Prime Factorization
I can write a number as a product of its prime factors using a factor tree.
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🎯 Content Objective / Objetivo de contenido
I can write a number as a product of its prime factors using a factor tree.
Today's Flow
Total pacing: ~50 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
The freighter docks with 48 cargo containers, and each storage pod only holds groups built from prime-numbered units. Is 48 prime or composite, and how can you tell before building a factor tree?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Space Station Cargo Breakdown
The space station received 60 supply crates. Mission Control needs to break this quantity into its prime components so the sorting robots can distribute them into equally sized pods. Help the crew find the prime factorization of 60!
Concept Launch
💡 What is prime factorization?
Prime factorization means writing a number as a product of prime numbers only. A prime number can be divided only by 1 and itself, like 2, 3, 5, and 7.
Keep breaking a number apart until every factor is a prime number you cannot split anymore.
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Now it's your turn
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| Prime number Número primo |
A number bigger than 1 that you can only divide by 1 and itself. Un número mayor que 1 que solo se puede dividir entre 1 y sí mismo. |
7 has only two factors: 1 × 7. So 7 is prime. | |
| Composite number Número compuesto |
A number bigger than 1 that you can divide by more than just 1 and itself. Un número mayor que 1 que se puede dividir entre más números, no solo 1 y sí mismo. |
12 = 1 × 12, 2 × 6, 3 × 4 — six factors, so 12 is composite | |
| Prime factorization Factorización prima |
Writing a number as prime numbers multiplied together. Escribir un número como números primos multiplicados. |
36 = 2 × 2 × 3 × 3 = 2² × 3² | |
| Factor tree Árbol de factores |
A picture that splits a number into its prime numbers, step by step. Un dibujo que separa un número en sus números primos, paso a paso. |
24 → 4 × 6 → (2 × 2) × (2 × 3) → 2 × 2 × 2 × 3 | |
| Exponent Exponente |
A small number that tells how many times to multiply a number by itself. Un número pequeño que dice cuántas veces multiplicar un número por sí mismo. |
2³ means 2 × 2 × 2 = 8 |
Vocabulary — True or False?
Which statements correctly use Prime factorization?
Fix the False One
Which Word Fits?
A whole number greater than 1 with exactly two factors, 1 and itself, is a ___ number.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
The freighter docks with 48 cargo containers, and each storage pod only holds groups built from prime-numbered units. Is 48 prime or composite, and how can you tell before building a factor tree?
👂 Listen For
Students say 48 is composite because it has more than two factors (divisible by 2, 3, 4, 6, ...), so it can be broken into smaller groups.
Extend: Why can a prime-numbered shipment, like 47 containers, NOT be split into smaller equal prime groups? Justify with the definition of prime.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
Sort these numbers — which are prime and which are composite?
✍️ Explore Discourse
How did you decide whether a number is prime or composite? What strategy did you use?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
Look at your factor tree for 48. How did you decide which numbers to break apart first, and how did you know when to stop?
👂 Listen For
Students explain they keep splitting composite branches and stop when every leaf is prime, ending with 48 = 2 × 2 × 2 × 2 × 3.
Extend: Write the prime factorization of 48 using exponents and explain what the exponent means in 2⁴ × 3.
Practice Check A
Two students found different factor trees for 60. Student A started with 2 × 30. Student B started with 6 × 10. Which statement is true?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
Which of the following is a prime number?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Ratio Table Builder
Fill the ratio table. Each row must be equivalent.
| Factor | A | B |
|---|---|---|
| ×1 | ||
| ×2 | ||
| ×3 |
✍️ Justify Your Thinking
Sort each number: can it be expressed as a product of exactly TWO prime factors, or does it need THREE or more?
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Keep breaking a number apart until every factor is a prime number you cannot split anymore." — and it works because ___.
Because Prime number means ___, but a tricky part is ___, so I have to ___.
A common mistake with Prime number is ___. It happens because ___, and the fix is ___.
I can prove my answer is correct by ___, using Composite number to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Keep breaking a number apart until every factor is a prime number you cannot split anymore. because ___
Keep breaking a number apart until every factor is a prime number you cannot split anymore. but ___
Keep breaking a number apart until every factor is a prime number you cannot split anymore. so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Sort these numbers — which are prime and which are composite?
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
Which of the following is a prime number?
Core practice aligned to the standard.
Extension with error analysis or multi-step reasoning.
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
Aboard Station Helios, the crew has 72 supply units and wants to know every possible way to arrange them into equal pods. They use prime factorization to find all the factors of 72.
✍️ Connection Reasoning
How does prime factorization help the crew find all the ways to arrange the supply units into equal pods?
The prime factorization of 72 is ___, which helps me find all factors by ___.
Turn & Talk — Connect
The station has 72 supply units to arrange into equal pods. How does the prime factorization of 72 help the crew find every possible equal-pod arrangement?
👂 Listen For
Students give 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² and reason that multiplying combinations of those primes produces all factors of 72, which are the possible equal-pod sizes.
Extend: Why does knowing the prime factorization of 72 make it faster to list all factors than checking every number from 1 to 72?
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
What is the prime factorization of 40?
Bonus Exit Check
What is the prime factorization of 30?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~50 min
🎯 Listen For · Common Errors
• Students say 48 is composite because it has more than two factors (divisible by 2, 3, 4, 6, ...), so it can be broken into smaller groups.
• Students explain they keep splitting composite branches and stop when every leaf is prime, ending with 48 = 2 × 2 × 2 × 2 × 3.
• Students give 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² and reason that multiplying combinations of those primes produces all factors of 72, which are the possible equal-pod sizes.
• Students explain that breaking down different starting splits still reduces to the same prime factors, so the prime factorization is unique (the Fundamental Theorem of Arithmetic in plain words).
Common mistake: A common mistake in Prime Factorization is skipping the key idea: "Keep breaking a number apart until every factor is a prime number you cannot split anymore." — always check your work against this rule before you submit.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: Both get the same prime factorization: 2 × 2 × 3 × 5 — The Fundamental Theorem of Arithmetic says every composite number has exactly one prime factorization. No matter how you start the factor tree, you always end with 2 × 2 × 3 × 5.
✓ Practice 2: 17 — 17 has exactly two factors: 1 and 17. 15 = 3 × 5, 21 = 3 × 7, and 9 = 3 × 3, so they are all composite.
✓ Practice 3: 2 × 3 × 5 — 30 = 2 × 15 = 2 × 3 × 5. All three factors (2, 3, 5) are prime, so 2 × 3 × 5 is the prime factorization.
✓ Practice 4: 2 × 3 × 3 — 18 = 2 × 9 = 2 × 3 × 3. Both 2 and 3 are prime, so 2 × 3 × 3 is the prime factorization.
✓ Exit ticket: 2 × 2 × 2 × 5 — 40 = 2 × 20 = 2 × 2 × 10 = 2 × 2 × 2 × 5. All factors (2, 2, 2, 5) are prime.