Grade 9 Math EQAO

1. Number Systems

• Natural Numbers: Integers greater than 0 (e.g., 1, 2, 3, …).

• Whole Numbers: Integers greater than or equal to 0 (e.g., 0, 1, 2, 3, …).

• Integers: Positive and negative whole numbers, including 0 (e.g., -2, -1, 0, 1, 2). Can be divided evenly into fractions.

• Rational Numbers: Numbers that can be expressed as fractions or terminating/repeating decimals (e.g., 12=0.5\frac{1}{2} = 0.521​=0.5, 23=0.666…\frac{2}{3} = 0.666…32​=0.666…).

• Irrational Numbers: Decimals that do not repeat or have no pattern (e.g., π,2\pi, \sqrt{2}π,2​).

• Real Numbers: Includes all rational and irrational numbers (e.g., 5, -1.3, π,2\pi, \sqrt{2}π,2​).

2. Density and Infinity

• Density: Refers to the spacing between numbers in a set.

  • Larger gaps = less dense (e.g., integers).

  • Smaller gaps = more dense (e.g., rational numbers).

• Infinity: A concept describing a limitless quantity, greater than any real number.

• Limits: A value a function approaches but never reaches.

3. Number Sets on a Number Line

• ∘\circ∘: Open circle → the value is excluded.

• ∙\bullet∙: Closed circle → the value is included.

4. Operations with Integers

• Addition:

  • Positive + Positive = Bigger positive number.

  • Negative + Negative = Bigger negative number.

• Subtraction:

  • Different signs → Subtract.

  • Same signs → Treat as addition and keep the sign.

• Multiplication/Division:

  • Same signs → Positive result.

  • Different signs → Negative result.

5. Fractions

• Equivalent Fractions: Different fractions representing the same value (e.g., 24=12\frac{2}{4} = \frac{1}{2}42​=21​).

• Adding/Subtracting Fractions: Find a common denominator and add/subtract the numerators. Simplify afterward.

• Multiplying Fractions: Multiply numerators and denominators. Simplify.

• Dividing Fractions: Flip the second fraction and multiply.

6. BEDMAS (Order of Operations)

• Brackets → Exponents → Division/Multiplication (left to right) → Addition/Subtraction (left to right).

7. Percents, Decimals, and Fractions

• Convert between forms:

  • Percent to Decimal: Divide by 100.

  • Decimal to Fraction: Write as number10n\frac{\text{number}}{10^n}10nnumber​.

  • Fraction to Percent: Convert to decimal first, then multiply by 100.

8. Collecting Like Terms

• Combine terms with the same variables and exponents.

  • Example: 3x+5x=8x3x + 5x = 8x3x+5x=8x.

• Terms with different variables or exponents cannot be combined.

9. Exponent Laws

• Product Rule: am⋅an=am+na^m \cdot a^n = a^{m+n}am⋅an=am+n.

• Quotient Rule: am/an=am−na^m / a^n = a^{m-n}am/an=am−n.

• Power Rule: (am)n=am⋅n(a^m)^n = a^{m \cdot n}(am)n=am⋅n.

• Zero Exponent Rule: a0=1a^0 = 1a0=1 (where a≠0a \neq 0a=0).

• Negative Exponent: a−m=1ama^{-m} = \frac{1}{a^m}a−m=am1​.

10. Scientific Notation

• Positive Exponents: Large numbers (e.g., 4.5×103=45004.5 \times 10^3 = 45004.5×103=4500).

• Negative Exponents: Small numbers (e.g., 4.5×10−3=0.00454.5 \times 10^{-3} = 0.00454.5×10−3=0.0045).

11. Solving Equations

1. Basic Equations:

   • Combine like terms and use inverse operations to isolate the variable.

   • Check by substitution.

2. Equations with Variables on Both Sides:

   • Move variable terms to one side and constants to the other.

3. Equations with Fractions:

   • Multiply through by the LCD to eliminate fractions.

12. Ratios and Proportions

• Ratio: Comparison of two quantities (e.g., 3:4 or 34\frac{3}{4}43​).

• Proportion: Two equal ratios (e.g., 34=68\frac{3}{4} = \frac{6}{8}43​=86​).