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Lesson plan Grade 8 Math

Solano Flat Reservoir Evaporation: Building a Linear Function

A 50-minute grade 8 lesson where students build a linear function for a reservoir losing depth in a dry spell and read its slope and starting value.

Objectives

  1. Students will construct a linear function from a described rate of change and starting value.
  2. Students will interpret the slope and initial value in terms of a draining reservoir.

Vocabulary

Evaporation
Evaporation happens when liquid water, often warmed by the sun, turns into an invisible gas called water vapor that rises into the air. It is how water leaves oceans, lakes, and puddles.

Also: linear function, rate of change, initial value

Teacher overview

During a dry spell, a reservoir loses depth at a fairly steady rate as water evaporates and small releases continue, which makes its depth over time a linear function. This lesson uses the fictional Solano Flat Reservoir so students can build the function, read its rate of change as the daily loss, and read its initial value as the starting depth. A common student misconception is treating any decreasing pattern as having a positive rate of change; the lesson keeps the negative sign attached to the loss so the slope's meaning stays clear.

Procedure

Hook (5 min): Ask students what happens to a puddle on a hot day, then to a whole reservoir over a rainless month. Record a few predictions about how fast the level might drop, without judging them yet.

Direct instruction (12 min): Introduce Solano Flat Reservoir: it begins a dry spell 32 feet deep and loses 0.4 feet each day. Model writing the function d = 32 - 0.4t, where t is days and d is depth in feet. Label 32 as the initial value (the depth at day 0) and -0.4 as the rate of change (feet lost per day). Substitute t = 10 together to show d = 28 feet.

Guided practice (13 min): Work Student Handout items 1 through 5 as a class, pausing after item 3 for a check for understanding: ask every student to write, on a mini-whiteboard or scrap paper, what the -0.4 means in words. Look for "the depth drops 0.4 feet each day." Re-teach if more than a few miss it.

Independent practice (15 min): Students complete Student Handout items 6 through 8 on their own, including solving for the day the reservoir reaches 20 feet and comparing a second, slower-losing reservoir.

Closure and exit ticket (5 min): On an index card, students use d = 32 - 0.4t to find the depth after 5 days. Collect the cards and scan for 30 feet, which is 32 - 0.4 × 5; this is the formative check that students can substitute correctly.

Solano Flat Reservoir Evaporation: Building a Linear Function — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

Solano Flat Reservoir is fictional. It begins a dry spell 32 feet deep and loses 0.4 feet of depth each day. Let t stand for the number of days.

  1. Write a function for the reservoir's depth d after t days.
  2. What is the rate of change, and what does it mean here?
  3. What is the initial value, and what does it mean here?
  4. Find the depth after 10 days.
  5. Find the depth after 25 days.
  6. The intake pipe stops working below 20 feet. On what day does the reservoir reach 20 feet? Solve 32 - 0.4t = 20.
  7. Make a table of the depth at t = 0, 15, and 30 days.
  8. A nearby reservoir also starts at 32 feet but loses only 0.25 feet each day. Write its function and find its depth after 30 days.

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 = 32 - 0.4t. The starting depth is 32 feet, and 0.4 feet is subtracted for each day t.
  2. Answer: -0.4 feet per day. The rate of change is negative because the depth drops 0.4 feet each day.
  3. Answer: 32 feet. The initial value is the depth at day 0, before any water is lost.
  4. Answer: 28 feet. d = 32 - 0.4(10) = 32 - 4 = 28 feet.
  5. Answer: 22 feet. d = 32 - 0.4(25) = 32 - 10 = 22 feet.
  6. Answer: 30 days. 32 - 0.4t = 20, so -0.4t = -12, and t = 30 days.
  7. Answer: 32, 26, and 20 feet. At t = 0 depth is 32; at t = 15 depth is 32 - 6 = 26; at t = 30 depth is 32 - 12 = 20 feet.
  8. Answer: 24.5 feet. The function is d = 32 - 0.25t; at t = 30, d = 32 - 0.25(30) = 32 - 7.5 = 24.5 feet.

Differentiation

Support

Provide the function d = 32 - 0.4t already written and a filled example (t = 10 gives 28 feet) so students focus on interpreting the slope and initial value. Allow calculators and a number line marked in feet for the substitution items.

Extension

Ask students to find the day the two reservoirs are 6 feet apart in depth, writing and solving an equation that sets the difference of the two functions equal to 6, then explain what that day means for the drier reservoir.

ELL supports

Pre-teach "rate of change," "initial value," and "evaporation" with a side-view sketch of a reservoir and a falling water line labeled day by day. Provide sentence frames: "The initial value is ___ feet, which means ___," and "The rate of change is ___, which means the depth ___ each day."