Math problem set Grade 7 Math
Marisco Canal Flow: A Grade 7 Proportional Reasoning Set
A grade 7 problem set where students use flow equals area times velocity on a fictional canal to explore proportional relationships and one flow-rate conversion.
Objectives
- Students will use flow equals area times velocity to find canal flow and velocity.
- Students will recognize flow as proportional to velocity, name the constant, and write its equation.
Vocabulary
Key words: flow, velocity, cross-section area, proportional
Teacher overview
Water moving through a channel follows a simple rule: flow equals the cross-section area times the water's velocity, written Q = A × V. On the fictional Marisco Canal the flow area stays fixed, so the flow becomes proportional to the velocity, with the area as the constant of proportionality. This set builds that idea and ends with one flow-rate conversion. A common error is treating flow and velocity as the same thing; velocity is a speed, while flow is how much water passes each second.
Problems
The fictional Marisco Canal has a cross-section flow area of 10 square feet. Use Q = A × V, where flow Q is in cubic feet per second and velocity V is in feet per second, for every problem.
- The water moves at 2 feet per second. Find the flow Q.
- The water speeds up to 3.5 feet per second. Find the flow Q.
- With the area fixed at 10 square feet, is flow Q proportional to velocity V? Find the constant of proportionality.
- Write an equation for the flow Q in terms of the velocity V for this canal.
- During a storm the canal must carry 45 cubic feet per second. Set up and solve an equation to find the velocity needed.
- Convert the storm flow of 45 cubic feet per second to gallons per minute, using 1 cubic foot per second = 448.8 gallons per minute.
- A second canal has a flow area of 15 square feet. At the same 2 feet per second as Problem 1, find its flow and compare it with the first canal's flow.
- Using Q = A × V, explain why doubling the velocity doubles the flow and why doubling the area also doubles the flow.
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: 20 cfs. Q = A × V = 10 × 2 = 20 cubic feet per second.
- Answer: 35 cfs. Q = 10 × 3.5 = 35 cubic feet per second.
- Answer: Yes; the constant is 10 square feet. For each pair, Q ÷ V = 20 ÷ 2 = 10 and 35 ÷ 3.5 = 10, a steady ratio, so flow is proportional to velocity with constant 10.
- Answer: Q = 10V. Multiply the velocity by the fixed area of 10 square feet to get the flow.
- Answer: 4.5 ft/s. Set up 45 = 10 × V, then divide both sides by 10: V = 45 ÷ 10 = 4.5 feet per second.
- Answer: 20,196 gpm. 45 cubic feet per second × 448.8 gallons per minute for each = 20,196 gallons per minute.
- Answer: 30 cfs, which is 10 cfs more. The second canal's flow is 15 × 2 = 30 cubic feet per second, compared with 20 for the first canal, so it carries 10 more, or 1.5 times as much.
- Answer: In Q = A × V, the flow is a product of area and velocity. If the area stays fixed and the velocity doubles, the product doubles. If the velocity stays fixed and the area doubles, the product doubles the same way, because doubling either factor doubles the result.
Differentiation
Support
Give students a filled-in table of velocity and flow for a few values so the proportional pattern is visible before Problem 3. Provide the conversion 1 cubic foot per second = 448.8 gallons per minute on a printed card for Problem 6.
Extension
Ask students to find what velocity the 15-square-foot second canal would need to carry the same storm flow of 45 cubic feet per second, compare it with the 4.5 feet per second from Problem 5, and explain what the comparison says about trading area for speed.
ELL supports
Provide a labeled cross-section drawing of a canal showing the flow area and an arrow for velocity, with the equation Q = A × V beside it. Allow students to answer Problems 3 and 8 with the labeled drawing plus numbers.