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Math problem set Grade 11 Math

Kelloway Water District Pay Steps and a Certification Payback

A grade 11 problem set comparing a fixed-step pay schedule with a percent-raise schedule at two fictional utilities, then pricing a license certification.

Objectives

  1. Students will construct a linear function and an exponential function from two described pay schedules.
  2. Students will compare the two schedules at six years and at twenty years and interpret the difference.
  3. Students will calculate how long a certification stipend takes to repay the cost of earning the license.

Vocabulary

Key words: step increase, percent raise, certification, stipend, median pay

Teacher overview

Utilities describe pay in two different ways, and the difference matters to a student deciding where to apply. Some post a step schedule that adds the same number of dollars each year, while others negotiate a percent raise that compounds. The Bureau of Labor Statistics reports a median of $60,020 per year for water and wastewater treatment plant and system operators as of May 2025, with about 10,000 openings projected each year, on average, from 2025 to 2035 for operators specifically. Total employment in the occupation is projected to decline 6 percent, so all of those openings come from replacing workers who retire or move to other occupations. This problem set has students build both schedules as functions for two fictional districts and compare them at six years and at twenty. The misconception it targets is the assumption that a percent raise always pays more: the fixed step is larger for the first several years here, and the exponential schedule only pulls ahead later, which is exactly what distinguishes equal differences from equal factors.

Kelloway Water District Pay Steps and a Certification Payback — Grade 11 Math
Name: ____________________________ Date: ______________

Problems

The Kelloway Water District, the neighboring Vellinge Regional Authority, and both pay schedules are fictional. The median pay figure in problem 8 is the reported national median for operators. Use these values throughout.

  • A new Grade I operator at Kelloway starts at $46,800 per year.
  • Kelloway adds a step of $1,950 to that salary at the end of each year worked.
  • A new Grade I operator at Vellinge starts at $45,200 per year.
  • Vellinge raises each salary by 3.6 percent at the end of each year worked.
  • Let t stand for the number of full years worked at either district.
  • Kelloway pays a $1,400 annual stipend to any operator who holds a Grade II license.
  • Earning the Grade II license costs $760 in course fees, the exam fee, and study materials.
  • Operators nationally earned a median of $60,020 per year as of May 2025.
  1. Write a function K(t) for the Kelloway salary after t full years.
  2. Write a function V(t) for the Vellinge salary after t full years.
  3. Find the Kelloway salary after 6 years.
  4. Find the Vellinge salary after 6 years, to the nearest dollar.
  5. Find the Kelloway salary after 20 years.
  6. Find the Vellinge salary after 20 years, to the nearest dollar.
  7. By what percent has the Vellinge starting salary grown over those 20 years? Round to one decimal place.
  8. How far above or below the reported national median of $60,020 is the Kelloway salary after 6 years?
  9. An operator pays the $760 certification cost and then collects the $1,400 annual stipend. How many months of the stipend does it take to repay that cost? Round to one decimal place.
  10. One schedule adds the same amount every year and the other multiplies by the same factor every year. Explain which schedule pays more in the early years, which pays more in the later years, and why the two behave differently.

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: K(t) = 46,800 + 1,950t.
    The starting salary is the constant term and the annual step is the constant rate of change, which makes the function linear.
  2. Answer: V(t) = 45,200(1.036)^t.
    A 3.6 percent raise multiplies the previous salary by 1.036, so the function is exponential with a base of 1.036.
  3. Answer: $58,500.
    K(6) = 46,800 + 1,950 x 6 = 46,800 + 11,700 = 58,500 dollars.
  4. Answer: $55,885.
    V(6) = 45,200 x 1.036^6. Since 1.036^6 = 1.23640, the salary is 45,200 x 1.23640 = 55,885 dollars. Accept answers from 55,884 to 55,886, since the rounding of the sixth power varies slightly by calculator.
  5. Answer: $85,800.
    K(20) = 46,800 + 1,950 x 20 = 46,800 + 39,000 = 85,800 dollars.
  6. Answer: $91,692.
    V(20) = 45,200 x 1.036^20. Since 1.036^20 = 2.02859, the salary is 45,200 x 2.02859 = 91,692 dollars. Accept answers from 91,690 to 91,695.
  7. Answer: 102.9 percent.
    The salary multiplied by 2.02859, so the increase is 2.02859 - 1 = 1.02859, or 102.9 percent above the starting salary.
  8. Answer: $1,520 below.
    60,020 - 58,500 = 1,520 dollars, so the six-year Kelloway salary sits 1,520 dollars below the reported national median for operators.
  9. Answer: 6.5 months.
    The stipend of 1,400 dollars per year is 1,400 divided by 12 = 116.67 dollars per month, and 760 divided by 116.67 = 6.51 months.
  10. Answer: Kelloway pays more early and Vellinge pays more later.
    The Kelloway schedule grows by equal differences, adding 1,950 dollars every year no matter how large the salary already is. The Vellinge schedule grows by equal factors, so its raise is 3.6 percent of a salary that keeps getting larger, which makes each raise bigger than the one before it. At six years Kelloway leads by 58,500 - 55,885 = 2,615 dollars, but by twenty years Vellinge leads by 91,692 - 85,800 = 5,892 dollars. Any exponential function with a base above 1 eventually passes any linear function, which is why the comparison flips.

Differentiation

Support

Give students a two-column table with the years 0 through 6 already listed and the first two salary rows filled in for each district, so the pattern of adding versus multiplying is visible before they write functions. Work problems 1 and 2 together, and provide a calculator reminder card showing how to enter a power such as 1.036 raised to the sixth.

Extension

Ask students to find the first whole year in which the Vellinge salary passes the Kelloway salary, then explain why a table is a more trustworthy tool than a graph for answering that question when the two curves cross at a shallow angle. A second extension is to build a third schedule that starts lower but raises by 5 percent a year, and to describe the conditions under which a lower starting offer is still the better long-run choice.

ELL supports

Pre-teach "step increase," "percent raise," "stipend," "certification," and "median" with a short labeled timeline showing one worker's salary at years 0, 1, and 2 under each schedule. Provide the sentence frame "In the early years ___ pays more because ___, but in the later years ___ pays more because ___" for problem 10 so students can present the comparison without composing the structure themselves.

Extensions

Operators typically enter the field with a high school diploma, on-the-job training, and a state license earned by examination, and licenses are issued in classes tied to plant size and experience. Other water-sector paths run through certificate programs, four-year degrees in civil or environmental engineering, and graduate study in hydrology or public health, and each of those leads to different pay structures worth comparing. Have students find a published salary schedule from a nearby city or utility, identify whether it uses steps, percent raises, or both, and rebuild problems 3 through 7 with the real schedule in place of the two fictional ones used here.