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: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Mission Control needs to break the 60 supply crates into prime parts. Is 60 a prime number or a composite number, and how can you tell before you even start the 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 Factor |
A number that divides evenly into another number, with no remainder. Un número que divide exactamente a otro número, sin dejar residuo. |
Factors of 12: 1, 2, 3, 4, 6, 12 — each divides 12 evenly. | |
| 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
Mission Control needs to break the 60 supply crates into prime parts. Is 60 a prime number or a composite number, and how can you tell before you even start the factor tree?
👂 Listen For
Students say 60 is composite because it has more than two factors (it is divisible by numbers other than 1 and 60, such as 2, 3, 5, 6), so it can be broken down.
Extend: Could Mission Control ever be asked to factor a prime number of crates, like 59? Explain what its factor tree would look like and why.
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 60. How did you decide which numbers to break apart first, and how do you know when to stop?
👂 Listen For
Students explain they keep splitting composite branches and stop only when every leaf is prime, ending with 60 = 2 × 2 × 3 × 5.
Extend: Two classmates started 60 as 6 × 10 and 4 × 15. Will they get the same prime factorization? Justify why or why not.
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
Which number is a prime number?
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
A station gardener is loading 72 seed pods into the greenhouse and wants to build a perfectly rectangular planting grid. Starting from the prime factorization of 72, she combines the prime factors to list every factor of 72 and choose a grid that uses all the pods.
✍️ Connection Reasoning
Starting from its prime factorization, how can the gardener use the factors of 72 to decide which rectangular grids will use all 72 seed pods?
Because 72 = ___, I can multiply combinations of those prime factors to get factors like ___, so a grid that works is ___ by ___.
Turn & Talk — Connect
The party planner has 72 balloons. How could the prime factorization of 72 help her find every way to arrange the balloons in equal rows?
👂 Listen For
Students give 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² and reason that multiplying different combinations of those primes produces all factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) and thus all equal-row arrangements.
Extend: Why does writing the prime factorization with exponents (2³ × 3²) make it easier to count how many factors 72 has than writing it the long way?
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: ~45 min
🎯 Listen For · Common Errors
• Students say 60 is composite because it has more than two factors (it is divisible by numbers other than 1 and 60, such as 2, 3, 5, 6), so it can be broken down.
• Students explain they keep splitting composite branches and stop only when every leaf is prime, ending with 60 = 2 × 2 × 3 × 5.
• Students give 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3² and reason that multiplying different combinations of those primes produces all factors (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72) and thus all equal-row arrangements.
• Students contrast exactly-two-factors (prime) with more-than-two-factors (composite) and explain that primes are the building blocks, so any composite splits into a unique product of primes.
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.