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Algebra 1 9-12 / Week 04 / Thursday
4/6
Week 04 · Linear Graphs and Rates of Change

Thursday

Three runs, three lines
// The cart on the ramp, one second at a time
⏱ about 20 min

Thursday: Three Runs, Three Lines

Wren pins three tables to the corkboard. "Run A, gentle plank. Run B, gentle plank with a push. Run C, middle plank."

"Same seconds, different centimeters," Comet says. "What can we make of them?"

"Three rules," Wren says. "What do you notice about run B's start?"

"Same start as A, 10 centimeters. But it gains 40 a second instead of 30. The push changed the rate, not the start."

Nova projects all three as lines on one grid. "Would you like a hint? Look at run C's first row."

"Zero at zero," Comet says. "Run C started right on the line, so its intercept is 0."

Then Wren flips a fourth card: the steep plank. 0, 5, 20, 45, 80. "That one is not a line at all."

"The rate keeps growing," Comet says. "Our engineer, what is the average rate between 1 and 3 seconds?"

The crew's Thursday data

Every number here is the crew's own made-up timing from the Loft, in centimeters past the start line.

SecondsRun A, gentle plankRun B, gentle plank, pushedRun C, middle plank
010100
1405050
27090100
3100130150
4130170200
A grid from 0 to 20 with three rising lines for runs A, B and C, drawn in tens of centimeters.

The graph is drawn in tens of centimeters so the lines fit the grid. The scale is written on the axis.

Run A: d = 30t + 10. Run B: d = 40t + 10. Run C: d = 50t. Same shape, different rate or start.

When the graph bends

On the steep plank the cart sped up. The readings 0, 5, 20, 45, 80 do not have equal steps.

You can still find an average rate of change over an interval: change in distance divided by change in time.

From 1 to 3 seconds: 45 - 5 = 40 centimeters over 2 seconds. That is the slope of the line joining those two dots.

This rule is the crew's own model from their stopwatch data, not a law. It only describes their plank.

SecondsSteep plank (centimeters)
00
15
220
345
480
COMPARE THE RUNS
  • Read the question.
  • Tap your answer.
Run B went from 10 at 0 seconds to 170 at 4 seconds. What is its rate in centimeters per second?
Run C passed (1, 50) and (3, 150). Where was it at t = 0?
Run C gains 50 centimeters a second from 0. Which equation is its line?
Run B passed (0, 10) and (2, 90). Which equation fits run B?
AVERAGE RATE OF CHANGE
  • Read the question.
  • Tap your answer.
The steep plank follows f(t) = 5t². What is the average rate of change from t = 1 to t = 3?
For f(t) = 5t², what is the average rate of change from t = 0 to t = 4?
Run A follows d = 30t + 10. What is its average rate of change from t = 1 to t = 4?
Run B follows d = 40t + 10. How many centimeters a second does it gain? Type the number.
WHY THIS EXERCISEThe push changed only the rate, so only the number in front of t changed.
Run C, d = 50t: the distance at 4 seconds, in centimeters.
Steep plank, 0, 5, 20, 45, 80: the average rate from 2 to 4 seconds, in centimeters per second.
StatementTrue or false?
Runs A and B have the same y-intercept.?
Run C has a larger rate of change than run B.?
The steep-plank readings have equal steps, so they make a line.?
The point (3, 130) is on the graph of y = 40t + 10.?
The point (2, 100) is on the graph of y = 50t.?
45 - 5 = 20 × 2?
WHY THIS EXERCISEThree rules, one shape: changing the rate tilts the line, changing the start slides it up or down.
Draw the steep-plank dots, seconds across and centimeters up. Join (1, 5) to (3, 45) with a ruler and label its slope.

Sharp reading, engineer. Tomorrow you find lines in everyday life and review the week.

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