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Expressions and Equations 6-8 / Week 02 / Puzzle Day
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Week 02 · Letters Stand for Numbers

Puzzle Day

Puzzle Day: The envelope game
// An envelope hides a number, and a letter stands in for it
⏱ about 20 min

Puzzle Day: The Envelope Game

Saturday at the Puzzle Post, Raven deals out envelopes. "The envelope game. Each envelope holds a secret number of beans."

"I give you an expression card," Rocket says, "and you build it with beans before you open the envelope. Then we check."

"What do you notice if your card is 12 - n and the envelope is big?" Raven asks.

"You run out of room to subtract," Rocket says. "So keep the envelopes small."

Nova hovers above the tray, her light soft. "Say the expression in words first," she says. "Then build it."

Rocket's cousin draws 2n + 1 and an envelope of four. Two rows of four, one more. "Nine," she says. Nova hums. "Check it."

  1. A variable is a letter standing for a number: an unknown one in an envelope, or any number from a set.
  2. "Subtract y from 5" is 5 - y. "3 more than twice n" is 2n + 3. Order matters in the words and in the expression.
  3. Terms are added or subtracted. The coefficient multiplies the letter. 2(8 + 7) is a product of two factors.
  4. To evaluate, replace every letter with its number, then follow the order of operations.
  5. Formulas are expressions with letters: perimeter 2l + 2w, area l × w, volume s³.
◇ THE ENVELOPE GAME WITH A GROWN-UP
Put 2 to 8 beans in each of three envelopes and seal them. Write 2n + 1, n + 5, 3n and 10 - n on cards.
One player draws a card and an envelope, says the expression in words, then opens the envelope to find n.
Build the expression with beans. Count, then check with arithmetic. A match scores a point.
Swap roles. End by measuring a rectangle at home and using 2l + 2w on it.
For a grown-up
This week your student wrote expressions with letters, named their parts and evaluated them with the order of operations.
Ask them why "subtract y from 5" is 5 - y and not y - 5. The answer should mention what you start with.
Beans are counters, never food. They stay away from children under three and pets.
Safety first
Beans stay on the tray. Nothing goes in a mouth.
Seal envelopes with the flap or tape, not with licking.
PUZZLE DAY CHECK
  • Read the question.
  • Tap your answer.
The card says 2n + 1 and the envelope holds n = 4. How many beans?
The card says 10 - n and the envelope holds n = 6. How many beans?
Which expression says "5 more than n"?
The perimeter of a rectangle is 2l + 2w. With l = 8 and w = 5, what is the perimeter?
Draw your three envelopes with their numbers and the card you built with beans for each one.

Great puzzling, crew. Next week: beans in rectangles show when two expressions are secretly the same.

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