6.AT.7 Lesson 6-8-group2 🟣 Group 2 · Set B
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

6.8 Small Group · Group 2

Second Challenge Form · Non-Routine Extension · Same Standard, New Problems

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

Interactive Workbench & Models →
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
  1. I have 3 guitarists, 5 drummers, and 2 keyboard players.
  2. I can add them as 3 + 5 + 2, which gives 10.
  3. The Commutative Property says order does not matter: 5 + 3 + 2 also gives 10.
  4. 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
  1. Look at 6 + 9 and 9 + 6. Do they give the same total?
  2. 6 + 9 = 15 and 9 + 6 = 15. Yes, the same.
  3. Only the order changed, so this shows the Commutative Property.
  4. Now prove it: say why that move had to work at all — not just that it did.
  5. 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 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 1 MULTIPLE CHOICE

    Which property is shown? (6 × 3) × 2 = 6 × (3 × 2)

    1. AAssociative Property
    2. BCommutative Property
    3. CIdentity Property
    4. DDistributive Property
    ✏️ Workspace & Solution Steps
  2. 2 MULTIPLE CHOICE

    Which property is shown? 47 + 0 = 47

    1. AIdentity Property
    2. BCommutative Property
    3. CAssociative Property
    4. DDistributive Property
    ✏️ Workspace & Solution Steps
SECTION 2 MATHEMATICAL WRITING & ERROR ANALYSIS [Reasoning & Critique]
  1. 3 ERROR ANALYSIS

    Fix our table's thinking

    1. 1The problem:Which property is shown? (6 × 3) × 2 = 6 × (3 × 2)
    2. 2A classmate at our table answered:Commutative Property

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  2. 4 ERROR ANALYSIS

    Spot the Common Mistake

    1. 1Statement:(2 + 3) + 4 = (3 + 2) + 4
    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
  3. 5 ERROR ANALYSIS

    Find the Property Error

    1. 1Statement:15 − 8 = 8 − 15
    2. 2Student's claim:This shows the Commutative Property of Subtraction.
    3. 3Reasoning:I switched the order, so it must be commutative.

    Which step contains the error? Explain the mathematical misconception and write the correct calculation below.

    ✏️ Corrected Mathematical Work & Explanation
  4. 6 OPEN RESPONSE

    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.

    ✏️ Mathematical Justification & Response
📐 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.
Student Mastery Self-Assessment:
1 · Need More Support
2 · Getting Closer
3 · Got It / Solid
4 · Master / Can Teach It