All the resources on this page come from the GADOE Framework 2nd Grade Unit 1 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 1 Overview

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview. In this unit, students will:

• understand the value placed on the digits within a three-digit number

• recognize that a hundred is created from ten groups of ten

• use skip counting strategies to skip count by 5s, 10s, and 100s within 1,000

• represent numbers to 1,000 by using numbers, number names, and expanded form

• compare two-digit number using >, =, <

Students extend their understanding of the base-ten system by viewing 10 tens as forming a new unit called a hundred. This lays the groundwork for understanding the structure of the base-ten system. 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).

The extension of place value also 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 begin to work towards developing an understanding of multiplication when they skip by 5’s, by 10’s, and by 100’s. While skip counting is not yet true multiplication, it does provide students with opportunities to connect counting to repeated addition. Providing students with contextual situations and opportunities to model these situations with arrays to quantify skip counting gives students opportunities to not only skip count, but also begin to quantify the groups they count. This is the missing link between skip counting and multiplication.

Representations such as manipulative materials, math drawings, and layered three-digit place value cards provide connections between written three-digit numbers and hundreds, tens, and ones. Numbers, number words, and expanded notation can be represented with drawings, place value cards, and by saying numbers aloud and in terms of their base-ten units, e.g. 456 is “four hundred fifty-six” and “four hundreds, five tens six ones.” Students should also develop flexible understanding of place value. For example, 456 can also be represented as 3 hundreds, 15 tens, 6 ones, etc.

Comparing magnitudes of two-digit numbers draws on the understanding that 1 ten is greater than any amount of ones represented by a one-digit number. Comparing magnitudes of three digit numbers draws on the understanding that 1 hundred (the smallest three-digit number) is greater than any amount of tens and ones represented by a two-digit number. For this reason, three-digit numbers are compared by first inspecting the hundreds place (e.g. 845 > 799; 849 < 855).

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.”

**When first beginning Number Talks in your classroom it is suggested that teachers consult the numerous resources found in the Grade Level Overview and in the Effective Instructional Practices guide, found here: https://www.georgiastandards.org/GeorgiaStandards/Documents/GSE-Effective-Instructional-Practices-Guide.pdf.

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 explain to themselves the meaning of a problem and look for ways to solve it. They may use concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking themselves, “Does this make sense?” They make conjectures about the solution and plan out a problem-solving approach.

2. Reason abstractly and quantitatively. Students are linking concrete representations of quantity (such as base 10 blocks or groupable models) to a variety of abstract representations, such as expanded form and multiple numerical representations of hundreds, tens, and ones.

3. Construct viable arguments and critique the reasoning of others. In this unit, teachers set the stage for students to be able to construct arguments, defend answers, and listen to the reasoning of others. Number Talks are an excellent way to set the stage for this.

4. Model with mathematics. In second grade, students will represent numbers in word form, expanded form, standard form, and with base ten blocks. They will understand that all of these represent the same number. Further, students understand that there can be multiple ways to represent the same number (19 tens is equal to 190 or 1 hundred and 9 tens).

5. Use appropriate tools strategically. Tools students use throughout this unit include number lines, hundreds charts, and base ten blocks. Students who use a number line strategically have progressed from counting by ones on a number line or hundreds chart to solving problems making leaps of tens. A further progression involves grouping tens and making leaps of 20, 30, or all the tens represented in a problem.

6. Attend to precision. Students will use vocabulary precisely. They will also be able to discuss and represent a number in multiple ways.

7. Look for and make use of structure. Students will look for patterns on a hundreds chart and use base ten blocks to make sense of numbers.

8. Look for and express regularity in repeated reasoning. Students will develop reasoning strategies for comparing three digit numbers. When children have multiple opportunities to add and subtract “ten” and multiples of “ten” they notice the pattern and gain a better understanding of place value. Students continually check their work by asking themselves, “Does this make sense?”

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

STANDARDS FOR MATHEMATICAL CONTENT

Understand Place Value
MGSE2.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. 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).

MGSE2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s.

MGSE2.NBT.3 Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.

MGSE2.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.

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:

• Use models, diagrams, and number sentences to represent numbers within 1,000.

• Write numbers in expanded form and standard form using words and numerals.

• Identify a digit’s place and value when given a number within 1,000.

• Compare two 3-digit numbers with appropriate symbols (<, =, and >).

• Understand and explain the difference between place and value.

• The value of a digit depends upon its place in a number.

• Understand the digit zero and what it represents in a given number.

• Numbers can be represented in many ways, such as with base ten blocks, words, pictures, number lines, and expanded form.

• Place value determines which numbers are larger or smaller than other numbers.

• Explain how place value helps us solve problems.

ESSENTIAL QUESTIONS

• Why should we understand place value?

• What is the difference between place and value?

• How does place value help us solve problems?

• How does the value of a digit change when its position in a number changes?

• What does “0” represent in a number?

CONCEPTS AND SKILLS TO MAINTAIN

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;

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

Second Grade Year Long Concepts:

• Organizing and graphing data as stated in MGSE2.MD.10 should be 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.

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

STRATEGIES FOR TEACHING AND LEARNING (Information adapted from Mathematics Common Core State Standards and Model Curriculum, Ohio Department of Education, Van de Walle’s Teaching Student-Centered Mathematics)

Place Value Instructional Strategies

The understanding that 100 is equal to 10 groups of ten or 100 ones is critical to understanding base-10 place value. Using proportional models such as base-ten blocks or bundles of tens along with place-value mats helps create connections between the physical and symbolic representations of a number and its magnitude. These models can build a stronger understanding when comparing two quantities and identifying the value of each place value position. Van de Walle (p.127) notes that “the models that most clearly reflect the relationship of ones, tens, and hundreds are those for which the ten can actually be made or grouped from single pieces.” Groupable base ten models can be made from beans and cups, bundled straws or craft sticks, unifix cubes, etc. If children are struggling with base ten blocks, you may consider using number cubes or inexpensive homemade manipulatives to help develop their understanding.

Model three-digit numbers using base-ten blocks in multiple ways. For example, 236 can be 236 ones, or 23 tens and 6 ones, or 2 hundreds, 3 tens and 6 ones, or 20 tens and 36 ones. Use activities and games that have students match different representations of the same quantity. Provide games and other situations that allow students to practice skip-counting. Students can use nickels, dimes and dollar bills to skip count by 5, 10 and 100. Pictures of the coins and bills can be attached to models familiar to students: a nickel on a five-frame with 5 dots or pennies and a dime on a ten-frame with 10 dots or pennies. This attachment supports development of the ability to unitize, or maintain the idea that a single item (a 10 dollar bill, for example) can represent 10 singles (one dollar bills).

On a number line, have students use a clothespin or marker to identify the number that is ten more than a given number or five more than a given number.

Have students create and compare all the three-digit numbers that can be made using digits from 0 to 9. For instance, using the numbers 1, 3, and 9, students will write the numbers 139, 193, 319, 391, 913 and 931. When students compare the digits in the hundreds place, they should conclude that the two numbers with 9 hundreds would be greater than the numbers showing 1 hundred or 3 hundreds. When two numbers have the same digit in the hundreds place, students need to compare their digits in the tens place to determine which number is larger.

Common Misconceptions with Place Value:
(Information adapted from Mathematics Navigator: Misconceptions and Errors, America’s Choice)

1. Some students may not move beyond thinking of the number 358 as 300 ones plus 50 ones plus 8 ones to the concept of 8 singles, 5 bundles of 10 singles or tens, and 3 bundles of 10 tens or hundreds. Use base-ten blocks to model the collecting of 10 ones (singles) to make a ten (a rod) or 10 tens to make a hundred (a flat). It is important that students connect a group of 10 ones with the word ten and a group of 10 tens with the word hundred. This unitizing understanding is critical to the use of $1, $10, and $100 bills as models.

2. When counting tens and ones (or hundreds, tens, and ones), the student misapplies the procedure for counting on and treats tens and ones (or hundreds, tens, and ones) as separate numbers. When asked to count collections of bundled tens and ones such as 32, student counts 10, 20, 30, 1, 2, instead of 10, 20, 30, 31, 32.

3. The student has alternative conception of multi-digit numbers and sees them as numbers independent of place value. Student reads the number 32 as “thirty-two” and can count out 32 objects to demonstrate the value of the number, but when asked to write the number in expanded form, she writes “3 + 2.” Student reads the number 32 as “thirty-two” and can count out 32 objects to demonstrate the value of the number, but when asked the value of the digits in the number, she responds that the values are “3” and “2.”

4. The student recognizes simple multi-digit numbers, such as thirty (30) or 400 (four hundred), but she does not understand that the position of a digit determines its value. Student mistakes the numeral 306 for thirty-six. Student writes 4008 when asked to record four hundred eight.

5. The student misapplies the rule for reading numbers from left to right. Student reads 81 as eighteen. The teen numbers often cause this difficulty.

6. The student orders numbers based on the value of the individual digits, instead of on the value of the digit based on its place in the number. 69 > 102, because 6 and 9 are bigger than 1 and 2.

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 Mathematics Glossary.

• >, =, and < comparison

• digit

• expanded form

• models

• number line

• number names

• place value

• skip-count

• base ten model

• flat

• rod

• units

Consider having a math word/vocabulary wall that includes both visuals (pictorial support) and definitions. Link to Math Vocabulary Word Wall Cards with pictorial support. If in doubt about any of these definitions, please consult the glossary found on www.georgiastandards.org.