Unit 6 · Standard 6.EE.3
Properties of Operations
Key Vocabulary Level 1 support
Picture first, then the word, then a plain-language meaning. Say each word out loud.
3 + 7 = 7 + 3 = 10 both ways; 4 × 5 = 5 × 4 = 20 both ways — order does not matter
Commutative Property
You can change the order and get the same answer.
(2 + 3) + 4 = 5 + 4 = 9 and 2 + (3 + 4) = 2 + 7 = 9 — same answer, different grouping
Associative Property
You can change the grouping and get the same answer.
9 + 0 = 9 (zero is the identity for addition); 6 × 1 = 6 (one is the identity for multiplication)
Identity Property
Adding 0 or multiplying by 1 keeps the same value.
Commutative works for + and ×, but NOT for − or ÷ (since 5 − 3 = 2 but 3 − 5 = −2)
Property
A rule that is always true in math.
In 3x, the 3 is the coefficient — it tells you to multiply x by 3
Coefficient
The number in front of a letter, like the 3 in 3x.
4x + 2x = 6x (like terms, both have x); 4x + 2y stays as 4x + 2y (unlike terms)
Like terms
Terms with the same letter, like 2x and 5x.
Key Ideas & Notes
- You're a band director arranging musicians on stage.
- You have 3 guitarists, 5 drummers, and 2 keyboard players.
- Whether you seat the guitarists first or the drummers first, you still have 3 + 5 + 2 = 10 musicians.
- The order you arrange them doesn't change the total!
- Read each pair of expressions. Identify which property of operations is being shown.
Think About It
- Does the order you add the musicians change the total?
- What if you group different sections together first — does the total change?
- What happens when you add 0 musicians to a group?
My Notes
Guided Examples
Example 1
Which property is shown? 5 + 13 = 13 + 5
Solution: The order of the addends changed, so this is the Commutative Property of Addition.
Answer: A. Commutative Property
Example 2
Which property is shown? (6 × 3) × 2 = 6 × (3 × 2)
Solution: The grouping (parentheses) changed but the order stayed the same, so this is the Associative Property of Multiplication.
Answer: A. Associative Property
Example 3
Which property is shown? 47 + 0 = 47
Solution: Adding 0 does not change the value, so this is the Identity Property of Addition.
Answer: A. Identity Property
Write About the Math The Writing Revolution
I can explain my reasoning using the words commutative property, associative property, and identity property.
1. Kernel Sentence subject + verb
Model: Associative Property is you can change the grouping and get the same answer.Propiedad asociativa es puedes cambiar la agrupación y obtener la misma respuesta.
Write a kernel sentence about associative property. Use a subject and a verb.Escribe una oración base sobre propiedad asociativa. Usa un sujeto y un verbo.
2. Sentence Expansion because · but · so
Kernel: Associative Property matters in mathPropiedad asociativa importa en matemáticas
Expand the kernel three ways. Add a reason, a contrast, and a result.
Associative Property matters in math because ___.Propiedad asociativa importa en matemáticas porque ___.
Associative Property matters in math, but ___.Propiedad asociativa importa en matemáticas, pero ___.
Associative Property matters in math, so ___.Propiedad asociativa importa en matemáticas, entonces ___.
3. Sentence Types 4 ways to write a math idea
Tell one true fact about associative property.Di un hecho verdadero sobre associative property.
Associative property ___.
Ask a question about associative property.Haz una pregunta sobre associative property.
How does ___ ?¿Cómo ___ ?
Show excitement about associative property.Muestra entusiasmo sobre associative property.
Wow, ___ !¡Guau, ___ !
Tell a partner what to do with associative property.Dile a un compañero qué hacer con associative property.
First, ___ .Primero, ___ .
4. Explain Your Reasoning use a sentence starter
I used the ___ property.Usé la propiedad ___.
It works because ___.Funciona porque ___.
This makes math easier when ___.Esto facilita las matemáticas cuando ___.
Try It
Solve on your own. Check the answer key when you are done.
1. Which property is shown? 47 + 0 = 47
- Identity Property
- Commutative Property
- Associative Property
- Distributive Property
2. Which property lets you rearrange 8 × 5 to 5 × 8?
- Commutative Property
- Associative Property
- Identity Property
- Distributive Property
Stretch Your Thinking Level 2 enrichment
Challenge task — explain your reasoning in full sentences.
Explain why the commutative property works for addition and multiplication but NOT for subtraction and division. Give a specific number example for each operation to prove your point.
Sentence starter: Addition: ___ + ___ = ___ + ___ = ___ (works). Subtraction: ___ − ___ ≠ ___ − ___ because ___. Multiplication: ___ × ___ = ___ × ___ = ___ (works). Division: ___ ÷ ___ ≠ ___ ÷ ___ because ___.
Reflect — Exit Ticket
Which property is shown? (8 + 5) + 2 = 8 + (5 + 2)
- Associative Property
- Commutative Property
- Identity Property
- Distributive Property
Answer Key & Teacher Guide
- Try It 1: A. Identity Property — Adding 0 does not change the value, so this is the Identity Property of Addition.
- Try It 2: A. Commutative Property — Changing the order of factors is the Commutative Property of Multiplication.
- Exit Ticket: A. Associative Property — The grouping (parentheses) changed but the numbers stayed in the same order, so this is the Associative Property of Addition.
Writing (TWR) — what to look for
- Kernel sentence: A complete sentence needs a subject and a verb. Example: Associative Property is you can change the grouping and get the same answer.
- Expansion: because gives a reason, but shows a contrast or exception, so shows a result. Answers vary; each must keep the kernel idea and add the correct kind of detail.
- Sentence types: Statement ends with a period, question with "?", exclamation with "!", and a command starts with an action verb (a "bossy" verb).