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.

1️⃣ **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!" 🎉