Wednesday
Wednesday: Puzzle Lab: Envelope Cards
Raven fans out four index cards on the bench: 2n + 1, n², 3n - 2 and 12 - n. "Expression cards. The envelope sets n."
Rocket shakes an envelope. "Five beans in this one. So I put 5 in for n on every card?"
"Every single n," says Raven. "What do you notice happens to the card 12 - n as n gets bigger?"
"It shrinks," Rocket says. "The others grow. The square grows fastest."
Nova hovers above the cards, her light moving from one to the next. "Build each one with beans," she says, "then count."
"And check the count against the arithmetic," Raven adds. "If they disagree, one of them is wrong."
Rocket is already laying out two rows of five beans and one more. "Eleven," he says. "The card agrees."
Your mission
Make expression cards, set the envelope number, and evaluate every card two ways: with beans and with arithmetic. Record it all in a table.
- Four index cards and a marker, for the expression cards.
- Three envelopes, each with a different secret number of beans (keep every number 8 or under).
- About 70 dried beans or buttons on a tray, and your Puzzle Notebook.
Seal envelopes with the flap or tape. No licking, no staples.
Keep the tray on the table so nothing rolls onto the floor.
- Write 2n + 1, n², 3n - 2 and 12 - n on four cards.
- Open the first envelope and count the beans. That count is n for this round.
- For each card, build the expression with beans: 2n + 1 is two groups of n and one more.
- Count the beans, then work the card out with arithmetic. Write both answers in your table.
- Repeat with the other two envelopes. Circle any row where the bean count and the arithmetic disagree, and find out why.
The crew's envelope log
| n in the envelope | 2n + 1 | n² | 3n - 2 | 12 - n |
|---|---|---|---|---|
| 3 | 7 | 9 | 7 | 9 |
| 5 | 11 | 25 | 13 | 7 |
| 8 | 17 | 64 | 22 | 4 |
These are the crew's own rounds, written in Raven's log. The same card gives a different number for each envelope.
With n = 8, the card n² needed 64 beans, almost the whole tray. The card 12 - n needed only 4.
- Read the question.
- Tap your answer.
| What the lab shows | True or false? |
|---|---|
| The same card can give different numbers for different envelopes. | ? |
| With n = 5, the card n² needs 25 beans. | ? |
| With n = 3, the card 3n - 2 needs 9 beans. | ? |
| The card 12 - n gets bigger as n gets bigger. | ? |
Three envelopes, four cards, twelve answers. Tomorrow the crew uses formulas to plan card frames and boxes.