Water for Schools
Menu

Lesson plan Grade 11 Math

Birch Hollow's Three Rate Plans: Break-Even Systems and Intervals

A grade 11 lesson where students solve three systems of equations comparing three water-rate plans and determine which plan is cheapest across each usage interval.

Objectives

  1. Students will write linear cost equations for three water-rate plans and solve the systems formed by each pair of plans to find three break-even usage levels.
  2. Students will determine, from three break-even points, which plan is the overall cheapest across each interval of usage, including a case where a break-even point does not change the overall cheapest plan.
  3. Students will graph three lines on axes they scale themselves and correctly restrict the model's domain to non-negative usage.

Vocabulary

Key words: system of equations, break-even point, flat fee, per-gallon rate, viable solution, usage interval

Teacher overview

Birch Hollow Water Utility (a fictional utility used for this lesson) has proposed three residential billing plans instead of the usual two, and residents have started arguing past each other because each person is picturing a different usage level without realizing it. A common misconception in this lesson is assuming that with three plans, there must be three separate "winning ranges," one for each plan; the lesson is built so students discover, from their own three systems of equations, that one of the three break-even points can fall entirely inside another plan's winning range without ever changing which plan is actually cheapest overall.

Procedure

Hook (5 min): Display all three of the fictional Birch Hollow Water Utility's proposed rate plans on the board with no calculations shown: Plan A charges a $48 flat monthly fee plus $2 per thousand gallons used; Plan B charges a $20 flat fee plus $4 per thousand gallons used; Plan C charges no flat fee and $6 per thousand gallons used. Ask students to rank the three plans from best deal to worst deal for a typical household, and note that the class will likely disagree, since the honest ranking depends on usage.

Direct instruction (18 min): Model writing each plan as an equation in x (thousand gallons used) and y (total monthly cost): Plan A is y = 48 + 2x; Plan B is y = 20 + 4x; Plan C is y = 6x. Explain that three plans create three possible pairs, and each pair forms its own system. Solve the first system, Plan A and Plan B, by substitution: 48 + 2x = 20 + 4x, so 28 = 2x, so x = 14, and y = 48 + 2(14) = 76. Solve the second system, Plan B and Plan C: 20 + 4x = 6x, so 20 = 2x, so x = 10, and y = 6(10) = 60. Solve the third system, Plan A and Plan C: 48 + 2x = 6x, so 48 = 4x, so x = 12, and y = 6(12) = 72. Mark all three break-even points on a number line in order of usage, x = 10, x = 12, x = 14, and note that the model only applies for x ≥ 0, since a household cannot use a negative amount of water.

Guided practice (17 min): In pairs, students graph all three lines on a single set of axes they scale themselves, choosing increments that fit costs up to about $120 and usage up to about 20 thousand gallons. Students mark all three break-even points on their graph and test a usage value inside each of the four regions the three points create (for example, x = 0, x = 11, x = 13, and x = 15), computing all three plans' cost at each test value and ranking them cheapest to most expensive. Working through this, students discover that Plan C is cheapest for x < 10, Plan B is cheapest for the entire stretch from x = 10 to x = 14 even though the Plan A-and-Plan-C break-even at x = 12 sits inside that stretch, and Plan A is cheapest for x > 14. Pairs mark these three winning regions directly on their graphed axes.

Independent practice (10 min): Each student answers three questions using the equations y = 48 + 2x, y = 20 + 4x, and y = 6x.

  1. A household uses 6,000 gallons in a month. Which plan is cheapest, and what is its monthly cost?
  2. A household uses 18,000 gallons in a month. Which plan is cheapest, and what is its monthly cost?
  3. Using the three break-even points, state the complete usage interval, in thousands of gallons, over which Plan B is the single overall-cheapest plan.

Closure / exit ticket (5 min): Students answer one question on an index card, continuing the numbered set:

  1. A household uses 13,000 gallons in a month. Identify the overall cheapest plan, then explain why the Plan A-and-Plan-C break-even at 12,000 gallons does not change which plan is cheapest overall at this usage level.

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: Plan C, $36.
    At x = 6: Plan A costs 48 + 2(6) = 60; Plan B costs 20 + 4(6) = 44; Plan C costs 6(6) = 36. Plan C is the cheapest.

  2. Answer: Plan A, $84.
    At x = 18: Plan A costs 48 + 2(18) = 84; Plan B costs 20 + 4(18) = 92; Plan C costs 6(18) = 108. Plan A is the cheapest.

  3. Answer: 10 to 14 thousand gallons.
    Plan B is the single cheapest plan for every usage level strictly between the Plan B-and-Plan-C break-even at x = 10 and the Plan A-and-Plan-B break-even at x = 14. At the two endpoints, Plan B ties instead of being the sole cheapest: with Plan C at x = 10, and with Plan A at x = 14.

  4. Answer: Plan B, $72.
    At x = 13: Plan A costs 48 + 2(13) = 74; Plan B costs 20 + 4(13) = 72; Plan C costs 6(13) = 78. Plan B is cheapest even though 13,000 gallons is past the Plan A-and-Plan-C break-even at 12,000 gallons, because that break-even only determines which of Plan A and Plan C costs more; it does not involve Plan B, so it cannot change that Plan B remains the overall minimum across the entire interval between the Plan B-and-Plan-C break-even (10) and the Plan A-and-Plan-B break-even (14).

Differentiation

Support

Provide the three equations already written and a partially completed table of costs at x = 0, 5, 10, 12, 14, 16, and 20 for all three plans, so students can see the ranking pattern before solving any systems algebraically. Allow students to locate each break-even point by finding where two columns of the table are closest, then require the algebraic solution only for the one system assigned to their pair.

Extension

Ask students to design a fourth plan, Plan D, whose line must cross all three existing lines somewhere within the graphed window (0 to 20 thousand gallons) yet never be the single cheapest option at any usage level in that window, then prove their plan meets both conditions using systems of equations against all three existing plans.

ELL supports

Pre-teach "flat fee," "per-gallon rate," and "break-even point" with a simple diagram of three ramps that start at different heights and rise at different steepnesses, meeting each other at three separate points. Provide a sentence frame for the independent-practice questions: "Plan ___ is cheapest at ___ gallons because it costs $___, which is less than Plan 's $ and Plan 's $."

Extensions

Have students collect the published flat fee and per-1,000-gallon rate from two or three real local water utilities' rate schedules and build the same three-way break-even analysis (or a two-way version if only two rate schedules are available) for their own community, then compare the resulting winning intervals to Birch Hollow's.