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Expressions and Equations 6-8 / Week 10 / Wednesday
3/6
Week 10 · Solving Linear Equations

Wednesday

Puzzle Lab: both cups loaded
// Envelopes on both sides
⏱ about 20 min

Wednesday: Puzzle Lab, Both Cups Loaded

Rocket counts beans into five envelopes while Raven seals them. "Same number in every envelope. Nobody peeks."

"Puzzle one," Raven says. "Three envelopes and two beans on the left, one envelope and ten beans on the right."

Rocket loads the cups. The hanger tips, then settles level. "Level! So the equation is 3x + 2 = x + 10."

"What do you notice when you take one envelope off each side?" Raven asks.

"Two envelopes and two beans against ten beans," Rocket says. "Take two beans off each: 2x = 8."

Nova hovers over the right cup, her light on the eight beans left. "How do eight beans share between two envelopes?"

"Four each," Rocket says, and opens an envelope. Four beans. "The balance agrees with the algebra."

"Write it in the log," Raven says. "Then puzzle two."

What you need

  • The Puzzle Post balance: a wire coat hanger, two paper cups and string, hung on a strong hook.
  • Six envelopes and a bag of dried beans or buttons.
  • A marker, index cards and your Puzzle Notebook.
  • A partner to fill the envelopes while you look away.
Safety first
A grown-up checks the hook or door handle. Hang the hanger at chest height, and never swing on it.
Only envelopes and beans go in the cups. Nothing heavy.
Small beans and buttons stay away from children under three and pets. Nothing goes in a mouth.

Run the lab

  1. Your partner puts the same secret number of beans in every envelope and seals them.
  2. Your partner loads both cups with envelopes and loose beans until the hanger hangs level.
  3. Write the equation: one side for each cup, with x for each envelope.
  4. Take the same number of envelopes off both cups. Then take the same number of beans off both cups.
  5. Share the beans left among the envelopes left. That is x.
  6. Open one envelope to check. Record the puzzle and your answer in the lab table.

The crew's lab log

Here are the five balance puzzles the crew ran. Every number is the crew's own, chosen for the lab.

PuzzleLeft cupRight cupEquationBeans in each envelope
12 envelopes, 3 beans1 envelope, 8 beans2x + 3 = x + 85
23 envelopes, 2 beans1 envelope, 10 beans3x + 2 = x + 104
32 envelopes, 6 beans4 envelopes2x + 6 = 4x3
43 envelopes, 1 beans2 envelopes, 7 beans3x + 1 = 2x + 7?
54 envelopes, 2 beans2 envelopes, 12 beans4x + 2 = 2x + 12?
A level coat-hanger balance: three envelopes and one bean on the left, two envelopes and seven beans on the right.
SOLVE THE CREW'S PUZZLES
  • Read the question.
  • Tap your answer.
Puzzle 4: 3x + 1 = 2x + 7. How many beans hide in each envelope?
A coat-hanger balance, level: the left cup holds 4 envelopes marked x and 2 beans, the right cup holds 2 envelopes marked x and 12 beansPuzzle 5: 4x + 2 = 2x + 12. How many beans hide in each envelope?
In puzzle 3, 2x + 6 = 4x, how many envelopes are left after you take two off each side?
Puzzle 1 was 2x + 3 = x + 8. How many beans were in each envelope? Type the number.
WHY THIS EXERCISEThe lab shows that algebra and the balance give the same answer every time.
StatementTrue or false?
Taking two envelopes off both cups keeps a level balance level.?
3 is a solution of 2x + 6 = 4x.?
4 is a solution of 4x + 2 = 2x + 12.?
The lab only works if every envelope hides the same number of beans.?
3 × 4 + 2 = 4 + 10?
WHY THIS EXERCISEEvery envelope is the same x. If the counts differed, one letter could not stand for them all.
Try it
Build a puzzle whose equation is 2x + 5 = x + 12. Load the cups, check it hangs level, then solve.
Now build one where the envelopes are only on one side. Which kind was quicker to solve?
Draw puzzle 5 as two cups. Cross off matching envelopes and beans until one envelope stands alone.

Great lab work. Tomorrow the crew finds puzzle cards that have no answer at all, and some with every answer.

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