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Lesson plan Grade 12 Math

Merristone Water Demand: Linear Versus Exponential Growth

A grade 12 math lesson where students build linear and exponential population models from two data points, project water demand, and test it against a plant's capacity.

Objectives

  1. Students will construct linear and exponential population models from two data points.
  2. Students will project water demand from each model and compare it to a plant's permitted capacity.

Vocabulary

Key words: linear growth, exponential growth, growth factor, per capita use, capacity

Teacher overview

A town that plans its water supply must first guess how many people it will serve, and the shape of that guess matters: a straight-line forecast and a compounding forecast pull apart over twenty years. This lesson has students build both models for the fictional town of Merristone from the same two data points, then turn each population into a water demand and test it against a treatment plant's permit. A common misconception to correct is assuming linear and exponential models that share two points stay close forever; they match only at those points and diverge everywhere else, which is exactly what makes the modeling choice a real decision.

Procedure

Hook (5 min): Show the two Merristone data points, a fictional town of 24,000 people that grew to 30,000 over ten years, and ask students to predict the population twenty years out. Record a few predictions and note that reasonable students disagree.

Direct instruction and model (13 min): Review that linear functions grow by equal differences over equal intervals and exponential functions grow by equal factors. Model, using a different set of numbers than the handout, how to build each function from two input-output pairs: a linear slope from the difference over the time span, and an exponential factor as the ratio raised to the reciprocal of the span. Emphasize that both models can pass through the same two points.

Guided practice (12 min): With the class, set up the linear slope for Merristone from 24,000 to 30,000 over ten years, and confirm the exponential ten-year factor is the ratio 30,000 over 24,000. Check for understanding by asking students to show thumbs up if the exponential model will predict more people at year 20 than the linear one and thumbs down if it will predict fewer; re-teach if fewer than about 80 percent are correct.

Independent practice (18 min): Students complete the Student Handout, constructing both models, projecting to year 20, converting population to water demand, and comparing demand to the plant's permitted capacity. Circulate and check that students raise the yearly factor to the correct power when projecting.

Closure and exit ticket (7 min): Each student writes one sentence answering: "For sizing a water plant twenty years out, would you trust the linear or the exponential forecast, and why?" Collect the responses and check that at least 80 percent justify their choice with the models' behavior before dismissal.

Merristone Water Demand: Linear Versus Exponential Growth — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

The town of Merristone is fictional. It had 24,000 people in year 0 and 30,000 people in year 10. Plan water use at 90 gallons per person per day, and take the treatment plant's permitted capacity as 3.3 MGD.

  1. Find the linear model's annual growth in people per year, using the two data points (24,000 at year 0 and 30,000 at year 10).
  2. Write the linear model P(t) equals 24,000 plus your slope times t, and use it to project the population at year 20.
  3. For the exponential model P(t) equals 24,000 times r to the t, find the yearly growth factor r, using r equals (30,000 divided by 24,000) raised to the one-tenth power. Round to four decimal places.
  4. Use the exponential model to project the population at year 20. (Hint: r to the 20th power is the square of r to the 10th power.)
  5. How many more people does the exponential model predict at year 20 than the linear model?
  6. Using the linear year-20 population and 90 gallons per person per day, find the projected daily demand in gallons per day.
  7. Convert your demand from question 6 to MGD.
  8. Using the exponential year-20 population and 90 gallons per person per day, find the projected demand in MGD. Round to the nearest hundredth.
  9. Under each model, does the year-20 demand stay under the plant's 3.3 MGD permitted capacity? State the margin for each.
  10. Explain which model a cautious water planner should use to size supply, and why.

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: 600 people per year. The slope is (30,000 minus 24,000) divided by 10 years, which is 6,000 divided by 10, or 600 people per year.
  2. Answer: 36,000 people. P(20) equals 24,000 plus 600 times 20, which is 24,000 plus 12,000, or 36,000 people.
  3. Answer: 1.0226. The factor r equals (30,000 divided by 24,000) to the one-tenth power, which is 1.25 to the 0.1 power, about 1.0226, roughly 2.26 percent growth per year.
  4. Answer: 37,500 people. Because r to the 10th power is 1.25, r to the 20th power is 1.25 squared, or 1.5625; P(20) equals 24,000 times 1.5625, which is 37,500 people.
  5. Answer: 1,500 people. The exponential model predicts 37,500 minus 36,000, which is 1,500 more people at year 20.
  6. Answer: 3,240,000 gallons per day. The linear year-20 population of 36,000 times 90 gallons per person per day equals 3,240,000 gallons per day.
  7. Answer: 3.24 MGD. 3,240,000 gallons per day divided by 1,000,000 equals 3.24 MGD.
  8. Answer: 3.38 MGD. The exponential population of 37,500 times 90 is 3,375,000 gallons per day, or 3.375, which rounds to 3.38 MGD.
  9. Answer: Linear yes, exponential no. The linear demand of 3.24 MGD is under the 3.3 MGD limit by 0.06 MGD, but the exponential demand of 3.38 MGD is over the limit by about 0.08 MGD.
  10. Answer: The exponential model is the safer choice for sizing supply. It predicts higher demand because growth compounds, so planning to it leaves margin if growth is faster than a straight line; sizing to the linear model risks a plant that is already over its permit by year 20.

Differentiation

Support

Provide the exponential factor from question 3 already computed so students who struggle with fractional exponents can still complete the projections. Give a filled-in table of both models at years 0, 10, and 20 for the fictional town to anchor the comparison.

Extension

Ask students to find the year in which the exponential model first pushes Merristone's demand past the 3.3 MGD permit, and to propose one demand-management step that could delay reaching that capacity.

ELL supports

Pre-teach "linear," "exponential," "factor," and "capacity" with a two-line graph showing a straight line and a curve crossing the same two points. Provide the sentence frame "The exponential model predicts more people because it grows by a ___ each year" for question 10.

Extensions

Have students pull a real population figure for their town from two census years, build both models, and project water demand at 90 gallons per person per day, then compare the spread between the models with the fictional Merristone result.