6.8 Small Group · Group 1

Tier 2 Intervention · Concrete-Representational-Abstract (CRA) · Dual-Language Anchors

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

Name: Date: Period: Learning Goal:

■ CONCEPT SUMMARY & GUIDED NOTES: Let's build it together — What are the properties of operations?

1Changing the order or the grouping of numbers being added or multiplied does not change the answer.

Properties are math rules that are always true. The Commutative Property lets you change the order. The Associative Property lets you change the grouping (the parentheses). We'll walk through a worked example, try one together, then you'll try a few with hints right there when you need them.

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.

3Mathematical 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.

4Watch 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
Sort each equation by the property it shows.

Target Categories: Commutative Associative Identity
2 DRAG SORT
Match each equation with the property it demonstrates.

Target Categories: Commutative Associative Identity

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

🗣️ 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

Starter: My mathematical claim is that the solution is...

E Evidence

Starter: The evidence from the model/table demonstrates that...

R Reasoning

Starter: This proves my answer because the standard mathematical definition of...

Student Mastery Self-Assessment:

1 · Need More Support

2 · Getting Closer

3 · Got It / Solid

4 · Master / Can Teach It