🔑 The Constant of Proportionality (k)
Finding the magic multiplier
🎯 Introduction
Remember how we kept dividing and getting the same number? That special number has a name, and it's super important!
The constant of proportionality is the number you ALWAYS multiply by to go from one quantity to the other. We often call it 'k' in math.
💡 Key Concepts
What is the Constant of Proportionality?
It's the 'magic multiplier' - the number that connects two proportional quantities. You multiply one quantity by k to get the other.
📌 Real Example:
If carpet costs $1.50 per square foot, then 1.50 is the constant of proportionality. You multiply square feet × 1.50 to get the cost.
💭 Think of k as a machine: you put in one number, multiply by k, and out comes the other number. Always the same k!
Finding k from a Table
To find k, just divide! Pick any row in your table and divide: k = y ÷ x (or output ÷ input).
📌 Real Example:
If 4 hours of work earns 15 per hour).
💭 k is the answer you get when you divide y by x in any row of a proportional relationship table.
What k Means in Real Life
The constant of proportionality is often a RATE - like dollars per item, miles per hour, or cups per batch.
📌 Real Example:
If k = 1.50 for carpet, it means '$1.50 per square foot.' If k = 65 for driving, it means '65 miles per hour.'
💭 k often answers the question 'how much of one thing per one of the other?'
📝 Step-by-Step Examples
Example 1: Finding k from a table
📋 Problem: Find the constant of proportionality: Square feet: 10, 20, 50 → Cost: 30, $75
Pick any pair from the table
Let's use 10 sq ft → $15
Divide: k = cost ÷ square feet
k = 15 ÷ 10 = 1.5
Check with another pair to make sure
30 ÷ 20 = 1.5 ✓ and 75 ÷ 50 = 1.5 ✓
State what k means
k = 1.5 means the carpet costs $1.50 per square foot
✅ Answer: k = 1.5 (or $1.50 per square foot)
Example 2: Using k to find missing values
📋 Problem: If k = 1.5, how much does 80 square feet of carpet cost?
Write the relationship
cost = k × square feet
Substitute the values you know
cost = 1.5 × 80
Multiply
cost = 120
✅ Answer: 80 square feet costs $120
Example 3: Finding k from a word problem
📋 Problem: A recipe uses 2 cups of peach juice for every 5 cups of grape juice. What is k?
Identify what's being compared
peach juice to grape juice
Write as a fraction (peach ÷ grape)
k = 2 ÷ 5 = 2/5 = 0.4
Interpret what this means
For every 1 cup of grape juice, you need 0.4 cups of peach juice
✅ Answer: k = 2/5 or 0.4
⚠️ Common Mistakes to Avoid
❌ Dividing in the wrong order
Why it's wrong: k = y ÷ x (not x ÷ y). The order matters! Getting it backwards gives you 1/k instead of k.
✅ How to avoid: Always think: 'What am I trying to find per what?' Cost per square foot means cost ÷ square feet.
❌ Thinking k must be a whole number
Why it's wrong: k can be any number! It can be a fraction (like 1/4), a decimal (like 3.14), or a whole number (like 5).
✅ How to avoid: Don't round unless told to. Fractions and decimals are perfectly fine for k.
You're doing awesome! Finding k is like finding a secret code that unlocks the relationship. Once you have k, you can find ANY value in the relationship! 🔑