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Algebra 1 9-12 / Week 04 / Loft Day
6/6
Week 04 · Linear Graphs and Rates of Change

Loft Day

Loft Day: the family ramp
// The cart on the ramp, one second at a time
⏱ about 20 min

Loft Day: The Family Ramp

Saturday the plank goes up in the backyard, barely tilted, with a towel at the bottom.

"Family race," Comet says. "Not the fastest cart. The one whose dots make the straightest line."

Each person releases the cart and a partner calls the marks every second. Wren writes the readings on a big sheet.

"What do you notice?" he asks when the sheet is full.

"Every run starts at mark 1," says Comet's cousin. "But the push runs climb faster."

Nova projects the dots in different colors. "Would you like a hint? Find each run's rate from two points."

"Three a second, four a second, three again," Comet reads. "The push changes the slope, not the start."

Wren hands the pen to the youngest. "Your turn to draw the lines."

  1. A linear rule like d = 3t + 1 graphs as a straight line with slope 3 and y-intercept 1.
  2. The horizontal intercept is where the output is 0: set the rule equal to 0 and solve.
  3. Positive slope rises, negative slope falls. Positive output sits above the axis.
  4. Average rate of change is change in output over change in input, on any interval, even for a curve.
  5. Every graph needs labeled axes and a scale.
◇ FAMILY RAMP READINGS
Set a board on two low supports with a gentle tilt and a towel at the end. Mark it every 10 centimeters.
Take turns: one person releases the cart, another calls the mark each second, a third writes.
Plot every run on one big sheet, seconds across and marks up, with labels and a scale.
Find each run's rate of change from two points. Whose line was straightest? Whose was steepest?
For a grown-up
Set up the board yourself and check the supports cannot tip. Keep the tilt gentle so the cart rolls, never flies.
Keep feet and fingers clear of the rolling cart, and keep the towel at the bottom.
If a run's dots wobble, that is a good moment: ask what changed between that run and the others.
LOFT DAY CHECK
  • Read the question.
  • Tap your answer.
The cousin's push run went from mark 1 at 0 seconds to mark 13 at 3 seconds. What was the rate?
The push run passed (0, 1) and (3, 13). Which equation fits it?
A family timer follows r = 9 - 3t. When does it reach zero?
A family run followed d = 4t + 1. Which mark did the cart pass at 3 seconds? Type the number.
WHY THIS EXERCISEThe rule predicts a reading. Checking it against the real mark tests the model.
Draw the family sheet: three lines from (0, 1) with slopes 3, 4 and 3, labeled with each person's name.
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