📐 What Are Proportional Relationships?
Understanding the basics
🎯 Introduction
Have you ever noticed that if you buy 2 candy bars instead of 1, you pay exactly twice as much? That's a proportional relationship in action!
A proportional relationship is when two things are connected in a very special way: when one thing doubles, the other doubles too. When one thing triples, the other triples too. They always grow (or shrink) together at the same rate.
💡 Key Concepts
Proportional Relationship Definition
Two quantities have a proportional relationship when you can always multiply one by the SAME number to get the other.
📌 Real Example:
If apples cost 2, 2 apples = 6. You ALWAYS multiply the number of apples by 2 to get the cost!
💭 Think of it like a recipe that you can make bigger or smaller. If you double all ingredients, you get double the food. The RATIO stays the same.
Identifying Proportional Relationships
To check if something is proportional, divide one quantity by the other. If you ALWAYS get the same answer, it's proportional!
📌 Real Example:
Gas costs 8 (8÷2=4). 5 gallons = 40 (40÷10=4). Same answer every time = PROPORTIONAL!
💭 Imagine you have a table with two columns. Divide each pair. If all your answers match, you have a proportional relationship.
NOT Proportional Relationships
Some relationships seem like they might be proportional, but they're not! Usually these have something added or subtracted, or involve multiplication by itself (like squaring).
📌 Real Example:
A pizza delivery fee is 10 per pizza. 1 pizza = 25. But 25÷2 = 12.5, not 15! The $5 fee messes it up - NOT proportional.
💭 If there's a 'starting amount' that doesn't change, it's usually NOT proportional.
📝 Step-by-Step Examples
Example 1: Checking if a relationship is proportional
📋 Problem: A movie theater charges $12 per ticket. Is the relationship between number of tickets and total cost proportional?
Make a table with some values
Tickets: 1, 2, 3, 4 → Cost: 24, 48
Divide cost by tickets for each row
12÷1=12, 24÷2=12, 36÷3=12, 48÷4=12
Check if all answers are the same
Yes! All equal 12.
State your conclusion
YES, this IS a proportional relationship!
✅ Answer: Yes, it's proportional because the cost is always 12 times the number of tickets.
Example 2: A NON-proportional relationship
📋 Problem: A gym membership costs 20 per month. Is this proportional?
Make a table with some values
Months: 1, 2, 3, 4 → Cost: 90, 130
Divide cost by months for each row
70÷1=70, 90÷2=45, 110÷3=36.67, 130÷4=32.5
Check if all answers are the same
NO! We got 70, 45, 36.67, 32.5 - all different!
State your conclusion
NOT proportional because of the $50 joining fee.
✅ Answer: No, it's NOT proportional because the $50 joining fee means you don't multiply by the same number each time.
⚠️ Common Mistakes to Avoid
❌ Thinking any relationship with multiplication is proportional
Why it's wrong: y = x² uses multiplication, but it's NOT proportional! When x doubles, y quadruples (not doubles).
✅ How to avoid: Always check: does doubling one quantity double the other? If not, it's not proportional.
❌ Forgetting to check if zero works
Why it's wrong: In a true proportional relationship, when one quantity is 0, the other must be 0 too.
✅ How to avoid: Ask yourself: 'If I have 0 of one thing, do I have 0 of the other?' If not, it's probably not proportional.
Great job learning about proportional relationships! This is the foundation for everything else in this unit. Take your time to really understand this before moving on. 🌟