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Expressions and Equations 6-8 / Week 03 / Puzzle Day
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Week 03 · Equivalent Expressions

Puzzle Day

Puzzle Day: The two-names game
// Two expressions that always give the same number
⏱ about 20 min

Puzzle Day: The Two-Names Game

Saturday at the Puzzle Post, Raven deals expression cards face down. "The two-names game. Draw a card, write a second name for it."

"Then the other player picks a number," Rocket says, "and we both work out our names. Match or miss."

"What do you notice if you get 3n + 2 and write 3(n + 2)?" Raven asks.

"Miss," Rocket says. "Those are the trap twins. Three n plus two is not three groups of n plus two."

Nova hovers above the cards, her light dim and patient. "Build the trap twins with beans," she says. "Then the trap is gone."

Rocket's cousin draws 4n + 8. "Four n plus eight, or four groups of n plus two," she says. Nova hums. "Test it."

  1. Equivalent expressions name the same number for every value of the letter.
  2. The distributive property opens a product into a sum: 3(n + 2) = 3n + 6. Factoring closes it again.
  3. Like terms combine by adding coefficients: n + n + n = 3n. Sorting gives 2n + 4 + n + 2 = 3n + 6.
  4. Expressions add and subtract term by term: (2n + 3) + (3n - 1) = 5n + 2. Subtract every term of the second.
  5. One miss proves a pair is not equivalent. A reason, not a match, proves that it is.
◇ THE TWO-NAMES GAME WITH A GROWN-UP
Write 3(n + 2), 4n + 8, 2n + 2n, 5(n + 1) and 6n + 3 on cards. Shuffle them face down.
Draw a card and write a second name for it. The other player picks a value for n from 1 to 6.
Both of you work out both names. A match scores a point. A miss means you go back and find the slip together.
Finish by building one pair with beans in rows, to show why the two names agree.
For a grown-up
This week your student learned the distributive property, combined like terms and factored expressions.
Ask them why 3(n + 2) is 3n + 6 and not 3n + 2. The answer should mention the 3 multiplying both parts.
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.
Keep the game at the table so nothing ends up on the floor.
PUZZLE DAY CHECK
  • Read the question.
  • Tap your answer.
Which expression is equivalent to 3(n + 2)?
Which expression is the same as 4n + 8?
Which expression is equivalent to 2n + 2n?
With the letter equal to 4, what number do 5(n + 1) and 5n + 5 both give?
Draw the trap twins 3n + 2 and 3(n + 2) with beans for n = 1. Circle the beans that differ.

Great puzzling, crew. Next week: an equation is a level balance, and the envelope finally opens.

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