Prealgebra Lesson for a 15-year-old: Arithmetic Rules (Day 1) — Foundation for AoPS Prealgebra Series

Clear, step-by-step Day 1 lesson on arithmetic rules for a 15-year-old. Covers commutative, associative, distributive laws, identity and inverse elements, order of operations, negatives, worked examples, practice problems and solutions. Aligned with ACARA v9 and AoPS Prealgebra objectives.

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Arithmetic Rules — Day 1 (for a 15-year-old)

This lesson builds the foundation you need for algebra: the key properties of arithmetic and the correct order for carrying out calculations. Work through the short explanations, then try the practice problems. Answers and step-by-step solutions are below.

Learning goals

  • Know and use the commutative, associative, and distributive laws.
  • Recognise identity and inverse elements for addition and multiplication.
  • Apply order of operations (PEMDAS) correctly, including exponents and negatives.
  • Solve mixed arithmetic expressions step-by-step and spot common mistakes.

Quick review of the rules

1. Commutative laws

Order doesn't matter for addition or multiplication.

  • a + b = b + a
  • a × b = b × a

2. Associative laws

When adding or multiplying, you can regroup terms without changing the result.

  • (a + b) + c = a + (b + c)
  • (a × b) × c = a × (b × c)

3. Distributive law

Multiplication distributes over addition or subtraction.

  • a × (b + c) = a × b + a × c
  • a × (b - c) = a × b - a × c

4. Identity elements

  • 0 is the additive identity: a + 0 = a
  • 1 is the multiplicative identity: a × 1 = a

5. Inverses

  • For addition: the inverse of a is -a because a + (-a) = 0
  • For multiplication: the inverse of a (nonzero) is 1/a because a × (1/a) = 1

6. Order of operations (PEMDAS)

Always follow this order when simplifying expressions:

  1. Parentheses (and other grouping symbols)
  2. Exponents (powers and roots)
  3. Multiplication and Division (left to right)
  4. Addition and Subtraction (left to right)

Note: Multiplication and division are on the same level; do them in the order they appear, left to right. Same for addition and subtraction.

7. Rules with negatives

  • Positive × Positive = Positive
  • Negative × Negative = Positive
  • Positive × Negative = Negative
  • Subtracting is adding the opposite: a - b = a + (-b)

Worked examples (step-by-step)

Example 1

Simplify: 3 + 4 × 2^2 - (6 ÷ 3)

Step 1: Parentheses first: inside parentheses 6 ÷ 3 = 2, so expression becomes 3 + 4 × 2^2 - 2.

Step 2: Exponents: 2^2 = 4, so 3 + 4 × 4 - 2.

Step 3: Multiplication: 4 × 4 = 16, so 3 + 16 - 2.

Step 4: Left to right: 3 + 16 = 19, then 19 - 2 = 17. Answer: 17.

Example 2 (use distributive law)

Simplify: 5(2x + 3) when x = 4

Either substitute first or distribute first. Distribute: 5(2x) + 5(3) = 10x + 15. Then substitute x = 4: 10(4) + 15 = 40 + 15 = 55.

Example 3 (negatives and order)

Simplify: -2(3 - 5) + 4

Step 1: Parentheses: 3 - 5 = -2. So -2 × (-2) + 4.

Step 2: -2 × -2 = 4. So 4 + 4 = 8. Answer: 8.

Common pitfalls

  • Doing addition before multiplication. Always check for multiplication and exponents first.
  • Forgetting to distribute the sign: - (a + b) = -a - b.
  • Mixing up order of left-to-right for multiplication/division and addition/subtraction.
  • Assuming subtraction is associative: (a - b) - c is not equal to a - (b - c) in general.

Practice problems (try these before checking solutions)

  1. Simplify: 7 + 3 × (2 + 5)
  2. Simplify: (4 + 6) × 2^2 - 5
  3. Simplify: -3(2 - 7) + 1
  4. Simplify using distributive law: 6(3x - 2) when x = 5
  5. Simplify: 18 ÷ 3 × 2 - 4
  6. Rewrite and simplify: 8 - (3 + 5) and explain why 8 - 3 + 5 is different.

Answers with steps

  1. 7 + 3 × (2 + 5)

    Parentheses: 2 + 5 = 7 → 7 + 3 × 7. Multiplication: 3 × 7 = 21 → 7 + 21 = 28.

  2. (4 + 6) × 2^2 - 5

    Parentheses: 4 + 6 = 10 → 10 × 2^2 - 5. Exponent: 2^2 = 4 → 10 × 4 - 5. Multiplication: 40 - 5 = 35.

  3. -3(2 - 7) + 1

    Parentheses: 2 - 7 = -5 → -3 × (-5) + 1. Multiply: 15 + 1 = 16.

  4. 6(3x - 2) when x = 5

    Substitute: 3(5) - 2 = 15 - 2 = 13 → 6 × 13 = 78. Or distribute: 18x - 12, then put x = 5 → 90 - 12 = 78.

  5. 18 ÷ 3 × 2 - 4

    Left to right for ÷ and ×: 18 ÷ 3 = 6 → 6 × 2 - 4 = 12 - 4 = 8.

  6. 8 - (3 + 5) and comparison with 8 - 3 + 5

    First: 3 + 5 = 8 → 8 - 8 = 0.

    Second: 8 - 3 + 5 (left to right): 8 - 3 = 5 → 5 + 5 = 10. They are different because parentheses change the grouping. Subtraction is not associative.

Extensions and next steps

  • Day 2: practice more problems focusing on tricky signs and nested parentheses.
  • Later: move to squares and exponent laws, then zero and negative exponents.
  • Use Alcumus and the video series for extra practice—work on timed problems that emphasize speed and accuracy.

How this maps to the course plan and ACARA v9

This lesson is the first step in the Prealgebra sequence: it prepares you for exponent laws, prime factorizations, fractions, linear equations and more. In ACARA terms it supports the Number and Algebra strand: applying properties of arithmetic, order of operations and working with integers and algebraic expressions.

If you want, I can now:

  • Give more practice problems at an appropriate difficulty level for you, including timed drills.
  • Prepare Day 2 focusing on tricky signs, nested parentheses and mental strategies.
  • Move on to the next topic: Squares and Exponent Laws.

Which would you like next?

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