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!"