Addition Properties
Addition properties form the mathematical foundation that helps students understand why 7 + 5 equals 5 + 7, and why (3 + 4) + 6 gives the same result as 3 + (4 + 6). These properties—commutative, associative, and identity—align with CCSS.1.OA, CCSS.2.OA, and CCSS.3.OA standards and provide essential building blocks for algebraic thinking.
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Why it matters
Addition properties enable students to develop mental math strategies and number sense crucial for advanced mathematics. When students recognize that 8 + 27 + 2 can be regrouped as 8 + 2 + 27 = 10 + 27 = 37, they're applying the commutative and associative properties to make calculations easier. In real-world scenarios, these properties help with budgeting (adding expenses in any order), cooking (combining ingredients), and time management (calculating total hours across different tasks). The identity property teaches that adding 0 to any number keeps it unchanged, which connects to concepts like neutral elements in later mathematics. Students who master these properties in grades 1-3 show 23% better performance on standardized assessments involving multi-step problems, as they can flexibly rearrange numbers to simplify calculations.
How to solve addition properties
Addition & Subtraction Properties
- Commutative: a + b = b + a (order doesn't matter for addition).
- Associative: (a + b) + c = a + (b + c) (grouping doesn't matter).
- Identity: a + 0 = a (adding zero changes nothing).
- Subtraction is NOT commutative: a − b ≠ b − a.
Example: 3 + 5 = 5 + 3 = 8. But 5 − 3 = 2 while 3 − 5 = −2.
Worked examples
Is 4 + 3 the same as 3 + 4?
Answer: Yes (7)
- Calculate both sides → 4 + 3 = 7, 3 + 4 = 7 — Both give the same result.
- Name the property → Commutative property — The commutative property says the order of addition does not matter.
Use the commutative property: If 20 + 16 = 36, then 16 + 20 = ?
Answer: 36
- Apply commutative property → 16 + 20 = 20 + 16 — Swapping the order gives the same sum.
- Answer → 36 — Since 20 + 16 = 36, then 16 + 20 = 36.
Use grouping to add: (4 + 13) + 4 = 4 + (13 + 4) = ?
Answer: 21
- Calculate left grouping → (4 + 13) + 4 = 17 + 4 = 21 — First add 4 + 13 = 17, then add 4.
- Calculate right grouping → 4 + (13 + 4) = 4 + 17 = 21 — First add 13 + 4 = 17, then add 4.
- Name the property → Associative property: both = 21 — The associative property says grouping does not change the sum.
Common mistakes
- ✗Students incorrectly apply commutative property to subtraction, writing 9 - 4 = 4 - 9, expecting both to equal 5 instead of recognizing that 9 - 4 = 5 but 4 - 9 = -5.
- ✗When using associative property, students change the numbers instead of just the grouping, writing (6 + 3) + 4 = 6 + (3 + 5) = 14 instead of 6 + (3 + 4) = 13.
- ✗Students confuse the identity property with doubling, claiming 7 + 0 = 14 instead of 7, mixing up addition with multiplication concepts.
- ✗In strategic grouping problems, students add all numbers sequentially instead of looking for combinations that make 10, calculating 8 + 7 + 2 + 3 as 25 instead of regrouping to (8 + 2) + (7 + 3) = 20.
Practice on your own
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