Lesson plan Grade 9 Math
Wrenfield's Two Water Rate Plans: A Systems of Equations Lesson
A grade 9 lesson where students set up and solve a system of linear equations to find the usage level where two water utility billing plans cost the same.
Objectives
- Students will write a linear equation for each of two water billing plans in terms of monthly usage.
- Students will solve a system of two linear equations both algebraically and graphically to find the usage level where the two plans cost the same.
Vocabulary
Key words: system of equations, break-even point, base fee, per-unit rate
Teacher overview
The fictional Wrenfield Water Utility offers residential customers a choice between two monthly billing plans: a flat base fee plus a low per-gallon rate, or no base fee with a higher per-gallon rate. Each plan is a linear function of monthly gallons used, and the question of which plan is cheaper depends entirely on how much water a household uses, which makes this a natural setting for solving a system of two linear equations. A common misconception in this lesson is assuming that one plan is simply "the better deal" in every case; students need the algebra to see that the answer actually depends on usage, and that there is one specific usage level where the two plans cost exactly the same.
Procedure
Hook (5 min): Present both plans from the fictional Wrenfield Water Utility on the board:
- Plan A: $18 base fee per month, plus $0.006 per gallon used.
- Plan B: no base fee, $0.009 per gallon used.
Ask students to guess, without calculating, which plan is the better deal for a typical household. Collect a few guesses, then note that the honest answer depends on usage.
Direct instruction (12 min): Model writing each plan as an equation, with y as total monthly cost and x as gallons used: y = 18 + 0.006x for Plan A, and y = 0.009x for Plan B. Explain that these two equations together form a system, and that solving the system means finding the (x, y) pair where both equations are true at once, the usage level and cost where the two plans match exactly. Demonstrate solving the system algebraically by substitution: since both equations equal y, set 18 + 0.006x = 0.009x, then solve for x. Show the full solution on the board: 18 = 0.003x, so x = 6,000 gallons, and the matching cost is y = 0.009 × 6,000 = $54.
Guided practice (15 min): In pairs, students graph both equations on the same set of axes using graph paper, choosing a sensible scale (for example, x from 0 to 10,000 gallons in increments of 1,000, and y from $0 to $100 in increments of $10). Students plot at least three points for each line, draw the lines, and mark the intersection point, then check that the intersection matches the algebraic solution of 6,000 gallons and $54 found in direct instruction.
Independent practice (13 min): Students answer two scenario questions individually, questions 1 and 2 on the handout.
- A Wrenfield household uses 4,000 gallons in a month. Which plan costs less, and by how much?
- A different Wrenfield household uses 9,000 gallons in a month. Which plan costs less, and by how much?
Circulate and check that students are substituting their usage value into both equations rather than guessing.
Closure / exit ticket (5 min): Students answer one question on an index card: "A different utility offers Plan C ($25 base fee plus $0.004 per gallon) and Plan D (no base fee, $0.008 per gallon). At what usage level do Plan C and Plan D cost the same, and what is that cost?" Collect and scan for correct system setup (25 + 0.004x = 0.008x) before dismissal.
Student Handout
Wrenfield Water Utility is fictional. Its two residential plans are: Plan A, an $18 base fee per month plus $0.006 per gallon used; and Plan B, no base fee and $0.009 per gallon used. Substitute each usage figure into both equations rather than guessing.
A Wrenfield household uses 4,000 gallons in a month. Under the Plan A and Plan B rates above, which plan costs less that month, and by how many dollars?
A different Wrenfield household uses 9,000 gallons in a month. Under the same Plan A and Plan B rates, which plan costs less that month, and by how many dollars?
Exit ticket: A different utility offers Plan C (a $25 base fee per month plus $0.004 per gallon) and Plan D (no base fee, $0.008 per gallon). At what monthly usage, in gallons, do Plan C and Plan D cost the same?
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 (Plan B is cheaper).
At 4,000 gallons, Plan A costs 18 + 0.006(4,000) = 18 + 24 = $42. Plan B costs 0.009(4,000) = $36. Plan B is $6 cheaper, because at this usage level the household has not yet reached the 6,000-gallon break-even point, so the plan without a base fee wins.Answer: $9 (Plan A is cheaper).
At 9,000 gallons, Plan A costs 18 + 0.006(9,000) = 18 + 54 = $72. Plan B costs 0.009(9,000) = $81. Plan A is $9 cheaper, because past the 6,000-gallon break-even point, the plan with the lower per-gallon rate and a base fee becomes the better deal.Answer: 6,250 gallons.
This is the exit-ticket answer. Setting the two equations equal: 25 + 0.004x = 0.008x. Subtracting 0.004x from both sides gives 25 = 0.004x, so x = 6,250 gallons. Substituting into Plan D's equation: 0.008 × 6,250 = $50, which is the cost where the two plans match.
Differentiation
Support
Provide a partially completed table of (usage, cost) pairs for both Plan A and Plan B, with several rows already filled in, so students can see the pattern of "Plan B cheaper below 6,000 gallons, Plan A cheaper above" before tackling the algebraic solution. Allow students to solve the system by guess-and-check using the table rather than algebraic substitution if needed.
Extension
Ask students to find a third billing plan (choosing their own base fee and per-gallon rate) that would break even with Plan A at exactly 3,000 gallons instead of 6,000, and have them show the algebra that proves their new plan meets that requirement.
ELL supports
Pre-teach "base fee," "per-gallon rate," and "break-even" using a simple diagram of a bill split into a flat starting amount and a growing amount that depends on usage. Provide a sentence frame for the independent practice questions: "Plan ___ costs less because at ___ gallons, Plan A costs $___ and Plan B costs $___."