Study guide: Solving Proportional Relationship Problems

📋 Course Outline

1. Recognizing Proportional Relationships
2. Solving with Tables
3. Solving with Graphs
4. Real-World Proportional Problems
5. Explaining and Checking Solutions

📖 1. Recognizing Proportional Relationships

🔑 Key Concepts & Definitions

• Proportional relationship: connects two quantities so that their values change at a consistent rate.

📝 Essential Points

• The relationship between the number of tests graded by a machine and the time is proportional when the machine grades 96 tests in 2 minutes.

💡 Memory Hook

Constant rate → proportional relationship

📖 2. Solving with Tables

★ Must-know

• 🔄 A table-solving method is to: organize corresponding values, extend the consistent relationship, use the completed table to find the unknown value

• A 6-ounce package of fruit snacks contains 45 pieces, so a proportional table gives 15 pieces for 2 ounces and can be used to find the number of pieces in a 10-ounce package.

Further detail

• Brooke pays $13.52 for 8 bottled teas, and the proportional relationship can be extended in a table to determine the cost of 12 bottled teas.

💡 Memory Hook

Given values → equivalent rate → unknown value

📖 3. Solving with Graphs

★ Must-know

• To solve a proportional problem with a graph, represent the relationship with plotted points and use the graph to determine the unknown quantity.

• A grading machine grades 96 tests in 2 minutes, and its relationship can be represented on a graph to determine how many minutes it takes to grade 300 tests.

Further detail

• The point (1, 48) on the grading-machine graph represents 48 tests graded in 1 minute.

💡 Memory Hook

A graph point marking the matching quantities

📖 4. Real-World Proportional Problems

★ Must-know

• If Ingrid types 3 pages in the same amount of time that Tanya types 4.5 pages, the proportional relationship can be used to determine Tanya's output when Ingrid has typed 11 pages.

Further detail

• A grading machine that grades 96 tests in 2 minutes can be used to solve a real-world question about the time required to grade 300 tests.

• The fruit-snack problem asks students to use a table to predict the number of pieces in a 10-ounce package from the information that 6 ounces contain 45 pieces.

📖 5. Explaining and Checking Solutions

★ Must-know

📌 In the typing problem, the claim that Tanya will type 15 pages must be checked against the proportional relationship and corrected if the reasoning is wrong.

Further detail

• 🔄 The collaborative problem-solving process includes: solve proportional relationship problems collaboratively, explain reasoning, analyze errors, use self-assessment to reflect on the success criteria

💡 Memory Hook

Correct reasoning versus an identified error

📊 Synthesis Tables

Methods for Solving Proportional Problems

Method	How it is used	Lesson example
Table	Organize corresponding values and extend the relationship	Fruit snacks and bottled teas
Graph	Plot the relationship and use points to find an unknown	Grading machine and typing problems