Equation Enigmas Worksheet
Equation Enigmas Packet
Algebraic Fluency Series
NAME:
DATE:
PART 1: GUIDED PROGRESSIONS (Q1 - Q2)
Follow the written guides below. Fill in your resulting equivalent equations on each line, utilizing the maximized vertical space for scratch calculations.
QUESTION 1 \(5(3 - 2x) + 14 = -31\)
1. DISTRIBUTE:
2. COMBINE CONSTANTS:
3. ISOLATE X TERM:
4. FINAL ANSWER:
x =
QUESTION 2 \(4(2w - 5) - 7 = 5\)
1. DISTRIBUTE:
2. COMBINE CONSTANTS:
3. ISOLATE W TERM:
4. FINAL ANSWER:
w =
Unit 2: Linear Equation Fluency Page 1 of 4
Equation Enigmas Challenge
STAGE 1 PROGRESSION (Q3 - Q4)
Student Assessment
QUESTION 3 \(7(1 - 3y) + 26 = 12\)
1. DISTRIBUTE:
2. COMBINE CONSTANTS:
3. ISOLATE Y TERM:
4. FINAL ANSWER:
y =
QUESTION 4 MAX WORKSPACE
Worth 4 Points
Solve the equation below. Utilize the massive workspace box below to detail every step.
\(-3(4k - 2) + 15 = 45\)
Step-by-Step Mathematical Pathway
Final Calculated Solution:
Unit 2: Linear Equation Fluency Page 2 of 4
Equation Enigmas Challenge
STAGE 2 PROGRESSION (Q5 - Q7)
Student Assessment
QUESTION 5 MAX WORKSPACE
Worth 4 Points
Solve the equation below, ensuring your mathematical justification is fully complete:
\(8(2 - 5z) + 13 = -11\)
Step-by-Step Mathematical Pathway
Final Calculated Solution:
PART 2: EQUIVALENT PATHS (SELECT TWO OPTIONS)
For each problem below, analyze the starting equation. Select the TWO options that represent equivalent pathways. Use the workspace scratchpad to test steps.
QUESTION 6 \(12(5 - 2x) + 14x = 44\)
\(60 - 2x + \dots\)
\(60 - 24x + 14x = 44\)
\(60 - 24x = 528\)
\(60 - 10x = 44\)
\(60 + 10x = 44\)
My Workspace (Distribute / Combine)
Unit 2: Linear Equation Fluency Page 3 of 4
Equivalent Path Analysis
STAGE 3 CHALLENGE (Q7 - Q10)
Standards Aligned
QUESTION 7 \(6(3 - 4p) - 8p = 82\)
\(18 - 4p - \dots\)
\(18 - 24p - 8p = 82\)
\(18 - 16p = 82\)
\(18 - 32p = 82\)
My Workspace
QUESTION 8 \(-3(4 + 5m) + 11m = -36\)
\(-12 + 5m\dots\)
\(-12 - 15m\dots\)
\(-12 + 15m\dots\)
\(-12 - 4m = \dots\)
My Workspace
QUESTION 9 \(15(2 - g) + 8g = 51\)
\(30 - g + \dots\)
\(30 - 15g\dots\)
\(30 - 23g = 51\)
\(30 - 7g = 51\)
My Workspace
QUESTION 10 \(-8(3 - 2b) - 9b = -3\)
\(-24 - 16b\dots\)
\(-24 + 16b - 9b\)
\(-24 - 25b = -3\)
\(-24 + 7b = -3\)
My Workspace
Synthesis Reflection Box
Explain in 2-3 complete sentences why distributing a negative multiplier outside parenthesis reverses the sign of terms inside. Give a solid mathematical justification:
Unit 2: Linear Equation Fluency Page 4 of 4
Equation Enigmas Answer Key
Algebraic Fluency Series • Teacher Resource
EQUATION ENIGMAS ANSWER KEY
FULL SOLUTIONS
PART 1 KEY: SCAFFOLDED PROGRESSIONS (Q1 - Q2)
Step-by-step key showing exact expected student equations for intermediate steps.
QUESTION 1 KEY \(5(3 - 2x) + 14 = -31\)
1. DISTRIBUTE: \(15 - 10x + 14 = -31\)
2. COMBINE CONSTANTS: \(29 - 10x = -31\)
3. ISOLATE X TERM: \(-10x = -60\)
4. FINAL ANSWER: x = 6
QUESTION 2 KEY \(4(2w - 5) - 7 = 5\)
1. DISTRIBUTE: \(8w - 20 - 7 = 5\)
2. COMBINE CONSTANTS: \(8w - 27 = 5\)
3. ISOLATE W TERM: \(8w = 32\)
4. FINAL ANSWER: w = 4
Unit 2: Linear Equation Fluency (Teacher Key) Page 1 of 4
Teacher Facilitation Guide
STAGE 1 SOLUTIONS (Q3 - Q4)
Standards Aligned
QUESTION 3 KEY \(7(1 - 3y) + 26 = 12\)
1. DISTRIBUTE: \(7 - 21y + 26 = 12\)
2. COMBINE CONSTANTS: \(33 - 21y = 12\)
3. ISOLATE Y TERM: \(-21y = -21\)
4. FINAL ANSWER: y = 1
QUESTION 4 KEY CORRECT VALUE: k = -2
Solve: \(-3(4k - 2) + 15 = 45\)
Expected Student Solution Pathway:
\(-3(4k - 2) + 15 = 45\)
\(\Rightarrow -12k + 6 + 15 = 45\) (Step 1: Distribute \(-3\); watch for \(-3 \times -2 = +6\))
\(\Rightarrow -12k + 21 = 45\) (Step 2: Combine constants \(6 + 15\))
\(\Rightarrow -12k = 24\) (Step 3: Subtract 21 from both sides)
\(\Rightarrow k = -2\) (Step 4: Divide by \(-12\))
Unit 2: Linear Equation Fluency (Teacher Key) Page 2 of 4
Teacher Facilitation Guide
STAGE 2 SOLUTIONS (Q5 - Q6)
Standards Aligned
QUESTION 5 KEY CORRECT VALUE: z = 1
Solve: \(8(2 - 5z) + 13 = -11\)
Expected Student Solution Pathway:
\(8(2 - 5z) + 13 = -11\)
\(\Rightarrow 16 - 40z + 13 = -11\) (Step 1: Distribute \(8\))
\(\Rightarrow 29 - 40z = -11\) (Step 2: Combine constants \(16 + 13\))
\(\Rightarrow -40z = -40\) (Step 3: Subtract 29 from both sides; watch for \(-11 - 29 = -40\))
\(\Rightarrow z = 1\) (Step 4: Divide by \(-40\))
Q6 Equation: \(12(5 - 2x) + 14x = 44\) CORRECT: B & D
