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Math problem set Grade 9 Science

Quill Lake's Algal Bloom: Exponential Growth and Linear Cleanup

A grade 9 math problem set modeling a fictional lake's algal bloom, contrasting exponential growth before treatment with linear decline after treatment starts.

Objectives

  1. Students will build a table and an equation modeling exponential growth from a doubling pattern.
  2. Students will build a linear equation modeling a constant rate of decline and solve it for a target value.
  3. Students will distinguish exponential growth from linear decline using the same dataset.

Vocabulary

Algal bloom
An algal bloom is a fast, thick growth of algae that can turn a lake green and scummy. Blooms happen when the water gets lots of nutrients and warm sunshine, and some blooms make toxins.

Also: exponential growth, linear model, average rate of change, algaecide

Teacher overview

Quill Lake (a fictional lake used for this problem set) develops a summer algal bloom fed by nutrients washing in from surrounding farmland. Left untreated, the bloom's surface area doubles every four days, a classic exponential growth pattern. Once county workers apply an algaecide treatment, the bloom instead shrinks by the same fixed number of acres every day, a linear pattern. This problem set has students build both models from one continuous dataset and use them to answer real management questions.

A common misconception in this math is assuming that any quantity which is increasing or decreasing must automatically be linear, since a growth pattern (doubling) and a decline pattern (subtracting a constant) can look superficially similar in a word problem. The key test is whether the change from one interval to the next is a constant ratio (exponential) or a constant difference (linear), and this set is built so students calculate both kinds of change directly from the same lake.

Quill Lake's Algal Bloom: Exponential Growth and Linear Cleanup — Grade 9 Science
Name: ____________________________ Date: ______________

Problems

On day 0, algae covers 2 acres of Quill Lake's surface. Left untreated, the bloom's area doubles every 4 days. Quill Lake's total surface area is 100 acres.

  1. Complete a table of the bloom's area, in acres, on days 0, 4, 8, and 12, assuming the doubling pattern holds and no treatment is applied.
  2. Write an exponential equation, A(t) = 2 × 2^(t/4), giving the bloom's area in acres on day t. Use your equation to confirm it gives the same value your table shows for day 12.
  3. Extend your table two more doubling periods, to days 16 and 20. Based on your extended table, on which of those two days does the bloom's area first exceed 50 acres, half of the lake's total surface?
  4. On day 12, county workers apply an algaecide treatment. From that point on, instead of doubling, the bloom shrinks by a constant 3 acres per day. Write a linear equation, B(d) = 16 − 3d, where d counts the number of days since day 12 (so d = 0 is day 12 itself, when the bloom measured 16 acres).
  5. Solve your equation from item 4 for the value of d where B(d) = 0. Convert that value of d back into a day number (day 12 + d), and state on which day the model predicts the bloom is fully cleared.
  6. Compare the bloom's growth from day 0 to day 12 with its decline from day 12 onward. Which period is exponential, and which is linear? Explain how you can tell using the pattern of change in each period (a constant ratio versus a constant difference).
  7. Calculate the average rate of change in the bloom's area from day 0 (2 acres) to day 12 (16 acres), in acres per day. Use: average rate of change = (change in area) ÷ (change in days).
  8. Challenge. Suppose county workers had instead started the same treatment (losing 3 acres per day) on day 8, when the bloom measured 8 acres (from your table in item 1). Write and solve a new linear equation for this earlier start date, and state on which day the model now predicts full clearance. Compare this day to your answer in item 5 and explain why treating earlier made a difference.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

  1. Answer: 2, 4, 8, 16 acres.
    Starting at 2 acres and doubling every 4 days: day 0 = 2, day 4 = 4, day 8 = 8, day 12 = 16 acres.

  2. Answer: 16 acres.
    A(12) = 2 × 2^(12/4) = 2 × 2^3 = 2 × 8 = 16 acres, matching the table in item 1.

  3. Answer: day 20.
    Extending the table: day 16 = 32 acres (2 × 2^4), day 20 = 64 acres (2 × 2^5). Day 16's value (32 acres) is still below 50 acres, so day 20 (64 acres) is the first day the bloom exceeds 50 acres.

  4. Answer: B(d) = 16 − 3d.
    The bloom starts at 16 acres on day 12 (d = 0) and loses 3 acres for every day, d, that passes, giving the linear equation B(d) = 16 − 3d.

  5. Answer: about day 17.
    Setting B(d) = 0: 0 = 16 − 3d, so 3d = 16, and d = 16 ÷ 3 ≈ 5.3 days after day 12. Day 12 + 5.3 ≈ day 17.3, so the model predicts the bloom is essentially cleared partway through day 17, with day 18 the first full day at 0 acres.

  6. Answer: exponential then linear.
    From day 0 to day 12, the bloom's area multiplies by the same factor (2) every 4-day interval, a constant ratio, which is exponential growth. From day 12 onward, the bloom loses the same fixed amount (3 acres) every day, a constant difference, which is linear decline.

  7. Answer: about 1.17 acres/day.
    Average rate of change = (16 − 2) ÷ (12 − 0) = 14 ÷ 12 ≈ 1.17 acres per day.

  8. Answer: about day 11.
    Starting from 8 acres on day 8, the new equation is C(d) = 8 − 3d, where d counts days since day 8. Setting C(d) = 0: 0 = 8 − 3d, so d = 8 ÷ 3 ≈ 2.7 days after day 8, giving roughly day 10.7, or day 11 for full clearance. This is about six days earlier than the day-17 clearance in item 5, showing that treating the bloom while it was still small let the same daily removal rate finish the job much sooner.

Differentiation

Support (below grade level)

Provide the table from item 1 already completed through day 12, and give the linear equation in item 4 already written, so students only need to substitute and solve for items 5 through 8.

Extension (above grade level)

Ask students to find the day (using the exponential equation from item 2, not the extended table) on which the untreated bloom would have first exceeded 90 acres, by testing values of t and comparing results, and to explain why this method works without formally solving for t using logarithms.

ELL supports

Pre-teach "exponential," "linear," and "average rate of change" using a simple pair of diagrams: a bar that doubles in height at each step (exponential) beside a bar that shrinks by the same amount at each step (linear), labeled in English and students' home languages where possible. Allow item 6 to be answered with the two diagrams plus a short phrase instead of a full paragraph explanation.