Worksheet Grade 8 Math
Windrow Farms Irrigation Rates: Comparing Proportional Slopes
A grade 8 worksheet where students find and compare the slopes of three proportional irrigation systems given as a table, an equation, and a graph.
Objectives
- Students will interpret the unit rate of a proportional relationship as the slope of its graph.
- Students will compare proportional relationships shown as a table, an equation, and a graph.
Vocabulary
Key words: proportional relationship, unit rate, slope
Teacher overview
When a sprinkler adds water at a steady rate, the gallons applied are proportional to the hours it runs, and the unit rate is the slope of the line. This worksheet gives three irrigation systems in three different forms, a table, an equation, and a graph, so students read a slope from each and compare them directly. A common mistake to watch for is comparing the systems by a single point instead of by their rates; a point that shows more gallons does not always belong to the faster system when the points are read at different hours. System C's point (3, 27,000) is higher than System A's point (2, 24,000), yet System C is slower.
Student Worksheet
Windrow Farms is a fictional farm that waters its fields with three sprinkler systems. Each system adds water at a steady rate, so gallons applied is proportional to hours run. Use gallons per hour as the unit rate (the slope) for each system.
System A is given by this table:
| Hours run | Gallons applied |
|---|---|
| 2 | 24,000 |
| 5 | 60,000 |
System B is given by the equation g = 15,000h, where g is gallons and h is hours.
System C is given by a graph that is a straight line through the points (0, 0) and (3, 27,000).
- Find the unit rate of System A in gallons per hour. This is the slope of its line.
- Write an equation for System A in the form g = (rate)h.
- What is the unit rate, or slope, of System B in gallons per hour?
- Find the slope of System C in gallons per hour using its two points.
- Rank Systems A, B, and C from the fastest rate to the slowest.
- If all three lines were graphed together, which one would be the steepest? Explain your choice.
- How many gallons does System B apply in 8 hours?
- System A must apply 90,000 gallons to a field. How many hours does it run?
- One acre-foot equals 325,851 gallons. How many gallons are in 2 acre-feet?
- Explain why a steeper line means a field gets its water in less time.
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: 12,000 gallons per hour.
Using the table, 24,000 / 2 = 12,000, and 60,000 / 5 = 12,000. Both give the same unit rate, so the slope is 12,000 gallons per hour. - Answer: g = 12,000h.
The rate is 12,000 gallons per hour, so gallons g equals 12,000 times hours h. - Answer: 15,000 gallons per hour.
In g = 15,000h, the number multiplied by h is the slope, so System B applies 15,000 gallons per hour. - Answer: 9,000 gallons per hour.
Slope = (27,000 - 0) / (3 - 0) = 27,000 / 3 = 9,000 gallons per hour. - Answer: B, A, C.
The rates are 15,000 for B, 12,000 for A, and 9,000 for C, so fastest to slowest is B, then A, then C. - Answer: System B.
System B has the greatest unit rate, 15,000 gallons per hour, so its line rises fastest and is the steepest. - Answer: 120,000 gallons.
15,000 gallons per hour x 8 hours = 120,000 gallons. - Answer: 7.5 hours.
90,000 gallons / 12,000 gallons per hour = 7.5 hours. - Answer: 651,702 gallons.
2 x 325,851 = 651,702 gallons in 2 acre-feet. - Answer: A steeper line applies more gallons each hour.
A steeper line has a larger slope, which is more gallons per hour, so the field reaches its target amount in fewer hours.
Differentiation
Support
Give students the slopes already labeled on a small graph of all three lines so they can see which is steepest before doing items 1 through 4 on their own. Provide the unit-rate reminder "slope = gallons divided by hours" at the top of the page.
Extension
Ask students to add a fourth system, System D, that is faster than A but slower than B, write its equation, and prove with a calculation that its rate falls between 12,000 and 15,000 gallons per hour. Have them place its line correctly among the other three.
ELL supports
Pre-teach "rate," "slope," and "steep" using a ramp picture that connects a steeper ramp to a faster climb. Provide a sentence frame for item 6: "System ___ is steepest because its rate is ___ gallons per hour."