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Algebra 1 9-12 / Week 05 / Tuesday
2/6
Week 05 · Linear Models and Sequences

Tuesday

Rules from two readings
// The cup tower and the filling jug
⏱ about 20 min

Tuesday: Rules From Two Readings

Comet props the clear jug under a tube and opens the clip. Water drips in. "Timing from now."

Wren reads the side at 2 minutes. "280 milliliters." At 5 minutes: "400."

"We missed the start," Comet says. "Can we still get the rule? What can we make from two readings?"

"Two points," Wren says. "(2, 280) and (5, 400). What do you notice? 120 milliliters in 3 minutes."

"Forty a minute." Comet counts back. "Two minutes before the first reading takes away 80. So it started at 200."

Nova projects the line through both points. "Would you like a hint? Say what the 40 and the 200 mean."

"Forty is the drip rate. Two hundred was already in the jug," Wren says. "V = 40t + 200."

"Our engineer, build the next rule from the table," Comet says.

From two input-output pairs

  1. Write the two pairs as points (input, output).
  2. Rate: change in output divided by change in input.
  3. Start: go back from one point to input 0 using the rate, or solve b = output - rate × input.
  4. Write y = rate × input + start, and check both points in it.

Jug: (2, 280) and (5, 400). Rate 40 milliliters a minute. Start 200. So V = 40t + 200.

The slope is the rate the jug fills. The intercept is how much water was there when the timing began.

A scatter plot of the jug, minutes across and milliliters up, with six points on one straight line.

From a table

A table is linear when equal steps in the input give equal steps in the output. Check the differences first.

If they are all the same, that difference is the rate. Then find the start the same way as before.

If the differences change, the table is not linear, and a line would be the wrong model.

Minutes (t)Jug A (milliliters)Jug B (milliliters)Differences, A
0200150
1240170+40
2280210+40
3320270+40
4360350+40

Two ways to the same rule

Way one: rate first, then step back to the start. Way two: write y = mx + b, put in the rate and one point, solve for b.

Jug, way two: 280 = 40 × 2 + b, so b = 200. Same rule. Use whichever feels clearer, and check with the other point.

BUILD THE RULE
  • Read the question.
  • Tap your answer.
The jug read 280 milliliters at 2 minutes and 400 at 5 minutes. Which rule fits?
From (2, 280) to (5, 400), what is the fill rate in milliliters per minute?
The jug passed (2, 280) and (5, 400). How much water was in it at t = 0?
IS THE TABLE LINEAR?
  • Read the question.
  • Tap your answer.
Jug A reads 200, 240, 280, 320 at minutes 0 to 3. Do these points lie on one straight line?
Jug B reads 150, 170, 210, 270 at minutes 0 to 3. Do these points lie on one straight line?
A table gives (1, 9) and (4, 21) with equal steps between. Which rule fits?
WHAT THE NUMBERS MEAN
  • Read the question.
  • Tap your answer.
In V = 40t + 200 for the jug, what does the 40 mean?
In V = 40t + 200, what does the 200 mean?
If the drip keeps going, V = 40t + 200. How many milliliters at 8 minutes?
BUILD A RULE FROM (2, 280) AND (5, 400)
  • ?Step back to t = 0: 280 - 2 × 40 = 200
  • ?Find the rate: (400 - 280) ÷ (5 - 2) = 40
  • ?Check the second point: 40 × 5 + 200 = 400
  • ?Write the rule: V = 40t + 200
WHY THIS EXERCISETwo readings are enough for a line because a line has one rate everywhere.
StatementTrue or false?
Two input-output pairs are enough to find a linear rule.?
In V = 40t + 200, the jug held 40 milliliters at the start.?
Equal input steps with equal output differences mean the table is linear.?
Jug B, 150, 170, 210, 270, is linear.?
40 × 5 + 200 = 400?
(400 - 280) ÷ (5 - 2) = 40?
WHY THIS EXERCISEKnowing what each number means keeps you from mixing up the rate and the start.
Try it
Pick any two rows from the jug A table and rebuild the rule. Do you get V = 40t + 200 again?
Now try two rows from jug B. Why does the rule fail to match the other rows?

Excellent modeling. Tomorrow is Loft Lab: you stack the tower and time the jug yourself.

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