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Algebra 1 9-12 / Week 05 / Loft Day
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Week 05 · Linear Models and Sequences

Loft Day

Loft Day: the family tower
// The cup tower and the filling jug
⏱ about 20 min

Loft Day: The Family Tower

Saturday the cups come out in the backyard, with a towel spread on the grass.

"Family tower," Comet says. "Everyone picks a design: cups per row and extras on top. Then predict before you stack."

Her cousin picks 5 a row with 2 on top. "So 7, 12, 17, 22. Rule 5n + 2."

"What do you notice when you stack it?" Wren asks.

"Row 4 used exactly 22. The prediction was right!"

Nova projects each design as dots on one grid. "Would you like a hint? The steeper the dots, the bigger the row."

"And the tallest dot at n = 1 is the design with the most extras," Comet says.

Wren hands the pen to the youngest. "Your design. Your rule. Then we stack it together."

  1. A linear model is rate × input + start. The slope is the rate in the story, the intercept is the start.
  2. Build a rule from two pairs: rate from the change, start by stepping back to input 0. Check with the other point.
  3. A table is linear when equal input steps give equal output differences.
  4. An arithmetic sequence is a linear function on whole numbers. Recursive: a(n) = a(n - 1) + d. Explicit: a(n) = dn + (a(1) - d).
  5. Geometric sequences multiply instead of add. They come back in week 7.
◇ FAMILY TOWER DESIGNS
Each person picks cups per row and extras on top, writes the first four terms and the explicit rule.
Stack each design on the floor, counting cups after every row. Compare with the prediction.
Plot every design as dots on one grid, rows across and cups up, with labels and a scale.
Whose dots are steepest? Whose start highest? Say what each number in the rule did to the picture.
For a grown-up
Towers stay on the floor and below shoulder height. Paper cups fall harmlessly, so let them.
If a prediction misses, count the stacked cups together and compare with the rule. Which number was off?
Keep the cups out of mouths for little ones.
LOFT DAY CHECK
  • Read the question.
  • Tap your answer.
The cousin's design is 7, 12, 17, 22, ... How many cups for 9 rows?
The cousin's design is 7, 12, 17, 22, ... Which explicit rule gives term n?
The cousin has 47 cups. Using 7, 12, 17, 22, ... how many full rows is that?
A family design follows a(n) = 2n + 3. How many cups for 6 rows? Type the number.
WHY THIS EXERCISEThe explicit rule jumps straight to any row without stacking the ones before.
Draw the family grid: three designs as dots, rows across and cups up, each labeled with its rule.
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