Lesson plan Grade 12 Math

Sable Ridge Pump Station: Pipe Diameter, Velocity, and Continuity

A grade 12 lesson using the continuity equation to model how pipe diameter changes affect flow velocity at a fictional pump station upgrade.

Grade: Grade 12
Subject: Math
Material: Lesson plan
Time: 50 min
Prep: Low prep
Materials: Calculators, Printed pipe-sizing handout, Chart paper for the recommendation write-up, Index cards for the exit ticket

Standards

HSG-MG.A.1 Common Core Math

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

How this material aligns: Students model the Sable Ridge main's cross-section as a circle to calculate pipe area from diameter before applying the continuity equation during guided and independent practice.

All materials for HSG-MG.A.1 →

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Students solve the continuity equation for the minimum pipe diameter that keeps velocity at or below the stated five feet-per-second design limit during the independent-practice pipe-sizing task.

All materials for HSA-CED.A.1 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-16.

Objectives

Students will calculate pipe cross-sectional area from diameter and use the continuity equation to find flow velocity.
Students will determine a minimum pipe diameter that keeps velocity within a stated design limit at a projected higher flow.

Vocabulary

Key words: continuity equation, cross-sectional area, velocity, design limit

Teacher overview

Sable Ridge Pump Station (a fictional facility used for this lesson) currently pushes water through a 12-inch main at a comfortable velocity, but the district's planners project that new development will roughly triple the flow this main must carry. Engineers rely on the continuity equation, Q = A × V, to answer a very practical question before a single foot of new pipe is bought: will the existing pipe still move water at a safe velocity once demand rises, or does the district need a larger pipe? This lesson walks students through that exact decision using Sable Ridge's projected growth.

A common misconception in this lesson is assuming that if flow triples, the pipe must become three times as wide to keep velocity in check. Cross-sectional area grows with the square of the diameter, so a modest increase in diameter absorbs a large increase in flow. Students see this during independent practice, when a 16-inch pipe, only one-third wider than the existing 12-inch main, carries three times today's flow within the 5 feet-per-second guideline.

Procedure

Hook (5 min): Show students a photo or sketch of a water main under a street and ask: if a town's water demand triples, does the pipe automatically need to be three times bigger? Collect a few predictions on the board without confirming or denying them.

Direct instruction (15 min): Introduce the continuity equation, Q = A × V (flow equals cross-sectional area times velocity), and the circle-area approximation used throughout the water industry, Area = 0.785 × Diameter². Model the current fictional Sable Ridge scenario: the main carries an average flow of 1.5 MGD, converted to 2.3205 cubic feet per second (cfs) using 1 MGD = 1.547 cfs, through a 12-inch (1-foot) diameter pipe with a cross-sectional area of 0.785 square feet. Solve together for the current velocity, V = Q ÷ A, showing that today's velocity is a comfortable 2.96 feet per second.

Guided practice (15 min): In pairs, students work with the projected flow of 4.5 MGD (6.9615 cfs) and answer:

What is the velocity in the existing 12-inch pipe at this projected flow?

Pairs discover a velocity of 8.87 feet per second, well above the district's engineering guideline that velocity in this main should not exceed 5 feet per second to limit pipe wear and water-hammer risk. Pairs then set up the equation A = Q ÷ V_max to find the minimum cross-sectional area needed at the 5 feet-per-second limit, and solve Area = 0.785 × Diameter² for the corresponding minimum diameter, finding that the main needs to be at least about 15.98 inches across.

Independent practice (10 min): Each student checks two candidate replacement pipe sizes against the 4.5 MGD projected flow and the 5 feet-per-second guideline, calculating the cross-sectional area from each diameter and then the resulting velocity:

For a 14-inch pipe, what is the resulting velocity, and does it meet the 5 feet-per-second guideline?
For a 16-inch pipe, what is the resulting velocity, and does it meet the 5 feet-per-second guideline?
Which of the two candidate pipe sizes should be recommended for the upgrade?

Students finish by writing one sentence justifying their recommendation using their velocity calculations.

Closure / exit ticket (5 min): Students answer one question on an index card: "If Sable Ridge's engineers instead raised the velocity guideline to 6 feet per second, would the existing 12-inch pipe now be acceptable at the 4.5 MGD projected flow? Use your earlier calculation to explain." Collect and scan for correct reasoning (comparing 8.87 feet per second to the new 6 feet-per-second limit) before dismissal.

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: 8.87 ft/s.
At 4.5 MGD, flow = 4.5 × 1.547 = 6.9615 cfs. In the existing 12-inch pipe, area = 0.785 × 1² = 0.785 sq ft. Velocity = 6.9615 ÷ 0.785 = 8.87 ft/s, which exceeds the 5 feet-per-second guideline.
Answer: 6.52 ft/s.
For a 14-inch pipe, diameter in feet = 14 ÷ 12 = 1.1667 ft. Area = 0.785 × 1.1667² = 1.0685 sq ft. Velocity = 6.9615 ÷ 1.0685 = 6.52 ft/s, which still exceeds the 5 feet-per-second guideline.
Answer: 4.99 ft/s.
For a 16-inch pipe, diameter in feet = 16 ÷ 12 = 1.3333 ft. Area = 0.785 × 1.3333² = 1.3956 sq ft. Velocity = 6.9615 ÷ 1.3956 = 4.99 ft/s, which meets the 5 feet-per-second guideline.
Answer: 16-inch pipe.
The 14-inch pipe still runs above the 5 feet-per-second guideline at the projected flow, while the 16-inch pipe brings velocity to just under the limit, so the 16-inch pipe is the smaller size that still satisfies the district's design guideline.

Exit ticket answer: No, the existing 12-inch pipe would still not be acceptable. Its calculated velocity at the projected flow is 8.87 feet per second, which is above even a relaxed 6 feet-per-second guideline, so raising the guideline from 5 to 6 feet per second is not enough to make the current pipe adequate; the district would still need to upsize the main, as in the 16-inch recommendation from problem 4.

Differentiation

Support (below grade level)

Provide the area formula pre-filled with the current 12-inch pipe's diameter already converted to feet (1.0 ft), and give students a table with columns for diameter (inches), diameter (feet), area (sq ft), and velocity (ft/s) so they only need to fill in one cell at a time rather than tracking every step from scratch.

Extension (above grade level)

Ask students to find the largest flow, in MGD, that the 16-inch pipe can carry at exactly 5.00 feet per second, and to explain what that result means for any growth in demand beyond the 4.5 MGD projection.

ELL supports

Pre-teach "diameter," "cross-sectional area," and "velocity" using a diagram of a pipe cut open to show a circular face, with each term labeled directly on the diagram. Provide the continuity equation and the area formula on a printed reference card that stays visible throughout the lesson, and allow students to record their calculations directly on a diagram rather than in full sentences.