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Algebra 1 9-12 / Week 08 / Monday
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Week 08 · Polynomials

Monday

Tiles for the launch pad
// Tiles for the launch pad
⏱ about 20 min

Monday: Tiles for the Launch Pad

Comet spreads cardboard tiles across the bench: a big square, long strips, small squares.

"The launch pad is a square, x by x," she says. "But the launcher is not built, so I do not know x yet."

"Then build the plan with tiles," Wren says. "The big square is x by x. A strip is x by 1. A small square is 1 by 1."

He lays a rectangle: the big square, five strips, six small squares. "Sides x + 2 and x + 3. What do you notice?"

Nova hovers over the layout, her light sorting the tiles by kind. "Would you like a hint? Count each kind separately."

"One x squared, five x, six ones," Comet reads. "So the area is x squared plus 5x plus 6."

"A polynomial," Wren says. "Our engineer, how many of each tile does (x + 1)(x + 4) need?"

Comet, Wren and Nova in the Loft with cardboard algebra tiles laid out as a launch-pad plan.

Three kinds of tiles

  • The x² tile is a square with sides x. Its area is x times x, written x².
  • The x tile is a strip x long and 1 wide. Its area is x.
  • The unit tile is a 1 by 1 square. Its area is 1.
  • Nobody knows x yet. The tiles keep the plan honest for whatever x turns out to be.

A polynomial

A polynomial is a sum of terms. Each term is a number times a power of x. In x² + 5x + 6, the terms are x², 5x and 6.

The number in front of a term is its coefficient. The coefficient of x in x² + 5x + 6 is 5.

The degree is the highest power. x² + 5x + 6 has degree 2. A plain number like 6 has degree 0.

A solved example to study: the Monday rectangle has sides x + 2 and x + 3. Its tiles are 1 x², 5 x and 6 units, so (x + 2)(x + 3) = x² + 5x + 6.

Algebra tiles for (x + 2)(x + 3): one x squared square, five x strips and six unit squares.
ExpressionTermsDegreeCoefficient of x
x² + 5x + 6325
2x² + 3x + 1323
x³ + 2x232
4x - 7214
9100
COUNT THE TILES
  • 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 expression equals (x + 2)(x + 3)?
Algebra tiles for (x + 1)(x + 4): one x² square, 5 x strips and 4 unit squares, so the area is x² + 5x + 4Which expression equals (x + 1)(x + 4)?
How many x strips does the rectangle (x + 3)(x + 3) need?
How many unit squares does the rectangle (x + 2)(x + 5) need?
TERMS AND DEGREE
  • Read the question.
  • Tap your answer.
What is the degree of 2x² + 3x + 1?
How many terms does 2x² + 3x + 1 have?
In 2x² + 3x + 1, what is the coefficient of x?
StatementTrue or false?
The degree of x² + 5x + 6 is 2.?
The degree of x³ + 2x is 2.?
(x + 2)(x + 3) = x² + 5x + 6?
(x + 2)(x + 3) = x² + 6?
An x tile is x long and 1 wide, so its area is x.?
WHY THIS EXERCISETiles make each term visible. A polynomial is just a tile count written as algebra.
Try it
Cut a paper square for x², then strips as long as its side, then small squares as wide as a strip.
Lay out (x + 1)(x + 2). Count the tiles of each kind and write the polynomial.
Scissors stay closed when you are not cutting, and a grown-up cuts anything thick.
Draw the tile rectangle for (x + 1)(x + 4). Label the sides and write the polynomial under it.

Clear counting. Tomorrow the crew adds, subtracts and multiplies polynomials, and finds that the answer is always another polynomial.