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Expressions and Equations 6-8 / Week 03 / Friday
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Week 03 · Equivalent Expressions

Friday

Same number, two names
// Two expressions that always give the same number
⏱ about 20 min

Friday: Same Number, Two Names

Raven holds up two cards: "4 chairs at each of t tables" and "4t". "Two names, one number."

"Like a nickname," Rocket says. "Puzzle three turned out to be about nicknames for numbers."

"What do you notice about which name is handier?" Raven asks.

"The short one for counting," Rocket says, "and the long one for seeing where it came from."

Nova hovers at the board, her light settling on 3(n + 2). "Keep both names," she says. "Each one shows something."

"Next week we find the number in the envelope for real," Raven says, "with a balance."

Rocket is already untangling a wire coat hanger. "I knew this would come in handy." Nova hums. "Carefully."

Everyday equivalents

SituationOne nameAnother name
4 chairs at each of t tables4tt + t + t + t
Each of 3 kids gets p pages and 2 stickers3(p + 2)3p + 6
Double the beans, then 6 more2b + 62(b + 3)
Half the cards plus half the cards again1/2c + 1/2cc
A week of m minutes a day7mm + m + m + m + m + m + m

The factored form often shows the story: 3(p + 2) says "three kids, each with p + 2 things."

The expanded form often makes counting easier: 3p + 6 says "3p pages and 6 stickers in all."

Both are right. Choosing the handier name is part of the skill.

How to check any claim

  1. Try one value in both expressions. A miss means not equivalent, and you are done.
  2. If they match, look for the reason: a distributive step, like terms, or a factor pulled out.
  3. Write the reason. That is the proof for every value, not just the ones you tried.
EVERYDAY EQUIVALENTS
  • Read the question.
  • Tap your answer.
Which expression is equivalent to 4t?
Which expression is equivalent to 3(p + 2)?
Which expression is the same as 2b + 6?
Which expression is equivalent to 7m, a week of m minutes a day?
WEEK REVIEW
  • Read the question.
  • Tap your answer.
Which expression is equivalent to 2(x + 4) + 3x?
Which expression is the same as 12n + 8?
Which expression is equivalent to (5k + 2) - (k + 6)?
With the letter equal to 2, what number do 2(x + 4) + 3x and 5x + 8 both give?
Week reviewTrue or false?
3(n + 2) and 3n + 6 are equivalent.?
y + y + y and 3y are equivalent.?
6x + 9 and 3(2x + 3) are equivalent.?
2n + 4 and 2n + 4 + n are equivalent.?
Equivalent expressions give the same number for every value of the letter.?
WHY THIS EXERCISE2n + 4 + n is 3n + 4, a different expression. One extra term changes the number for every n except 0.
OUR WEEK, IN ORDER
  • ?We combined like terms and factored out common factors.
  • ?We counted bean rectangles two ways and met the distributive property.
  • ?We added and subtracted Rocket's and Raven's card expressions, fractions included.
  • ?We ran Puzzle Lab and tested four pairs at three values each.
WHY THIS EXERCISEEach day built on the one before: open, sort, test, then combine whole expressions.
Try it
Find a situation at home with equal groups and extras: plates per table, pencils per cup.
Write it two ways, factored and expanded, in your Puzzle Notebook.
Draw 3(p + 2) as three rows of pages and stickers, then write 3p + 6 beside it.

A week of nicknames for numbers. Tomorrow is Puzzle Day: the two-names game with your family.

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