Chapter 1: Numbers & Values

• Whole Number: A whole number is a number without fractions or decimals. Examples: 0, 1, 2, 3, 10, 100.
• Place Value: Place value is the value of a digit based on its position in a number. For example, in 345, the 3 is in the hundreds place, so it's worth 300.
• Absolute Value: The absolute value of a number is its distance from zero on a number line, always positive. For example, \( |-5| = 5 \).
• Distance: In math, distance means how far apart two points or numbers are. For numbers, it's the absolute value of their difference. For example, the distance between 2 and −3 is \( |2 - (-3)| = |5| = 5 \).

Chapter 2: Fractions & Mixed Numbers

• Changing Improper Fractions to Mixed Numbers:
  Divide the numerator by the denominator. The quotient is the whole number, the remainder is the new numerator.
  Example:
  \( \frac{9}{4} = 2 \frac{1}{4} \) because \( 9 \div 4 = 2 \) remainder 1.
• Changing Mixed Numbers to Improper Fractions:
  Multiply the whole number by the denominator and add the numerator. Put that over the original denominator.
  Example:
  \( 3 \frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5} \)

Chapter 3: Adding & Subtracting Fractions

• Adding Fractions with Like Denominators: Add the numerators; keep the denominator.
  
  \( \frac{2}{7} + \frac{3}{7} = \frac{5}{7} \)
• Subtracting Fractions with Like Denominators: Subtract the numerators; keep the denominator.
  
  \( \frac{5}{8} - \frac{1}{8} = \frac{4}{8} = \frac{1}{2} \)
• Adding or Subtracting Fractions with Unlike Denominators: Find a common denominator, convert, then add or subtract.
  \( \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \)
• Adding Mixed Numbers with Unlike Denominators: Convert to improper fractions, find common denominator, add, and simplify.
• Subtracting Mixed Numbers with Unlike Denominators: Convert to improper fractions, find common denominator, subtract, and simplify.
• Estimating Sums and Differences of Fractions and Mixed Numbers: Round fractions to 0, ½, or 1; round mixed numbers to nearest whole. Then add or subtract to estimate.

Chapter 4: Multiplying & Dividing Fractions

• Multiplying Fractions and Whole Numbers: Multiply the whole number by the numerator; keep the denominator.
  \( 3 \times \frac{2}{5} = \frac{6}{5} = 1\frac{1}{5} \)
• Multiplying Fractions: Reciprocals: The reciprocal of a fraction is obtained by swapping numerator and denominator. Multiplying a number by its reciprocal gives 1.
  Reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). \( \frac{3}{4} \times \frac{4}{3} = 1 \)
• Multiplying Fractions and Mixed Numbers: Reducing: Convert mixed numbers to improper fractions, multiply, then simplify.
• Dividing Fractions by Whole Numbers: Multiply by the reciprocal of the whole number.
  \( \frac{5}{6} \div 2 = \frac{5}{6} \times \frac{1}{2} = \frac{5}{12} \)
• Dividing Whole Numbers by Fractions: Multiply the whole number by the reciprocal of the fraction.
  \( 3 \div \frac{2}{5} = 3 \times \frac{5}{2} = \frac{15}{2} \)
• Dividing Fractions by Fractions: Multiply by the reciprocal of the divisor.
  \( \frac{3}{4} \div \frac{1}{2} = \frac{3}{4} \times \frac{2}{1} = \frac{6}{4} = \frac{3}{2} \)
• Dividing Mixed Numbers: Convert to improper fractions, then follow the steps above.

Chapter 5: Ratios & Proportions

• Ratios: A ratio compares two quantities. Written as a:b, a to b, or a/b.
• Proportions and Cross-Multiplying: A proportion is two equal ratios. To solve: cross-multiply and solve for the unknown.
  If \( \frac{2}{3} = \frac{x}{6} \), then \( 2 \times 6 = 3 \times x \implies x = 4 \)
• Ratio Tables: A table that shows equivalent ratios.
  2:3, 4:6, 6:9, ...
• Rates: A rate is a ratio comparing two quantities with different units (e.g. miles per hour).
• Problem-Solving with Proportions: Use proportions to solve real-world problems (e.g. recipes, maps, scaling).

Chapter 6: Examples

Chapter 7: Pre-Test (40 Questions)

Chapter 8: Step-by-Step Solutions for Pre-Test Errors

Chapter 9: Post-Test (40 Questions)