Slope Table Worksheet
TEKS AR.2A • 8.4C • A.3A Rate of change from finite differences
Rate of Change from Tables
Find the constant rate of change (slope) \( m \) for each table representation.
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Reference: Slope Formula: \( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \)
\( m = \frac{\text{Change in } y}{\text{Change in } x} \)
1
Determine the rate of change (slope) from the table below.
m =
Positive Negative
Show Work / Finite Differences:
2
Determine the rate of change (slope) from the table below.
m =
Positive Negative
Show Work / Finite Differences:
3
Determine the rate of change (slope) from the table below.
m =
Positive Negative
Show Work / Finite Differences:
Algebra 1 • Linear Functions • Constant Rate of Change Remember: \( m = \frac{\Delta y}{\Delta x} \), change in \( y \) always belongs in the numerator!
Slope Table Answer Key
Teacher Resource • TEKS AR.2A • 8.4C • A.3A Official Key & Solutions
Rate of Change from Tables Answer Key
Complete finite difference annotations, step-by-step calculations, and common misconceptions.
Course: Algebra 1 / Math 8
Topic: Linear Functions
1 Problem 1 Solution Positive Slope
Final Answer: m = 2
Finite differences: Δx: +2 • +3 • +4
Change in y: Δy: +4 • +6 • +8
✓ Positive □ Negative
Method 1 • Rate of Change Formula:
\( m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{2 - 0} = \frac{4}{2} = 2 \)
Check Non-Consecutive Intervals:
\( \frac{13 - 7}{5 - 2} = \frac{6}{3} = 2 \quad \text{and} \quad \frac{21 - 13}{9 - 5} = \frac{8}{4} = 2 \implies \) Constant rate verified
2 Problem 2 Solution Negative Slope
Final Answer: m = -3
Finite differences: Δx: +3 • +3 • +2
Change in y: Δy: -9 • -9 • -6
□ Positive ✓ Negative
Method 1 • Rate of Change Formula:
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 18}{1 - (-2)} = \frac{-9}{1 + 2} = \frac{-9}{3} = -3 \)
Check with Later Points (4, 0) & (6, -6):
\( m = \frac{-6 - 0}{6 - 4} = \frac{-6}{2} = -3 \implies \) Confirmed negative slope
3 Problem 3 Solution Negative Fraction
Final Answer: m = -½ (or -0.5)
Finite differences: Δx: +4 • +6 • +4
Change in y: Δy: -2 • -3 • -2
□ Positive ✓ Negative
Method 1 • Rate of Change Formula:
\( m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{5 - 7}{0 - (-4)} = \frac{-2}{0 + 4} = \frac{-2}{4} = -\frac{1}{2} \)
Check Across Extended Interval (0, 5) & (6, 2):
\( m = \frac{2 - 5}{6 - 0} = \frac{-3}{6} = -\frac{1}{2} = -0.5 \implies \) Simplifies consistently
Teacher Facilitation & Common Student Pitfalls
Inverting Δx and Δy: Students often compute \( \frac{\Delta x}{\Delta y} \) (e.g., getting \( \frac{1}{2} \) or \( -\frac{1}{3} \)). Remind them \( y \) is always on top (rise over run).
Double Negative Arithmetic: In Problem 2, \( 1 - (-2) \) must become \( 1 + 2 = 3 \). Students frequently calculate \( 1 - 2 = -1 \), leading to incorrect positive slopes.
Non-Unit x-Intervals: Point out that \( x \) does not always increase by \( 1 \). Students must divide \( \Delta y \) by \( \Delta x \), not simply read the change in \( y \).
