🕵️ Secret Agent Math: Cracking Multi-Step Inequalities
📋 Materials Needed
- Small dry-erase board & marker (or paper and dark marker)
- Ruler or straightedge (for drawing number lines)
- 1 red crayon/highlighter and 1 green crayon/highlighter
- Printout or quick sketch of a "Top Secret Laser Vault" grid (a basic -10 to 10 number line)
🎯 Target Objectives & Success Criteria
Learning Objective: By the end of this 15-minute session, the student will solve two-step inequalities and correctly graph the solution set on a number line, including flipping the inequality symbol when multiplying/dividing by a negative number.
Success Criteria:
- I can isolate the variable using inverse operations.
- I can spot the "Negative Trap" and flip the inequality sign.
- I can accurately plot open vs. closed circles and direction arrows on a number line.
1. Mission Briefing (Hook & Rules) 2 Minutes
Scenario: "Agent, your mission is to bypass a laser alarm system protecting a vault. Unlike standard keypads that need one exact number, this laser security grid stays disarmed only for a range of numbers. To find the safe zone, you must solve multi-step inequalities!"
⚠️ Top-Secret Field Rules
- Open Circle ⭕ (< or >): The boundary number is not included (laser trips if touched!).
- Closed Circle 🔴 (≤ or ≥): The boundary number is safe to stand on.
- 🚨 THE NEGATIVE TRAP: If you multiply or divide by a negative number, the laser changes direction! (Flip the inequality symbol!)
2. The Demonstration: "I Do" 3 Minutes
Mission Target: Solve and graph $-3x + 5 \le 11$
- Isolate the variable term: Subtract 5 from both sides.
$-3x \le 6$ - Isolate the variable: Divide both sides by $-3$.
ALERT! Divided by a negative! Flip the sign!
$x \ge -2$ - Plot on the Laser Grid:
- Draw a closed circle at $-2$ (because it is $\ge$).
- Shade the ray pointing to the right (towards all numbers greater than $-2$).
3. Co-Op Mission: "We Do" 4 Minutes
Work through this problem together on the dry-erase board. Prompt the student to make key decisions.
Joint Challenge: $4x - 7 > 5$
- Instructor: "What is our first move to unmask $x$?"
Student Action: Adds 7 to both sides $\rightarrow 4x > 12$. - Instructor: "Now, divide by 4. Do we trigger the Negative Trap?"
Student Action: Identifies that 4 is positive, so the sign stays $\rightarrow x > 3$. - Instructor: "Graph the safe zone!"
Student Action: Draws an open circle at 3 and shades to the right using the green marker.
4. Field Agent Solo Challenge: "You Do" 4 Minutes
Hand the student the "Vault Code Sheet" to complete independently. Observe quietly and step in only if the alarm (trap) is triggered!
🔐 Final Security Code Challenge:
$-2x + 3 \ge 11$
Student Tasks:
- Solve for $x$ on your whiteboard.
- Draw the number line grid from $-6$ to $2$.
- Plot the circle and shade the safe path.
Answer Key (Click to reveal)
1. Subtract 3: $-2x \ge 8$
2. Divide by $-2$ & FLIP: $x \le -4$
3. Graph: Closed circle at $-4$, arrow pointing to the left.
5. Mission Debrief & Assessment 2 Minutes
Quick Check Questions:
- "Agent, if you end up with $x < 5$, is $5$ a safe number to step on?" (No, open circle!)
- "What is the single mathematical move that forces us to flip our security sign?" (Multiplying or dividing by a negative number.)
Summative Check: Inspect the student's solution to the Solo Challenge against the success criteria.
⚙️ Adaptations & Differentiation
Support (Scaffolding)
Keep a index card visual cue: "Did I divide/multiply by a negative? ➔ YES? Flip ↕️ ➔ NO? Keep ➡️". Use physical painter's tape on the floor as a physical number line to jump on.
Extension (Advanced)
Introduce a double-sided security breach (Variables on both sides):
$3x - 2 < 5x + 6$.