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@@ -3,48 +3,62 @@ Module 7: Understanding Non-Proportional Relationships
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  Task Introduction
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  "Welcome to this module on understanding non-proportional relationships! In this module, you’ll explore why certain relationships are not proportional, identify key characteristics, and connect these ideas to algebraic thinking. Let’s dive into some problems to analyze!"
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- Problems:
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- Problem 1: Ali drives at an average rate of 25 miles per hour for 3 hours to get to his house from work. How long will it take him if he is able to average 50 miles per hour?
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- Problem 2: Tugce’s cell phone service charges her $22.50 per month for phone service, plus $0.35 for each text she sends or receives. Last month, she sent or received 30 texts, and her bill was $33. How much will she pay if she sends or receives 60 texts this month?
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- Problem 3: Ali and Deniz both go for a run. When they run, both run at the same rate. Today, they started at different times. Ali had run 3 miles when Deniz had run 2 miles. How many miles had Deniz run when Ali had run 6 miles?
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- Step-by-Step Prompts for Analysis
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- 1. Problem 1: Inverse Proportionality
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- Initial Prompt:
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- "Let’s start with Problem 1. Is the relationship between speed and time proportional? Why or why not?"
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- Hints for Teachers:
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- - "Think about the relationship between speed and time. If Ali increases his speed, what happens to the time taken?"
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- - "Consider whether the ratio of miles to hours remains constant."
 
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- 2. Problem 2: Non-Proportional Linear Relationship
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- Initial Prompt:
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- "Is the relationship between the number of texts and the total bill proportional? Why or why not?"
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- Hints for Teachers:
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- - "Think about the initial cost of $22.50. Does this fixed amount affect whether the relationship is proportional?"
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- - "Consider if doubling the number of texts would double the total bill."
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- 3. Problem 3: Additive Relationship
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- Initial Prompt:
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- "Now, let’s look at Problem 3. Is the relationship between the miles Ali and Deniz run proportional? Why or why not?"
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- Hints for Teachers:
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- - "Think about whether the difference in miles remains constant as they run."
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- - "If Ali always has a one-mile lead, does that suggest a proportional relationship or a consistent additive difference?"
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- Reflection and Discussion:
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- - "What are the key characteristics that distinguish proportional relationships from non-proportional ones?"
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- - "How can graphing these relationships help students understand proportionality?"
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- - "Why is it important to expose students to both proportional and non-proportional relationships?"
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- Problem Posing Activity:
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- - "Now it’s your turn to create three non-proportional problems similar to the ones we explored. Write each problem and explain why the relationship is not proportional."
 
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- Summary:
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- - "We explored non-proportional relationships, distinguishing them from proportional ones by analyzing characteristics like inverse relationships, fixed costs, and additive differences."
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- - "We applied mathematical generalization and extension, thinking creatively about different relationships."
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- - "We discussed how to guide students in understanding proportionality by exploring non-examples."
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- """
 
 
 
 
 
 
 
 
 
 
 
 
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  Task Introduction
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  "Welcome to this module on understanding non-proportional relationships! In this module, you’ll explore why certain relationships are not proportional, identify key characteristics, and connect these ideas to algebraic thinking. Let’s dive into some problems to analyze!"
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+ 🚀 **Problems:**
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+ 1️⃣ **Problem 1:** Ali drives at an average rate of 25 miles per hour for 3 hours to get to his house from work. How long will it take him if he is able to average 50 miles per hour?
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+ 2️⃣ **Problem 2:** Tugce’s cell phone service charges her $22.50 per month for phone service, plus $0.35 for each text she sends or receives. Last month, she sent or received 30 texts, and her bill was $33. How much will she pay if she sends or receives 60 texts this month?
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+ 3️⃣ **Problem 3:** Ali and Deniz both go for a run. When they run, both run at the same rate. Today, they started at different times. Ali had run 3 miles when Deniz had run 2 miles. How many miles had Deniz run when Ali had run 6 miles?
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+ ---
 
 
 
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+ ### **Step-by-Step Prompts for Analysis**
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+ #### **Problem 1: Inverse Proportionality**
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+ **Initial Prompt:**
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+ "Let’s start with Problem 1. Is the relationship between speed and time proportional? Why or why not?"
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+ 💡 **Hints if Teachers Are Stuck:**
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+ - "Think about what happens when speed increases. Does time increase or decrease?"
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+ - "If the product of two quantities remains constant, what kind of relationship is that?"
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+ ✏️ **If Teachers Provide an Answer:**
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+ - ✅ Correct: "Great! Now, can you explain in detail why this is the case? Let’s go step by step."
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+ - ❌ Incorrect: "Not quite. Think about how speed and time interact. Would doubling speed double the time?"
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+ ---
 
 
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+ #### **Problem 2: Non-Proportional Linear Relationship**
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+ **Initial Prompt:**
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+ "Is the relationship between the number of texts and the total bill proportional? Why or why not?"
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+ 💡 **Hints if Teachers Are Stuck:**
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+ - "Does doubling the number of texts double the total cost?"
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+ - "What happens when a fixed cost is involved?"
 
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+ ✏️ **If Teachers Provide an Answer:**
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+ - ✅ Correct: "That’s right! Now, explain your reasoning in more detail. How does the fixed cost affect proportionality?"
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+ - ❌ Incorrect: "Hmm, not quite. Remember, proportional relationships pass through the origin. Does this one?"
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+ ---
 
 
 
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+ #### **Problem 3: Additive Relationship**
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+ **Initial Prompt:**
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+ "Now, let’s look at Problem 3. Is the relationship between the miles Ali and Deniz run proportional? Why or why not?"
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+
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+ 💡 **Hints if Teachers Are Stuck:**
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+ - "What remains constant in this situation: the ratio or the difference?"
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+ - "How does their different starting times affect proportionality?"
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+
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+ ✏️ **If Teachers Provide an Answer:**
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+ - ✅ Correct: "Exactly! Now, take me through your thought process. What patterns do you see?"
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+ - ❌ Incorrect: "Not quite. In a proportional relationship, the ratio stays the same. Is that the case here?"
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+
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+ ---
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+ ### **Problem Posing Activity**
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+ 📌 "Now, let’s take this a step further! Can you create a problem similar to the ones we explored? Make sure it includes a fixed cost, an additive difference, or an inverse relationship."
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+
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+ ---
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+
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+ ### **Summary and Reflection**
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+ 📌 "To wrap up, let’s reflect: Which **Common Core practice standards** did we apply in this module? How did **creativity** play a role in problem-solving?"
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+ 📌 "How might you guide your students in reasoning through proportional and non-proportional relationships?"
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+ """