Week 06 · Two-Step Equations
Puzzle Day
Puzzle Day: The two-envelope challenge
// Undo in reverse order, rewrite to see the puzzle, estimate to check
⏱ about 20 min
Puzzle Day: The Two-Envelope Challenge
Saturday morning, Rocket hangs the balance and lines up envelopes in pairs. "Puzzle Day. Two envelopes every round."
"Each visitor writes px + q = r, undoes in reverse order, and says the number," Raven says. "What do you notice about the hard ones?"
"The hard ones have a bundle," Rocket says. "Then you pick a road: divide first or untie first."
Nova hovers above the jar of beans, her light warm. "Estimate before you share," she says. "Hints only."
"Hints only," Rocket agrees, hiding the first pair. Nova hums. "That includes the second step."
- A two-step equation like 2x + 3 = 11 undoes in reverse order: subtract 3, then divide by 2.
- An equation like 3(x + 2) = 15 can be divided first or distributed first. Both roads meet.
- Rewriting an expression, like a + 0.5a = 1.5a, shows what a puzzle really says.
- With decimals and fractions, estimate first with friendly numbers, then solve exactly and compare.
- Check every answer by substitution.
◇ RUN THE TWO-ENVELOPE CHALLENGE WITH YOUR FAMILY
Hang the balance and set out beans, envelopes, string and index cards.
Hide the same number in two envelopes, add loose beans, and level the other cup.
Your family member writes the equation, undoes the loose beans first, then shares, and says the number.
Open an envelope together and count. Then try a bundle round and let them pick a road.
Keep score of how many rounds were solved before the envelope opened.
Hide the same number in two envelopes, add loose beans, and level the other cup.
Your family member writes the equation, undoes the loose beans first, then shares, and says the number.
Open an envelope together and count. Then try a bundle round and let them pick a road.
Keep score of how many rounds were solved before the envelope opened.
For a grown-up
This week your student solved two-step equations like 2x + 3 = 11 and 3(x + 2) = 15.
Ask them why the added number is undone first. The answer should mention reverse order or "last on, first off".
Check the hook before the hanger goes up. Beans and buttons stay away from children under three and pets.
Ask them why the added number is undone first. The answer should mention reverse order or "last on, first off".
Check the hook before the hanger goes up. Beans and buttons stay away from children under three and pets.
Safety first
A grown-up checks the hook. The hanger hangs at chest height and nothing heavy goes in a cup.
Small beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
Pick up dropped beans right away so no one slips.
Small beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
Pick up dropped beans right away so no one slips.
PUZZLE DAY CHECK
- Read the question.
- Tap your answer.
Two envelopes and 3 beans balance 9 beans. How many beans are in each envelope?Two bundles, each an envelope and 1 bean, balance 12 beans. How many beans are in each envelope?
Which value of x makes 3x + 2 = 14 true: 3, 4, 12?
Which expression is equivalent to c + 0.5c?
Draw your Puzzle Day balance with two envelopes and loose beans set up, and the equation card beside it.
Great puzzling, crew. Next week a cardboard box with a slot turns every number you feed it into a new one.