Week 03 · Rational Functions
Dock Day
Dock Day: the family average
// Lap times that climb toward a wall
⏱ about 20 min
Dock Day: The Family Average
Saturday the stopwatch comes to the kitchen with a stack of index cards and a clear path around the table.
"Walk six laps," Comet tells the family. "Someone clicks at the start line. Someone writes the totals."
"The first lap is always slow," Wren says. "Watch the average per lap fall toward one number and stop short."
Nova dims to a hint. "Would you like a hint? Write the total as a rule, then divide by the lap number."
"Ours came out 50n + 30," someone says. "So the average is 50 + 30/n. The wall is 50."
"Now build a rule," Comet says. "Laps times seconds per lap. Compose it with laps per week."
"Add, multiply, compose," Wren says. "The whole week around one kitchen table."
- A rational function is p(x)/q(x). It has no value where q(x) = 0.
- Zeros are where the top is 0. Vertical asymptotes are where only the bottom is 0. A shared zero is a hole.
- The horizontal asymptote comes from degrees. Bottom higher gives y = 0, equal gives the ratio of leading coefficients, top higher gives none.
- Dividing top and bottom by x proves the wall for (ax + b)/(cx + d) is y = a/c.
- Build new rules by adding, multiplying, dividing or composing old ones. Compose inside first.
◇ FAMILY LAPS AND THE WALL
Mark a start line and clear the path. One person walks six laps while another clicks the stopwatch at each pass.
Record the total after each lap. Divide by the lap number to get the average. Watch it settle.
Fit a rule: total = (seconds per lap) × n + (start delay). Write the average as a rational function and name its wall.
Trade roles so everyone walks, everyone clicks and everyone computes one average.
Record the total after each lap. Divide by the lap number to get the average. Watch it settle.
Fit a rule: total = (seconds per lap) × n + (start delay). Write the average as a rational function and name its wall.
Trade roles so everyone walks, everyone clicks and everyone computes one average.
For a grown-up
This is a walking activity. Nobody runs, and the path is clear of cords, toys and pets.
A surprising average is a good moment: check the total, then divide again together.
The lap times and rules are the family's own made-up model, not a record of any real race.
A surprising average is a good moment: check the total, then divide again together.
The lap times and rules are the family's own made-up model, not a record of any real race.
DOCK DAY CHECK
- Read the question.
- Tap your answer.
A family total rule is 50n + 30. What is the average per lap after 3 laps?
The family's average rule is y = (50x + 30)/x. What wall does it approach?L(x) = 2x + 1 laps in week x, and M(n) = x + 4 seconds for n laps. What is M(L(2))?
A family member walks laps with a total rule of 40n + 20. The average per lap falls toward what number? Type it.
Divide top and bottom by n. The 20/n shrinks away.
WHY THIS EXERCISEThe wall is always the ratio of the leading coefficients, whatever the start delay.
Draw your family's six average points with the dashed wall under them. Label the first and sixth averages.