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Lesson plan Grade 9 Math

Ostgard Tiered Water Rates: A Block-Pricing Equations Lesson

A 55-minute grade 9 lesson where students write and graph a tiered water-rate equation for the Ostgard utility and test bills against an affordability threshold.

Objectives

  1. Students will write a two-piece tiered equation relating a water bill to the amount used.
  2. Students will interpret the slope of each tier as a price per 1,000 gallons.
  3. Students will compare a monthly water bill to an affordability threshold.

Vocabulary

Key words: tiered rate, base charge, slope, affordability, block pricing

Teacher overview

The fictional Ostgard utility charges for water in tiers: a flat base charge, a lower price for the first block of water, and a higher price above it. US water and sewer bills rose about 24 percent from 2019 to 2024, outpacing inflation, so how a bill is built matters to families. In this lesson students write a two-piece equation for the Ostgard bill, graph it, and check a sample bill against a widely used rule of thumb that treats water and sewer above 4.5 percent of household income as a high burden. A common misconception is applying the higher tier-2 price to all the water used, instead of only the water above the first block.

Procedure

Launch (8 min): Project a sample water bill for the fictional Ostgard utility and ask students why the price per 1,000 gallons might change as a household uses more. Collect ideas about fairness and conservation without resolving them yet.

Model the rate structure (15 min): Present Ostgard's fictional rates: an $18 monthly base charge, the first 4,000 gallons at $4 per 1,000 gallons, and any use above 4,000 gallons at $7 per 1,000 gallons. Let u stand for usage in thousands of gallons. Build the first piece with the class: for u at or below 4, the cost is C = 18 + 4u. Then work one example above the first block. A home that uses 6,000 gallons pays the $18 base, plus $16 for the first 4,000 gallons (4 x $4), plus $14 for the next 2,000 gallons (2 x $7), for a total of $48. Show that this matches the second-piece equation C = 7u + 6, since 7(6) + 6 = 48.

Guided practice (12 min): With the class, graph both pieces on graph paper and confirm the two pieces meet at 4,000 gallons: the first piece gives 18 + 4(4) = 34, and the second gives 7(4) + 6 = 34. Discuss why a steeper second tier can encourage households to conserve.

Independent practice (15 min): Students complete questions 1 through 8 on the handout, writing the tiered equation and using it to find bills and usages for the fictional Ostgard utility.

Closure / exit ticket (5 min): Students answer handout question 9 on a slip: find the bill for a household that uses 5,000 gallons. Collect and scan for the correct tier choice before dismissal.

Ostgard Tiered Water Rates: A Block-Pricing Equations Lesson — Grade 9 Math
Name: ____________________________ Date: ______________

Student Handout

All dollar amounts below describe the fictional Ostgard utility. Let u be usage in thousands of gallons.

  1. A household uses 3,000 gallons in a month. Find its water bill.
  2. A household uses exactly 4,000 gallons. Find its bill.
  3. A household uses 7,000 gallons. Find its bill.
  4. Write the equation for the monthly cost C when usage u (in thousands of gallons) is more than 4.
  5. What does the slope 7 in that equation represent?
  6. A household's bill is $69. How many thousand gallons did it use?
  7. A widely used rule of thumb, built by adding two EPA benchmarks, treats combined water and sewer costs above 4.5 percent of household income as a high burden. A household earns $2,000 a month. Find 4.5 percent of that income, in dollars.
  8. Compare the $55 bill from question 3 with the $90 threshold from question 7. Is the water bill alone within the threshold? Explain.
  9. Find the bill for a household that uses 5,000 gallons.

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: $30. Usage u = 3, so C = 18 + 4(3) = 18 + 12 = $30.
  2. Answer: $34. Usage u = 4, so C = 18 + 4(4) = 18 + 16 = $34.
  3. Answer: $55. Usage u = 7, so C = 7(7) + 6 = 49 + 6 = $55.
  4. Answer: C = 7u + 6. Above 4,000 gallons the base plus first block total $34, then each extra unit adds $7, giving C = 34 + 7(u - 4) = 7u + 6.
  5. Answer: $7 per 1,000 gallons. The slope is the tier-2 price, so each extra 1,000 gallons adds $7 to the bill.
  6. Answer: 9,000 gallons. 69 = 7u + 6, so 7u = 63 and u = 9, meaning 9,000 gallons.
  7. Answer: $90. 4.5 percent of $2,000 is 0.045 x 2,000 = $90 per month.
  8. Answer: Yes. The $55 water bill is below the $90 threshold, though adding a sewer charge could push the combined cost closer to it.
  9. Answer: $41. Usage u = 5, so C = 7(5) + 6 = 35 + 6 = $41.

Differentiation

Support

Give students the two-piece structure already set up, with the base charge and first-block cost filled in, so they only compute the tier-2 part. Provide the meeting point at 4,000 gallons worked out, and allow a calculator throughout.

Extension

Ask students to graph both pieces of the cost equation on one set of axes, mark where the pieces meet, and describe how the graph would change if the utility raised only the tier-2 price.

ELL supports

Pre-teach "base charge," "tier," "slope," and "threshold" with a simple two-step staircase picture, where each step is a price per 1,000 gallons. Provide a sentence frame for question 5.