Lesson plan Grade 9 Math
Grayhaven Reservoir: Linear Leak Loss Versus Exponential Algae Growth
A grade 9 lesson comparing a reservoir's constant-rate leak loss to a warm cove's percent-based algae growth, distinguishing linear from exponential models.
Objectives
- Students will distinguish a linear growth pattern from an exponential growth pattern using a table of values.
- Students will construct a linear function and an exponential function from a table's starting value and constant rate or factor.
Vocabulary
Key words: linear function, exponential function, growth factor, constant rate
Teacher overview
At the fictional Grayhaven Reservoir, two very different situations are unfolding at the same time. A small pipe leak on the reservoir's outflow line loses the same number of gallons every week, a steady, predictable pattern. Meanwhile, in a warm, sheltered cove of the same reservoir, an algae population is growing by the same percentage every week during a summer bloom, a pattern that accelerates as the population gets larger. This lesson uses both situations side by side so students can see, in a single water-system setting, the difference between a quantity that grows by equal differences and one that grows by equal factors.
A common misconception is expecting both patterns to "look the same" early on, since a slow week of algae growth can resemble the leak's steady loss for the first week or two. The lesson is built to correct this directly by having students extend both tables far enough to see the algae bloom's growth accelerate while the leak's loss stays perfectly steady.
Procedure
Hook (5 min): Present the fictional Grayhaven Reservoir's two tables on the board, showing only Weeks 0 through 2 of each:
| Week | Leak loss (gal) | Algae index |
|---|---|---|
| 0 | 0 | 100 |
| 1 | 40 | 125 |
| 2 | 80 | 156.25 |
Ask students what pattern they notice in each column and whether the two columns are growing "the same way."
Direct instruction (12 min): Extend the leak column by hand, showing that it always adds exactly 40 gallons per week (0, 40, 80, 120, 160...), an equal-difference pattern. Extend the algae column, showing that each week's value is 1.25 times the week before (100, 125, 156.25, 195.31...), an equal-factor pattern. Define linear growth as growth by equal differences over equal intervals, and exponential growth as growth by equal factors over equal intervals. Write the leak's equation, y = 40x, and the algae bloom's equation, y = 100(1.25)^x, showing how each equation comes directly from the table: 40 is the constant weekly difference, 100 is the Week 0 starting value, and 1.25 is the constant weekly growth factor.
Guided practice (15 min): In pairs, students extend both tables through Week 6, then answer: which pattern shows equal differences, and which shows equal factors? Students check their extended tables against a partner's and correct any arithmetic errors before moving on.
Independent practice (8 min): Each student answers two questions individually.
- Using the leak's equation, how many gallons will the leak have lost after 12 weeks if it goes unnoticed?
- Using the algae bloom's equation, what is the algae index after 8 weeks? Round to the nearest whole number.
Closure / exit ticket (5 min): Students answer one question on an index card: "A second leak at Grayhaven starts at Week 0 and loses 60 gallons every week instead of 40. Write this leak's equation, and state how many total gallons it loses over a 10-week billing period." Collect and scan for a correctly written linear equation before dismissal.
From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.
Answer Key
Answer: 480 gal.
Using y = 40x with x = 12 weeks: 40 × 12 = 480 gallons lost.Answer: 596 (algae index).
Using y = 100(1.25)^8: 100 × 1.25^8 ≈ 100 × 5.9605 ≈ 596.05, which rounds to 596.Exit ticket answer: y = 60x; 600 gal over 10 weeks.
The second leak's constant weekly loss of 60 gallons gives the linear equation y = 60x. Over 10 weeks: 60 × 10 = 600 gallons lost in total.
Differentiation
Support
Provide the extended tables through Week 6 already filled in for both the leak and the algae bloom, so students can focus guided practice on identifying which pattern is which and writing the equations rather than on repeated arithmetic. Highlight the constant 40 and the constant 1.25 directly on the printed table.
Extension
Ask students to find the week at which the algae index would first exceed 1,000, using their equation from direct instruction and either repeated calculation or systematic guess-and-check, and to explain why this "crossing point" happens abruptly rather than gradually once exponential growth accelerates.
ELL supports
Pre-teach "constant rate" and "growth factor" using a simple visual: a line of evenly spaced dots for the leak (equal jumps) next to a curve of dots that spread farther apart each week for the algae (equal multiplying). Provide the equation templates "y = ___x" and "y = ()^x" with blanks for students to fill in from the table rather than deriving from scratch.