6.12 Small Group · Group 2

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

Least Common Multiple · Multiple · Procedural Fluency · Real-World Applications

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — What is the least common multiple (LCM)?

1The LCM is the FIRST number that shows up in both numbers' multiple lists — 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 want the LCM of 4 and 6. I skip count the multiples of 4: 4, 8, 12, 16, 20, 24.
Then I skip count the multiples of 6: 6, 12, 18, 24.
I look for the first number in BOTH lists. 12 is in both lists, and it comes before 24.
So the LCM of 4 and 6 is 12. I check: 12 ÷ 4 = 3 and 12 ÷ 6 = 2, so both go into 12 evenly.

3Second Model — Try it together — then prove it

Let's find the LCM of 3 and 5. What are the multiples of 3? (3, 6, 9, 12, 15.)
What are the multiples of 5? (5, 10, 15.) What is the first number in both lists? (15.)
So the LCM of 3 and 5 is 15.
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

Least Common Multiple (Mínimo común múltiplo) — The least positive number that is a multiple of two or more numbers.
Multiple (Múltiplo) — What you get when you multiply a number by 1, 2, 3, and so on.
Common multiple (Múltiplo común) — A number that two or more numbers both go into.
Skip counting (Conteo salteado) — Counting by a number, like 2, 4, 6, to list its multiples.
Prime factorization (Factorización prima) — Writing a number as prime numbers multiplied together. It helps you find the LCM.

5Watch out

A common mistake in Least Common Multiple is multiplying the two numbers together instead of listing multiples to find the smallest shared one. For example, for 4 and 6, a student calculates 4 × 6 = 24 and calls that the LCM, but the actual least common multiple is 12 (multiples of 4: 4, 8, 12; multiples of 6: 6, 12 — 12 is the first number that appears in both lists). This shortcut only works when the two numbers share no common factors (like 4 and 9, where the LCM really is 36 = 4 × 9); for numbers like 4 and 6 that share a factor of 2, multiplying overshoots the true LCM. Always list multiples of both numbers and pick the first one that appears in both lists.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 MATCHING GAME
Match each pair of numbers to their LCM.
1. LCM(2, 5)
2. LCM(4, 10)
3. LCM(3, 8)
4. LCM(6, 9)
5. LCM(5, 7)
6. LCM(4, 6)
- A10
- B20
- C24
- D18
- E35
- F12
2 FILL TABLE

List multiples of each number until you find the LCM.

Numbers Multiples of First Multiples of Second LCM
6 and 10
8 and 12
9 and 12

✏️ Scratchpad / Reasoning

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

3 MULTIPLE CHOICE
One machine runs every 4 min, another every 6 min. They just ran together. In how many minutes do they run together again?
1. A12
2. B10
3. C24
4. D6
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
One ship docks every 8 min, another every 12 min. They just docked together. In how many minutes do they dock together again?
1. A24
2. B20
3. C48
4. D16
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
What is the LCM of 3 and 5?
1. A15
2. B3
3. C5
4. D30
✏️ Workspace & Solution Steps

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

6 ERROR ANALYSIS
Level 2 Extension — Crew Member's Schedule Sync Mistake
1. 1Problem:The water recycler cycles every 8 minutes and the heat exchanger cycles every 12 minutes. They just ran together. When do they next sync?
2. 2List multiples of 8:8, 16, 24, 32
3. 3List multiples of 12:12, 24, 36
4. 4Pick the number to use:The greatest factor that 8 and 12 share is 4.
5. 5Answer:They next sync in 4 minutes.
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 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

Numbers	Multiples of First	Multiples of Second	LCM
6 and 10
8 and 12
9 and 12