Math problem set Grade 8 Math
Redwater Pump Test: Comparing Two Pump Rates as Slopes
A grade 8 problem set comparing two fictional pumps, one shown as a table and one as an equation, by reading each flow rate as the slope of its graph.
Objectives
- Students will find the unit rate of a proportional relationship as the slope of its graph.
- Students will compare two proportional relationships shown in different forms.
Vocabulary
Key words: proportional, slope, unit rate
Teacher overview
The Redwater station is a fictional pumping station testing two new pumps. Each pump moves water at a steady rate, so the gallons pumped and the minutes running form a proportional relationship whose graph is a straight line through the origin. This set gives one pump as a table and the other as an equation, so students compare rates shown in different forms. A common mistake is reading the slope backward, as minutes per gallon; students should keep the rate as gallons per minute, the amount pumped divided by the time.
Problems
The fictional Redwater station tests two pumps. Pump A is shown by the table below. Pump B is shown by the equation g = 75t, where g is gallons pumped and t is minutes.
| Pump A: minutes (t) | Pump A: gallons (g) |
|---|---|
| 2 | 120 |
| 4 | 240 |
| 6 | 360 |
| 8 | 480 |
- Find Pump A's rate in gallons per minute from the table.
- Write an equation for Pump A in the form g = (rate) times t.
- What is Pump B's rate in gallons per minute? Read it from the equation g = 75t.
- Which pump is faster, and by how many gallons per minute?
- If you graphed both pumps with minutes across and gallons up, which pump's line would be steeper, and why?
- How many gallons does Pump A move in 10 minutes?
- How many gallons does Pump B move in 10 minutes?
- Pump A fills a 3,000-gallon tank. How many minutes does that take?
- Pump B fills the same 3,000-gallon tank. How many minutes does that take?
- Explain your reasoning: Both graphs pass through the point (0, 0). What does that point mean for these pumps?
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: 60 gallons per minute.
Divide gallons by minutes from any row: 120 / 2 = 60, and 240 / 4 = 60, so the rate is 60 gallons per minute. - Answer: g = 60t.
Pump A moves 60 gallons each minute, so gallons equal 60 times the minutes: g = 60t. - Answer: 75 gallons per minute.
In g = 75t, the number multiplied by t is the rate, so Pump B moves 75 gallons per minute. - Answer: Pump B by 15 gallons per minute.
Pump B moves 75 and Pump A moves 60, so Pump B is faster by 75 - 60 = 15 gallons per minute. - Answer: Pump B, because it has the larger slope.
Slope is the rate. Pump B's rate of 75 is greater than Pump A's rate of 60, so Pump B's line rises faster and is steeper. - Answer: 600 gallons. g = 60 times 10 = 600 gallons.
- Answer: 750 gallons. g = 75 times 10 = 750 gallons.
- Answer: 50 minutes.
Divide the tank size by the rate: 3,000 / 60 = 50 minutes. - Answer: 40 minutes.
Divide the tank size by the rate: 3,000 / 75 = 40 minutes. - Answer: The point (0, 0) means that at 0 minutes each pump has moved 0 gallons. Pumping starts from empty, which is why both proportional lines begin at the origin.
Differentiation
Support
Fill in the first row of ratio work for item 1 (120 divided by 2) and provide the equation form g = (rate) times t already started for item 2. Allow a calculator for the division in items 8 and 9.
Extension
Ask students to find a third pump's rate so that it fills the 3,000-gallon tank in exactly 30 minutes, write its equation, and state where its line would sit compared with Pump A and Pump B.
ELL supports
Pre-teach "rate," "slope," and "proportional" with a picture of a straight line rising from the origin. Provide a sentence frame for item 5: "Pump ___ has the steeper line because its rate is ___."