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Algebra 1 9-12 / Week 08 / Loft Day
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Week 08 · Polynomials

Loft Day

Loft Day: tile pictures
// Tiles for the launch pad
⏱ about 20 min

Loft Day: Tile Pictures

Saturday the tile piles come out and the family gathers around the bench.

"Everyone gets one x squared tile, six strips and eight units," Comet says. "Build any rectangle you can."

Rectangles grow across the bench. Some are long and thin, some nearly square.

"What do you notice about the sides?" Wren asks. "Say each side as x plus a number."

"Mine is x + 2 by x + 4," someone says. "One square, six strips, eight units. All of them!"

Nova hovers above the busiest corner, her light dimming to a hint. "Could the same tiles make a different rectangle?"

Someone tries x + 1 by x + 5 and runs out of strips. Laughter, then counting.

"Write every rectangle as a polynomial," Comet says. "Then we swap and check each other."

  1. A polynomial is a sum of terms, each a number times a power of x. Its degree is the highest power.
  2. Tiles: an x² square, x strips and unit squares. A rectangle of tiles is a product of two binomials.
  3. Add and subtract by combining like terms. Multiply by taking every term through every term.
  4. The answer is always a polynomial, just as the whole numbers are closed under adding, subtracting and multiplying.
  5. (x + a)² = x² + 2ax + a², and (x + a)(x - a) = x² - a². Seeing the shape lets you rewrite.
◇ FAMILY TILE RECTANGLES
Give each person one x² tile, some strips and some units. Ask them to build a rectangle using all their tiles.
Read each rectangle's sides as x plus a number, and write its polynomial together.
Swap tile sets and try to build a different rectangle from the same tiles. Count what runs out.
For a twist, hand out tiles for (x + 3)² and ask what makes that rectangle special.
For a grown-up
A grown-up cuts the cardboard tiles ahead of time with a craft blade. Children handle only the finished tiles.
Small tiles go back in a box at the end so nobody slips on them.
A rectangle that will not close is a good moment: ask which tile count does not match the sides.
LOFT DAY CHECK
  • Read the question.
  • Tap your answer.
Algebra tiles for (x + 2)(x + 4): one x² square, 6 x strips and 8 unit squares, so the area is x² + 6x + 8A family rectangle has sides x + 2 and x + 4. Which polynomial is its area?
Algebra tiles for (x + 1)(x + 2): one x² square, 3 x strips and 2 unit squares, so the area is x² + 3x + 2Which expression equals (x + 1)(x + 2)?
Two family rectangles are x² + 3x + 2 and x² + 6x + 8. What are their tiles together?
Which expression equals (x + 2)²?
How many x strips does the rectangle (x + 3)(x + 4) use? Type the number.
WHY THIS EXERCISEThe strip count is the middle coefficient, the sum of the two numbers in the sides.
Draw the family's best rectangle with its tiles outlined and its polynomial written below.
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