Tuesday
Tuesday: A Group, Multiplied
Raven lays out two identical bundles on the workbench. Each bundle is one envelope and 2 beans tied with string.
"Two bundles balance 10 beans," she says. "What do you notice about how to write it?"
"Two of the same group," Rocket says. "2(x + 2) = 10. Everything in the bundle got doubled."
"So you can split the beans two ways," Raven says. "Share the 10 first, or untie the bundles first."
Rocket shares. "10 ÷ 2 = 5. So one bundle is 5. An envelope plus 2 is 5. The envelope is 3."
Nova hovers above the bundles, her light flicking between them. "Now untie them instead," she says. "Same answer, other road."
"2x + 4 = 10," Rocket says, writing. "Subtract 4, divide by 2. Three." Nova hums. "Two roads, one check."
p(x + q) = r: two ways to undo
2(x + 2) = 10 says two equal groups, each an envelope and 2 beans, balance 10 beans.
Road one: divide both sides by 2 first. x + 2 = 5, then subtract 2, so x = 3.
Road two: distribute first. 2(x + 2) = 2x + 4, so 2x + 4 = 10. Subtract 4: 2x = 6. Divide by 2: x = 3.
Both roads reach the same number. Choose the road where the numbers divide evenly.
| Equation | Divide first | Or distribute first | Solution |
|---|---|---|---|
| 2(x + 2) = 10 | x + 2 = 5 | 2x + 4 = 10 | x = 3 |
| 3(x + 2) = 15 | x + 2 = 5 | 3x + 6 = 15 | x = 3 |
| 4(x - 1) = 10 | x - 1 = 2.5 | 4x - 4 = 10 | x = 3.5 |
| 0.5(x + 6) = 7 | x + 6 = 14 | 0.5x + 3 = 7 | x = 8 |
Arithmetic road, algebra road
A puzzle can be solved with arithmetic or with algebra, and the operations match up.
The Puzzle Post sign is a rectangle with a perimeter of 54 cm and a length of 6 cm. Arithmetic: 54 ÷ 2 = 27, then 27 - 6 = 21.
Algebra: 2(6 + w) = 54. Divide by 2: 6 + w = 27. Subtract 6: w = 21. The same two operations, in the same order.
- Read the question.
- Tap your answer.
- Read the question.
- Tap your answer.
- ?Write the equation: 3(x + 2) = 15.
- ?Check: 3 × (3 + 2) = 15.
- ?Divide both sides by 3: x + 2 = 5.
- ?Subtract 2 from both sides: x = 3.
Circle the road that felt easier and write one sentence saying why.
Two roads, one answer. Tomorrow is Puzzle Lab: two envelopes on the balance and a log of every undo.