LCM Investigation Slides 7th Grade Remedial Math
LCM Error
Investigation
"Is this right?"
LCM(10, 15, 18) = 10 × 15 × 18 = 2,700
Is this a common multiple?
Is it the LEAST?
The Expert Method
Watch carefully: How does she build the multiple?
Stop at 1:54
Embedded media
Look For:
The "Factor Trees" setup.
The "Sharing" rule (avoiding duplicates).
The Visual Brackets at 1:54.
"It needs to represent my 15, which is 3 and a 5, but I already have a 5..."
The "Representation" Proof
Building LCM(10, 15, 18)
2 × 5 × 3 × 3
Represents 10
Represents 15
Represents 18
2 × 5 = 10
3 × 3 = 9
10 × 9 = 90
The Big Debate
Critical Thinking
Why is the LCM usually much smaller than the product of all three numbers?
Example: LCM(10, 15, 18) is 90.
Key Insight
Shared factors are "reused." We don't need to keep adding copies of the same prime number!
Reality Check
If the numbers have NO shared factors, then (and only then) is the LCM equal to the product.
Two Truths and a Lie
Find the false statement about LCM!
A
The LCM must contain enough prime factors to "build" every number in the set.
B
If a factor is shared between two numbers, we only need to write it once.
C
The LCM is always the result of multiplying the original three numbers together.
LIE?
Ready to Investigate?
Error Analysis Worksheet LCM Error Analysis
Math Detective Lab
Student Name
Date
Mission: Three students tried to find the LCM, but they all made mistakes. Use the "Factor Tree & Brackets" method to find where they went wrong and fix it!
1 Case of "The Multiplier"
LCM(4, 6, 10)
Student Work (Incorrect):
LCM = 4 × 6 × 10 = 240
What went wrong?
Correct Solution (Use Brackets!):
2 Case of "The Double-Counter"
LCM(6, 9, 12)
Student Work (Incorrect):
"I listed every prime factor from every tree!"
Trees: (2×3), (3×3), (2×2×3)
LCM = 2 × 3 × 3 × 3 × 2 × 2 × 3 = 648
What went wrong?
Correct Solution (Use Brackets!):
3 Case of "The GCF Confuser"
LCM(8, 12, 20)
Student Work (Incorrect):
"I only picked the factors they all share."
8: 2×2×2
LCM = 2 × 2 = 4
What went wrong?
Correct Solution (Use Brackets!):
Reflect:
Compare the LCM you found in Case 1 (4, 6, 10) to the original student's answer (240). Why is the actual LCM always smaller than or equal to the product of the numbers?
LCM Visual Calculator Reference LCM Visual Calculator
A Step-by-Step Representation Guide
Your Tools
First 10 Primes:
2
3
5
7
11
13
17
19
23
29
The Rule
"Every number must be represented in the final list, but we never invite a factor twice if they can share."
1 Step 1: The Factory
Draw a factor tree for each number. Circle the Prime Factors at the end of every branch.
Tree for #1
Tree for #2
Tree for #3
2 Step 2: The Assembly Line
Build your expression using the "Represented" method.
Number A
Write its factors ...
Number B
Does Number A already have what I need? Only add what is missing!
Number C
Check the whole list. Add only what is still missing!
3 Step 3: The Proof
Draw your brackets to prove every number is "hiding" in your string. Then, multiply!
__ × __ × __ × __ × __
Pro Tip:
If you multiply and get a giant number, check if you counted a "shared" factor twice!
Method From Video:
Kaylee's Prime Brackets
Teacher Facilitator Guide Facilitator Guide
LCM Error Investigation
7th Grade Remedial Math
Objective
Students will identify and correct common misconceptions in Least Common Multiple (LCM) calculations for three numbers using the prime factorization "representation" method.
Materials Needed
LCM Slides
Error Analysis Worksheet
Visual Calculator Sheet
Video: "LCM: Three Numbers"
Instructional Flow
5 MIN
The Hook
Present LCM(10, 15, 18) = 2700 on board. Ask: "Is this a common multiple?" (Yes). "Is it the least?" (No). Challenge them to guess a smaller one.
10 MIN
Video Viewing
Watch Least Common Multiple: Three Numbers . CRITICAL: Pause at 1:54. Ask: "How do the brackets prove we have everything we need?" Discuss the 'sharing' of the factor 5 between 10 and 15.
15 MIN
Error Analysis Lab
Distribute the Error Analysis Worksheet . Students act as "Math Detectives." They must use the Visual Calculator Sheet to fix the three cases. Circulate and check for the "Case of the Double-Counter"—it's the most common remedial error.
10 MIN
Size Debate
Debate: "Why is the LCM usually smaller than the product?" Use physical blocks or circles on the board to show 'overlapping' prime factors.
Expert Facilitation Tips
Common Misconceptions
The "All-In" Mistake
Students often want to put every factor from every tree into the LCM. Remind them: "If they share a factor, they are sharing a room. You don't need two rooms for one person!"
Vocabulary Support
Multiple: Think "Multi-ply." It gets bigger.Factor: "Small parts." They build the number.Least: Smallest, but must be big enough for everyone. Error Analysis Answer Key Answer Key
LCM Error Analysis Lab
Teacher Resource
Case 1: The Multiplier LCM(4, 6, 10)
Error Identified:
The student simply multiplied the original numbers (4 × 6 × 10). This finds a common multiple , but not the least because it doesn't account for shared factors like 2.
Solution:
4 = 2 × 2
Case 2: The Double-Counter LCM(6, 9, 12)
Error Identified:
The student listed every single prime factor from every tree and multiplied them all. They did not "share" factors. For example, they included four 2s and four 3s, rather than just what was necessary to "represent" the numbers.
Solution:
6 = 2 × 3
Case 3: The GCF Confuser LCM(8, 12, 20)
Error Identified:
The student found the Greatest Common Factor (GCF) instead of the LCM. They only looked at what factors all three numbers had in common, rather than building a multiple that could contain all three numbers.
Solution:
8 = 2 × 2 × 2
Discussion Reflection Answer
The actual LCM is usually smaller than the product because many numbers share prime factors. When numbers share factors, we only need to write that factor once in the LCM string to "represent" both numbers. Multiplying the original numbers together is like buying a separate car for three people who are all driving to the same place; finding the LCM is like having them carpool.
