All the resources on this page come from the GADOE Framework 2nd Grade Unit 6 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 6 Overview

In this unit students will:

• Understand and model multiplication as repeated addition and as rectangular arrays.

• Determine if a number is odd or even (within twenty).

• Create and interpret picture graphs and bar graphs.

The standard M.2.OA.3 calls for students to apply their work with doubles addition facts to the concept of odd or even numbers. Van de Walle states, “All too often students are simply told that the even numbers are those that end in 0, 2, 4, 6 or 8 and odd numbers are those that end in 1, 3, 5, 7 or 9. While of course this is true, it is only an attribute of even and odd numbers rather than a definition that explains what even or not even really means” (Teaching Student Centered Mathematics, page 291).

Students should have ample experiences exploring the concept that if a number can be decomposed (broken apart) into two equal addends (e.g., 10 = 5 +5), then that number (10 in this case) is an even number. Students should explore this concept with concrete objects (e.g., counters, place value cubes, etc.) before moving towards pictorial representations such as circles or arrays.

The standard calls for students to use rectangular arrays to work with repeated addition. This is a building block for multiplication in 3rd Grade. Students should explore this concept with concrete objects (e.g., counters, bears, square tiles, etc.) as well as pictorial representations on grid paper or other drawings. Based on the commutative property of addition, students can add either the rows or the columns and still arrive at the same solution. The standard calls for students to work with categorical data by organizing, representing and interpreting data using four categories. Students should have experiences with interpreting and gaining meaning from picture and bar graphs.

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, tallying, and graphing should be addressed on an ongoing basis through the use of calendar, centers, and games. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources can be utilized to supplement this unit. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources.

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. “For additional information on Number Talks please see the Grade Level Overview.

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 develop an understanding of odd/even numbers, arrays, and repeated addition and use that knowledge to solve mathematical problems.

2. Reason abstractly and quantitatively. Students make connections between equal groups of objects and arrays and the concept of adding equal addends.

3. Construct viable arguments and critique the reasoning of others. Students develop and explain strategies for using arrays to solve a variety of mathematical problems.

4. Model with mathematics. Students use a growing understanding of odd/even numbers and model to determine solutions for various mathematical problems.

5. Use appropriate tools strategically. Students use mathematical tools such as number lines, graphs, arrays, and pictures to solve an assortment of problems.

6. Attend to precision. Students use precise mathematical language to communicate an understanding of odd/even numbers, rows, columns, arrays, equal addends, repeated addition, and graphs.

7. Look for and make use of structure. Students look for mathematical patterns using odd/ even numbers, arrays, and repeated addition to create strategies for solving problems.

8. Look for and express regularity in repeated reasoning. Students make connections between how odd/even numbers, arrays, and repeated addition can be used to solve math problems.

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

STANDARDS FOR MATHEMATICAL CONTENT

Work with equal groups of objects to gain foundations for multiplication.
MGSE2.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.

MGSE2.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.

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.

BIG IDEAS

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

• Understand the similarities between skip counting, repeated addition, and multiplication.

• Construct arrays for a given repeated addition sentence.

• Write a repeated addition equation for a given array.

• Determine how the addition sentence for a given array changes when the array is rotated ¼ turn.

• Understand that multiplication is repeated addition.

• Write an equation to express an even number.

• Identify if a number is even or odd by modeling the number in pairs.

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

• Repeatedly adding the same quantity, using a grouping picture or forming a rectangular array are strategies for representing repeated addition equations.

• Arrays are a way of representing both repeated addition and skip counting.

• Arrays should be identified in rows and then columns.

• Explore and be able to explain even and odd numbers while using manipulatives.

• An even number can be decomposed into two equal addends.

• Double addition facts assist in recognizing even numbers.

• Tables and charts can help make solving problems easier.

• Questions can be solved by collecting and interpreting data.

ESSENTIAL QUESTIONS

• How are odd and even number lines identified on the number line?

• How do I determine if a number is odd or even?

• What strategies can I use to tell if a number is odd or even?

• What is odd? What is even?

• How are arrays and repeated addition related?

• How can rectangular arrays help us with repeated addition?

• How can we model repeated addition on the number line?

• How can we a model repeated addition equation with an array?

• How does skip counting help us solve repeated addition problems?

• What is an array?

• What is repeated addition?

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 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 Mathematics Georgia Standards of Excellence State Standards and Model Curriculum, Ohio Department of Education Teaching)

Work with equal groups of objects to gain foundations for multiplication.
MGSE2.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.

MGSE2.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.

Instructional Strategies

Students need to understand that a collection of objects can be one thing (a group), and a group contains a given number of objects. Investigate separating no more than 20 objects into two equal groups. Find objects (the total number of objects in collections up to 20 members) that will have some objects and no objects remaining after separating the collections into two equal groups. Odd numbers will have one object remaining, while even numbers will not. For an even number of objects in a collection, show students the total as the sum of equal addends (repeated addition). For example, 10 objects separated into two equal groups can be represented as 5 + 5 = 10.

Another strategy is for students to think of numbers as a collection of objects. If each object can be paired, has a partner, then it is an even number. If not, the number is odd. For example, students represent the number 6 (even) as XX XX XX (every X has a partner); whereas 7 (odd) is represented as XX XX XX X (one X does not have a partner).

A rectangular array is an arrangement of objects in horizontal rows and vertical columns. Arrays can be made out of any number of objects that can be put into equal rows and columns. Making a connection to real world objects will aid students in differentiating between rows (rows in a garden) and columns (columns holding a roof up). All rows contain the same number of items and all columns contain an equal number of items.

Have students use objects to build all the arrays possible with no more than 25 objects. Their arrays should have up to 5 rows and up to 5 columns. Ask students to draw the arrays on grid paper and write two different repeated addition equations under the arrays: one showing the total as a sum by rows (how many are in each row added by using the number of rows) and the other showing the total as a sum by columns (how many are in each column added by using the number of columns). Both equations will show the total as a sum of equal addends (repeated addition).

To build understanding, teachers should ask students questions such as:

• What direction do rows go? (across)

• What direction do columns go? (up and down)

• How many rows do you see? How many ____ are in each row? What number is repeated? How many times?

• How many columns do you see? How many ____ are in each column? What number is repeated? How many times?

Build on knowledge of composing and decomposing numbers to investigate arrays with up to 5 rows and up to 5 columns in different orientations. For example, form an array with 3 rows and 4 objects in each row. Represent the total number of objects with equations showing a sum of equal addends (repeated addition) two different ways: by rows, 12 = 4 + 4 + 4; by columns, 12 = 3 + 3 + 3 + 3. Rotate the array 90° to form 4 rows with 3 objects in each row. Write two different equations to represent 12 as a sum of equal addends: by rows, 12 = 3 + 3 + 3 + 3; by columns, 12 = 4 + 4 + 4. Have students discuss this statement and explain their reasoning: The two arrays are different (the equations) and yet the same (the sum).

Ask students to think of a full ten-frame showing 10 circles as an array. One view of the ten frame is 5 rows with 2 circles in each row. Students count by rows to 10 and write the equation 10 = 2 + 2 + 2 + 2 + 2, or students count by columns to 10 and write the equation 5 + 5 = 10

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.

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. • addends • addition • array • bar graph • columns • data • equal sharing/forming equal sized groups • equation • even • odd • pairing • picture graph • product • rectangular • rows • scale • sum • total

COMMON MISCONCEPTIONS

(As stated in Teaching Student-Centered Mathematics Developmentally Appropriate Instruction for Grades K-2, Van de Walle, Lovin, Karp, Bay-Williams): “Children must come to understand that errors provide opportunities for growth as they are uncovered and explained. Trust must be established with an understanding that it is okay to make mistakes. Without this trust, many ideas will never be shared.”

Regarding odd and even numbers, students may initially have difficulty “proving” how they know numbers are odd or even. Students may struggle with communicating their growing knowledge using precise mathematical language. Teachers should provide multiple opportunities for students manipulate numbers of objects and express their observations.

With regard to an understanding of arrays, students may at first confuse rows and columns. Children should be provided with numerous chances to make real life connections to examples of rows and columns (some picture cards are provided in this unit). To further support an understanding, teachers may encourage students to use their hands to show the direction rows and columns go. As students move on to explore arrays, they may struggle with creating arrays. For example, students may not know where to begin as they attempt to create an array with 18 objects. Teachers should encourage students to look for equal groups (repeating number patterns) that can be added to make a total of 18. Multiple opportunities will increase student fluency with these tasks. When relating arrays to repeated addition equations, students who do not naturally make connections between the two may be guided towards discovery with questions such as “How many rows do you see in the array” “How many objects are in each row?” “Does each row have the same amount of objects?” “How can you add these numbers to find the total number of objects?”