New Second Grade Common Core Standards
This year, 2nd grade has adopted all of the new Common Core standards. Kindergarten began using them in 2011. First Grade began using them in 2012, and now we will be using them. The rest of the school will be adopting them next year. In the meantime, those upper grades will be using IN State Standards plus whatever Common Core Standards that are not already state standards. Please note...the list below is LONG!!!!
List of 2nd Grade Common Core Standards
Reading Standards: Foundational Skills RF
These standards are directed toward fostering students’ understanding and working knowledge of concepts of print, the alphabetic principle, and other basic conventions of the English writing system. These foundational skills are not an end in and of themselves; rather, they are necessary and important components of an effective, comprehensive reading program designed to develop proficient readers with the capacity to comprehend texts across a range of types and disciplines. Instruction should be differentiated: good readers will need much less practice with these concepts than struggling readers will. The point is to teach students what they need to learn and not what they already know – to discern when particular children or activities warrant more or less attention.
Print Concepts
2.RF.1 (There is not a grade 2 standard for this concept. Please see preceding grades for more information).
Phonological Awareness
2.RF.2 (There is not a grade 2 standard for this concept. Please see preceding grades for more information).
Phonics and Word Recognition
2.RF.3 Know and apply grade-level phonics and word analysis skills in decoding words.
a. Distinguish long and short vowels when reading regularly spelled one-syllable words.
b. Know spelling-sound correspondences for additional common vowel teams.
c. Decode regularly spelled two-syllable words with long vowels.
d. Decode words with common prefixes and suffixes.
e. Identify words with inconsistent but common spelling-sound correspondences.
f. Recognize and read grade-appropriate irregularly spelled words.
Fluency
2.RF.4 Read with sufficient accuracy and fluency to support comprehension.
a. Read on-level text with purpose and understanding.
b. Read on-level text orally with accuracy, appropriate rate, and expression on successive readings.
c. Use context to confirm or self-correct word recognition and understanding, rereading as necessary.
Reading Standards for Informational Text:
Literary Nonfiction and Historical, Scientific, and Technical Texts RI
The following standards offer a focus for instruction each year and help ensure that students gain adequate exposure to a range of texts and tasks. Rigor is also infused through the requirement that students read increasingly complex texts through the grades. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Key Ideas and Details
2.RI.1 Ask and answer such questions as who, what, where, when, why, and how to demonstrate
understanding of key details in a text.
2.RI.2 Identify the main topic of a multi-paragraph text as well as the focus of specific paragraphs within
the text.
2.RI.3 Describe the connection between a series of historical events, scientific ideas or concepts, or steps in
technical procedures in a text.
Craft and Structure
2.RI.4 Determine the meaning of words and phrases in a text relevant to a grade 2 topic or subject area. 2.RI.5 Know and use various text features (e.g., captions, bold print, subheadings, glossaries, indexes,
electronic menus, icons) to locate key facts or information in a text efficiently.
2.RI.6 Identify the main purpose of a text, including what the author wants to answer, explain, or describe.
Integration of Knowledge and Ideas
2.RI.7 Explain how specific images (e.g., a diagram showing how a machine works) contribute to and
clarify a text.
2.RI.8 Describe how reasons support specific points the author makes in a text.
2.RI.9 Compare and contrast the most important points presented by two texts on the same topic.
Range of Reading and Level of Text Complexity
2.RI.10 By the end of year, read and comprehend informational texts, including history/social studies,
science, and technical texts, in the grades 2-3 text complexity band proficiently, with scaffolding as
needed at the high end of the range.
Reading Standards for Literature: Stories, Novels, Dramas, and Poetry RL
The following standards offer a focus for instruction and help ensure that students gain adequate exposure to a range of texts and tasks. Rigor is also infused through the requirement that students read increasingly complex texts through the grades. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Key Ideas and Details
2.RL.1 Ask and answer such questions as who, what, where, when, why, and how to demonstrate
understanding of key details in a text.
2.RL.2 Recount stories, including fables and folktales from diverse cultures, and determine their central
message, lesson, or moral.
2.RL.3 Describe how characters in a story respond to major events and challenges.
Craft and Structure
2.RL.4 Describe how words and phrases (e.g., regular beats, alliteration, rhymes, repeated lines) supply
rhythm and meaning in a story, poem, or song.
2.RL.5 Describe the overall structure of a story, including describing how the beginning introduces the
story and the ending concludes the action.
2.RL.6 Acknowledge differences in the points of view of characters, including by speaking in a different
voice for each character when reading dialogue aloud.
Integration of Knowledge and Ideas
2.RL.7 Use information gained from the illustrations and words in a print or digital text to demonstrate
understanding of its characters, setting, or plot.
2.RL.8 (This College and Career Readiness Standard does not have a literature component)
2.RL.9 Compare and contrast two or more versions of the same story (e.g., Cinderella stories) by different
authors or from different cultures.
Range of Reading and Level of Text Complexity
2.RL.10 By the end of the year, read and comprehend literature, including stories and poetry, in the grades
2-3 text complexity band proficiently, with scaffolding as needed at the high end of the range.
Writing Standards W
The following standards offer a focus for instruction to help ensure that students gain adequate mastery of a range of skills and applications. Each year in their writing, students should demonstrate increasing sophistication in all aspects of language use, from vocabulary and syntax to the development and organization of ideas, and they should address increasingly demanding content and sources. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Text Types and Purposes
2.W.1 Write opinion pieces in which they introduce the topic or book they are writing about, state an
opinion, supply reasons that support the opinion, use linking words (e.g., because, and, also) to
connect opinion and reasons, and provide a concluding statement or section.
2.W.2 Write informative/explanatory texts in which they introduce a topic, use facts and definitions to
develop points, and provide a concluding statement or section.
2.W.3 Write narratives in which they recount a well-elaborated event or short sequence of events, include
details to describe actions, thoughts, and feelings, use temporal words to signal event order, and
provide a sense of closure.
Production and Distribution of Writing
2.W.4 (Begins in grade 3)
2.W.5 With guidance and support from adults and peers, focus on a topic and strengthen writing as needed
by revising and editing.
2.W.6 With guidance and support from adults, use a variety of digital tools to produce and publish writing,
including in collaboration with peers.
Research to Build and Present Knowledge
2.W.7 Participate in shared research and writing projects (e.g., read a number of books on a single topic to
produce a report; record science observations).
2.W.8 Recall information from experiences or gather information from provided sources to answer a
question.
2.W.9 (Begins in grade 4)
Range of Writing
2.W.10 (Begins in grade 3)
Language Standards L
The following standards offer a focus for instruction to help ensure that students gain adequate mastery of a range of skills and applications. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Conventions of Standard English
2.L.1 Demonstrate command of the conventions of Standard English grammar and usage when writing or
speaking.
a. Use collective nouns (e.g., group).
b. Form and use frequently occurring irregular plural nouns (e.g., feet, children, teeth, mice, fish).
c. Use reflexive pronouns (e.g., myself, ourselves).
d. Form and use the past tense of frequently occurring irregular verbs (e.g., sat, hid, told).
e. Use adjectives and adverbs, and choose between them depending on what is to be modified.
f. Produce, expand, and rearrange complete simple and compound sentences (e.g., The boy
watched the movie; The little boy watched the movie; The action movie was watched by the little
boy).
2.L.2 Demonstrate command of the conventions of Standard English capitalization, punctuation, and
spelling when writing.
a. Capitalize holidays, product names, and geographic names.
b. Use commas in greetings and closings of letters.
c. Use an apostrophe to form contractions and frequently occurring possessives.
d. Generalize learned spelling patterns when writing words (e.g., cage -> badge; boy -> boil).
e. Consult reference materials, including beginning dictionaries, as needed to check and correct spellings.
Knowledge of Language
2.L.3 Use knowledge of language and its conventions when writing, speaking, reading, or listening.
a. Compare formal and informal uses of English.
Vocabulary Acquisition and Use
2.L.4 Determine or clarify the meaning of unknown and multiple-meaning words and phrases based on
grade 2 reading and content, choosing flexibly from an array of strategies.
a. Use sentence-level context as a clue to the meaning of a word or phrase.
b. Determine the meaning of the new word formed when a known prefix is added to a known word (e.g., happy/unhappy, tell/retell).
c. Use a known root word as a clue to the meaning of an unknown word with the same root (e.g., addition, additional).
d. Use knowledge of the meaning of individual words to predict the meaning of compound words (e.g., birdhouse, lighthouse, housefly; bookshelf, notebook, bookmark).
e. Use glossaries and beginning dictionaries, both print and digital, to determine or clarify the meaning of words and phrases.
2.L.5 Demonstrate understanding of word relationships and nuances in word meanings.
a. Identify real-life connections between words and their use (e.g., describe foods that are spicy or juicy).
b. Distinguish shades of meaning among closely related verbs (e.g., toss, throw, hurl) and closely related adjectives (e.g., thin, slender, skinny, scrawny).
2.L.6 Use words and phrases acquired through conversations, reading and being read to, and responding to
texts, including using adjectives and adverbs to describe (e.g., When other kids are happy, that makes
me happy).
Speaking and Listening Standards SL
The following standards offer a focus for instruction to help ensure that students gain adequate mastery of a range of skills and applications. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Comprehension and Collaboration
2.SL.1 Participate in collaborative conversations with diverse partners about grade 2 topics and texts with
peers and adults in small and larger groups.
a. Follow agreed-upon rules for discussions (e.g., gaining the floor in respectful ways, listening to others with care, speaking one at a time about the topics and texts under discussion).
b. Build on others' talk in conversations by linking their comments to the remarks of others.
c. Ask for clarification and further explanation as needed about the topics and texts under discussion.
2.SL.2 Recount or describe key ideas or details from a text read aloud or information presented orally or
through other media.
2.SL.3 Ask and answer questions about what a speaker says in order to clarify comprehension, gather
additional information, or deepen understanding of a topic or issue.
Presentation of Knowledge and Ideas
2.SL.4 Tell a story or recount an experience with appropriate facts and relevant, descriptive details,
speaking audibly in coherent sentences.
2.SL.5 Create audio recordings of stories or poems; add drawings or other visual displays to stories or
recounts of experiences when appropriate to clarify ideas, thoughts, and feelings.
2.SL.6 Produce complete sentences when appropriate to task and situation in order to provide requested
detail or clarification.
INDIANA COMMON CORE STATE STANDARDS for MATHEMATICS
INDIANA COMMON CORE STATE STANDARDS
GRADE 2 MATHEMATICS
Introduction
The world we live in continues to change. For students to succeed in school, at work, and in
the community, they will need more skills and knowledge than ever before. To ensure all
students have every opportunity to succeed, Indiana adopted the Common Core State
Standards in the area of Mathematics, as well as English Language Arts and Literacy in
History/Social Studies, Science, and Technical Subjects.
Common Core State Standards Philosophy
The Common Core State Standards:
are aligned with college and work expectations.
include rigorous content and application of knowledge through high-order skills.
build upon strengths and lessons of current state standards.
are internationally benchmarked.
are evidence and/or research-based.
The Common Core State Standards will
help prepare all students with the knowledge and skills needed to succeed in college
and careers.
assist students who move between states.
create the opportunity for America to compete for high-wage, high-skill jobs in a
knowledge-based economy.
allow for more focused pre-service education and professional development.
create economies of scale around areas such as curriculum, instructional resources,
and assessment.
Role of the Teacher
While the standards set expectations for student learning for each grade level, they do not
prescribe how the standards should be taught. Teachers should use their skills, experience,
talents, and resources to design standards-based classroom lessons that meet the needs of
individual students.
Assessment Horizon
Indiana is a governing state in the Partnership for Assessment of Readiness for College and
Careers (PARCC). Through PARCC a new generation of assessments will be developed and
used throughout the nation, allowing us to compare and evaluate policies that affect student
achievement across states and school districts.
The Common Core assessment includes a summative test, which will be administered online
beginning with the 2014-15 school year. The assessment also includes “through-course
assessments” that take place over the course of the year, allowing teachers to adjust their
instruction based on students’ results. As a governing state, Indiana will pilot Common Core
items in 2012-13 and 2013-14 and provide schools with data regarding student performance
on the Common Core State Standards.
Impact on Instruction
Grades K, 1, and 2
In 2011-12, kindergarten teachers will be the first to teach only the Common Core. Those
kindergarteners will be the first class of 3rd graders to participate in the Common Core
assessment. First-grade and second-grade students will learn Indiana Academic Standards
and identified elements of Common Core in 2011-12. First-grade students will learn only
Common Core in 2012-13. Second-grade students will learn only Common Core in 2013-14.
Grades 3-12
Students in grades 3-12 will receive an intentional introduction to the Common Core
beginning with the 2011-12 school year to ensure they have the foundation for future
learning with no instructional gaps. In 2014-15, all ELA and math teachers will only teach
Common Core.
Final Notes
Indiana Department of Education staff members are working diligently, both externally with
other states, and internally across the IDOE, to ensure the development of a common scope
and sequence for Common Core. This work is critical as we begin the transition to the
Common Core State Standards.
Equally as important, efforts are underway to assist teachers in understanding what to teach
next year (and in subsequent years). A tight alignment between the content which is “taught”
and that which is “tested” on ISTEP+ and ECAs, together with the initiation of Common
Core State Standards instruction, facilitates the important work of developing the foundation
for future learning and assessment.
Mathematics Academic Standards: Grade 2
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the
discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical
maturity and expertise throughout the elementary, middle and high school years. Designers of curricula,
assessments, and professional development should all attend to the need to connect the mathematical practices
to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding.
Expectations that begin with the word “understand” are often especially good opportunities to connect the
practices to the content. Students who lack understanding of a topic may rely on procedures too heavily.
Without a flexible base from which to work, they may be less likely to consider analogous problems, represent
problems coherently, justify conclusions, apply the mathematics to practical situations, use technology
mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an
overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively
prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of
intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice.
These points of intersection are intended to be weighted toward central and generative concepts in the school
mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to
qualitatively improve the curriculum, instruction, assessment, professional development, and student
achievement in mathematics.
Mathematics Academic Standards: Grade 2
Grade 2 Overview
In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten
notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4)
describing and analyzing shapes.
1. Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens,
and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including
comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that
the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens
+ 3 ones).
2. Students use their understanding of addition to develop fluency with addition and subtraction within 100.
They solve problems within 1000 by applying their understanding of models for addition and subtraction, and
they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences
of whole numbers in base-ten notation, using their understanding of place value and the properties of
operations. They select and accurately apply methods that are appropriate for the context and the numbers
involved to mentally calculate sums and differences for numbers with only tens or only hundreds.
3. Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and
other measurement tools with the understanding that linear measure involves an iteration of units. They
recognize that the smaller the unit, the more iterations they need to cover a given length.
4. Students describe and analyze shapes by examining their sides and angles. Students investigate, describe,
and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and
analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume,
congruence, similarity, and symmetry in later grades.
Mathematics Academic Standards: Grade 2
Operations and Algebraic Thinking OA
Represent and solve problems involving addition and subtraction.
2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving
situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in
all positions, e.g., by using drawings and equations with a symbol for the unknown number to
represent the problem.1
Add and subtract within 20.
2.OA.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory
sums all of two one-digit numbers.
Work with equal groups of objects to gain foundations for multiplication.
2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by
pairing objects or counting them by 2s; write an equation to express an even number as a sum of two
equal addends.
2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and
up to 5 columns; write an equation to express the total as a sum of equal addends.
1 See Glossary, Table 1.
2 See standard 1.OA.6 for a list of mental strategies.
Mathematics Academic Standards: Grade 2
Number and Operations in Base Ten NBT
Understand place value.
2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and
ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens - called a "hundred."
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six,
seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >,
=, and < symbols to record the results of comparisons.
Use place value understanding and properties of operations to add and subtract.
2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction.
2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.
2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy
to a written method. Understand that in adding or subtracting three-digit numbers, one adds or
subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to
compose or decompose tens or hundreds.
2.NBT.8 Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given
number 100-900.
2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of
operations. 3
3 Explanations may be supported by drawing or objects.
Mathematics Academic Standards: Grade 2
Measurement and Data MD
Measure and estimate lengths in standard units.
2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks,
meter sticks, and measuring tapes.
2.MD.2 Measure the length of an object twice, using length units of different lengths for the two
measurements; describe how the two measurements relate to the size of the unit chosen.
2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.4 Measure to determine how much longer one object is than another, expressing the length difference in
terms of a standard length unit.
Relate addition and subtraction to length.
2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the
same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the
unknown number to represent the problem.
2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points
corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100
on a number line diagram.
Work with money and time.
2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢
symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?
Represent and interpret data.
2.MD.9 Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by
making repeated measurements of the same object. Show the measurements by making a line plot,
where the horizontal scale is marked off in whole-number units.
2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four
categories. Solve simple put-together, take-apart, and compare problems using information presented
in a bar graph.4
4 See Glossary, Table 1.
Mathematics Academic Standards: Grade 2
Geometry G
Reason with shapes and their attributes.
2.G.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.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of
them.
2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words
halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths.
Recognize that equal shares of identical wholes need not have the same shape.
5 Sizes are compared directly or visually, not compared by measuring.
Mathematics Academic Standards: Grade 2
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels
should seek to develop in their students. These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education.
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make
conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping
into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the
original problem in order to gain insight into its solution. They monitor and evaluate their progress and change
course if necessary. Older students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get the information they need.
Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables,
and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or
trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a
problem. Mathematically proficient students check their answers to problems using a different method, and they
continually ask themselves, Does this make sense? They can understand the approaches of others to
solving complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—and the
ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents
for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the
problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute
them; and knowing and flexibly using different properties of operations and objects.
Mathematics Academic Standards: Grade 2
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into
cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,
and respond to the arguments of others. They reason inductively about data, making plausible arguments that
take into account the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is
flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments
using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and
be correct, even though they are not generalized or made formal until later grades. Later, students learn to
determine domains to which an argument applies. Students at all grades can listen or read the arguments of
others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school
event or analyze a problem in the community. By high school, a student might use geometry to solve a design
problem or use a function to describe how one quantity of interest depends on another. Mathematically
proficient students who can apply what they know are comfortable making assumptions and approximations to
simplify a complicated situation, realizing that these may need revision later. They are able to identify important
quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables,
graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.
They routinely interpret their mathematical results in the context of the situation and reflect on whether the
results make sense, possibly improving the model if it has not served its purpose.
Mathematics Academic Standards: Grade 2
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software. Proficient students are
sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each
of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,
mathematically proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other mathematical
knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically
proficient students at various grade levels are able to identify relevant external mathematical resources, such as
digital content located on a website, and use them to pose or solve problems. They are able to use technological
tools to explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they
choose, including using the equal sign consistently and appropriately. They are careful about specifying units of
measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately
and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the
elementary grades, students give carefully formulated explanations to each other. By the time they reach high
school they have learned to examine claims and make explicit use of definitions.
Mathematics Academic Standards: Grade 2
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may sort a
collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well
remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x
+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing
line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also
can step back for an overview and shift perspective. They can see complicated things, such as some algebraic
expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2
as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any
real numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the
same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the
calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle
school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel
when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general
formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students
maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness
of their intermediate results.
These standards are directed toward fostering students’ understanding and working knowledge of concepts of print, the alphabetic principle, and other basic conventions of the English writing system. These foundational skills are not an end in and of themselves; rather, they are necessary and important components of an effective, comprehensive reading program designed to develop proficient readers with the capacity to comprehend texts across a range of types and disciplines. Instruction should be differentiated: good readers will need much less practice with these concepts than struggling readers will. The point is to teach students what they need to learn and not what they already know – to discern when particular children or activities warrant more or less attention.
Print Concepts
2.RF.1 (There is not a grade 2 standard for this concept. Please see preceding grades for more information).
Phonological Awareness
2.RF.2 (There is not a grade 2 standard for this concept. Please see preceding grades for more information).
Phonics and Word Recognition
2.RF.3 Know and apply grade-level phonics and word analysis skills in decoding words.
a. Distinguish long and short vowels when reading regularly spelled one-syllable words.
b. Know spelling-sound correspondences for additional common vowel teams.
c. Decode regularly spelled two-syllable words with long vowels.
d. Decode words with common prefixes and suffixes.
e. Identify words with inconsistent but common spelling-sound correspondences.
f. Recognize and read grade-appropriate irregularly spelled words.
Fluency
2.RF.4 Read with sufficient accuracy and fluency to support comprehension.
a. Read on-level text with purpose and understanding.
b. Read on-level text orally with accuracy, appropriate rate, and expression on successive readings.
c. Use context to confirm or self-correct word recognition and understanding, rereading as necessary.
Reading Standards for Informational Text:
Literary Nonfiction and Historical, Scientific, and Technical Texts RI
The following standards offer a focus for instruction each year and help ensure that students gain adequate exposure to a range of texts and tasks. Rigor is also infused through the requirement that students read increasingly complex texts through the grades. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Key Ideas and Details
2.RI.1 Ask and answer such questions as who, what, where, when, why, and how to demonstrate
understanding of key details in a text.
2.RI.2 Identify the main topic of a multi-paragraph text as well as the focus of specific paragraphs within
the text.
2.RI.3 Describe the connection between a series of historical events, scientific ideas or concepts, or steps in
technical procedures in a text.
Craft and Structure
2.RI.4 Determine the meaning of words and phrases in a text relevant to a grade 2 topic or subject area. 2.RI.5 Know and use various text features (e.g., captions, bold print, subheadings, glossaries, indexes,
electronic menus, icons) to locate key facts or information in a text efficiently.
2.RI.6 Identify the main purpose of a text, including what the author wants to answer, explain, or describe.
Integration of Knowledge and Ideas
2.RI.7 Explain how specific images (e.g., a diagram showing how a machine works) contribute to and
clarify a text.
2.RI.8 Describe how reasons support specific points the author makes in a text.
2.RI.9 Compare and contrast the most important points presented by two texts on the same topic.
Range of Reading and Level of Text Complexity
2.RI.10 By the end of year, read and comprehend informational texts, including history/social studies,
science, and technical texts, in the grades 2-3 text complexity band proficiently, with scaffolding as
needed at the high end of the range.
Reading Standards for Literature: Stories, Novels, Dramas, and Poetry RL
The following standards offer a focus for instruction and help ensure that students gain adequate exposure to a range of texts and tasks. Rigor is also infused through the requirement that students read increasingly complex texts through the grades. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Key Ideas and Details
2.RL.1 Ask and answer such questions as who, what, where, when, why, and how to demonstrate
understanding of key details in a text.
2.RL.2 Recount stories, including fables and folktales from diverse cultures, and determine their central
message, lesson, or moral.
2.RL.3 Describe how characters in a story respond to major events and challenges.
Craft and Structure
2.RL.4 Describe how words and phrases (e.g., regular beats, alliteration, rhymes, repeated lines) supply
rhythm and meaning in a story, poem, or song.
2.RL.5 Describe the overall structure of a story, including describing how the beginning introduces the
story and the ending concludes the action.
2.RL.6 Acknowledge differences in the points of view of characters, including by speaking in a different
voice for each character when reading dialogue aloud.
Integration of Knowledge and Ideas
2.RL.7 Use information gained from the illustrations and words in a print or digital text to demonstrate
understanding of its characters, setting, or plot.
2.RL.8 (This College and Career Readiness Standard does not have a literature component)
2.RL.9 Compare and contrast two or more versions of the same story (e.g., Cinderella stories) by different
authors or from different cultures.
Range of Reading and Level of Text Complexity
2.RL.10 By the end of the year, read and comprehend literature, including stories and poetry, in the grades
2-3 text complexity band proficiently, with scaffolding as needed at the high end of the range.
Writing Standards W
The following standards offer a focus for instruction to help ensure that students gain adequate mastery of a range of skills and applications. Each year in their writing, students should demonstrate increasing sophistication in all aspects of language use, from vocabulary and syntax to the development and organization of ideas, and they should address increasingly demanding content and sources. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Text Types and Purposes
2.W.1 Write opinion pieces in which they introduce the topic or book they are writing about, state an
opinion, supply reasons that support the opinion, use linking words (e.g., because, and, also) to
connect opinion and reasons, and provide a concluding statement or section.
2.W.2 Write informative/explanatory texts in which they introduce a topic, use facts and definitions to
develop points, and provide a concluding statement or section.
2.W.3 Write narratives in which they recount a well-elaborated event or short sequence of events, include
details to describe actions, thoughts, and feelings, use temporal words to signal event order, and
provide a sense of closure.
Production and Distribution of Writing
2.W.4 (Begins in grade 3)
2.W.5 With guidance and support from adults and peers, focus on a topic and strengthen writing as needed
by revising and editing.
2.W.6 With guidance and support from adults, use a variety of digital tools to produce and publish writing,
including in collaboration with peers.
Research to Build and Present Knowledge
2.W.7 Participate in shared research and writing projects (e.g., read a number of books on a single topic to
produce a report; record science observations).
2.W.8 Recall information from experiences or gather information from provided sources to answer a
question.
2.W.9 (Begins in grade 4)
Range of Writing
2.W.10 (Begins in grade 3)
Language Standards L
The following standards offer a focus for instruction to help ensure that students gain adequate mastery of a range of skills and applications. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Conventions of Standard English
2.L.1 Demonstrate command of the conventions of Standard English grammar and usage when writing or
speaking.
a. Use collective nouns (e.g., group).
b. Form and use frequently occurring irregular plural nouns (e.g., feet, children, teeth, mice, fish).
c. Use reflexive pronouns (e.g., myself, ourselves).
d. Form and use the past tense of frequently occurring irregular verbs (e.g., sat, hid, told).
e. Use adjectives and adverbs, and choose between them depending on what is to be modified.
f. Produce, expand, and rearrange complete simple and compound sentences (e.g., The boy
watched the movie; The little boy watched the movie; The action movie was watched by the little
boy).
2.L.2 Demonstrate command of the conventions of Standard English capitalization, punctuation, and
spelling when writing.
a. Capitalize holidays, product names, and geographic names.
b. Use commas in greetings and closings of letters.
c. Use an apostrophe to form contractions and frequently occurring possessives.
d. Generalize learned spelling patterns when writing words (e.g., cage -> badge; boy -> boil).
e. Consult reference materials, including beginning dictionaries, as needed to check and correct spellings.
Knowledge of Language
2.L.3 Use knowledge of language and its conventions when writing, speaking, reading, or listening.
a. Compare formal and informal uses of English.
Vocabulary Acquisition and Use
2.L.4 Determine or clarify the meaning of unknown and multiple-meaning words and phrases based on
grade 2 reading and content, choosing flexibly from an array of strategies.
a. Use sentence-level context as a clue to the meaning of a word or phrase.
b. Determine the meaning of the new word formed when a known prefix is added to a known word (e.g., happy/unhappy, tell/retell).
c. Use a known root word as a clue to the meaning of an unknown word with the same root (e.g., addition, additional).
d. Use knowledge of the meaning of individual words to predict the meaning of compound words (e.g., birdhouse, lighthouse, housefly; bookshelf, notebook, bookmark).
e. Use glossaries and beginning dictionaries, both print and digital, to determine or clarify the meaning of words and phrases.
2.L.5 Demonstrate understanding of word relationships and nuances in word meanings.
a. Identify real-life connections between words and their use (e.g., describe foods that are spicy or juicy).
b. Distinguish shades of meaning among closely related verbs (e.g., toss, throw, hurl) and closely related adjectives (e.g., thin, slender, skinny, scrawny).
2.L.6 Use words and phrases acquired through conversations, reading and being read to, and responding to
texts, including using adjectives and adverbs to describe (e.g., When other kids are happy, that makes
me happy).
Speaking and Listening Standards SL
The following standards offer a focus for instruction to help ensure that students gain adequate mastery of a range of skills and applications. Students advancing through the grades are expected to meet each year’s grade-specific standards and retain or further develop skills and understandings mastered in preceding grades.
Comprehension and Collaboration
2.SL.1 Participate in collaborative conversations with diverse partners about grade 2 topics and texts with
peers and adults in small and larger groups.
a. Follow agreed-upon rules for discussions (e.g., gaining the floor in respectful ways, listening to others with care, speaking one at a time about the topics and texts under discussion).
b. Build on others' talk in conversations by linking their comments to the remarks of others.
c. Ask for clarification and further explanation as needed about the topics and texts under discussion.
2.SL.2 Recount or describe key ideas or details from a text read aloud or information presented orally or
through other media.
2.SL.3 Ask and answer questions about what a speaker says in order to clarify comprehension, gather
additional information, or deepen understanding of a topic or issue.
Presentation of Knowledge and Ideas
2.SL.4 Tell a story or recount an experience with appropriate facts and relevant, descriptive details,
speaking audibly in coherent sentences.
2.SL.5 Create audio recordings of stories or poems; add drawings or other visual displays to stories or
recounts of experiences when appropriate to clarify ideas, thoughts, and feelings.
2.SL.6 Produce complete sentences when appropriate to task and situation in order to provide requested
detail or clarification.
INDIANA COMMON CORE STATE STANDARDS for MATHEMATICS
INDIANA COMMON CORE STATE STANDARDS
GRADE 2 MATHEMATICS
Introduction
The world we live in continues to change. For students to succeed in school, at work, and in
the community, they will need more skills and knowledge than ever before. To ensure all
students have every opportunity to succeed, Indiana adopted the Common Core State
Standards in the area of Mathematics, as well as English Language Arts and Literacy in
History/Social Studies, Science, and Technical Subjects.
Common Core State Standards Philosophy
The Common Core State Standards:
are aligned with college and work expectations.
include rigorous content and application of knowledge through high-order skills.
build upon strengths and lessons of current state standards.
are internationally benchmarked.
are evidence and/or research-based.
The Common Core State Standards will
help prepare all students with the knowledge and skills needed to succeed in college
and careers.
assist students who move between states.
create the opportunity for America to compete for high-wage, high-skill jobs in a
knowledge-based economy.
allow for more focused pre-service education and professional development.
create economies of scale around areas such as curriculum, instructional resources,
and assessment.
Role of the Teacher
While the standards set expectations for student learning for each grade level, they do not
prescribe how the standards should be taught. Teachers should use their skills, experience,
talents, and resources to design standards-based classroom lessons that meet the needs of
individual students.
Assessment Horizon
Indiana is a governing state in the Partnership for Assessment of Readiness for College and
Careers (PARCC). Through PARCC a new generation of assessments will be developed and
used throughout the nation, allowing us to compare and evaluate policies that affect student
achievement across states and school districts.
The Common Core assessment includes a summative test, which will be administered online
beginning with the 2014-15 school year. The assessment also includes “through-course
assessments” that take place over the course of the year, allowing teachers to adjust their
instruction based on students’ results. As a governing state, Indiana will pilot Common Core
items in 2012-13 and 2013-14 and provide schools with data regarding student performance
on the Common Core State Standards.
Impact on Instruction
Grades K, 1, and 2
In 2011-12, kindergarten teachers will be the first to teach only the Common Core. Those
kindergarteners will be the first class of 3rd graders to participate in the Common Core
assessment. First-grade and second-grade students will learn Indiana Academic Standards
and identified elements of Common Core in 2011-12. First-grade students will learn only
Common Core in 2012-13. Second-grade students will learn only Common Core in 2013-14.
Grades 3-12
Students in grades 3-12 will receive an intentional introduction to the Common Core
beginning with the 2011-12 school year to ensure they have the foundation for future
learning with no instructional gaps. In 2014-15, all ELA and math teachers will only teach
Common Core.
Final Notes
Indiana Department of Education staff members are working diligently, both externally with
other states, and internally across the IDOE, to ensure the development of a common scope
and sequence for Common Core. This work is critical as we begin the transition to the
Common Core State Standards.
Equally as important, efforts are underway to assist teachers in understanding what to teach
next year (and in subsequent years). A tight alignment between the content which is “taught”
and that which is “tested” on ISTEP+ and ECAs, together with the initiation of Common
Core State Standards instruction, facilitates the important work of developing the foundation
for future learning and assessment.
Mathematics Academic Standards: Grade 2
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content
The Standards for Mathematical Practice describe ways in which developing student practitioners of the
discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical
maturity and expertise throughout the elementary, middle and high school years. Designers of curricula,
assessments, and professional development should all attend to the need to connect the mathematical practices
to mathematical content in mathematics instruction.
The Standards for Mathematical Content are a balanced combination of procedure and understanding.
Expectations that begin with the word “understand” are often especially good opportunities to connect the
practices to the content. Students who lack understanding of a topic may rely on procedures too heavily.
Without a flexible base from which to work, they may be less likely to consider analogous problems, represent
problems coherently, justify conclusions, apply the mathematics to practical situations, use technology
mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an
overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively
prevents a student from engaging in the mathematical practices.
In this respect, those content standards which set an expectation of understanding are potential “points of
intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice.
These points of intersection are intended to be weighted toward central and generative concepts in the school
mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to
qualitatively improve the curriculum, instruction, assessment, professional development, and student
achievement in mathematics.
Mathematics Academic Standards: Grade 2
Grade 2 Overview
In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten
notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4)
describing and analyzing shapes.
1. Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens,
and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including
comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that
the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens
+ 3 ones).
2. Students use their understanding of addition to develop fluency with addition and subtraction within 100.
They solve problems within 1000 by applying their understanding of models for addition and subtraction, and
they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences
of whole numbers in base-ten notation, using their understanding of place value and the properties of
operations. They select and accurately apply methods that are appropriate for the context and the numbers
involved to mentally calculate sums and differences for numbers with only tens or only hundreds.
3. Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and
other measurement tools with the understanding that linear measure involves an iteration of units. They
recognize that the smaller the unit, the more iterations they need to cover a given length.
4. Students describe and analyze shapes by examining their sides and angles. Students investigate, describe,
and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and
analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume,
congruence, similarity, and symmetry in later grades.
Mathematics Academic Standards: Grade 2
Operations and Algebraic Thinking OA
Represent and solve problems involving addition and subtraction.
2.OA.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving
situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in
all positions, e.g., by using drawings and equations with a symbol for the unknown number to
represent the problem.1
Add and subtract within 20.
2.OA.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory
sums all of two one-digit numbers.
Work with equal groups of objects to gain foundations for multiplication.
2.OA.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by
pairing objects or counting them by 2s; write an equation to express an even number as a sum of two
equal addends.
2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and
up to 5 columns; write an equation to express the total as a sum of equal addends.
1 See Glossary, Table 1.
2 See standard 1.OA.6 for a list of mental strategies.
Mathematics Academic Standards: Grade 2
Number and Operations in Base Ten NBT
Understand place value.
2.NBT.1 Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and
ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens - called a "hundred."
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six,
seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s.
2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.NBT.4 Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >,
=, and < symbols to record the results of comparisons.
Use place value understanding and properties of operations to add and subtract.
2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations,
and/or the relationship between addition and subtraction.
2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.
2.NBT.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value,
properties of operations, and/or the relationship between addition and subtraction; relate the strategy
to a written method. Understand that in adding or subtracting three-digit numbers, one adds or
subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to
compose or decompose tens or hundreds.
2.NBT.8 Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given
number 100-900.
2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of
operations. 3
3 Explanations may be supported by drawing or objects.
Mathematics Academic Standards: Grade 2
Measurement and Data MD
Measure and estimate lengths in standard units.
2.MD.1 Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks,
meter sticks, and measuring tapes.
2.MD.2 Measure the length of an object twice, using length units of different lengths for the two
measurements; describe how the two measurements relate to the size of the unit chosen.
2.MD.3 Estimate lengths using units of inches, feet, centimeters, and meters.
2.MD.4 Measure to determine how much longer one object is than another, expressing the length difference in
terms of a standard length unit.
Relate addition and subtraction to length.
2.MD.5 Use addition and subtraction within 100 to solve word problems involving lengths that are given in the
same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the
unknown number to represent the problem.
2.MD.6 Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points
corresponding to the numbers 0, 1, 2,..., and represent whole-number sums and differences within 100
on a number line diagram.
Work with money and time.
2.MD.7 Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.
2.MD.8 Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢
symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?
Represent and interpret data.
2.MD.9 Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by
making repeated measurements of the same object. Show the measurements by making a line plot,
where the horizontal scale is marked off in whole-number units.
2.MD.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four
categories. Solve simple put-together, take-apart, and compare problems using information presented
in a bar graph.4
4 See Glossary, Table 1.
Mathematics Academic Standards: Grade 2
Geometry G
Reason with shapes and their attributes.
2.G.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.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of
them.
2.G.3 Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words
halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths.
Recognize that equal shares of identical wholes need not have the same shape.
5 Sizes are compared directly or visually, not compared by measuring.
Mathematics Academic Standards: Grade 2
Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels
should seek to develop in their students. These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education.
1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and
looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make
conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping
into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the
original problem in order to gain insight into its solution. They monitor and evaluate their progress and change
course if necessary. Older students might, depending on the context of the problem, transform algebraic
expressions or change the viewing window on their graphing calculator to get the information they need.
Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables,
and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or
trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a
problem. Mathematically proficient students check their answers to problems using a different method, and they
continually ask themselves, Does this make sense? They can understand the approaches of others to
solving complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the
representing symbols as if they have a life of their own, without necessarily attending to their referents—and the
ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents
for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the
problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute
them; and knowing and flexibly using different properties of operations and objects.
Mathematics Academic Standards: Grade 2
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously
established results in constructing arguments. They make conjectures and build a logical progression of
statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into
cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others,
and respond to the arguments of others. They reason inductively about data, making plausible arguments that
take into account the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is
flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments
using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and
be correct, even though they are not generalized or made formal until later grades. Later, students learn to
determine domains to which an argument applies. Students at all grades can listen or read the arguments of
others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in
everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school
event or analyze a problem in the community. By high school, a student might use geometry to solve a design
problem or use a function to describe how one quantity of interest depends on another. Mathematically
proficient students who can apply what they know are comfortable making assumptions and approximations to
simplify a complicated situation, realizing that these may need revision later. They are able to identify important
quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables,
graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions.
They routinely interpret their mathematical results in the context of the situation and reflect on whether the
results make sense, possibly improving the model if it has not served its purpose.
Mathematics Academic Standards: Grade 2
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software. Proficient students are
sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each
of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example,
mathematically proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other mathematical
knowledge. When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data. Mathematically
proficient students at various grade levels are able to identify relevant external mathematical resources, such as
digital content located on a website, and use them to pose or solve problems. They are able to use technological
tools to explore and deepen their understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they
choose, including using the equal sign consistently and appropriately. They are careful about specifying units of
measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately
and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the
elementary grades, students give carefully formulated explanations to each other. By the time they reach high
school they have learned to examine claims and make explicit use of definitions.
Mathematics Academic Standards: Grade 2
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they may sort a
collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well
remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x
+ 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing
line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also
can step back for an overview and shift perspective. They can see complicated things, such as some algebraic
expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2
as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any
real numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general methods
and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the
same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the
calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle
school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel
when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general
formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students
maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness
of their intermediate results.