Feed the Fish: A GCD/LCM Cards Activity

Here’s a brief, not-so-fun two-player game intended to build intuition about the greatest common divisor and least common multiple of two numbers, as well as to predict their product formula.

Materials Needed

• Two decks of playing cards (ideally with different colors or designs)
• Papers reading “Pond” and “Picnic” ( DOCX or PDF)

Setting

Two fishers are sitting down to eat their packed lunches. Having eaten a big breakfast, neither is as hungry as they’d hoped, so they agree to share some of their food and toss the rest into the lake for the fish to eat.

Directions

1. Each player is dealt a hand of six* playing cards, of prime denomination (2, 3, 5, or 7), each from a separate deck. It’s helpful if the cards are easily distinguishable. We will call them “red” and “blue”. *During practice runs, use smaller hands.
2. Players alternate turns. On each turn:
   1. Player A selects a card from their own hand, placing it face-up on the “Picnic” sheet. Player A asks Player B if they have a matching card (same denomination; suit does not matter) by saying
      “Can we share a          ?”
      For example, “Can we share a five?”
   2. If Player B does have the requested card:
      The fishers share Player A’s card in the picnic.
      Player B discards their matching card in the Pond, saying
      “Yes, we can share yours. I’ll feed mine to the fish.”
      If Player B does not have the requested card:
        The fishers do not include Player A’s card in the picnic.
      Player B instructs A to move A’s card from Picnic to Pond, saying
      “No, we don’t share that. Feed it to the fish.”
3. Play continues until all cards have been divided among Picnic and Pond. Nobody wins… until everyone learns something.

Discussion Prompts, 1

After playing several rounds of the game, students may be asked to discuss:

1. What kinds of initial hands give the fishers the most to eat (maximum number of Picnic cards)?
2. What kinds of initial hands give the fishers the least to eat (maximum number of Pond cards)?
3. Fill in the blank: The Pond cannot end up with a (lesser/equal/greater) number of fives than does the Picnic.
4. In the end, how much of the lunches get eaten, whether by fishers or by fish? How do you know?

Sealing the Metaphor

Set up for another round, but this time, ask students to multiply their cards’ values together before beginning, writing their product down but keeping it secret from the other player.

After the round has concluded, have students multiply the Picnic cards’ values together, and multiply the Pond cards’ values together, writing each on their respective sheet. (Players’ original products should remain secret.)

Discussion Prompts, 2

Now that the cards have metaphorical value as prime factors, players may be asked:

1. Do you have enough information to determine what your opponent’s original product was? How can you tell? If yes, do so. If no, what additional information would be necessary?
2. Now, disclose your original product to your opponent. Multiply your original number with theirs. What other product must be equal to this one? How can you be sure?
3. Explain why the Pond product is necessarily a common factor of your and your opponent’s individual products.
4. Explain why the Picnic product is necessarily a common multiple of your and your opponent’s individual products.
5. (Subtle!) Your cards were all prime numbers. Explain why this guarantees that there cannot be a greater common divisor than the Pond product, nor a lesser common multiple than the Picnic product.
6. (Alternative to 4) What would happen if you played this game with cards of denomination 2, 3, 4, 6 instead? Find initial hands which, when played, could make the conclusion of questions #3 and/or #4 false. What goes wrong in this case, and why?