Factor Forge

Hook 5 min

1. "Every number has a secret recipe made only of primes, and each number has exactly one." Write 84 on the board.
2. "Can anyone break 84 into smaller pieces?" Take ideas like 4 × 21. "Can we keep breaking?"

Teach 10 min

1. In Smash, ask: "Which prime goes into 84?" Strike with the 2 prime hammer. A prime coin breaks off and the factor tree grows.
2. Strike the wrong hammer on purpose. It clangs and shows why: 84 ÷ 5 = 16.8, not a whole number.
3. Finish the tree: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
4. Try 60, starting with 5 this time. "Does it matter which prime we strike first?" (No. The coins at the end are the same.)
5. Switch to GCF & LCM and model one pair. Smash both numbers, then tap a coin from the first and the same prime from the second. Press Melt it! The GCF is the shared primes multiplied. The LCM is the shared primes once, times every leftover coin.

Practice 15 min

1. Pairs do three more GCF & LCM pairs. Before each Melt it!, they predict the GCF on a whiteboard.
2. Point out the check that appears every time: GCF × LCM = the two numbers multiplied. "Why would that always work?"
3. If the tool warns that a shared prime is left unpaired, use Undo last pair and look again.
4. Finish with one round of Forge Rush: 60 seconds, and a wrong hammer costs 2 seconds. "Who used a divisibility rule to go faster?"

Apply 10 min

1. Pose a GCF problem: "You have 36 pencils and 48 erasers and want identical kits with nothing left over. What's the most kits you can make?" (12, the GCF.)
2. Pose an LCM problem: "Hot dogs come in packs of 8 and buns in packs of 6. What's the fewest of each that come out even?" (24, the LCM.)
3. Pairs explain which tool they used and why: GCF for splitting into equal groups, LCM for things lining up again.

Exit check 5 min

On a card: write 90 as a product of primes (2 × 3² × 5), then find the GCF and LCM of 12 and 18 (6 and 36). Cards that mix up GCF and LCM show who needs the crucible again.