Key Concepts
- Exponents: Repeated multiplication (e.g. 24 = 2×2×2×2 = 16).
- Scientific Notation: Writing numbers as a product of a decimal 1–10 and a power of ten. Example: 3,500 = 3.5 × 103.
- Order of Operations: PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
- Commutative Property: Order doesn't matter for addition/multiplication. a+b = b+a; ab = ba.
- Associative Property: Grouping doesn't matter for addition/multiplication. (a+b)+c = a+(b+c).
- Distributive Property: a(b+c) = ab + ac.
- Identity Property: a+0=a; a×1=a.
- Zero Property: a×0=0.
- Equality Properties: If a=b, then a+c=b+c, a−c=b−c, a×c=b×c, and if c≠0, a/c=b/c.
- Factors: Numbers that multiply to give another number (e.g. factors of 6: 1,2,3,6).
- Multiples: Products of a number and integers (e.g. multiples of 4: 4,8,12,...).
- Variable Expressions: Expressions with variables (e.g. 2x+3).
- Solving Equations: Finding variable values. Use inverse operations (add/subtract/multiply/divide).
- Inequalities: Mathematical sentences using <, >, ≤, ≥. Example: x > 5.
- Solving by Substitution: Replace a variable with a number or another expression to check or solve.
Concepts to Focus On
- Order of operations and applying properties
- Translating and simplifying variable expressions
- Solving basic equations and inequalities
- Working with exponents and scientific notation
- Recognizing and using factors and multiples
Worked Examples
Exponents
Example: 32 = 3 × 3 = 9
Example: 32 = 3 × 3 = 9
Scientific Notation
Example: 45,000 = 4.5 × 104
Example: 45,000 = 4.5 × 104
Order of Operations
Example: 2 + 3 × 4 = 2 + 12 = 14
Example: 2 + 3 × 4 = 2 + 12 = 14
Commutative Property
Example: 5 + 7 = 7 + 5; 3 × 8 = 8 × 3
Example: 5 + 7 = 7 + 5; 3 × 8 = 8 × 3
Associative Property
Example: (2 + 3) + 4 = 2 + (3 + 4) = 9
Example: (2 + 3) + 4 = 2 + (3 + 4) = 9
Distributive Property
Example: 2(3 + 5) = 2×3 + 2×5 = 6 + 10 = 16
Example: 2(3 + 5) = 2×3 + 2×5 = 6 + 10 = 16
Identity and Zero Properties
Example: 7 + 0 = 7 (identity), 9 × 0 = 0 (zero)
Example: 7 + 0 = 7 (identity), 9 × 0 = 0 (zero)
Factors and Multiples
Example: Factors of 12: 1, 2, 3, 4, 6, 12; Multiples of 3: 3, 6, 9, 12, ...
Example: Factors of 12: 1, 2, 3, 4, 6, 12; Multiples of 3: 3, 6, 9, 12, ...
Variable Expressions
Example: If x = 4, then 2x + 1 = 2×4 + 1 = 9
Example: If x = 4, then 2x + 1 = 2×4 + 1 = 9
Solving Equations by Addition
Example: x - 5 = 7 → x = 12
Example: x - 5 = 7 → x = 12
Solving Equations by Multiplication
Example: 3x = 21 → x = 7
Example: 3x = 21 → x = 7
Inequalities
Example: x + 2 < 7 → x < 5
Example: x + 2 < 7 → x < 5
Solving by Substitution
Example: If y = 2, then 4y = 8
Example: If y = 2, then 4y = 8