TEKS A3B Prerequisites Teacher Guide
SpEd & Co-Teaching Resource TEKS A.3(B) Prerequisite Toolkit
Rate of Change Prerequisite Guide
Instructional Scaffolds, Diagnostic Matrix, & CRA Strategies for High-Need Learners
Target Course
Algebra 1 / Inclusion
Tier 2/3 Support
TEKS A.3(B) Focus: Calculate the rate of change of a linear function represented tabularly, graphically, or algebraically. Mastery requires solidifying four foundational prerequisite bridges before applying the slope formula.
Prerequisite Diagnostic Screener Matrix
| Prerequisite Skill | Observable Student Stumble | Immediate Tier 2/3 SpEd Intervention |
|---|
| 1. Integer Subtraction | Treats \(-3 - 5\) as \(2\) or \(5 - (-2)\) as \(3\) (drops signs). | Vertical number line strip; "Add the Opposite" highlight protocol. |
| 2. Coordinate Orientation | Reverses \((x,y)\) as \((y,x)\); plots \(y\) on horizontal axis. | Color-coding anchor: Green \(x\) (crawl) & Blue \(y\) (climb). |
| 3. Ratio Orientation | Inverts rate as \(\frac{\Delta x}{\Delta y}\) because \(x\) is first in tables. | "Rise Over Run" elevator visual; \(\Delta y\) stamped physically in numerator. |
| 4. Unit Simplification | Leaves \(\frac{-12}{3}\) unreduced or writes decimal without context. | Unit label matching brackets: \(\frac{\text{dollars}}{\text{hour}}\); fraction tile check. |
1 Prerequisite Bridge: Signed Integer Differences in \(y_2 - y_1\)
CRA Protocol
The slope numerator requires computing differences between integers. SpEd students frequently drop subtraction signs when subtracting negative coordinates (e.g., \(4 - (-6)\)).
Concrete Stage
Two-color counters or walking a floor number line. Physical movement: turn opposite direction for subtraction.
Representational
Vertical thermometer: Mark \(y_2\) on top and count down/up to \(y_1\). Box double negatives: \(4 \;\boxed{- \;(-)}\; 6 \rightarrow 4 + 6\).
Abstract Rule
Formula template with built-in parentheses: \(m = \frac{(\quad) - (\quad)}{(\quad) - (\quad)}\). Insert coordinates before simplifying signs.
2 Prerequisite Bridge: Coordinate Pair Tracking & Variable Roles
Visual Scaffolding
Color-Coding Coordinate Anchor
Mandate dual-color highlighters for all ordered pairs during guided practice:
x-coordinate → Independent / Input / Horizontal
y-coordinate → Dependent / Output / Vertical
Order Consistency: "Stack & Subtract"
Prevents crossing pairs (\(y_2 - y_1\) over \(x_1 - x_2\)):
Point B: ( 6 , 15 ) | Point A: -( 2 , 7 )
Δx = 4 • Δy = 8 → m = 8 / 4 = 2
Rule: Blue \(\Delta y\) always goes into the numerator!
Algebra 1 TEKS A.3(B) Special Education Teacher Resource Guide Page 1 of 3 • Prerequisite Diagnostics & Foundations
Representational Prerequisite Bridges & Co-Teaching
Connecting Tables, Graphs, and Multi-Tiered Co-Teaching Delivery
Bridges 3 & 4
3 Prerequisite Bridge: Table Differences & Delta Identification (\(\Delta y / \Delta x\))
Tabular Scaffold
The "Reading Left-to-Right" Trap: In standard tables, the \(x\) column is on the left. Students naturally write \(\frac{\text{Left}}{\text{Right}} = \frac{\Delta x}{\Delta y}\), causing an immediate inverted rate error.
Visual "Side-Car" Annotation Protocol
- Highlight the \(y\) column in blue and draw curved delta brackets on the right side (\(+6\)).
- Highlight the \(x\) column in green and draw delta brackets on the left side (\(+2\)).
- Transfer directly to the Fraction Staging Pad: Blue on top, Green on bottom.
Non-Uniform Table Intervals
When \(\Delta x\) is not \(1\), students assume non-linearity. Teach the "Ratio Check" routine:
Interval 1: \(\Delta y / \Delta x = 6 / 2 = 3\)
Interval 2: \(\Delta y / \Delta x = 12 / 4 = 3\)
✓ Constant Rate of Change = 3
SpEd Teacher Cue: "Rate of change is not just the difference in \(y\); it is the difference in \(y\) divided by the difference in \(x\)!"
4 Prerequisite Bridge: Graph Reading & Lattice Point Selection
Graphical Scaffold
1. Slope Direction Check
Before calculating, students must label line trajectory using Mr. Slope Face:
- • Uphill L → R: Positive (+)
- • Downhill L → R: Negative (−)
- • Horizontal line: Zero (0)
- • Vertical line: Undefined (U)
2. "Crosshairs" Strategy
Students guess coordinates in the middle of grid squares instead of exact intersections.
Remediation: Use a transparent grid overlay or circular "magnifier" to find exact grid intersection corners ("crosshairs" only).
3. Right-Triangle Staircase
Count vertical rise first, then horizontal run.
Count grid steps, not axis numbers (especially when axis scales are \(1\) unit per square).
Co-Teaching Implementation Protocol for A.3(B)
Parallel Teaching
Tier 1 Concept Launch
Split class in half. SpEd teacher leads group using tactile vertical number lines & colored highlighters, while GenEd teacher delivers pacing with digital graphing tools.
Alternative Teaching
Pre-Teach & Re-Teach
While GenEd conducts warmup, SpEd pulls a targeted 4-6 student cluster to pre-calculate integer differences or review coordinate reading for 8 minutes before whole-group starts.
Station Rotation
Independent Guided Practice
Station A: SpEd Teacher (Prerequisite table scaffolds & formula template). Station B: GenEd (STAAR contextual problem solving). Station C: Self-checking digital practice.
Algebra 1 TEKS A.3(B) Special Education Teacher Resource Guide Page 2 of 3 • Visual Representations & Co-Teaching
Accommodations, Error Analysis & IEP Toolkit
Printable Scaffolds, Corrective Feedback Scripts, and Progress Monitoring
Classroom Scaffolds
Reproducible Graphic Organizer: Rate of Change Calculation Mat
Provide this laminated reference template to students receiving supplemental visual supports (accommodated testing / daily instruction):
Point 1 \((x_1, y_1)\):
( x₁ , y₁ )
Point 2 \((x_2, y_2)\):
( x₂ , y₂ )
Step 1: Vertical Change (\(\Delta y\))
( y₂ − y₁ ) = Δy
Step 2: Horizontal Change (\(\Delta x\))
( x₂ − x₁ ) = Δx
Step 3: Rate Ratio
Rate =
Δy Δx
Contextual Language Stems for Real-World Problems (TEKS A.3B Context)
Standard Contextual Sentence Frame
“For every 1 additional [x-unit], the [y-context] [increases / decreases] by [|rate| y-units].”
Non-Unit Rate Context Frame
“The rate of change is [value] [y-units] per [x-unit], representing how fast the [y-variable] changes.”
Targeted Error Correction Scripts
| Observed Student Error | Root Misconception | Teacher Prompt & Corrective Script |
|---|
| Calculates \(\frac{4 - 8}{6 - 2} = \frac{4}{4} = 1\) | Dropped negative sign on \(4 - 8\) | “Point to your starting temperature of 4 degrees. If it drops 8 degrees, are we above or below zero? What sign must your numerator have?” |
| Finds rate as \(\frac{2}{5}\) instead of \(\frac{5}{2}\) from table | Inverted ratio (\(\Delta x / \Delta y\)) | “Look at your elevator rule: Do we walk down the hallway before the elevator moves, or does the elevator go up/down first? Place your blue \(y\) on top!” |
| States rate of change is zero for a vertical line | Confuses zero slope with undefined slope | “What is \(\Delta x\) between these points? Can we divide a number by zero pizzas among zero friends? It is undefined / no slope!” |
Sample IEP Goal & Objective Formulation for A.3(B) Prerequisites
Measurable Annual Goal: By the end of the instructional period, given a linear relationship in a table or graph, the student will correctly identify \(\Delta y\) and \(\Delta x\) and calculate the rate of change using a visual template with 80% accuracy across 4 out of 5 consecutive trials as measured by weekly probe data.
Algebra 1 TEKS A.3(B) Special Education Teacher Resource Guide Page 3 of 3 • Scaffolds, Error Analysis & IEP Monitoring
