Pre-Algebra Honors Study Guide & Practice Review

Unit 1: Signed-Number Arithmetic & Order of Operations

Key Strategies & Cheat Sheet

• Order of Operations (PEMDAS): Parentheses/Brackets → Exponents → Multiplication/Division (Left to Right) → Addition/Subtraction (Left to Right).

• Signed Exponents: Pay close attention to parentheses! (-3)² = (-3) × (-3) = 9, whereas -3² = -(3 × 3) = -9.

• Absolute Value: Always evaluate the expression inside the absolute value bars first. |-5| = 5. Absolute value measures distance from 0, so it is always non-negative.

• Fraction Operations: Find a common denominator before adding or subtracting fractions. Dividing fractions means multiplying by the reciprocal (Keep-Change-Flip).

Guided Example

Problem: Evaluate [-4]² - 3 × (-2) + |-6 + 1|

Step 1   Inside absolute value: |-6 + 1| = |-5| = 5.

Step 2   Exponent:     [-4]² = 16.

Step 3  Multiplication: 3 × (-2) = -6.

Step 4  Combine:   16 - (-6) + 5 = 16 + 6 + 5 = 27.

Practice Problems (Do on separate paper)

1. Evaluate: -9 + 5 × (-4) - (-8)

2. Evaluate: (-3)³ - 4(-2) + 7

3. Simplify: |-10 + 4| - |5 - 12|

4. Compute: (3/4) - (-1/3) + (5/12)

Unit 2: Expressions & Multi-Step Equations

Key Strategies & Cheat Sheet

• Evaluating Expressions: Substitute negative numbers using parentheses to avoid sign mistakes.

• Solving Multi-Step Equations: Distribute → Combine Like Terms → Move variables to one side → Isolate the variable using inverse operations.

• Literal Equations: Treat all other letters as constants when solving for a target variable.

• Translating Words to Algebra:

• "Less than" flips the order: "4 less than 3x" becomes 3x - 4.

• "Increased by" means addition. "Is" means equals (=).

Guided Example

Problem: Solve for x: 4(2x - 3) = 3x + 13

Step 1 Distribute: 8x - 12 = 3x + 13.

Step 2 Subtract 3x from both sides: 5x - 12 = 13.

Step 3 Add 12 to both sides:       5x = 25.

Step 4  Divide by 5:   x = 5.

Practice Problems(Do on separate paper)

5. Evaluate 3x² - 5x - 2 when x = -3.

6. Solve for x: 4(3x - 2) = 7x + 12

7. Solve for x: (3/5)x - 2 = 7

8. Solve the literal equation for y: 3x - 4y = 24

9. Translate and solve: "Five less than twice a number, increased by three times the number, is 40."

Unit 3: Ratios, Rates & Proportional Reasoning

Key Strategies & Cheat Sheet

• Proportions: Use cross-multiplication (a/b = c/d → a × d = b × c) to solve for missing variables.

• Scaling Quantities: Find the scale factor by dividing the target amount by the original amount.

• Inverse Proportions (Work Rate): If work rate is constant, total work done = (Workers) × (Time). If workers decrease, time increases proportionally!

Guided Example

Problem: If 8 workers can complete a project in 6 hours, how long will it take 3 workers?

Step 1   Calculate total worker-hours required: 8 workers × 6 hours = 48 worker-hours.

Step 2    Divide total hours by the new number of workers: 48 worker-hours / 3 workers    

    =  16 hrs

Practice Problems(Do on separate paper)

10. Simplify the ratio 24:36 and express it as a unit rate in miles per gallon assuming 24 miles and 36 gallons.

11. Comparison: Car X travels 315 miles on 9 gallons. Car Y travels 210 miles on 6 gallons. Compare their fuel economies.

12. Solve the proportion: 3x - 24 = 108

13. A recipe for 6 servings requires 2.5 cups of flour. How many cups of flour are needed for 15 servings?

14. Honors Application: If 5 technicians configure a server room in 12 hours, how many hours will it take 4 technicians working at the exact same pace?

Unit 4: The Coordinate Plane & Linear Relationships

Key Strategies & Cheat Sheet

• Quadrants: Quadrant I (+, +), Quadrant II (-, +), Quadrant III (-, -), Quadrant IV (+, -). Points on axes (e.g., (0,0)) are on an axis/origin, not in a quadrant.

• Slope Formula: m = (y₂ - y₁) / (x₂ - x₁). (Rise over Run).

• Finding Additional Points: Use the slope m = Δy / Δx to move from a known point (x, y).

• Collinearity Check: Points are collinear (on the same straight line) if the slope between any two pairs of points is equal!

Guided Example

Problem: Check if points (1, 3), (4, 9), and (6, 13) are collinear.

Step 1      Slope between (1,3) and (4,9): m₁ = (9 - 3) / (4 - 1) = 6 / 3 = 2.

Step 2      Slope between (4,9) and (6,13): m₂ = (13 - 9) / (6 - 4) = 4 / 2 = 2.

Step 3      Since m₁ = m₂ = 2, the points are collinear!

Practice Problems(Do on separate paper)

15. Identify the quadrant or location for: P(-5, 2), Q(3, -6), R(0, -4), and S(-2, -3).

16. Find the slope of the line passing through (-2, 7) and (3, -3).

17. A line has slope -2/3 and passes through (4, 1). Find two other points on this line.

18. Determine if (2, 4), (5, 10), and (8, 15) lie on the same line using slope.

19. Verification: Is the point (-4, 9) on the line y = -2x + 1? Show your work.

Unit 5: Data Analysis & Statistical Reasoning

Key Strategies & Cheat Sheet

• Mean: (Sum of all data values) / (Total count of data values).

• Median: Middle value when data is ordered least to greatest. (If even count, average the middle two)

• Range: Maximum value - Minimum value (measures spread/variability).

• Outlier Effect: Extreme outliers pull the mean towards them, but the median remains resistant and stable.

Guided Example

Problem: A dataset of 5 values has a mean of 14. Four values are 10, 12, 15, and 18. Find the missing value.

Step 1    Total sum needed = 5 × 14 = 70.

Step 2    Sum of 4 known values = 10 + 12 + 15 + 18 = 55.

Step 3    Missing value = 70 - 55 = 15.

Practice Problems (Do on separate paper)

20. Find the mean, median, and range for: {14, 22, 8, 19, 12, 27}.

21. A data set of 5 values has a mean of 22. Four values are 18, 24, 20, and 28. Find the 5th value.

22. Compare Set A {15, 16, 17, 18, 19} and Set B {10, 14, 17, 20, 24}:

• Calculate the mean and range for each set.

• Which set has greater variability? Explain.

23. Write a short explanation of why an outlier affects the mean more than the median, and create your own 5-number dataset showing this effect.

Section 6: Practice Problem Answer Key