Intervention Roadmap Teacher Guide Boundary Blueprints
Teacher Facilitation Guide | Tier 2 Intervention
Standard
HS.A-REI.D.12
Learning Objective
Students will represent the solution sets of linear inequalities and systems of inequalities as half-planes and overlapping regions on a coordinate plane.
Common Misconceptions
Boundary Confusion: Using solid lines for \( < \) or \( > \) and dashed for \( \le \) or \( \ge \).Shading Error: Shading based on the inequality sign without checking if 'y' is isolated (e.g., in standard form).Intersection Blindness: Thinking the solution to a system is just the lines, rather than the overlapping shaded area.
Tier 2 Strategies
Concrete-Representational-Abstract (CRA):
1 Use physical highlighters (two different colors) to show the overlap for systems.
2 The "Test Point" Ritual: Always use (0,0) unless the line passes through the origin.
3 Vocalize the inequality: "y is less than or equal to..." means "below or on the line."
Intervention Protocol
1
The "Fence" Check (5 mins)
Display the Boundary Line slide. Ask: "If you can stand on the fence, is it solid or dashed?"
Prompt: "Look at the symbol. Does it have the 'equal to' bar? That's our 'ground' to stand on. No bar? It's a cliff edge!"
2
Point of Truth (10 mins)
Model using (0,0) as a test point on the Point of Truth slide. Show both a "True" result and a "False" result.
Prompt: "Plug in the origin. Is it a lie or the truth? Truth stays, Lies go!"
3
The Overlap Zone (15 mins)
Transition to systems. Give students two different colored highlighters. Use the Overlap Zone slide.
Strategy: Shade the first inequality lightly with one color. Shade the second with another. The "Winner" area is where the colors mix.
Formative Checkpoints
Observation Intervention Response Student shades the wrong side using "above/below" logic. Require the Test Point calculation. "Let the math tell you where to go." Student draws all solid lines regardless of symbols. Use a "Line Key" visual. Trace the symbol—if it has a flat bottom (≤), the line is solid. Student struggles with the intersection. Ask: "Is this point in the Blue zone? Is it in the Yellow zone? It must be in BOTH."
Boundary Blueprint Slides Boundary Blueprints
Mapping Solutions to Linear Inequalities
The Boundary Line
Dashed Line
\( < \)
\( > \)
Points on the line are NOT solutions.
Think: A cliff edge—don't step on it!
Solid Line
\( \le \)
\( \ge \)
Points on the line ARE solutions.
Think: A solid fence—you can lean on it!
The Point of Truth
Pick a point not on the line.
We love (0, 0) because it makes math easy!
If Statement is TRUE:
Shade the side WITH that point.
If Statement is FALSE:
Shade the OTHER side.
Example: \( y > x + 1 \)
Test \( (0, 0) \)
\( 0 > 0 + 1 \)
\( 0 > 1 \)
FALSE! Shade the other side!
The Overlap Zone
Rules for Systems:
Graph each inequality separately.
The solution is where the shadings overlap .
If they don't overlap? No Solution.
INTERSECTION ZONE
Blueprint Check
Look at the graph. What symbol was used?
Dashed Line Boundary
Is it:
\( \le \)
\( > \)
Discuss: How do you know?
Shading Shortcuts Worksheet Boundary Blueprints
Student Resource
Algebra 1 | Tier 2 Intervention
Name:
Date:
1
Level 1: Choose Your Side
Inequality: \( y \ge x - 2 \)
1. Boundary Line Type:
Solid
Dashed
2. Test Point (0,0):
Show substitution here...
Is the statement TRUE or FALSE?
Shade the Solution Side
2
Level 2: The Overlap Zone
Graph the System:
\( y < -x + 4 \)
\( y \ge 2x - 1 \)
LINE A: \( y < -x + 4 \)
Line Type
Test (0,0)
LINE B: \( y \ge 2x - 1 \)
Line Type
Test (0,0)
Final Checklist
Graph Line A (Dashed or Solid?)
Graph Line B (Dashed or Solid?)
Shade the OVERLAP (Intersection)
Workspace: Blueprint Grid
Blueprint Challenge
A city planner is zoning a park. The region must follow these rules:
Rule 1: Boundary
The edge is defined by \( y \le \frac{1}{2}x + 3 \). Is the edge included in the solution zone? Explain why.
Response...
Rule 2: Point Check
Is the point (4, 1) part of the solution zone for Rule 1? Show your math check below.
Calculations...
Progress Tracker Exit Ticket Post-Intervention
Exit Blueprint
Formative Check: HS.A-REI.D.12
Engineer Name
Class Code
1
Identify the Boundary: \( y > -2x + 1 \)
Line Type:
Solid
Dashed
Test Point (0,0):
Show math here...
Sketch Shading
2
System Visual Analysis
A student graphed a system and shaded one area that was blue and another area that was yellow. What does the green (blue + yellow) area represent?
The solution to only the first inequality.
The solution to only the second inequality.
The solution set for the system (both inequalities).
Self-Rating
How confident are you with shading systems?
1
2
3
4
5
PROGRESS MONITORING TOOL // COMPLETED BY LEARNER
