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

Friday

Models all around
// The cup tower and the filling jug
⏱ about 20 min

Friday: Models All Around

Comet flops onto the Loft couch with a notebook. "Sequences are everywhere. Stairs: each step 18 centimeters higher."

"Step n is at 18n centimeters," Wren says. "An arithmetic sequence with no offset."

"And my reading plan: 20 pages a night from page 35," Comet says. "Page 20n + 35 after n nights."

Nova projects a strip of paper folding in half, again and again. "Would you like a hint? Count the layers."

"One, two, four, eight," Comet says. "That doubles. It is not adding the same number."

"A geometric sequence," Wren says. "Multiply each time. That is week 7. What do you notice about today?"

"Adding makes lines. Multiplying makes something steeper." Comet grins. "Our engineer, review first. Then systems next week."

Linear models in a day

  • Stairs: step n is at 18n centimeters. Rate 18, start 0.
  • A reading plan: 20 pages a night from page 35. After n nights you are at page 20n + 35.
  • A jug filling at a steady drip: milliliters = rate × minutes + start.
  • A cup tower: cups = cups per row × rows + the extras on top.

Arithmetic or geometric?

An arithmetic sequence adds the same number each time: 4, 7, 10, 13. Its graph is dots on a line.

A geometric sequence multiplies by the same number each time: 1, 2, 4, 8. Its dots curve upward.

Check the differences. If they are equal, it is arithmetic. If the ratios are equal instead, it is geometric.

SequenceDifferencesRatiosKind
4, 7, 10, 133, 3, 3not equalarithmetic
1, 2, 4, 81, 2, 42, 2, 2geometric
18, 36, 54, 7218, 18, 18not equalarithmetic

The week in five lines

  • A linear model is y = rate × input + start. Name the input and output first.
  • Two input-output pairs are enough: rate from the change, start by stepping back to input 0.
  • A table is linear when equal input steps give equal output differences.
  • An arithmetic sequence is a linear function with whole-number inputs. Recursive: a(n) = a(n - 1) + d. Explicit: a(n) = dn + (a(1) - d).
  • The slope means the rate in the story. The intercept means the start.
EVERYDAY MODELS
  • Read the question.
  • Tap your answer.
Each stair step is 18 centimeters higher, starting at 18. How high is step 12?
Comet starts a book at page 35 and reads 20 pages a night. What page after night 6?
Comet reads 20 pages a night starting from page 35. Which rule gives her page after n nights?
A strip starts as 1 layer and doubles each fold. How many layers after 5 folds (term 6)?
MIXED REVIEW
  • Read the question.
  • Tap your answer.
The stairs sequence is 18, 36, 54, 72, ... Which explicit rule gives step n?
The flag string is 5, 9, 13, 17, ... Which recursive rule fits?
A table gives (1, 11) and (3, 19). Which rule fits the line?
A table gives 2, 4, 8 at inputs 1, 2, 3. Do these points lie on one straight line?
StatementTrue or false?
An arithmetic sequence adds the same number each time.?
1, 2, 4, 8 is an arithmetic sequence.?
The domain of a sequence is the whole numbers 1, 2, 3 and so on.?
In y = 20n + 35, the 35 is the rate.?
In the sequence starting 18 and adding 18, term 5 is 90.?
20 × 6 + 35 = 155?
WHY THIS EXERCISEThese lines sum up the week. Next week two lines meet, and you solve for the meeting point.
Try it
Find a staircase at home. Measure one step's height and write the sequence of heights for steps 1 to 5.
Measure with a tape measure while standing on the floor, never leaning over the stairs.
Draw 4, 7, 10, 13 and 1, 2, 4, 8 as dots on two small grids. Label which is arithmetic and which is geometric.

Terrific week. You can build a rule from a story, from two readings or from a table. Tomorrow is Loft Day.

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