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

Monday

Beans in rectangles
// Two expressions that always give the same number
⏱ about 20 min

Monday: Beans in Rectangles

Rocket lays beans in three rows on the Puzzle Post bench. "Each row is an envelope's worth plus two more. Three rows."

"So three groups of n plus 2," says Raven. "3(n + 2). What do you notice if you look down the columns instead?"

Rocket squints. "All the envelope beans are together on the left. Three n. And the extras are together on the right. Six."

"3n + 6," Raven writes. "Same beans, two expressions."

Nova hovers above the rectangle, her light splitting it down the middle. "Two names for one number," she says. "Whatever n is."

"Puzzle three for the street," Rocket says. "Write both names, then try to find an n where they disagree."

Nova hums. "They will be trying for a long time."

Rocket and Raven lay beans in three rows on the garage bench, split by a tape line, while Nova hovers above.

The distributive property

Three rows of squares split into 4 and 2: three rows of 4, plus three rows of 2

Three rows of (4 + 2) squares is 3 × (4 + 2) = 18. Count the columns instead: 3 × 4 + 3 × 2 = 18. Same squares.

With a letter it works the same way. 3(n + 2) means 3 rows of n beans and 2 more. Column by column that is 3n + 6.

This is the distributive property: multiplying a sum multiplies each part. 3(n + 2) = 3n + 6, always.

Equivalent means always

Two expressions are equivalent when they name the same number for every value of the letter.

Try n = 4. Then 3(n + 2) = 3 × 6 = 18, and 3n + 6 = 12 + 6 = 18. Try n = 10: 36 both times.

One matching value is not proof. The bean rectangle is the proof: the same beans, counted two ways, for any n at all.

COUNT THE RECTANGLE TWO WAYS
  • Read the question.
  • Tap your answer.
A rectangle of unit squares, three rows of six, split into 4 and 2 columnsA bean rectangle has 3 rows, each with 4 beans and then 2 more. How many beans in all?
A rectangle of unit squares, two rows of eight, split into 5 and 3 columnsA bean rectangle has 2 rows, each with 5 beans and then 3 more. How many beans in all?
A rectangle of unit squares, four rows of four, split into 3 and 1 columnsA bean rectangle has 4 rows, each with 3 beans and then 1 more. How many beans in all?
A rectangle of unit squares, five rows of four, split into 2 and 2 columnsA bean rectangle has 5 rows, each with 2 beans and then 2 more. How many beans in all?
DISTRIBUTE
  • Read the question.
  • Tap your answer.
Which expression is equivalent to 3(n + 2)?
Which expression is equivalent to 2(x + 5)?
Which expression is equivalent to 4(y + 3)?
Which expression is equivalent to 5(2k + 1)?
StatementTrue or false?
3(n + 2) and 3n + 6 are equivalent.?
3(n + 2) and 3n + 2 are equivalent.?
2(x + 5) and 2x + 10 are equivalent.?
Two expressions are equivalent if they match for one value of the letter.?
WHY THIS EXERCISE3(n + 2) = 3n + 6, not 3n + 2: the 3 multiplies the 2 as well as the n. One match can be luck; every match is equivalence.
What is 3(n + 2) written without parentheses? Type the number that comes after 3n +.
WHY THIS EXERCISEThe 3 multiplies both the n and the 2, so the extras total 3 × 2 = 6.
Try it
Lay out 2 rows of beans, each row 3 beans then 4 more. Count by rows, then by columns.
Write both expressions, 2(3 + 4) and 2 × 3 + 2 × 4, in your Puzzle Notebook.
Safety first
Beans are counters, never food. They stay on the tray, away from children under three and pets.
Tape marks the split line. Press it on the bench, not on the floor where it can trip someone.
Draw 3 rows of beans with n beans and then 2 more in each row. Write 3(n + 2) and 3n + 6 under it.

Two names for one number. Tomorrow you combine like terms and pull common factors back out.