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Lesson plan Grade 8 Math

Pinewick Well Drawdown: Building a Linear Function

A 50-minute grade 8 lesson where students build and interpret a linear function for the water depth in a fictional well as its pump draws the level down.

Objectives

  1. Students will construct a linear function for a well's water depth from a rate and a starting value.
  2. Students will interpret the rate of change and initial value of the function in context.

Vocabulary

Drawdown
Drawdown is how far the water level in a well drops when a pump pulls water out. The harder and longer you pump, the bigger the drawdown gets.

Also: linear function, rate of change, initial value

Teacher overview

The fictional Pinewick well field draws drinking water from an aquifer. Before a pump starts, the water sits at a resting depth called the static level. Once the pump runs, the water level drops a little deeper each hour, a change called drawdown, which makes the depth a linear function of pumping time. In a real well the drop is fastest at first and then slows, so the steady rate used here is a simplified model. This lesson has students build that function and read its parts. A common misconception is treating the starting depth as the rate; students should see the static level as the initial value and the steady per-hour drop as the rate of change.

Procedure

Hook (5 min): Ask students what they think happens to the water level inside a well while a pump is running. Collect a few predictions, then confirm that the level drops, which is why operators watch it closely, and explain that this lesson treats the drop as steady to build a simple model.

Direct instruction (12 min): Model a linear function with a demonstration well that is not on the handout. This demo well has a static depth of 20 feet, and the water drops 1 foot each hour of pumping. Build the function D = 20 + t together, and label 20 as the initial value (the static level) and 1 as the rate of change (the drawdown rate). Evaluate it at 4 hours to get 24 feet.

Guided practice (13 min): As a class, change the demo well's drawdown rate to 2 feet per hour and rewrite the function as D = 20 + 2t. Ask a check-for-understanding question: "Which number changed, the initial value or the rate of change?" Expect students to identify that only the rate of change changed. Evaluate the new function at 5 hours together.

Independent practice (15 min): Students complete the Student Handout for the Pinewick well and a second well, building and interpreting each linear function.

Closure and exit ticket (5 min): On an index card, students write the function for a well with a static depth of 35 feet that drops 2 feet per hour, then find the depth after 3 hours. Collect and check that the function is D = 35 + 2t and the depth is 41 feet.

Pinewick Well Drawdown: Building a Linear Function — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

Use the fictional Pinewick well. Its static depth to water is 40 feet. While the pump runs, the water drops 2 feet deeper each hour. Let t stand for hours of pumping and D for the depth to water in feet.

  1. Write a linear function for the depth to water, D, after t hours of pumping.
  2. State the rate of change and the initial value of your function, and explain what each one means for the well.
  3. What is the depth to water after 6 hours of pumping?
  4. The pump must shut off before the water reaches 70 feet deep. After how many hours does the water reach 70 feet?
  5. A second well, Well B, has a static depth of 30 feet and drops 3 feet each hour, so its function is D = 30 + 3t. What is Well B's depth after 6 hours?
  6. After 6 hours, which well has water closer to the surface, and by how many feet?
  7. Explain your reasoning: What does a larger drawdown rate tell you about how quickly a well's water level is falling?

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: D = 40 + 2t.
    Start at the static depth of 40 feet and add 2 feet for each hour t, so D = 40 + 2t.
  2. Answer: rate of change 2 feet per hour, initial value 40 feet.
    The rate of change is 2 feet per hour, the drawdown rate the water drops while pumping. The initial value is 40 feet, the static depth before the pump starts.
  3. Answer: 52 feet.
    D = 40 + 2(6) = 40 + 12 = 52 feet.
  4. Answer: 15 hours.
    40 + 2t = 70. Subtract 40: 2t = 30. Divide by 2: t = 15 hours.
  5. Answer: 48 feet.
    D = 30 + 3(6) = 30 + 18 = 48 feet.
  6. Answer: Well B by 4 feet.
    Well B is at 48 feet and Pinewick is at 52 feet, so Well B's water is closer to the surface by 52 - 48 = 4 feet.
  7. Answer: A larger drawdown rate means the water level is falling faster, so the depth increases more for each hour of pumping and the well reaches its shut-off depth sooner.

Differentiation

Support

Provide the function form D = (initial value) + (rate) times t as a fill-in template, and pre-label the static level and drawdown rate for the Pinewick well before item 1. Allow a calculator for every computation item.

Extension

Ask students to find the drawdown rate a well would need so that, starting from a static depth of 40 feet, it reaches 70 feet in exactly 10 hours, then write and check the function.

ELL supports

Pre-teach "static level," "drawdown," and "depth to water" with a labeled cross-section of a well showing the resting water line and a lower pumping water line. Provide a sentence frame for item 2: "The rate of change is ___, which means ___; the initial value is ___, which means ___."