Mathematics Grade 8 Playing with Numbers Divisibility

Grade 8 Mathematics Study Guide: Playing with Numbers — Divisibility

1. The Driving Question

"If you have 120 cookies to pack into boxes, how do you know if you can split them evenly into 2 boxes, 3 boxes, 5 boxes — or any number of boxes — without breaking a single cookie? And why do some numbers, like 12, let you split them in so many ways, while others, like 13, seem to resist every attempt?"

2. The Core Idea — Built, Not Listed

Imagine you’re organizing a school field trip. You have 120 students, and you need to divide them into equal groups for buses, lunch tables, or activity stations. If you try to split them into 2 buses, you get 60 students per bus — no leftovers. But if you try 7 buses, you’d have 17 students on most buses and 1 left over (120 ÷ 7 = 17 R1). That leftover student is the clue: divisibility is about whether a number can be split into equal parts without leftovers.

Now, think of numbers like Lego structures. Some numbers, like 12, are built from smaller blocks (2, 3, 4, 6) — you can rearrange them into neat rows. Others, like 13, are like a single tall tower: no matter how you try to break them down, you’ll always have a leftover piece. The rules of divisibility are shortcuts to predict which numbers are "Lego-friendly" without doing the full division.

Key Vocabulary:
- Divisible – A number is divisible by another if it can be divided evenly (no remainder).
Example: 24 is divisible by 3 (24 ÷ 3 = 8), but not by 5 (24 ÷ 5 = 4 R4).
- Factor – A number that divides another number evenly.
Example: 4 is a factor of 20 because 20 ÷ 4 = 5. (Not the usual "6 is a factor of 12.") - Prime Number – A number greater than 1 with exactly two factors: 1 and itself.
Example: 17 is prime (factors: 1, 17). College note: Primes are the "atoms" of number theory — every number is built from them.
- Composite Number – A number with more than two factors.
Example: 18 is composite (factors: 1, 2, 3, 6, 9, 18).

3. Assessment Translation

How this appears in assessments:
- Classroom (Formative): Exit tickets with questions like "Is 168 divisible by 3? Explain how you know without dividing." - State Tests (Grade 8): Multiple-choice or short-answer questions testing divisibility rules (e.g., "Which number is divisible by both 2 and 9? A) 126 B) 135 C) 144 D) 153"). Distractors often include numbers that almost fit the rule (e.g., 135 is divisible by 9 but not 2).
- Proficient Response: A student explains why a rule works, not just applies it. For example: "168 is divisible by 3 because the sum of its digits (1 + 6 + 8 = 15) is divisible by 3. This works because 100, 10, and 1 are all 1 more than a multiple of 3, so their remainders add up to the total remainder."

Model Student Response (Short Answer):
Prompt: "A bakery has 252 cupcakes to pack into boxes. Can they pack them into boxes of 6 with none left over? Explain." Response: "Yes, 252 is divisible by 6. First, check if it’s divisible by 2: the last digit is 2, so yes. Then check if it’s divisible by 3: 2 + 5 + 2 = 9, which is divisible by 3. Since 6 = 2 × 3, 252 is divisible by 6. The bakery can pack 42 boxes of 6 cupcakes each."

4. Mistake Taxonomy

Mistake 1: Misapplying the Divisibility Rule for 3
Question: "Is 247 divisible by 3? Explain." Common Wrong Response: "Yes, because 2 + 4 + 7 = 13, and 13 is a prime number." Why It Loses Credit: The student confuses the rule (sum of digits must be divisible by 3) with unrelated facts (13 is prime). They also don’t check if 13 is divisible by 3 (it’s not).
Correct Approach: 1. Add the digits: 2 + 4 + 7 = 13.
2. Check if 13 is divisible by 3: 13 ÷ 3 = 4 R1 → no.
3. Conclude: 247 is not divisible by 3.

Mistake 2: Ignoring the "No Remainder" Rule
Question: "A teacher has 45 pencils to give to 4 students equally. Can they do this without breaking pencils? Show your work." Common Wrong Response: "Yes, 45 ÷ 4 = 11.25, so each student gets 11 pencils." Why It Loses Credit: The student performs the division but ignores the "no leftovers" requirement. The question asks for equal distribution without breaking pencils.
Correct Approach: 1. Divide: 45 ÷ 4 = 11 R1.
2. There’s a remainder of 1, so the pencils cannot be split equally.
3. Answer: No, because 45 is not divisible by 4.

Mistake 3: Overgeneralizing Divisibility Rules
Question: "Which of these numbers is divisible by 8? A) 120 B) 128 C) 136 D) 144" Common Wrong Response: "A) 120, because it’s divisible by 2 and 4." Why It Loses Credit: The student assumes divisibility by 2 and 4 guarantees divisibility by 8 (it doesn’t — 12 is divisible by 2 and 4 but not 8). They didn’t check the last three digits (120 ÷ 8 = 15, but 120 ÷ 4 = 30 doesn’t confirm 8).
Correct Approach: 1. Check the last three digits of each number:
- 120: 120 ÷ 8 = 15 → yes.
- 128: 128 ÷ 8 = 16 → yes.
- 136: 136 ÷ 8 = 17 → yes.
- 144: 144 ÷ 8 = 18 → yes.
2. All options are divisible by 8, but if the question had a non-divisible option (e.g., 124), the rule would catch it.

5. Connection Layer

1. Within Math: Divisibility → Prime Factorization
   Why it matters: Divisibility rules help you break numbers into primes (e.g., 60 is divisible by 2, 3, and 5, so its prime factors are 2² × 3 × 5). This is the foundation for simplifying fractions and finding common denominators.

2. Across Subjects: Divisibility → Cryptography (Computer Science)
   Why it matters: Modern encryption (like the RSA algorithm) relies on the fact that multiplying two large primes is easy, but factoring the result back into primes is hard. Divisibility rules are the first step in understanding why some numbers are "uncrackable."

3. Outside School: Divisibility → Barcodes and Check Digits
   Why it matters: The last digit of a UPC barcode (like on a cereal box) is a "check digit" calculated using divisibility by 10. If the barcode is damaged, the check digit helps detect errors — just like how your brain uses divisibility rules to spot math mistakes.

6. The Stretch Question

"Why does the divisibility rule for 7 work the way it does? (Hint: It’s not as simple as the rules for 2 or 3.) Can you invent a new rule for 7 that’s easier to remember?"

Pointer Toward the Answer: The standard rule for 7 is clunky: double the last digit, subtract it from the rest of the number, and repeat until you get a number you recognize (e.g., 343 → 34 – 6 = 28 → 28 is divisible by 7). This works because 7 is a prime number, and its relationship to 10 (our base system) isn’t as clean as 2 or 5. To invent a better rule, think about how 7 relates to numbers like 20 or 50 — for example, 7 × 3 = 21, which is 1 more than 20. Could you use that to create a pattern? (Spoiler: Some mathematicians use a rule where you add/subtract multiples of 7 based on the last digit — but no one’s found a perfect shortcut yet!)