Course Information

1. View an Introductory Video
   Watch a short video about the Mathematics Intervention Toolkit and PD Course.

2. Read a Course Overview
   The Toolkit Overview (1 MB) provides a 2-page introduction to the course.
3. Get Answers to FAQs
   Select the lettered sections below to read answers to frequently asked questions.

   A. Who should take this PD course?

   The primary audience is teachers of mathematics intervention in grades 3–6. This includes teachers with different roles, such as interventionists, Title I teachers, mathematics specialists, general educators, and special educators. Participants will be able to apply the strategies in a variety of intervention settings, including separate intervention classes; intervention blocks, such WIN (What I Need) blocks; and designated times for intervention during general education classes.
   
   The course is designed for groups of teachers from the same district to take the course together so they can collaborate and support each other's learning. We strongly encourage mathematics coaches to take the course with teachers so that they can support planning and implementation of strategies.

   B. What will participants do in the PD course?

   The PD course has a hybrid format that combines online sessions, Professional Learning Community (PLC) sessions, and classroom implementation.

   • Learn individually in the online sessions about the WWC Practice Guide's recommendations by using online activities and resources, including videos, readings, and math apps.
   • Learn together with colleagues by participating in the PLC sessions (in-person or virtual). Discuss the recommendations, try strategies, and prepare to use instructional routines with students.
   • Implement with students by using instructional routines that incorporate evidence-based strategies. These Try It! Routines provide ready-to-use resources and options to support implementation with your students.
   • Debrief together with colleagues about your teaching experiences with the routines and discuss ways to strengthen implementation at the PLC sessions.
   • Reflect individually about your learning and plan actions to strengthen knowledge and practices.

   C. How does the course support teachers in implementing strategies with students?

   A top priority of the course is to support teachers in implementing evidence-based strategies with their students. The course provides five ready-to-use instructional routines that incorporate recommended strategies and focus on key mathematics content. The Professional Learning Community (PLC) sessions provide a supportive, collaborative structure for teachers to learn the routines, prepare to implement them with students, and debrief their experiences.

   D. How many PD hours does the course provide?

   The course is designed to provide about 28 PD hours. Districts can choose to implement six modules during one school year or three modules each year for two years. This design reflects research recommendations that effective PD involves active learning and should be sustained over time to support classroom implementation. The Introductory Module and Module 5 are each about 1 week long. Modules 1-4 are each about 5 weeks long.

   • Introductory Module     2 hours
   • Modules 1 – 4              24 hours (6 hrs. per module)
   • Module 5                      2 hours
   •                           Total: 28 hours

   Note that the exact number of PD hours will vary depending on district requirements and how the course is implemented. For example, a district might choose to increase or decrease the time for the modules' PLC sessions to fit their needs. Check with your course facilitator for information about the exact number of PD hours.

   E. What mathematics content does the course focus on?

   The course focuses on key Number and Operations topics, such as fractions, that are a high priority for mathematics intervention. Building a strong foundation with fractions is critical for students' success with grade-level mathematics and future classes. The modules provide example activities for supporting students' learning of fraction magnitude, representations, equivalence, comparison, and operations. In addition, there are activities on place value, multiplication and division of whole numbers and on decimal concepts and operations.

   F. Will the course work with different intervention programs?

   The course is designed to work with a wide variety of mathematics intervention programs. Teachers will learn strategies that can be applied with their intervention program. They will discuss how and when to implement the strategies in their sequence of instruction to support their students' learning.
   
   It's not necessary for districts to have a specific intervention program or curriculum to implement the course. The course is designed to support districts that have different intervention curricula and those that have not adopted a program. The modules focus on recommended strategies that are applicable across mathematics content topics.

   G. How can an individual educator use the PD modules?

   While the recommended course format is to have district-based PLC groups, this may not be an option for some educators. Individual educators can use the modules' Online Sessions to learn about the recommendations on their own. The Online Sessions have readings, videos, math activities, check-for-understanding questions, and other resources.
   
   Use the Teachers' PD Course Modules menu to select a module: 1) Mathematical Language, 2) Representations, 3) Number Lines, or 4) Word Problems. Then select the Explore-A and Explore-B tabs to access the online activities.

Kick-Off Session

KEY ACTIVITIES

The handouts are available in the Introductory Module Participant Workbook (1 MB). Resources for key activities are provided below for access during the session and/or to revisit them afterwards.

1. Learn about the Goals
   Write goals for your professional learning on the handout Course Goals (H1).
2. Preview the Recommendations
   Use the handout Overview of Recommendations (H2) to start learning about the five recommendations you will explore in the course.
3. Try an Instructional Routine
   Try an instructional routine called Sort, Explain, and Generalize that has strategies from three recommendations: Mathematical Language, Representations, and Systematic Instruction. Sort fractions by comparing them to benchmark numbers. The directions are below and on the handout Try an Example Routine (H3). Use printed fraction cards or a virtual version (button below).

   First, get to know the routine in the role of students by sorting the cards and using the sentence starters, and then discuss it from a teaching perspective.

   • Work in pairs and take turns sorting fraction cards into three categories: Closer to 0, Closer to , or Closer to 1.
   • When you place a card explain your reasons by using this sentence starter:
     • "The fraction is closer to ___ because…"
   • Then, your partner responds:
     • "I agree because…" OR: "I disagree because…"

   Start the SortingSorting: Try Again

4. Watch a Classroom Example
   View a video of an intervention teacher and students using the routine. Take notes on the handout Video Observations (H4) and then discuss:
   • How did the teacher support students in sorting the fractions and explaining their thinking?
   • What did you notice about students' understanding of and challenges with comparing fractions to benchmark numbers?
   • What ideas from the video would you like to apply?

Wrap-Up

Summarize key ideas and reflect on your learning.

1. Reflect on Learning
   Reflect on your learning at the Kick-Off Session by completing the Reflection handout (H7). This form is designed to support professional learning and is not evaluative.

Resources

Get links to resources for the module.

Explore the Recommendation (Part A)

Activities

Download the Module 1 Participant Workbook (4 MB) for access to the handouts. The handouts are labeled with a letter H and a number, such as H1.

1. What Is the Recommendation?
   Learn about the recommendation from Dr. Karen Karp, professor of mathematics education and a member of the expert author panel that wrote the Practice Guide. Write notes on the Reference Sheet (H1) to create a helpful resource to use in the future.

2. Why Is Mathematical Language Important?
   Read about the importance of mathematical language by selecting the lettered sections below. Then, add your ideas to the Reference Sheet (H1), question 2.

   A. Why is understanding mathematical language important for student learning?

   • Helps students to build understanding of mathematical concepts/processes.
   • Helps teachers and students communicate more clearly with each other.
   • Helps students to access language used in instruction during intervention and in their core mathematics class.
   • Prepares students for the language they will be using in future mathematics instruction.
   • Add your ideas on the Reference Sheet (H1).

   B. Standards for mathematical practice

   Understanding mathematical language is integral to most states' Standards for Mathematical Practice or Process Standards, particularly related to attending to precision. Here's what many states use: "Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately."

   Source: corestandards.org

   C. Standards for mathematical content

   Throughout most states' standards, using mathematical language in writing and speaking is essential to building understanding of concepts and processes. Students are expected to describe, compare, categorize, reason, justify, and explain their understanding, approaches, and solutions.

   Source: corestandards.org

3. What Are Recommended Strategies?
   Learn about evidence-based strategies by reading the handout How to Carry Out the Recommendation (H2). This handout includes an excerpt from the WWC Practice Guide.
4. View a Classroom Example
   Watch a classroom video to see an example of strategies in action. As you watch, take notes on the handout Video Observations (H3) about the focus questions:
   1. What strategies for mathematical language did you notice in the video?
   2. What are one or two ideas from the video that stood out for you?

5. Check for Understanding
   Answer two questions about the recommendation and get feedback.

   Start the Knowledge CheckKnowledge Check: Download Summary

Explore the Recommendation (Part B)

1. Try Graphic Organizers
   Graphic organizers are a helpful tool for depicting and reinforcing the meaning of vocabulary terms. Try this approach yourself by completing the examples on the handout Vocabulary Strategy: Use Graphic Organizers (H4).

   

2. Sort Mathematical Terms
   Teachers need to use clear, concise, and correct mathematical terms throughout instruction to build and reinforce students' understanding of mathematical language. It's helpful to identify: Which words should you use? Which words should you avoid?

   Do a PD activity for teachers by selecting the button below.

   Start the SortingUse Precise Mathematical Language: Try Again

   Sources: Terms from WWC Practice Guide and Karp et al., 2014; Activity from Toolkit Authors

3. Consider Potential Challenges
   Read about possible obstacles and the expert panel's advice in the handout Challenges and Suggestions (H5).
4. Check for Understanding
   Do a short activity to check your understanding of the recommendation. Answer two multiple choice questions and get immediate feedback.

   Start the Knowledge CheckKnowledge Check: Download Summary

5. Reflect
   Reflect on your learning about the recommendation for mathematical language. Write responses to the prompts on the handout Reflection for Online Session (H6).

PLC Session A

BEFORE THE SESSION

1. Get Ready for the Session
   • Make sure to complete the online activities (Explore-A and Explore-B), so that you will be prepared to discuss the recommendation.
   • You will need to access handouts (H7 - 10 and R1 - 3) from the Module 1 Participant Workbook (4 MB)

AT THE SESSION

Resources for key activities are provided below for access during the session and/or to revisit them afterwards.

2. Build and Reinforce Vocabulary
   Use the card sorting routine to build students' understanding of the terms, "unit fraction" and "non-unit fraction." Directions are on the handout Card Sorting Routine: Unit Fractions (H7).
   
   First, get to know the activity in the role of students by sorting the fraction cards and using the sentence starters, and then discuss it from a teaching perspective.
3. Discuss a Video Example
   You and your PLC colleagues will rewatch the classroom video together. Take notes on the Video Discussion handout (H8) and then discuss.
   • How does the teacher engage and support students in talking about their ideas and using math vocabulary words?
   • What do you notice about the students' explanations?
   • What are one or two ideas from the video that you would like to try with students?

4. Walk Through the Instructional Routine
   You and your colleagues will try the routine together. Use the handout Walk-Through of Routine: Script (H9) to take turns in the roles of "teacher" and "students."
5. Discuss Ideas and Start Planning
   Share suggestions for using the routine with students. Use these handouts in the Routine Teaching Guide section of the Participant Workbook. Resources for the routine are labeled with the letter R and a number.
   • 2-Page Overview of Routine (R1).
   • Planner (R2).
   • Suggestions for Addressing Potential Challenges (R3).

Use the Try It! Routine

You've reached a key step in the module: implementing Mathematical Language strategies with your students.

1. Read about the Instructional Routine
   In this routine, the teacher presents a Which One Doesn't Belong (WODB) puzzle to engage students in communicating mathematical ideas. The WODB puzzles show four related but different numbers. Students are asked to consider the four possibilities, select one that doesn't belong with the others, and explain their reasoning for why it is different. The puzzles are designed so that there are reasons why each choice doesn't belong, which allows students to identify different reasons and add to the discussion.

   Read answers to Frequently Asked Questions (FAQs) by selecting the lettered sections below.

   A. What recommended strategies does the routine incorporate?

   The routine focuses on the WWC Practice Guide recommendation for mathematical language, implementation step 3: "Support students in using mathematically precise language during their verbal and written explanations of their problem solving." The routine incorporates these strategies:

   • Sentence starters and frames provide a starting point for students to explain their ideas.
   • Questioning and rephrasing to elicit and reinforce students' use of mathematical language.
   • Mathematical language chart supports students' use of relevant vocabulary terms.
   • Partner work and different discussion formats provide opportunities and support for students to share ideas.
   • Puzzles that have multiple responses so students can share different ideas.
   • Response cards for students to choose an answer (A, B, C, or D), hold it up, and then explain their reasons.

   B. What mathematics content does it focus on?

   This versatile routine can be used with a variety of math topics. The Routine Teaching Guide offers a choice of seven puzzles on the topics of whole number place value, fractions, and decimals. Alternatively, you can focus on different math topics by using puzzles from other sources or by creating your own. 

   C. How do you integrate the routine with your intervention program?

   The routine is designed to reinforce the use of mathematical language and to support students in strengthening their communication skills. Opportunities for students to discuss their ideas in pairs, small groups, and with the whole group are important and powerful for math intervention class.

   The routine is about 30 minutes long and can be used in one lesson. It works well to integrate the routine into a unit by choosing a puzzle on a relevant math topic. The puzzles can be used to reinforce and review content from prior lessons. The Routine Teaching Guide provides puzzles on the topics of whole number place value, fractions, and decimals. If your curriculum includes WODB puzzles, you can use those puzzles with the routine.

2. Prepare to Use the Routine
   Use the following resources to prepare for teaching the routine:
   • Routine Teaching Guide in the Module 1 Participant Workbook (4 MB).
   • Classroom Video Example on the Explore-A tab or PLC-A tab.
   • Optional: Module 1 Optional Teaching Slides for Routine (2 MB)
3. Implement with Students
   Use the routine one or more times with students. Collect student work samples to show or describe in your slides when you share at PLC Session B.
4. Make Slides to Share at PLC Session B
   Use the Debriefing Slides Template (264 KB) to prepare six slides about your teaching experiences. Make sure to respond to all five questions on the slides template.

PLC Session B

Share experiences using the routine and discuss ideas for strengthening implementation.

BEFORE THE SESSION

1. Get Ready for PLC Session B
   Make sure to prepare a short presentation to share your teaching experiences.
   • Use the Debriefing Slides Template (264 KB) to create 6 slides on the debriefing questions.
   • At the session, you will need to access handouts (H11-13) from the Module 1 Participant Workbook (4 MB).

AT THE SESSION

2. Use the Debriefing Protocol
   You and your colleagues will take turns sharing your teaching experiences. The Debriefing Protocol (H11) provides a structured process for collaborative debriefing.

Debriefing Protocol: Overview

Goal: Debrief and learn from our collective experiences using an instructional routine.

Part 1: Share Experiences

• Presenter describes experiences and shows slides (~6 minutes).
• Timekeeper gives a 1-minute warning to presenter and says when time is up.
• Group members listen carefully during the presentation (avoid interruptions). Then, they can ask questions and note ideas to discuss in Part 2.

Part 2: Group Discussions

• Discuss common themes.
• Identify ideas to apply.
3. Recap Strategies in Module
   Write down ideas for each question. Then, select the lettered bars to see example lists*. For future reference, a copy of the lists is on the handout Recap Strategies for Mathematical Language (H12).

   A. What are recommended strategies for mathematical language?

   • Provide a variety of examples of new math vocabulary words.
   • Use concrete and semi-concrete representations to build understanding of math vocabulary words and to help students explain their ideas.
   • Connect math vocabulary words to concrete, semi-concrete, and abstract representations.
   • Use graphic organizers for math vocabulary (such as to link definitions, characteristics, examples, and/or non-examples).
   • Use clear, correct, and consistent mathematical language during instruction.
   • Use sentence starters to support students in explaining their ideas orally and in writing.
   • Use math vocabulary charts and/or word walls to support students' use of math language.
   • Use questioning strategies to support students in explaining their ideas.
   • Provide consistent opportunities for students to discuss and explain math concepts and strategies.
   • Use partner discussions to support students in sharing ideas and listening to others' ideas.

   B. What approaches should you avoid?

   • Avoid teaching vocabulary words in isolation. Instead, integrate vocabulary into math instruction to help students build meaning through context.
   • Avoid using informal, catchy terms in place of formal math vocabulary. Instead, consistently use precise, formal mathematics vocabulary and support students in building understanding of this terminology. The use of imprecise, informal vocabulary can cause students confusion when they encounter the precise language in other classes or contexts.

   *These lists are not exhaustive.

4. Reflect and Plan Next Steps
   Discuss ways to strengthen current practices and identify new ideas to apply. Use the handout Strengthen Strategies (H13).

Wrap-Up

Summarize key ideas and reflect on your learning about Mathematical Language.

1. Summarize and Synthesize
   Think about: What ideas do you want to make sure to remember and apply? Add ideas to the Reference Sheet for Mathematical Language (H1) so that you will leave the module with a helpful resource for future use.
2. Reflect on Learning
   Complete the Self-Reflection Form (H14) to reflect on your learning about the recommendation for mathematical language. This form's purpose is to provide opportunities for self-reflection to support professional learning and is not evaluative.

Resources

Get links to resources for the module and to extend your learning.

Explore the Recommendation (Part A)

Activities

Download the Module 2 Participant Workbook (4 MB) for access to the handouts. The handouts are labeled with a letter H and a number, such as H1.

1. What Is the Recommendation?
   Learn about the recommendation from Dr. Barbara Dougherty, a member of the expert author panel that wrote the Practice Guide. Write notes on the Reference Sheet (H1).

2. Why Is It Important?
   Read about reasons for using concrete, semi-concrete, and abstract representations by selecting the lettered sections below. Then, add your ideas to the Reference Sheet (H1), question 2.

   A. Why is understanding representations important?

   • Helps students to build understanding of mathematical concepts and procedures
   • Helps students to represent and make sense of problem situations
   • Helps students to show and explain their approaches
   • Add your ideas on the Reference Sheet (H1)

   B. Standards for mathematical practice

   Understanding representations is integral to most states’ Standards for Mathematical Practice or Process Standards, particularly the following:

   • Make sense of problems and persevere in solving them;
   • Model with mathematics; and
   • Use appropriate tools strategically.

   Source: corestandards.org

   C. Standards for mathematical content

   Throughout most states’ content standards, using representations is essential to building understanding of concepts and processes. Here are some examples:

   • Illustrate and explain calculations for multiplication and division problems by using equations, rectangular arrays, and/or area models;
   • Represent and compare fractions by using visual fraction models and numeric fractions;
   • Represent and solve word problems involving fraction operations by using visual fraction models and equations; and
   • Add, subtract, multiply, and divide decimals using concrete models or drawings and strategies based on place value.

   Source: corestandards.org

3. What Are Recommended Strategies?
   Learn about evidence-based strategies by reading the handout How to Carry Out the Recommendation (H2). This handout includes an excerpt from the WWC Practice Guide.
4. View a Classroom Example
   Watch an example of the strategies in action. Take notes on the Video Observations handout (H3) about the focus questions:
   1. What instructional strategies for representations do you notice in the video?
   2. What are one or two ideas from the video that stood out for you?

5. Check for Understanding
   Answer three multiple-choice questions about the recommendation.

   Start the Knowledge CheckKnowledge Check: Download Summary

Select Next to continue learning about the recommendation on the Explore-B tab.

Explore the Recommendation (Part B)

1. Explore Decimal Addition with Base Ten Blocks
   Do two activities to learn about this strategy:
   1. Watch a video about the benefits of base ten blocks and how to use them.

   2. Solve problems on the Decimal Addition handout (H4).
2. Explore Decimal Subtraction with Base Ten Blocks
   Apply similar strategies to subtract decimals.
   1. Watch a short demonstration video.

   2. Solve problems on the Decimal Subtraction handout (H5).
3. Consider Potential Challenges
   Read about possible obstacles and the expert panel's advice in the handout Challenges and Suggestions (H6).
4. Check for Understanding
   Answer two multiple-choice questions about the recommendation.

   Start the Knowledge CheckKnowledge Check: Download Summary

5. Reflect
   Reflect on your learning by completing the Reflection for Online Session (H7).

PLC Session A

BEFORE THE SESSION

1. Get Ready for the Session
   • Make sure to complete the online activities (Explore A & B Tabs), so that you will be prepared to discuss the recommendation for representations.
   • At the session, you will need to access handouts (H8-11 and R1-3) from the Module 2 Participant Workbook (4 MB).

AT THE SESSION

Resources for key activities are provided below for access during the session and/or to revisit them afterwards.

2. Discuss a Video Example
   You and your PLC colleagues will rewatch and discuss the classroom video example. Take notes on the Video Discussion handout (H9) for these focus questions:
   • What do you notice about how the teacher supports students with explaining their ideas and using the representations?
   • What do you notice about how students: a) explain why the statement is false? b) change the statement to make it true?
   • What are one or two ideas from the classroom video that you would like to try with your students? Why?

3. Walk Through the Instructional Routine
   You and your colleagues will try the routine together. Use the handouts Example Problem (H10) and Script (H11).
4. Discuss Ideas and Start Planning
   Discuss suggestions for using the routine. The Routine Teaching Guide (in the Participant Workbook) has resources to help you plan:
   • 2-Page Overview of Routine (R1)
   • Planner (R2)
   • Suggestions for Addressing Potential Challenges (R3)

Use the Try It! Routine

You’ve reached a key step in the module: implementing strategies with your students.

1. Read about the Instructional Routine
   This routine supports students in using concrete, semi-concrete, and abstract representations to determine if a statement about quantities is true and to explain and record their reasoning. For example, in this routine, students are asked “Is it True? 0.26 > 0.3.” First, they can compare the two quantities by using base ten blocks or by shading decimal grids. Next, they can use their concrete or semi-concrete representation to show and explain why it is false. Last, they are asked to “Make it True” by changing a symbol and/or number, such as changing 0.26 > 0.3 to 0.26 < 0.3 or to 0.46 > 0.3.

   Read answers to Frequently Asked Questions (FAQs) by selecting the lettered sections below.

   A. What strategies does the routine incorporate?

   This routine incorporates the following strategies to support students:

   • Modeling quantities with concrete, semi-concrete, and abstract representations.
   • Connecting concrete and/or semi-concrete representations to abstract representations of mathematics concepts and processes.
   • Using representations as thinking tools to make sense of, solve, and revise problems.
   • Explaining ideas by using visual representations (also helpful for listeners).
   • Asking questions to support creating and connecting representations.
   • Using mini-whiteboards for students to create, show, and explain their representations.
   • Using response cards, sentence starters, and vocabulary charts to support student communication.
   • Using strategically designed problems to focus on and address misconceptions and difficulties with fraction and decimal topics.

   B. What mathematics content does it focus on?

   The Is it True? routine provides flexibility for using the same format with a variety of mathematics topics. There are problem sets for the topics of fraction comparison and equivalence, fraction addition, fraction subtraction, decimal place value, decimal comparison, decimal addition, and decimal subtraction.

   C. How do you integrate the routine with your curriculum?

   The routine is designed to reinforce and deepen understanding of mathematical concepts and processes that have already been introduced to students. For example, the problem sets on fraction addition would be best used after students are familiar with the operation. Look over the problem sets for fractions and decimals to choose relevant content and items to fit the sequence of instruction in your curriculum and the needs of your students.

2. Prepare to Use the Routine
   Use the following resources to prepare for teaching the routine:
   • Routine Teaching Guide in the Module 2 Participant Workbook (4 MB)
   • Classroom Video Example on the Explore-A tab or PLC-A tab.
   • Optional: Module 2 Optional Teaching Slides for Instructional Routine (181 KB)
3. Implement with Students
   Use the routine one or more times with students. Collect student work samples to show or describe in your slides when you share at PLC Session B.
4. Make Slides to Share at PLC Session B
   Use the Debriefing Slides Template (264 KB) to prepare 6 slides about your teaching experiences.

PLC Session B

Share experiences using the routine and discuss ideas for strengthening implementation.

BEFORE THE SESSION

1. Get Ready for PLC Session B
   Prepare a short presentation to share your teaching experiences using the instructional routine.
   • Use the Debriefing Slides Template (264 KB) to create 6 slides on the debriefing questions.
   • At the session, you will need to access handouts (H12-14) from the Module 2 Participant Workbook (4 MB).

AT THE SESSION

2. Use the Debriefing Protocol
   You and your colleagues will take turns sharing your teaching experiences. The Debriefing Protocol (H12) provides a structured process for collaborative debriefing.

   Debriefing Protocol: Overview

   Goal: Debrief and learn from our collective experiences using an instructional routine.

   Part 1: Share Experiences

   • Presenter describes experiences and shows slides (~6 minutes).
   • Timekeeper gives a 1-minute warning to presenter and says when time is up.
   • Group members listen carefully during the presentation (avoid interruptions). Then, they can ask questions and note ideas to discuss in Part 2.

   Part 2: Group Discussion

   • Discuss common themes.
   • Identify ideas to apply.
3. Recap Strategies in Module
   Write down ideas for each question. Then, select the lettered bars to see example lists*. For future reference, a copy of the lists is on the handout Recap Strategies for Representations (H13).

   A. What are recommended strategies for representations?

   • Use concrete representations, such as fraction tiles or base ten blocks, to build understanding of mathematical ideas.
   • Use concrete materials that are proportional for place value concepts, fractions, and decimals.
   • Use semi-concrete representations, such as drawings or diagrams, to build understanding of mathematical ideas.
   • Explicitly connect concrete, semi-concrete representations, and abstract representations to build understanding of mathematical ideas.
   • Use questioning to help students understand how a model represents a mathematical concept and to support students in creating and explaining representations.
   • Provide ample and meaningful opportunities for students to work with concrete and semi-concrete representations to model concepts and procedures.
   • Support students in using representations as thinking tools to solve problems.
   • Have students use their representations to explain their solution or approach, such as by pointing to parts of their model.

   B. What approaches should you avoid?

   • Avoid using representations in a rigid sequence of concrete, then semi-concrete, then abstract. Instead, flexibly use two or more representations in ways that support the learning goals.
   • Avoid putting concrete materials away too quickly or not using them at all in higher grades. Instead, keep concrete materials available and emphasize connections between the representations. The use of concrete materials is appropriate at all grade levels.

   *These lists are not exhaustive.

4. Reflect and Plan Next Steps
   Discuss ways to strengthen current practices and identify new ideas to apply. Use the handout Strengthen Strategies (H14).

Wrap-Up

Summarize key ideas and reflect on your learning about Representations.

1. Summarize and Synthesize
   Think about: What ideas do you want to make sure to remember and apply when you use multiple representations with students? Add ideas to the Reference Sheet for Representations (H1) so that you will leave the module with a helpful resource for future use.
2. Reflect on Learning
   Complete the Self-Reflection Form (H15) to reflect on your learning about the recommendation for representations. This form's purpose is to provide opportunities for self-reflection to support professional learning and is not evaluative.

Resources

Get links to resources for the module and to extend your learning.

Explore the Recommendation (Part A)

Activities

Download the Module 3 Participant Workbook (3 MB) for access to the handouts. The handouts are labeled with a letter H and a number, such as H1.

1. What Is the Recommendation?
   Learn about the recommendation from Dr. Nancy Jordan, a member of the expert author panel that wrote the Practice Guide. Write notes on the Reference Sheet (H1).

2. Why Is Using Number Lines Important?
   Read about reasons for using number lines by selecting the lettered sections below. Then, add your ideas to the Reference Sheet (H1), question 2.

   A. Why is using number lines important for student learning?

   Understanding and using number lines helps students to:

   • Represent and understand the magnitudes of whole numbers, fractions, and other numbers.
   • Compare quantities and determine equivalencies.
   • Be prepared for more advanced mathematical work, such as with positive and negative numbers and coordinate grids.
   • Add your ideas on the Reference Sheet (H1)

   B. Standards for mathematical practice

   Understanding and using the number line representation is integral to most states’ Standards for Mathematical Practice or Process Standards. For example, students use number lines to conceptualize, model, and solve problems. Here are several relevant standards:

   • Make sense of problems and persevere in solving them.
   • Model with mathematics.
   • Use appropriate tools strategically.

   Source: corestandards.org

   C. Standards for mathematical content

   Throughout most states’ content standards, using number lines is essential to building understanding of number and operations concepts and processes. Here are some examples:

   • Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2.
   • Represent whole-number sums and differences on a number line diagram.
   • Understand a fraction as a number on a number line.
   • Represent fractions and decimals on a number line diagram.
   • Solve word problems by representing the problems on a number line diagram.

   Source: corestandards.org

3. What Are Recommended Strategies?
   Learn about evidence-based strategies by reading the handout How to Carry Out the Recommendation for Number Lines (H2). It includes an excerpt from the WWC Practice Guide.
4. View a Classroom Example
   Watch a teacher and students use an instructional routine for estimating the locations of fractions on a number line. Take notes on the handout Video Observations (H3) for these focus questions:
   1. What instructional strategies for number lines do you notice in the video?
   2. What are one or two ideas from the video that stand out for you?

5. Use Strategies for Number Lines
   Look at activities for students on the Explore Example Activities handout (H4). Choose one activity to try yourself.
6. Check for Understanding
   Answer nine questions about key characteristrics of number lines and get feedback.

   Start the Knowledge CheckKnowledge Check: Download Summary

Select Next to continue learning about the recommendation on the Explore-B tab.

Explore the Recommendation (Part B)

1. Locate Fractions on a Number Line
   Use a math app to place fractions on a number line and get feedback. Try the activity yourself and consider ways to use the app with students.

   Select this link to go to the app: Locating Fractions on a Number Line
   
   
   
   Source: App created on GeoGebra by Education Development Center

2. Model Fraction Addition by Using Number Lines
   Learn about recommended strategies by doing two activities:
   1. Watch a video demonstration of ways to add fractions on a number line.

   2. Try the strategies yourself by completing the activity handout Explore Fraction Addition on the Number Line (H5).
3. Model Fraction Subtraction by Using Number Lines
   Explore strategies to model subtracting fractions on a number line.
   1. Watch a short demonstration video.

   2. Solve problems on the activity handout Explore Fraction Subtraction on the Number Line (H6).
4. Consider Potential Challenges
   Read about possible obstacles and the expert panel's advice in the handout Challenges and Suggestions (H7).
5. Check for Understanding
   Answer three multiple-choice questions about the number lines recommendation and get feedback.

   Start the Knowledge CheckKnowledge Check: Download Summary

6. Reflect
   Reflect on your learning about the recommendation for number lines. Write responses to the prompts on the Reflection for Online Session handout (H8).

PLC Session A

BEFORE THE SESSION

1. Get Ready for the Session
   • Make sure to complete the online activities (Explore A & B Tabs), so that you will be prepared to discuss the recommendation for number lines.
   • You will need to access handouts (H9-12 and R1-3) from the Module 3 Participant Workbook (3 MB).

AT THE SESSION

Resources for key activities are provided below for access during the session and/or to revisit them afterwards.

2. Sort Representations of Number Lines
   Work with colleagues to sort cards that show number lines into two categories: Accurate Representations and Not Accurate Representations. The "Accurate" category is for number lines that have no errors. The "Not Accurate" category is for number lines that have one or more errors. The directions are listed below and on the handout Card Sorting Routine: Number Lines (H10). Use printed cards or a virtual version (button below).
   
   First, get to know the activity in the role of students by sorting the cards and using the sentence frames, and then discuss it from a teaching perspective.
   
   • Take turns placing cards.
   • Explain your reasons by using this sentence frame:
     • “I placed the card in the ____ category because …”
   • Then, your partner responds:
     • “I agree because …” OR: “I disagree because…”

   Start the SortingSort Representations of Number Lines: Try Again

3. Watch Video Examples
   You and your colleagues will view and discuss classroom video clips of the instructional routine. Take notes on the handout Video Discussion (H11) and then discuss these questions:
   • How does the teacher support students with 1) placing fractions on the number line and 2) explaining their ideas?
   • What do you notice about the students' understandings and/or difficulties with fractions and the number line representation?
   • What ideas from the videos do you want to apply when you use the routine with students?

   Video, Part 1 (Watch excerpt: 0:49–4:12.)

   
   

   Video, Part 2

4. Walk Through the Instructional Routine
   You and your colleagues will try the routine together. Use the Walk-Through of Routine: Script handout (H12) to take turns in the roles of "teacher" and "students."
5. Discuss Ideas and Start Planning
   Discuss suggestions for using the routine. The Routine Teaching Guide (in the Participant Workbook) has resources to help you plan:
   • 2-Page Overview of Routine (R1)
   • Planner (R2)
   • Suggestions for Addressing Potential Challenges (R3)

Use the Try It! Routine

You’ve reached a key step in the module: implementing strategies with your students.

1. Read about the Instructional Routine
   

   In this routine, students place fractions on a number line by using a benchmark strategy, estimation, and reasoning. The number line is marked with the benchmark numbers of 0, , and 1, and students are given cards with fractions. For each fraction, the teacher asks questions, such as: “Is  greater than or less than ?”  Students compare the fraction to the benchmark numbers and estimate where to place it on the number line. They explain their reasons by completing a sentence frame: “I placed the fraction ____ at this location because….” After the cards are placed, students discuss the fractions’ locations on the number line. Then, they are asked to create new fractions to add to the number line, such as a fraction that is closer to 1 than .

   Read answers to Frequently Asked Questions (FAQs) by selecting the labeled sections below.

   A. What strategies does the routine incorporate?

   The routine incorporates the following recommended strategies to support students:

   • Number line representation for locating and comparing fractions.
   • Benchmark strategy for placing fractions on the number line by comparing them to benchmark
     numbers of 0, , and 1.
   • Focused questions to help students compare and place fractions.
   • Sentence frames to support students in explaining their reasons.
   • Fraction card sets with scaffolded sequences of fractions to place.
   • Fraction tiles are used as needed to provide concrete linear representations.

   B. What mathematics content does the routine focus on?

   The routine focuses on locating and comparing fractions on the number line. You have a choice of three card sets that have different fractions: 1) unit fractions; 2) non-unit fractions; and 3) non-unit fractions with equivalent fractions.

   C. How do you integrate the routine with your curriculum?

   The routine is designed to reinforce and deepen understanding of fractions and the number line representation. As prerequisites, students should have prior knowledge of fractions and experience working with concrete linear representations, such as fraction tiles. In addition, students should have prior experience using number lines with whole numbers. Look over the card sets to select one that is a good fit for your students and the instructional sequence in your program. If you are unsure, begin with the unit fractions card set to gather information on students’ foundational understanding. Then, you can adjust the level of challenge as needed for subsequent card sets.

2. Prepare to Use the Routine
   Use the Planner (R2) and other resources in the Routine Teaching Guide (in the Module 3 Participant Workbook (3 MB)) to prepare for using the routine with students.

   Optional: You may want to revisit the classroom videos on the PLC-A tab. You also have the option to use the Module 3 Optional Teaching Slides for Instructional Routine (982 KB)

3. Use the Routine One or More Times with Your Students
   Collect student work samples to show or describe in your slides when you share at PLC Session B.
4. Make Slides to Share at PLC Session B
   Use the Debriefing Slides Template (264 KB) to prepare about 6 slides about your teaching experiences. Make sure to respond to the five debriefing questions in the template.

PLC Session B

Share experiences using the routine and discuss ideas for strengthening implementation.

BEFORE THE SESSION

1. Get Ready for PLC Session B
   Make sure to prepare a short presentation to share your experiences using the routine.
   • As mentioned on the Try It! Routine tab, use the Debriefing Slides Template (264 KB) to create about 6 slides. Include a slide that shows or describes at least one example of student work.
   • At the session, you will need to access handouts (H13-15) from the Module 3 Participant Workbook (3 MB).

AT THE SESSION

2. Use the Debriefing Protocol
   You and your colleagues will take turns sharing your teaching experiences. The Debriefing Protocol (H13) provides a structured process for collaborative debriefing.

Debriefing Protocol: Overview

Goal: Debrief and learn from our collective experiences using an instructional routine

Part 1: Share Experiences

• Presenter describes experiences and shows slides (~6 minutes).
• Timekeeper gives a 1-minute warning to presenter and says when time is up.
• Group members listen carefully during the presentation (avoid interruptions). Then, they can ask questions and note ideas to discuss in Part 2.

Part 2: Group Discussion

• Discuss common themes.
• Identify ideas to apply.
3. Recap Strategies in Module
   Write down ideas for each question. Then, select the lettered bars to see example lists*. For future reference, a copy of the lists is on the handout Recap Strategies for Number Lines (H14).

   A. What strategies are recommended for teaching number lines?

   • Use concrete linear representations, such as fraction tiles, to build number lines and locate fractions.
   • Have students fold strips of paper into equal parts to build understanding of partitioning. Use the folded strips to mark the location of fractions on a number line.
   • Use kinesthetic activities, such as walking on the number line, to build understanding of directionality and that a number's magnitude or size is represented by its distance from 0.
   • Use a variety of number lines that have different labeled endpoints, such as 0–3, to avoid having students overgeneralize that all fraction number lines are labeled 0–1.
   • Use benchmark numbers, estimation, and reasoning to locate and compare fractions on a number line.
   • Use questioning strategies to focus students on key aspects of number lines and have them explain their ideas.
   • Place fraction tiles on the number line to model fraction addition and subtraction and connect the concrete, semi-concrete, and numeric representations.
   • Use multiple number lines to identify common denominators and equivalent fractions for adding and subtracting fractions with unlike denominators.

   B. What approaches should you avoid?

   • Always using just one type of number line, such as with labeled endpoints of 0 and 1.
   • Always using a number line that is partitioned to match the fraction's denominator, because students may assume that they should not make more partitions on number lines.

   *These lists are not exhaustive.

4. Reflect and Plan Next Steps
   Discuss ways to strengthen current practices and new ideas to apply. Use the Strengthen Strategies (H15) handout.

Wrap-Up

Summarize key ideas and reflect on your learning about Number Lines.

1. Summarize and Synthesize
   Think about: What ideas do you want to make sure to remember and implement when you use number lines with students? Add ideas to the Reference Sheet for Number Lines (H1) so that you will leave the module with a helpful resource for future use.
2. Reflect on Learning
   Complete the Self-Reflection Form (H16) to reflect on your learning about the number lines recommendation. This form's purpose is to provide opportunities for self-reflection to support professional learning and is not evaluative.

Resources

Get links to resources for the module and to extend your learning.

Explore the Recommendation (Part A)

Activities

Download the Module 4 Participant Workbook (5 MB) for access to the handouts. The handouts are labeled with a letter H and a number, such as H1.

1. Why Is Understanding Word Problems Important?
   Read about the importance of word problems by selecting the lettered sections below. Then, add your ideas to the Reference Sheet (H1), question 2.

A. Why is understanding word problems important for student learning?

Understanding and solving word problems helps students to:

• Apply the mathematics they are learning to a variety of contexts.
• Deepen understanding of the meaning of mathematical operations.
• Create representations of problem situations and model operations.
• Build an important foundation for future mathematics topics.
• Add your ideas on the Reference Sheet (H1)

B. Standards for mathematical practice

Understanding and solving word problems is integral to most states’ Standards for Mathematical Practice or Process Standards. Here are two relevant example standards:

• Make sense of problems and persevere in solving them, such as making sense of a problem situation and identifying operations to solve it.
• Model with mathematics, such as writing an equation to describe a situation, and interpreting results to determine if they are reasonable and make sense for a given context.

Source: corestandards.org

C. Standards for mathematical content

In all states’ content standards, representing and solving word problems is essential to building understanding of number and operations concepts and processes. Here are some examples:

• Solve multi-step word problems for whole numbers that involve the four operations. Represent these problems by using concrete visual models, drawings, and equations.
• Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations to represent the problem, distinguishing multiplicative comparison from additive comparison.
• Solve word problems involving addition and subtraction of fractions, e.g., by using visual fraction models or equations to represent the problem.
• Assess the reasonableness of answers to word problems.

Source: corestandards.org

2. What Are Recommended Strategies?
   Learn about evidence-based strategies by reading the handout How to Carry Out the Recommendation for Word Problems (H2). This handout includes an excerpt from the WWC Practice Guide.
3. View a Classroom Example
   Watch an intervention teacher and students use an instructional routine for representing and solving word problems. Take notes on the handout Video Observations (H3) for these focus questions:
   1. What instructional strategies for word problems do you notice in the video? 
   2. What are one or two ideas from the video that stood out for you?

   Video 1

   Optional: Video 2
4. Explore Word Problems
   Complete activities on the handout Word Problem Types (H4), to deepen your own understanding of different problem types.
5. Match Word Problems and Representations
   The goal of this PD activity is for teachers to deepen understanding of representations for word problems. Match each visual representation to a word problem. Select the button below to begin.

   Start the MatchingMatching: Try Again

6. Check for Understanding
   Answer four questions about the word problems recommendation and get feedback.

Start the Knowledge CheckKnowledge Check: Download Summary

Select Next to continue learning about the recommendation on the Explore-B tab.

Explore the Recommendation (Part B)

1. Explore Strategies for Multiplicative Comparison Problems
   Complete two activities:
   1. Watch a video demonstration of ways to represent and solve multiplicative comparison problems.

   2. Try the strategies yourself by completing the handout Solve Multiplicative Comparison Problems (H5).
2. Identify Problem Types and Unknowns
   Examine word problems to identify their problem types and what is unknown.

   Start the Word ProblemsWord Problems: Download Summary

3. Why Should the Key Word Method be Avoided?
   Learn about this ineffective strategy by reading the handout Avoid the Key Word Method (H6). Find out about strategies to use instead.
4. Consider Potential Challenges
   Read about potential obstacles and ways to address them on the handout Challenges and Suggestions (H7).
5. Check for Understanding
   Answer four new questions about the word problems recommendation and get feedback.

   Start the Knowledge CheckKnowledge Check: Download Summary

6. Reflect
   Reflect on your learning about the word problems recommendation. Write responses to the prompts on the Reflection for Online Session handout (H8).

PLC Session A

BEFORE THE SESSION

1. Get Ready for the Session
   • Make sure to complete the online activities (Explore A & B Tabs), so that you will be prepared to discuss the recommendation for word problems.
   • You will need to access handouts (H9-12 and R1-3) from the Module 4 Participant Workbook (5 MB).

AT THE SESSION

Resources for key activities are provided below for access during the session and/or to revisit them afterwards.

2. Sort Problem Types
   Work with colleagues to sort word problems into two categories: Change Problems and Compare Problems. The directions are listed below and on the handout Card Sorting Routine: Word Problems (H10). Use printed cards or a virtual version (button below).
   
   First, get to know the activity in the role of students by sorting the cards and using the sentence frames, and then discuss it from a teaching perspective.
   • Take turns placing cards.
   • Explain your reasons by using this sentence frame:
     This problem is a ____________ problem because . . .
   • Then your partner will respond by saying:
     I agree because…       OR:      I disagree because...

   Start the SortingSort Problem Types: Try Again

3. Discuss a Video Example
   You and your PLC colleagues will rewatch the classroom video together. Take notes on the handout Video Discussion (H11) and then discuss:
   • How did the teacher support students in representing and solving word problems?
   • What did you notice about the students' understanding of the word problems?
   • What ideas from the video would you like to apply when you use the routine?

4. Walk Through the Instructional Routine
   You and your colleagues will try the routine together. Use the handout Walk-through of Routine: Script (H12) to take turns in the roles of “teacher” and “students.”
5. Discuss Ideas and Start Planning
   Share suggestions and start planning for implementation. Use these handouts:
   • 2-Page Overview of Routine (R1).
   • Planner (R2).
   • Suggestions for Addressing Potential Challenges (R3).

Use the Try It! Routine

You’ve reached a key step in the module: implementing strategies with your students.

1. Read about the Instructional Routine
   

   This routine is designed to support students with understanding, representing, and solving equal groups word problems. This problem type is central to the upper elementary grades' major focus on multiplication and division.

   In the routine, students first act out a word problem by taking on roles and using concrete materials (counters and paper plates). This interactive experience helps students to make sense of the problem situation and solve it. Then, students work on new problems, write equations, and determine if answers are reasonable. They compare word problems to identify similarities and differences, helping them to generalize about the characteristics of equal groups problems.

   Read answers to Frequently Asked Questions (FAQs) by selecting the labeled sections below.

   A. What strategies does the routine incorporate?

   The routine incorporates the following recommended strategies to support students:

   • Act out word problems to visualize the situations.
   • Use concrete representations to model and solve problems.
   • Use focused questions to make sense of a problem situation.
   • Write equations to represent and solve problems.
   • Connect representations of the same problem situation.
   • Determine whether an answer makes sense and is reasonable.
   • Use sentence starters and frames to explain solution methods and reasoning.
   • Explore one problem type (i.e., equal groups) in depth.

   B. What mathematics content does the routine focus on?

   The routine focuses on equal groups problems which are a key mathematics topic for the upper elementary grades. Students model the problem situations by acting them out, using manipulatives, and writing equations. The routine helps students to build understanding of the underlying mathematical structure of the equal groups problems, and also develop the meaning of the operations of multiplication and division.

   C. How do you integrate the routine with your curriculum?

   The Routine Teaching Guide provides a selection of word problems so that you can choose ones that match your students' prior knowledge. It is important that students have prerequisite knowledge and skills in multiplying and dividing whole numbers. If you have an intervention curriculum, we suggest incorporating the routine in a unit or lessons that focus on equal groups word problems. You can choose to use the example problems for the routine to replace or supplement similar problems in your program. Students also need to have prior experience using the manipulatives, such as tiles or base ten blocks, that you plan to use to model the problems.

2. Prepare to Use the Routine
   Use the following resources to prepare for teaching the routine:
   • Routine Teaching Guide in the Module 4 Participant Workbook (5 MB) to prepare for using the routine with students.
   • Optional: Module 4 Optional Teaching Slides for Instructional Routine (570 KB)
   • Classroom Video Example on the Explore-A tab or PLC-A tab.
3. Implement with Students
   Use the routine one or more times with your students. Collect student work samples to show or describe in your slides when you share at the next PLC session.
4. Make Slides to Share at PLC Session B
   Use the Debriefing Slides Template (264 KB) to prepare 6 slides about your teaching experiences. Make sure to respond to the five debriefing questions in the template.

PLC Session B

Share experiences using the routine and discuss ideas for strengthening implementation.

BEFORE THE SESSION

1. Get Ready for PLC Session B
   Make sure to prepare a short presentation to share your experiences using the routine.
   • As mentioned on the Try It! tab, use the Debriefing Slides Template (264 KB) to create 6 slides. Include a slide that shows or describes one or two examples of student work.
   • At the session, you will need to access handouts (H13–15) from the Module 4 Participant Workbook (5 MB).

AT THE SESSION

2. Use the Debriefing Protocol
   You and your colleagues will take turns sharing your teaching experiences. The handout Debriefing Protocol (H13) provides a structured process for collaborative debriefing.

   Debriefing Protocol: Overview

   Goal: Debrief and learn from our collective experiences using an instructional routine

   Part 1: Share Experiences

   • Presenter describes experiences and shows slides (~6 minutes).
   • Timekeeper gives a 1-minute warning to presenter and says when time is up.
   • Group members listen carefully during the presentation (avoid interruptions). Then, they can ask questions and note ideas to discuss in Part 2.

   Part 2: Group Discussion

   • Discuss common themes.
   • Identify ideas to apply.
3. Recap Strategies in Module
   Write down ideas for each question. Then, select the lettered bars to see example lists. For future reference, a copy of the lists is on the handout Recap Strategies (H14).

   A. What are recommended strategies for teaching word problems?

   Here is a list of some strategies* recommended by the WWC Practice Guide.

   • Teach students to identify word problem types.
   • Introduce a new problem type with a story that includes all quantities and then present the same story with a missing quantity.
   • Act out or role play problem situations.
   • Read problems more than one time.
   • Ask guiding questions.
   • Use concrete and semi-concrete representations.
   • Connect multiple representations of the same problem.
   • Teach solution methods by using worked-out examples, diagrams, and other strategies.
   • Teach essential vocabulary used in word problems.

   B. What approaches should you avoid?

   The WWC Practice Guide recommends avoiding these approaches:

   • Teaching a key word method of linking specific words with specific operations.
   • Calling a word problem type by an operation, such as "subtraction problems" because more than one operation can be used.
   • Giving students just one kind of problem with the same unknown.
   • Having students draw detailed pictures that focus on specific aspects of word problem contexts.

   C. What are suggestions for choosing word problems to use with students?

   • Introduce one problem type at a time.
   • Provide many problems of the same type to build understanding of the underlying structure.
   • Provide problems that have unknowns in different positions.
   • Start with accessible problems that use familiar contexts and vocabulary.
   • Periodically revisit problem types to reinforce understanding.
   • Provide a mix of previously learned and newly learned problem types.

   *Note these lists are not exhaustive.

4. Reflect and Plan Next Steps
   Discuss ways to strengthen current practices and identify new ideas to apply. Use the Strengthen Strategies (H15) handout.

Wrap-Up

Summarize key ideas and reflect on your learning about the word problems recommendation.

1. Summarize and Synthesize
   Think about: What ideas do you want to make sure to remember and implement when you teach word problems with students? Add ideas to the Reference Sheet for Word Problems (H1) so that you will leave the module with a helpful resource for future use.
2. Reflect on Learning
   Complete the Self-Reflection Form (H16) to reflect on your learning about the word problems recommendation. This form's purpose is to provide opportunities for self-reflection to support professional learning and is not evaluative.

Resources

Get links to resources for the module and to extend your learning.

PLC Session

Key Activities

The handouts are available in the Module 5 Participant Workbook (2 MB). Resources for key activities are provided below for access during the session and/or to revisit them afterwards.

1. Play Fractions Games
   Play two fractions games to focus on comparing and adding fractions.
   • Use the handouts (H4–6) for the Aim Low, Aim High Games.
   • Discuss: What strategies do the games incorporate to support student learning?
2. Discuss Feedback Strategies
   Read the handout Provide Feedback to Students (H7) and star* ideas that stand out for you. Discuss:
   • What is one idea that stood out for you or that you would add?
   • What is important to keep in mind when giving feedback to students struggling with math?
3. Use a Card Sort to Review Content with Students
   Try a card sorting routine that focuses on equivalent fractions. If you are meeting in person, use printed cards. If you are meeting virtually, use an online version by selecting the button below.

   First, get to know the routine in the role of students by sorting the cards and using the sentence starters, and then discuss it from a teaching perspective. The directions are listed below and on the handout Card Sorting Routine: Fraction Equivalence (H8).

   • Each card has two fractions. Decide if they are equivalent or not equivalent.
   • Place each card in a category. Explain your reasons by completing a sentence starter:
     • The fractions are equivalent because ...
     • The fractions are not equivalent because...
     • Then your partner will respond by saying:
       I agree because…       OR:      I disagree because...

   Start the SortingSorting: Try Again

4. Plan a Review Activity
   Plan ways to use the card sorting routine to review math content with students. Discuss the questions on the Planner (H9).
5. Recap Strategies
   Write down ideas for each question. Then, select the lettered sections to see example lists*. The handout Recap Strategies for Systematic Instruction (H10) has a copy of these lists for future reference.

   A. What are recommended strategies for systematic instruction?

   Here is a list of some strategies* recommended by the WWC Practice Guide.

   • Use an intentional sequence of instruction to build towards specific learning outcomes.
   • When introducing concepts/procedures, use accessible numbers.
   • Provide visual supports, such as manipulatives and diagrams.
   • Provide verbal supports, such as sentence starters and vocabulary charts.
   • Use worked-out examples with missing parts for students to complete.
   • Use a mix of previously and newly learned content.
   • Provide periodic reviews of content by using engaging activities.
   • Provide opportunities for students to explain and discuss their ideas throughout lessons.
   • Find out about students' understanding by asking probing questions.
   • Identify students' strengths and build on them to address areas of challenge.
   • Ask questions to give students opportunities to identify errors and self-correct.
   • Provide supportive feedback that is tailored to individual students.

   B. What approaches should you avoid?

   The WWC Practice Guide recommends avoiding these approaches*:

   • Moving to more complex concepts too quickly.
   • Teaching lesson topics in isolation.
   • Focusing only on new content and not integrating reviews of prior content.
   • Focusing only on one type of problem, which may cause overgeneralizations.
   • Putting opportunities for students to discuss ideas only at the end of lessons.

   *These lists are not exhaustive.

Wrap-Up

Use this opportunity to reflect on your learning and strategies.

1. Reflect on Learning
   Complete the Self-Reflection Form (H11) to reflect on your learning about the systematic instruction recommendation. This form's purpose is to provide opportunities for self-reflection to support professional learning and is not evaluative.

Resources

Get links to resources for the module and to extend your learning.