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

Wednesday

Loft Lab: cut your own tiles
// Tiles for the launch pad
⏱ about 20 min

Wednesday: Loft Lab, Cut Your Own Tiles

Comet rules lines on a flat cardboard box. "Lab day. Unit is 2 centimeters. x is 9 centimeters."

"Why 9?" Wren asks. "So nobody can mistake x for a stack of units," she says. "Nine is not a multiple of 2."

A grown-up runs the craft blade along the lines. Comet and Wren sort the pieces: squares, strips, units.

"Build (x + 1)(x + 2)," Wren says. They slide tiles into a rectangle. "One square, three strips, two units."

Nova hovers above the bench, her light outlining the rectangle. "Would you like a hint? Check the side lengths."

"Top is x + 2. Side is x + 1. It fits," Comet says. "What do you notice if we try (x + 2)(x + 2)?"

"A perfect square," Wren says. "Our engineer, build four rectangles and log the tiles for each."

What you need

  • A flat piece of cardboard or thick paper, a ruler, a pencil and scissors.
  • Tiles to cut: 2 big squares (x by x), 10 strips (x by 1) and 16 unit squares (1 by 1).
  • Choose x so it is not a whole number of units, such as 9 centimeters with a 2 centimeter unit.
  • Your Loft Log and a flat surface to build on.
Safety first
A grown-up does any cutting with a craft blade. You may cut thin card with scissors, slowly, away from your fingers.
Keep blades closed or capped when not in use and off the floor.
Pick up small tiles at the end so nobody slips on them.

Run the lab

  1. Rule the tiles on the cardboard with the ruler. Keep every unit exactly the same width.
  2. Cut the tiles, or have a grown-up cut them. Sort them into three piles.
  3. Build (x + 1)(x + 2) as a rectangle with sides x + 1 and x + 2. Count each kind of tile.
  4. Build (x + 2)(x + 2), then (x + 1)(x + 3), then (x + 2)(x + 4).
  5. For each rectangle, record the x², x and unit counts, and write the polynomial.
  6. Check one rectangle by distributing on paper. The counts and the algebra should agree.

The crew's lab log

Here are the four rectangles the crew built, with the tile counts they recorded. The numbers are the crew's own.

Rectanglex² tilesx tilesUnit tilesPolynomial
(x + 1)(x + 2)132x² + 3x + 2
(x + 2)(x + 2)144x² + 4x + 4
(x + 1)(x + 3)14??
(x + 2)(x + 4)16??
Algebra tiles for (x + 2)(x + 2): one x squared tile, four x strips and four unit squares.
READ THE CREW'S RECTANGLES
  • Read the question.
  • Tap your answer.
Algebra tiles for (x + 2)(x + 2): one x² square, 4 x strips and 4 unit squares, so the area is x² + 4x + 4Which expression equals (x + 2)(x + 2)?
Algebra tiles for (x + 1)(x + 3): one x² square, 4 x strips and 3 unit squares, so the area is x² + 4x + 3Which expression equals (x + 1)(x + 3)?
How many tiles in all does the rectangle (x + 2)(x + 4) use?
How many unit tiles does the rectangle (x + 2)(x + 4) need? Type the number.
WHY THIS EXERCISEThe unit count is the product of the two numbers, which is the last term of the polynomial.
StatementTrue or false?
(x + 1)(x + 2) = x² + 3x + 2?
(x + 2)(x + 2) = x² + 4?
(x + 1)(x + 3) = x² + 4x + 3?
Choosing x as a whole number of units would make x look like a stack of units.?
The strips in a rectangle (x + a)(x + b) number a + b.?
WHY THIS EXERCISEBuilding the rectangle by hand shows where every term of the product comes from.
Try it
Build a rectangle with sides 2x + 1 and x + 2 using two x² tiles. Count and write the polynomial.
Then distribute on paper and compare. Did the tiles and the algebra agree?
Draw your (x + 2)(x + 4) rectangle with each tile outlined. Label both sides.

Careful building. Tomorrow the crew puts numbers into the pad plans and checks that the algebra holds.

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