Wednesday
Wednesday: Puzzle Lab: Test Every Claim
Raven pins four pairs of cards to the board. "Each pair claims to be equivalent. Puzzle Lab tests the claims."
"Build each one with beans for three different n," Rocket says. "If the counts match every time, the claim stands."
"And if they match twice and miss once?" Raven asks. "What do you notice about that?"
"Then it was never equivalent," Rocket says. "One miss is enough to sink it."
Nova hovers over the first pair, 3(n + 2) and 3n + 6. "Choose your n before you build," she says. "No peeking at the arithmetic."
Rocket picks n = 1, 3 and 5. "Small, medium, large," he says, and starts laying rows.
Your mission
Test four pairs of expressions at three values each. Build with beans, count, and write the counts in a table. Then say which pairs are truly equivalent.
- Eight index cards with the expressions from the crew's pairs, or pairs of your own.
- About 60 dried beans or buttons on a tray.
- Three small cards marked 1, 3 and 5 for the values of n, and your Puzzle Notebook.
- Masking tape to mark rows on the bench.
Tape goes on the bench or table, not across the floor.
Work sitting down so beans do not scatter underfoot.
- Pick a pair of cards. Pick a value card for n.
- Build the first expression with beans: rows for groups, loose beans for extras. Count.
- Build the second expression the same way. Count.
- Write both counts in your table. Repeat for the other two values of n.
- If all three rows match, mark the pair "equivalent". If any row misses, mark it "not equivalent".
The crew's lab table
These are the crew's own counts, written in Raven's log for the four pairs on the board.
| Pair | First | Second | n = 1 | n = 3 | n = 5 | Verdict |
|---|---|---|---|---|---|---|
| 1 | 3(n + 2) | 3n + 6 | 9 and 9 | 15 and 15 | 21 and 21 | equivalent |
| 2 | n + n + n | 3n | 3 and 3 | 9 and 9 | 15 and 15 | equivalent |
| 3 | 2(n + 1) + n | 3n + 2 | 5 and 5 | 11 and 11 | 17 and 17 | equivalent |
| 4 | 4n + 2 | 2(2n + 1) | 6 and 6 | 14 and 14 | 22 and 22 | equivalent |
Every pair on the board matched at every value, so all four claims stand. Pair 3 took two steps: distribute 2(n + 1), then combine with the extra n to get 3n + 2.
A pair that fails even once is not equivalent. Rocket tested 3(n + 2) against 3n + 2 on the side, and it missed at every value.
- Read the question.
- Tap your answer.
| What the lab shows | True or false? |
|---|---|
| 3(n + 2) and 3n + 6 are equivalent. | ? |
| 3(n + 2) and 3n + 2 are equivalent. | ? |
| Matching at three values proves two expressions are equivalent for every value. | ? |
| 4n + 2 and 2(2n + 1) are equivalent. | ? |
Every claim tested. Tomorrow the crew adds and subtracts whole expressions from the card log.