Module 10: Developing Conceptual Understanding through Tables and Proportional Reasoning

Task Introduction
"Welcome to the final module in this series! In this module, you’ll watch a video of a lesson on proportional reasoning involving tables. You’ll reflect on the teacher’s practices, how students connect their reasoning, and the ways these practices address Common Core standards. Let’s dive in!"

Video:
"Watch the video provided at this link. Before watching how students approach the task, solve it yourself to reflect on your own reasoning."

Pre-Video Task Prompt
"Before watching the video, let's start by solving the problem."

1. How did you approach solving the problem?
   - What strategies did you use?
   - Did you recognize proportional relationships within the table?

Hints if Needed:
- "Think about the relationships both horizontally (within rows) and vertically (between columns) in the table."
- "How might unit rate play a role in reasoning proportionally?"

"After you solve the problem, let me know, and we’ll move to the next step!"

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Post-Video Reflection Prompts  
"Now that you’ve watched the video and solved the problem, let’s reflect on different aspects of the lesson one by one."

Step 1: Observing Creativity-Directed Practices
- "What creativity-directed practices did you notice the teacher implementing during the lesson?"
- "Reflect on how these practices supported students’ reasoning and collaboration."

Hints if Needed:
- "Consider whether the teacher encouraged mathematical connections, collaborative problem-solving, or extended students’ thinking beyond the unit rate."

"When you're ready, share your thoughts, and we'll move to the next reflection."

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Step 2: Student Reasoning and Connections
- "How did students connect the relationship between price and container size?"
- "How did their reasoning evolve as they worked through the task?"

Hints if Needed:
- "Did students start with the given information (e.g., the 24-ounce container costing $3)?"
- "How did they use this information to reason proportionally?"

"Once you respond, we’ll move on!"

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Step 3: Teacher Actions in Small Groups
- "How did the teacher’s actions during small group interactions reflect the students’ reasoning?"
- "How did the teacher use these interactions to inform whole-class discussions?"

Hints if Needed:
- "Think about how the teacher listened to students’ reasoning and used their ideas to guide the next steps."
- "What types of questions did the teacher ask?"

"Once you're ready, let’s move forward!"

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Step 4: Initial Prompts and Sense-Making
- "How did the teacher prompt students to initially make sense of the task?"
- "What role did these prompts play in guiding students’ reasoning?"

Hints if Needed:
- "Did the teacher ask open-ended questions?"
- "How did these prompts help students engage with the task?"

"Share your response, and we’ll continue!"

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Step 5: Common Core Practice Standards
- "What Common Core practice standards do you think the teacher emphasized during the lesson?"
- "Choose four and explain how you observed these practices in action."

Hints if Needed:
- "Consider whether the teacher emphasized reasoning, collaboration, or modeling with mathematics."
- "How did the students demonstrate these practices?"

"When you’re ready, let’s move to the final steps!"

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Step 6: Problem Posing Activity
"Based on what you observed, pose a problem that encourages students to use visuals and proportional reasoning."
- "What real-world context will you use?"
- "How will students use visuals like bar models or tables to represent proportional relationships?"
- "Does your problem encourage multiple solution paths?"

Hints if Needed:
- "Try to design a problem where students can approach it differently but still apply proportional reasoning."

"Once you've created your problem, let me know!"

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Step 7: Summary and Final Reflection
- "What’s one change you will make in your own teaching based on this module?"
- "Reflect on a specific strategy, question type, or approach to representation that you want to implement."

Encouraging Closing Statement:  
"Great work completing all the modules! We hope you’ve gained valuable insights into fostering creativity, connecting mathematical ideas, and engaging students in meaningful learning experiences. It was a pleasure working with you—see you in the next professional development series!"