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Algebra 1 9-12 / Week 05 / Thursday
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Week 05 · Linear Models and Sequences

Thursday

The crew's Loft data
// The cup tower and the filling jug
⏱ about 20 min

Thursday: The Crew's Loft Data

Wren unrolls a long string of paper flags across the bench. "For the fair booth. I taped one every 4 centimeters."

Comet measures. "The first flag is at 5 centimeters, not zero. Then 9, 13, 17."

"A sequence," Wren says. "What do you notice about term 1 and the step?"

"Start 5, add 4. So flag n is at 4n + 1." She checks flag 3: 13. "Yes."

Nova projects a second table beside it. "Would you like a hint? This pitcher started with more water and filled faster."

"100 at the start, up 60 each minute," Wren reads. "P = 60t + 100."

"Two rules, two stories," Comet says. "Our engineer, read the tables and say what each number means."

The crew's Thursday tables

All of these are the crew's own made-up measurements from the Loft, in centimeters and milliliters.

Flag number (n)Position (centimeters)
15
29
313
417
521
625
Minutes (t)Jug A (milliliters)Pitcher B (milliliters)
0200100
1240160
2280220
3320280
4360340

Flags: a(n) = 4n + 1, or a(1) = 5, a(n) = a(n - 1) + 4. The 4 is the spacing, the 1 is the offset of the first flag.

Pitcher B: P = 60t + 100. Its slope 60 beats jug A's 40, so B fills faster. Its intercept 100 is below A's 200.

A grid from 0 to 20 with two rising lines for jug A and pitcher B, drawn in tens of milliliters.

The graph is drawn in tens of milliliters so both lines fit. The steeper line is the faster fill.

READ THE FLAG STRING
  • Read the question.
  • Tap your answer.
Flags sit at 5, 9, 13, 17, ... centimeters. Where is flag 15?
The flag positions are 5, 9, 13, 17, ... Which explicit rule gives flag n?
A flag sits at 41 centimeters. Using 5, 9, 13, 17, ... which flag number is it?
READ THE PITCHERS
  • Read the question.
  • Tap your answer.
Pitcher B read 100 at 0 minutes and 220 at 2 minutes. Which rule fits?
Pitcher B read 160 at 1 minute and 340 at 4 minutes. What is its rate?
Jug A is V = 40t + 200 and pitcher B is P = 60t + 100. Which fills faster?
The flag rule is a(n) = 4n + 1. How many centimeters apart are neighboring flags? Type the number.
WHY THIS EXERCISEIn a sequence rule, the coefficient of n is the common difference.
Flag string, a(n) = 4n + 1: the position of flag 10, in centimeters.
Pitcher B, P = 60t + 100: the milliliters at 6 minutes.
Jug A, V = 40t + 200: the milliliters at t = 0.
StatementTrue or false?
Pitcher B starts with more water than jug A.?
Pitcher B fills faster than jug A.?
In the sequence starting 5 and adding 4, term 7 is 29.?
In the sequence starting 5 and adding 4, term 12 is 50.?
The flag string is an arithmetic sequence.?
60 × 4 + 100 = 340?
WHY THIS EXERCISETwo rules side by side show what the slope and the intercept each control.
WHICH NUMBER CHANGED?
  • Read the question.
  • Tap your answer.
Jug A is V = 40t + 200. Pitcher B is P = 60t + 100. Which is true?
The flags are at 5, 9, 13, 17, ... If Wren taped them every 6 centimeters instead, what changes in a(n) = 4n + 1?
Draw the flag string as dots, flag number across and centimeters up. Label the axes and the scale.

Sharp reading, engineer. Tomorrow you find sequences and linear models in everyday life and review the week.

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