Equation Enigma Task Sheet
Rich Task Investigation Multi-Step Equations (Single-Side Variable)
THE EQUATION ENIGMA
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Scenario: The Master Scale Calibration
An archivist must determine the exact mass \(w\) of an ancient relic. The relic items are placed on the left platform of a precision dual-pan beam balance, which is balanced against a certified solid brass counterweight of exactly \(97\text{ kg}\) on the right platform.
Left Pan \(4\) sealed crates (each holding \(2\) relics of weight \(w\) plus \(5\text{ kg}\) padding) plus \(3\) loose relics of weight \(w\).
Right Pan A single certified standard brass counterweight measuring exactly \(97\text{ kg}\).
1
Formulate the Mathematical Model
Define your variable clearly, write the algebraic expression for the left pan, and set up an equation equal to the known counterweight.
Variable Definition:
Left Pan Expression:
Complete Equation:
2
Solve with Step-by-Step Mathematical Justification
State each property applied (e.g., Distributive, Combine Like Terms, Subtraction POE)
Algebraic Steps
Mathematical Reason / Property Applied
Solution Verification: Substitute your value of \(w\) back into the left pan expression to verify equality with \(97\text{ kg}\).
Left Pan Value:
Relic Mass \(w\) =
Part II: Strategy Showdown
COMPARING SOLUTION PATHS
Target Equation: \(4(2x - 3) + 5 = 41\)
Maya: Distribute & Combine First Standard Expansion
\(4(2x - 3) + 5 = 41\)
\(8x - 12 + 5 = 41\) (Distributes 4)
\(8x - 7 = 41\) (Combines \(-12 + 5\))
\(8x = 48\) (Adds 7 to both sides)
\(x = 6\) (Divides by 8)
Devon: Isolate Grouped Term First Inverse Operations
\(4(2x - 3) + 5 = 41\)
\(4(2x - 3) = 36\) (Subtracts 5 first)
\(2x - 3 = 9\) (Divides both sides by 4)
\(2x = 12\) (Adds 3 to both sides)
\(x = 6\) (Divides by 2)
A. Mathematical Justification: Explain why Devon's first step (subtracting 5 before distributing 4) is mathematically sound. How does it align with inverse operations?
B. Strategic Evaluation: If the equation were \(4(2x - 3) + 5 = 40\), which method would be more convenient and lead to cleaner arithmetic? Explain why.
Part III: Constraint Puzzle
THE COEFFICIENT INVESTIGATION
Consider the equation with variables only on the left side: \(5(x - 3) - \mathbf{A}x = \mathbf{B}\). Determine values for \(\mathbf{A}\) and \(\mathbf{B}\) to satisfy each condition.
Infinite Solutions Identity
True for all real values of \(x\).
\(\mathbf{A} =\)
\(\mathbf{B} =\)
No Solution Contradiction
No real number makes this true.
\(\mathbf{A} =\)
\(\mathbf{B} \neq\)
Unique: \(x = 4\) One Solution
Choose an integer pair for \(x = 4\).
\(\mathbf{A} =\)
\(\mathbf{B} =\)
Architect Challenge: Design a Single-Sided Variable Equation Constraint: Variable only on one side • Requires distributing • Solution \(x = 3\)
Your Equation:
Proof / Check Step:
Focus: CCSS.MATH.CONTENT.8.EE.C.7.B & HSA.REI.A.1, B.3 The Equation Enigma • Rich Task
Equation Enigma Answer Key
Teacher Answer Key & Guide Multi-Step Equations (Single-Side Variable)
THE EQUATION ENIGMA • KEY
TOTAL SCORE: 20 PTS
Standards: 8.EE.C.7b, HSA-REI.A.1
Task Objective: Model a physical balance scenario into a single-sided variable equation and justify each inverse operation.
Target: \(w = 7\text{ kg}\)
1
Formulate the Mathematical Model (3 pts)
1 pt each field
Variable Definition:
Let \(w =\) mass of one relic (in kg)
Left Pan Expression:
\(4(2w + 5) + 3w\)
Complete Equation:
\(4(2w + 5) + 3w = 97\)
2
Step-by-Step Algebraic Justification (8 pts)
4 pts steps • 4 pts reasons
Algebraic Steps
Mathematical Reason / Property Applied
\(4(2w + 5) + 3w = 97\)
Given equation (balanced scale scenario)
\(8w + 20 + 3w = 97\)
Distributive Property of Multiplication
\(11w + 20 = 97\)
Combine Like Terms (\(8w + 3w = 11w\))
\(11w = 77\)
Subtraction Property of Equality (subtract 20)
\(w = 7\)
Division Property of Equality (divide by 11)
Substitution Check: Final Answer: \(w = 7\text{ kg}\)
\(4(2(7) + 5) + 3(7) = 4(14 + 5) + 21 = 4(19) + 21 = 76 + 21 = 97\text{ kg}\quad\checkmark\text{ (Equal to Right Pan)}\)
Common Misconceptions to Watch For:
- Over-distributing: Students multiplying 4 by \(3w\) as well: \(8w + 20 + 12w = 97\) (warn that parenthesis closes before \(+ 3w\)).
- Premature addition: Adding \(5 + 3w\) inside parentheses before distributing.
- Property naming: Confusing the Subtraction Property of Equality with combining like terms.
Equation Enigma • Teacher Solutions Page 1 of 2
Teacher Answer Key
STRATEGY ANALYSIS & PUZZLE SOLUTIONS
Target Equation: \(4(2x - 3) + 5 = 41\)
Part II: Strategy Showdown Model Answers (4 pts)
2 pts each prompt
A. Mathematical Justification (Why Devon's move is valid):
Exemplar Response: Devon treats the grouped binomial \((2x - 3)\) as a single composite quantity. Subtracting 5 from both sides uses the Subtraction Property of Equality (\(4(2x - 3) = 36\)). Then dividing both sides by 4 uses the Division Property of Equality (\(2x - 3 = 9\)). This rigorously follows the reverse order of operations (SADMEP) to peel away external operations before entering the group.