Linear Comparison Worksheet AR.2C • AR.3A Linear Analysis
Linear Table Showdown
Comparing Vertical Tables, Slopes, Intercepts, & Extending Patterns
Name:
Date: Period:
Directions: Examine the two vertical tables. Find the \(y\)-intercept, calculate the slope (\(m = \frac{\Delta y}{\Delta x}\)), write each equation (\(y = mx + b\)), and fill in the two empty rows with new coordinate points that follow each rule.
A Function A (Table) Unit Step (\(\Delta x = +1\))
↓ Write 2 new \((x, y)\) pairs above ↓
B Function B (From Points) Varying Step (\(\Delta x \ne 1\))
↓ Write 2 new \((x, y)\) pairs above ↓
1. Coordinates of the \(y\)-intercept:
Where input \(x = 0\)
Function A: ( , )
Function B: ( , )
How can you identify the \(y\)-intercept immediately from any vertical table?
2. Calculate the Rate of Change (Slope, \(m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}\)):
Function A Work:
Slope (\(m_A\)):
Function B Work:
Slope (\(m_B\)):
3. Write each linear equation in slope-intercept form (\(y = mx + b\)):
Equation A: \(y =\)
Equation B: \(y =\)
4. Compare and contrast the key attributes of the two lines:
Which graph has a steeper rate of change? Which graph crosses the vertical axis higher? Explain how your calculations prove this.
Unit 3: Linear Functions • Comparative Analysis Single-Page Worksheet
Linear Thinking Teacher Guide Instructional Resource • TEKS AR.2C & AR.3A
Linear Comparison Facilitation Guide
High-Level Inquiry & Strategic Questioning Framework
Teacher Reference
Core Instructional Focus: Breaking the “Pattern Only in \(y\)” Misconception
Students often determine slope simply by looking at the consecutive difference in the \(y\)-column, assuming \(\Delta x = 1\). This side-by-side contrast juxtaposes Function A (uniform unit step \(\Delta x = +1\)) against Function B (non-consecutive points with varying \(\Delta x = +8, +4, +12\)), compelling students to apply the true definition of rate of change: \(m = \frac{\Delta y}{\Delta x}\).
Structural Contrast: Why These Two Tables?
Function A: Constant Unit Step
Input pattern: \(x\) increments uniformly by \(+1\) (\(-2, -1, 0, 1\)).Slope recognition: \(\Delta y / \Delta x = 1 / 1 = 1\).Intercepts visible: Both the \(y\)-intercept \((0, 1)\) and the \(x\)-intercept \((-1, 0)\) appear explicitly in the table.Pedagogical role: Serves as an accessible anchor baseline.
Function B: Irregular Step (From Pairs)
Input pattern: Irregular jumps: \(\Delta x = +8, +4, +12\).Slope recognition: \(\Delta y / \Delta x = 2/8 = 1/4 = 3/12 = 1/4\). Requires fraction simplification.Intercepts visible: \(x\)-intercept \((-12, 0)\) and \(y\)-intercept \((0, 3)\) are both present but separated by gaps.Pedagogical role: Disables superficial counting tricks.
High-Level Questioning Protocol for Page 1 Analysis
Q1 • Identifying the \(y\)-Intercept (DOK 1–2) Coordinate Identification
Guiding Question: “When we say a graph intercepts the vertical axis, what must always be true about its horizontal location (\(x\)-coordinate)?”Probing Question: “Table A lists \((0, 1)\) and \((-1, 0)\). How do you know without graphing which is the \(x\)-intercept and which is the \(y\)-intercept?”
Q2 • Calculating Rate of Change (DOK 2–3) Ratio & Covariation
Guiding Question: “In Table B, look at the change in \(y\) from row 1 to row 2 (\(+2\)), and then from row 2 to row 3 (\(+1\)). Does this mean Table B is not linear? Why or why not?”High-Level Prompt: “What ratio remains constant in Table B even though the differences between rows keep changing? Show how \(\frac{2}{8}\) and \(\frac{1}{4}\) express the same rate.”
Q4 • Comparing Key Attributes (DOK 3) Cross-Model Synthesis
Inquiry Prompt: “Function B starts higher on the \(y\)-axis (\(b = 3\) vs \(b = 1\)). Does this guarantee that Function B will always have higher values than Function A? How does slope impact their relationship as \(x\) grows?”
Unit 3: Linear Functions • Teacher Facilitation Guide Page 1 of 2
Instructional Strategies • Higher-Order Thinking
Probing the Independent Problems
Tiered Questioning, Solution Discussion, & Common Pitfalls
DOK 3–4 Focus
Use these targeted questions while circulating during student work and during the whole-class debrief.
1 Problem 1: \(x\)-Intercepts
Targeted Question: “Table B explicitly shows \((-12, 0)\). Table A shows \((-1, 0)\). Why do many students mistakenly say Table B has no \(x\)-intercept when looking at only the positive coordinates?”
Key Insight: Emphasize that finding intercepts means setting the opposite variable to zero (\(y=0\) for \(x\)-intercept).
2 Problem 2: Point \((8, 5)\)
Targeted Question: “When you plug \(x = 8\) into Function B, you get \(\frac{1}{4}(8) + 3 = 2 + 3 = 5\). But in Function A, \(8 + 1 = 9 \ne 5\). What does this tell you graphically?”
Key Insight: Connect algebraic substitution to whether a point lies on the line.
3 Problem 3: Long-Term Growth
Targeted Question: “At \(x=0\), Function B was ahead by 2 units (\(3\) vs \(1\)). But at \(x=24\), Function A is 16 units higher (\(25\) vs \(9\)). Which attribute caused Function A to overtake Function B?”
Key Insight: Slope represents rate of accumulation; steeper rate always dominates over time.
4 Problem 4: Intersection Point
Targeted Question: “Setting \(x + 1 = \frac{1}{4}x + 3\) yields \(x = \frac{8}{3} \approx 2.67\). What does this fractional value tell you about why this point didn't appear as an integer in either table?”
Key Insight: Tables only sample discrete points; lines represent infinite continuous solutions.
Anticipated Student Misconceptions & Diagnostic Pivots
Student Misconception Evidence / Error Sign Teacher Pivot Question Inverting the slope ratio (\(\Delta x / \Delta y\)) Student writes \(m = 4\) or \(m = \frac{8}{2}\) for Function B. “Does slope measure vertical change over horizontal change, or horizontal over vertical? Remember: \(\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}\).” Ignoring \(\Delta x\) when it varies Student looks at \(0 \to 2\) in \(y\) and states slope is \(+2\). “How much did \(x\) change when \(y\) increased by 2? Is that 1 unit of \(x\) or 8 units?” Confusing \(x\)- and \(y\)-intercepts Student says the \(y\)-intercept of Table A is \((-1, 0)\). “On which axis does the point \((-1, 0)\) land? What coordinate is zero at the \(y\)-axis?”
Extension Challenge (DOK 4):
“Create a new 4-row vertical table for Function C that has a \(y\)-intercept located between Function A and Function B (\(1 < b < 3\)) and is steeper than Function A (\(m > 1\)). Ensure your table does not use a constant step of \(+1\) in \(x\).”
Unit 3: Linear Functions • Teacher Facilitation Guide Page 2 of 2
Linear Comparison Answer Key Answer Key • TEKS AR.2C & AR.3A
Linear Table Showdown • Solutions
Includes Completed Blank Rows, Slopes, Intercepts, & Synthesis
Teacher Key
Table A (with extensions)
Summary:
\(y = x + 1\)
Accept any points where \(y = x + 1\) (e.g., \((2, 3), (3, 4), (4, 5)\)).
Table B (with extensions)
Summary:
\(y = \frac{1}{4}x + 3\)
Accept any points where \(y = \frac{1}{4}x + 3\) (e.g., \((4, 4), (8, 5), (16, 7)\)).
1. Coordinates of the \(y\)-intercept: 2 Points
Function A: ( 0 , 1 ) Found at row 3 where \(x = 0\)
Function B: ( 0 , 3 ) Found at row 3 where \(x = 0\)
Explanation Criteria: In any vertical table, the \(y\)-intercept is always the row where the input \(x = 0\). Students locate \(0\) in the \(x\)-column and write the paired \(y\)-value.
2. Rate of Change / Slope (\(m = \frac{\Delta y}{\Delta x}\)): 4 Points
Function A Solution:
Using \((0, 1)\) and \((1, 2)\):
Slope (\(m_A\)) = 1
Function B Solution:
Using \((-4, 2)\) and \((0, 3)\):
Slope (\(m_B\)) = \(\frac{1}{4}\) (or 0.25)
3. Slope-Intercept Form (\(y = mx + b\)): 2 Points
Equation A: \(y = x + 1\) (Also accept \(y = 1x + 1\))
Equation B: \(y = \frac{1}{4}x + 3\) (Also accept \(y = 0.25x + 3\))
4. Attribute Comparison Justification: 2 Points
• Steepness: Function A is steeper because its rate of change (\(m = 1\)) is greater than Function B's rate of change (\(m = 0.25\)). Function A increases 4 times faster per unit of \(x\).
Table Mastery Practice Worksheet TEKS AR.2C • AR.3A Skills Practice
Linear Table Mastery Practice
Finding Slope, Intercept, & Equation Across Positive & Negative Slopes
Name:
Date: Period:
Directions: For each vertical table, find the rate of change (slope \(m = \frac{\Delta y}{\Delta x}\)), write the \(y\)-intercept as an ordered pair \((0, b)\), and write the equation in slope-intercept form (\(y = mx + b\)).
Set 1 Positive Slopes Comparison
Rising Lines (\(m > 0\))
Type A (Unit Step)
Slope Work (\(\frac{\Delta y}{\Delta x}\)):
Slope (\(m\)):
\(y\)-Intercept:
Equation: \(y =\)
Type B (Irregular)
Slope Work (\(\frac{\Delta y}{\Delta x}\)):
Slope (\(m\)):
\(y\)-Intercept:
Equation: \(y =\)
Set 2 Negative Slopes Comparison
Falling Lines (\(m < 0\))
Type A (Unit Step)
Slope Work (\(\frac{\Delta y}{\Delta x}\)):
Slope (\(m\)):
\(y\)-Intercept:
Equation: \(y =\)
Type B (Irregular)
Slope Work (\(\frac{\Delta y}{\Delta x}\)):
Slope (\(m\)):
\(y\)-Intercept:
Equation: \(y =\)
Table Mastery Answer Key Answer Key • TEKS AR.2C & AR.3A
Table Mastery Practice • Solutions
Sets 1 & 2: Positive & Negative Slope Analysis
Teacher Key
Complete step-by-step solutions for each table. Award 1 point for slope work/value, 1 point for \(y\)-intercept coordinate, and 1 point for the final equation (3 points per table).
Set 1 Solutions Positive Slopes (\(m > 0\))
6 Points Total
Table 1A (Type A • Unit Step) \(\Delta x = +1\)
Slope Work: \(m = \frac{-1 - (-3)}{1 - 0} = \frac{2}{1} = \mathbf{2}\)
Slope (\(m\)): 2
\(y\)-Intercept: ( 0 , -3 )
Equation: \(y = 2x - 3\)
Table 1B (Type B • Irregular) Points Given
Slope Work: \(m = \frac{4 - 2}{0 - (-3)} = \frac{2}{3}\)
Slope (\(m\)): \(\frac{2}{3}\)
\(y\)-Intercept: ( 0 , 4 )
Equation: \(y = \frac{2}{3}x + 4\)
Set 2 Solutions Negative Slopes (\(m < 0\))
6 Points Total
Table 2A (Type A • Unit Step) \(\Delta x = +1\)
Slope Work: \(m = \frac{2 - 5}{1 - 0} = \frac{-3}{1} = \mathbf{-3}\)
Slope (\(m\)): -3
\(y\)-Intercept: ( 0 , 5 )
Equation: \(y = -3x + 5\)
Table 2B (Type B • Irregular) Points Given
Slope Work: \(m = \frac{-1 - 0}{0 - (-2)} = -\frac{1}{2}\)
Slope (\(m\)): \(-\frac{1}{2}\) (or -0.5)
\(y\)-Intercept: ( 0 , -1 )
Equation: \(y = -\frac{1}{2}x - 1\)
Teacher Alert: In Table 2B, watch for students writing \(m = +1/2\) by missing that \(y\) decreases as \(x\) increases (\(2 \to 0 \to -1 \to -3\)).
Table Mastery Answer Key • Sets 1 & 2 Page 1 of 2
Answer Key • TEKS AR.2C & AR.3A
Table Mastery Practice • Solutions
Sets 3 & 4: Mixed Integer & Fractional Slopes
Teacher Key
Complete worked answers for Sets 3 and 4. Emphasize simplified fractional slopes rather than unrounded decimals.
Set 3 Solutions Positive Slopes (\(m > 0\))
6 Points Total
Table 3A (Type A • Unit Step) \(\Delta x = +1\)
Slope Work: \(m = \frac{2 - (-2)}{1 - 0} = \frac{4}{1} = \mathbf{4}\)
Slope (\(m\)): 4
\(y\)-Intercept: ( 0 , -2 )
Equation: \(y = 4x - 2\)
Table 3B (Type B • Irregular) Points Given
Slope Work: \(m = \frac{1 - (-2)}{0 - (-5)} = \frac{3}{5}\)
Slope (\(m\)): \(\frac{3}{5}\) (or 0.6)
\(y\)-Intercept: ( 0 , 1 )
Equation: \(y = \frac{3}{5}x + 1\)
