All the resources on this page come from the GADOE Framework 2nd Grade Unit 2 Overview. Reviewing this information before or during planning can help ensure your instruction is aligned with the standards and is at the depth expected.

Unit 2 Overview

In this unit students will:

• cultivate an understanding of how addition and subtraction affect quantities and are related to each other

• will reinforce the multiple meanings for addition (combine, join, and count on) and subtraction (take away, remove, count back, and compare)

• further develop their understanding of the relationships between addition and subtraction

• recognize how the digits 0-9 are used in our place value system to create numbers and manipulate amounts

• continue to develop their understanding solving problems with money

• At the beginning of Unit 2, it is recommended that students practice counting money collections daily during Number Corner or as part of daily Math Maintenance in order to be prepared for the tasks. For more detailed information regarding unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the grade level overview.

As students in second grade begin to count larger amounts, they should group concrete materials into tens and ones to keep track of what they have counted. This is the introduction of our place value system where students must learn that the digits (0-9) have different values depending on their position in a number.

Students in second grade now build on their work with one-step problems to solve two-step problems. Second graders need to model and solve problems and represent their solutions with equations. The problems should involve sums and differences less than or equal to 100 using the numbers 0 to 100. Picture Graphs and Bar Graphs are also introduced in second grade. Investigations and experiences with graphing should take place all year long.

Addition and Subtraction in Elementary School
(Information adapted from North Carolina DPI Instructional Support Tools)

• The strategies that students use to solve problems provide important information concerning number sense, and place value.

• It is important to look at more than answers students get. The strategies used provide useful information about what problems to give the next day, and how to differentiate instruction.

• It is important to relate addition and subtraction.

• Student-created strategies provide reinforcement of place value concepts. Traditional algorithms can actually “unteach” place value.

• Student created strategies are built on a student’s actual understanding, instead of on what the book says or what we think/hope they know!

• Students make fewer errors with invented strategies, because they are built on understanding rather than memorization.

Students use various counting strategies, including counting all, counting on, and counting back with numbers up to 20. This standard calls for students to move beyond counting all and become comfortable at counting on and counting back. The counting all strategy requires students to count an entire set. The counting and counting back strategies occur when students are able to hold the start number in their head and count on from that number. Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis. Additionally, the required fluency expectations for second grade students (knowing from memory all sums of two one digit numbers) should be a gradual progression. The word fluency is used judiciously in the standards to mark the endpoints of progressions of learning that begin with solid underpinnings and then pass upward through stages of growing maturity. By doing this we are allowing students to gradually enhance their understanding of the concept of number and to develop computational proficiency.

NUMBER TALKS

Between 5 and 15 minutes each day should be dedicated to “Number Talks” in order to build students’ mental math capabilities and reasoning skills. Sherry Parrish’s book Number Talks provides examples of K-5 number talks. The following video clip from Math Solutions is an excellent example of a number talk in action. https://www.teachingchannel.org/video/numbertalk-math-lesson-2nd-grade

During the Number Talk, the teacher is not the definitive authority. The teacher is the facilitator and is listening for and building on the students’ natural mathematical thinking. The teacher writes a problem horizontally on the board in whole group or a small setting. The students mentally solve the problem and share with the whole group how they derived the answer. They must justify and defend their reasoning. The teacher simply records the students’ thinking and poses extended questions to draw out deeper understanding for all.

The effectiveness of Numbers Talks depends on the routines and environment that is established by the teacher. Students must be given time to think quietly without pressure from their peers. To develop this, the teacher should establish a signal, other than a raised hand, of some sort to identify that one has a strategy to share. One way to do this is to place a finger on their chest indicating that they have one strategy to share. If they have two strategies to share, they place out two fingers on their chest and so on.

Number Talk problem possible student responses:

Number talks often have a focus strategy such as “making tens” or “compensation.” Providing students with a string of related problems, allows students to apply a strategy from a previous problem to subsequent problems. Some units lend themselves well to certain Number Talk topics. For example, the place value unit may coordinate well with the Number Talk strategy of “making ten.”

Note: Most of the concepts in this unit can be supported through 10-minute daily Number Talks. If you are doing effective daily Number Talks, your students may move more quickly through this unit and/or provide you with excellent formative information regarding your future daily Number Talks and implementation of the tasks.

STANDARDS FOR MATHEMATICAL PRACTICE

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students have multiple opportunities to develop strategies for mental math addition and subtraction as well as solving story problems, riddles, and graphs.

2. Reason abstractly and quantitatively. Students use number lines, base ten blocks, money, and other manipulatives to connect quantities to written symbols. Students compare numbers and discover the commutative property of addition.

3. Construct viable arguments and critique the reasoning of others. Students develop strategies for mental math as well as solving story problems and interpreting graphs. The students share and defend their thinking.

4. Model with mathematics. Students use words, pictures, graphs, money, and manipulatives to express addition and subtraction problems.

5. Use appropriate tools strategically. Students use estimation, pictures, and manipulatives to solve addition and subtraction computation as well as story problems.

6. Attend to precision. Students have daily practice in number talks and tasks to use mathematical language to explain their own reasoning.

7. Look for and make use of structure. Students look for patterns developing mental strategies: making tens, repeated addition, fact families, and doubles.

8. Look for and express regularity in repeated reasoning. Students look for shortcuts in addition mental math: such as rounding up, then adjusting; repeated addition; riddles; and reasonableness of answers.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson. ***

STANDARDS FOR MATHEMATICAL CONTENT

Represent and solve problems involving addition and subtraction.
MGSE2.OA.1 Use addition and subtraction within 100 to solve one and two step word problems by using drawings and equations with a symbol for the unknown number to represent the problem. Problems include contexts that involve adding to, taking from, putting together/taking apart (part/part/whole) and comparing with unknowns in all positions.

Add and subtract within 20.
MGSE2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Use place value understanding and properties of operations to add and subtract.
MGSE2.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.

Work with time and money.
MGSE2.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
MGSE2.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 problems1 using information presented in a bar graph

BIG IDEAS

By the conclusion of this unit, students should be able to demonstrate the following competencies:

• Represent and solve problems involving addition and subtraction.

• Solve a variety of word problems involving money using $ and ¢ symbols.

• Understand and apply properties of operations and the relationship between addition and subtraction.

• Recognize how the digits 0-9 are used in our place value system to create numbers and manipulate amounts.

• Understand how addition and subtraction affect quantities and are related to each other.

• Know the multiple meanings for addition (combine, join, and count on) and subtraction (take away, remove, count back, and compare)

• Use the inverse operation to check that they have correctly solved the problem.

• Solve problems using mental math strategies.

• Draw and interpret picture and bar graphs to represent a data set with up to four categories.

Essential Questions

• How do we represent a collection of objects using tens and ones?

• How do I express money amounts?

• When will estimating be helpful to us?

• How can we use skip counting to help us solve problems?

• Can we change the order of numbers if we subtract? Why or why not?

• Can we change the order of numbers when we add (or subtract)? Why or why not?

• How can estimation strategies help us build our addition skills?

• How do we use addition to tell number stories?

• How can benchmark numbers help us add?

• How does using ten as a benchmark number help us add and subtract?

• What strategies can help us when adding and subtracting with regrouping?

• What strategies will help me add multiple numbers quickly and accurately?

• How can we solve addition problems with and without regrouping?

• How can addition help us know we subtracted two numbers correctly?

• How can we solve subtraction problems with and without regrouping?

• How can strategies help us when adding and subtracting with regrouping?

• How can we model and solve subtraction problems with and without regrouping? How can mental math strategies, for example estimation and benchmark numbers, help us when adding and subtracting with regrouping?

• How can I use a number line to help me model how I combine and compare numbers?

• How are addition and subtraction alike and how are they different?

• What is a number sentence and how can I use it to solve word problems?

• How do we solve problems in different ways?

• How can we solve problems mentally? What strategies help us with this?

• How can we show/represent problems in different ways?

• How can problem situations and problem-solving strategies be represented?

• How are problem-solving strategies alike and different?

• How can different combinations of numbers and operations be used to represent the same quantity?

CONCEPTS AND SKILLS TO MAINTAIN

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, and make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between varieties of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wp-content/uploads/2015/03/FluencyWithoutFear-2015.pdf and: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timedtests.pdf

Skills from Grade 1:

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20;

• Developing understanding of whole number relationships and place value, including grouping in tens and ones;

Second Grade Year Long Concepts:

• Organizing and graphing data as stated in MGSE2.MD.10 should be regularly incorporated in activities throughout the year. Students should be able to draw a picture graph and a bar graph to represent a data set with up to four categories as well as solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

• Routine topics such as counting, time, money, positional words, patterns, and tallying should be addressed on an ongoing basis throughout instructional time. These topics that should also be addressed daily through Number Corner or Math Maintenance.

• Students will be asked to use estimation and benchmark numbers throughout the year in a variety of mathematical situations.

STRATEGIES FOR TEACHING AND LEARNING
(Information adapted from North Carolina DPI Instructional Support Tools)

Represent and solve problems involving addition and subtraction.
MGSE2.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.

Instructional Strategies
This standard calls for students to add and subtract numbers within 100 in the context of one and two step word problems. Students should have ample experiences working on various types of problems that have unknowns in all positions, including Result Unknown, Change Unknown, and Start Unknown. See Table 1 on page 11 for further examples. The problems should involve sums and differences less than or equal to 100 using the numbers 0 to 100. It is vital that students develop the habit of checking their answer to a problem to determine if it makes sense for the situation and the questions being asked.

This standard also calls for students to solve one- and two-step problems using drawings, objects and equations. Students can use place value blocks or hundreds charts, or create drawings of place value blocks or number lines to support their work. Examples of one-step problems with unknowns in different places are provided in Table 1. Two step-problems include situations where students have to add and subtract within the same problem.

Example:

In the morning there are 25 students in the cafeteria. 18 more students come in. After a few minutes, some students leave. If there are 14 students still in the cafeteria, how many students left the cafeteria? Write an equation for your problem.

Working on addition and subtraction simultaneously, continually relating the two operations is important for helping students recognize and understand the (inverse) relationship of these two operations. It is also vital that students develop the habit of checking their answer to a problem to determine if it makes sense for the situation and the questions being asked. An excellent way to do this is to ask students to write word problems for their classmates to solve. A good place to start is by giving students the answer to a problem. Then tell students whether you want them to write an addition or subtraction problem situation. Also let them know that the sums and differences can be less than or equal to 100. For example, ask students to write an addition word problem for their classmates to solve which requires adding four two-digit numbers with 100 as the answer. Students then share, discuss and compare their solution strategies after they solve the problems.

The strategies that students use to solve problems provide important information concerning number sense and place value understandings therefore it is important to look at more than answers students get. The strategies students use provide useful information about what problems to give the next day and how to differentiate instruction. Student-created strategies provide reinforcement of place value concepts. Teaching traditional algorithms can actually hinder the development of conceptual knowledge of our place value system; whereas student created strategies are built on a student’s actual understanding, instead of on what the book says or what we think/hope they know. Students make fewer errors with their own invented strategies because they are built on their own understanding rather than memorization.

Add and Subtract within 20.
MGSE2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.

Instructional Strategies
This standard mentions the word fluently when students are adding and subtracting numbers within 20. Fluency means accuracy (attending to precision), efficiency (using well-understood strategies with ease), and flexibility (using strategies such as making 10 or breaking apart numbers). Research indicates that teachers can best support students’ development of automaticity with sums and differences through varied experiences making 10, breaking numbers apart and working on mental strategies, rather than timed tests. Evidence from research has indicated that timed tests cause unhealthy math anxiety with learners as they are developing a solid foundation in numeracy: https://www.youcubed.org/resources/new-evidence-timed-testteaching-children-mathematics-april-2014/.

Provide many activities that will help students develop a strong understanding of number relationships, addition and subtraction so they can develop, share and use efficient strategies for mental computation. An efficient strategy is one that can be done mentally and quickly. Students gain computational fluency, using efficient and accurate methods for computing, as they come to understand the role and meaning of arithmetic operations in number systems. Efficient mental processes become automatic with use.

Provide activities in which students apply the commutative and associative properties to their mental strategies for sums less or equal to 20 using the numbers 0 to 20. Have students study how numbers are related to 5 and 10 so they can apply these relationships to their strategies for knowing 5 + 4 or 8 + 3. Students might picture 5 + 4 on a ten-frame to mentally see 9 as the answer. For remembering 8 + 7, students might think: since 8 is 2 away from 10, take 2 away from 7 to make 10 + 5 = 15. Activities such as these will provide good opportunities to use “number talks” as described in the 2nd Grade Overview.

Example:

When presented the problem, 4 + 8 + 6, the student uses number talk to say “I know 6 + 4 = 10, so I can add 4 + 8 + 6 by adding 4 + 6 to make 10 and then add 8 to make 18.” Make anchor charts/posters for student-developed mental strategies for addition and subtraction within 20. Use names for the strategies that make sense to the students and include examples of the strategies. Present a particular strategy along with the specific addition and subtraction facts relevant to the strategy. Have students use objects and drawings to explore how these facts are alike.

Provide simple word problems designed for students to invent and try a particular strategy as they solve it. Have students explain their strategies so that their classmates can understand it. Guide the discussion so that the focus is on the methods that are most useful. Encourage students to try the strategies that were shared so they can eventually adopt efficient strategies that work for them. Use anchor charts/posters illustrating the various student strategies to use as reference as the students develop their toolbox of strategies.

Use place value understanding and properties of operations to add and subtract.
MGSE2.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. (For additional information please see the Grade Level Overview)

**This standard mentions the word fluently, just as stated with MGSE2.OA.2, when students are adding and subtracting numbers within 100. Fluency means accuracy (attending to precision), efficiency (using well-understood strategies with ease), and flexibility (using strategies such as making 10 or breaking apart numbers).

This standard calls for students to use pictorial representations and/or strategies to find the solution. Students who are struggling may benefit from further work with concrete objects (e.g., place value blocks).

Provide many activities that will help students develop a strong understanding of number relationships, addition and subtraction so they can develop, share and use efficient strategies for mental computation. An efficient strategy is one that can be done mentally and quickly. Students gain computational fluency, using efficient and accurate methods for computing, as they come to understand the role and meaning of arithmetic operations in number systems. Efficient mental processes become automatic with use.

Students need to build on their flexible strategies for adding within 100 in Grade 1 to fluently add and subtract within 100, add up to four two-digit numbers, and find sums and differences less than or equal to 1000 using numbers 0 to 1000.

Initially, students apply base-ten concepts and use direct modeling with physical objects or drawings to find different ways to solve problems. They move to inventing strategies that do not involve physical materials or counting by ones to solve problems. Student-invented strategies likely will be based on place-value concepts, the commutative and associative properties, and the relationship between addition and subtraction. These strategies should be done mentally or with a written record for support.

It is vital that student-invented strategies be shared, explored, recorded and tried by others. Recording the expressions and equations in the strategies horizontally encourages students to think about the numbers and the quantities they represent instead of the digits. Not every student will invent strategies, but all students can and will try strategies they have seen that make sense to them. Different students will prefer different strategies.

Students will decompose and compose tens and hundreds when they develop their own strategies for solving problems where regrouping is necessary. They might use the make-ten strategy (37 + 8 = 40 + 5 = 45, add 3 to 37 then 5) or (62 - 9 = 60 – 7 = 53, take off 2 to get 60, then 7 more) because no ones are exchanged for a ten or a ten for ones.

Have students analyze problems before they solve them. Present a variety of subtraction problems within 1000. Ask students to identify the problems requiring them to decompose the tens or hundreds to find a solution and explain their reasoning.

Work with Money (For additional information please refer to the Grade Level Overview)
MGSE2.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?

• Relate to whole-number place value and base-ten understandings. For example, 23¢ = 2 dimes and 3 pennies.

• Understand the relationship between quantity and value. For example, 1 dime = 10¢. Help students to understand that the relationship between coin size and value is inconsistent.

• Limit problems to the use of just dollar and cents symbols. There should be no decimal notation for money at this point.

This standard calls for students to solve word problems involving either dollars or cents. Since students have not been introduced to decimals, problems should either have only dollars or only cents.

Example: What are some possible combinations of coins (pennies, nickels, dimes, and quarters) that equal 37 cents?

Example: What are some possible combinations of dollar bills ($1, $5 and $10) that equal 12 dollars?

The topic of money begins at Grade 2 and builds on the work in other clusters in this and previous grades. Help students learn money concepts and solidify their understanding of other topics by providing activities where students make connections between them. For instance, link the value of a dollar bill as 100 cents to the concept of 100 and counting within 1000. Use play money - nickels, dimes, and dollar bills to skip count by 5s, 10s, and 100s. Reinforce place value concepts with the values of dollar bills, dimes, and pennies. Students use the context of money to find sums and differences less than or equal to 100 using the numbers 0 to 100. They add and subtract to solve one- and two-step word problems involving money situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. Students use drawings and equations with a symbol for the unknown number to represent the problem. The dollar sign, $, is used for labeling whole-dollar amounts without decimals, such as $29. Students need to learn the relationships between the values of a penny, nickel, dime, quarter and dollar bill.

Represent and Interpret Data
MGSE2.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.

Instructional Strategies
At first students should create real object and picture graphs so each row or bar consists of countable parts. These graphs show items in a category and do not have a numerical scale. For example, a real object graph could show the students’ shoes (one shoe per student) lined end to end in horizontal or vertical rows by their color. Students would simply count to find how many shoes are in each row or bar. The graphs should be limited to 2 to 4 rows or bars. Students would then move to making horizontal or vertical bar graphs with two to four categories and a single unit scale.

As students continue to develop their use of reading and interpreting data, it is highly suggested to incorporate these standards into daily routines. It is not merely the making or filling out of the graph, but the connections made from the date represented that builds and strengthens mathematical reasoning.

Use the information in the graphs to pose and solve simple put together, take-apart, and compare problems illustrated in Table 1 located on page 12.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are not an inclusive list and should not be taught in isolation. Instructors should pay particular attention to them and how their students are able to explain and apply them (i.e. students should not be told to memorize these terms).

Teachers should present these concepts to students with models and real-life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers. For specific definitions, please reference the Georgia Standards of Excellence State Standards Glossary. • add • addition and subtraction within 5, 10, 20, 100, or 1000 • associative property for addition • bar graph • commutative property for addition • comparing • counting strategy • difference • doubles plus one • equations • estimating: fluency • fluently • identity property for addition • join • line plot • picture graph • place value • quantity • recalling facts • re-grouping • remove • scale • strategies for addition • strategy • subtract • unknowns