6.13 Small Group · Group 2

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

Prime Factorization · Prime number · Procedural Fluency · Reasoning & Critique

Name: Date: Period: Learning Goal:

■ ADVANCED CONCEPT ANCHOR: Push further — What is prime factorization?

1Keep breaking a number apart until every factor is a prime number you cannot split anymore — 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 prime factorization of 60. I start by splitting it: 60 = 6 × 10.
6 is not prime, so I break it: 6 = 2 × 3. Both 2 and 3 are prime, so I stop those branches.
10 is not prime, so I break it: 10 = 2 × 5. Both 2 and 5 are prime, so I stop.
Now every factor is prime: 60 = 2 × 2 × 3 × 5.
Using exponents, I write it as 2² × 3 × 5. I check: 2 × 2 × 3 × 5 = 60. Correct.

3Second Model — Try it together — then prove it

Let's factor 12 together. What is one way to split 12 into two factors? (You might say 3 × 4.)
Is 3 prime? Yes, so we keep it. Is 4 prime? No — what is 4 split apart? (4 = 2 × 2.)
So put the prime factors together: 12 = 2 × 2 × 3. Every factor is prime, so we are done.
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

Prime Factorization (Factorización prima) — Writing a whole number as a product of only prime numbers.
Prime number (Número primo) — A number bigger than 1 that you can only divide by 1 and itself.
Composite number (Número compuesto) — A number bigger than 1 that you can divide by more than just 1 and itself.
Factor (Factor) — A factor of a whole number divides it evenly, with no remainder. For example, 3 is a factor of 12 because 12 ÷ 3 = 4.
Factor tree (Árbol de factores) — A picture that splits a number into its prime numbers, step by step.
Exponent (Exponente) — A small number that tells how many times to multiply a number by itself.

5Watch out

A common mistake in Prime Factorization is stopping the factor tree too soon — leaving a composite number like 4, 6, 8, or 9 in the final answer instead of continuing to split it. For example, writing 20 = 4 × 5 as the prime factorization (4 is composite) instead of finishing with 20 = 2 × 2 × 5. Before you submit, check every number in your final answer: if any factor can still be broken into smaller factors, the tree isn't finished yet.

SECTION 1 CONCEPTUAL UNDERSTANDING & VISUAL MODELS [Visual Modeling]

1 MATCHING GAME
Match each number to its prime factorization.
1. 12
2. 18
3. 20
4. 28
5. 45
6. 50
- A2 × 2 × 3
- B2 × 3 × 3
- C2 × 2 × 5
- D2 × 2 × 7
- E3 × 3 × 5
- F2 × 5 × 5
2 FILL TABLE

Build the factor tree. Write each number as a product of primes.

Number First Split Prime Factorization
24
36
40
56

✏️ Scratchpad / Reasoning

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

3 MULTIPLE CHOICE
Factor Tree for 90

The first lock panel shows 90. What is the prime factorization of 90?
1. A2 × 3 × 3 × 5
2. B2 × 45
3. C9 × 10
4. D3 × 30
✏️ Workspace & Solution Steps
4 MULTIPLE CHOICE
The second panel needs a number whose prime factorization is 2² × 3². Which number unlocks it?
1. A36
2. B24
3. C12
4. D18
✏️ Workspace & Solution Steps
5 MULTIPLE CHOICE
Factor Tree for 78

The final lock asks for the prime factorization of 78 written as a product of primes. Which is correct?
1. A2 × 3 × 13
2. B2 × 39
3. C3 × 26
4. D6 × 13
✏️ Workspace & Solution Steps

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

6 ERROR ANALYSIS
Level 2 Extension — Fix the Broken Factor Tree
1. 1Find prime factorization of 96:Start: 96
2. 2First split:96 = 8 × 12
3. 3Break down 8:8 = 2 × 2 × 2
4. 4Break down 12:12 = 2 × 6
5. 5Final answer:96 = 2 × 2 × 2 × 2 × 6
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

Number	First Split	Prime Factorization
24
36
40
56