K-12 Math (US): 3-5 Number & Operations K-12 Math Number Theory Factors and multiples

Study Guide: Factors and Multiples (Grade 3–5, Number & Operations)

1. The Driving Question

"If you have 24 cupcakes and want to pack them into boxes so every box has the same number—no leftovers—how many different ways can you do it? And why does this same idea help you figure out when two Ferris wheels with different numbers of seats will line up again at the starting point?"

2. The Core Idea — Built, Not Listed

Imagine you’re at a birthday party at the park, and you have 24 cupcakes to pack into boxes for guests. You want every box to have the exact same number of cupcakes—no half-cupcakes, no extras rolling around. How many cupcakes can go in each box?

• 4 cupcakes per box? 24 ÷ 4 = 6 boxes. Works!
• 5 cupcakes per box? 24 ÷ 5 = 4 boxes with 4 left over. Doesn’t work.
• 8 cupcakes per box? 24 ÷ 8 = 3 boxes. Works!

The numbers that work (1, 2, 3, 4, 6, 8, 12, 24) are called factors of 24—they’re the numbers that divide 24 evenly, like puzzle pieces that fit perfectly. Now, flip it around: if you start with 3 cupcakes per box, how many boxes do you need to pack 24 cupcakes? 3 × 8 = 24. The 8 is a multiple of 3—it’s what you get when you skip-count by 3s (3, 6, 9, 12, 15, 18, 21, 24…).

Key Vocabulary:
- Factor: A number that divides another number evenly (no remainder).
Example: 7 is a factor of 35 because 35 ÷ 7 = 5 (no leftovers). Not the usual "12 ÷ 3 = 4"! - Multiple: The result of multiplying a number by an integer (whole number).
Example: 50 is a multiple of 10 because 10 × 5 = 50. Not just "5, 10, 15…" but also 50, 100, etc. - Prime number: A number with exactly two factors—1 and itself.
Example: 13 is prime (only 1 × 13 works). Not 2 or 3—pick a bigger one like 17! - Composite number: A number with more than two factors.
Example: 15 is composite (1, 3, 5, 15). Not 4 or 6—pick something like 21.

3. Assessment Translation (Grades 3–5)

How this appears in class:
- Exit tickets: "List all the factors of 18. Explain how you know you found them all." - Show-your-work problems: "Liam has 36 stickers. He wants to arrange them in rows with the same number of stickers in each row. What are all the possible ways he can do this?" - Short constructed response: "Is 49 a prime number? Explain your answer using factors."

Proficient vs. Developing Responses:
| Proficient | Developing | |----------------|----------------| | "Factors of 18: 1, 2, 3, 6, 9, 18. I know I’m done because 1 × 18 = 18, 2 × 9 = 18, and 3 × 6 = 18—no other pairs work." | "Factors of 18: 1, 2, 3, 6, 9. (Missing 18.)" | | "Liam can arrange 36 stickers in rows of 1, 2, 3, 4, 6, 9, 12, 18, or 36. I checked by dividing 36 by each number." | "Rows of 6 or 9. (Missing other options.)" | | "49 is not prime because 7 × 7 = 49, so it has three factors: 1, 7, and 49." | "49 is prime because it’s odd. (Ignores factor pairs.)" |

Model Proficient Response:
"Question: What are all the factors of 20? Answer: The factors of 20 are 1, 2, 4, 5, 10, and 20. I found them by pairing numbers that multiply to 20: 1 × 20, 2 × 10, and 4 × 5. I stopped when the pairs started repeating."

What teachers look for:
- Systematic listing (not random guesses).
- Explanation (not just a list).
- All factors included (no missing pairs).

4. Mistake Taxonomy

Mistake 1: Missing Factor Pairs
Prompt: "List all the factors of 30." Common Wrong Answer: "1, 2, 3, 5, 6, 10, 15." Why It Loses Credit: Missing 30 (every number is a factor of itself) and 15’s pair (2).
Correct Approach: 1. Start with 1: 1 × 30 = 30.
2. Next, 2: 2 × 15 = 30.
3. Then, 3: 3 × 10 = 30.
4. Then, 5: 5 × 6 = 30.
5. Stop when pairs repeat (6 × 5 is the same as 5 × 6).

Mistake 2: Confusing Factors and Multiples
Prompt: "Is 12 a factor or a multiple of 6?" Common Wrong Answer: "Factor, because 6 × 2 = 12." Why It Loses Credit: The student mixed up the terms. 12 is a multiple of 6 (6 × 2 = 12). 6 is a factor of 12 (12 ÷ 6 = 2).
Correct Approach: - Ask: "Does 6 divide 12 evenly?" Yes → 6 is a factor of 12.
- Ask: "Is 12 in the skip-counting list for 6?" Yes (6, 12, 18…) → 12 is a multiple of 6.

Mistake 3: Forgetting 1 and the Number Itself
Prompt: "Is 17 a prime number? Explain." Common Wrong Answer: "No, because 1 × 17 = 17." Why It Loses Credit: The student forgot that prime numbers must have exactly two factors. Listing 1 and 17 proves it’s prime.
Correct Approach: 1. List all factors: 1 and 17.
2. Count them: only two → prime.
3. Explain: "17 is prime because its only factors are 1 and itself."

5. Connection Layer

1. Within Math: Factors and multiples → Fractions (simplifying and finding common denominators)
   Why? To simplify 12/18, you find the greatest common factor (GCF) of 12 and 18 (6). To add 1/4 + 1/6, you find the least common multiple (LCM) of 4 and 6 (12).

2. Across Subjects: Factors and multiples → Music (rhythm and time signatures)
   Why? A 4/4 time signature means there are 4 beats per measure, and each beat is a quarter note. If you have 12 eighth notes, they divide evenly into 3 measures (12 ÷ 4 = 3)—just like factors!

3. Outside School: Factors and multiples → Sports (basketball court lines and player rotations)
   Why? A basketball court is 94 feet long. If you’re marking equal sections for drills, you’d use factors of 94 (1, 2, 47, 94). If players rotate every 3 minutes in a 12-minute drill, they’ll switch 4 times (12 ÷ 3 = 4)—multiples in action!

6. The Stretch Question

"If you multiply two prime numbers together, is the result always a composite number? What if you multiply three primes? Can you ever get a prime number as the result?"

Pointer Toward the Answer:
- Start with small primes: 2 × 3 = 6 (composite). 5 × 7 = 35 (composite).
- The product of any two primes will always have at least three factors: 1, the two primes, and itself. That makes it composite.
- For three primes: 2 × 3 × 5 = 30 (composite). The product will always have more than two factors.
- Trick question: What if one of the "primes" is 1? (Spoiler: 1 is not a prime number—it only has one factor!)