Operations and Algebraic Thinking 2.OA
Add and subtract within 20.
2. Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all
sums of two one-digit numbers.
Students may still need to support the development of their fluency with math drawings when solving
problems. Math drawings represent the number of objects counted (using dots and sticks) and do not
need to represent the context of the problem. Thinking about numbers by using 10-frames or making
drawings using groups of fives and tens may be helpful ways to understand single-digit additions and
subtractions. The National Council of Teachers of Mathematics Illuminations project (NCTM Illumi-
nations 2013a) offers examples of interactive games that students can play to develop counting and
addition skills. 1
2. See Standard 1.0A.6 for a list of mental strategies.
130 Grade Two California Mathematics Framework
14.
FLUENCY
California’s Common Core State Standards for Mathematics (K–6) set expectations for fluency in computation
(e.g., “Fluently add and subtract within 20 . . .”) [2.OA.2 ]. Such standards are culminations of progressions of
learning, often spanning several grades, involving conceptual understanding, thoughtful practice, and extra
support where necessary. The word fluent is used in the standards to mean “reasonably fast and accurate”
and possessing the ability to use certain facts and procedures with enough facility that using such knowledge
does not slow down or derail the problem solver as he or she works on more complex problems. Procedural
fluency requires skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Developing
fluency in each grade may involve a mixture of knowing some answers, knowing some answers from patterns,
and knowing some answers through the use of strategies.
Adapted from UA Progressions Documents 2011a.
Mental strategies, such as those listed in table 2-6, help students develop fluency in adding and sub-
tracting within 20 as they make sense of number relationships. Table 2-6 presents the mental strategies
listed with standard 1.OA.6 as well as two additional strategies.
Table 2-6. Mental Strategies
l Counting on
l Making tens (9 + 7 = [9 + 1] + 6 = 10 + 6)
l Decomposing a number leading to a ten (14 – 6 = 14 – 4 – 2 = 10 – 2 = 8)
l Related facts (8 + 5 = 13 and 13 – 8 = 5)
l Doubles (1 + 1, 2 + 2, 3 + 3, and so on)
l Doubles plus one (7 + 8 = 7 + 7 + 1)
l Relationship between addition and subtraction (e.g., by knowing that 8 + 4 = 12,
one also knows that 12 – 8 = 4)
l Equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known
equivalent 6 + 6 + 1 = 12 + 1 = 13)
Adapted from NCDPI 2013b.
Grade-two students build important foundations for multiplication as they explore odd and even num-
bers in a variety of ways (2.OA.3). They use concrete objects (e.g., counters or place-value cubes) and
move toward pictorial representations such as circles or arrays (MP.1). Through investigations, students
realize that an even number of objects can be separated into two equal groups (without extra objects
remaining), while an odd number of objects will have one object remaining (MP.7 and MP.8).
Operations and Algebraic Thinking 2.OA
Work with equal groups of objects to gain foundations for multiplication.
3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing
objects or counting them by 2s; write an equation to express an even number as a sum of two equal
addends.
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up
to 5 columns; write an equation to express the total as a sum of equal addends.
California Mathematics Framework Grade Two 131
15.
Students also apply their work with doubles addition facts and decomposition of numbers (breaking
them apart) into two equal addends (e.g., 10 = 5 + 5) to understand the concept of even numbers.
Students reinforce this concept as they write equations representing sums of two equal addends, such
as 2 + 2 = 4, 3 + 3 = 6, 5 + 5 = 10, 6 + 6 = 12, or 8 + 8 = 16. Students are encouraged to explain how
they determined if a number is odd or even and what strategies they used (MP.3).
With standard 2.OA.4, second-grade students use rectangular arrays to work with repeated addition—
a building block for multiplication in grade three—using concrete objects (e.g., counters, buttons,
square tiles) as well as pictorial representations on grid paper or other drawings of arrays (MP.1). Using
the commutative property of multiplication,
students add either the rows or the columns
and arrive at the same solution (MP.2). Students
write equations that represent the total as
the sum of equal addends, as shown in the
examples at right.
The first example helps students to understand 4 + 4 + 4 = 12
that 3 × 4 = 4 × 3; the second example sup- 3 + 3 + 3 + 3 = 12 5 + 5 + 5 + 5 = 20
ports the fact that 4 × 5 = 5 × 4 (ADE 2010). 4 + 4 + 4 + 4 + 4 = 20
Focus, Coherence, and Rigor
In the cluster “Work with equal groups of objects to gain foundations for multiplica-
tion,” student work reinforces addition skills and understandings and is connected
to work in the major clusters “Represent and solve problems involving addition and
subtraction” (2.OA.1 ) and “Add and subtract within 20” (2.OA.2 ). Also, as students
work with odd and even groups (2.OA.3) they build a conceptual understanding of
equal groups, which supports their introduction to multiplication and division in
grade three.
Domain: Number and Operations in Base Ten
In grade one, students viewed two-digit numbers as amounts of tens and ones. A critical area of
instruction in grade two is to extend students’ understanding of base-ten notation to include hundreds.
Second-grade students understand multi-digit numbers (up to 1000). They add and subtract within
1000 and become fluent with addition and subtraction within 100 using place-value strategies (UA
Progressions Documents 2012b).
132 Grade Two California Mathematics Framework
16.
Number and Operations in Base Ten 2.NBT
Understand place value.
1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones;
e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of 10 tens—called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six,
seven, eight, or nine hundreds (and 0 tens and 0 ones).
2. Count within 1000; skip-count by 2s, 5s, 10s, and 100s. CA
3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits,
using >, =, and < symbols to record the results of comparisons.
Second-grade students build on their previous work with groups of tens to make bundles of hundreds,
with or without leftovers, using base-ten blocks, cubes in towers of 10, 10-frames, and so forth, as well
as math drawings that initially show the 10 tens within 1 hundred, but then move to a quick-hundred
version that is a drawn square in which students visualize 10 tens; see figure 2-2 for examples. Bundling
hundreds will support students’ discovery of place-value patterns (MP.7). Students explore the idea that
numbers such as 100, 200, 300, and so on are groups of hundreds that have “0” in the tens and ones
places. Students might represent numbers using place-value (base-ten) blocks or math drawings (MP.1).
Figure 2-2. Recognizing 10 Tens as 1 Hundred
Using Base-Ten Blocks 2.NBT.1
These have the same value: Six (6) hundreds is the same as 600:
Using Math Drawings
When I bundle 10 “ten-sticks,” I get 1 “hundred The picture shows 3 hundreds, or 300.
square.”
Adapted from KATM 2012 (2nd Grade Flipbook).
C alifornia Mathematics Framework Grade Two 133
17.
As students represent various numbers, they associate number names with number quantities (MP.2).
For example, 243 can be expressed as both “2 groups of hundred, 4 groups of ten, and 3 ones” and
“24 tens and 3 ones.” Students can read number names as well as place-value concepts to say a num-
ber. For example, 243 should be read as “two hundred forty-three” as well as “2 hundreds, 4 tens, and
3 ones.” Flexibility with seeing a number like 240 as “2 hundreds and 4 tens” as well as “24 tens” is an
important indicator of place-value understanding (KATM 2012, 2nd Grade Flipbook).
In kindergarten, students were introduced to counting by tens. In second grade they extend this to
skip-count by twos, fives, tens, and hundreds (2.NBT.2 ). Exploring number patterns can help students
skip-count. For example, when skip-counting by fives, the ones digit alternates between 5 and 0, and
when skip-counting by tens and hundreds, only the tens and hundreds digits change, increasing by one
each time. In this way, skip-counting can reinforce students’ understanding of place value. Work with
skip-counting lays a foundation for multiplication; however, because students do not keep track of the
number of groups they have counted, they are not yet learning true multiplication. The ultimate goal is
for grade-two students to count in multiple ways without visual support.
Focus, Coherence, and Rigor
As students explore number patterns by skip-counting, they also develop mathemat-
ical practices such as understanding the meaning of written quantities (MP.2) and
recognizing number patterns and structures in the number system (MP.7).
Grade-two students need opportunities to read and represent numerals in various ways (2.NBT.3 ). An
example adapted from KATM (2012, 2nd Grade Flipbook) illustrates different ways for second-graders to
represent numerals:
l Standard form (e.g., 637)
l Base-ten numerals in standard form (e.g., 6 hundreds, 3 tens, and 7 ones)
l Number names in word form (e.g., six hundred thirty-seven)
l Expanded form (e.g., 600 + 30 + 7)
l Equivalent representations (e.g., 500 + 130 + 7; 600 + 20 + 17; 30 + 600 + 7)
When students read the expanded form for a number, they might say “6 hundreds plus 3 tens plus
7 ones” or “600 plus 30 plus 7.” Understanding the expanded form is valuable when students use
place-value strategies to add and subtract large numbers (see also 2.NBT.7).
Second-grade students use the symbols for greater than (>), less than (<), and equal to (=) to compare
numbers within 1000 (2.NBT.4 ). Students build on work in standards (2.NBT.1 and 2.NBT.3 ) by
examining the amounts of hundreds, tens, and ones in each number. To compare numbers, students
apply their understanding of place value. The goal is for students to understand that they look at the
numerals in the hundreds place first, then the tens place, and if necessary, the ones place. Students
should have experience communicating their comparisons in words before using only symbols to
indicate greater than, less than, and equal to.
134 Grade Two California Mathematics Framework
18.
Example: Compare 452 and 455. 2.NBT.4
Student 1 explains that 452 has 4 hundreds, 5 tens, and 2 ones and that 455 has 4 hundreds, 5 tens, and
5 ones. “They have the same number of hundreds and the same number of tens, but 455 has 5 ones and 452
only has 2 ones. So, 452 is less than 455, or 452 < 455.”
Student 2 might think that 452 is less than 455. “I know this because when I count up, I say 452 before I say
455.”
Adapted from KATM 2012 (2nd Grade Flipbook).
As students compare numbers, they also develop mathematical practices such as making sense of
quantities (MP.2), understanding the meaning of symbols (MP.6), and making use of number patterns
and structures in the number system (MP.7).2
Number and Operations in Base Ten 2.NBT
Use place-value understanding and properties of operations to add and subtract.
5. Fluently add and subtract within 100 using strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction.
6. Add up to four two-digit numbers using strategies based on place value and properties of operations.
7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy to
a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts
hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or
decompose tens or hundreds.
7.1 Use estimation strategies to make reasonable estimates in problem solving. CA
8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given
number 100–900.
9. Explain why addition and subtraction strategies work, using place value and the properties of operations.3
Place-value understanding is central to multi-digit computations. In grade two, students develop,
discuss, and later use efficient, accurate, and generalizable methods to compute sums and differences
of whole numbers in base-ten notation. While students become fluent in such methods within 100 at
grade two, they also use these methods for sums and differences within 1000 (2.NBT.5–7 ).
General written methods for numbers within 1000 are discussed in the chapter first, as these strategies
are merely extensions of those for numbers within 100. Of course, all methods for adding and subtract-
ing two- and three-digit numbers should be based on place value and should be learned by students
with an emphasis on understanding. Math drawings can support student understanding, and as stu-
dents become familiar with math drawings, these drawings should accompany written methods.
3. Explanations may be supported by drawings or objects.
California Mathematics Framework Grade Two 135
19.
Written methods for recording addition and subtraction are based on two important features of
the base-ten number system:
l When numbers are added or subtracted in the base-ten system, like units are added or subtracted
(e.g., ones are added to ones, tens to tens, hundreds to hundreds).
l Adding and subtracting multi-digit numbers written in base-ten can be facilitated by composing
and decomposing units appropriately, so as to reduce the calculations to adding and subtracting
within 20 (e.g., 10 ones make 1 ten, 100 ones make 1 hundred, 1 hundred makes 10 tens).
Figure 2-3 presents two written methods for addition, with accompanying illustrations (base-ten blocks
can also be used to illustrate). Students initially work with math drawings or manipulatives alongside
the written methods, but they will eventually use written methods exclusively, mentally constructing
pictures as necessary and using other strategies. Teachers should note the importance of these written
methods as students generalize to larger numbers and decimals and emphasize the regrouping nature
of combining units. These two methods are given only as examples and are not meant to represent all
such place-value methods.
Figure 2-3. Addition Methods Supported with Math Drawings
Examples 2.NBT.7
Addition Method 1: In this written addition method,
all partial sums are recorded underneath the addition
bar. Addition is performed from left to right in this
example, but students can also work from right to left.
In the accompanying drawing, it is clear that hundreds
are added to hundreds, tens to tens, and ones to ones,
which are eventually grouped into larger units where
possible to represent the total, 623.
4 5 6 4 5 6 4 5 6 4 5 6
+ 1 6 7 + 1 6 7 + 1 6 7 + 1 6 7
5 0 0 5 0 0 5 0 0
1 1 0 1 1 0
1 3
6 2 3
136 Grade Two California Mathematics Framework
20.
Figure 2-3. (continued)
Examples 2.NBT.7
Addition Method 2: In this written addition method, digits representing newly composed units are placed
below the addends from which they were derived, to the right to indicate that they are represented as a
larger, newly composed unit. The addition proceeds right to left. The advantage to placing the composed
units as shown is that it is clearer where
they came from—e.g., the 1 and 3 that came
from the sum of the ones-place digits (6 + 7)
are close to each other. This eliminates
confusion that can arise from traditional
methods involving “carrying,”which tends
to separate the two digits that came from 13
and obscure the meaning of the numbers.
4 5 6 4 5 6 4 5 6 4 5 6
+ 1 6 7 + 1 6 7 + 1 6 7 + 1 6 7
1 1 1 1 1
3 2 3 6 2 3
Add the ones, 6 + 7, Add the tens, 5 + 6 + 1, Add the hundreds,
and record these as 13, and record these 12 tens 4 + 1 + 1, and record
with 3 in the ones place with 2 in the tens place these 6 hundreds in
and a 1 underneath the and 1 under the the hundreds column.
tens column. hundreds column.
Adapted from Fuson and Beckmann 2013 and UA Progressions Documents 2012b.
In grade one, students were not expected to compute differences of two-digit numbers other than
multiples of 10. In grade two, students subtract two-digit numbers, with and without decomposing,
which highlights the similarity between these two cases.
Figure 2-4 presents two methods for subtraction, one where all decomposing is done first, the other
where decomposing is done as needed. Students will encounter situations in which they “don’t have
enough” to subtract. This is more precise than saying, “You can’t subtract a larger number from a
smaller number,” or the like, as the latter assertion is a false mathematical statement. In later grades,
students will subtract larger numbers from smaller ones, and that will result in negative numbers as
answers (for example, 9 – 15 = −6).
Note that the accompanying illustrations show the decomposing steps in each written subtraction
method. Again, these methods generalize to numbers of all sizes and are based on decomposing larger
units into smaller units when necessary.
California Mathematics Framework Grade Two 137
21.
Figure 2-4. Subtraction Methods Supported with Math Drawings
Examples 2.NBT.7
Subtraction Method 1: In this written subtraction method, all necessary decompositions are done first.
Decomposing can start from the left or the right with this method. Students may be less likely to erroneously
subtract the top number from the bottom in this method.
11 11
3 12 15 3 12 15
4 2 5 4 2 5 4 2 5
− 2 7 8 − 2 7 8 − 2 7 8
1 4 7
Decomposing left to right, 1 hundred, then 1 ten
Subtraction Method 2: In this written subtraction method, decomposing is done as needed. Students first
ungroup a ten so they can subtract 8 from 15. They may erroneously try to subtract the tens as well, getting
7 – 1 = 6. Led to see their error, students find they need to ungroup hundreds first to subtract the tens, then
the hundreds.
11
1 15 3 1 15
4 2 5 4 2 5 4 2 5
− 2 7 8 − 2 7 8 − 2 7 8
7 1 4 7
Adapted from Fuson and Beckmann 2013 and UA Progressions Documents 2012b.
When developing fluency with adding and subtracting within 100 (2.NBT.5 ), second-grade students
use the methods just discussed, as well as other strategies, without the support of drawings.
Strategies for Addition and Subtraction
Examples of addition strategies based on place value for 48 + 37:
l Adding by place value: 40 + 30 = 70, 8 + 7 = 15, and 70 + 15 = 85
l Incremental adding (by tens and ones): 48 + 10 = 58, 58 + 10 = 68, 68 + 10 = 78, and 78 + 7 = 85
l Composing and decomposing (making a “friendly” number): 48 + 2 = 50, 37 – 2 = 35, and 50 + 35 = 85
Examples of subtraction strategies based on place value for 81 – 37:
l Adding up (from smaller number to larger number): 37 + 3 = 40, 40 + 40 = 80, 80 + 1 = 81, and
3 + 40 + 1 = 44
l Incremental subtracting: 81 – 10 = 71, 71 – 10 = 61, 61 – 10 = 51, 51 – 7 = 44
l Subtracting by place value: 81 – 30 = 51, 51 – 7 = 44
Adapted from ADE 2010.
138 Grade Two California Mathematics Framework
22.
As students develop fluency with adding and subtracting within 100, they also support mathematical
practices such as making sense of quantities (MP.2), calculating accurately (MP.6), and making use of
number patterns and structures in the number system (MP.7).
Example: Find the sum of 43 + 34 + 57 + 24. 2.NBT.6
Student A (Commutative and Associative Properties). “I saw the 43 and 57 and added them first. I know
3 plus 7 equals 10, so when I added them, 100 was my answer. Then I added 34 and had 134. Then I added
24 and had 158. So 43 + 57 + 34 + 24 = 158.”
Student B (Place-Value Strategies). “I broke up all of the numbers into tens and ones. First I added the tens:
40 + 30 + 50 + 20 = 140. Then I added the ones: 3 + 4 + 7 + 4 =18. That meant I had 1 ten and 8 ones. So,
140 + 10 is 150. 150 and 8 more is 158. So, 43 + 34 + 57 + 24 = 158.”
Student C (Place-Value Strategies and Commutative and Associative Property). “I broke up all the numbers
into tens and ones. First I added up the tens: 40 + 30 + 50 + 20. I changed the order of the numbers to make
adding easier. I know that 30 plus 20 equals 50, and 50 more equals 100. Then I added the 40 and got 140.
Then I added up the ones: 3 + 4 + 7 + 4. I changed the order of the numbers to make adding easier. I know
that 3 plus 7 equals 10 and 4 plus 4 equals 8. I also know that 10 plus 8 equals 18. I then combined my tens
and my ones: 140 plus 18 (1 ten and 8 ones) equals 158.”
Adapted from NCDPI 2013b.
Finally, students explain why addition and subtraction strategies work, using place value and the
properties of operations (2.NBT.9 ). Second-grade students need multiple opportunities to explain
their addition and subtraction thinking (MP.2). For example, students use place-value understanding,
properties of operations, number names, words (including mathematical language), math drawings,
number lines, and physical objects to explain why and how they solve a problem (MP.1, MP.6). Students
can also critique the work of other students (MP.3) to deepen their understanding of addition and sub-
traction strategies.
Example 2.NBT.9
There are 36 birds in the park. Suddenly, 25 more birds arrive. How many birds are there? Solve the problem
and show your work.
Student A. “I broke 36 and 25 into tens and ones (30 + 6) + (20 + 5). I can change the order of my numbers,
since it doesn’t change any amounts, so I added 30 + 20 and got 50. Then I added 5 and 5 to make 10 and
added it to the 50. So, 50 and 10 more is 60. I added the one that was left over and got 61. So there are 61
birds in the park.”
Student B. “I used a math drawing and made a pile of 36 and a pile of 25.
Altogether, I had 5 tens and 11 ones. 11 ones is the same as one ten and
one left over. So, I really had 6 tens and 1 one. That makes 61.”
Adapted from NCDPI 2013b.
California Mathematics Framework Grade Two 139
23.
Focus, Coherence, and Rigor
When students explain why addition and subtraction strategies work (2.NBT.9 ), they
reinforce foundations for solving one- and two-step word problems (2.OA.1 ) and
extend their understanding and use of various strategies and models, drawings, and a
written method to add and subtract (2.NBT.5 and 2.NBT.7 ).
Students are to fluently add and subtract within 100 in grade two (using place-value strategies, prop-
erties of operations, and/or the relationship between addition and subtraction) (2.NBT.5 ). In grade
one, students added within 100 using concrete models or drawings and used at least one method that
is generalizable to larger numbers (such as between 101 and 1000). In grade two, students extend
addition to within 1000 using these generalizable concrete methods. This extension could be connect-
ed first to adding all two-digit numbers (e.g., 78 + 47) so that students can see and discuss composing
both ones and tens without the complexity of hundreds in the drawings or numbers.
After solving addition problems that compose both ones and tens for all two-digit numbers and then
three-digit numbers within 1000, the fluency problems for grade two seem easy: 28 + 47 requires com-
posing only the ones. This is now easier to do without drawings: one just records the new ten before it
is added in, or adds it in mentally. Fluent adding means adding without drawings.
The same approach may be taken for subtraction, first solving with concrete models or drawings of
subtractions within 100 that involve decomposing 1 ten to make 10 ones and then solving subtraction
problems that require two decompositions, of 1 hundred to make 10 tens and of 1 ten to make 10
ones. Spending a long time subtracting within 100 initially can stimulate students to count on or count
down, methods that become considerably more difficult above 100. Problems with all possibilities of
decompositions should be mixed in so that students solve problems requiring two, one, and no decom-
positions. Then students can spend time on subtractions that include multiple hundreds (totals from
201 to 1000). After this experience, focusing within 100 just on the two cases of one decomposition
(e.g., 73 – 28) or no decomposition (e.g., 78 – 23) is relatively easy to do without drawings.
Mental math as an instructional tool. Many teachers incorporate an activity known as “mental math” into
their classrooms. The teacher typically writes a problem on the board (such as 45 + 47) and asks students to
solve the problem only through a mental process. The teacher then records all answers given by students,
whether correct or incorrect, without judgment. A class discussion follows; students explain how they got
their answers and decide which answer is correct. The class may agree or disagree with a particular method,
find out where another student made an error, or compare different solution methods (e.g., how finding
45 + 45 + 2 is similar to finding 40 + 40 + 12). In mental math, multiple strategies often emerge naturally
from the students, and opportunities to explore these strategies arise. When students do not have more
than one strategy for solving a problem, this can be an indication to the teacher that students have a limited
repertoire of such strategies, and therefore mental math can be used as a valuable instructional tool. Mental
math supports several Mathematical Practice standards, including MP.1, MP.2, MP.3, MP.7, and MP.8. (Stan-
dard 2.NBT.8 calls for second-grade students to practice mental math by adding and subtracting multiples
of 10 and 100 from a given number between 100 and 900.)
140 Grade Two California Mathematics Framework
24.
Domain: Measurement and Data
Grade-two students transition from measuring lengths with informal or non-standard units to mea-
suring with standard units—inches, feet, centimeters, and meters—and using standard measurement
tools (2.MD.1 ). Students learn the measure of length as a count of how many units are needed to
match the length of the object or distance being measured. Using both customary units (inches and
feet) and metric units (centimeters and meters), students measure the length of objects with rulers,
yardsticks, meter sticks, and tape measures. Students become familiar with standard units (e.g.,
12 inches in a foot, 3 feet in a yard, and 100 centimeters in a meter) and how to estimate lengths
(adapted from KATM 2012, 2nd Grade Flipbook).
Measurement and Data 2.MD
Measure and estimate lengths in standard units.
1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter
sticks, and measuring tapes.
2. Measure the length of an object twice, using length units of different lengths for the two measurements;
describe how the two measurements relate to the size of the unit chosen.
3. Estimate lengths using units of inches, feet, centimeters, and meters.
4. Measure to determine how much longer one object is than another, expressing the length difference in
terms of a standard length unit.
Students also can learn accu-
rate measurement procedures Figure 2-5. Example of a Student-Created Ruler
and concepts by constructing Using a unit to draw a ruler
simple unit rulers (see figure
2-5). Using copies of a standard
unit, such as manipulatives that
measure one inch,
students mark off unit lengths
on strips of paper, explicitly
Students use a standard unit (shown below the ruler) to make rulers,
connecting the process of mea- helping them to understand the meaning of the marks on rulers.
suring with a ruler to measur-
ing by iterating physical units. Adapted from UA Progressions Documents 2012a.
Thus, students’ first rulers are simple tools to help count the iteration of unit lengths. Frequently
comparing results of measuring the same object with manipulatives of standard unit length (e.g.,
a block that is one inch long) and with student-created rulers can help students connect their expe-
riences and ideas. As they build and use these tools, they develop the ideas of unit length iteration
(unit lengths are all of equal size), correct alignment (with a ruler), measurement of the length between
hashmarks on the ruler, and the zero-point concept (the idea that the zero of the ruler indicates one
endpoint of a length).
California Mathematics Framework Grade Two 141
25.
Grade-two students learn the concept of the inverse relationship between the size of the unit of length
and the number of units required to cover a definite length or distance—specifically, that the larger
the unit, the fewer units are needed to measure something, and vice versa (2.MD.2 ). Students mea-
sure the length of the same object using units of different lengths (ruler with inches versus ruler with
centimeters, or a foot ruler versus a yardstick) and discuss the relationship between the size of the units
and the measurements.
Example 2.MD.2
A student measured the length of a desk in both feet and inches. The student found
that the desk was 3 feet long and that it was 36 inches long.
Teacher: “Why do you think you have two different measurements for the same desk?”
Student: “It only took 3 feet because the feet are so big. It took 36 inches because an
inch is much smaller than a foot.”
Students use this information to understand how to select appropriate tools for measuring a given
object. For instance, a student might think, “The longer the unit, the fewer units I need.” Measurement
problems also support mathematical practices such as reasoning quantitatively (MP.2), justifying
conclusions (MP.3), using appropriate tools (MP.5), attending to precision (MP.6), and making use of
structure or patterns (MP.7).
After gaining experience with measurement, students learn to estimate lengths using units of inches,
feet, centimeters, and meters (2.MD.3 ). Students estimate lengths before they measure. After mea-
suring an object, students discuss their estimations, measurement procedures, and the differences
between their estimates and the measurements. Students should begin by estimating measurements of
familiar items (e.g., the length of a desk, pencil, favorite book, and so forth). Estimation helps students
focus on the attribute to be measured, the length units, and the process. Students need many experi-
ences with the use of measurement tools to develop their understanding of linear measurement; an
example is provided below.
Example 2.MD.3
Teacher: “How many inches do you think this string is if you measure it with a ruler?”
Student: “An inch is pretty small. I’m thinking it will be somewhere between 8 and 9
inches.”
Teacher: “Measure it and see.”
Student: “It is 9 inches. I thought that it would be somewhere around there.”
This example also supports mathematical practices such as making sense of quantities (MP.2) and using
appropriate tools strategically (MP.5).
142 Grade Two California Mathematics Framework
26.
Students also measure to determine the difference in length between one object and another, express-
ing the difference in terms of a standard length unit (2.MD.4 ). Grade-two students use inches, feet,
yards, centimeters, and meters to compare the lengths of two objects. They use comparative phrases
such as “It is 2 inches longer” or “It is shorter by 5 centimeters” to describe the difference in length be-
tween the two objects. Students use both the quantity and the unit name to precisely compare length
(ADE 2010 and NCDPI 2013b).
Focus, Coherence, and Rigor
As students compare objects by their lengths, they also reinforce skills and under-
standing related to solving “compare” problems in the major cluster “Represent
and solve problems involving addition and subtraction.” Drawing comparison
bars to represent the different measurements helps make this link explicit (see
standard 2.OA.1 ).
Students apply the concept of length to solve addition and subtraction problems. Word problems
should refer to the same unit of measure (2.MD.5 ).
Measurement and Data 2.MD
Relate addition and subtraction to length.
5. Use addition and subtraction within 100 to solve word problems involving lengths that are given in
the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the
unknown number to represent the problem.
6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corre-
sponding to the numbers 0, 1, 2, . . ., and represent whole-number sums and differences within 100 on a
number line diagram.
In grade two, students also connect the concept of the ruler to the concept of the number line. These
understandings are essential to supporting work with number line diagrams.
California Mathematics Framework Grade Two 143
27.
Example 2.MD.5
Kate jumped 14 inches in gym class. Lilly jumped 23 inches. How much farther did Lilly jump than Kate?
Solve the problem and then write an equation.
Student A: My equation is 14 + — = 23. I thought,
“14 and what makes 23?” I used cubes. I made a train
of 14. Then I made a train of 23. When I put them
side by side, I saw that Kate would need 9 more cubes
to be the same as Lilly. So, Lilly jumped 9 more inches
than Kate.
14 + 9 = 23. (MP.1, MP.2, MP.4)
Student B: My equation is 23 – 14 = —. I thought about what the difference was between Kate and Lilly.
I broke up 14 into 10 and 4. I know that 23 minus 10 is 13. Then, I broke up the 4 into 3 and 1. 13 minus 3 is
10. Then, I took one more away. That left me with 9. So, Lilly jumped 9 inches more than Kate. That seems to
make sense, since 23 is almost 10 more than 14.
23 – 14 = 9. (MP.2, MP.7, MP.8)
Focus, Coherence, and Rigor
Addition and subtraction word problems involving lengths develop mathematical
practices such as making sense of problems (MP.1), reasoning quantitatively (MP.2),
justifying conclusions (MP.3), using appropriate tools strategically (MP.5), attending
to precision (MP.6), and evaluating the reasonableness of results (MP.8). Similar word
problems also support students’ ability to fluently add and subtract, which is part
of the major work at the grade (refer to fluency expectations in standards 2.OA.1
and 2.NBT.5 ).
Using a number line diagram to understand number and number operations requires students to
comprehend that number line diagrams have specific conventions: namely, that a single position is
used to represent a whole number and that marks are used to indicate those positions. Students need
to understand that a number line diagram is like a ruler in that consecutive whole numbers are one
unit apart; thus, they need to consider the distances between positions and segments when identifying
missing numbers. These understandings underlie the successful use of number line diagrams. Students
think of a number line diagram as a measurement model and use strategies relating to distance, prox-
imity of numbers, and reference points (UA Progressions Documents 2012a).
144 Grade Two California Mathematics Framework
28.
Example 2.MD.6
There were 27 students on the bus. Nineteen (19) students got off the bus. How many students are on the bus?
Student: I used a number line. I saw that 19 is really close to 20. Since 20 is a lot easier to work with, I took
a jump of 20. But, that was one too many. So, I took a jump of 1 to make up for the extra. I landed on 8. So,
there are 8 students on the bus. What I did was 27 – 20 = 7, and then 7 + 1 = 8.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Adapted from ADE 2010 and NCDPI 2013b.
Teachers should ensure that students make the connection between problems involving measuring
with a ruler and those involving a number line as a problem-solving strategy.
Focus, Coherence, and Rigor
Using addition and subtraction within 100 to solve word problems involving length
(2.MD.5) and representing sums and differences on a number line (2.MD.6) reinforces
the use of models to add and subtract and supports major work at grade two (see
standards 2.OA.A.1 and 2.NBT.7 ). Similar problems also develop mathematical
practices such as making sense of problems (MP.2), justifying conclusions (MP.3), and
modeling mathematics (MP.4).
In grade one, students learned to tell time to the nearest hour and half-hour. Second-grade students
tell time to the nearest five minutes (2.MD.7 ).
Measurement and Data 2.MD
Work with time and money.
7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. Know
relationships of time (e.g., minutes in an hour, days in a month, weeks in a year). CA
8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols
appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?
California Mathematics Framework Grade Two 145
29.
Students can make connections between skip-counting by fives (2.NBT.2 ) and five-minute intervals on
the clock. Students work with both digital and analog clocks. They recognize time in both formats and
communicate their understanding of time using both numbers and language.
Second-grade students also understand that there are two 12-hour cycles in a day—a.m. and p.m.
A daily journal can help students make real-world connections and understand the differences
between these two cycles.
Focus, Coherence, and Rigor
Students’ understanding and use of skip-counting by fives and tens (2.NBT.2 ) can
also support telling and writing time to the nearest five minutes (2.MD.7 ). Students
notice the pattern of numbers and apply this understanding to time (MP.7).
In grade two, students solve word problems involving dollars or cents (2.MD.8). They identify, count,
recognize, and use coins and bills in and out of context. Second-grade students should have opportuni-
ties to make equivalent amounts using both coins and bills. Dollar bills should include denominations
up to one hundred ($1, $5, $10, $20, $50, $100). Note that students in second grade do not express
money amounts using decimal points.
Just as students learn that a number may be represented in different ways and still remain the same
amount—e.g., 38 can be 3 tens and 8 ones or 2 tens and 18 ones—students can apply this understand-
ing to money. For example, 25 cents could be represented as a quarter, two dimes and a nickel, or 25
pennies, all of which have the same value. Building the concept of equivalent worth takes time, and
students will need numerous opportunities to create and count different sets of coins and to recognize
the “purchasing power” of coins (e.g., a girl can buy the same things with a nickel that she can pur-
chase with 5 pennies).
As teachers provide students with opportunities to explore coin values (e.g., 25 cents), actual coins
(e.g., 2 dimes and 1 nickel), and drawings of circles that have values indicated, students gradually learn
to mentally assign a value to each coin in a set, place a random set of coins in order, use mental math,
add on to find differences, and skip-count to determine the total amount.
Examples 2.MD.8
Using pennies, nickels, dimes, and quarters, how many different ways can you make 37
cents?
Using $1, $5, and $10 bills, how many different ways can you make $12?
Adapted from ADE 2010 and NCDPI 2013b.
146 Grade Two California Mathematics Framework
30.
Measurement and Data 2.MD
Represent and interpret data.
9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by
making repeated measurements of the same object. Show the measurements by making a line plot,
where the horizontal scale is marked off in whole-number units.
10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four
categories. Solve simple put-together, take-apart, and compare problems4 using information presented
in a bar graph.
Students use the measurement skills described in previous standards (2.MD.1–4 ) to measure objects
and create measurement data (2.MD.9). For example, they measure objects in their desk to the nearest
inch, display the data collected on a line plot, and answer related questions. Line plots are first intro-
duced in grade two. A line plot can be thought of as plotting data on a number line (see figure 2-6).3
In grade one, students worked with three categories of
Figure 2-6. Example of a Line Plot
data. In grade two, students work with data that have up
Number of Pencils Measured
to four categories. Students organize and represent data
on a picture graph or bar graph (with single-unit scale)
and interpret the results. They solve simple put-together,
take-apart, and “compare” problems using information
presented in a bar graph (2.MD.10). In grade two, picture
graphs (pictographs) include symbols that represent single
units. Pictographs should include a title, categories,
category label, key, and data.
0 1 2 3 4 5 6
Length of Pencils (in inches)
Focus, Coherence, and Rigor
Students use data to pose and solve simple one-step addition and subtraction prob-
lems. The use of picture graphs and bar graphs to represent a data set (2.MD.10)
reinforces grade-level work in the major cluster “Represent and solve problems in-
volving addition and subtraction” and provides a context for students to solve related
addition and subtraction problems (2.OA.1 ).
4. See glossary, table GL-4.
California Mathematics Framework Grade Two 147
31.
Example 2.MD.10
Students are responsible for purchasing ice cream Team A: Bar Graph
for an event at school. They decide to collect data
13
to determine which flavors to buy for the event. 12
12
Students decide on the question to ask their 11
peers—“What is your favorite flavor of ice cream?” 10
—and four likely responses: chocolate, vanilla, 9
Number of People
9
strawberry, and cherry. Students form two teams 8
and collect information from different classes in 7
6
their school. Each team decides how to keep track 6
5
of its data (e.g., with tally marks, check marks, or in 5
a table). Each team selects either a picture graph or 4
a bar graph to display its data. Graphs are created 3
2
using paper or a computer.
1
The teacher facilitates a discussion about the data Chocolate Vanilla Strawberry Cherry
collected, asking questions such as these:
Flavors of Ice Cream
l Based on the graph from Team A, how many
students voted for cherry, strawberry, vanilla,
or chocolate ice cream?
Team B: Picture Graph
l Based on the graph from Team B, how many
Favorite Ice Cream Flavor
students voted for cherry, strawberry, vanilla,
or chocolate ice cream? Chocolate
l Based on the data from both teams, which
flavor received the most votes? Which flavor Vanilla
received the fewest votes?
l What was the second-favorite flavor? Strawberry
l Based on the data collected, what flavors of
ice cream do you think we should purchase Cherry
for our event, and why do you think that?
represents 1 student
Adapted from NCDPI 2013b.
Representing and interpreting data to solve problems also develops mathematical practices such as
making sense of problems (MP.1), reasoning quantitatively (MP.2), justifying conclusions (MP.3), using
appropriate tools strategically (MP.5), attending to precision (MP.6), and evaluating the reasonableness
of results (MP.8).
148 Grade Two California Mathematics Framework
32.
Domain: Geometry
In grade one, students reasoned about attributes of geometric shapes. A critical area of instruction in
second grade is for students to describe and analyze shapes by examining their sides and angles. This
work develops a foundation for understanding area, volume, congruence, similarity, and symmetry in
later grades.
Geometry 2.G
Reason with shapes and their attributes.
1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given
number of equal faces.5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of
them.
3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words
halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths.
Recognize that equal shares of identical wholes need not have the same shape.
Students identify, describe, and draw triangles, quadrilaterals (squares, rectangles and parallelograms,
and trapezoids), pentagons, hexagons, and cubes (2.G.1); see figure 2-7. Pentagons, triangles, and hexa-
gons should appear as both regular (having equal sides and equal angles) and irregular. Second-grade
students recognize all four-sided shapes as quadrilaterals. They use the vocabulary word angle in place
of corner, but they do not need to name angle types (e.g., right, acute, obtuse). Shapes should be pre-
sented in a variety of orientations and configurations. 4
Figure 2-7. Examples of the Presentation of Various Shapes
triangles quadrilaterals pentagons hexagons
Source: ADE 2010.
As students use attributes to identify and describe shapes, they also develop mathematical practices
such as analyzing givens and constraints (MP.1), justifying conclusions (MP.3), modeling with mathemat-
ics (MP.4), using appropriate tools strategically (MP.5), attending to precision (MP.6), and looking for a
pattern or structure (MP.7).
Students partition a rectangle into rows and columns of same-size squares and count to find the total
number of squares (2.G.2). As students partition rectangles into rows and columns, they build a foun-
dation for learning about the area of a rectangle and using arrays for multiplication.
5. Sizes are compared directly or visually, not by measuring.
California Mathematics Framework Grade Two 149
33.
Example 2.G.2
Teacher: Partition this rectangle into 3 equal rows and 4 equal columns. How can you partition into 3 equal
rows? Then into 4 equal columns? Can you do it in the other order? How many small squares did you make?
Student: I counted 12 squares in the rectangle. This is a lot like
when we counted arrays by counting 4 + 4 + 4 = 12.
An interactive whiteboard or manipulatives such as square tiles, cubes, or other square-shaped objects
can be used to help students partition rectangles (MP.5).
In grade one, students partitioned shapes into halves, fourths, and quarters. Second-grade students
partition circles and rectangles into two, three, or four equal shares (regions). Students explore this
concept with paper strips and pictorial representations and work with the vocabulary terms halves,
thirds, and fourths (2.G.3). Students recognize that when they cut a circle into three equal pieces, each
piece will equal one-third of its original whole and the whole may be described as three-thirds. If a
circle is cut into four equal pieces, each piece will equal one-fourth of its original whole, and the whole
is described as four-fourths.
Circle cut Circle cut Circle not cut Circle cut
into halves into thirds into thirds into fourths
Students should see circles and rectangles partitioned in multiple ways so they learn to recognize that
equal shares can be different shapes within the same whole.
halves fourths
150 Grade Two California Mathematics Framework
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