Worksheet Grade 8 Careers & CTE

Two Water-Career Pay Paths: Linear Functions and Break-Even

A grade 8 worksheet where students model two fictional water-career pay paths as linear functions and solve a system to find the year the salaries become equal.

Grade: Grade 8
Subject: Careers & CTE
Material: Worksheet
Time: 40 min
Prep: No prep
Materials: None

Standards

8.F.B.4 Common Core Math

Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description, a table, or a graph, and interpret them in terms of the situation modeled.

How this material aligns: Items 1 through 3 have students construct a linear function for each pay path and interpret the initial value as starting pay and the rate of change as the annual raise.

All materials for 8.F.B.4 →

8.EE.C.8 Common Core Math

Analyze and solve pairs of simultaneous linear equations, understanding that solutions correspond to points of intersection, and solving systems by graphing and algebraically in the context of real-world problems.

How this material aligns: Item 6 has students solve the system of the two pay equations to find the exact year both paths pay the same, the point-of-intersection reasoning this standard targets.

All materials for 8.EE.C.8 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-17.

Objectives

Students will write and interpret linear functions that model two water-career pay paths.
Students will solve a system of two linear equations to find the year the two pay paths are equal.

Vocabulary

Key words: linear function, rate of change, initial value, break-even

Teacher overview

This worksheet turns a career-planning question into linear-function practice, using a fictional pair of friends who enter two different water careers. Students write a linear equation for each pay path, then solve the system to find the year both friends earn the same amount. A common mistake this set targets is assuming the path with the bigger yearly raise is always ahead; in the early years, the higher starting pay can keep the other path in the lead.

Two Water-Career Pay Paths: Linear Functions and Break-Even — Grade 8 Careers & CTE

Name: ____________________________ Date: ______________

Student Worksheet

Suppose two friends both start water-sector careers in the same year. These salary paths are fictional, built only for this worksheet. Let t be the number of years each person has worked.

Devon trains as a treatment operator through an apprenticeship, starting at $41,000 per year and earning a raise of $2,000 each year.
Priya works as a laboratory analyst after a two-year degree, starting at $50,000 per year and earning a raise of $1,000 each year.

Write a linear equation for Devon's pay, y, after t years. Identify the initial value and the rate of change.
Write a linear equation for Priya's pay, y, after t years.
Explain in words what the rate of change in Devon's equation means for his career.
What is Devon's pay after 5 years? Show your substitution.
What is Priya's pay after 5 years? Show your substitution.
Solve the system of the two equations to find the year t when Devon and Priya earn the same pay. Show your steps.
Before the break-even year, whose pay is higher? After it, whose pay is higher? Explain using the two rates of change.
At the break-even year, both friends earn the same amount. Compare that amount to the real median pay for operators of $60,020. Which is higher, and by about how much?
A classmate says, "The path with the bigger raise is always the better deal." Use both equations to explain why that is not true in the first few years.

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

Answer Key

Answer: y = 41,000 + 2,000t. The initial value is $41,000 (Devon's starting pay) and the rate of change is $2,000 per year (his annual raise).
Answer: y = 50,000 + 1,000t. The initial value is $50,000 and the rate of change is $1,000 per year.
Answer: The rate of change, $2,000 per year, means Devon's pay increases by $2,000 for each year he works.
Answer: $51,000. y = 41,000 + 2,000(5) = 41,000 + 10,000 = 51,000.
Answer: $55,000. y = 50,000 + 1,000(5) = 50,000 + 5,000 = 55,000.
Answer: 9 years. Set 41,000 + 2,000t = 50,000 + 1,000t. Subtract 1,000t from both sides: 41,000 + 1,000t = 50,000. Subtract 41,000: 1,000t = 9,000, so t = 9.
Answer: Before year 9, Priya's pay is higher because her starting pay is larger. After year 9, Devon's pay is higher because his raise of $2,000 per year outpaces Priya's $1,000 per year, so his total keeps growing faster.
Answer: The median is higher, by $1,020. At t = 9, both earn 41,000 + 2,000(9) = 59,000, so the break-even pay is $59,000, which is lower than the $60,020 median by 60,020 - 59,000 = $1,020.
Answer: In the early years the higher starting pay matters more than the raise. At t = 5, Priya ($55,000) still earns more than Devon ($51,000) even though Devon's raise is larger, so a bigger raise is only "better" once enough years pass to overcome the lower start.

Differentiation

Support

Provide the equation frame "y = (starting pay) + (raise per year) times t" for items 1 and 2, and give a filled-in substitution example for a different starting pay before students attempt items 4 and 5.

Extension

Ask students to write a third pay path that would break even with Devon's at exactly year 6, state its starting pay and yearly raise, and prove it with the system.

ELL supports

Pre-teach "initial value," "rate of change," and "break-even" with a simple two-line graph showing where the lines cross. Allow items 3, 7, and 9 to be answered with short phrases plus the equations rather than full paragraphs.