Chapter 2: Arithmetic Expressions
Punctuation of Mathematics: Structure of terms, removing brackets, and the distributive property.
📖 Core Vocabulary
A phrase like 13 + 2 or 5 × 4 that evaluates to a specific number.
Parts separated by '+' signs. Subtraction is converted to adding the inverse: 83 − 14 = (83) + (−14).
Specifies evaluation priority before other operations: 30 + (5 × 4) = 50.
✂️ Bracket Removal Rules
Rule 1: Minus Preceding Brackets
All internal signs flip to their opposite:
• a − (b + c) = a − b − c
• a − (b − c) = a − b + c
Ex: 500 − (250 − 100) = 500 − 250 + 100 = 350
Rule 2: Plus Preceding Brackets
Internal terms retain their exact signs:
• a + (b − c) = a + b − c
Ex: 28 + (35 − 10) = 28 + 35 − 10 = 53
🧩 Distributive Property
Multiple of a sum equals the sum of multiples.
a × (b + c) = a × b + a × c
Ex: 2 × (43 + 24) = 2×43 + 2×24 = 134
a × (b − c) = a × b − a × c
Ex: 97 × 25 = (100 − 3)×25 = 2500 − 75 = 2425
⭐ Structural Laws of Addition
• Commutative Property: Swapping terms does not change the total sum:
Term 1 + Term 2 = Term 2 + Term 1 (Holds for negative tokens too).
• Associative Property: Grouping terms in any order preserves value:
(Term 1 + Term 2) + Term 3 = Term 1 + (Term 2 + Term 3).
• Tinkering Terms: Increasing one term by 1 increases the sum by 1. Subtracting 9 is equivalent to subtracting 10 and adding 1 back.
⚠️ Common Exam Traps
Trap 1: Left-to-Right Blind Addition
In 30 + 5 × 4, you cannot add 30 + 5 first! Multiplication forms a single term (5 × 4 = 20) which is evaluated before addition: 30 + 20 = 50.
Trap 2: Ignoring Sign Flipping
Writing 100 − (15 + 56) as 100 − 15 + 56 gives 141 (incorrect). Removing brackets with a preceding minus gives 100 − 15 − 56 = 29.