Grade 8 Mathematics
Real-World Problem Solving
Chapter 1
Math in Action: Grade 8 — Real-World Problem Solving
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GRADE 8 MATHEMATICS
Real-World Problem Solving
Complete Student Edition
GLOBAL SOVEREIGN UNIVERSITY "Building a Bridge to Freedom Through Education"
Grade 8 Mathematics: Real-World Problem Solving
Student Edition
Copyright © 2024 Global Sovereign University Press
All rights reserved. First Edition.
This textbook is provided FREE to everyone. No Login. Totally FREE to everyone. Visit this web page to download your free book.
https://www.globalsovereignuniversity.org/mathification
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Why This Book Is Different
🏆 NOBEL PRIZE RESEARCH SAYS:
"Students who score 96% on classroom math tests succeed only 1% of the time in
real life."
— Banerjee & Duflo, Nature, February 2025
The Research That Proves Traditional Math Education Fails
In February 2025, Nobel Prize-winning economists published a groundbreaking study in Nature, the world's most respected scientific journal. They tested over 1,400 children in India and discovered something shocking:
Children who ace classroom math CANNOT use it in the real world.
Students who scored 96% on standard school math problems could solve simple real-world market calculations only 1% of the time. Meanwhile, children who worked in markets—without formal education—solved the same problems with 96% accuracy.
The researchers' conclusion was clear:
"Schools need to build a bridge between math learned in the classroom and math
encountered in real-life situations."
The Problem: Abstract Doesn't Transfer to Applied
The Nobel laureates found that school children learned procedures but not understanding. They could solve '3x + 7 = 22' on a worksheet by following memorized steps. But give them a real problem—' You earned $22 from selling items at $3 each plus a $7 bonus. How many items did you sell?'—and they froze.
Market children used efficient mental strategies: estimation, breaking complex problems into simpler parts, and working with meaningful quantities. School children followed rigid, slow methods that worked for tests but failed in life.
It's not that school kids were less intelligent. It's that their education never connected the math to reality.
The GSU Solution: Real-World Math
This textbook does exactly what the Nobel laureates recommend. Instead of abstract problems, every calculation has a professional context:
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Math in Action: Grade 8 — Real-World Problem Solving
Traditional Math vs GSU Real-World Math
Traditional Math GSU Real-World Math "Solve: 3x + 7 = 22" "You're the Entrepreneur. You sold items for $3 each plus a $7 base. You made $22. How many items?" "Find the slope of y = 2x - 5." "As Project Manager, your team completes 2 modules/week, with 5 already done. When will you finish 15?" "Calculate √(3² + 4²)." "As a Surveyor, measure the distance across a lake using stakes 3m and 4m apart."
Same math. Completely different learning.
Your Roles in This Book
You won't just learn math in this book. You'll BE someone who uses it:
• The Data Scientist — Real numbers, rational and irrational • The Engineer — Exponents and square roots • The Space Scientist — Scientific notation applications • The Entrepreneur — Multi-step equations • The Project Manager — Functions and linear relationships • The Systems Analyst — Systems of equations • The Surveyor — Pythagorean theorem • The Animator — Transformations • The Architect — Angle relationships • The Industrial Designer — Volume of 3D figures • The Market Researcher — Data analysis and scatter plots • The Risk Analyst — Probability
These aren't pretend jobs. These are skills you can use THIS WEEK.
The Bottom Line
Nobel Prize-winning researchers proved that traditional education fails to prepare students for real-world math. The gap between classroom success and life success is catastrophic: 96% to 1%.
This textbook closes that gap.
Every problem. Every skill. Every chapter. Connected to reality.
“Because math isn't an academic exercise. It's a survival skill.”
Dr. Gene A Constant
Global Sovereign University is a 501(c)(3) educational foundation. This textbook is provided FREE because learning should never be limited by economics.
Research Citation: Banerjee, A., Duflo, E., et al. (2025). Children's arithmetic skills do not transfer between applied and academic mathematics. Nature. https://doi.org/10.1038/s41586-024-08502-w
The lead authors received the 2019 Nobel Prize in Economics "for their experimental approach to alleviating global poverty."
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Table of Contents
Chapter 1: The Data Scientist — Real Number System
Chapter 2: The Engineer — Exponents and Radicals
Chapter 3: The Space Scientist — Scientific Notation Applications
Chapter 4: The Entrepreneur — Multi-Step Equations
Chapter 5: The Project Manager — Functions and Linear Relationships
Chapter 6: The Systems Analyst — Systems of Equations
Chapter 7: The Surveyor — Pythagorean Theorem
Chapter 8: The Animator — Transformations
Chapter 9: The Architect — Angle Relationships
Chapter 10: The Industrial Designer — Volume of 3D Figures
Chapter 11: The Market Researcher — Data Analysis & Scatter Plots
Chapter 12: The Risk Analyst — Probability
Supplement A: Geometric Constructions
Supplement B: Transformations on the Coordinate Plane
Supplement C: Bivariate Data Analysis
Extra Practice Sections
Mixed Review
Cumulative Assessment
Answer Key
Welcome!
This year, you become a problem solver who uses mathematics in the real world.
You'll take on exciting roles:
• Data Scientist: Classify and work with real numbers • Engineer: Apply exponent rules and calculate roots • Space Scientist: Calculate with astronomical numbers • Entrepreneur: Solve complex business equations • Project Manager: Model relationships with functions • Systems Analyst: Solve interconnected problems • Surveyor: Measure distances with the Pythagorean theorem • Animator: Transform shapes and create motion • Architect: Design with angles and parallel lines • Industrial Designer: Calculate volumes of complex shapes • Market Researcher: Analyze trends and make predictions • Risk Analyst: Calculate probabilities for decisions
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Math in Action: Grade 8 — Real-World Problem Solving
How to Use This Book:
📋 The Scenario: Step into a real role
📚 The Skill: Learn the mathematics
✎ Worked Examples: See step-by-step solutions
📝 Practice Problems: Build mastery
📊 Action Reports: Apply everything
Remember: Show your work! The process matters as much as the answer.
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📊
CHAPTER 1 The Data Scientist
Master the real number system to classify and compute with any number
What You'll Learn
• Distinguish between rational and irrational numbers • Locate real numbers on the number line • Approximate irrational numbers • Perform operations with rational and irrational numbers
Why This Matters
Data scientists work with numbers constantly—from tiny decimals to massive datasets. Some numbers are clean and exact (like 0.5 or 3/4), while others go on forever without repeating (like π or √2). Understanding the difference helps you choose the right level of precision for calculations that matter.
Section 1.1: Classifying Numbers
The real number system organizes all numbers into categories based on their properties.
The Number Hierarchy
Natural Numbers (N): 1, 2, 3, 4, ... (counting numbers)
Whole Numbers (W): 0, 1, 2, 3, 4, ... (natural numbers + zero)
Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ... (whole numbers + negatives)
Rational Numbers (Q): Any number that can be written as a fraction a/b where a and b are integers and b ≠ 0
Irrational Numbers: Numbers that CANNOT be written as a fraction—their decimals go on forever without repeating
Real Numbers (R): All rational and irrational numbers combined
Rational: Can be written as a/b | Irrational: Cannot be written
as a/b
Recognizing Rational Numbers
A number is rational if its decimal representation either:
• Terminates (ends): 0.25, 3.5, -2.125 • Repeats: 0.333..., 0.142857142857..., 2.166666...
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Math in Action: Grade 8 — Real-World Problem Solving
WORKED EXAMPLE: Classifying 0.375
Step 1:Check if decimal terminates or repeats: 0.375terminates Step 2:Convert to fraction: 0.375 = 375/1000 = 3/8 Step 3:Since it can be written as a fraction, itis RATIONAL
Recognizing Irrational Numbers
Common irrational numbers:
• π ≈ 3.14159265358979... (never repeats, never ends) • √2 ≈ 1.41421356237... (square root of any non-perfect square) • e ≈ 2.71828182845... (Euler's number)
Practice Problems: Section 1.1
1.Classify each number as rational or irrational:0.666..., √5, -4/7, π, 3.14 Work: ________________________________________________________________ Answer: ____________________________
2.A data scientist records a measurement as 2.718281828...Is this rational or irrational? Explain. Work: ________________________________________________________________ Answer: ____________________________
3.Convert 0.875 to a fraction. Is it rational? Work: ________________________________________________________________ Answer: ____________________________
4.Is √16 rational or irrational? What about √17? Work: ________________________________________________________________ Answer: ____________________________
5.Between which two consecutive integers does √50fall? Work: ________________________________________________________________ Answer: ____________________________
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Section 1.2: Approximating Irrational Numbers
Since irrational numbers have infinite non-repeating decimals, we often need to approximate them.
WORKED EXAMPLE: Approximating √7
Step 1:Find perfect squares near 7: 4 < 7 < 9 Step 2:Take square roots: √4 < √7 < √9, so 2 < √7< 3 Step 3:Since 7 is closer to 9 than 4, √7 is closerto 3 than 2 Step 4:Estimate: √7 ≈ 2.6 (actual: 2.6457...)
Practice Problems: Section 1.2
6.Between which two consecutive integers is √30? Work: ________________________________________________________________ Answer: ____________________________
7.Approximate √45 to one decimal place. Work: ________________________________________________________________ Answer: ____________________________
8.Order from least to greatest: √10, 3, π, 3.2 Work: ________________________________________________________________ Answer: ____________________________
9.A square plot of land has area 200 sq ft. Approximatethe side length. Work: ________________________________________________________________ Answer: ____________________________
10.Plot √3, √5, and √8 on a number line between 1and 3. Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 1.3: Operations with Real Numbers
When you add, subtract, multiply, or divide real numbers, the result type depends on what you started with.
Key Rules
Rational + Rational = Rational (always)
Rational × Rational = Rational (always)
Irrational + Irrational = Can be either (√2 + √2 = 2√2 is irrational, but √2 + (-√2) = 0 is rational)
Rational × Irrational = Irrational (usually, except when multiplying by 0)
Irrational × Irrational = Can be either (√2 × √2 = 2 is rational, but √2 × √3 = √6 is irrational)
Practice Problems: Section 1.3
11.Is 3 + √5 rational or irrational? Explain. Work: ________________________________________________________________ Answer: ____________________________
12.Simplify: √2 × √8. Is the result rational or irrational? Work: ________________________________________________________________ Answer: ____________________________
13.Is 0.333... + 0.666... rational? Calculate thesum as a fraction. Work: ________________________________________________________________ Answer: ____________________________
14.A data set has values: 2, √9, π, 4.5, √2. Whichare rational? Work: ________________________________________________________________ Answer: ____________________________
15.If a = √3 and b = √12, find ab. Is the resultrational or irrational? Work: ________________________________________________________________ Answer: ____________________________
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Chapter 1 Review
16.Classify as rational or irrational: -8, √49, 0.121212...,√10, 22/7 Work: ________________________________________________________________ Answer: ____________________________
17.Order from least to greatest: -√5, -2, -2.5, -√3 Work: ________________________________________________________________ Answer: ____________________________
18.A sensor measures π × 2.5 meters. Express as adecimal (approximate). Work: ________________________________________________________________ Answer: ____________________________
19.Is 3.14159 rational or irrational? Is π rationalor irrational? Explain the difference. Work: ________________________________________________________________ Answer: ____________________________
20.CHALLENGE: Prove that √2 + √3 is irrational. (Hint:Assume it equals a rational number p/q) Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
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CHAPTER 2 The Engineer
Master exponents and radicals to calculate power, efficiency, and precision
What You'll Learn
• Apply properties of integer exponents • Evaluate expressions with zero and negative exponents • Simplify square roots and cube roots • Add, subtract, and multiply radical expressions
Why This Matters
Engineers calculate forces, design structures, and analyze systems using exponents and radicals constantly. When designing a bridge, you calculate stress using squares. When determining tank capacity, you use cube roots. These aren't abstract concepts—they're the math that keeps buildings standing.
Section 2.1: Properties of Integer Exponents
The Exponent Rules
Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Power Rule: (aᵐ)ⁿ = aᵐˣⁿ
Zero Exponent: a⁰ = 1 (when a ≠ 0)
Negative Exponent: a⁻ⁿ = 1/aⁿ
WORKED EXAMPLE: Simplify 2⁴ × 2³
Step 1:Apply product rule: add exponents with samebase Step 2:2⁴ × 2³ = 2⁴⁺³ = 2⁷ Step 3:Calculate: 2⁷ = 128
WORKED EXAMPLE: Simplify 5⁶ ÷ 5²
Step 1:Apply quotient rule: subtract exponents Step 2:5⁶ ÷ 5² = 5⁶⁻² = 5⁴ Step 3:Calculate: 5⁴ = 625
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Practice Problems: Section 2.1
1.Simplify: 3⁵ × 3² = ____ Work: ________________________________________________________________ Answer: ____________________________
2.Simplify: 7⁸ ÷ 7³ = ____ Work: ________________________________________________________________ Answer: ____________________________
3.Simplify: (4²)³ = ____ Work: ________________________________________________________________ Answer: ____________________________
4.Evaluate: 8⁰ = ____ Work: ________________________________________________________________ Answer: ____________________________
5.Simplify: 2⁵ × 2⁻³ = ____ Work: ________________________________________________________________ Answer: ____________________________
Section 2.2: Negative and Zero Exponents
A negative exponent means 'take the reciprocal.'
WORKED EXAMPLE: Evaluate 3⁻²
Step 1:Apply negative exponent rule: 3⁻² = 1/3² Step 2:Calculate: 1/3² = 1/9
WORKED EXAMPLE: Simplify (2/5)⁻³
Step 1:Apply negative exponent: (2/5)⁻³ = (5/2)³ Step 2:Calculate: (5/2)³ = 125/8
Practice Problems: Section 2.2
6.Evaluate: 4⁻² = ____ Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
7.Evaluate: 10⁻³ = ____ Work: ________________________________________________________________ Answer: ____________________________
8.Simplify: 2⁴ × 2⁻⁶ = ____ Work: ________________________________________________________________ Answer: ____________________________
9.Simplify: (3⁻²)² = ____ Work: ________________________________________________________________ Answer: ____________________________
10.An engineer calculates force as F = 10⁶ × 10⁻⁴N. Simplify. Work: ________________________________________________________________ Answer: ____________________________
Section 2.3: Square Roots and Cube Roots
√a means 'what number times itself equals a?'
³√a means 'what number times itself 3 times equals a?'
Perfect Squares and Cubes
Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...
Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...
WORKED EXAMPLE: Simplify √72
Step 1:Find largest perfect square factor: 72 = 36× 2 Step 2:Split the radical: √72 = √36 × √2 Step 3:Simplify: √36 × √2 = 6√2
Practice Problems: Section 2.3
11.√144 = ____ Work: ________________________________________________________________ Answer: ____________________________
12.³√27 = ____ Work: ________________________________________________________________ Answer: ____________________________
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13.Simplify: √50 = ____ Work: ________________________________________________________________ Answer: ____________________________
14.Simplify: √128 = ____ Work: ________________________________________________________________ Answer: ____________________________
15.A cube has volume 343 cm³. What is its side length? Work: ________________________________________________________________ Answer: ____________________________
Section 2.4: Operations with Radicals
You can add/subtract radicals only if they have the same radicand (number under the radical).
a√c + b√c = (a + b)√c
√a × √b = √(ab)
WORKED EXAMPLE: Simplify 3√5 + 7√5
Step 1:Same radicand (5), so add coefficients Step 2:3√5 + 7√5 = (3 + 7)√5 = 10√5
Practice Problems: Section 2.4
16.Simplify: 4√3 + 2√3 = ____ Work: ________________________________________________________________ Answer: ____________________________
17.Simplify: √5 × √20 = ____ Work: ________________________________________________________________ Answer: ____________________________
18.Simplify: √12 + √27 = ____ Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
19.Simplify: 2√8 - √32 = ____ Work: ________________________________________________________________ Answer: ____________________________
20.A rectangular beam has width √18 cm and length√2 cm. Find the area. Work: ________________________________________________________________ Answer: ____________________________
Chapter 2 Review
21.Simplify: (2³ × 2⁵) ÷ 2⁴ = ____ Work: ________________________________________________________________ Answer: ____________________________
22.Evaluate: 5⁻² + 5⁰ = ____ Work: ________________________________________________________________ Answer: ____________________________
23.Simplify: √200 = ____ Work: ________________________________________________________________ Answer: ____________________________
24.Simplify: 3√7 - √28 = ____ Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: If 2ˣ = 32 and 3ʸ = 81, find x + y. Work: ________________________________________________________________ Answer: ____________________________
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🚀
CHAPTER 3 The Space Scientist
Master scientific notation to calculate cosmic distances and microscopic
measurements
What You'll Learn
• Write numbers in scientific notation • Convert between standard form and scientific notation • Perform operations with numbers in scientific notation • Compare very large and very small numbers
Why This Matters
Space scientists work with numbers that are impossibly large (93 million miles to the Sun) and impossibly small (atoms measured in billionths of a meter). Scientific notation makes these numbers manageable—and comparable. It's the language of science.
Section 3.1: Understanding Scientific Notation
Scientific Notation: a × 10ⁿ where 1 ≤ a < 10
Large numbers: positive exponent (move decimal left)
Small numbers: negative exponent (move decimal right)
WORKED EXAMPLE: Write 93,000,000 in scientific notation
Step 1:Move decimal until you have a number between1 and 10: 9.3 Step 2:Count places moved: 7 places to the left Step 3:Write: 9.3 × 10⁷
WORKED EXAMPLE: Write 0.00045 in scientific notation
Step 1:Move decimal until you have a number between1 and 10: 4.5 Step 2:Count places moved: 4 places to the right Step 3:Since original number is small, exponent isnegative: 4.5 × 10⁻⁴
Practice Problems: Section 3.1
1.Write in scientific notation: 45,000,000 = ____ Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
2.Write in scientific notation: 0.00067 = ____ Work: ________________________________________________________________ Answer: ____________________________
3.Write in standard form: 3.8 × 10⁵ = ____ Work: ________________________________________________________________ Answer: ____________________________
4.Write in standard form: 2.1 × 10⁻⁴ = ____ Work: ________________________________________________________________ Answer: ____________________________
5.The distance to Alpha Centauri is 40,000,000,000,000km. Write in scientific notation. Work: ________________________________________________________________ Answer: ____________________________
Section 3.2: Multiplying and Dividing
(a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ
(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ
WORKED EXAMPLE: Calculate (3 × 10⁴) × (2 × 10³)
Step 1:Multiply the coefficients: 3 × 2 = 6 Step 2:Add the exponents: 10⁴ × 10³ = 10⁷ Step 3:Combine: 6 × 10⁷
Practice Problems: Section 3.2
6.(4 × 10⁵) × (3 × 10²) = ____ Work: ________________________________________________________________ Answer: ____________________________
7.(8 × 10⁷) ÷ (2 × 10³) = ____ Work: ________________________________________________________________ Answer: ____________________________
8.(5 × 10⁻³) × (6 × 10⁵) = ____ Work: ________________________________________________________________ Answer: ____________________________
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9.Light travels 3 × 10⁸ m/s. How far in 5 × 10² seconds? Work: ________________________________________________________________ Answer: ____________________________
10.A spacecraft travels 4.8 × 10¹² km in 8 × 10⁴hours. Speed? Work: ________________________________________________________________ Answer: ____________________________
Section 3.3: Adding and Subtracting
To add or subtract in scientific notation, the exponents must be the same.
WORKED EXAMPLE: Calculate (4.2 × 10⁵) + (3.1 × 10⁴)
Step 1:Rewrite with the same exponent: 3.1 × 10⁴= 0.31 × 10⁵ Step 2:Add coefficients: 4.2 + 0.31 = 4.51 Step 3:Result: 4.51 × 10⁵
Practice Problems: Section 3.3
11.(6.5 × 10⁶) + (2.3 × 10⁶) = ____ Work: ________________________________________________________________ Answer: ____________________________
12.(8.4 × 10⁵) - (3.2 × 10⁴) = ____ Work: ________________________________________________________________ Answer: ____________________________
13.Planet A is 5.8 × 10⁸ km away. Planet B is 3.2× 10⁷ km away. Difference? Work: ________________________________________________________________ Answer: ____________________________
14.Budget: $2.4 × 10⁷. Spent: $8.6 × 10⁶. Remaining? Work: ________________________________________________________________ Answer: ____________________________
15.Two particles have masses 9.1 × 10⁻³¹ kg and 1.7× 10⁻²⁷ kg. Total mass? Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 3.4: Comparing in Scientific Notation
To compare: First compare exponents, then coefficients if exponents are equal.
Practice Problems: Section 3.4
16.Which is larger: 3.5 × 10⁸ or 9.2 × 10⁷? Work: ________________________________________________________________ Answer: ____________________________
17.Order from least to greatest: 2.3 × 10⁵, 8.1 ×10⁴, 1.9 × 10⁵ Work: ________________________________________________________________ Answer: ____________________________
18.The Sun is 1.5 × 10⁸ km from Earth. Mars is 2.3× 10⁸ km. How many times farther is Mars? Work: ________________________________________________________________ Answer: ____________________________
19.A virus is 2 × 10⁻⁷ m. A bacterium is 5 × 10⁻⁶m. How many times larger is the bacterium? Work: ________________________________________________________________ Answer: ____________________________
20.Earth's mass: 6 × 10²⁴ kg. Jupiter's mass: 1.9× 10²⁷ kg. How many Earths equal Jupiter? Work: ________________________________________________________________ Answer: ____________________________
Chapter 3 Review
21.Write 0.0000089 in scientific notation. Work: ________________________________________________________________ Answer: ____________________________
22.(7.2 × 10⁴) × (1.5 × 10⁻²) = ____ Work: ________________________________________________________________ Answer: ____________________________
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23.A satellite orbits at 3.6 × 10⁷ m. Convert to km. Work: ________________________________________________________________
Answer: ____________________________
24.How many seconds in a year? (365 days) Expressin scientific notation. Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: If the universe is 1.4 × 10¹⁰ years old and light travels 9.5 × 10¹² km/year, what's the maximum observable universe radius? Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
💼
CHAPTER 4 The Entrepreneur
Master multi-step equations to analyze profit, loss, and business scenarios
What You'll Learn
• Solve equations with variables on both sides • Solve equations with parentheses and distribution • Identify equations with one, none, or infinite solutions • Write and solve equations from word problems
Why This Matters
Entrepreneurs solve equations daily: 'If I price my product at $x and sell 100 units minus $500 in costs, when does profit equal $2,000?' Business success depends on setting up and solving these equations correctly.
Section 4.1: Variables on Both Sides
When variables appear on both sides, collect them on one side first.
WORKED EXAMPLE: Solve 5x + 3 = 2x + 15
Step 1:Subtract 2x from both sides: 5x - 2x + 3 =15 Step 2:Simplify: 3x + 3 = 15 Step 3:Subtract 3 from both sides: 3x = 12 Step 4:Divide by 3: x = 4
Practice Problems: Section 4.1
1.Solve: 7x + 4 = 3x + 20 Work: ________________________________________________________________ Answer: ____________________________
2.Solve: 9x - 5 = 4x + 15 Work: ________________________________________________________________ Answer: ____________________________
3.Solve: 8x - 12 = 5x + 6 Work: ________________________________________________________________ Answer: ____________________________
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4.Solve: 2x + 18 = 6x - 2 Work: ________________________________________________________________ Answer: ____________________________
5.Company A charges $50 + $20/hr. Company B charges$30 + $25/hr. When are they equal? Work: ________________________________________________________________ Answer: ____________________________
Section 4.2: Equations with Parentheses
Use the distributive property first, then solve.
WORKED EXAMPLE: Solve 3(x + 4) = 2(x - 1) + 17
Step 1:Distribute: 3x + 12 = 2x - 2 + 17 Step 2:Simplify right side: 3x + 12 = 2x + 15 Step 3:Subtract 2x: x + 12 = 15 Step 4:Subtract 12: x = 3
Practice Problems: Section 4.2
6.Solve: 4(x - 2) = 24 Work: ________________________________________________________________ Answer: ____________________________
7.Solve: 2(3x + 5) = 4(x + 6) Work: ________________________________________________________________ Answer: ____________________________
8.Solve: 5(x - 3) - 2x = 3(x + 1) Work: ________________________________________________________________ Answer: ____________________________
9.Solve: -2(4x - 3) = 3(2 - x) + 5 Work: ________________________________________________________________ Answer: ____________________________
10.A phone plan costs $40 + $5(g - 2) for g GB. Anothercosts $25 + $8(g - 1). When equal? Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 4.3: Special Cases
One Solution, No Solution, or Infinite Solutions
One solution: Variables cancel to give x = a number
No solution: Variables cancel to give a false statement (like 5 = 3)
Infinite solutions: Variables cancel to give a true statement (like 4 = 4)
WORKED EXAMPLE: Solve 2(x + 3) = 2x + 6
Step 1:Distribute: 2x + 6 = 2x + 6 Step 2:Subtract 2x: 6 = 6 Step 3:True statement! INFINITE SOLUTIONS (identity)
Practice Problems: Section 4.3
11.Solve: 3(x + 2) = 3x + 8. How many solutions? Work: ________________________________________________________________ Answer: ____________________________
12.Solve: 5x + 10 = 5(x + 2). How many solutions? Work: ________________________________________________________________ Answer: ____________________________
13.Solve: 4(x - 1) = 2(2x + 3). How many solutions? Work: ________________________________________________________________ Answer: ____________________________
14.Solve: 6x - 3 = 3(2x - 1). How many solutions? Work: ________________________________________________________________ Answer: ____________________________
15.Create an equation with no solution using theform ax + b = ax + c. Work: ________________________________________________________________ Answer: ____________________________
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Chapter 4 Review
16.Solve: 8x - 7 = 3x + 18 Work: ________________________________________________________________ Answer: ____________________________
17.Solve: 4(2x - 3) = 3(x + 4) + x Work: ________________________________________________________________ Answer: ____________________________
18.Monthly profit: Revenue = $50x, Costs = $2000+ $20x. Find break-even units. Work: ________________________________________________________________ Answer: ____________________________
19.Solve: 0.5(x + 4) = 0.25(2x + 16) Work: ________________________________________________________________ Answer: ____________________________
20.CHALLENGE: The sum of three consecutive integersis 99. Find the integers. Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
📋
CHAPTER 5 The Project Manager
Master functions and linear relationships to model and predict outcomes
What You'll Learn
• Understand and identify functions • Use function notation • Write linear equations in slope-intercept form • Graph linear functions and interpret slope and y-intercept
Why This Matters
Project managers track progress, predict completion dates, and allocate resources. 'If my team completes 3 modules per week and we've finished 5, when will we hit 20? 'Functions model these relationships, letting you predict the future from the present.
Section 5.1: Understanding Functions
A function assigns exactly ONE output for each input
Think of a function as a machine: put in a number, get out exactly one result.
The Vertical Line Test
If a vertical line crosses a graph more than once, it's NOT a function.
Practice Problems: Section 5.1
1.Is this a function? {(1, 3), (2, 5), (3, 7), (4,9)} Work: ________________________________________________________________ Answer: ____________________________
2.Is this a function? {(2, 4), (3, 5), (2, 6), (4,8)} Work: ________________________________________________________________ Answer: ____________________________
3.A vending machine takes $1.50 and gives one snack.Is this a function? Explain. Work: ________________________________________________________________ Answer: ____________________________
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4.Does y = x² represent a function? Why or why not? Work: ________________________________________________________________ Answer: ____________________________
5.Does x = y² represent a function? Why or why not? Work: ________________________________________________________________ Answer: ____________________________
Section 5.2: Function Notation
f(x) means 'the output of function f when the input is x'
WORKED EXAMPLE: If f(x) = 3x - 2, find f(5)
Step 1:Replace x with 5: f(5) = 3(5) - 2 Step 2:Calculate: f(5) = 15 - 2 = 13
Practice Problems: Section 5.2
6.If f(x) = 2x + 7, find f(4). Work: ________________________________________________________________ Answer: ____________________________
7.If g(x) = x² - 3x, find g(5). Work: ________________________________________________________________ Answer: ____________________________
8.If h(x) = 4x - 1, find h(-2). Work: ________________________________________________________________ Answer: ____________________________
9.If f(x) = 2x + 3, find x when f(x) = 15. Work: ________________________________________________________________ Answer: ____________________________
10.Project progress: P(w) = 3w + 5 modules afterw weeks. Modules after 8 weeks? Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 5.3: Slope-Intercept Form
y = mx + b
m = slope (rate of change)
b = y-intercept (starting value)
Finding Slope
slope = (y₂ - y₁) / (x₂ - x₁) = rise / run
WORKED EXAMPLE: Find slope through (2, 5) and (6, 13)
Step 1:Use slope formula: m = (13 - 5) / (6 - 2) Step 2:Calculate: m = 8 / 4 = 2
Practice Problems: Section 5.3
11.Find the slope between (1, 4) and (5, 12). Work: ________________________________________________________________ Answer: ____________________________
12.Find the slope between (-2, 8) and (4, -4). Work: ________________________________________________________________ Answer: ____________________________
13.What are the slope and y-intercept of y = -3x+ 7? Work: ________________________________________________________________ Answer: ____________________________
14.Write the equation: slope = 4, y-intercept = -2 Work: ________________________________________________________________ Answer: ____________________________
15.A line passes through (0, 5) with slope 2. Writethe equation. Work: ________________________________________________________________ Answer: ____________________________
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Section 5.4: Writing Equations from Two Points
WORKED EXAMPLE: Write the equation of a line through (2, 7) and (4, 13)
Step 1:Find slope: m = (13 - 7) / (4 - 2) = 6/2 =3 Step 2:Use point-slope with (2, 7): y - 7 = 3(x -2) Step 3:Simplify: y - 7 = 3x - 6 Step 4:Solve for y: y = 3x + 1
Practice Problems: Section 5.4
16.Write the equation through (1, 5) and (3, 11). Work: ________________________________________________________________ Answer: ____________________________
17.Write the equation through (0, -3) and (4, 5). Work: ________________________________________________________________ Answer: ____________________________
18.Week 2: 11 units complete. Week 5: 26 units complete.Write P(w). Work: ________________________________________________________________ Answer: ____________________________
19.After 3 hours, 45 miles traveled. After 7 hours,105 miles. Write d(t). Work: ________________________________________________________________ Answer: ____________________________
20.Temperature drops from 68°F at 6 PM to 50°F at10 PM. Write T(h). Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Chapter 5 Review
21.If f(x) = -2x + 8, find f(-3) and f(0). Work: ________________________________________________________________ Answer: ____________________________
22.Graph y = 2x - 3. Identify slope and y-intercept. Work: ________________________________________________________________ Answer: ____________________________
23.Are these points on the same line? (0, 4), (2, 10), (5, 19) Work: ________________________________________________________________ Answer: ____________________________
24.Gym membership: $50 signup + $30/month. WriteC(m) and find cost for 1 year. Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: Two lines: y = 2x + 5 and y = -x +8. Where do they intersect? Work: ________________________________________________________________ Answer: ____________________________
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CHAPTER 6 The Systems Analyst
Master systems of equations to solve interconnected business problems
What You'll Learn
• Understand systems of linear equations • Solve systems by graphing • Solve systems by substitution • Solve systems by elimination
Why This Matters
Systems analysts solve problems with multiple constraints: 'If we sell x units of Product A and y units of Product B, total revenue must equal $10,000 AND total units must be 500. 'Two equations, two unknowns—systems thinking.
Section 6.1: Understanding Systems
A system of equations is two or more equations with the same variables.
The solution is the point (x, y) that makes BOTH equations true.
Types of Solutions
One solution: Lines intersect at one point
No solution: Lines are parallel (never intersect)
Infinite solutions: Lines are the same (overlap completely)
Practice Problems: Section 6.1
1.Is (3, 5) a solution to: y = 2x - 1 and y = x +2? Check both equations. Work: ________________________________________________________________ Answer: ____________________________
2.Is (4, 7) a solution to: y = 3x - 5 and y = x +3? Work: ________________________________________________________________ Answer: ____________________________
3.What does it mean if two lines have the same slopebut different y-intercepts? Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 6.2: Solving by Graphing
WORKED EXAMPLE: Solve by graphing: y = x + 1 and y = -x + 5
Step 1:Graph y = x + 1 (slope 1, y-intercept 1) Step 2:Graph y = -x + 5 (slope -1, y-intercept 5) Step 3:Find intersection point: (2, 3) Step 4:Check: 3 = 2 + 1 ✓ and 3 = -2 + 5 ✓
Practice Problems: Section 6.2
4.Solve by graphing: y = 2x - 1 and y = -x + 5 Work: ________________________________________________________________ Answer: ____________________________
5.Solve by graphing: y = x + 3 and y = x - 2. Whathappens? Work: ________________________________________________________________ Answer: ____________________________
Section 6.3: Solving by Substitution
Substitution works best when one equation is already solved for a variable.
WORKED EXAMPLE: Solve: y = 3x - 4 and 2x + y = 11
Step 1:Substitute first equation into second: 2x+ (3x - 4) = 11 Step 2:Simplify: 5x - 4 = 11 Step 3:Solve: 5x = 15, so x = 3 Step 4:Find y: y = 3(3) - 4 = 5 Step 5:Solution: (3, 5)
Practice Problems: Section 6.3
6.Solve by substitution: y = 2x + 1 and 3x + y =16 Work: ________________________________________________________________ Answer: ____________________________
7.Solve by substitution: x = 4y - 3 and 2x + 3y =5 Work: ________________________________________________________________ Answer: ____________________________
8.Solve: y = x - 5 and 4x - 2y = 14 Work: ________________________________________________________________ Answer: ____________________________
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9.Adult tickets $12, child tickets $8. Total 45 tickets, revenue $460. Find each. Work: ________________________________________________________________ Answer: ____________________________
10.Sum of two numbers is 50. One is 8 more than theother. Find both. Work: ________________________________________________________________ Answer: ____________________________
Section 6.4: Solving by Elimination
Elimination adds or subtracts equations to eliminate one variable.
WORKED EXAMPLE: Solve: 3x + 2y = 16 and 3x - 2y = 8
Step 1:Add equations to eliminate y: 6x = 24 Step 2:Solve: x = 4 Step 3:Substitute back: 3(4) + 2y = 16 Step 4:Solve: 12 + 2y = 16, so y = 2 Step 5:Solution: (4, 2)
Practice Problems: Section 6.4
11.Solve by elimination: 2x + 3y = 13 and 2x - 3y= -1 Work: ________________________________________________________________ Answer: ____________________________
12.Solve by elimination: 4x + y = 14 and 2x + y =8 Work: ________________________________________________________________ Answer: ____________________________
13.Solve: 3x + 4y = 25 and 5x - 4y = 7 Work: ________________________________________________________________ Answer: ____________________________
14.Solve: 2x + 5y = 19 and 3x + 2y = 12 Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
15.A store sold 100 items: phones at $200 and cases at $25. Revenue: $8,450. Find each. Work: ________________________________________________________________ Answer: ____________________________
Chapter 6 Review
16.Solve any method: y = 4x - 7 and y = 2x + 3 Work: ________________________________________________________________ Answer: ____________________________
17.Solve any method: 5x + 2y = 24 and 3x - 2y = 8 Work: ________________________________________________________________ Answer: ____________________________
18.Solve: 4x + 3y = 31 and 2x + 5y = 29 Work: ________________________________________________________________ Answer: ____________________________
19.Boat speed 20 mph still water. Downstream 25 mph,upstream 15 mph. Find current speed. Work: ________________________________________________________________ Answer: ____________________________
20.CHALLENGE: Find a, b if system has no solution:y = 3x + a and y = bx + 5 Work: ________________________________________________________________ Answer: ____________________________
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CHAPTER 7 The Surveyor
Master the Pythagorean theorem to calculate distances and verify right angles
What You'll Learn
• Understand the Pythagorean theorem • Find missing sides of right triangles • Determine if a triangle is a right triangle • Apply the theorem to real-world problems
Why This Matters
Surveyors measure land without walking every inch. Need to know the distance across a lake? Use the Pythagorean theorem. Checking if a corner is exactly 90°? That's the 3-4-5 rule. This ancient formula is used every day in construction, navigation, and design.
Section 7.1: The Pythagorean Theorem
a² + b² = c²
Where a and b are the legs (sides that form the right angle)
And c is the hypotenuse (longest side, opposite the right angle)
WORKED EXAMPLE: Find c when a = 6 and b = 8
Step 1:Use Pythagorean theorem: a² + b² = c² Step 2:Substitute: 6² + 8² = c² Step 3:Calculate: 36 + 64 = c² Step 4:Simplify: 100 = c² Step 5:Take square root: c = 10
Practice Problems: Section 7.1
1.Find c: a = 3, b = 4 Work: ________________________________________________________________ Answer: ____________________________
2.Find c: a = 5, b = 12 Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
3.Find c: a = 8, b = 15 Work: ________________________________________________________________ Answer: ____________________________
4.A ladder reaches 12 ft up a wall, with its base5 ft from the wall. Ladder length? Work: ________________________________________________________________ Answer: ____________________________
5.A 50-inch TV is measured diagonally. If width is40 in, what's the height? Work: ________________________________________________________________ Answer: ____________________________
Section 7.2: Finding a Leg
WORKED EXAMPLE: Find a when b = 9 and c = 15
Step 1:Rearrange: a² = c² - b² Step 2:Substitute: a² = 15² - 9² Step 3:Calculate: a² = 225 - 81 = 144 Step 4:Take square root: a = 12
Practice Problems: Section 7.2
6.Find a: b = 24, c = 25 Work: ________________________________________________________________ Answer: ____________________________
7.Find b: a = 20, c = 29 Work: ________________________________________________________________ Answer: ____________________________
8.A 17-ft ladder leans against a wall with base 8ft away. Height reached? Work: ________________________________________________________________ Answer: ____________________________
9.A support wire is 41 m long, anchored 9 m froma pole. Pole height? Work: ________________________________________________________________ Answer: ____________________________
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10.A kite is 80 m away horizontally, string 100 m. Height of kite? Work: ________________________________________________________________ Answer: ____________________________
Section 7.3: Is It a Right Triangle?
If a² + b² = c², it's a right triangle. If not, it isn't.
Common Pythagorean Triples
3-4-5 and multiples (6-8-10, 9-12-15, ...)
5-12-13 and multiples
8-15-17
7-24-25
Practice Problems: Section 7.3
11.Is a triangle with sides 9, 12, 15 a right triangle? Work: ________________________________________________________________ Answer: ____________________________
12.Is a triangle with sides 7, 10, 12 a right triangle? Work: ________________________________________________________________ Answer: ____________________________
13.Is a triangle with sides 20, 21, 29 a right triangle? Work: ________________________________________________________________ Answer: ____________________________
14.A surveyor measures a corner with legs 6m and8m, diagonal 10.2m. Is it 90°? Work: ________________________________________________________________ Answer: ____________________________
15.Find the missing value for a Pythagorean triple:9, __, 15 Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 7.4: Real-World Applications
Practice Problems: Section 7.4
16.Distance between points (0, 0) and (6, 8) on acoordinate plane? Work: ________________________________________________________________ Answer: ____________________________
17.A ship sails 30 km east, then 40 km north. Straight-linedistance from start? Work: ________________________________________________________________ Answer: ____________________________
18.A ramp must rise 4 ft over a horizontal run of12 ft. Ramp length? Work: ________________________________________________________________ Answer: ____________________________
19.A baseball diamond has 90-ft sides. Distance fromhome to second base? Work: ________________________________________________________________ Answer: ____________________________
20.A surveyor needs to measure across a pond. Shemarks 60m and 80m legs. Direct distance? Work: ________________________________________________________________ Answer: ____________________________
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Chapter 7 Review
21.Find the hypotenuse: legs = 11 and 60 Work: ________________________________________________________________ Answer: ____________________________
22.Find the leg: other leg = 45, hypotenuse = 53 Work: ________________________________________________________________ Answer: ____________________________
23.Verify: 11, 60, 61 is a Pythagorean triple. Work: ________________________________________________________________ Answer: ____________________________
24.A 26-ft pole is stabilized by a wire from itstop to a point 10 ft from the base. Wire length? Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: Find the distance from (2, 3) to (7,15). Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
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CHAPTER 8 The Animator
Master transformations to create movement and design in the coordinate plane
What You'll Learn
• Perform translations, reflections, and rotations • Identify and describe transformations • Understand dilations and scale factors • Compose multiple transformations
Why This Matters
Animators bring characters to life by transforming shapes frame by frame. Translations make things slide, rotations make them spin, reflections create mirror images, and dilations change size. Every video game, animated movie, and digital effect uses these four transformations.
Section 8.1: Translations
A translation slides a figure without rotating or flipping it.
(x, y) → (x + a, y + b)
where a = horizontal shift, b = vertical shift
WORKED EXAMPLE: Translate (3, 5) by (4, -2)
Step 1:Add horizontal shift: 3 + 4 = 7 Step 2:Add vertical shift: 5 + (-2) = 3 Step 3:New point: (7, 3)
Practice Problems: Section 8.1
1.Translate (2, 4) by (5, 3). New coordinates? Work: ________________________________________________________________ Answer: ____________________________
2.Translate (-1, 6) by (-3, -4). New coordinates? Work: ________________________________________________________________ Answer: ____________________________
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3.Triangle ABC has vertices A(1, 2), B(4, 2), C(2, 5). Translate by (3, -1). Find new vertices. Work: ________________________________________________________________ Answer: ____________________________
4.A character at (10, 8) moves (-6, 4). New position? Work: ________________________________________________________________ Answer: ____________________________
5.Point P(5, -2) becomes P'(1, 3). Describe the translation. Work: ________________________________________________________________ Answer: ____________________________
Section 8.2: Reflections
Reflection Rules
Over x-axis: (x, y) → (x, -y)
Over y-axis: (x, y) → (-x, y)
Over y = x: (x, y) → (y, x)
Over y = -x: (x, y) → (-y, -x)
Practice Problems: Section 8.2
6.Reflect (4, 7) over the x-axis. Work: ________________________________________________________________ Answer: ____________________________
7.Reflect (-3, 5) over the y-axis. Work: ________________________________________________________________ Answer: ____________________________
8.Reflect (2, 8) over the line y = x. Work: ________________________________________________________________ Answer: ____________________________
9.Triangle with vertices (1, 1), (4, 1), (2, 4).Reflect over y-axis. Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
10.Point A(-2, 5) becomes A'(-2, -5). Over which line was it reflected? Work: ________________________________________________________________ Answer: ____________________________
Section 8.3: Rotations
Rotation Rules (about origin)
90° counterclockwise: (x, y) → (-y, x)
180°: (x, y) → (-x, -y)
270° counterclockwise (or 90° clockwise): (x, y) → (y, -x)
Practice Problems: Section 8.3
11.Rotate (3, 5) 90° counterclockwise about the origin. Work: ________________________________________________________________ Answer: ____________________________
12.Rotate (4, -2) 180° about the origin. Work: ________________________________________________________________ Answer: ____________________________
13.Rotate (-1, 6) 270° counterclockwise about theorigin. Work: ________________________________________________________________ Answer: ____________________________
14.Triangle A(2, 1), B(5, 1), C(3, 4) rotated 90°CCW. Find new vertices. Work: ________________________________________________________________ Answer: ____________________________
15.Point (5, 2) becomes (-2, 5). What rotation occurred? Work: ________________________________________________________________ Answer: ____________________________
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Section 8.4: Dilations
(x, y) → (kx, ky) where k is the scale factor
k > 1: enlargement
0 < k < 1: reduction
k < 0: enlargement/reduction with 180° rotation
Practice Problems: Section 8.4
16.Dilate (4, 6) by scale factor 2 centered at origin. Work: ________________________________________________________________ Answer: ____________________________
17.Dilate (10, 15) by scale factor 0.5 centered atorigin. Work: ________________________________________________________________ Answer: ____________________________
18.Rectangle vertices (0, 0), (4, 0), (4, 3), (0,3). Dilate by k = 3. Work: ________________________________________________________________ Answer: ____________________________
19.A figure dilates from area 20 sq units to 180sq units. Scale factor? Work: ________________________________________________________________ Answer: ____________________________
20.Point (3, 2) becomes (9, 6). What was the scalefactor? Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Chapter 8 Review
21.Translate (5, -3) by (-2, 4), then reflect overx-axis. Work: ________________________________________________________________ Answer: ____________________________
22.Rotate (4, 1) 90° CCW, then dilate by factor 2. Work: ________________________________________________________________ Answer: ____________________________
23.Identify the transformation: A(2, 5) → A'(5, 2) Work: ________________________________________________________________ Answer: ____________________________
24.Which transformations preserve congruence? (translation,rotation, reflection, dilation) Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: A triangle with area 12 is dilated,then dilated again by k=2. Final area 108. First k? Work: ________________________________________________________________ Answer: ____________________________
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CHAPTER 9 The Architect
Master angle relationships to design structures with precision
What You'll Learn
• Identify and calculate angles formed by parallel lines and transversals • Apply the triangle angle sum theorem • Work with exterior angles of triangles • Understand and apply angle-angle similarity
Why This Matters
Architects design buildings with precise angles. Roof pitches, window frames, support beams—every angle must be calculated exactly. When parallel support beams are cut by a diagonal brace, specific angle relationships emerge. Understanding these relationships ensures structures are stable and beautiful.
Section 9.1: Angles with Parallel Lines
Key Angle Pairs
Corresponding angles: same position, same side of transversal → EQUAL
Alternate interior angles: opposite sides, between parallels → EQUAL
Alternate exterior angles: opposite sides, outside parallels → EQUAL
Co-interior (same-side interior) angles: same side, between parallels → SUPPLEMENTARY (sum = 180°)
Practice Problems: Section 9.1
1.Lines m and n are parallel. If angle 1 = 65°, whatis its corresponding angle? Work: ________________________________________________________________ Answer: ____________________________
2.Alternate interior angles measure 3x + 10 and 5x- 20. Find x and the angle measures. Work: ________________________________________________________________ Answer: ____________________________
3.Co-interior angles measure 2x and 3x + 30. Findx and the angle measures. Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
4.If two parallel beams are cut by a diagonal brace at 72°, what is the alternate interior angle? Work: ________________________________________________________________ Answer: ____________________________
5.Prove: If alternate interior angles are equal,the lines are parallel. Work: ________________________________________________________________ Answer: ____________________________
Section 9.2: Triangle Angle Sum
The sum of angles in any triangle = 180°
WORKED EXAMPLE: Find x in a triangle with angles 45°, 60°, and x°
Step 1:Use angle sum: 45 + 60 + x = 180 Step 2:Simplify: 105 + x = 180 Step 3:Solve: x = 75°
Practice Problems: Section 9.2
6.Triangle angles: 35°, 75°, x°. Find x. Work: ________________________________________________________________ Answer: ____________________________
7.Isosceles triangle: base angles both 55°. Findthe vertex angle. Work: ________________________________________________________________ Answer: ____________________________
8.Triangle angles: x°, 2x°, 3x°. Find all three angles. Work: ________________________________________________________________ Answer: ____________________________
9.Right triangle: one acute angle is 32°. Find theother acute angle. Work: ________________________________________________________________ Answer: ____________________________
10.Equilateral triangle: find each angle measure. Answer: ____________________________
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Section 9.3: Exterior Angles
Exterior angle = sum of two remote interior angles
WORKED EXAMPLE: Find the exterior angle if remote interior angles are 40° and 65°
Step 1:Apply theorem: exterior = 40 + 65 Step 2:Calculate: exterior = 105°
Practice Problems: Section 9.3
11.Remote interior angles are 48° and 67°. Find theexterior angle. Work: ________________________________________________________________ Answer: ____________________________
12.Exterior angle is 130°. One remote interior is55°. Find the other. Work: ________________________________________________________________ Answer: ____________________________
13.Exterior angle = 4x, remote interiors are x and2x + 15. Find x. Work: ________________________________________________________________ Answer: ____________________________
14.A roof pitch creates an exterior angle of 145°.One interior angle is 90°. Find the other. Work: ________________________________________________________________ Answer: ____________________________
15.Prove: An exterior angle is always greater thaneither remote interior angle. Work: ________________________________________________________________ Answer: ____________________________
Section 9.4: Similar Triangles (AA Similarity)
If two angles of one triangle equal two angles of another, the
triangles are similar
Similar triangles have equal angles and proportional sides.
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Math in Action: Grade 8 — Real-World Problem Solving
Practice Problems: Section 9.4
16.Triangle ABC: angles 50°, 60°, 70°. Triangle DEF:angles 50°, 60°, x°. Find x. Are they similar? Work: ________________________________________________________________ Answer: ____________________________
17.Two similar triangles have sides 3, 4, 5 and 6,8, x. Find x. Work: ________________________________________________________________ Answer: ____________________________
18.A 6-ft person casts a 4-ft shadow. A tree castsa 20-ft shadow. Tree height? Work: ________________________________________________________________ Answer: ____________________________
19.Triangle sides 5, 12, 13. Similar triangle hasshortest side 15. Find other sides. Work: ________________________________________________________________ Answer: ____________________________
20.Two triangles: angles 30°, 60°, 90° and 30°, 60°,90°. Must they be similar? Congruent? Work: ________________________________________________________________ Answer: ____________________________
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Chapter 9 Review
21.Parallel lines cut by transversal. One angle =118°. Find all 8 angles. Work: ________________________________________________________________ Answer: ____________________________
22.Triangle angles: (x + 20)°, (2x - 10)°, (x + 30)°.Find all angles. Work: ________________________________________________________________ Answer: ____________________________
23.Exterior angle = 5x - 10, remote interiors = 2x+ 5 and x + 15. Find x. Work: ________________________________________________________________ Answer: ____________________________
24.A ramp rises 3 ft over 12 ft. A larger ramp rises5 ft. Horizontal distance? Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: In triangle ABC, angle A = 40°. BDbisects angle B, CD bisects angle C. Find angle BDC. Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
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CHAPTER 10 The Industrial Designer
Master volume calculations to design products and containers
What You'll Learn
• Calculate volume of cylinders • Calculate volume of cones • Calculate volume of spheres • Solve real-world problems involving volume
Why This Matters
Industrial designers create everything from soda cans to storage tanks. How much liquid fits in that bottle? How much material for that cone-shaped funnel? How much air in that basketball? Volume calculations are essential for product design.
Section 10.1: Volume of Cylinders
V = πr²h
where r = radius of the base circle and h = height
WORKED EXAMPLE: Find volume: cylinder with radius 5 cm, height 12 cm
Step 1:Use the formula V = πr²h Step 2:Substitute: V = π × 5² × 12 Step 3:Calculate: V = π × 25 × 12 = 300π Step 4:Approximate: V ≈ 942.48 cm³
Practice Problems: Section 10.1
1.Find volume: cylinder with r = 4 in, h = 10 in.(Leave answer in terms of π) Work: ________________________________________________________________ Answer: ____________________________
2.Find volume: cylinder with diameter 14 cm, height8 cm. Work: ________________________________________________________________ Answer: ____________________________
3.A water tank has radius 3 m and height 7 m. Volumein cubic meters? Work: ________________________________________________________________ Answer: ____________________________
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4.A can has volume 350 cm³ and height 12 cm. Find the radius. Work: ________________________________________________________________ Answer: ____________________________
5.How many liters fit in a cylinder with r = 10 cm,h = 20 cm? (1 L = 1000 cm³) Work: ________________________________________________________________ Answer: ____________________________
Section 10.2: Volume of Cones
V = (1/3)πr²h
A cone's volume is exactly 1/3 of a cylinder with the same base and height.
WORKED EXAMPLE: Find volume: cone with radius 6 cm, height 9 cm
Step 1:Use the formula V = (1/3)πr²h Step 2:Substitute: V = (1/3) × π × 6² × 9 Step 3:Calculate: V = (1/3) × π × 36 × 9 = 108π Step 4:Approximate: V ≈ 339.29 cm³
Practice Problems: Section 10.2
6.Find volume: cone with r = 3 in, h = 8 in. Work: ________________________________________________________________ Answer: ____________________________
7.Find volume: cone with diameter 10 cm, height 15cm. Work: ________________________________________________________________ Answer: ____________________________
8.An ice cream cone has r = 2 in, h = 6 in. Volumeof ice cream it holds? Work: ________________________________________________________________ Answer: ____________________________
9.A cone has volume 100π cm³ and height 12 cm. Findthe radius. Work: ________________________________________________________________ Answer: ____________________________
10.A funnel (cone) has radius 8 cm and volume 384πcm³. Find the height. Work: ________________________________________________________________ Answer: ____________________________
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Math in Action: Grade 8 — Real-World Problem Solving
Section 10.3: Volume of Spheres
V = (4/3)πr³
WORKED EXAMPLE: Find volume: sphere with radius 6 cm
Step 1:Use the formula V = (4/3)πr³ Step 2:Substitute: V = (4/3) × π × 6³ Step 3:Calculate: V = (4/3) × π × 216 = 288π Step 4:Approximate: V ≈ 904.78 cm³
Practice Problems: Section 10.3
11.Find volume: sphere with r = 9 in. Work: ________________________________________________________________ Answer: ____________________________
12.Find volume: sphere with diameter 20 cm. Work: ________________________________________________________________ Answer: ____________________________
13.A basketball has diameter 24 cm. Volume of airinside? Work: ________________________________________________________________ Answer: ____________________________
14.A sphere has volume 36π in³. Find the radius. Work: ________________________________________________________________ Answer: ____________________________
15.Compare: sphere r = 3 vs. cube side = 6. Whichhas greater volume? Work: ________________________________________________________________ Answer: ____________________________
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Section 10.4: Composite Figures
For composite figures, break them into basic shapes and add/subtract volumes.
Practice Problems: Section 10.4
16.A cone (r = 4, h = 6) sits on a cylinder (r =4, h = 10). Total volume? Work: ________________________________________________________________ Answer: ____________________________
17.A hemisphere (r = 5) tops a cylinder (r = 5, h= 8). Total volume? Work: ________________________________________________________________ Answer: ____________________________
18.A cylinder (r = 6, h = 15) has a cone-shaped hole(r = 6, h = 15) drilled out. Remaining volume? Work: ________________________________________________________________ Answer: ____________________________
19.An ice cream cone holds 8π in³ of cone + hemispherescoop (r = 2). Total? Work: ________________________________________________________________ Answer: ____________________________
20.A capsule is a cylinder (r = 3, h = 10) with hemisphereends. Total volume? Work: ________________________________________________________________ Answer: ____________________________
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Chapter 10 Review
21.A silo is a cylinder (r = 8m, h = 20m) toppedwith a hemisphere. Total volume? Work: ________________________________________________________________ Answer: ____________________________
22.Water fills a cone at 5 cm³/sec. How long to filla cone with r = 6cm, h = 9cm? Work: ________________________________________________________________ Answer: ____________________________
23.A ball (sphere r = 6) is dropped in a cylinder(r = 6, h = 20). Water rises. How much? Work: ________________________________________________________________ Answer: ____________________________
24.Three tennis balls (r = 3.5 cm each) fit in acylindrical can. Can dimensions and volume? Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: A cone, cylinder, and sphere all haveradius 4 and height/diameter 8. Order by volume. Work: ________________________________________________________________ Answer: ____________________________
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CHAPTER 11 The Market Researcher
Master data analysis to find patterns and make predictions
What You'll Learn
• Construct and interpret scatter plots • Identify correlation (positive, negative, none) • Draw and use lines of best fit • Use two-way tables to analyze categorical data
Why This Matters
Market researchers analyze data to predict consumer behavior. Is there a relationship between advertising spending and sales? Do age and product preference correlate? Scatter plots and trend lines reveal patterns that guide million-dollar decisions.
Section 11.1: Scatter Plots
A scatter plot shows the relationship between two quantitative variables.
Each point represents one data pair (x, y).
Types of Correlation
Positive correlation: As x increases, y tends to increase
Negative correlation: As x increases, y tends to decrease
No correlation: No clear pattern between x and y
Practice Problems: Section 11.1
1.Study hours vs test scores: Positive, negative,or no correlation? Explain. Work: ________________________________________________________________ Answer: ____________________________
2.Temperature vs ice cream sales: Positive, negative,or no correlation? Work: ________________________________________________________________ Answer: ____________________________
3.Shoe size vs IQ: Positive, negative, or no correlation? Work: ________________________________________________________________ Answer: ____________________________
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4.Car age vs resale value: Positive, negative, or no correlation? Work: ________________________________________________________________ Answer: ____________________________
5.Data: (1,3), (2,5), (3,4), (4,7), (5,8). Plot anddescribe correlation. Work: ________________________________________________________________ Answer: ____________________________
Section 11.2: Lines of Best Fit
A line of best fit (trend line) approximates the overall trend in a scatter plot.
It passes through the 'middle' of the data points, minimizing distances.
WORKED EXAMPLE: Data suggests trend line y = 2x + 1. Predict y when x = 10.
Step 1:Substitute x = 10 into equation: y = 2(10)+ 1 Step 2:Calculate: y = 21
Practice Problems: Section 11.2
6.Line of best fit: y = 3x + 5. Predict y when x= 8. Work: ________________________________________________________________ Answer: ____________________________
7.Line of best fit: y = -2x + 50. Predict y whenx = 15. Work: ________________________________________________________________ Answer: ____________________________
8.Ad spending (x thousands) vs sales (y thousands):y = 4.5x + 10. Predict sales for $5000 ad spend. Work: ________________________________________________________________ Answer: ____________________________
9.Data points: (2, 9), (4, 15), (6, 21), (8, 27).Find a line of best fit. Work: ________________________________________________________________ Answer: ____________________________
10.Using your line from #9, predict y when x = 12. Work: ________________________________________________________________ Answer: ____________________________
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Section 11.3: Interpreting Slope and Intercept
In context, slope represents the rate of change.
The y-intercept represents the starting value when x = 0.
Practice Problems: Section 11.3
11.Sales = 500 + 20(advertising). What does the 20mean? The 500? Work: ________________________________________________________________ Answer: ____________________________
12.Cost = 2.50x + 150. x = units produced. Interpretslope and y-intercept. Work: ________________________________________________________________ Answer: ____________________________
13.Test score = 60 + 5(hours studied). Interpretand predict score for 8 hours. Work: ________________________________________________________________ Answer: ____________________________
14.Weight loss = -0.5(weeks) + 200. Starting weight?Rate of loss? Weight after 10 weeks? Work: ________________________________________________________________ Answer: ____________________________
15.A researcher finds height (cm) = 75 + 6(age inyears) for children ages 2-12. Interpret. Work: ________________________________________________________________ Answer: ____________________________
Section 11.4: Two-Way Tables
Two-way tables organize categorical data by two variables.
They help identify patterns and associations.
Practice Problems: Section 11.4
16.100 students surveyed: 40 prefer pizza, 35 prefertacos, 25 prefer burgers. 60 are girls. 25 girls prefer pizza. Complete a two-way table. Work: ________________________________________________________________ Answer: ____________________________
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17.From your table in #16, what percent of girls prefer pizza? Work: ________________________________________________________________ Answer: ____________________________
18.From your table, what percent of pizza-loversare girls? Work: ________________________________________________________________ Answer: ____________________________
19.200 customers: 120 bought Product A, 80 boughtB. 150 were repeat customers. 90 repeat customers bought A. Make a two-way table. Work: ________________________________________________________________ Answer: ____________________________
20.From #19, are repeat customers more likely tobuy A or B? Work: ________________________________________________________________ Answer: ____________________________
Chapter 11 Review
21.Describe the correlation: as exercise increases,resting heart rate tends to decrease. Answer: ____________________________
22.Line of best fit: y = -3x + 100. Predict y whenx = 25. Work: ________________________________________________________________ Answer: ____________________________
23.Revenue = 1000 + 150(employees). Interpret slope.Predict revenue with 20 employees. Work: ________________________________________________________________ Answer: ____________________________
24.Why might correlation not imply causation? Givean example. Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: Data shows strong positive correlationbetween firefighters sent and fire damage. Does sending fewer firefighters reduce damage? Explain. Work: ________________________________________________________________ Answer: ____________________________
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CHAPTER 12 The Risk Analyst
Master probability to assess risks and make informed decisions
What You'll Learn
• Calculate theoretical and experimental probability • Use sample spaces and organized lists • Calculate compound probability (independent and dependent events) • Use simulations to estimate probability
Why This Matters
Risk analysts help companies make decisions under uncertainty. What's the probability a product will fail? What are the odds of two system failures on the same day? Insurance, investment, and business strategy all depend on understanding probability.
Section 12.1: Basic Probability
P(event) = favorable outcomes / total possible outcomes
Probability ranges from 0 (impossible) to 1 (certain).
Can also be expressed as a percent (0% to 100%).
Practice Problems: Section 12.1
1.A standard die is rolled. P(rolling a 5)? Work: ________________________________________________________________ Answer: ____________________________
2.A standard die is rolled. P(rolling even)? Work: ________________________________________________________________ Answer: ____________________________
3.A bag has 5 red, 3 blue, 2 green marbles. P(red)? Work: ________________________________________________________________ Answer: ____________________________
4.A spinner has 8 equal sections numbered 1-8. P(primenumber)? Work: ________________________________________________________________ Answer: ____________________________
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5.A card is drawn from a standard 52-card deck. P(heart)? Work: ________________________________________________________________ Answer: ____________________________
Section 12.2: Compound Events - Independent
Independent events: The outcome of one doesn't affect the other.
P(A and B) = P(A) × P(B)
WORKED EXAMPLE: Two coins are flipped. P(both heads)?
Step 1:Events are independent Step 2:P(head on first) = 1/2 Step 3:P(head on second) = 1/2 Step 4:P(both heads) = 1/2 × 1/2 = 1/4
Practice Problems: Section 12.2
6.A die is rolled twice. P(6, then 6)? Work: ________________________________________________________________ Answer: ____________________________
7.A coin is flipped 3 times. P(all heads)? Work: ________________________________________________________________ Answer: ____________________________
8.P(rain today) = 0.3, P(rain tomorrow) = 0.4. P(rainboth days)? Work: ________________________________________________________________ Answer: ____________________________
9.A machine has 2% defect rate. Two items checked.P(both defective)? Work: ________________________________________________________________ Answer: ____________________________
10.Spinner A: 3 sections (1,2,3). Spinner B: 4 sections(1,2,3,4). P(both show 2)? Work: ________________________________________________________________ Answer: ____________________________
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Section 12.3: Compound Events - Dependent
Dependent events: The outcome of one affects the probability of the other.
P(A and B) = P(A) × P(B|A)
P(B|A) means 'probability of B given that A already happened'
WORKED EXAMPLE: Draw 2 cards without replacement. P(both aces)?
Step 1:P(first ace) = 4/52 Step 2:After drawing an ace, P(second ace) = 3/51 Step 3:P(both aces) = (4/52) × (3/51) = 12/2652 =1/221
Practice Problems: Section 12.3
11.Bag: 6 red, 4 blue marbles. Draw 2 without replacement.P(both red)? Work: ________________________________________________________________ Answer: ____________________________
12.Bag: 6 red, 4 blue. Draw 2 without replacement.P(first red, then blue)? Work: ________________________________________________________________ Answer: ____________________________
13.5 boys, 7 girls in a group. Two picked randomly.P(both girls)? Work: ________________________________________________________________ Answer: ____________________________
14.A deck of cards. Draw 2 without replacement. P(bothhearts)? Work: ________________________________________________________________ Answer: ____________________________
15.10 tickets in a hat, 3 winners. You draw 2. P(winningat least one)? Work: ________________________________________________________________ Answer: ____________________________
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Section 12.4: Sample Spaces and Simulations
A sample space lists all possible outcomes.
Simulations use random trials to estimate probability.
Practice Problems: Section 12.4
16.List the sample space for flipping a coin twice. Work: ________________________________________________________________ Answer: ____________________________
17.List the sample space for rolling a die and flippinga coin. Work: ________________________________________________________________ Answer: ____________________________
18.How many outcomes in the sample space for rolling3 dice? Work: ________________________________________________________________ Answer: ____________________________
19.A simulation runs 1000 trials of an event. 230successes. Estimated P(success)? Work: ________________________________________________________________ Answer: ____________________________
20.Theoretical P(event) = 0.25. After 200 trials,expected successes? Work: ________________________________________________________________ Answer: ____________________________
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Chapter 12 Review
21.A bag has 8 white, 5 black, 7 gray marbles. P(notwhite)? Work: ________________________________________________________________ Answer: ____________________________
22.Two dice rolled. P(sum = 7)? Work: ________________________________________________________________ Answer: ____________________________
23.3 coins flipped. P(exactly 2 heads)? Work: ________________________________________________________________ Answer: ____________________________
24.System A has 95% reliability, System B has 90%.P(both work)? Work: ________________________________________________________________ Answer: ____________________________
25.CHALLENGE: A password is 4 digits (0-9, repeatsallowed). P(all same digit)? Work: ________________________________________________________________ Answer: ____________________________
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EXTRA PRACTICE
Additional problems for each chapter to build mastery
Chapter 1 Extra Practice: Real Numbers
1.Classify: √64, √65, 0.123123..., π/2 Work: ________________________________________________________________ Answer: ____________________________
2.Order from least to greatest: √12, 3.5, √15, 4 Work: ________________________________________________________________ Answer: ____________________________
3.Simplify: √18 + √50 Work: ________________________________________________________________ Answer: ____________________________
4.Between which integers is √85? Work: ________________________________________________________________ Answer: ____________________________
5.Is 3√2 + 5 rational or irrational? Work: ________________________________________________________________ Answer: ____________________________
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Chapter 2 Extra Practice: Exponents and Radicals
6.Simplify: (3⁴ × 3⁻²) ÷ 3³ Work: ________________________________________________________________ Answer: ____________________________
7.Simplify: √180 Work: ________________________________________________________________ Answer: ____________________________
8.Evaluate: 2⁻³ + 4⁻¹ Work: ________________________________________________________________ Answer: ____________________________
9.Simplify: 5√3 × 2√6 Work: ________________________________________________________________ Answer: ____________________________
10.Simplify: ³√64 + √81 Work: ________________________________________________________________ Answer: ____________________________
Chapter 3 Extra Practice: Scientific Notation
11.Write in scientific notation: 0.0000456 Work: ________________________________________________________________ Answer: ____________________________
12.(2.4 × 10⁶) × (5 × 10⁻³) = ____ Work: ________________________________________________________________ Answer: ____________________________
13.(9.6 × 10⁸) ÷ (3.2 × 10⁵) = ____ Work: ________________________________________________________________ Answer: ____________________________
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14.Compare: 5.2 × 10⁷ vs 8.1 × 10⁶ Work: ________________________________________________________________ Answer: ____________________________
15.A light-year is 9.46 × 10¹² km. Alpha Centauriis 4.37 light-years away. Distance in km? Work: ________________________________________________________________ Answer: ____________________________
Chapter 4 Extra Practice: Multi-Step Equations
16.Solve: 8x - 5 = 3x + 20 Work: ________________________________________________________________ Answer: ____________________________
17.Solve: 4(2x - 3) = 3(x + 4) - 1 Work: ________________________________________________________________ Answer: ____________________________
18.Solve: 6x + 10 = 2(3x + 5). How many solutions? Work: ________________________________________________________________ Answer: ____________________________
19.Cell plan A: $45 + $0.05/text. Plan B: $30 + $0.08/text.When equal? Work: ________________________________________________________________ Answer: ____________________________
20.Solve: ½(4x - 8) = x + 6 Work: ________________________________________________________________ Answer: ____________________________
Chapters 5-6 Extra Practice: Functions and Systems
21.If f(x) = -3x + 10, find f(-2). Work: ________________________________________________________________ Answer: ____________________________
22.Find slope and y-intercept: 4x + 2y = 10 Work: ________________________________________________________________ Answer: ____________________________
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23.Write equation through (2, 5) and (6, 13). Work: ________________________________________________________________ Answer: ____________________________
24.Solve by substitution: y = 3x - 1 and 2x + y =14 Work: ________________________________________________________________ Answer: ____________________________
25.Solve by elimination: 3x + 4y = 18 and 5x - 4y= 6 Work: ________________________________________________________________ Answer: ____________________________
Chapters 7-8 Extra Practice: Pythagorean Theorem and Transformations
26.Find c: a = 15, b = 20 Work: ________________________________________________________________ Answer: ____________________________
27.Is 9, 40, 41 a Pythagorean triple? Work: ________________________________________________________________ Answer: ____________________________
28.Distance from (1, 2) to (7, 10)? Work: ________________________________________________________________ Answer: ____________________________
29.Rotate (5, -3) 90° counterclockwise about origin. Work: ________________________________________________________________ Answer: ____________________________
30.Reflect (4, 7) over y = x. Work: ________________________________________________________________ Answer: ____________________________
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Chapters 9-10 Extra Practice: Angles and Volume
31.Parallel lines cut by transversal. One angle =72°. Find its alternate interior angle. Work: ________________________________________________________________ Answer: ____________________________
32.Triangle angles: 2x, 3x, 4x. Find all angles. Work: ________________________________________________________________ Answer: ____________________________
33.Volume of cylinder: r = 7, h = 10. (Leave in termsof π) Work: ________________________________________________________________ Answer: ____________________________
34.Volume of cone: r = 9, h = 12. Work: ________________________________________________________________ Answer: ____________________________
35.Volume of sphere: r = 6. Work: ________________________________________________________________ Answer: ____________________________
Chapters 11-12 Extra Practice: Data and Probability
36.Line of best fit: y = 2.5x + 8. Predict y whenx = 12. Work: ________________________________________________________________ Answer: ____________________________
37.Describe correlation: As study time increases,test scores tend to increase. Work: ________________________________________________________________ Answer: ____________________________
38.P(rolling two dice and getting sum of 9)? Work: ________________________________________________________________ Answer: ____________________________
39.Bag: 7 red, 5 blue. Draw 2 without replacement.P(both blue)? Work: ________________________________________________________________ Answer: ____________________________
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40.A coin is flipped 4 times. P(exactly 3 heads)? Work: ________________________________________________________________ Answer: ____________________________
ANSWER KEY
Check your work and learn from mistakes
Chapter 1: Real Numbers
1. 0.666... = rational (repeating), √5 = irrational, -4/7 = rational, π = irrational, 3.14 = rational (terminating)
2. Irrational (non-repeating, non-terminating decimal suggesting e)
3. 0.875 = 7/8, rational
4. √16 = 4 (rational, perfect square); √17 ≈ 4.12 (irrational)
5. Between 7 and 8 (since 49 < 50 < 64)
6. Between 5 and 6 (since 25 < 30 < 36)
7. √45 ≈ 6.7
8. √10 ≈ 3.16, so the order is 3, π (3.14), √10, 3.2
9. √200 ≈ 14.1 ft
10. √3 ≈ 1.7, √5 ≈ 2.2, √8 ≈ 2.8
11-15. [See detailed solutions in teacher guide]
16-20. [See detailed solutions in teacher guide]
Chapter 2: Exponents and Radicals
1. 3⁷ = 2187
2. 7⁵ = 16807
3. 4⁶ = 4096
4. 8⁰ = 1
5. 2² = 4
6. 4⁻² = 1/16
7. 10⁻³ = 0.001
8. 2⁻² = 1/4
9. 3⁻⁴ = 1/81
10. 10² = 100 N
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11. √144 = 12
12. ³√27 = 3
13. √50 = 5√2
14. √128 = 8√2
15. ³√343 = 7 cm
16-20. 6√3; 10; 5√3; 0; 6 cm²
21-25. 2⁴ = 16; 1.04; 10√2; √7; x + y = 9
Chapter 3: Scientific Notation
1. 4.5 × 10⁷
2. 6.7 × 10⁻⁴
3. 380,000
4. 0.00021
5. 4 × 10¹³ km
6-10. 1.2 × 10⁸; 4 × 10⁴; 3 × 10³; 1.5 × 10¹¹ m; 6 × 10⁷ km/hr
11-15. 8.8 × 10⁶; 8.08 × 10⁵; 5.48 × 10⁸ km; 1.54 × 10⁷; ~3 × 10⁻²⁷ kg
16-20. 3.5 × 10⁸; 8.1 × 10⁴, 1.9 × 10⁵, 2.3 × 10⁵; ~1.5×; 25×; ~316 Earths
21-25. 8.9 × 10⁻⁶; 1.08 × 10³; 3.6 × 10⁴ km; ~3.15 × 10⁷ s; ~1.33 × 10²³ km
Chapter 4: Multi-Step Equations
1. x = 4
2. x = 4
3. x = 6
4. x = 5
5. 4 hours
6. x = 8
7. x = 7
8. No solution
9. x = 1
10. 5 GB
11-15. No solution; Infinite; No solution; Infinite; Answers vary
16-20. x = 5; x = 6; 67 units; Infinite solutions; 32, 33, 34
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Chapters 5-6: Functions and Systems
Chapter 5: 1. Yes; 2. No (2 maps to 4 and 6); 6. 15; 7. 10; 8. -9; 9. x = 6; 10. 29 modules
11. m = 2; 12. m = -2; 13. m = -3, b = 7; 14. y = 4x - 2; 15. y = 2x + 5
16-20. y = 3x + 2; y = 2x - 3; P(w) = 5w + 1; d(t) = 15t; T(h) = -4.5h + 68
21-25. f(-3) = 14, f(0) = 8; m = 2, b = -3; No; C(m) = 50 + 30m, $410; (1, 7)
Chapter 6: 1. Yes; 2. Yes; 4. (2, 3); 5. No solution (parallel)
6-10. (3, 7); (1, 2); (2, 3); 25 adults, 20 children; 21 and 29
11-15. (3, 2); (3, 2); (4, 3); (2, 3); 35 phones, 65 cases
16-20. (5, 13); (4, 2); (4, 3); 2.5 mph; b = 3, any a ≠ 5
Chapters 7-8: Pythagorean Theorem and Transformations
Chapter 7: 1. c = 5; 2. c = 13; 3. c = 17; 4. 13 ft; 5. 30 in
6-10. a = 7; b = 21; 15 ft; 40 m; 60 m
11-15. Yes; No; Yes; No (should be 10); 12
16-20. 10 units; 50 km; ~12.6 ft; ~127.3 ft; 100 m
21-25. 61; 28; Yes (11² + 60² = 61²); ~24.2 ft; 13 units
Chapter 8: 1. (7, 7); 2. (-4, 2); 3. A'(4, 1), B'(7, 1), C'(5, 4); 4. (4, 12); 5. (-4, 5)
6-10. (4, -7); (3, 5); (8, 2); (-1, 1), (-4, 1), (-2, 4); x-axis
11-15. (-5, 3); (-4, 2); (6, 1); (-1, 2), (-1, 5), (-4, 3); 90° CCW
16-20. (8, 12); (5, 7.5); All × 3; k = 3; k = 3
21-25. (3, -1); (2, 8); Reflection over y = x; T, R, R (not dilation); k = 3
Chapters 9-10: Angles and Volume
Chapter 9: 1. 65°; 2. x = 15, angles = 55°; 3. x = 30, angles = 60° and 120°; 4. 72°
6-10. 70°; 70°; 30°, 60°, 90°; 58°; 60°
11-15. 115°; 75°; x = 25; 55°; Exterior > either remote (since exterior = sum)
16-20. x = 70°, yes; x = 10; 30 ft; 36, 48; Similar but not necessarily congruent
21-25. 118°, 62°, etc.; 35°, 60°, 85°; x = 10; 20 ft; 110°
Chapter 10: 1. 160π in³; 2. 392π cm³; 3. 63π m³; 4. r ≈ 3.04 cm; 5. ~6.28 L
6-10. 24π in³; 125π/3 cm³; 8π in³; r = 5 cm; h = 18 cm
11-15. 972π in³; 4000π/3 cm³; 2304π cm³; r = 3 in; Cube (216 > 113.1)
16-20. 192π; 250π/3 + 200π; 360π; 8π + 16π/3; 90π + 36π
21-25. 1408π/3 m³; ~113 sec; 8 cm rise; r = 3.5, h = 21, V = 257.25π; Sphere < Cone < Cylinder
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Chapters 11-12: Data and Probability
Chapter 11: 1-5. Positive; Positive; None; Negative; Positive moderate
6-10. 29; 20; $32,500; y = 3x + 3; 39
11-15. Slope = revenue per $ of advertising, intercept = base revenue; Per-unit cost, fixed cost; ~100; 195 lbs; 6 cm/year growth rate
16-20. [Table solutions]; ~42%; ~63%; [Table solutions]; Yes, 90/120 = 75% vs 60/80 = 75%
21-25. Negative; 25; 4000; Correlation ≠ causation; Larger fires need more firefighters
Chapter 12: 1. 1/6; 2. 1/2; 3. 1/2; 4. 4/8 = 1/2 (primes: 2,3,5,7); 5. 13/52 = 1/4
6-10. 1/36; 1/8; 0.12; 0.0004; 1/12
11-15. 30/90 = 1/3; 24/90 = 4/15; 42/132 = 7/22; 156/2652 = 1/17; ~47%
16-20. {HH, HT, TH, TT}; 12 outcomes; 216; 0.23; 50
21-25. 12/20 = 3/5; 6/36 = 1/6; 4/8 = 1/2; 0.855; 1/1000
Extra Practice Answers
1. √64 = 8 (rational), √65 (irrational), 0.123123... (rational), π/2 (irrational)
2. √12, 3.5, √15, 4
3. 3√2 + 5√2 = 8√2
4. Between 9 and 10
5. Irrational
6. 3⁻¹ = 1/3
7. 6√5
8. 1/8 + 1/4 = 3/8
9. 10√18 = 30√2
10. 4 + 9 = 13
11-40. [See teacher guide for complete solutions.]
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