Week 02 · Equations and Inequalities in One Variable
Loft Day
Loft Day: the envelope game
// Keep both sides equal, and know why
⏱ about 20 min
Loft Day: The Envelope Game
Saturday the coat hanger swings from the Loft rafters and the family crowds around it.
"Envelope game," Comet announces. "Someone seals beans in an envelope. The rest of us solve for it."
A cousin loads two envelopes and 1 bean on the left, 9 beans on the right. The hanger settles level.
"2x + 1 = 9," Wren says. "What do you notice? Take 1 bean off each side, then split 8 beans between 2 envelopes."
"Four each!" The envelopes open. Four and four.
Nova hovers above the hook, her light dimmed to a hint. "Would you like a hint? Try a limit next: at most 12 beans."
"An inequality game," Comet says. "What can we make? A whole table of them."
- Name the unknown, write what the story says is true, then solve.
- Every step keeps both sides equal: equal things treated the same stay equal.
- Check a solution by substitution, in the equation and in the story.
- An inequality gives a range. Flip the sign only when you multiply or divide by a negative.
- A formula can be solved for any one letter with the same moves.
◇ FAMILY ENVELOPE GAME
Hang a coat hanger with a paper cup on each end. One person seals a secret number of beans in an envelope.
They set up a level balance with the envelope, loose beans and beans on the other side. Everyone writes the equation.
Solve it together, saying the reason for each move. Then open the envelope and check.
For a twist, set a limit such as "at most 12 beans on the left" and find every envelope count that fits.
They set up a level balance with the envelope, loose beans and beans on the other side. Everyone writes the equation.
Solve it together, saying the reason for each move. Then open the envelope and check.
For a twist, set a limit such as "at most 12 beans on the left" and find every envelope count that fits.
For a grown-up
Hang the hanger low and over a table or the floor. Beans stay out of mouths and away from pets.
A grown-up covers any sharp wire end with tape and handles the scissors.
If a guess is off, test it in the equation first. Finding which side is too heavy is the learning.
A grown-up covers any sharp wire end with tape and handles the scissors.
If a guess is off, test it in the equation first. Finding which side is too heavy is the learning.
LOFT DAY CHECK
- Read the question.
- Tap your answer.
The cousin's balance showed 2x + 1 = 9. How many beans were in each envelope?Another round shows 3x + 2 = 14. How many beans are in each envelope?
The twist round: an envelope and 4 beans on the left must total at most 12. Which x fit?
A round shows 2x + 5 = 17. How many beans are in each envelope? Type the number.
Take 5 beans off each side, then share what is left between 2 envelopes.
WHY THIS EXERCISEA guess becomes a solution only when it passes the check in the equation.
Draw the family balance for 2x + 1 = 9 and show the two steps that find x.