Lesson plan Grade 10 Math
Amberlin Water District: Comparing Two Billing Plans by Equation
A grade 10 lesson where students solve and graph a system of linear equations to find the break-even usage point between two fictional water utility billing plans.
Objectives
- Students will write a linear equation representing a water utility billing plan.
- Students will solve a system of two linear equations algebraically to find a break-even usage point.
- Students will graph two billing plans on the same axes and interpret the point where the lines cross.
Vocabulary
Key words: system of equations, break-even point, flat fee, per-unit rate
Teacher overview
Amberlin Water District (a fictional utility used for this lesson) is proposing two different monthly billing structures, and residents want to know which one costs less. This lesson has students write each plan as a linear equation, solve the resulting system to find the usage level where both plans cost the same, and graph the two lines to see why one plan is cheaper below that point and the other is cheaper above it. A common misconception is assuming the plan with the lower flat fee is always cheaper; in fact, the per-unit rate determines which plan wins at higher usage levels, which is exactly what solving the system reveals.
Procedure
Hook (5 min): Show two billing flyers from Amberlin Water District's fictional rate proposal on the board: Plan A charges an $18 monthly base fee plus $4.50 per 1,000 gallons used, and Plan B charges a $30 monthly base fee plus $3.00 per 1,000 gallons used. Ask students which plan they would choose and why, without doing any calculation yet.
Direct instruction (15 min): Model writing each plan as an equation, with x representing usage in thousands of gallons and y representing total monthly cost: Plan A is y = 18 + 4.5x, and Plan B is y = 30 + 3x. Solve the system by substitution: set 18 + 4.5x equal to 30 + 3x, giving 1.5x = 12, so x = 8 (8,000 gallons). Substitute back to find the cost at that point: y = 18 + 4.5(8) = 54. Sketch both lines on labeled, scaled axes (usage in thousands of gallons on the x-axis, cost in dollars on the y-axis) and mark the crossing point at (8, 54).
Guided practice (15 min): In pairs, students set up and solve a second system for two more plans: Plan C, y = 22 + 4x, and Plan D, y = 10 + 6x. They solve algebraically for the break-even usage and cost, then graph both lines on grid paper to confirm their algebraic answer matches where the lines cross.
Independent practice (10 min): Each student answers two questions on the handout: at a usage of 12,000 gallons (x = 12), which of Plan A or Plan B costs less, and by how much? Then, using the graph from direct instruction, students explain in one sentence which plan is cheaper for a household that uses very little water each month.
Closure / exit ticket (5 min): Students answer one question on an index card: "If a household's usage is below the break-even point you found for Plan A and Plan B, which plan costs less? Use the slopes of the two lines to justify your answer, without recalculating." Collect and scan for correct reasoning (the plan with the lower per-unit rate costs more per gallon at low usage because of its flat fee, so the plan with the lower flat fee wins below the break-even point) before dismissal.
Student Handout
Amberlin Water District is a fictional utility. Show your work for every question.
Amberlin is comparing two more billing plans. Plan C's total monthly cost in dollars is y = 22 + 4x, and Plan D's total monthly cost in dollars is y = 10 + 6x, where x is monthly usage in thousands of gallons. Solve the system algebraically to find the usage at which the two plans cost the same, then substitute back to check the cost at that point. Give the break-even usage in gallons.
Plan A charges an $18 monthly base fee plus $4.50 per 1,000 gallons used. Plan B charges a $30 monthly base fee plus $3.00 per 1,000 gallons used. A household uses 12,000 gallons in one month (x = 12). Which of the two plans costs less that month, and by how many dollars?
Using the same Plan A ($18 base fee plus $4.50 per 1,000 gallons) and Plan B ($30 base fee plus $3.00 per 1,000 gallons), which plan is cheaper for a household that uses very little water each month? Answer in one sentence.
Exit ticket: Plan A and Plan B cost exactly the same at a usage of 8,000 gallons. If a household's usage is below that break-even point, which plan costs less? Use the slopes of the two lines to justify your answer, without recalculating.
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: 6,000 gallons.
Setting 22 + 4x = 10 + 6x gives 12 = 2x, so x = 6 (6,000 gallons). Substituting back: y = 22 + 4(6) = 46.Answer: $6 (Plan B costs less).
At x = 12: Plan A costs 18 + 4.5(12) = 72. Plan B costs 30 + 3(12) = 66. Plan B is cheaper by 72 − 66 = 6 dollars.Answer: Plan A is cheaper at very low usage.
Plan A's lower flat fee ($18 versus $30) means it costs less than Plan B until usage climbs high enough for Plan B's lower per-unit rate to close the gap, which happens at the 8,000-gallon break-even point.Answer: Plan A costs less below the break-even point.
This is the exit-ticket answer. Plan A has the lower flat fee (its y-intercept), so at low usage, before Plan B's lower per-gallon rate has a chance to make up the difference, Plan A's total cost stays lower.
Differentiation
Support (below grade level)
Provide the Plan A and Plan B equations already written, so students focus on the substitution and solving steps, and give students a printed coordinate grid with the axes already labeled and scaled from 0 to 15 (usage) and 0 to 90 (cost).
Extension (above grade level)
Ask students to design a third plan, Plan E, using a flat fee and per-unit rate of their choosing, that would break even with Plan A at exactly 10,000 gallons, and to show the algebra that proves their plan meets that condition.
ELL supports
Pre-teach "flat fee," "per-unit rate," and "break-even point" using a simple diagram of a bill showing a fixed starting amount plus an amount that grows with usage. Provide sentence starters for the exit ticket: "Plan ___ costs less below the break-even point because ___."