LESSON PLAN

Factors, Multiples, GCF, and LCM

Apothem Team

Grade 6 · Number

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Pose the locker-decoration problem cold: "Cupcakes come in packs of 6, napkins in packs of 8. What's the smallest number of each pack so nothing is left over?" Let pairs wrestle for three minutes with any method — lists, drawings, guess-and-check.

Most pairs list multiples of 6 and 8 until they collide at 24. Name what they did: they hunted a common multiple — and today builds the machinery (factors, primes, GCF, LCM) that answers such questions without hoping for a collision.

Explore

Factor-rainbow workshop: each pair takes two numbers (say 36 and 48) and finds ALL factors of each by testing divisors in order, recording pairs as arcs (1–36, 2–18, 3–12, 4–9, 6–6). Then they circle the factors the two numbers share and star the greatest.

Flip to multiples: list the first ten multiples of each number, circle common ones, star the least. Close with the noticing prompt: "What's the relationship between your starred GCF and starred LCM and the original numbers?" (For 36 and 48: GCF 12, LCM 144, and 12 × 144 = 36 × 48 — a pattern worth flagging even if it isn't proven today.)

Formalize

Formalize the four terms: a factor divides a number exactly; a multiple is the number times any whole number; the GCF is the largest shared factor; the LCM the smallest shared nonzero multiple. Prime factorization is the master key — every number is a unique product of primes:

$36 = 2^2 \times 3^2 \qquad 48 = 2^4 \times 3 \qquad \text{GCF} = 2^2 \times 3 = 12 \qquad \text{LCM} = 2^4 \times 3^2 = 144$

From the prime factorizations: GCF takes the LOWER power of each shared prime; LCM takes the HIGHER power of every prime that appears. GCF answers "split into equal groups" questions; LCM answers "when do cycles line up" questions — matching the tool to the story is half the skill.

Practice

Practice: factor rainbows for 40 and 72; prime factor trees for 90 and 84; GCF/LCM by both listing and prime methods; two word problems (one grouping/GCF, one cycles/LCM) where students must first NAME which tool the story wants.

Exit ticket: "Buses on route A leave every 12 minutes, route B every 18. They just left together — when next?" (LCM 36 min.)

Exit ticket

Exit ticket: "Buses on route A leave every 12 minutes, route B every 18. They just left together — when next?" (LCM 36 min.)

TIP Students default to LCM for every word problem because it came last. Drill the diagnosis question — "is this story about splitting things up (GCF) or waiting for things to line up (LCM)?" — before any computation.

WORKED EXAMPLES

Example 1 — GCF two ways: the 24-pencil, 36-eraser gift bags

The problem: 24 pencils and 36 erasers must fill identical gift bags with nothing left over. What's the greatest number of bags?

Way 1 — factor lists: factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Common: 1, 2, 3, 4, 6, 12 → greatest is 12.

Way 2 — primes: $24 = 2^3\times3$ , $36 = 2^2\times3^2$ . Shared: $2^2 \times 3 = 12$ .

Step 3: Answer the story: 12 bags, each holding $24\div12 = 2$ pencils and $36\div12 = 3$ erasers.

Step 4: Why GCF was the right tool: the story SPLITS two quantities into identical groups — the splitting word is the tell.

Example 2 — LCM without listing forever: hot dogs (10) and buns (8)

The classic: hot dogs in packs of 10, buns in packs of 8. Fewest of each so none are lonely?

Step 1: Prime factorize: $10 = 2\times5$ , $8 = 2^3$ .

Step 2: LCM takes the highest power of every prime present: $2^3 \times 5 = 40$ .

Step 3: Translate back: 40 hot dogs = 4 packs; 40 buns = 5 packs.

Step 4: Compare with brute listing (10, 20, 30, 40 / 8, 16, 24, 32, 40) — same answer, and fine here. But for 84 and 90, listing crawls while primes fly: $84 = 2^2\cdot3\cdot7$ , $90=2\cdot3^2\cdot5$ → LCM $= 2^2\cdot3^2\cdot5\cdot7 = 1260$ . Choose method by the size of the numbers.

Example 3 — A student claims "bigger numbers have bigger GCFs" — audit it

Step 1: Test the claim with a fair pair: GCF of 100 and 99. Factors shared? 99 = 9×11, 100 = 4×25 — the only common factor is 1. GCF(100, 99) = 1, tiny, despite big numbers.

Step 2: Counter-pair: GCF(12, 18) = 6 — small numbers, healthy GCF. The claim is dead by counterexample.

Step 3: Extract the real driver: GCF measures SHARED STRUCTURE, not size. Consecutive numbers like 99 and 100 share nothing (any common factor would have to divide their difference, 1 — a genuinely lovely argument worth showing).

Step 4: Name the special case: pairs with GCF 1 are called relatively prime — neither needs to be prime itself (99 and 100 are both composite). Ask for another relatively prime composite pair (8 and 15, 9 and 16 …).

MATERIALS

•Hundred charts for multiple-marking

•Factor rainbow templates

•Prime factor tree sheets

•Story-sort cards (GCF vs LCM)

•Practice set (PDF)

WATCH FOR

!1 called prime (it isn't — primes have exactly two distinct factors) and 2 called composite (it's the only even prime).

!GCF and LCM answers swapped in word problems. The story-sort drill (split vs line-up) targets exactly this.

!Factor and multiple used interchangeably. Anchor: factors of 12 fit INSIDE 12 (≤ 12); multiples of 12 grow FROM 12 (≥ 12).