6.NOS.4 Lesson 6-7-group2 🟣 Group 2 · Challenge & Extension
MASTERY CHECK
☐ Exceeds☐ Meets Target☐ Needs Practice

6.7 Small Group · Group 2

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

Find Factors and Multiples · Factor · Procedural Fluency · Real-World Applications

Interactive Workbench & Models →
Name: Date: Period: Learning Goal:
■ ADVANCED CONCEPT ANCHOR: Push further — Factors split things up; multiples count forward
1GCF = the biggest number that divides BOTH — use it to split into equal groups. LCM = the smallest number BOTH divide into — use it to find when cycles meet again — 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 build the combo packs
  1. The store has 48 pencils and 36 notebooks. Every combo pack must hold the same number of each, with nothing left over — so the number of packs has to be a factor of BOTH quantities.
  2. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Listing them in order is what makes the comparison easy.
  3. The numbers in both lists are the common factors: 1, 2, 3, 4, 6, and 12.
  4. The greatest of those is 12, so the GCF of 48 and 36 is 12. The manager can make 12 combo packs — each holding 48 ÷ 12 = 4 pencils and 36 ÷ 12 = 3 notebooks.
3Second Model — Try it together — then prove it
  1. A factor tree breaks each number into primes. 48 = 2 × 2 × 2 × 2 × 3, and 36 = 2 × 2 × 3 × 3.
  2. Now look at what the two share: two 2s and one 3.
  3. Multiply the shared primes: 2 × 2 × 3 = 12.
  4. Same answer as the lists, 12 — which is the check. For big numbers, the factor tree is usually the faster road.
  5. Now prove it: say why that move had to work at all — not just that it did.
  6. Sentence frame — convince a skeptic: "My method works because ___ . It would stop working if ___ ."
4Mathematical Word Bank
  • Find Factors and Multiples (Hallar factores y múltiplos) — Listing what divides a number evenly (its factors) and what it divides into evenly (its multiples), then finding what two numbers share.
  • Factor (Factor) — A whole number that divides another number evenly, with no remainder.
  • Multiple (Múltiplo) — The result of multiplying a number by 1, 2, 3, and so on — where you land when you skip-count by it.
  • Greatest common factor (GCF) (Máximo común divisor (MCD)) — The largest number that is a factor of two or more numbers. It answers “how many equal groups can I make?”
  • Least common multiple (LCM) (Mínimo común múltiplo (mcm)) — The smallest number that is a multiple of two or more numbers. It answers “when will these line up again?”
  • Factor pair (Par de factores) — Two factors that multiply to make a number. In a word problem, one is the number of groups and the other is the size of each group.
5Watch out
  • A common mistake is reaching for the wrong tool: listing FACTORS for a problem about repeating cycles. Ayasha's classes repeat every 6 and 8 weeks, so the weeks they land on are multiples — answering with the GCF of 2 claims a class that meets every 8 weeks happens in week 2. Use the size of the answer as a check: a GCF is never larger than the smaller number, and an LCM is never smaller than the larger one.
SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]
  1. 1 FILL TABLE

    The store has 48 pencils and 36 notebooks. Complete the factor lists, then find the GCF.

    NumberFactorsGCF
    48
    36
    ✏️ Scratchpad / Reasoning
  2. 2 DRAG SORT

    Sort each real problem by the tool that solves it.

    Target Categories: Use the GCF Use the LCM
    • Split 48 pencils and 36 notebooks into identical packs
    • Two classes repeat every 6 and 8 weeks — when do they meet?
    • Cut two ribbons into the longest equal pieces with none left over
    • Hot dogs in packs of 10, buns in packs of 8 — when do they match?
    • The answer is at most the smaller number
    • The answer is at least the larger number
SECTION 2 REAL-WORLD CONTEXTS & PROBLEM SOLVING [Applications]
  1. 3 MULTIPLE CHOICE

    Claim: “The GCF of two numbers is always smaller than both of them.” Always, sometimes, or never true?

    1. ASometimes — it equals the smaller number whenever that number divides the larger one
    2. BAlways — a common factor has to be smaller than what it divides
    3. CNever — the GCF is always one of the two numbers
    4. DSometimes — it depends whether the numbers are even
    ✏️ Workspace & Solution Steps
  2. 4 MULTIPLE CHOICE

    Two buses leave the depot together. One returns every 12 minutes, the other every 18. A student wants to know when they next leave together and answers 6. What went wrong?

    1. AThey found the GCF. Meeting again needs the LCM, 36 minutes
    2. BNothing — 6 minutes is correct
    3. CThey should have added 12 + 18 to get 30 minutes
    4. DThey should have multiplied 12 × 18 to get 216 minutes
    ✏️ Workspace & Solution Steps
  3. 5 MULTIPLE CHOICE

    Which list shows all the factors of 12?

    1. A1, 2, 3, 4, 6, 12
    2. B12, 24, 36, 48
    3. C1, 12
    4. D2, 3, 4, 6
    ✏️ Workspace & Solution Steps
  4. 6 MULTIPLE CHOICE

    Which list shows the first four multiples of 8?

    1. A8, 16, 24, 32
    2. B1, 2, 4, 8
    3. C8, 9, 10, 11
    4. D2, 4, 6, 8
    ✏️ Workspace & Solution Steps
🗣️ 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 mathematical model proves the solution because each step preserves quantitative equivalence.
BUT An estimate provides a quick benchmark, but an exact proof is required for precision.
SO The quantities follow the standard rule, so the final result is verified with certainty.
📐 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