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Module 5: Proportional Reasoning and Developing Efficient Procedures

Task Introduction
"Welcome to this module on proportional reasoning and the development of efficient mathematical procedures! In this module, we’ll explore how to solve proportional reasoning tasks using conceptual tools like the double number line, ratio tables, and the cross-product strategy. You’ll also reflect on how to help students develop efficient strategies after building conceptual understanding."

Problem:
"Ali is planning an end-of-semester party for 60 pre-service teachers. He decides to make a fruit punch using 1 1/2 pints of orange juice for every 8 servings. If he wants to prepare 1 serving for each person, how much orange juice will he need?"

"Let's start by solving this problem using the double number line method and one additional strategy of your choice. As you work, explain your reasoning step by step. I will ask follow-up questions to guide you, rather than providing the answer immediately. Afterward, we will explore the cross-product strategy together and discuss why it works."

Step-by-Step Prompts for Representations

Double Number Line
Initial Prompt:
"Let’s begin with the double number line method. How would you set up your double number line to represent the relationship between servings and pints of orange juice? Can you explain your process?"

If the Teacher Provides an Initial Response:
"That’s a great start! Can you walk me through how you placed the values on the number line and why you made those choices?"

Hints for Teachers Who Are Stuck:
First Hint: "Try marking 0, 1 1/2, and other multiples of 1 1/2 on the line for orange juice. Align these with 0, 8, and corresponding multiples on the line for servings. How far can you extend the lines to reach 60 servings?"
Second Hint: "If 64 servings require 12 pints of orange juice, how can you adjust to find the amount for exactly 60 servings? What happens if you remove the portion for 4 servings?"

If the Teacher Provides a Partially Correct Answer:
"You’ve aligned some of the values correctly—good start! Can you check how much orange juice is needed for 4 servings and adjust accordingly?"

If the Teacher Provides an Incorrect Answer:
"It seems like there’s a mistake in how the intervals align. Remember, 1 1/2 pints corresponds to 8 servings. How can you adjust the spacing on the number line to reflect this proportion correctly?"

If the Teacher Provides a Correct Answer:
"Great work! Your double number line accurately represents the proportional relationship. How might you help students visualize this method effectively?"

Ratio Table
Initial Prompt:
"Now, let’s try solving this problem using a ratio table. How would you structure your table to track the relationship between pints of orange juice and servings? Describe your process."

Hints for Teachers Who Are Stuck:
First Hint: "Start with 1 1/2 pints for 8 servings. Can you extend the table for 16, 24, and so on until reaching 60 servings?"
Second Hint: "If 60 isn’t a direct multiple of 8, consider working with 64 servings first and then adjusting. How much juice would you subtract for 4 servings?"

If the Teacher Provides a Partially Correct Answer:
"You’re making progress! Now, how can you refine the calculation by considering the ratio for 4 servings (3/4 pint)?"

If the Teacher Provides an Incorrect Answer:
"Check your ratios—do they remain consistent across the table? Try revising the numbers to maintain the proportional relationship."

If the Teacher Provides a Correct Answer:
"Excellent! Your ratio table effectively shows the proportional relationships. How could you use this method to help students connect ratios to multiplication and division?"

Cross-Product Strategy
Initial Prompt:
"Now, solve this problem using the cross-product strategy. Set up a proportion as (1 1/2)/8 = x/60, where x represents the pints of orange juice needed. What’s your first step?"

Hints for Teachers Who Are Stuck:
First Hint: "Use cross-multiplication: (1 1/2) x 60 = 8x. Can you simplify and solve for x?"
Second Hint: "Rewrite 1 1/2 as an improper fraction (3/2) and calculate: (3/2) x 60 = 8x. What do you get for x?"

If the Teacher Provides a Partially Correct Answer:
"You’re on the right track! How can you simplify and solve for x? Double-check your arithmetic."

If the Teacher Provides an Incorrect Answer:
"It looks like there’s a small mistake. Can you rework the calculation using cross-multiplication?"

If the Teacher Provides a Correct Answer:
"Well done! You’ve accurately solved for x. How would you explain why the cross-product strategy works to students who are more comfortable with visual representations?"

Reflection Prompts
"How do the double number line, ratio table, and cross-product strategies highlight different aspects of proportional reasoning?"
"Why is it important for students to build conceptual understanding before transitioning to efficient procedures?"
"How might you guide students to connect these strategies and see the meaning behind mathematical procedures?"

Problem Posing Activity
Task Introduction
"Now it’s your turn! Create a proportional reasoning problem that involves fractions and requires students to solve it using at least two different strategies. Write your problem and explain how you would solve it."

Prompts to Guide Problem Posing:
"What real-world context will you use? Consider situations like recipes, scaling projects, or adjusting quantities."
"Does your problem support multiple solution strategies, including the cross-product approach?"

AI Evaluation Prompts
Evaluating Problem Feasibility:
"Does your problem align with proportional reasoning? Can students solve it using multiple methods?"

Feedback:
If Feasible: "Great problem! It supports proportional reasoning and allows students to explore multiple strategies."
If Not Feasible: "Your problem might need revision. Consider adjusting the numbers to ensure a proportional relationship. How could you modify it?"

Evaluating Solution Processes:
"Does your explanation connect conceptual understanding (e.g., number lines, tables) with efficient procedures (e.g., cross-product)? How might you clarify these connections?"

Summary Prompts
Content Knowledge
"You explored proportional reasoning through conceptual tools and efficient procedures, strengthening your problem-solving strategies."

Creativity-Directed Practices
"You applied generalization to connect equivalent fractions and develop explanations for the cross-product strategy."

Pedagogical Content Knowledge
"You examined ways to transition students from conceptual understanding to efficient procedures, ensuring they grasp the reasoning behind mathematical strategies."
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