Variable Vortex Worksheet
Math Lab Blueprint
VARIABLE VORTEX
Name:
Date:
MISSION BRIEFING
Master the Vortex of variables, parentheses, and decimals. Distribute negative values carefully, gather your like terms, and isolate the variable. All solutions are clean integers!
Model Analysis: Problem 0 Example Solution
\(4(0.5n - 3) = n - 0.25(12 - 8n)\)
Step 1: Expand Parentheses
Distribute carefully. Watch signs!
\(2n - 12 = n - 3 + 2n\)
Step 2: Combine Like Terms
Combine \(n + 2n\) on the right.
\(2n - 12 = 3n - 3\)
Step 3: Collect Variables
Subtract \(2n\) from both sides.
\(-12 = n - 3\)
Step 4: Isolate & Solve
Add \(3\) to both sides to finish.
\(n = -9\)
Active Blueprint 1 Solve for \(x\)
\(6(0.5x - 2) = 2x - 0.2(10 - 15x)\)
Show Work Here
\(x =\)
Active Blueprint 2 Solve for \(y\)
\(8(0.25y + 3) = y - 0.5(12 - 6y)\)
Show Work Here
\(y =\)
UNIT 3: MULTI-STEP EQUATIONS PAGE 1 OF 2
Math Lab Blueprint | Variable Vortex Worksheet
Student Copy
Active Blueprint 3 Solve for \(z\)
\(5(0.8z - 3) = z - 0.4(15 - 10z)\)
Show Work Here
\(z =\)
Active Blueprint 4 Solve for \(w\)
\(10(0.3w + 2) = w - 0.25(16 - 12w)\)
Show Work Here
\(w =\)
Vortex Trap Analysis
A student made a common sign mistake when solving the following equation. Look closely at their work below:
Equation
\(2(0.5k - 4) = k - 0.1(30 - 20k)\)
Student's Steps
Step 1: \(k - 8 = k - 3 - 2k\)
Step 3: \(2k - 8 = -3\)
Step 2: \(k - 8 = -k - 3\)
Step 4: \(k = 2.5\)
Identify the trap: Which step contains the error? Explain what was done wrong and provide the correct step.
UNIT 3: MULTI-STEP EQUATIONS PAGE 2 OF 2
Variable Vortex Answer Key
Teacher Edition
ANSWER KEY & GUIDE
Unit 3: Linear Equations
PEDAGOGICAL INSIGHT
This lesson targets the intersection of multi-step expansion and decimal arithmetic. The primary misconception occurs when distributing negative rational numbers. Remind students that multiplying a negative coefficient by a subtraction term inside parentheses yields a positive product.
Active Blueprint 1 Solved Target Variable: \(x\)
Steps
\(6(0.5x - 2) = 2x - 0.2(10 - 15x)\)
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Expand: \(3x - 12 = 2x - 2 + 3x\)
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Combine: \(3x - 12 = 5x - 2\)
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Collect: \(-12 = 2x - 2\)
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Isolate: \(-10 = 2x\)
Final Ans: \(x = -5\)
Vortex Trap Highlight
Students often write -3x instead of +3x when distributing -0.2(-15x).
Quick Check
LHS = RHS = -27
Active Blueprint 2 Solved Target Variable: \(y\)
Steps
\(8(0.25y + 3) = y - 0.5(12 - 6y)\)
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Expand: \(2y + 24 = y - 6 + 3y\)
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Combine: \(2y + 24 = 4y - 6\)
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Collect: \(24 = 2y - 6\)
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Isolate: \(30 = 2y\)
Final Ans: \(y = 15\)
Vortex Trap Highlight
Students fail to distribute to BOTH terms. Watch for errors yielding -6y (failing to distribute) or -3y (sign error).
Quick Check
LHS = RHS = 54
UNIT 3: MULTI-STEP EQUATIONS ANSWER KEY PAGE 1 OF 2
Teacher Edition | Variable Vortex Answer Key
Answer Key
Active Blueprint 3 Solved Target Variable: \(z\)
Steps
\(5(0.8z - 3) = z - 0.4(15 - 10z)\)
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Expand: \(4z - 15 = z - 6 + 4z\)
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Combine: \(4z - 15 = 5z - 6\)
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Collect: \(-15 = z - 6\)
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Isolate: \(z = -9\)
Final Ans: \(z = -9\)
Vortex Trap Highlight
Decimals like 0.8 and 0.4 distribute to clean integers. If students have decimals in their intermediate steps, they made a computational mistake.
Quick Check
LHS = RHS = -51
Active Blueprint 4 Solved Target Variable: \(w\)
Steps
\(10(0.3w + 2) = w - 0.25(16 - 12w)\)
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Expand: \(3w + 20 = w - 4 + 3w\)
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Combine: \(3w + 20 = 4w - 4\)
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Collect: \(20 = w - 4\)
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Isolate: \(w = 24\)
Final Ans: \(w = 24\)
Vortex Trap Highlight
Remind students that multiplying by 0.25 is equivalent to dividing by 4, helping them compute 0.25(16) and 0.25(12w) easily.
Quick Check
LHS = RHS = 92
Vortex Trap Analysis Solution
Expected Student Response:
Step Containing Error: Step 1
Explanation: The student incorrectly distributed the negative coefficient. When distributing \(-0.1\) to \(-20k\), they wrote \(-2k\). Since a negative times a negative is positive, it should be \(+2k\).
Correct Equation Sequence:
• Step 1: \(k - 8 = k - 3 + 2k\)
• Step 2: \(k - 8 = 3k - 3\)
• Step 3: \(-8 = 2k - 3\)
• Step 4: \(-5 = 2k \implies k = -2.5\)
UNIT 3: MULTI-STEP EQUATIONS ANSWER KEY PAGE 2 OF 2
