Frankenstein Equations Slides Experiment Log #10-A
Frankenstein Equations
Identifying the structural limits of algebra and the power of numerical approximation.
The Sorting Challenge
With your lab group, sort the provided cards into two distinct containment zones:
Zone A: Algebraic. $x$ can be isolated using standard operations.Zone B: Numerical. $x$ is trapped; requires "guessing" or graphing.
"Can you get x alone on one side without it appearing on the other?"
[ Place Cards Here ]
The Dead End
0:00 - 8:05
Embedded media
Watch for:
"We get stuck pretty quickly... we're not going to get anywhere anytime soon."
Key Method: Bracketing
Successive approximation is just a systematic way of narrowing down where the solution lives.
Low Guess Solution High Guess
x = 4
Example 1
x ≈ 2.5
Example 2
The "Hard" Checklist
Check your card sort. Did you spot these Mixed Function Families ?
The Hybrid
Exponentials + Linear/Quadratic
\(5^x = 25x + 51\)
The Radical
Polynomials + Roots
\(\frac{1}{2}x^2 - 20 = \sqrt{20x + 5}\)
The Transcendental
Logarithms + Rational Terms
\(\ln(x) = \frac{1}{x}\)
Activity: Frankenstein's Lab
1 Create Your Creature
Mix two different function families (Trig, Log, Exp, Root, Polynomial) to create a "unsolvable" equation.
2 The Mutation Log
Explain exactly why algebra fails. Which term is "trapping" the variable?
3 Peer Containment
Swap papers. Find a whole-number approximation that gets within 1 unit of the solution.
Pro-Tip:
Choose families that grow at different rates (like $2^x$ vs $x^2$) to ensure they cross each other!
Equation Cards Deck Equation Cards Deck
Activity: Frankenstein Equations Card Sort
Cut along dashed lines
Instructions:
Sort these 12 equations into two piles: Solvable by Algebra (can isolate \(x\)) and Needs Approximation (mixed function families).
#001
\(3x - 7 = 14 + x\)
#002
\(2^x = \frac{30}{x-2}\)
#003
\(x^2 - 5x + 6 = 0\)
#004
\(x + \cos(x) = 0\)
#005
\(\log_2(x) = 5\)
#006
\(5^x = 25x + 51\)
#007
\(2^{x+1} = 16\)
#008
\(\ln(x) = \frac{1}{x}\)
#009
\(\frac{x}{x-1} = 2\)
#010
\(\frac{1}{2}x^2 - 20 = \sqrt{20x + 5}\)
#011
\(x^2 = 25\)
#012
\(x \cdot 3^x = 10\)
End of card deck. Ensure students have scissors and two envelopes or sorting mats.
Card Sort Answer Key Card Sort Answer Key
Teacher Reference • Frankenstein Equations
Math 10
Algebraic Methods
\(3x - 7 = 14 + x\)
Linear: Combine like terms and isolate.
\(x^2 - 5x + 6 = 0\)
Quadratic: Factor into \((x-2)(x-3)\).
\(\log_2(x) = 5\)
Logarithmic: Rewrite as \(2^5 = x\).
\(2^{x+1} = 16\)
Exponential: Same base (\(16 = 2^4\)).
\(\frac{x}{x-1} = 2\)
Rational: Multiply by denominator.
\(x^2 = 25\)
Square Root: \(\pm\sqrt{25}\).
Numerical Methods
\(2^x = \frac{30}{x-2}\)
Mixed: Exp + Rational (Trapped x).
\(x + \cos(x) = 0\)
Mixed: Polynomial + Trig.
\(5^x = 25x + 51\)
Mixed: Exp + Polynomial.
\(\ln(x) = \frac{1}{x}\)
Mixed: Log + Rational.
\(\frac{1}{2}x^2 - 20 = \sqrt{20x + 5}\)
Mixed: Polynomial + Root.
\(x \cdot 3^x = 10\)
Mixed: Polynomial factor + Exp.
The "Trap" Rule
A key identifier for students: If the variable \(x\) appears in two different "families" (e.g., inside an exponent AND on the base level), standard algebraic operations that isolate one \(x\) will usually bury the other deeper into a new function. This is what necessitates Successive Approximation .
Frankenstein Lab Worksheet Frankenstein's Lab
Scientist Name
Date
OBJECTIVE: Construct and approximate a transcendental creature.
1
The Monstrous Equation
Mix two function families (e.g., Trig + Linear, Log + Root, Exponential + Polynomial) to create your own "unsolvable" equation. Make sure both sides of the equation have different types of growth!
Left Side Family
Right Side Family
2
Why Algebra Fails
Describe the "trap." If you tried to isolate \(x\), what would happen? Why can't you get all the \(x\)'s together in one happy family?
3
Peer Containment Protocol
Swap papers with a partner. Their creature is loose! Use **Successive Approximation** (whole numbers only) to find the nearest integer solution. Show your guesses below.
Guess #1
Result: Too high / Too low?
Guess #2
Result: Too high / Too low?
Guess #3
Result: Too high / Too low?
The Containment Value:
x ≈
