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Lesson plan Grade 12 Math

Draywick Road Culvert: Design Storm Risk Over 50 Years

A 60-minute grade 12 lesson building the exponential risk model for a fictional town's culvert, then sizing it with the rational method and the continuity equation.

Objectives

  1. Students will express a design storm's return period as an annual exceedance probability.
  2. Students will construct and evaluate an exponential model for cumulative risk across a culvert's service life.
  3. Students will size a culvert using the rational method and the continuity equation.

Vocabulary

Key words: return period, annual exceedance probability, rational method, runoff coefficient, rainfall intensity, continuity equation

Teacher overview

A culvert built for a 25-year storm does not flood once every 25 years on a schedule; it carries a 4 percent chance of being overtopped in any given year, and across a 50-year service life those small annual chances accumulate into a large total risk. This lesson works through the fictional town of Pellham, where the Draywick Road culvert over Vosswick Creek is due for replacement, and pairs the risk model with the sizing math that decides how large the replacement pipe must be. The misconception to correct is the belief that a 25-year storm arrives on a schedule, or that having just had one makes another unlikely for years.

Procedure

Opening and the meaning of a return period (8 min): Present the fictional Pellham situation: the Draywick Road culvert washed out during a heavy storm, and residents are asking why a culvert designed for a 25-year storm failed after 11 years. Ask students what they think a 25-year storm means, and record the answers. Introduce the working definition used by engineers, which is that a 25-year storm has a 1 in 25 chance of being equaled or exceeded in any single year.

Building the risk model (12 min): Model the reasoning at the board with a different structure than the handout uses. Suppose a culvert is designed for a 10-year storm, so its annual exceedance probability is 0.10 and the chance of no exceedance in one year is 0.90. Ask what the chance of surviving two years untouched is, and draw out the multiplication, 0.90 times 0.90. Extend to 30 years, 0.90 raised to the 30th power, which is about 0.042, so the chance of at least one exceedance is about 95.8 percent. Name the assumption the model rests on, that each year is independent of the last, and ask the class where that assumption might fail.

Handout items 1 through 5 (15 min): Students work the risk items independently, then compare answers with a partner. Watch for two errors: computing 0.04 raised to the 50th power instead of 0.96 raised to the 50th power, and reporting the chance of no exceedance when the item asks for at least one.

Sizing the replacement, teacher-led (8 min): Introduce the rational method, peak flow Q equals the runoff coefficient C times rainfall intensity i times drainage area A, with Q in cubic feet per second when i is in inches per hour and A is in acres. Then introduce the continuity equation, Q equals area times velocity, for finding a round pipe's capacity. Work one quick example with numbers unlike the handout's, such as C of 0.30, i of 3.0 inches per hour, and A of 20 acres, giving 18 cubic feet per second.

Handout items 6 through 10 (12 min): Students compute the two design flows, the existing culvert's capacity, and the diameter the replacement needs, then answer the closing probability item.

Formative check and close (5 min): Collect the handout at item 10 as an exit ticket and scan the responses. Move on only if at least 80 percent of the class gives 0.04 with independence as the reason. If fewer do, reteach the independence assumption at the start of the next class using the item 10 wording.

Draywick Road Culvert: Design Storm Risk Over 50 Years — Grade 12 Math
Name: ____________________________ Date: ______________

Student Handout

All values below describe the fictional town of Pellham and its Draywick Road culvert.

The existing culvert is a round pipe 48 inches, or 4.0 feet, in diameter, and it was designed for a 25-year storm. Engineers estimate that water moves through a culvert of this slope and roughness at 8.5 feet per second when it runs full. The drainage area above the crossing is 46 acres with a runoff coefficient of 0.55. The 25-year rainfall intensity for the design storm duration is 4.8 inches per hour, and the 100-year intensity is 6.5 inches per hour. The town plans for a 50-year service life on the replacement. Use area equals 0.785 times diameter squared, peak flow Q equals C times i times A, and Q equals area times velocity.

  1. Write the annual exceedance probability for a 25-year design storm as a decimal.

  2. Find the probability that the culvert sees no exceedance in 50 years. Round to three decimal places.

  3. Find the probability that the culvert sees at least one exceedance in 50 years. Report it as a percent, rounded to the nearest tenth.

  4. Pellham could instead build for a 100-year storm, with an annual exceedance probability of 0.01. Find the probability of at least one exceedance of that larger culvert in 50 years, as a percent rounded to the nearest tenth.

  5. For the 25-year design, solve for the number of years at which the chance of at least one exceedance first reaches 50 percent. Round to the nearest tenth of a year.

  6. Use the rational method to find the peak flow from the 46-acre drainage area during the 25-year storm, in cubic feet per second. Round to the nearest tenth.

  7. Use the rational method again to find the peak flow during the 100-year storm, in cubic feet per second. Round to the nearest tenth.

  8. Find the capacity of the existing 4.0-foot culvert running full at 8.5 feet per second, in cubic feet per second. Round to the nearest tenth.

  9. Pellham wants the replacement to carry the full 100-year flow at the same 8.5 feet per second. Find the smallest pipe diameter that will do it, rounded up to the nearest half foot.

  10. Pellham had a storm last year that exceeded the culvert's 25-year design. What is the probability of another exceedance this year? Explain your answer in one sentence.

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: 0.04.
    A 25-year return period means a 1 in 25 chance in any single year: 1 / 25 = 0.04.

  2. Answer: 0.130.
    The chance of no exceedance in one year is 1 - 0.04 = 0.96. Across 50 independent years: 0.96 raised to the 50th power = 0.1299, which rounds to 0.130.

  3. Answer: 87.0 percent.
    The chance of at least one exceedance is 1 - 0.130 = 0.870, or 87.0 percent. Over a 50-year life, a 25-year culvert is far more likely than not to be overtopped at least once.

  4. Answer: 39.5 percent.
    No exceedance in one year has probability 0.99. Across 50 years: 0.99 raised to the 50th power = 0.605. At least one exceedance = 1 - 0.605 = 0.395, or 39.5 percent. Building for the larger storm drops the 50-year risk from 87.0 percent to 39.5 percent, yet a roughly two in five chance of overtopping remains over the same service life.

  5. Answer: 17.0 years.
    Solve 0.96 raised to the power n = 0.50. Taking natural logs, n = ln(0.50) / ln(0.96) = (-0.69315) / (-0.04082) = 16.98, which rounds to 17.0 years. Within 17 years the culvert is as likely as not to have been overtopped.

  6. Answer: 121.4 cfs.
    Q = C x i x A = 0.55 x 4.8 x 46 = 121.44, which rounds to 121.4 cubic feet per second.

  7. Answer: 164.5 cfs.
    Q = 0.55 x 6.5 x 46 = 164.45, which rounds to 164.5 cubic feet per second.

  8. Answer: 106.8 cfs.
    Area = 0.785 x 4.0 x 4.0 = 12.56 square feet. Q = 12.56 x 8.5 = 106.76, which rounds to 106.8 cubic feet per second. The existing culvert cannot pass even the 25-year flow of 121.4, so it was undersized for its own design storm.

  9. Answer: 5.0 ft.
    Required area = 164.45 / 8.5 = 19.35 square feet. Solve 0.785 x diameter squared = 19.35, so diameter squared = 24.65 and diameter = 4.96 feet. Rounded up to the nearest half foot, the replacement is a 5.0-foot, or 60-inch, pipe, which carries 0.785 x 25 x 8.5 = 166.8 cubic feet per second.

  10. Answer: 0.04.
    The model treats each year as independent of the last, so last year's storm does not change this year's chance, which remains 1 in 25. Naming that assumption, and stating where it might fail, is what this item is checking.

Differentiation

Support

Give students a printed table of powers of 0.96 for the fictional Pellham culvert at 10, 20, 30, 40, and 50 years so item 2 becomes a lookup and item 5 can be estimated from the table before the logarithm is introduced. Provide items 6 and 7 with the rational method formula already substituted.

Extension

Ask students to find the service life at which the fictional town's 100-year culvert reaches the same 87 percent cumulative risk that the 25-year culvert reaches in 50 years, then write two sentences on what that comparison says about designing for a longer return period rather than replacing a pipe sooner.

ELL supports

Pre-teach "return period," "exceedance," "runoff coefficient," and "intensity" with a labeled sketch of the fictional crossing showing the drainage area, the pipe, and the road above it. Provide a two-row frame for items 2 and 3 labeled "chance of no exceedance" and "chance of at least one," so students see that the two answers sum to 1.

Extensions

Have students bring cost into the fictional Pellham decision. Give teams an installed price per linear foot for 48-inch, 60-inch, and 72-inch pipe, and a stated cost for one road washout, then ask them to compare the expected 50-year cost of each option using the cumulative risk figures they computed. The comparison shows why a public works department sometimes builds larger than the minimum code requirement, and it gives students a defensible number to put in front of a town board.