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Lesson plan Grade 9 Math

Ashcombe's Growing Water Demand: Choosing the Right Quantity to Model

A grade 9 lesson where students build a linear function from two years of population and water-demand data and decide which quantity best models growth.

Objectives

  1. Students will calculate gallons per person per day from population and total-demand data and evaluate whether it stays stable across two years.
  2. Students will construct a linear function relating population to total daily water demand from two data points.

Vocabulary

Key words: per-capita demand, linear function, slope, y-intercept

Teacher overview

The fictional suburb of Ashcombe has been growing steadily, and its water department has two years of records: population and total daily water demand for each year. Before building any model, planners have to decide which quantity actually tells the useful story: the total number of gallons used each day, or the number of gallons used per person each day. This lesson has students calculate both, notice which one stays more stable as the town grows, and then build a linear function that predicts future total demand from projected population.

A common misconception is assuming that a bigger total number is automatically more useful for planning. A rising total demand could simply mean a growing population using water at the same rate per person, which is a very different planning situation than a stable population suddenly using far more water per person. Comparing the per-person quantity across both years directly addresses this.

Procedure

Hook (5 min): Present the fictional suburb of Ashcombe's two years of data on the board:

  • Year A: population 8,000; total demand 760,000 gallons per day.
  • Year B: population 10,000; total demand 900,000 gallons per day.

Ask students which year "used more water" and whether that answer changes anything about how worried the water department should be.

Direct instruction (15 min): Calculate gallons per person per day (gpcd) for each year: Year A is 760,000 ÷ 8,000 = 95 gpcd, and Year B is 900,000 ÷ 10,000 = 90 gpcd. Point out that the per-person figure actually dropped slightly even though the total rose, meaning the growth in total demand is explained mostly by population growth, not by each person using more water. This is the kind of quantity choice HSN-Q.A.2 asks students to make: total gallons tells the department how large a system to build, while gallons per person tells them whether individual behavior is changing. Then model finding the linear function for total demand as a function of population, using the two data points (8,000, 760,000) and (10,000, 900,000): slope = (900,000 − 760,000) ÷ (10,000 − 8,000) = 140,000 ÷ 2,000 = 70 gallons per additional person, and y-intercept = 760,000 − 70(8,000) = 200,000, giving the equation D(p) = 70p + 200,000, where the 200,000-gallon intercept represents demand not tied directly to residents, such as schools, businesses, and system losses.

Guided practice (15 min): In pairs, students use D(p) = 70p + 200,000 to answer two questions for a population of 9,000: 1. What is the predicted total demand? 2. Dividing that answer by 9,000, what is that year's projected gpcd, and does it fall between the 90 and 95 gpcd already found for Years A and B?

Independent practice (5 min): Each student uses the equation to answer two questions for the water department's projected population of 13,000 people, several years in the future: 3. What is the predicted total demand? 4. What is the projected gpcd?

Closure / exit ticket (5 min): Students answer one question (5) on an index card: "A new industrial customer adds a flat 50,000 gallons per day of demand that does not depend on population. How would this change the y-intercept of Ashcombe's equation, and what would the new equation be?" Collect and scan for the correct new intercept 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: 830,000 gal/day.
    D(9,000) = 70(9,000) + 200,000 = 630,000 + 200,000 = 830,000 gallons per day.

  2. Answer: about 92.2 gpcd.
    830,000 ÷ 9,000 ≈ 92.2 gallons per person per day, which falls between the 90 and 95 gpcd found for Years A and B, as expected.

  3. Answer: 1,110,000 gal/day.
    D(13,000) = 70(13,000) + 200,000 = 910,000 + 200,000 = 1,110,000 gallons per day.

  4. Answer: about 85.4 gpcd.
    1,110,000 ÷ 13,000 ≈ 85.4 gallons per person per day.

  5. Exit ticket answer: new intercept 250,000; new equation D(p) = 70p + 250,000.
    A flat addition of 50,000 gallons per day that does not depend on population adds directly to the y-intercept, since the intercept represents the demand that exists even before any residents are counted: 200,000 + 50,000 = 250,000.

Differentiation

Support

Provide the gpcd calculations for Years A and B already completed, so guided practice can focus on building and using the linear function rather than repeating the per-capita division. Give the slope formula written out with the numbers already substituted, leaving only the final arithmetic for students to complete.

Extension

Ask students to determine, using only the data given in this lesson, what population Ashcombe would need to reach for its projected gpcd, using D(p) = 70p + 200,000, to drop to exactly 80 gallons per person per day, and to show the algebra that solves for that population.

ELL supports

Pre-teach "per-capita" using a simple picture: a large gallon jug labeled "total demand" being poured into many small cups labeled "per person." Provide the equation D(p) = 70p + 200,000 as a fill-in template where students substitute numbers into labeled blanks rather than building the equation from scratch during independent practice.