Week 10 · Solving Linear Equations
Puzzle Day
Puzzle Day: the two-cup puzzle
// Envelopes on both sides
⏱ about 20 min
Puzzle Day: The Two-Cup Puzzle
Saturday the family crowds into the garage. The balance hangs level with envelopes in both cups.
"Two envelopes and four beans against one envelope and nine," Rocket announces. "Every envelope hides the same count."
Raven hands out index cards. "Write the equation first. No peeking in the envelopes."
Someone writes 2x + 4 = x + 9. "What do you notice you can take off both sides?" Raven asks.
"One envelope each, then four beans each," comes the answer. "x is five."
Nova hovers over the sealed envelopes, her light dimming to a hint. "Check before you open," she says.
Two times five plus four is fourteen. Five plus nine is fourteen. The envelope opens: five beans.
"Now you build one for us," Rocket says, handing over the beans.
- An equation can have x on both sides. Take the same envelopes off both cups, then the same beans.
- Five moves solve any linear equation: distribute, collect like terms, letters to one side, numbers to the other, divide.
- Fraction and decimal coefficients use the same moves.
- Three endings: x = a is one solution, a = a is infinitely many, a = b is none.
- Check every answer in the original equation.
◇ FAMILY TWO-CUP PUZZLE
Hang the balance. One person fills three or more envelopes with the same secret count and seals them.
Load both cups with envelopes and loose beans until it hangs level. Everyone writes the equation.
Solve on paper, then open one envelope to check. Trade roles and build a new puzzle.
Try one impossible puzzle: load matching envelopes with different bean counts. Talk about why it will not hang level.
Load both cups with envelopes and loose beans until it hangs level. Everyone writes the equation.
Solve on paper, then open one envelope to check. Trade roles and build a new puzzle.
Try one impossible puzzle: load matching envelopes with different bean counts. Talk about why it will not hang level.
For a grown-up
Check the hook or door handle before hanging the hanger, and keep it at chest height.
Beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
If the hanger will not level, uneven cups or string are the usual cause. Equal sides should hang level.
Beans and buttons stay away from children under three and pets. Nothing goes in a mouth.
If the hanger will not level, uneven cups or string are the usual cause. Equal sides should hang level.
PUZZLE DAY CHECK
- Read the question.
- Tap your answer.
The family puzzle shows 2x + 4 = x + 9. How many beans are in each envelope?
The balance shows 3x + 5 = x + 11. How many beans are in each envelope?Two envelopes and 5 beans against two envelopes and 9 beans. How many solutions does 2x + 5 = 2x + 9 have?
The puzzle 2x + 4 = x + 9 was solved. Type the number of beans in each envelope.
Take one envelope and four beans off both sides.
WHY THIS EXERCISEA puzzle you can build and check is a puzzle you understand.
Draw the family puzzle 2x + 4 = x + 9 as two cups, then draw the cups after each move.