6.8 Small Group · Group 2

Tier 1 Extension · Non-Routine Synthesis · Misconception Traps · Rigorous CER Proofs

Properties of Operations · Commutative Property · Procedural Fluency · Reasoning & Critique

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — What are the properties of operations?

1Changing the order or the grouping of numbers being added or multiplied does not change the answer — and you can say why it is true, and where it would stop being true.

You already can do this. So we go up a level, not up a number: find what is always true, show it a second way, and be ready to defend it with a reason instead of an answer.

2Worked Example — Watch me

I have 3 guitarists, 5 drummers, and 2 keyboard players.
I can add them as 3 + 5 + 2, which gives 10.
The Commutative Property says order does not matter: 5 + 3 + 2 also gives 10.
The Associative Property says grouping does not matter: (3 + 5) + 2 = 3 + (5 + 2), both equal 10.

3Second Model — Try it together — then prove it

Look at 6 + 9 and 9 + 6. Do they give the same total?
6 + 9 = 15 and 9 + 6 = 15. Yes, the same.
Only the order changed, so this shows the Commutative Property.
Now prove it: say why that move had to work at all — not just that it did.
Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."

4Mathematical Word Bank

Properties of Operations (Propiedades de las operaciones) — Rules that let you reorder or regroup numbers without changing an expression's value.
Commutative Property (Propiedad conmutativa) — You can change the order and get the same answer.
Associative Property (Propiedad asociativa) — You can change the grouping and get the same answer.
Identity Property (Propiedad de identidad) — Adding 0 or multiplying by 1 keeps the same value.
Property (Propiedad) — A rule that is always true in math.
Distributive Property (Propiedad distributiva) — Multiplying a number by everything inside the parentheses: a(b + c) = ab + ac.

5Watch out

A common mistake in Properties of Operations is applying the commutative property where it does not belong, or confusing it with the associative property. Students often assume 15 − 8 = 8 − 15 (it does not: 7 ≠ −7 — commutative only works for addition and multiplication), or they see parentheses move and call it "associative" even when the numbers inside just swapped places, like (2 + 3) + 4 = (3 + 2) + 4 (that is commutative — the order inside the parentheses changed, not the grouping). Before you label a property, ask: did only the order change (commutative), did only the grouping change (associative), or was a 0 or 1 involved (identity)?

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 DRAG SORT
Match each equation with the property it demonstrates.

Target Categories: Commutative Associative Identity
2 MATCHING GAME
Match each equation to its property.
1. 7 + 9 = 9 + 7
2. (2 × 5) × 3 = 2 × (5 × 3)
3. 15 × 1 = 15
4. 0 + 23 = 23
5. 6 × 4 = 4 × 6
6. (8 + 3) + 7 = 8 + (3 + 7)
- ACommutative (Add)
- BAssociative (Mult)
- CIdentity (Mult)
- DIdentity (Add)
- ECommutative (Mult)
- FAssociative (Add)

SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]

3 MULTIPLE CHOICE
Track 3 plays (7 + 3) + 5 = 7 + (3 + 5). Only the parentheses moved. Which property is this?
1. AAssociative Property
2. BCommutative Property
3. CDistributive Property
4. DIdentity Property
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
Track 4 plays 1 × y = y. Which property is this?
1. AIdentity Property
2. BDistributive Property
3. CCommutative Property
4. DAssociative Property
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Which property is shown? 5 + 13 = 13 + 5
1. ACommutative Property
2. BAssociative Property
3. CIdentity Property
4. DDistributive Property
✏️ Workspace & Solution Steps

SECTION 3 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]

6 ERROR ANALYSIS
Level 2 Extension — Catch the Mislabeled Track
1. 1Track:3(a + b) = 3a + 3b
2. 2Student's label:This shows the Associative Property.
3. 3Reasoning:There are parentheses, so the grouping must have changed.
Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

✏️ Corrected Mathematical Work & Explanation

🗣️ MATHEMATICAL DISCOURSE & TALK MOVES SMP.3 / Construct Viable Arguments

Group Discussion Prompt: How does your visual model justify your mathematical solution?

🗣️ Partner A:
Partner A: "I modeled this by identifying the relationship and..."

👂 Partner B:
Partner B: "I agree with your step because the standard mathematical rule states..."

✍️ The Writing Revolution (TWR) · Sentence Expansion Because · But · So

Complete each sentence stem to demonstrate precise mathematical reasoning:

BECAUSE The ratio remains equivalent because both quantities are scaled by the exact same multiplicative factor.

BUT Two ratios may look similar, but inverting the order of terms fundamentally changes the comparison.

SO The recipe requires 3 parts flour to 2 parts water, so the unit rate is 1.5 cups of flour per cup of water.

📐 CER MATHEMATICAL PROOF MATRIX (CER 2.0) SMP.3 / Proof & Justification

Writing Task: Justify why your mathematical solution is accurate and complete.

C Claim

State your direct mathematical answer/claim with units.

E Evidence

Cite exact numbers, calculations, dimensions, or graph data.

R Reasoning

Explain the mathematical theorem, property, or definition connecting evidence to claim.

✍️ AUTHOR YOUR OWN EXTENSION CHALLENGE

Create an original multi-step word problem aligned to this standard. Include constraints, and write the complete step-by-step mathematical proof below.

✏️ Author Workspace & Complete Solution Key

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It