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Expressions and Equations 6-8 / Week 04 / Wednesday
3/6
Week 04 · The Balance: One-Step Equations

Wednesday

Puzzle Lab: Build the balance
// An equation is a level balance, and a solution keeps it level
⏱ about 20 min

Wednesday: Puzzle Lab: Build the Balance

Rocket checks the hook twice, then hangs the coat hanger at chest height. Two paper cups swing from its ends on string.

"Puzzle Lab," he says. "Raven hides beans in an envelope. I find the number without opening it."

Raven seals the envelope and drops it in the left cup with 4 loose beans. Then she counts beans into the right cup until it levels.

"Eleven on the right," Rocket reads. "What do you notice?" Raven asks. "You already know what to do."

"x + 4 = 11. Take 4 from both sides." Rocket lifts out beans. "x = 7."

Raven opens the envelope. Seven beans. Nova hovers above, her light pulsing. "Now swap. And write every result in the log."

Your mission

Build the Puzzle Post balance and play the envelope game. One person hides beans, the other writes and solves the equation.

Keep a log of every round: beans in the cup, beans on the right, the equation, your answer and the real count.

  • A wire coat hanger and a sturdy hook or a door handle, checked by a grown-up.
  • Two paper cups and string to tie one to each end of the hanger.
  • About 40 dried beans or buttons, and three or four envelopes.
  • Your Puzzle Notebook and a pencil.
Safety first
A grown-up checks the hook. Hang the hanger at chest height, and nothing heavy goes in a cup.
No one swings on the balance or hangs anything from the hook but the hanger.
Small beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
Shoes on in the garage, and pick up any beans that fall so no one slips.
  1. Tie a cup to each end of the hanger with string. Hang it and check that it sits level when empty.
  2. One player hides some beans in an envelope and puts it in the left cup with a few loose beans.
  3. The other player adds beans to the right cup until the hanger is level, then counts them.
  4. Write the equation, undo on both sides, and say the hidden number. Open the envelope to check.
  5. Play a round with two or three equal envelopes, all hiding the same number. Write px = q and share.
  6. Swap roles after every round. Log at least six rounds.

The crew's log

RoundLeft cupRight cupEquationSolved
1envelope + 4 beans11 beansx + 4 = 11x = 7
2envelope + 6 beans9 beansx + 6 = 9x = 3
32 equal envelopes12 beans2x = 12x = 6
4envelope + 1 bean13 beansx + 1 = 13x = 12
53 equal envelopes21 beans3x = 21x = 7
64 equal envelopes10 beans4x = 10x = 2.5

These are the crew's own rounds from Raven's log. Round 6 was a trick: four envelopes and 10 beans gave a solution that is not whole.

The envelopes could not hold 2.5 beans each, so Raven had written a half-bean puzzle on paper instead.

READ THE CREW'S LOG
  • Read the question.
  • Tap your answer.
A level balance: 1 envelope and 4 beans on the left, 11 beans on the right.The balance is level: one envelope and 4 beans on the left, 11 beans on the right. How many beans are in each envelope?
A level balance: 2 envelopes on the left, 12 beans on the right.The balance is level: 2 envelopes on the left, 12 beans on the right. How many beans are in each envelope?
A level balance: 1 envelope and 6 beans on the left, 9 beans on the right.The balance is level: one envelope and 6 beans on the left, 9 beans on the right. How many beans are in each envelope?
Round 6 on paper: four equal envelopes balance 10. What does each envelope hold?
StatementTrue or false?
x = 7 is a solution of x + 4 = 11.?
x = 7 is a solution of 3x = 21.?
x = 15 is a solution of x + 6 = 9.?
x = 6 is a solution of 2x = 12.?
WHY THIS EXERCISERound 1 and round 5 both hid 7 beans, found by different undo steps. Substitution confirms each one.
Draw your balance with the cups labeled, and one round from your log shown as beans and an envelope.

Six rounds solved and checked. Tomorrow the crew turns their puzzle cards into equations.

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