Direct & Inverse Proportion
The Linker: Understanding How Variables Connect
The Proportion Lab
Phase 1: The Unitary Method (Foundation)
Step 1: Find Cost of 1 Item
This is the "k" value - the constant of proportionality
Step 2: Multiply by New Quantity
Cost per item × number of items = Total cost
Examiner Trap: Direct Proportion
Students often forget to divide by the first quantity. You MUST find the cost of 1 item first. Direct proportion means: if x doubles, y doubles. The formula is y = kx where k is constant.
The Logic Track
The "k" Constant
Always find 'k' first! k is the DNA of the relationship. It never changes for that proportion.
Notice how...
When you change the number of items from 5 to 8, the cost per item (k = £4.00) stays constant. This is the definition of direct proportion.
Real-World Example
The Unitary Method works for: speed (miles per hour), density (grams per cm³), or price per kilogram. The pattern is always the same: divide to find "per one unit", then multiply.
The Proportion Engine
Phase 3: Grade 8-9 Killer Questions
Given Information:
1Write the Formula
2Substitute & Find k
20 = k × 2
k = 20 ÷ 2
k = 10.000
3Use Formula with New Value
y = 10.000 × 5
y = 50.000
Grade 9 Killer Trap: Mixed Proportion
Some exam questions give you "y is directly proportional to x²" or "inversely proportional to the square root of x". The method is IDENTICAL:
- 1. Write the formula with k
- 2. Substitute the given values to find k
- 3. Write the full formula with your k value
- 4. Substitute the new x to find the new y
Mastery Checklist: Can you...
- ✓ Use the Unitary Method for Foundation questions?
- ✓ Write y = kx or y = k/x from a word problem?
- ✓ Find k from given values?
- ✓ Solve proportion problems with x² or √x?
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