📊 Estimation Sprints

Master essential estimation and approximation techniques through focused practice drills. These exercises will build speed and accuracy in quick approximation and bounds calculations.

🎯 Core Estimation Skills

Order of Magnitude

  • Powers of 10: $10^0 = 1$, $10^1 = 10$, $10^2 = 100$, $10^3 = 1,000$
  • Scientific notation: $2.5 \times 10^3 = 2,500$
  • Quick estimates: $47 \times 23 \approx 50 \times 20 = 1,000$ (actual: 1,081)
  • Magnitude checks: Is answer too big or too small?

Rounding Strategies

  • Round to nearest 10: $47 + 38 \approx 50 + 40 = 90$ (actual: 85)
  • Round to nearest 100: $298 + 156 \approx 300 + 200 = 500$ (actual: 454)
  • Round to nearest 1000: $1,234 + 2,567 \approx 1,000 + 3,000 = 4,000$ (actual: 3,801)
  • Round to nearest power of 10: $47 \times 23 \approx 50 \times 20 = 1,000$

Approximation Techniques

  • Linear approximation: $f(x) \approx f(a) + f’(a)(x-a)$
  • Taylor series: Use first few terms
  • Binomial approximation: $(1+x)^n \approx 1 + nx$ for small $x$
  • Exponential approximation: $e^x \approx 1 + x$ for small $x$

⚡ Quick Approximation Methods

Common Approximations

  • $\sqrt{2} \approx 1.414$
  • $\sqrt{3} \approx 1.732$
  • $\sqrt{5} \approx 2.236$
  • $\pi \approx 3.1416$
  • $e \approx 2.718$
  • $\ln 2 \approx 0.693$
  • $\ln 3 \approx 1.099$

Fraction Approximations

  • $\frac{1}{8} = 0.125$
  • $\frac{3}{8} = 0.375$
  • $\frac{5}{8} = 0.625$
  • $\frac{7}{8} = 0.875$
  • $\frac{1}{3} \approx 0.333$
  • $\frac{2}{3} \approx 0.667$

Power Approximations

  • $2^{10} \approx 1,000$
  • $3^4 = 81$
  • $5^3 = 125$
  • $7^2 = 49$
  • $2^{20} \approx 1,000,000$

🎯 Practice Drills

5-Minute Sprint: Order of Magnitude

Target: 20 problems in 5 minutes (90%+ accuracy)

  1. What is the order of magnitude of $47 \times 23$?
  2. What is the order of magnitude of $156 \div 12$?
  3. What is the order of magnitude of $2^{10}$?
  4. What is the order of magnitude of $3^5$?
  5. What is the order of magnitude of $5^4$?

5-Minute Sprint: Rounding and Estimation

Target: 25 problems in 5 minutes (85%+ accuracy)

  1. Estimate $47 + 38$ by rounding to nearest 10
  2. Estimate $298 + 156$ by rounding to nearest 100
  3. Estimate $1,234 + 2,567$ by rounding to nearest 1000
  4. Estimate $47 \times 23$ by rounding to nearest 10
  5. Estimate $156 \div 12$ by rounding to nearest 10

5-Minute Sprint: Approximation Techniques

Target: 15 problems in 5 minutes (80%+ accuracy)

  1. Approximate $\sqrt{101}$ using linear approximation
  2. Approximate $\sqrt{99}$ using linear approximation
  3. Approximate $\sqrt{102}$ using linear approximation
  4. Approximate $\sqrt{98}$ using linear approximation
  5. Approximate $\sqrt{103}$ using linear approximation

🔢 Advanced Estimation Techniques

Bounds and Inequalities

  • Upper bounds: Find maximum possible value
  • Lower bounds: Find minimum possible value
  • Arithmetic-Geometric Mean: $\frac{a+b}{2} \geq \sqrt{ab}$ for $a, b > 0$
  • Cauchy-Schwarz: $(a_1b_1 + a_2b_2)^2 \leq (a_1^2 + a_2^2)(b_1^2 + b_2^2)$
  • Triangle inequality: $|a + b| \leq |a| + |b|$

Extreme Value Testing

  • Boundary cases: Test at limits and extremes
  • Special values: Test $x = 0, 1, -1$
  • Large values: Test with large numbers
  • Small values: Test with small numbers
  • Zero cases: Test when variables equal zero

Monotonicity

  • Increasing functions: Larger input → larger output
  • Decreasing functions: Larger input → smaller output
  • Test two values: Compare function values
  • Use derivatives: If applicable
  • Apply properties: Use known monotonicity

🎯 Advanced Practice Drills

5-Minute Sprint: Bounds and Inequalities

Target: 12 problems in 5 minutes (75%+ accuracy)

  1. Find upper and lower bounds for $x^2 - 4x + 3$ on $[0, 4]$
  2. Find upper and lower bounds for $x^3 - 3x + 1$ on $[-2, 2]$
  3. Find upper and lower bounds for $x^4 - 2x^2 + 1$ on $[-1, 1]$
  4. Find upper and lower bounds for $x^2 + 2x - 3$ on $[-3, 1]$
  5. Find upper and lower bounds for $x^3 - 6x + 4$ on $[-2, 2]$

5-Minute Sprint: Extreme Value Testing

Target: 15 problems in 5 minutes (80%+ accuracy)

  1. Test $x = 0$ for $f(x) = x^2 - 4x + 3$
  2. Test $x = 1$ for $f(x) = x^2 - 4x + 3$
  3. Test $x = -1$ for $f(x) = x^2 - 4x + 3$
  4. Test $x = 2$ for $f(x) = x^2 - 4x + 3$
  5. Test $x = 3$ for $f(x) = x^2 - 4x + 3$

📊 Progress Tracking

Accuracy Targets

  • Order of magnitude: 90%+ accuracy
  • Rounding and estimation: 85%+ accuracy
  • Approximation techniques: 80%+ accuracy
  • Advanced techniques: 75%+ accuracy

Speed Targets

  • Order of magnitude: 4 problems per minute
  • Rounding and estimation: 5 problems per minute
  • Approximation techniques: 3 problems per minute
  • Advanced techniques: 2 problems per minute

Weekly Goals

  • Week 1: Master basic estimation and rounding
  • Week 2: Add approximation techniques, maintain accuracy
  • Week 3: Add advanced techniques, increase speed
  • Week 4: Master all areas, optimize speed

⚡ Quick Reference

Essential Approximations:

  • $\sqrt{2} \approx 1.414$
  • $\sqrt{3} \approx 1.732$
  • $\sqrt{5} \approx 2.236$
  • $\pi \approx 3.1416$
  • $e \approx 2.718$

Common Rounding:

  • Round to nearest 10: $47 \approx 50$
  • Round to nearest 100: $298 \approx 300$
  • Round to nearest 1000: $1,234 \approx 1,000$
  • Round to nearest power of 10: $47 \approx 50$

Order of Magnitude:

  • $10^0 = 1$
  • $10^1 = 10$
  • $10^2 = 100$
  • $10^3 = 1,000$
  • $10^6 = 1,000,000$

Practice Schedule:

  • Daily: 10 minutes of estimation practice
  • Focus areas: Work on your weakest skills
  • Progressive difficulty: Increase complexity over time
  • Time pressure: Practice under time constraints
  • Accuracy first: Speed comes with accuracy

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