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Lesson plan Grade 8 Health & Consumer Science

Merrow Falls Water Rate Plans: A Grade 8 Systems of Equations Lesson

A grade 8 consumer-science lesson where students write each of two town water-rate plans as a linear equation and solve the system to find the break-even usage.

Objectives

  1. Students will model each of two water-rate plans as a linear equation from a verbal description.
  2. Students will solve a system of two linear equations to find the break-even usage level.
  3. Students will decide which rate plan is cheaper for a given monthly water use and justify the choice.

Vocabulary

Key words: base fee, rate of change, break-even point, system of equations

Teacher overview

Choosing between two water billing plans is a real consumer decision, and it is also a clean use of a system of two linear equations. In this lesson, students work with a fictional utility that offers two plans: one with a high monthly base fee and a low per-gallon charge, and one with a low base fee and a high per-gallon charge. Each plan becomes a linear cost equation, and the point where the two equations are equal is the usage level at which the plans cost the same. A common student misconception is that the plan with the lower base fee is always cheaper; the break-even math shows that the better plan depends on how much water a household actually uses.

Procedure

Hook (5 min): Ask students to imagine two households, one that waters a large garden all summer and one that travels most of the month. Would the same water plan be the best deal for both? Take a few predictions and record them without confirming.

Direct instruction (12 min): Introduce the town of Merrow Falls and its two plans. Model the method on a different pair of plans than the handout uses so the handout stays fresh: Plan X charges a 20 dollar base fee plus 5 dollars per thousand gallons, giving C = 20 + 5x, and Plan Y charges a 10 dollar base fee plus 9 dollars per thousand gallons, giving C = 10 + 9x. Set the two equal, 20 + 5x = 10 + 9x, solve to x = 2.5 thousand gallons, and substitute back to get a cost of 32.50 dollars. Graph both lines on the board and point to the intersection as the break-even point.

Guided practice (10 min): In pairs, students complete handout items 1 through 4, writing the two Merrow Falls equations and solving the system for its break-even point. Circulate and check that students set the two cost expressions equal rather than setting each to zero.

Independent practice (13 min): Students complete handout items 5 through 7 on their own, testing two specific usage levels and explaining which plan wins in each case.

Closure and exit ticket (5 min): Students complete handout item 8, a third plan evaluated at a stated usage level, and hand it in. Scan for the correct substitution before dismissal; if more than a few students miss it, reteach the substitution step next class.

Merrow Falls Water Rate Plans: A Grade 8 Systems of Equations Lesson — Grade 8 Health & Consumer Science
Name: ____________________________ Date: ______________

Student Handout

Suppose the town of Merrow Falls offers two monthly water plans. Let x stand for the number of thousand-gallon units a household uses in a month.

  • Steady Plan: a 30 dollar monthly base fee, plus 4 dollars for every thousand gallons used.
  • Saver Plan: a 12 dollar monthly base fee, plus 8 dollars for every thousand gallons used.
  1. Write a linear equation for the Steady Plan's monthly cost C in dollars, in terms of x.
  2. Write a linear equation for the Saver Plan's monthly cost C in dollars, in terms of x.
  3. Set your two equations equal and solve for x to find the break-even point, in thousand gallons.
  4. What is the monthly cost, in dollars, at that break-even point? Show that both plans give the same amount.
  5. A household uses 3,000 gallons in a month, so x = 3. Find the cost under each plan and state which plan is cheaper and by how much.
  6. A household uses 7,000 gallons in a month, so x = 7. Find the cost under each plan and state which plan is cheaper and by how much.
  7. Explain, using the base fees and the per-thousand-gallon charges, why the Saver Plan is the better deal for low water users while the Steady Plan is better for high users.
  8. A third plan, the Basic Plan, charges a 24 dollar base fee plus 6 dollars for every thousand gallons. Find its monthly cost when a household uses 5,000 gallons, so x = 5.

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: C = 30 + 4x.
    The 30 dollar base fee is the initial value and 4 dollars per thousand gallons is the rate of change.
  2. Answer: C = 12 + 8x.
    The 12 dollar base fee is the initial value and 8 dollars per thousand gallons is the rate of change.
  3. Answer: 4.5 thousand gallons.
    Setting 30 + 4x = 12 + 8x gives 18 = 4x, so x = 4.5, which is 4,500 gallons.
  4. Answer: 48 dollars.
    Steady Plan: 30 + 4(4.5) = 48. Saver Plan: 12 + 8(4.5) = 48. Both equal 48 dollars, confirming the break-even point.
  5. Answer: Saver Plan, cheaper by 6 dollars.
    Steady Plan: 30 + 4(3) = 42 dollars. Saver Plan: 12 + 8(3) = 36 dollars. The Saver Plan costs 6 dollars less.
  6. Answer: Steady Plan, cheaper by 10 dollars.
    Steady Plan: 30 + 4(7) = 58 dollars. Saver Plan: 12 + 8(7) = 68 dollars. The Steady Plan costs 10 dollars less.
  7. Answer: The Saver Plan starts lower but climbs faster.
    Its low 12 dollar base fee helps light users, but its 8 dollar per-thousand charge adds up quickly, so heavy users pay less under the Steady Plan's cheaper 4 dollar per-thousand rate despite its higher base fee.
  8. Answer: 54 dollars.
    Basic Plan: 24 + 6(5) = 24 + 30 = 54 dollars.

Differentiation

Support

Provide the two Merrow Falls equations already written for items 1 and 2 so that students who struggle with translation can begin at the solving step. Offer a filled-in example of setting two expressions equal, and allow calculator use for the arithmetic in items 5 and 6.

Extension

Ask students to find, algebraically, the base fee a fourth plan would need if it charged 7 dollars per thousand gallons and had to break even with the Steady Plan at exactly 6,000 gallons. Have them verify their answer by substitution and explain what the result means for a household deciding among plans.

ELL supports

Pre-teach base fee, rate, and break-even with a simple labeled picture of a bill showing a fixed part and a per-use part. Provide sentence frames for item 7, such as "The Saver Plan is cheaper when a household uses ___ because ___."

Extensions

Invite students to look at a real water bill from home, with a family member's permission, and identify which part is a fixed service charge and which part changes with the amount of water used. Students can then sketch their own household's cost as a linear equation and describe, in one sentence, whether their household is a light or heavy water user.