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Lesson plan Grade 10 Math

Duskmoor Pond Algae Bloom: Linear vs. Exponential Growth Lesson

A grade 10 lesson comparing linear phosphorus loading to exponential algae-coverage growth in a fictional pond, using explicit formulas to distinguish the two models.

Objectives

  1. Students will distinguish linear growth from exponential growth using two datasets from the same pond.
  2. Students will write explicit formulas for a constant weekly increase and a weekly doubling pattern.
  3. Students will explain why an exponential growth model eventually produces an unrealistic result for a bounded real-world quantity.

Vocabulary

Key words: linear growth, exponential growth, explicit formula, rate of change

Teacher overview

Nutrient runoff, especially phosphorus and nitrogen from fertilized land, can trigger algae blooms in ponds and reservoirs, and the two processes often grow at very different rates. This lesson uses a fictional pond to compare a phosphorus load that grows by the same amount every week (linear growth) with an algae bloom that grows by the same factor every week (exponential growth). A common misconception is that both patterns "grow fast," when in fact linear growth adds a constant amount while exponential growth compounds, producing dramatically larger results after only a few more weeks, a distinction this lesson is built to make visible.

Procedure

Hook (5 min): Show two unlabeled tables side by side: one where a quantity increases by 5 each week (20, 25, 30, 35...) and one where a quantity doubles each week (4, 8, 16, 32...). Ask students to predict each table's next two values and to describe, in their own words, how each pattern is growing.

Direct instruction (15 min): Reveal that both tables come from Duskmoor Pond (a fictional pond used for this lesson). The first table tracks the total pounds of phosphorus in the pond, starting at 20 pounds already present and adding 5 pounds every week from a nearby drainage ditch: P(w) = 20 + 5w. The second table tracks the percent of the pond's surface covered by algae, starting at 4 percent and doubling every week: A(w) = 4 × 2^w. Define linear growth as adding the same amount each interval and exponential growth as multiplying by the same factor each interval, and connect each definition to its formula.

Guided practice (15 min): In pairs, students use both formulas to extend each table through week 6, then calculate the average rate of change of the algae coverage between consecutive weeks: 1. Extend the phosphorus formula to find P(6). 2. Extend the algae formula to find A(6). 3. What is the average rate of change of algae coverage between week 3 and week 4? 4. What is the average rate of change of algae coverage between week 4 and week 5, and how does that rate compare with the phosphorus model's constant rate of 5 pounds per week?

Independent practice (5 min): Students answer question 5: using A(w) = 4 × 2^w, what does the formula predict for week 5, and why is that result not physically possible for a real pond?

Closure / exit ticket (5 min): Students answer one question (6) on an index card: "A pond's algae coverage is 64 percent in week 4. If it keeps doubling, what does the model predict for week 5? Is a real pond likely to reach that value? Explain in one sentence." Collect and scan for whether students correctly compute 128 percent and recognize it as impossible 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

  1. Answer: 50 lb.
    Extending the phosphorus formula to week 6: P(6) = 20 + 5(6) = 20 + 30 = 50 pounds.

  2. Answer: 256 percent.
    Extending the algae formula to week 6: A(6) = 4 × 2^6 = 4 × 64 = 256 percent, which is impossible for a real surface and shows the model breaking down.

  3. Answer: 32 percentage points per week.
    Between weeks 3 and 4, algae coverage rises from 32% to 64%, a change of 32 points per week (A(4) − A(3) = 64 − 32).

  4. Answer: 64 percentage points per week.
    Between weeks 4 and 5, algae coverage rises from 64% to 128%, a change of 64 points per week (A(5) − A(4) = 128 − 64). The exponential model's rate of change increases week over week, while the phosphorus model's rate of change stays fixed at 5 pounds per week throughout.

  5. Answer: 128 percent, which is not possible.
    A(5) = 4 × 2^5 = 4 × 32 = 128 percent. A real pond cannot have more than 100 percent of its surface covered, so the exponential model has already become unrealistic by week 5.

  6. Exit ticket answer: 128 percent; not likely reached in reality.
    Doubling from 64% predicts 128%, which exceeds the maximum possible coverage of 100%. Real ponds run out of open water surface and available nutrients, so growth slows well before doubling could continue unchecked.

Differentiation

Support (below grade level)

Provide a partially completed table for both formulas through week 4, and have students verify the given values before extending the tables themselves, so the pattern (add 5, or multiply by 2) is visible before they compute new rows.

Extension (above grade level)

Ask students to find the week at which the algae model (exponential) first produces a larger value than the phosphorus model (linear), given that the algae model starts smaller, and to explain why the exponential model eventually overtakes the linear one even though it started lower.

ELL supports

Pre-teach "constant" and "doubling" using physical objects: add 5 blocks to a pile each round for linear growth, and double a stack of blocks each round for exponential growth, before introducing the formulas in symbolic form.