Least Common Multiple
I can find the least common multiple (LCM) of two numbers by listing or comparing their multiples.
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🎯 Content Objective / Objetivo de contenido
I can find the least common multiple (LCM) of two numbers by listing or comparing their multiples.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Astronaut Reyes checks oxygen every 4 hours and Chen checks water every 6 hours, both starting at 6:00 AM. Why isn't the next shared check simply 10 hours later (4 + 6)?
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Space Station Shift Scheduler
Astronaut Reyes checks the oxygen system every 4 hours. Astronaut Chen checks the water recycler every 6 hours. They both start their shifts at 6:00 AM. When is the next time both astronauts will be checking their systems at the same time?
Concept Launch
💡 What is the least common multiple (LCM)?
A multiple is what you get when you count by a number: 4, 8, 12, 16. The least common multiple is the smallest multiple that two numbers share.
The LCM is the FIRST number that shows up in both numbers' multiple lists.
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 |
|---|---|---|---|
| Multiple Múltiplo |
What you get when you multiply a number by 1, 2, 3, and so on. Lo que obtienes al multiplicar un número por 1, 2, 3, y así. |
Multiples of 4: 4, 8, 12, 16, 20 — found by multiplying 4 × 1, 4 × 2, 4 × 3, ... | |
| Least common multiple Mínimo común múltiplo |
The smallest number that two or more numbers both go into. El número más pequeño al que llegan dos o más números al multiplicar. |
Multiples of 4: 4, 8, 12, 16, 20, 24. Multiples of 6: 6, 12, 18, 24. First match: 12 → LCM = 12 | |
| Common multiple Múltiplo común |
A number that two or more numbers both go into. Un número al que llegan dos o más números al multiplicar. |
12, 24, and 36 are all common multiples of both 3 and 4 | |
| Skip counting Conteo salteado |
Counting by a number, like 2, 4, 6, to list its multiples. Contar de un número en un número, como 2, 4, 6, para listar sus múltiplos. |
Skip count by 6: 6, 12, 18, 24, 30 — each jump adds 6 | |
| Prime factorization Factorización prima |
Writing a number as prime numbers multiplied together. It helps you find the LCM. Escribir un número como números primos multiplicados. Ayuda a encontrar el mcm. |
4 = 2 × 2, 6 = 2 × 3. LCM uses the highest power of each prime: 2² × 3 = 12 |
Vocabulary — True or False?
Which statements correctly use Least common multiple?
Fix the False One
Which Word Fits?
The product of a number and any whole number is a ___ of that 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
Astronaut Reyes checks oxygen every 4 hours and Chen checks water every 6 hours, both starting at 6:00 AM. Why isn't the next shared check simply 10 hours later (4 + 6)?
👂 Listen For
Students reason that each astronaut repeats on multiples of their own number, so they only overlap at a common multiple, and adding 4 + 6 does not land on both schedules.
Extend: If Reyes checked every 4 hours and Chen every 8 hours, when would they next line up? Explain why the answer is not 4 + 8.
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
Here are the first several multiples of 4 and 6 mixed together. Sort them — which are common multiples of BOTH 4 and 6, and which are multiples of only one?
✍️ Explore Discourse
What is the least common multiple of 4 and 6? How did sorting the multiples help you find it?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
You skip counted the multiples of 4 and 6. How did you find the least common multiple from your lists?
👂 Listen For
Students skip count 4, 8, 12 and 6, 12 and identify 12 as the LCM because it is the smallest number appearing in both lists.
Extend: Will the LCM of two numbers ever be smaller than the larger number? Justify using 4 and 6.
Practice Check A
Two buses leave the station at 8:00 AM. Bus A returns every 15 minutes and Bus B returns every 20 minutes. When is the next time both buses are at the station together?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
What is the LCM of 3 and 5?
✍️ 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 pair: does the LCM equal the product of the two numbers, or is the LCM less than the product?
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: "The LCM is the FIRST number that shows up in both numbers' multiple lists." — and it works because ___.
Because Multiple means ___, but a tricky part is ___, so I have to ___.
A common mistake with Multiple is ___. It happens because ___, and the fix is ___.
I can prove my answer is correct by ___, using Least common multiple to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
The LCM is the FIRST number that shows up in both numbers' multiple lists. because ___
The LCM is the FIRST number that shows up in both numbers' multiple lists. but ___
The LCM is the FIRST number that shows up in both numbers' multiple lists. so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Here are the first several multiples of 4 and 6 mixed together. Sort them — which are common multiples of BOTH 4 and 6, and which are multiples of only one?
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
What is the LCM of 3 and 5?
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
Hot dogs come in packs of 8 and buns come in packs of 6. You want to buy the exact same number of hot dogs and buns with none left over.
✍️ Connection Reasoning
How does the LCM help you figure out the fewest hot dogs and buns to buy so they come out even?
I need to buy ___ hot dogs and ___ buns because the LCM of 8 and 6 is ___.
Turn & Talk — Connect
Hot dogs come in packs of 8 and buns in packs of 6. How does the LCM help you buy the fewest of each so none are left over?
👂 Listen For
Students compute LCM(8, 6) = 24, so 3 packs of hot dogs and 4 packs of buns give 24 each with nothing left over.
Extend: Why is the LCM the FEWEST you can buy and not just any common multiple like 48? Explain the trade-off.
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
What is the LCM of 9 and 12?
Bonus Exit Check
What is the LCM of 6 and 8?
✍️ 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 reason that each astronaut repeats on multiples of their own number, so they only overlap at a common multiple, and adding 4 + 6 does not land on both schedules.
• Students skip count 4, 8, 12 and 6, 12 and identify 12 as the LCM because it is the smallest number appearing in both lists.
• Students compute LCM(8, 6) = 24, so 3 packs of hot dogs and 4 packs of buns give 24 each with nothing left over.
• Students explain LCM is for repeating/lining-up events (schedules, packs) while GCF is for splitting into equal groups, and correctly tag the schedule as an LCM problem.
Common mistake: A common mistake in Least Common Multiple is skipping the key idea: "The LCM is the FIRST number that shows up in both numbers' multiple lists." — 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: 9:00 AM (60 minutes later) — LCM(15, 20) = 60 minutes. Both buses return together 60 minutes after 8:00 AM, at 9:00 AM.
✓ Practice 2: 15 — Multiples of 3: 3, 6, 9, 12, 15, ... Multiples of 5: 5, 10, 15, ... The smallest common multiple is 15.
✓ Practice 3: 24 — Multiples of 6: 6, 12, 18, 24, ... Multiples of 8: 8, 16, 24, ... The smallest common multiple is 24.
✓ Practice 4: 20 — Multiples of 4: 4, 8, 12, 16, 20, ... Multiples of 10: 10, 20, ... The smallest common multiple is 20.
✓ Exit ticket: 36 — Multiples of 9: 9, 18, 27, 36, ... Multiples of 12: 12, 24, 36, ... The smallest common multiple is 36.