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Algebra 2 9-12 / Week 08 / Wednesday
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Week 08 · The Function Toolkit

Wednesday

Glass House Lab: the filling jug
// Build, move, compare, cross
⏱ about 20 min

Wednesday: Glass House Lab, the Filling Jug

Two clear jugs sit on the potting bench, one straight-sided and one wide at the bottom. Comet holds a measuring cup.

"One cup every pour," Wren says, tape measure against the glass. "I read the height after each. What do you notice?"

The straight jug climbs 2 centimeters a cup, every cup. The wide jug starts slow and speeds up.

"Same water, different jugs," Comet says. "The straight one is a line. The wide one bends."

Nova projects both tables side by side. "Would you like a hint? Divide the change in height by the cups poured."

"Average rate of change," Wren says. "From cup 2 to cup 4 the wide jug rises 2.25 a cup."

"And from cup 4 to cup 6 it rises faster," Comet says. "The rate itself is changing."

What you need

  • Two clear jugs or jars of different shapes, a measuring cup and a tape measure or ruler.
  • Water, a towel, and your Glass House Log with two tables ready.
  • A steady table. Nothing is heated, and the jugs stay on the table while you measure.
Safety first
Pour slowly and keep the jugs on the table. Nothing heavy is lifted alone.
Wipe up any spill at once so no one slips. Water only, nothing hot.
If a jug is glass, a grown-up handles it and you read the tape.

Run the lab

  1. Mark the starting height of each empty jug and write it in the cup 0 row.
  2. Pour one measuring cup into the first jug. Read the height and record it.
  3. Repeat until six cups are in. Then do the same for the second jug.
  4. For each jug, find the average rate of change on cups 0 to 2, 2 to 4 and 4 to 6.
  5. Write a rule for the straight jug: height = rate × cups + start.
  6. Compare the rule with the wide jug's table. Where is the wide jug ahead, and where behind?

The crew's readings

The crew's heights are made up for the Glass House. Yours will differ, and that is the point of a lab.

Cups pouredStraight jug (cm)Wide jug (cm)
011
132
253.5
375.5
498
51111
61314.5

Average rate of change

Average rate of change over an interval is the change in output divided by the change in input.

Straight jug, cups 0 to 6: (13 - 1) ÷ 6 = 2 cm per cup. The same on every interval, so its rule is a line.

Wide jug, cups 0 to 2: 1.25 cm per cup. Cups 4 to 6: 3.25 cm per cup. The rate grows, so the graph bends upward.

IntervalStraight jug rate (cm per cup)Wide jug rate (cm per cup)
cups 0 to 221.25
cups 2 to 422.25
cups 4 to 623.25

The straight jug follows the rule h(c) = 2c + 1 exactly. Check: h(4) = 9, and the table says 9.

Compare the two jugs at cup 3: the rule gives 7 and the wide jug's table gives 5.5. The straight jug is ahead.

RATES FROM THE LAB
  • Read the question.
  • Tap your answer.
The straight jug follows h(c) = 2c + 1. What is its average rate of change from c = 0 to c = 6?
The wide jug reads 3.5 cm at cup 2 and 8 cm at cup 4. What is its average rate of change, in cm per cup?
The wide jug reads 8 cm at cup 4 and 14.5 cm at cup 6. What is its average rate of change, in cm per cup?
The straight jug follows f(c) = 2c + 1. The wide jug's table reads g(4) = 8, g(5) = 11, g(6) = 14.5. At c = 5, which is taller?
The straight jug rule is h(c) = 2c + 1. What is its height after 6 cups? Type the number.
WHY THIS EXERCISEA rule and a table are two ways to give the same function. The rule predicts the row you have not poured yet.
StatementTrue or false?
A line has the same average rate of change on every interval.?
(13 - 1) ÷ 6 = 2?
The wide jug's rate of change is the same on every interval.?
(8 - 3.5) ÷ 2 = 2.25?
Average rate of change is the change in input divided by the change in output.?
WHY THIS EXERCISEAverage rate of change is the slope idea stretched to any function, line or not.
Draw both jugs' tables as dots on one grid, cups across and height up. Join each set with a smooth line.

Careful measuring. Tomorrow a line meets a parabola on the wall, and the crew finds exactly where.

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