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Algebra 1 9-12 / Week 09 / Monday
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Week 09 · Factoring

Monday

Slide the tiles into a rectangle
// Sliding the tiles back into a rectangle
⏱ about 20 min

Monday: Slide the Tiles Into a Rectangle

Comet tips a bag of tiles onto the bench: one x squared, five strips, six units. "This week we go backwards."

"x squared plus 5x plus 6," Wren reads. "What rectangle made these? What do you notice about the units?"

"Six units," Comet says. "They have to fill a corner. A 1 by 6 corner, or a 2 by 3 corner."

She slides the units into a 2 by 3 block and lays the strips around the big square. Two down, three along.

Nova hovers over the rectangle, her light running along both edges. "Would you like a hint? Read the sides."

"x + 2 and x + 3," Wren says. "So x squared plus 5x plus 6 equals (x + 2)(x + 3). Factoring undoes multiplying."

"Our engineer," Comet says, "the next bag holds x squared, seven strips and twelve units. What rectangle do they make?"

Comet slides cardboard algebra tiles into a rectangle in the Loft while Wren reads the sides and Nova hovers.

Factoring undoes the area model

Last week two sides gave a polynomial. This week a polynomial gives back its two sides. That is factoring.

A solved example to study: x² + 5x + 6. The units, 6, must form a corner. The strips, 5, split between two sides.

Two numbers that multiply to 6 and add to 5: 2 and 3. So the sides are x + 2 and x + 3.

Check by multiplying back: (x + 2)(x + 3) = x² + 5x + 6. The tiles and the algebra agree.

Algebra tiles for x squared plus 5x plus 6 arranged as a rectangle with sides x + 2 and x + 3.

The two-number search

  1. Write the polynomial as x² + bx + c. Note b, the strips, and c, the units.
  2. List the pairs of numbers that multiply to c.
  3. Pick the pair that adds to b.
  4. Write the sides: (x + first number)(x + second number).
  5. Multiply back to check.
Polynomialc (units)b (strips)The pairFactored form
x² + 5x + 6652 and 3(x + 2)(x + 3)
x² + 7x + 121273 and 4(x + 3)(x + 4)
x² + 6x + 886??
x² + 8x + 15158??
SLIDE INTO A RECTANGLE
  • Read the question.
  • Tap your answer.
Algebra tiles for (x + 2)(x + 3): one x² square, 5 x strips and 6 unit squares, so the area is x² + 5x + 6Which is the factored form of x² + 5x + 6?
Algebra tiles for (x + 3)(x + 4): one x² square, 7 x strips and 12 unit squares, so the area is x² + 7x + 12Which is the factored form of x² + 7x + 12?
Which is the factored form of x² + 6x + 8?
Which is the factored form of x² + 8x + 15?
FIND THE PAIR
  • Read the question.
  • Tap your answer.
Which two numbers multiply to 12 and add to 7?
Which two numbers multiply to 8 and add to 6?
Which two numbers multiply to 15 and add to 8?
StatementTrue or false?
x² + 5x + 6 = (x + 2)(x + 3)?
x² + 7x + 12 = (x + 2)(x + 6)?
x² + 6x + 8 = (x + 2)(x + 4)?
The unit tiles must form a rectangular corner for the big rectangle to close.?
Factoring and multiplying are opposite moves.?
WHY THIS EXERCISEEvery factoring can be checked by multiplying. That check is the habit this week builds.
Try it
Lay out your tiles for x² + 6x + 8 loose on the table. Slide them into a rectangle without looking at the answer.
Read the sides, then multiply back on paper to check.
Pick the small tiles up at the end so nobody slips on them.
Draw the tiles for x² + 7x + 12 arranged as a rectangle. Label both sides.

Clean reversal. Tomorrow four kinds of factoring, and the rule that comes first every time: pull out the common factor.