Vertical Rate Tables Worksheet
TEKS AR.2A • Finite Differences & Linear Models
Linear Relationships from Tables
Name:
Date:
Pd:
Directions: For each table, determine: a. First differences | b. Slope | c. \(y\)-intercept | d. Equation
\(y = mx + b\)
1 Table 1 Given (0, 2)
a.
b.
c.
d.
2 Table 2 Solve for \(b\)
a.
b.
c.
d.
3 Table 3 Given (0, -5)
a.
b.
c.
d.
4 Table 4 Solve for \(b\)
a.
b.
c.
d.
Vertical Rate Tables Answer Key
TEACHER ANSWER KEY TEKS AR.2A
Linear Relationships from Tables
SOLUTIONS MASTER
Key: a. 1st differences | b. Slope (\(m\)) | c. \(y\)-intercept (\(b\)) | d. Equation
Red Text = Correct Student Responses
1 Table 1 Solutions \(m = 2\)
+3 +3 +4
+6 +6 +8
a. \(\frac{6}{3} = 2\) (\(\frac{8}{4} = 2\))
b. \(m = 2\)
c. \(b = 2\) from \((0, 2)\)
d. \(y = 2x + 2\)
2 Table 2 Solutions \(m = -3\)
+2 +3 +4
-6 -9 -12
a. \(\frac{-6}{2} = -3\) (\(\frac{-12}{4} = -3\))
b. \(m = -3\)
c. \(b = 28\) (\(25 = -3(1) + b\))
d. \(y = -3x + 28\)
3 Table 3 Solutions \(m = 4\)
+2 +3 +4
+8 +12 +16
a. \(\frac{8}{2} = 4\) (\(\frac{16}{4} = 4\))
b. \(m = 4\)
c. \(b = -5\) from \((0, -5)\)
d. \(y = 4x - 5\)
4 Table 4 Solutions \(m = -5\)
+2 +3 +4
-10 -15 -20
a. \(\frac{-10}{2} = -5\) (\(\frac{-20}{4} = -5\))
b. \(m = -5\)
c. \(b = 40\) (\(30 = -5(2) + b\))
d. \(y = -5x + 40\)
Table Rate Questioning Guide
Teacher Facilitation • TEKS AR.2A
High-Level Discussion & Questioning Guide
DOK Levels 2 & 3
Purpose of These Questions:
Move students beyond mechanical arithmetic by prompting them to justify rate invariance across uneven intervals, explain the meaning of initial values, and analyze common finite-difference pitfalls.
1
Invariance & Ratio Thinking (DOK 2)
“Why can't we simply look at the differences in \(y\) to find the slope?”
Look for: Students recognizing that the \(x\)-values skip by different increments (+2, +3, +4). A rate is a ratio of two changes (\(\Delta y / \Delta x\)), not just \(\Delta y\).
“In Table 1, the changes in \(y\) are 6, 6, and 8. How can this still represent a constant rate of change?”
Look for: Explaining that \(6/3 = 2\) and \(8/4 = 2\); the simplified ratio remains constant even when step sizes vary.
2
Finding Intercepts Without \(x = 0\) (DOK 3)
“In Tables 2 and 4, \(x = 0\) is not in the table. What are two different strategies to find \(b\)?”
Look for: (1) Working backwards from \(x=1\) using the slope pattern, and (2) algebraic substitution of any coordinate into \(y = mx + b\).
“If you calculate the slope using the first point and the last point, do you get the same result as adjacent points? Why?”
Look for: Justification that all points on a linear function share the exact same constant rate of change.
3
Error Analysis & Non-Examples (DOK 3)
“A student claims Table 1's slope is \(\frac{3}{6} = \frac{1}{2}\). What mistake did they make, and what mnemonic prevents this?”
Pitfall: Inverted ratio (\(\Delta x / \Delta y\) instead of \(\Delta y / \Delta x\)). Prompt: “Rise over Run” or “y goes in the sky.”
“Can a table have equal second differences instead of first differences? What kind of function would that be?”
Extension: Connects TEKS AR.2A to quadratic functions, reinforcing that constant first differences uniquely identify linear models.
