Lead Investigator Teacher Guide Lead Investigator Guide
Lesson: Operation Investigation (PEMDAS)
8TH GRADE PRE-ALGEBRA
Learning Objective
Students will apply the Order of Operations (PEMDAS) to evaluate complex algebraic expressions including exponents and parentheses, with a focus on identifying and correcting common substitution errors.
Materials Needed
Operation Investigation Slides
Case File: Spot the Error Worksheet (1 per student)
Scientific Calculators
Red pens for "investigation" marks
Timeline
Warm-up 5 min
Video Analysis 10 min
Main Activity 25 min
Closing Case 5 min
Facilitation Notes
1. Warm-up: The Perimeter Pitfall
Display the slide with the error: \(2 + 3 \times 4 = 20\). Ask students to identify why this is wrong. They should recognize that multiplication must occur before addition.
2. Video Viewing: Strategic Pausing
Watch the provided video from 4:29 to 8:23. Pause at the following timestamps for discussion:
5:00: Ask: "Before the narrator solves it, what is the first step according to PEMDAS?"7:23: The narrator arrives at \(36 - 135\). Ask: "Will our answer be positive or negative? Why?"
3. Main Activity: Spot the Error
Hand out the "Case File" worksheets. Instruct students to act as "Forensic Math Investigators." They must find the specific PEMDAS rule that was broken in each "faulty" solution.
Answer Key: Case File Worksheet
Case #1: The Exponent Jump
Expression: \(2(x+3)^2 - 4\) for \(x=2\)
THE ERROR:
The student multiplied \(2 \times 5\) before squaring. (Multiplication before Exponents).
CORRECT SOLUTION:
\(2(2+3)^2 - 4 = 2(5)^2 - 4 = 2(25) - 4 = 50 - 4 = 46\)
Case #2: The Stealth Subtracter
Expression: \(x^2 - 5(x-y)^3\) for \(x=6, y=3\)
THE ERROR:
The student subtracted \(36 - 5\) before multiplying by \(27\). (Subtraction before Multiplication).
CORRECT SOLUTION:
\(36 - 5(3)^3 = 36 - 5(27) = 36 - 135 = -99\)
Case #3: The Parentheses Panic
Expression: \(3x^2 + 2y\) for \(x=4, y=5\)
THE ERROR:
The student calculated \((3 \times 4)^2\) as \(12^2 = 144\). They should have squared \(4\) first.
CORRECT SOLUTION:
\(3(4)^2 + 2(5) = 3(16) + 10 = 48 + 10 = 58\)
Closure Discussion Prompt
"Why is \((x-y)^3\) different from \(x^3 - y^3\)? Try it with \(x=4, y=2\)."
Answer: \((4-2)^3 = 2^3 = 8\). But \(4^3 - 2^3 = 64 - 8 = 56\). Parentheses mandate subtraction *before* the exponent.
Case File Worksheet Case File: Operation Investigation
Classification: Top Secret / Pre-Algebra Math Forensics
Investigator: _____________________
Date: _________________________
Mission Briefing
We have intercepted five solved algebraic expressions. Intelligence suggests that three of these solutions contain critical Order of Operations (PEMDAS) violations. Your mission: Identify the violation, explain the mistake, and provide the correct mathematical result.
CASE #1
The Exponent Jump
Subject Expression
\(2(x+3)^2 - 4\) for \(x=2\)
Suspect Solution
Step 1: \(2(2+3)^2 - 4\)
Step 2: \(2(5)^2 - 4\)
Step 3: \((10)^2 - 4\)
Step 4: \(100 - 4 = 96\)
Identify the Violation (Explain the error)
The Correct Result (Show work)
CASE #2
The Rational Route
Subject Expression
\(x^2 - 4(x-1)\) for \(x=5\)
Suspect Solution
Step 1: \(5^2 - 4(5-1)\)
Step 2: \(25 - 4(4)\)
Step 3: \(25 - 16\)
Step 4: \(9\)
Is this solution clean or compromised?
The Correct Result (Show work)
CASE #3
The Stealth Subtracter
Subject Expression
\(x^2 - 5(x-y)^3\) for \(x=6, y=3\)
Suspect Solution
Step 1: \(6^2 - 5(6-3)^3\)
Step 2: \(36 - 5(3)^3\)
Step 3: \(31(27)\)
Step 4: \(837\)
Identify the Violation (Explain the error)
The Correct Result (Show work)
CASE #4
The Parentheses Panic
Subject Expression
\(3x^2 + 2y\) for \(x=4, y=5\)
Suspect Solution
Step 1: \(3(4)^2 + 2(5)\)
Step 2: \((12)^2 + 10\)
Step 3: \(144 + 10\)
Step 4: \(154\)
Identify the Violation (Explain the error)
The Correct Result (Show work)
CASE #5
The Square Secure
Subject Expression
\(x + y(x-2)^2\) for \(x=4, y=3\)
Suspect Solution
Step 1: \(4 + 3(4-2)^2\)
Step 2: \(4 + 3(2)^2\)
Step 3: \(4 + 3(4)\)
Step 4: \(4 + 12 = 16\)
Is this solution clean or compromised?
The Correct Result (Show work)
Operation Investigation Slides Top Secret: Math Forensics
Operation
Investigation
Applying PEMDAS to evaluate complex algebraic expressions.
Analyze
Identify
Resolve
Warm-up Analysis
05:00 MIN
Review: PEMDAS
Parentheses
Exponents
M/D Mult & Div (L to R)
A/S Add & Sub (L to R)
Spot the Violation:
\(2 + 3 \times 4 = 20\)
"Why is this solution illegal in the world of algebra? What rule was broken?"
Surveillance Footage
4:29 - 8:23
Embedded media
Investigation Notes
Ex 4: Watch for Parentheses priority.
Ex 5: Focus on negative signs and cubing.
Pro Tip: Substitute using parentheses.
Strategic Pause Points:
5:00 - Predict the next step.
The "Common Pitfall" Files
Multiplication First?
\(2x^2\)
ERROR: Multiplying 2 by x, then squaring the result.
REMEDY: Square the variable before multiplying.
Premature Subtraction
\(36 - 5(27)\)
ERROR: Doing \(36 - 5 = 31\), then multiplying by 27.
REMEDY: Multiply 5 and 27 before subtracting from 36.
Active Mission:
Spot the Error
Review the Case File worksheet.
Scan each solution for PEMDAS violations.
Explain the Error in the box.
Solve the case with the Correct Result.
FIELD RULES
Use your Scientific Calculator to verify.
Be specific! Mention the step where it failed.
Consult with your partner on Case #3.
TIME ALLOTTED: 25 MINUTES
Work through all 5 cases. Accuracy is vital.
Case Closed Discussion
"Are these expressions equal? Why or why not?"
\((x-y)^3\)
vs
\(x^3 - y^3\)
Test it with \(x=4\) and \(y=2\). Show your evidence!
