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Algebra 1 9-12 / Week 09 / Loft Day
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Week 09 · Factoring

Loft Day

Loft Day: the factor race
// Sliding the tiles back into a rectangle
⏱ about 20 min

Loft Day: The Factor Race

Saturday the tile piles and a stack of polynomial cards wait on the bench.

"Teams of two," Comet says. "Draw a card, slide the tiles into a rectangle, read the sides. Fastest correct rectangle wins."

Cards flip. Tiles slide. Someone shouts "x + 2 by x + 5!" and someone else says "Check it!"

"What do you notice when a team multiplies back?" Wren asks. "They catch their own mistakes."

Nova hovers above the race, her light dimming to a hint. "One card has no rectangle at all."

The team with that card stares, then grins. "Does not factor! Do we win?"

"You win for noticing," Comet says. "Next round: write a card that will fool the other team."

  1. Factoring undoes multiplying: slide the tiles for a polynomial back into a rectangle and read its sides.
  2. Pull out the greatest common factor first, always.
  3. For x² + bx + c, find two numbers that multiply to c and add to b. For ax² + bx + c, split the middle.
  4. A difference of squares factors as (x - a)(x + a). A perfect square factors as (x + a)² or (x - a)².
  5. The factored form shows the zeros, and the zeros plus the midpoint give a rough graph.
◇ FAMILY FACTOR RACE
Write six polynomials on cards, each one a rectangle you have already built. Add one card that does not factor.
In pairs, draw a card, slide tiles into a rectangle and read the sides. Multiply back to check before claiming the point.
Trade roles so everyone writes a card and everyone solves one.
For a twist, time how long the "does not factor" card takes before someone calls it.
For a grown-up
Tiles are already cut. If more are needed, a grown-up handles the craft blade.
A wrong rectangle is a good moment: ask what the strips and units say about the sides.
Keep tiles on the table and off the floor.
LOFT DAY CHECK
  • Read the question.
  • Tap your answer.
Algebra tiles for (x + 2)(x + 5): one x² square, 7 x strips and 10 unit squares, so the area is x² + 7x + 10A race card reads x² + 7x + 10. Which rectangle do the tiles make?
Which is the factored form of x² + 8x + 12?
A race card reads x² - 25. Which is its factored form?
For the card x² + 7x + 10, which x make the polynomial zero?
A race card reads x² + 8x + 12. The two numbers multiply to 12 and add to 8. What is the larger one? Type the number.
WHY THIS EXERCISEFinding the pair is the heart of factoring x² + bx + c, and the tiles confirm it.
Draw the fastest rectangle your team built today, with tiles outlined and sides labeled.
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