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Expressions and Equations 6-8 / Week 03 / Tuesday
2/6
Week 03 · Equivalent Expressions

Tuesday

Combine and factor
// Two expressions that always give the same number
⏱ about 20 min

Tuesday: Combine and Factor

Raven sets three envelopes in a row. "n plus n plus n. Puzzle three has a shorter name for that."

"Three n," Rocket says. "Three envelopes, 3n. Easy."

"Now this one." Raven adds two envelopes and four loose beans, then one more envelope and two beans. "What do you notice?"

Rocket sorts the bench: envelopes left, beans right. "Three envelopes, six beans. 3n + 6."

Nova hovers over the sorted bench. "Like with like," she says. "Envelopes with envelopes. Beans with beans."

"And 3n + 6 can go back into rows," Raven says. "Three rows of n + 2. The rectangle again."

Rocket draws the rows. "Distribute to open it, factor to close it," he says. "Two directions, one rectangle."

Combine like terms

Like terms have the same letter part. 2n and n are like terms. 2n and 4 are not: one counts envelopes, the other counts beans.

To combine, add the coefficients: n + n + n = 3n. In the same way, 2n + 4 + n + 2 = 3n + 6. Envelopes with envelopes, beans with beans.

Subtraction works the same way: 5n - 2n = 3n, and 7 - 3 + 4n = 4n + 4.

Factor out a common factor

Factoring runs the distributive property backward. Find the greatest common factor of the coefficients and pull it out front.

6x + 9: the greatest common factor of 6 and 9 is 3, so 6x + 9 = 3(2x + 3). Check by distributing: 3 × 2x + 3 × 3 = 6x + 9.

24x + 18 = 6(4x + 3). The parentheses hold what is left after dividing each term by 6.

DirectionStartFinishPicture
Distribute3(n + 2)3n + 6Open the rows into columns
Combinen + n + n3nThree envelopes side by side
Combine2n + 4 + n + 23n + 6Sort envelopes left, beans right
Factor6x + 93(2x + 3)Fold the columns back into 3 rows
COMBINE LIKE TERMS
  • Read the question.
  • Tap your answer.
Which expression is equivalent to n + n + n?
Which expression is equivalent to 2n + 4 + n + 2?
Which expression is equivalent to 5y - 2y + 7?
Which expression is equivalent to 4k + 3 + 2k - 1?
FACTOR IT
  • Read the question.
  • Tap your answer.
Which expression is the same as 6x + 9?
Which expression is the same as 24x + 18?
Which expression is the same as 4n + 10?
Which expression is the same as 15y + 5?
SPOT THE EQUIVALENT ONE
  • Read the question.
  • Tap your answer.
Which expression is equivalent to y + y + y?
Which expression is equivalent to 2(n + 4)?
Which expression is equivalent to 5k + 10?
SIMPLIFY 2(N + 3) + N IN ORDER
  • ?Distribute: 2(n + 3) = 2n + 6.
  • ?Rewrite the whole thing: 2n + 6 + n.
  • ?Combine like terms: 3n + 6.
  • ?Check with n = 1: both give 9.
WHY THIS EXERCISEDistribute, combine, check is the routine for every expression this week.
Try it
Lay out 2 envelopes and 4 beans, then 1 envelope and 2 beans. Sort them, and write the combined expression.
Now put the sorted bench back into equal rows. Write the factored form.
Draw 6x + 9 as 3 rows: each row has 2 envelopes and 3 beans. Write 3(2x + 3) beside it.

Combined and factored. Tomorrow is Puzzle Lab: build the rectangles and test every claim with three values.

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