How to Explain Factors and Multiples to Sixth Graders

Factors and multiples are fundamental number concepts that students use constantly—from simplifying fractions to finding common denominators. This guide helps you teach these concepts clearly so they stick.

Why Factors and Multiples Matter for Sixth Graders

Understanding factors and multiples enables students to:

• Simplify fractions by finding the greatest common factor
• Add unlike fractions by finding the least common multiple
• Solve division problems understanding divisibility
• Recognize number patterns and relationships
• Factor algebraic expressions in later math courses

Real-world applications include:

• Splitting items into equal groups
• Finding when events repeat together
• Understanding gear ratios and rhythms
• Solving scheduling problems

Key Concepts Broken Down Simply

What Are Factors?

Factors are numbers that divide evenly into another number (no remainder).

Factors of 12:
  12 ÷ 1 = 12  ✓    1 and 12 are factors
  12 ÷ 2 = 6   ✓    2 and 6 are factors
  12 ÷ 3 = 4   ✓    3 and 4 are factors
  12 ÷ 4 = 3   ✓    (already found)
  12 ÷ 5 = 2.4 ✗    5 is NOT a factor
  12 ÷ 6 = 2   ✓    (already found)

Factors of 12: {1, 2, 3, 4, 6, 12}

Factor pairs come in partnerships:

    1  ×  12  =  12
    2  ×  6   =  12
    3  ×  4   =  12

    Factor pairs: (1,12), (2,6), (3,4)

What Are Multiples?

Multiples are the results of multiplying a number by whole numbers.

Multiples of 4:
  4 × 1 = 4
  4 × 2 = 8
  4 × 3 = 12
  4 × 4 = 16
  4 × 5 = 20
  ...and so on forever!

Multiples of 4: {4, 8, 12, 16, 20, 24, 28, ...}

Factors vs. Multiples

┌─────────────────────────────────────────────────────┐
│            FACTORS vs MULTIPLES                     │
├─────────────────────────────────────────────────────┤
│                                                     │
│   FACTORS                    MULTIPLES              │
│   ─────────                  ─────────              │
│   Divide INTO the number     Multiply BY the number │
│   Limited quantity           Infinite quantity      │
│   Smaller or equal           Larger or equal        │
│   "What divides 12?"        "What's 12 times...?"  │
│                                                     │
│   12's factors:              12's multiples:        │
│   {1,2,3,4,6,12}            {12,24,36,48,60...}    │
│       ↑                          ↑                  │
│   stops at 12                never ends            │
│                                                     │
└─────────────────────────────────────────────────────┘

Memory trick:

• Factors are Fewer
• Multiples are Many

Prime and Composite Numbers

PRIME: Only factors are 1 and itself
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31...

COMPOSITE: Has factors besides 1 and itself
  4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20...

SPECIAL CASES:
  1 is neither prime nor composite
  2 is the only even prime number

Prime Factorization

Every composite number can be written as a product of primes.

Prime factorization of 60:

Method 1: Factor Tree

         60
        /  \
       6    10
      / \   / \
     2   3 2   5

60 = 2 × 3 × 2 × 5 = 2² × 3 × 5

Method 2: Division Ladder

     60 ÷ 2 = 30
     30 ÷ 2 = 15
     15 ÷ 3 = 5
      5 ÷ 5 = 1

60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Greatest Common Factor (GCF)

The GCF is the largest factor shared by two or more numbers.

Find GCF of 24 and 36:

Method 1: List factors
  Factors of 24: {1, 2, 3, 4, 6, 8, 12, 24}
  Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}
  Common factors: {1, 2, 3, 4, 6, 12}
  GCF = 12

Method 2: Prime factorization
  24 = 2³ × 3
  36 = 2² × 3²

  GCF = multiply common primes with LOWEST power
      = 2² × 3 = 4 × 3 = 12

Least Common Multiple (LCM)

The LCM is the smallest multiple shared by two or more numbers.

Find LCM of 6 and 8:

Method 1: List multiples
  Multiples of 6: {6, 12, 18, 24, 30, 36...}
  Multiples of 8: {8, 16, 24, 32, 40, 48...}
  Common multiples: {24, 48, 72...}
  LCM = 24

Method 2: Prime factorization
  6 = 2 × 3
  8 = 2³

  LCM = multiply ALL primes with HIGHEST power
      = 2³ × 3 = 8 × 3 = 24

Visual Examples and Diagrams

Factor Rainbow

Find all factors of 36:

    1 ─────────────────────────────────── 36
       2 ─────────────────────────── 18
          3 ─────────────────── 12
             4 ─────────── 9
                6 ─── 6

Factors of 36: {1, 2, 3, 4, 6, 9, 12, 18, 36}

Venn Diagram for GCF and LCM

Factors of 12 and 18:

         12                    18
    ┌──────────┐          ┌──────────┐
    │          │          │          │
    │    4     │          │    9     │
    │          │   ┌──────┤          │
    │   12     ├───┤ 1,2, │   18     │
    │          │   │ 3, 6 │          │
    └──────────┘   └──────┴──────────┘
                      ↑
               Common factors

GCF = largest in overlap = 6

Multiple Number Line

Multiples of 3 and 4:

3:  3   6   9  12  15  18  21  24  27  30  33  36
    ├───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼───┼
    │       │           │           │           │
4:  4       8          16          20          28
    │       │       │   │       │   │       │   │
    └───────┴───────┴───┴───────┴───┴───────┴───┘
    4   8  12  16  20  24  28  32  36

Common multiples: 12, 24, 36...
LCM = 12 (first common one)

Factor Tree Variations

Same number, different trees—same result!

      36                36               36
     /  \              /  \             /  \
    4    9            2   18           6    6
   / \  / \            \  / \         / \  / \
  2  2 3  3            2 2  9        2  3 2  3
                          / \
                         3   3

All give: 36 = 2² × 3²

Hands-On Activities

Activity 1: Factor Rectangle Drawing

Materials: Graph paper, colored pencils

Instructions:

1. Choose a number (like 24)
2. Draw ALL possible rectangles with that area
3. Each rectangle shows a factor pair

Rectangles with area 24:

1 × 24: ■ (1 row, 24 columns)
2 × 12: ■■ (2 rows, 12 columns)
3 × 8:  ■■■ (3 rows, 8 columns)
4 × 6:  ■■■■ (4 rows, 6 columns)

Factor pairs: (1,24), (2,12), (3,8), (4,6)

Activity 2: LCM Pattern Finder

Materials: Two different colored stamps or stickers

Instructions:

1. Mark color A every 3 spaces: 3, 6, 9, 12...
2. Mark color B every 4 spaces: 4, 8, 12...
3. Find where both colors appear together
4. That's the LCM!

Activity 3: Prime Number Sieve

Materials: Hundred chart, colored pencils

Instructions:

1. Cross out 1 (not prime)
2. Circle 2, cross out all other multiples of 2
3. Circle 3, cross out all other multiples of 3
4. Continue with 5, 7, 11...
5. Remaining numbers are prime!

  1  ②  ③  ✗  ⑤  ✗  ⑦  ✗  ✗  ✗
  ⑪ ✗  ⑬ ✗  ✗  ✗  ⑰ ✗  ⑲ ✗
  ✗  ✗  ㉓ ✗  ✗  ✗  ✗  ✗  ㉙ ✗
  ㉛ ✗  ✗  ✗  ✗  ✗  ㊲ ✗  ✗  ✗
  ㊶ ✗  ㊸ ✗  ✗  ✗  ㊻ ✗  ✗  ✗

Activity 4: GCF Gift Bags

Scenario: You have 24 candy bars and 36 lollipops. You want to make gift bags with the same number of each candy, using all candy, with as many bags as possible.

Solution:

GCF of 24 and 36 = 12 bags
Each bag: 24÷12 = 2 candy bars
          36÷12 = 3 lollipops

Activity 5: LCM Scheduling

Scenario: Bus A comes every 12 minutes. Bus B comes every 8 minutes. Both just arrived. When will they arrive together again?

Solution:

LCM of 12 and 8 = 24 minutes

Bus A: 12, 24, 36, 48...
Bus B: 8, 16, 24, 32...

Both arrive at minute 24, 48, 72...

Common Mistakes and How to Fix Them

Mistake 1: Confusing Factors and Multiples

Wrong: "Multiples of 6 are 1, 2, 3, 6"

Correct: Those are FACTORS. Multiples of 6 are 6, 12, 18, 24...

Fix: Use the memory trick:

• Factors are Fewer (limited list)
• Multiples are More (infinite list)

Mistake 2: Missing Factor Pairs

Wrong: Factors of 36 are 1, 2, 3, 4, 6, 36

Correct: Also include 9, 12, 18!

Fix: Use factor rainbows—connect pairs until you meet in the middle.

Mistake 3: Stopping Prime Factorization Too Early

Wrong: 36 = 4 × 9

Correct: 36 = 2² × 3² (4 and 9 aren't prime)

Fix: Keep breaking down until ALL factors are prime.

Mistake 4: GCF/LCM Confusion

Wrong: "The GCF of 6 and 8 is 24"

Correct: GCF = 2, LCM = 24

Fix: Remember:

• GCF = Greatest (largest) that's a Factor (divides into both)
• LCM = Least (smallest) that's a Multiple (both divide into it)

Mistake 5: Wrong Operation for GCF/LCM from Prime Factorization

Wrong: For GCF, taking highest powers

Correct:

• GCF: Take lowest powers of common primes
• LCM: Take highest powers of all primes

Fix: Remember "GCF = Low and Common, LCM = High and All"

Practice Ideas for Home

Quick Factor Practice

Find all factors of:
1. 18    {1, 2, 3, 6, 9, 18}
2. 28    {1, 2, 4, 7, 14, 28}
3. 45    {1, 3, 5, 9, 15, 45}
4. 100   {1, 2, 4, 5, 10, 20, 25, 50, 100}

Multiple Listing

List the first 8 multiples of:
1. 7:  {7, 14, 21, 28, 35, 42, 49, 56}
2. 9:  {9, 18, 27, 36, 45, 54, 63, 72}
3. 11: {11, 22, 33, 44, 55, 66, 77, 88}

GCF Problems

Find the GCF:
1. GCF(16, 24) = 8
2. GCF(35, 49) = 7
3. GCF(48, 72) = 24
4. GCF(15, 28) = 1 (relatively prime!)

LCM Problems

Find the LCM:
1. LCM(6, 9) = 18
2. LCM(4, 10) = 20
3. LCM(8, 12) = 24
4. LCM(5, 7) = 35

Real-World Problems

1. Tile Floor: A room is 12 feet by 18 feet. What's the largest square tile (in whole feet) that fits perfectly? (GCF = 6 feet)

2. Track Running: Track A takes 4 minutes to complete. Track B takes 6 minutes. Starting together, when do you finish a lap together? (LCM = 12 minutes)

3. Party Planning: 24 cupcakes and 18 juice boxes. Maximum equal party bags? (GCF = 6 bags, with 4 cupcakes and 3 juice boxes each)

Connection to Future Math Concepts

Simplifying Fractions

Simplify 24/36:
GCF of 24 and 36 = 12
24/36 = (24÷12)/(36÷12) = 2/3

Adding Fractions with Unlike Denominators

Add 5/6 + 3/8:
LCM of 6 and 8 = 24

5/6 = 20/24
3/8 = 9/24
Sum = 29/24

Algebra: Factoring Expressions

Factor: 12x + 18

GCF of 12 and 18 = 6
12x + 18 = 6(2x + 3)

Number Theory

Relatively prime numbers: GCF = 1
15 and 28 are relatively prime
(share no common factors except 1)

Quick Reference

┌────────────────────────────────────────────────────┐
│       FACTORS & MULTIPLES QUICK REFERENCE          │
├────────────────────────────────────────────────────┤
│ FACTORS: Numbers that divide evenly into n         │
│   - Finite list, ≤ n                              │
│   - Come in pairs that multiply to n              │
│                                                    │
│ MULTIPLES: Results of n × whole numbers           │
│   - Infinite list, ≥ n                            │
│                                                    │
│ PRIME: Only factors are 1 and itself              │
│ COMPOSITE: Has other factors                       │
│                                                    │
│ GCF: Greatest Common Factor                        │
│   - Largest shared factor                         │
│   - Prime factorization: LOWEST powers, COMMON    │
│                                                    │
│ LCM: Least Common Multiple                        │
│   - Smallest shared multiple                      │
│   - Prime factorization: HIGHEST powers, ALL      │
└────────────────────────────────────────────────────┘

Understanding factors and multiples gives students power over numbers. These concepts unlock fraction operations, prepare students for algebra, and appear in countless real-world situations. With practice identifying factors and multiples, your sixth grader builds essential mathematical foundations.