Proportion Blueprint Reference Sheet
Grade 7 Math Anchor Standards 7.RP.A.2 • 7.G.A.1
Proportionality & Scale Factor
Architectural Reference Guide
Constant of Proportionality
\(k = \frac{y}{x}\) Equation: \(y = kx\)
Unit rate; the fixed multiplier connecting \(x\) and \(y\).
Scale Factor (\(k\))
\(k = \frac{\text{New}}{\text{Original}}\) New: \(\text{Orig} \times k\)
Multiplier applied to every side of the original figure.
Size Change Rules
- Enlargement: \(k > 1\)
- Reduction: \(0 < k < 1\)
- Congruent: \(k = 1\)
Visual Models for Proportional Relationships
Multiple representations of constant rates
Model 1: Ratio Table \(k = 6\)
| Time, \(x\) (hr) | Distance, \(y\) (mi) | Ratio (\(y \div x\)) |
|---|
| 1 | 6 | \(6 \div 1 = 6\) |
| 2 | 12 | \(12 \div 2 = 6\) |
| 5 | 30 | \(30 \div 5 = 6\) |
Every single pair has the exact same quotient: \(k = 6\).
Model 2: Double Number Line Unit Rate = 15
Cost ($)
$0 $15 $30 $45
Shirts
0 1 2 3
Aligned zero markers show equivalent ratios scale together.
Model 3: Coordinate Graph Origin & Line
(0,0) (1, k)
1. Straight line: Constant rate.
2. Origin (0, 0): Must pass through origin!
3. Point (1, k): Unit rate sits at \(x = 1\).
Non-origin lines are NOT proportional.
Model 4: Scaled Figures Scale Factor = 2.5
Original (A)
2 cm 4 cm
× 2.5
New / Model (B)
5 cm 10 cm
\(k = \frac{\text{New}}{\text{Orig}} = \frac{10}{4} = 2.5\) • Angles are identical!
Step-by-Step Solving Procedures
Standard problem algorithms
A. Find \(k\) & Write Equation
- 1 Identify \((x, y)\) pair from table or text.
- 2 Compute: \(k = \frac{y}{x}\). Simplify completely.
- 3 Write equation: write \(\mathbf{y = kx}\).
Example: \((3, 15) \rightarrow k = \frac{15}{3} = 5\)
Equation: \(y = 5x\)
B. Find Scale Factor (\(k\))
- 1 Pick corresponding sides on figures.
- 2 Set up ratio: \(\frac{\text{New}}{\text{Original}}\).
- 3 Simplify to fraction or decimal.
Example: Orig = 12 cm, New = 8 cm
\(k = \frac{8}{12} = \mathbf{\frac{2}{3}}\) (Reduction!)
C. Calculate Missing Length
- 1 Identify: Original side and factor (\(k\)).
- 2 Multiply: \(\text{New} = \text{Original} \times k\).
- 3 Check & Label: Check units carefully.
Example: Orig = 14 m, \(k = 0.5\)
\(\text{New} = 14 \times 0.5 = \mathbf{7\text{ m}}\)
Pitfall Detector • Top 3 Traps to Avoid on Tests
× The Additive Trap: Scale factor is ALWAYS multiplication, never adding or subtracting to sides!
× Flipped Fraction Trap: Do NOT automatically put the larger number on top. It is strictly \(\frac{\text{New}}{\text{Original}}\).
× Origin Bypass: A line crossing at \((0, 4)\) is linear, but not proportional because \(0 \neq 4k\).
Proportion Blueprint Guided Notes
Name:
Date:
Period:
Guided Notes • Part 1
Constant of Proportionality (\(k\))
Architectural Studio Standards 7.RP.A.2a-d
Core Concept: What is a Proportional Relationship?
1. Two quantities \(x\) and \(y\) are in a proportional relationship if the ratio \(\frac{y}{x}\) is always .
2. This fixed ratio is called the constant of proportionality, represented by the variable: .
Ratio Formula: \(k = \frac{\quad\quad}{x}\) Direct Equation: \(y =\) \(x\)
Model 1: Finding \(k\) From a Table Step-by-Step Guided Analysis
Step 1: Select an \((x, y)\) row.
Step 2: Compute ratio: \(k = \frac{y}{x}\).
Step 3: Test remaining rows to verify.
| Flour, \(x\) (cups) | Cookies, \(y\) | Calculate Ratio (\(y \div x\)) |
|---|
| 2 | 24 | \(24 \div 2 = 12\) |
| 3 | 36 | \(36 \div 3 = \) |
| 5 | | \(y \div 5 = 12\) |
| 84 | \(84 \div x = 12\) |
Findings:
Constant: \(k =\)
Each cup makes cookies.
Equation: \(y =\)
Model 2: Complete the Double Number Line Unit rate alignment
A blueprint printer prints sheets at a steady rate. Fill in the missing values on both lines:
Sheets, \(y\)
0 18 36 [ ] 90
Time, \(x\) (min)
0 1 2 3 [ ]
Unit rate: \(k =\) sheets/min Equation: \(y =\)
Model 3: The Coordinate Plane Rules Origin & Unit Rate
Two Non-Negotiable Test Rules:
1. Must be a perfectly line.
2. Must pass through the , point (0, 0).
Key Point: The point (1, r) shows the rate.
(0,0) (1, 8)
From graph:
Point: (1, 8)
\(k = \)
Equation: \(y =\)
Check for Understanding #1 • Show Your Work Independent Application
A 3D printer creates 15 prototype gears in 2.5 hours. If the relationship is proportional, find the constant of proportionality (\(k\)) and write an equation to predict the number of gears (\(y\)) printed in \(x\) hours.
