Table Proportions Anchor Chart Anchor Chart • 7.RP.A.2 Representation 1 of 3
Tables Focus
Proportions in Tables
Determine whether data in a table represents a proportional relationship and calculate the constant of proportionality (\(k\)).
The Golden Rule
Constant Ratio Test
A table shows a proportional relationship if and only if the ratio of output to input, \(\frac{y}{x}\), yields the exact same constant for every single row (where \(x \neq 0\)).
Must also have \((0, 0)\) as a baseline point Multiplicative relationship (\(y = k \cdot x\))
Constant of Proportionality
\(k = \frac{y}{x}\)
Unit Rate \(\rightarrow\) \(y = kx\)
✓
PROPORTIONAL TABLE
Constant \(k = 4\)
\(x\) (Hours) \(y\) (Miles) Ratio \(\frac{y}{x}\) \(1\) \(4\) \(\frac{4}{1} = \mathbf{4}\) \(2\) \(8\) \(\frac{8}{2} = \mathbf{4}\) \(3\) \(12\) \(\frac{12}{3} = \mathbf{4}\) \(5\) \(20\) \(\frac{20}{5} = \mathbf{4}\)
Why it works:
Every ratio \(\frac{y}{x} = 4\). The constant of proportionality is \(k = 4\) .
Equation: \(y = 4x\) At \(x=0\), \(y=0\)
✗
NOT PROPORTIONAL
Ratios Vary
\(x\) (Days) \(y\) (Fee $) Ratio \(\frac{y}{x}\) \(1\) \(6\) \(\frac{6}{1} = \mathbf{6}\) \(2\) \(9\) \(\frac{9}{2} = \mathbf{4.5}\) \(3\) \(12\) \(\frac{12}{3} = \mathbf{4}\) \(4\) \(15\) \(\frac{15}{4} = \mathbf{3.75}\)
Why it fails:
The ratios \(6 \neq 4.5 \neq 4\) are not constant. No single \(k\) exists!
Pattern: \(y = 3x + 3\) At \(x=0\), \(y=3 \neq 0\)
Step-by-Step Table Testing Protocol
1
Label Columns
Identify input \(x\) and output \(y\) clearly.
2
Compute \(\frac{y}{x}\)
Divide each \(y\) by its corresponding \(x\).
3
Verify All Rows
Check every row! All quotients must be identical.
4
Identify \(k\)
The constant ratio is your \(k\)! Write \(y = kx\).
Watch Out For the "Additive Trap": A constant rate of change (e.g. \(+3\) each row) does NOT mean the relationship is proportional! Proportions must be strictly multiplicative from \(x\) to \(y\). Always compute \(\frac{y}{x}\), not just row differences!
Graph Proportions Anchor Chart Anchor Chart • 7.RP.A.2 Representation 2 of 3
Graphs Focus
Proportions in Graphs
Recognize proportional graphs instantly with the two golden criteria and locate the constant of proportionality (\(k\)).
1
Criterion 1
Must Be a Straight Line
The graph must be strictly linear . No curves, no parabolas, no sharp bends, and no steps.
2
Criterion 2
Must Pass Through the Origin
The line must cross directly through \((0, 0)\) . When input is \(0\), output must be \(0\).
Anatomy of a Proportional Graph & Finding \(k\)
Key Point: \((1, k)\)
y x 0 1 2 3 4 2 4 6 8 Run = +1 Rise = +2 Origin (0, 0) (1, 2) ★ Unit Rate (2, 4) Straight Line + Starts at Origin = Proportional!
Method 1 • Shortcut
Look at Point \((1, k)\)
Find where the graph crosses \(x = 1\) . The corresponding \(y\)-value is always the constant of proportionality!
Point \((1, 2) \implies k = 2\)
Method 2 • Any Point
Ratio \(\frac{y}{x}\) (Slope)
Pick any clear point \((x, y)\) on the line and calculate \(k = \frac{\text{rise}}{\text{run}} = \frac{y}{x}\).
Using \((3, 6) \implies k = \frac{6}{3} = 2\)
Non-Proportional Visual Traps (Instant Disqualifiers)
Fails Origin
Straight Line, Misses \((0,0)\)
Has a non-zero \(y\)-intercept. Starts above or below the origin!
Fails Straight
Starts at \((0,0)\), but Curved
Rate of change is not constant (e.g. \(y = x^2\)). Not linear!
Bent Line
Changes Slope / Has Bends
Rate of change shifts midway. Must be one single continuous straight line!
The 5-Second Graph Inspection Rule:
1. Does it touch \((0,0)\)? • 2. Is it straight with no bends? • 3. Read \((1, k)\) for your constant!
Unit Rate Form \((1, k) \implies y = kx\)
Equation Proportions Anchor Chart Anchor Chart • 7.RP.A.2 Representation 3 of 3
Equations Focus
Proportions in Equations
Identify proportional equations using standard form, isolate \(y\) to reveal \(k\), and analyze key features.
The Universal Proportional Standard Form
\(y\) Output (Dep.)
=
\(k\) Constant (\(k\))
•
\(x\) Input (Indep.)
Pure Multiplication: \(x\) is multiplied only by a constant coefficient \(k\).
No Added Constant (\(+ b\)): If anything is added or subtracted, it is NOT proportional!
✓
PROPORTIONAL EQUATIONS
Form: \(y = kx\)
\(y = 6x\) \(k = 6\)
\(y = \frac{2}{5}x\) \(k = \frac{2}{5}\)
\(y = 0.75x\) \(k = 0.75\)
\(y = x\) (\(1 \cdot x\)) \(k = 1\)
\(\frac{y}{x} = 8\) (\(\times x\)) \(k = 8\)
When \(x = 0\), output \(y = 0\) every time!
✗
NOT PROPORTIONAL
Disqualified
\(y = 3x + 4\) Added constant (\(+4\))
\(y = 5x - 2\) Subtracted value (\(-2\))
\(y = x^2\) Exponent / Nonlinear
\(y = \frac{12}{x}\) \(x\) in denominator (Inverse)
\(y = 7\) No \(x\) term (Constant)
If \(x = 0\) produces \(y \neq 0\), it fails the origin test!
Finding \(k\) When \(y\) Is Not Alone (Algebraic Rearranging)
Coefficient on \(y\)
\(4y = 20x\)
Divide both sides by \(4\):
\(y = 5x \implies \mathbf{k = 5}\)
Divisor Under \(y\)
\(\frac{y}{3} = 4x\)
Multiply both sides by \(3\):
\(y = 12x \implies \mathbf{k = 12}\)
Ratio Form
\(\frac{y}{x} = 3.5\)
Multiply both sides by \(x\):
\(y = 3.5x \implies \mathbf{k = 3.5}\)
The 3-Step Equation Litmus Test:
1. Isolate \(y\) completely. • 2. Confirm no added/subtracted terms or powers. • 3. Read the number multiplying \(x\) as \(k\)!
Proportion Rule \(y = kx \iff k = \frac{y}{x}\)
