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Lesson plan Grade 10 Math

Wrenhaven Tiered Water Rates: Writing and Solving Bill Equations

A 50-minute grade 10 lesson where students model a fictional town's tiered water rates as equations, compute monthly bills, and solve for usage at a target bill.

Objectives

  1. Students will compute a monthly water bill from a tiered rate schedule.
  2. Students will write a two-variable equation for cost as a function of gallons used within a tier.
  3. Students will solve a one-variable equation to find the usage that produces a target bill.

Vocabulary

Key words: tiered rate, service charge, conservation rate, break-even usage

Teacher overview

The town of Wrenhaven, a fictional town invented for this lesson, charges an increasing-block (tiered) water rate, so each additional block of water costs more per gallon than the last. This lesson has students compute bills from that schedule, write an equation for cost within a single tier, and solve for the usage that hits a target bill. A common misconception is that a tiered bill is one flat rate times total gallons; students often forget that only the gallons inside each tier are charged at that tier's price, which the worked example is built to correct.

Procedure

Hook (5 min): Project the fictional Wrenhaven rate schedule and ask students to predict whether a household that doubles its water use will exactly double its bill. Collect a show of hands for "yes" and "no" without revealing the answer, and tell students the lesson will test their prediction. The Wrenhaven schedule is a fixed monthly service charge of eighteen dollars, plus 4.50 dollars per 1,000 gallons for the first 4,000 gallons, 7.00 dollars per 1,000 gallons for the next 6,000 gallons, and 11.00 dollars per 1,000 gallons for everything above 10,000 gallons.

Direct instruction (12 min): Model a full bill for a household in the fictional town of Wrenhaven that uses 8,500 gallons. Work it in blocks on the board: the service charge is 18 dollars; the first 4,000 gallons cost 4 × 4.50 = 18 dollars; the remaining 4,500 gallons fall in the second tier and cost 4.5 × 7.00 = 31.50 dollars. The total is 18 + 18 + 31.50 = 67.50 dollars. Emphasize that only the gallons inside each tier are priced at that tier's rate.

Guided practice (12 min): With the class, build an equation for the monthly cost y (dollars) of a household whose use x (gallons) falls between 4,000 and 10,000. The first tier is always fully used, contributing 18 dollars, plus the service charge of 18 dollars, so the fixed part is 36 dollars. The tier-two gallons number (x − 4,000), priced at 7.00 dollars per 1,000, add 0.007(x − 4,000). Combine to y = 0.007x + 8. Check it at x = 8,500: 0.007 × 8,500 + 8 = 67.50 dollars, matching the worked example. Have students verify it at x = 10,000.

Independent practice (16 min): Students complete the Student Handout, computing three full bills from the schedule, writing the tier-two equation, and solving it for the usage that produces two target bills. Circulate and check that students charge only the in-tier gallons at each tier's rate.

Closure and exit ticket (5 min): Return to the hook prediction. Ask students to answer one exit-ticket question on an index card: a household in Wrenhaven uses 5,000 gallons; find its bill. The answer is 18 + 18 + (1 × 7.00) = 43.00 dollars. Collect the cards and scan for students who correctly split the 5,000 gallons into 4,000 tier-one gallons and 1,000 tier-two gallons before dismissal.

Wrenhaven Tiered Water Rates: Writing and Solving Bill Equations — Grade 10 Math
Name: ____________________________ Date: ______________

Student Handout

Use the fictional Wrenhaven rate schedule for every question: a fixed monthly service charge of 18 dollars, plus 4.50 dollars per 1,000 gallons for the first 4,000 gallons, 7.00 dollars per 1,000 gallons for the next 6,000 gallons (from 4,001 to 10,000), and 11.00 dollars per 1,000 gallons above 10,000 gallons.

  1. Household A uses 3,000 gallons in a month. Find its bill.

  2. Household B uses 10,000 gallons in a month. Find its bill.

  3. Household C uses 14,000 gallons in a month. Find its bill.

  4. Write an equation for the monthly cost y, in dollars, for a household whose use x, in gallons, falls between 4,000 and 10,000.

  5. Using your equation from item 4, find the water use x that produces a monthly bill of 60.00 dollars. Round to the nearest gallon.

  6. A household wants to keep its monthly bill at or below 50.00 dollars. Using the equation from item 4, find the most water it can use in a month.

  7. Explain why this tiered rate structure encourages conservation more strongly than a single flat rate per gallon would.

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: 31.50 dollars.
    Service charge 18 + first 3,000 gallons at 4.50 per 1,000 (3 × 4.50 = 13.50) = 18 + 13.50 = 31.50 dollars. All 3,000 gallons fall inside tier one.

  2. Answer: 78.00 dollars.
    Service charge 18 + tier one (4 × 4.50 = 18) + tier two (6 × 7.00 = 42) = 18 + 18 + 42 = 78.00 dollars. The 10,000 gallons exactly fill tiers one and two.

  3. Answer: 122.00 dollars.
    Service charge 18 + tier one (18) + tier two (42) + tier three (4 × 11.00 = 44) = 18 + 18 + 42 + 44 = 122.00 dollars. The last 4,000 gallons fall in tier three.

  4. Answer: y = 0.007x + 8.
    The service charge (18) plus the full first tier (18) is a fixed 36 dollars. The tier-two gallons (x − 4,000) cost 7.00 per 1,000, or 0.007(x − 4,000). So y = 36 + 0.007(x − 4,000) = 0.007x + 8, valid for x between 4,000 and 10,000.

  5. Answer: 7,429 gal.
    Set 0.007x + 8 = 60. Then 0.007x = 52, so x = 52 ÷ 0.007 = 7,428.57, which rounds to about 7,429 gallons. This falls between 4,000 and 10,000, so the tier-two equation applies.

  6. Answer: 6,000 gal.
    Set 0.007x + 8 = 50. Then 0.007x = 42, so x = 42 ÷ 0.007 = 6,000 gallons. A household staying at or below 50.00 dollars can use at most 6,000 gallons.

  7. Answer: Higher use is priced at a higher rate.
    Under a tiered rate the price per gallon rises as a household uses more, so extra water in the top tiers is expensive. A single flat rate charges the same price for every gallon, so it gives no added reason to cut back once service is on. The rising tiers reward households that keep their use low.

Differentiation

Support

Provide a partially worked bill (service charge and tier-one total already filled in) for the direct-instruction example so students can focus on the tier-two gallons. Give a printed strip that restates the rule "charge only the gallons inside a tier at that tier's rate," and allow calculators throughout.

Extension

Ask students to write the cost equation for the third tier (use above 10,000 gallons) and use it to find the usage that produces a 150.00-dollar bill, then explain why the third-tier equation has a steeper slope than the tier-two equation.

ELL supports

Pre-teach "service charge," "tier," and "block" using a labeled diagram of three stacked blocks of water priced at rising rates. Provide sentence frames for item 7: "A tiered rate charges ___ for extra water, so a household has a reason to ___."