Power Play WorksheetPower Play Algebra Architect Recovery • Lesson 1: Exponents Project Record Name: Structural Integrity: TEKS A.11.B Simplify numeric and algebraic expressions using the laws of exponents. DRAFTING NOTES: Multiplication Product Rule: \(x^a \cdot x^b = x^{a+b}\) Power to Power: \((x^a)^b = x^{a \cdot b}\) DRAFTING NOTES: Division Quotient Rule: \(\frac{x^a}{x^b} = x^{a-b}\) Negative Power: \(x^{-a} = \frac{1}{x^a}\) Phase 1: Foundation Practice 1 \(x^5 \cdot x^8\) 2 \((m^4)^3\) 3 \(\frac{b^{12}}{b^4}\) 4 \(\frac{24k^8}{6k^2}\) Phase 2: Complex Assemblies 5. Simplify: \((2a^3b^4)^2 \cdot (3a^2b)\) Level: Structural 6. Simplify: \(\frac{15x^5y^2}{3x^8y^{-4}}\) Level: Balance Safety Rule: Zero Power Any base raised to the power of 0 equals 1. (where base \(\neq 0\)) \(x^0 = 1\) Master Architect Challenge \[\left(\frac{4x^2y^3}{2x^5y}\right)^3\]
Growth Blueprints WorksheetGrowth Blueprints Algebra Architect Recovery • Lesson 1: Exponentials Name: Architectural Specs: TEKS A.9.B-D Identify features and construct equations: \(f(x) = ab^x\). \(f(x) = a \cdot b^x\) a: Start Value (y-int) | b: Factor Site Analysis: Visual Survey (0, 8) y-intercept: Asymptote: Domain: Range: Growth Decay Phase 2: Materials Inventory \(x\)\(y\)021102503250 a = b = \(y = \) _________________ \(x\)\(y\)06413221638 a = b = \(y = \) _________________ Phase 3: Project Narratives 1. A construction project starts with 500 workers and grows by 12% each month. Equation: \(w(t) = \) 2. An architect's digital drawing tablet costs $1,200 and loses 15% of its value per year. Equation: \(v(t) = \) Final Site Analysis Expert Valuation Function: \(V(t) = 45,000(1.035)^t\) A. Initial property value: $ B. Percentage rate of increase: % C. Projected valuation after 5 years (Round to nearest cent): $
Site Survey QuizSite Survey Quiz Algebra Architect Recovery • Lesson 1 NAME: DATE: Part I: Exponent Integrations 1. Simplify: \( (5x^3y^4) \cdot (2x^2y^6) \) 2. Simplify: \( \frac{36a^8b^{10}}{9a^2b^5} \) 3. Simplify: \( (4m^2n^5)^3 \) 4. Simplify: \( \frac{x^{-4}y^3}{z^0} \) Part II: Growth Analysis 5. Specifications for: \( g(x) = 50(2)^x \) y-intercept: Asymptote: Growth/Decay? Colony starts at 100 members and triples every hour. Construct equation (P) for hours (h). \( P = \) 6. Which represents exponential decay with a y-intercept of 12? \( f(x)=12(1.05)^x \) \( f(x)=12(0.85)^x \) \( f(x)=0.85(12)^x \) \( f(x)=5(12)^x \)
Architect Key Lesson 1Master Key Algebra Architect Recovery • Teacher Resource • Lesson 1 Teacher Use Only Power Play (Exponents) Answers 1. x⁵ · x⁸: x¹³ 2. (m⁴)³: m¹² 3. b¹²/b⁴: b⁸ 4. 24k⁸/6k²: 4k⁶ 5. (2a³b⁴)² · (3a²b): 12a⁸b⁹ Process: 4a⁶b⁸ · 3a²b 6. (15x⁵y²)/(3x⁸y⁻⁴): 5y⁶/x³ Process: 5x⁻³y⁶ Challenge Result: 8y⁶ / x⁹ Growth Blueprints Answers Visual Survey Analysis: y-intercept: (0, 8) Asymptote: y = 0 Domain: All Real Numbers Range: y > 0 Type: Decay Phase 2 Table Equations: Record 01: y = 2(5)ˣ Record 02: y = 64(0.5)ˣ Scenario 1 (Workers): w(t) = 500(1.12)ᵗ Scenario 2 (Tablet): v(t) = 1200(0.85)ᵗ Final Analysis (V=45000(1.035)ᵗ): A. Initial Value: $45,000 B. Growth Rate: 3.5% C. 5-Yr Valuation: $53,445.82 Site Survey Quiz Answers 10x⁵y¹⁰ 4a⁶b⁵ 64m⁶n¹⁵ y³ / x⁴ y-int: (0, 50); Asymp: y=0; Growth 6. Decay Model: Choice B Bacteria Calc: P = 100(3)ʰ
Polynomial Assembly WorksheetPolynomial Assembly Algebra Architect Recovery • Lesson 2: Operations Project Record Name: Structural Specs: TEKS A.10.A, A.10.D Combine materials and expand layouts using addition, subtraction, and the Box Method. Addition/Subtraction: Combine terms with the same variable and exponent. \( (3x^2 + 5x) + (2x^2 - 8x) = 5x^2 - 3x \) Multiplication: \(x\) \(+3\) \(x\) \(x^2\) \(3x\) \(+4\) \(4x\) \(12\) Organize distributions with a grid. Phase 1: Combinations (Add/Sub) 1. \( (4x^2 - 3x + 10) + (2x^2 + 8x - 5) \) 2. \( (x^2 + 7x) - (3x^2 - 4x + 2) \) Negative Load Phase 2: Expansion (Box Method) 3. \( 3x(4x^2 - 5x + 2) \) \(3x\) 4. \( (x + 5)(x - 3) \) \(x\)\(-3\) \(x\)\(+5\) 5. \( (2x + 4)(3x - 1) \) \(3x\)\(-1\) \(2x\)\(+4\) Result... Phase 3: Material Reduction Architect's Tip: Divide every term in the numerator by the denominator base. 6. \( \frac{20x^2 + 15x}{5x} \) 7. \( \frac{18x^4 - 12x^3 + 6x^2}{6x^2} \) Building Site Challenge Find total Area for width x + 4 and length 3x - 2. <table class="border-separate border-spacing-1.5"><tbody><tr><td></td><td class="text-center font-bold text-xs text-emerald-400 font-mono">\(3x\)</td><td class="text-center font-bold text-xs text-emerald-400 font-mono">\(-2\)</td></tr><tr><td class="pr-3 font-bold text-xs text-emerald-400 font-mono text-right">\(x\)</td><td class="w-12 h-12 border-2 border-emerald-600 bg-white/5 rounded-lg shadow-inner"></td><td class="w-12 h-12 border-2 border-emerald-600 bg-white/5 rounded-lg shadow-inner"></td></tr><tr><td class="pr-3 font-bold text-xs text-emerald-400 font-mono text-right">\(+4\)</td><td class="w-12 h-12 border-2 border-emerald-600 bg-white/5 rounded-lg shadow-inner"></td><td class="w-12 h-12 border-2 border-emerald-600 bg-white/5 rounded-lg shadow-inner"></td></tr></tbody></table> Area =
Factoring Blueprints WorksheetFactoring Blueprints Algebra Architect Recovery • Lesson 2: Factoring NAME: Deconstruction Analysis: TEKS A.10.E Factor \(ax^2 + bx + c\) using the AC Method and Box GCF Method. a·c ?? b 1. Factor Hunt: Find numbers that multiply to \(a \cdot c\) and add to \(b\). 2. Grid GCF: Pull out the GCF from each row and column. 1. Factor: \(x^2 + 7x + 10\) AC / b <table class="border-separate border-spacing-1"><tbody><tr><td></td><td class="w-12 h-6 border-b-2 border-slate-300"></td><td class="w-12 h-6 border-b-2 border-slate-300"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300 pr-1 text-right text-[7px] text-slate-300 font-bold italic">GCF</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">x²</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white shadow-inner"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300 pr-1 text-right text-[7px] text-slate-300 font-bold italic">GCF</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white shadow-inner"></td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">10</td></tr></tbody></table> (x + __)(x + __) 2. Factor: \(2x^2 + 11x + 5\) AC / b <table class="border-separate border-spacing-1"><tbody><tr><td></td><td class="w-12 h-6 border-b-2 border-slate-300"></td><td class="w-12 h-6 border-b-2 border-slate-300"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300 pr-1 text-right text-[7px] text-slate-300 font-bold italic">GCF</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">2x²</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300 pr-1 text-right text-[7px] text-slate-300 font-bold italic">GCF</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white"></td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">5</td></tr></tbody></table> 3. Factor: \(x^2 - 10x + 25\) AC / b <table class="border-separate border-spacing-1"><tbody><tr><td></td><td class="w-12 h-6 border-b-2 border-slate-300"></td><td class="w-12 h-6 border-b-2 border-slate-300"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300 pr-1 text-right text-[7px] text-slate-300 font-bold italic">GCF</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">x²</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white shadow-inner"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300 pr-1 text-right text-[7px] text-slate-300 font-bold italic">GCF</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white shadow-inner"></td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">25</td></tr></tbody></table> <table class="border-separate border-spacing-1"><tbody><tr><td></td><td class="w-12 h-6 border-b-2 border-slate-300"></td><td class="w-12 h-6 border-b-2 border-slate-300"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300"></td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">x²</td><td class="w-12 h-12 border-2 border-emerald-800 bg-white shadow-inner"></td></tr><tr><td class="w-8 h-12 border-r-2 border-slate-300"></td><td class="w-12 h-12 border-2 border-emerald-800 bg-white shadow-inner"></td><td class="w-12 h-12 border-2 border-emerald-800 bg-white flex items-center justify-center font-mono text-xs text-slate-300 font-bold">-24</td></tr></tbody></table>
Structural Integrity QuizStructural Integrity Algebra Architect Recovery • Quiz 02: Lesson 2 NAME: DATE: Ensure all algebraic work is displayed for verification credit. Part I: Operational Assembly 1. Simplify (Subtract): \( (6x^2 - 4x + 9) - (2x^2 + 5x - 3) \) 2. Multiply: \( (3x + 4)(2x - 5) \) <table class="border-separate border-spacing-0.5"><tbody><tr><td></td><td class="w-8 h-3 border-b border-red-200 bg-red-50"></td><td class="w-8 h-3 border-b border-red-200 bg-red-50"></td></tr><tr><td class="w-6 h-8 border-r border-red-200"></td><td class="w-8 h-8 border border-red-800 bg-white"></td><td class="w-8 h-8 border border-red-800 bg-white"></td></tr><tr><td class="w-6 h-8 border-r border-red-200"></td><td class="w-8 h-8 border border-red-800 bg-white"></td><td class="w-8 h-8 border border-red-800 bg-white"></td></tr></tbody></table> 3. Reduce (Division): \( \frac{24x^5 + 16x^4 - 8x^2}{4x^2} \) 4. Distribute: \( 4x^2(3x - 7) \) Part II: Deconstruction 5. Factor completely: \( x^2 - 11x + 24 \) AC / b <table class="border-separate border-spacing-1"><tbody><tr><td></td><td class="w-8 h-4 border-b border-red-300 bg-white"></td><td class="w-8 h-4 border-b border-red-300 bg-white"></td></tr><tr><td class="w-6 h-8 border-r border-red-300"></td><td class="w-8 h-8 border border-red-800 bg-white flex items-center justify-center font-mono text-[8px] text-slate-300">\( x^2 \)</td><td class="w-8 h-8 border border-red-800 bg-white"></td></tr><tr><td class="w-6 h-8 border-r border-red-300"></td><td class="w-8 h-8 border border-red-800 bg-white"></td><td class="w-8 h-8 border border-red-800 bg-white flex items-center justify-center font-mono text-[8px] text-slate-300">24</td></tr></tbody></table> Factors = 6. Factored identification: 25x² - 64 (5x - 8)(5x - 8) (5x + 8)(5x + 8) (5x - 8)(5x + 8) (25x - 1)(x + 64)
Architect Key Lesson 2Master Key Algebra Architect Recovery • Teacher Resource • Lesson 2 Teacher Use Only Polynomial Assembly Answers 1. Addition:6x² + 5x + 5 2. Subtraction:-2x² + 11x - 2 3. 3x(4x²-5x+2):12x³ - 15x² + 6x 4. (x+5)(x-3):x² + 2x - 15 5. (2x+4)(3x-1):6x² + 10x - 4 6. Div by 5x:4x + 3 7. Div by 6x²:3x² - 2x + 1 Final Site Challenge: Area = 3x² + 10x - 8 Factoring Blueprints Answers 1. x²+7x+10 (x+2)(x+5) 2. 2x²+11x+5 (2x+1)(x+5) 3. x²-10x+25 (x-5)² 4. x²-5x-24 (x-8)(x+3) 5. x²-81 (x-9)(x+9) 6. 4x²-49 (2x-7)(2x+7) Master Challenge (x⁴-16): (x²+4)(x-2)(x+2) Structural Integrity Quiz Answers 4x² - 9x + 12 6x² - 7x - 20 6x³ + 4x² - 2 12x³ - 28x² (x - 3)(x - 8) 6. Binomial ID: Choice C
