Common Core State Standards © 2011



Teaching for Conceptual Understanding: Number and Operations in Base Ten

Professional Development

Facilitator Handbook

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Pearson School Achievement Services

Teaching for Conceptual Understanding: Number and Operations in Base Ten

Facilitator Handbook

Pearson provides these materials for the expressed purpose of training district and school personnel on the effective implementation of Pearson products within classrooms, and other professional development topics. These materials may not be used for any other purpose, and may not be reproduced, distributed, or stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without Pearson’s express written permission.

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© 2013 Pearson, Inc.

All rights reserved.

Printed in the United States of America.

ISBN 115500

Facilitator Agenda

Teaching for Conceptual Understanding: Number and Operations in Base Ten

|Section |Time |Agenda Items |

|Introduction |5 minutes |Slides 1–3 |

| | |Welcome |

| | |Agenda |

| | |Outcomes |

|1: Outlining the Progression Story |30 minutes |Slides 4–10 |

| | |Section 1 Big Question |

| | |Mathematical Task |

| | |Activity: What understanding allows you to answer this? |

| | |Relational Understanding |

| | |Outlining the Base Ten Progression |

| | |Activity: What is the story? |

| | |Revisit Section 1 Big Question |

|2: Exploring Number and Operations in Base Ten |60 minutes |Slides 11–22 |

| | |Section 2 Big Questions |

| | |Introducing the Base Ten Number System |

| | |Activity: Base Ten: Agree or disagree? |

| | |Activity: Representing 4 |

| | |Activity: How do you count? |

| | |Activity: Counting Levels of Thinking |

|Break |15 minutes | |

|2: Exploring Number and Operations in Base Ten (continued) |90 minutes |Slides 23–41 |

| | |Progression of Conceptual Understanding of Place Value |

| | |Activity: Conceptual Understanding of Base Ten Representations |

| | |Activity: Teaching Tool: Hundreds Charts |

| | |Video: Estephania’s Place Value Misconceptions |

| | |Video: Procedural Fluency without Conceptual Understanding |

| | |Activity: Summarizing the Place Value Standards |

| | |Continued Progression of Place Value After Grade 2 |

|Lunch |30 minutes | |

|Section |Time |Agenda Items |

|2: Exploring Number and Operations in Base Ten (continued) |115 minutes |Slides 42–58 |

| | |Tools for Teaching Base Ten Conceptual Understanding |

| | |Activity: Tools for Teaching Base Ten Conceptual Understanding |

| | |Reflecting on Place Value in Base Ten |

| | |The Progression of Operations in Base Ten: Addition and Subtraction |

| | |Activity: Invented Strategies and the Standard Algorithms |

| | |Exploring the Operations |

| | |Video: Connor’s Invented Strategies for Addition and Subtraction |

| | |Activity: Linking Invented Strategies to the Standard Algorithms |

| | |Developing Strategies for Multiplication and Division |

| | |The Progression of Operations in Base Ten: Multiplication and Division |

| | |Activity: Build a Deck for Multiplication and Division |

| | |Operations with Decimals |

| | |Reflections: Things to Consider |

| | |Revisit Section 2 Big Questions |

|Break |15 minutes | |

|3: Planning Lessons with Number and Operations in Base Ten |50 minutes |Slides 59–66 |

|Standards | |Section 3 Big Questions |

| | |Strategy-Friendly Environment |

| | |Standards for Mathematical Practice |

| | |Misconceptions |

| | |Video: Freddie’s Misconceptions |

| | |Video: Digging Deeper to Confirm Conceptual Understanding |

| | |Planning Lessons with Numbers and Operations in Base Ten Standards |

| | |Activity: Four-Step Planning Process |

| | |Revisit Section 3 Big Questions |

|Reflection and Closing |10 minutes |Slides 67–73 |

| | |Review Outcomes |

| | |Final Reflection: Taking Action |

| | |Evaluations |

| | |References |

| | |Pearson Legal Statement |

|Total |6 hours | |

Preparation and Background

Workshop Information

Big Ideas

• Within the CCSSM, every standard teachers teach at each grade level directly affects student learning in subsequent grade levels.

• Students must have a solid understanding of place value and base ten to be successful with operations in base ten.

• Teachers should not teach the Number and Operations in Base Ten standards as stand-alone standards; rather, they should teach them in conjunction with other domain standards that are closely related.

• Successful planning involves several key elements so that students reach a conceptual understanding of the content.

Big Questions

• How will the Number and Operations in Base Ten standards progression affect your current teaching practice?

• What ideas are going to be most challenging for your students to master?

• What changes might you need to make in your classroom that will allow your students to feel comfortable creating their own strategies for problems involving operations in base ten?

• What key elements must you be mindful of when planning lessons that involve base ten operations?

• How will your lesson-planning process change to incorporate these key elements?

Assessments of Participants’ Learning during the Workshop

• Creation of the Progression Outline

• Base Ten Reflection

• Things to Consider Reflection

• Tools for Teaching Instruction Planning

• Connections within Base Ten

• Grade-Level Connections

• Section 2 Big Questions

• Making Connections within the CCSSM Chart

• Completion of the Lesson-Planning Template

Assessment in the School or Classroom

• Implement the instructional strategies identified as necessary for the conceptual understanding of base ten operations.

Outcomes

• Articulate the learning progressions necessary for students to conceptually understand base ten operations.

• Identify strategies for helping students build their mathematical understanding of base ten operations.

• Use a planning template to build lessons that strategically support the conceptual development of base ten operations.

• Identify strategies that support simultaneous development of conceptual understanding and problem-solving skills with the intentional use of purposeful student struggle, flexible grouping, and ongoing assessments.

• Articulate common misconceptions as opportunities for students’ conceptual understanding of base ten operations.

Facilitator Goals

• Assist participants in understanding the progression of Number and Operations in Base Ten concepts as outlined in the CCSSM.

• Deepen participants’ conceptual understanding of operations in base ten.

• Guide participants in developing the instructional strategies needed for the conceptual understanding of base ten operations. The strategies include the following:

o The use of manipulatives to create models for learning

o Being mindful of instructional implications and classroom implications for creating a classroom environment for strategy creation

o Using student-invented strategies

o Making connections between and within domain standards

Section 2: Exploring Number and Operations in Base Ten (Slides 11–58)

Time: 265 minutes

Big Questions

• What ideas are going to be most challenging for your students to master?

• What changes might you need to make in your classroom that will allow your students to feel comfortable creating their own strategies for problems involving operations in base ten?

Learning Objectives

• Articulate the learning progressions necessary for students to conceptually understand base ten operations.

• Identify strategies for helping students build their mathematical understanding of base ten operations.

• Identify strategies that support simultaneous development of conceptual understanding and problem-solving skills with the intentional use of purposeful student struggle, flexible grouping, and ongoing assessments.

• Articulate common misconceptions as opportunities for students’ conceptual understanding of base ten operations.

Materials per Section

• Counters

• Brown paper bags

• Highlighters

• Layered Place Value Cards (Layered_Place_Value_Cards.doc in the Additional Resources folder)

• Colored chips

• Dominoes

• Base ten blocks

• Straws

• Rubber bands

• Craft sticks

• Paper cups

• Blank paper

• Markers

• Sticky tack

• Chart paper

• Index cards

• CCSSM documents (CCSSM.pdf and ccss_progression_nbt_2011_04_073.pdf located in the Additional Resources folder)

• Participant Workbook, pages 8–31

|Topic |Presentation Points |Presentation Preview |

|Section 2 Big Questions |Display Slide 11. |[pic] |

| |Explain to participants that they are now going to begin to examine in detail the Number and | |

| |Operations in Base Ten progression of the CCSSM for Grades K–5 and that by the end of this section, | |

| |you would like them to be able to answer the following Big Questions: | |

| |What ideas are going to be most challenging for your students to master? | |

| |What changes might you need to make in your classroom that will allow your students to feel | |

| |comfortable creating their own strategies for problems involving operations in base ten? | |

|Introducing the Base Ten Number System |Display Slide 12. |[pic] |

| |Activity: Base Ten: Agree or disagree? |PW: Pages 8–9 |

| |Direct participants to the Participant Workbook to read the statements and write Agree or Disagree | |

| |in the appropriate column of the Understanding Place Value Anticipation and Reflection Guide. | |

| |When the majority of the group is finished, direct participants to the Participant Workbook to read | |

| |the article titled, “Understanding Place Value” (Van de Walle, Karp, and Bay-Williams 2013). | |

| |Direct participants to complete the Reflection column of the Understanding Place Value Anticipation | |

| |and Reflection Guide in the Participant Workbook. | |

| |Display Slide 13. |[pic] |

| |Activity: Representing 4 |PW: Page 10 |

| |Direct participants to the Participant Workbook to answer the following question: |Chart: Representing 4 |

| |What are some ways you can represent 4? | |

| |Ask for volunteers to share their answers, and record these on chart paper. As you record | |

| |participants’ responses, separate the symbolic and non-symbolic representations into two columns; | |

| |however, do not tell participants why you are recording their responses in one column or another. | |

| | | |

| |Typical responses to this question will include numerals, tally marks, sets of four objects, the | |

| |word four, and geometric quantities that have a measure of four such as 4 inches; a grid with four | |

| |unit squares; a gallon that has a column of four quarts, and so on. These representations will be | |

| |either symbolic or non-symbolic. Non-symbolic representations include manipulatives or pictures of a| |

| |sets of four objects. Symbolic representations include the following: four, 4, 6 – 2, IV, and so on.| |

| | | |

| |After you have captured a large sample of responses, ask participants to tell you the difference | |

| |between the representations in the two columns. If they are unable to tell you the difference, | |

| |explain that one is symbolic and the other is non-symbolic representations. | |

| |Point to the non-symbolic representations and ask participants the following question: | |

| |How do you know that all of these examples are representations of 4? | |

| |Explain to participants that when they created non-symbolic representations of 4, they drew four | |

| |things and then demonstrated that each drawing represented four by counting each of the four things | |

| |that they drew. Point out that to be able to construct the representation and show that it | |

| |represented 4, they needed to be able to classify the things they were counting into a one-to-one | |

| |correspondence. The following slide is an example of this idea. | |

| |Display Slide 14. |[pic] |

| |Direct participants to think about the question on the slide: | |

| |What is the first math concept that students usually encounter at home or in school? | |

| |A participant is likely to answer “counting.” When this occurs, make the segue into the next | |

| |activity. | |

| |Display Slide 15. |[pic] |

| |Point to each of the flowers and say, “One flower, one flower, one flower, one flower. When you | |

| |combine all of these into one group, you get a group of four flowers.” | |

| |Display Slide 16. |[pic] |

| |Activity: How do you count? |PW: Page 10 |

| |Provide each participant with a paper bag with a random quantity of beans or other counters. | |

| |NOTE: Although the quantity is random, there should be at least thirty counters in the bag so as to | |

| |discourage counting by ones and to encourage grouping. | |

| |Direct participants to count their counters using whatever strategies they wish. | |

| |Direct participants to the Participant Workbook to write their final numbers and an explanation of | |

| |their strategies. | |

| |As a whole group, discuss the strategies that they used to count the numbers. Discuss the grouping | |

| |strategies that they used and why participants chose to group the counters in certain ways over | |

| |alternate ways of counting. | |

| |Display Slide 17. |[pic] |

| |Activity: Counting Levels of Thinking |PW: Page 11 |

| |Tell participants that, as in other content areas, there are identifiable levels of thinking related| |

| |to counting, and each level has a set of defining characteristics. | |

| |Direct participants to the Participant Workbook to look at the Learning Trajectory for Counting | |

| |chart. | |

| |Direct participants to do a sort of the Levels of Thinking (on the left side of the chart) by | |

| |working at their tables. | |

| |Explain that you have completed the first two as examples and they should complete the chart. | |

| |Pass out only the “Levels of Thinking” cards at this time (one set per table). Give participants | |

| |time to work with partners to identify the order for the levels of thinking. | |

| |When the majority of the participants have completed sorting the levels of thinking cards, pass out | |

| |the “Characteristics” cards (one set per table), and direct participants to match the | |

| |characteristics with the appropriate level of thinking. | |

| |Remind participants that once they read the characteristics, they may choose to reorder their levels| |

| |of thinking cards. | |

| |The key for this activity is on the following slide (Levels of Thinking Counting_Key.pdf), so that | |

| |participants can check their sorts. This is the same key that you use to make the “Levels of | |

| |Thinking” and “Characteristics” cards. | |

| |Display Slide 18. |[pic] |

| |Discuss the key for this activity (Levels of Thinking Counting_Key.pdf), so participants can check |PW: Page 11 |

| |their sorts. At this point, participants can write the correct levels of thinking and the | |

| |characteristics of each in the Participant Workbook. | |

| |Display Slide 19. |[pic] |

| |Ask participants the following question: |PW: Page 12 |

| |What happens when you want to do more than count quantities? | |

| |Direct participants to the Participant Workbook. | |

| |Give participants a few moments to solve the problem in the Participant Workbook, without using | |

| |paper or pencil, and then have them share their answers with the rest of their table groups. | |

| | | |

| |Briefly discuss with participants what they needed to know and be able to do to answer this problem.| |

| |Explain to participants that when solving this problem, they were able to work the problem because | |

| |they knew how to do mental math. And, to do mental math, they were able mentally to represent | |

| |numbers symbolically and knew something about place value.  | |

| |Historians of mathematics point out that the shift from non-symbolic representations of numbers to | |

| |symbolic representations of numbers was a major advance in mathematical thinking. Symbolic | |

| |representations allowed people to represent larger quantities more easily. For example, in early | |

| |societies, people paid taxes in the form of crops and animals and thus required a system to keep | |

| |inventory of all of the goods collected. Simply drawing the items or using tally marks became | |

| |unwieldy quickly. Indeed, as the need for more sophisticated counting and measuring emerged, more | |

| |sophisticated methods of expressing quantities developed. | |

| |Have participants think about the problem that they just worked. Have them think about a way to | |

| |solve this problem by using non-symbolic representations. Ask them to think for only a brief moment | |

| |about this, because a non-symbolic representation would be unwieldy. The purpose here is to make the| |

| |point that the base ten number system was built for efficiency. | |

| |Display Slide 20. |[pic] |

| |The symbolic number system is the base ten system. Ask participants to summarize what they now know | |

| |about the base ten number system: | |

| |Why it is called a base ten system? | |

| |If the answers do not come to light in the discussion, bring up the following (The Common Core | |

| |Standards Writing Team 2011, 2–3): | |

| |This system represents every number in the base ten system by using only ten digits: 0, 1, 2, 3, 4, | |

| |5, 6, 7, 8, and 9. | |

| |The numbers are represented as a string of digits, each digit representing a value that depends on | |

| |its placement in the string. | |

| |The relationship between the values represented in the base ten system is the same for whole numbers| |

| |and decimals. | |

| |Display Slide 21. |[pic] |

| |Discuss with participants that the values of 4, 40, 400, and 4,000 are distinguished by the | |

| |placement of the 4 in each numeral. Participants can do this because they know that each place value| |

| |has a value 10 times greater than the one to its immediate right or that each place has a value 10 | |

| |times less than the one to its immediate left. For example | |

| |10 is ten times greater than 1; | |

| |100 is ten times greater than 10; and | |

| |1000 is ten times greater than 100. | |

| |Display Slide 22. |[pic] |

| |Explain that, “Each place of a base ten numeral represents a base ten unit” (The Common Core |PW: Page 12 |

| |Standards Writing Team 2011, 2–3). |Chart: Base Ten Units Examples |

| |Ask participants for examples of base ten units, and record responses on chart paper. For example, |Chart: True Statements about Base Ten |

| |ones, tens, tenths, hundreds, hundredths, and so on. | |

| |Direct participants to the Participant Workbook to: | |

| |Write as many true statements about base ten as you can in two minutes. | |

| |NOTE: An example of a statement would be, “One hundred is ten times ten.” | |

| |Have pairs share some of their statements, and record their the statements on chart paper. | |

| |Wrap up the discussion by using the statement examples to drive home the point that, “The digit in | |

| |the place represents 0 to 9 of those units. Because ten like units make a unit of the next highest | |

| |value, only ten digits are needed to represent any quantity in base ten” (The Common Core Standards | |

| |Writing Team 2011, 2). | |

| | | |

| |Explain that although the base-ten system is very efficient and uniform, it can be a very complex | |

| |topic for young students to grasp. Explain further that the progression of the CCSSM pertaining to | |

| |Number and Operations in Base Ten will help them unravel the complexity for students and make it | |

| |easier to understand and to build success with its operations. | |

| |To lead into the next topic, ask the questions, rhetorically, “What, at a specific grade level, do | |

| |students need to understand? How can teachers help students gain a conceptual understanding that | |

| |will build a strong foundation for working within the base ten number system?” | |

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