All the resources on this page come from the GADOE Framework 2nd Grade Unit 4 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 4 Overview

In this unit students will:

• continue to develop their understanding of and facility with addition and subtraction

• add up to 4 two-digit numbers. • use a variety of models (base ten blocks- ones, tens, and hundreds only; diagrams; number lines; place value strategies; etc.) to add and subtract within one thousand.

• become fluent with mentally adding or subtracting 10 or 100 to a given three-digit number.

• demonstrate fluency with addition and subtraction.

• understand the relationship between addition and subtraction (inverse operations).

• represent three digit numbers with a variety of different models (base ten blocks- ones, tens, and hundreds only; diagrams; number lines; place value strategies; etc.).

• recognize and use place value to manipulate numbers.

• continue to develop their understanding of, and facility with, money.

• count with pennies, nickels, dimes, and dollar bills.

• represent a money amount with words or digits and symbols (either cent or dollar signs).

• represent and interpret data in picture and bar graphs.

• use information from a bar graph to solve addition and subtraction equations.

At the beginning of Unit 4, it is recommended that students practice counting money collections, telling time, estimation, review geometric shapes studied in first grade, patterning, etc. daily. This could take place during Math Maintenance time in order to be prepared for future tasks. For additional information on unpacking standards and math maintenance, please see your Grade Level Overview.

Children in second grade are usually familiar with numbers to one hundred and can count and write them with a degree of accuracy. They are beginning to understand the place value system. An important item to facilitate this understanding is the relationship between the numbers and groups of hundreds, tens and ones (for example, the number 142 means one group of one hundred, four groups of ten and two ones). However, students need to understand that place value is not simply how many ones, tens and hundreds there are in a given number. Literally speaking, place value refers to the notion that where a digit is placed in a given number will determine the number’s value. As students understand the significance of the positions of digits in numbers, they can explain the meaning of each digit and its assigned value in each place.

Having a thorough understanding of place value in this manner provides a foundation for operations with numbers. Also, when students know the same number can be represented by different equivalent groupings, they become more flexible with their use of numbers in operations (for example, fifty-three can be represented by five tens and three ones; four tens and thirteen ones; three tens and twenty-three ones; etc.). Taking numbers apart (decomposing) and recombining (composing) them in different ways is a significant skill for computation. Important tools used to develop and extend place value understandings include base ten blocks, tens frames, and 99s charts.

Students need to build on their flexible strategies for adding within 20 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.

A large portion of the second grade standards emphasizes the importance of students developing a solid understanding of the relationship between addition and subtraction. An example of this is when a child uses an addition strategy (counting on) to solve a subtraction problem. For example, how far is it from 16 to 75? You could add 4 to 16 to make 20, and then add 50 to get to 70, and finally 5 more to make the total of 75. The total added to 16 to make 75 is 59 (4 + 50 + 5 = 59). This process of adding on from 16 to get to 75 helps students focus on the distance between the two amounts. Using a linear model of an “open number line” (meaning a line that does not have designated numbers already on it) can help students act out the scenario described above. They can begin at 16, make a jump of 4 to land on 20; make a jump of 50 to land on 70; then a jump of 5 to finally arrive at 75! Totaling up the “jumps” produces the answer of 59. Using this model also helps students develop an understanding and recognize that subtraction can also be thought of as a comparison and not just as taking away, separating, or “subtracting” something.

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.”

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.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

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

STANDARDS FOR MATHEMATICAL CONTENT

Use place value understanding and properties of operations to add and subtract
MGSE2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.

MGSE2.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.

MGSE2.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.

MGSE2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

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:

• Know how to add up to 4 two-digit numbers

• Be able to use a variety of models (base ten blocks- ones, tens, and hundreds only; diagrams; number lines; place value strategies; etc.) to add and subtract within one thousand

• Mentally add or subtract 10 or 100 to a given three-digit number

• Understand the relationship between addition and subtraction (inverse operations)

• Represent three-digit numbers with a variety of different models (base ten blocks- ones, tens, and hundreds only; diagrams; number lines; place value strategies; etc.)

• Recognize and use place value to manipulate numbers

• We can verify the results of our computation by using the inverse operation.

• Estimation helps us see whether or not our answers are reasonable.

• A numeral’s meaning and value is based upon where digits are placed to write the numeral.

• Adding or subtracting ten from a given number changes the digit in the tens place of a given number but not the digit in the ones place of a given number. It also changes the value of the given number by either increasing or decreasing it in increments of ten.

• Adding or subtracting 100 from a given number changes the digit in the hundreds place of that given number but not the digits in the tens and ones places of that given number. It also changes the value of the given number by either increasing or decreasing it in increments of 100.

• Addition means the joining of two or more sets that may or may not be the same size. There are several types of addition problems, see the chart below.

• Subtraction has more than one meaning. It not only means the typical “take away” operation, but also can denote finding the distance between two amounts, i.e. comparison. Different subtraction situations are described in the chart below.

• Numbers may be represented in a variety of ways such as base ten blocks, diagrams, number lines, and expanded form.

• Place value can help to determine which numbers are larger or smaller than other numbers.

• Counting dollars is just like counting by ones and tens in our place value system.

• Counting coins can be connected to how we count by ones, fives, and tens.

• Count with pennies, nickels, dimes, and dollar bills

• Represent a money amount with words or digits and symbols (either cent or dollar signs)

• Interpret data in picture and bar graphs

• Use information from a bar graph to solve addition and subtraction questions and equations

ESSENTIAL QUESTIONS

• How can I keep track of an amount?

• How can I learn to quickly calculate sums in my head?

• How can I use a number line to add or subtract?

• How can I use a number line to figure out 10 more or less than a number?

• How can I use data to help me understand the answers to the questions posed?

• How can place value help us locate a number on the number line?

• How can we select among the most useful mental math strategies for the task we are trying to solve?

• How do we know if we have enough money to buy something?

• How does mental math help us calculate more quickly and develop an internal sense of numbers?

• If we have two or more numbers, how do we know which is greater?

• In what type of situations do we add? In what type of situations do we add?

• In what type of situations do we subtract?

• What are the different ways we can represent an amount of money?

• What are the different ways we can show or make (represent) a number?

• What estimation and mental math strategies can I use to help me solve real world problems?

• What happens to the value of a number when we add 10 to it or subtract 10 from it? What digits change? What digits stay the same? Why?

• What happens to the value of a number when we add or subtract 100 from it? What digits change, what digits stay the same? Why?

• What is an effective way to estimate numbers?

• What is mental math?

• What is the difference between place and value?

• What mental math strategies can we use?

• What strategies are helpful when estimating sums in the hundreds?

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

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

• What type of graph should I use to display data?

• Why do I need to ask questions and collect data?

• Why is it important to be able to count amounts of money?

• Why should we understand place value?

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 MGSE.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.

• 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)

For additional information on the Arc of a Lesson and how to set up an environment to support mathematical thinking see the Grade Level Overview.

In general:

• Students should be actively engaged by providing them with multiple opportunities to develop their own understanding, and encouraged to share their thinking on a regular basis.

• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words. The tasks that address the MGSE for data in 2nd grade are embedded within each of the 2nd grade units.

• Appropriate manipulatives and technology should be used to enhance student learning.

• Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.

• Math journals are an excellent way for students to show what they are learning about a concept. These could be spiral bound notebooks that students draw or write in to describe the day’s math lesson. Second graders love to go back and look at things they have done in the past, so journals could also serve as a tool for a nine-week review, parent conferencing, as well as a tool for assessment.

Specific to the Georgia Standards of Excellence Standards:

Use place value understanding and properties of operations to add and subtract
MGSE2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations.

MGSE2.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.

MGSE2.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.

MGSE2.NBT.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

Instructional Strategies
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 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?

Instructional Strategies
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 problems2 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. For more information on daily routines please see the Grade Level Overview.

Students would then move to making horizontal or vertical bar graphs with two to four categories and a single-unit scale. Use the information in the graphs to pose and solve simple put together, take-apart, and compare problems illustrated in Table 1.

COMMON MISCONCEPTIONS:

Students may think that the 4 in 46 represents 4, not 40. Students need many experiences representing two-and three-digit numbers with manipulatives that group (base ten blocks) and those that do NOT group, such as counters, etc. When adding two-digit numbers, some students might start with the digits in the ones place and record the entire sum. Then they add the digits in the tens place and record this sum. Assess students’ understanding of a ten and provide more experiences modeling addition with grouped and pre-grouped base-ten materials as mentioned above. When subtracting two-digit numbers, students might start with the digits in the ones place and subtract the smaller digit from the greater digit. Then they move to the tens and the hundreds places and subtract the smaller digits from the greater digits. Assess students’ understanding of a ten and provide more experiences modeling subtraction with grouped and pre-grouped base-ten materials.

Students might overgeneralize the value of coins when they count them. They might count them as individual objects. Also some students think that the value of a coin is directly related to its size, so the bigger the coin, the more it is worth. Place pictures of a nickel on the top of five frames that are filled with pictures of pennies. In like manner, attach pictures of dimes and pennies to ten-frames and pictures of quarters to 5 x 5 grids filled with pennies. Have students use these materials to determine the value of a set of coins in cents.

Sometimes students will record twenty-nine dollars as 29$. Remind them that the dollar sign goes in front. The cent sign goes after the number and there is no decimal point used with the cent sign.

The attributes for the same kind of object can vary. This will cause equal values in an object graph to appear unequal. For example, when making an object graph using shoes for boys and girls, five adjacent boy shoes would likely appear longer than five adjacent girl shoes. To standardize the objects, place the objects on the same-sized construction paper or sticky-note, then make the object graph.

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 Standards Glossary. • addition • associative property • bar graph • commutative property • comparing • compose • concrete model counting strategy • decompose • difference • dime• dollar bill • estimate • expanded form • fluency • hundreds • identity property • join • line plot • mental math • model • nickel • ones • penny • picture graph • place value • properties of operations • quantity • quarter • remove • scale • strategy • subtraction • tens