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Lesson plan Grade 8 Math

Wickerstone Plant Inflow: The Shape of a 24-Hour Day

A grade 8 lesson on describing and sketching qualitative graphs, using a fictional water reclamation plant's twelve inflow readings across one day.

Objectives

  1. Students will describe where a graph increases, decreases, and levels off using a plant's inflow record.
  2. Students will sketch a graph that matches a written description of a tank filling, holding, and draining.
  3. Students will explain why the inflow record is a function of the time of day.

Vocabulary

Key words: inflow, peak, interval, increasing, decreasing, function

Teacher overview

Water arriving at a reclamation plant does not arrive evenly. Flow is lowest in the early morning while a town sleeps, rises sharply once people wake and shower, eases at midday, and rises again through the evening. Operators watch that shape on a screen all day, because the shape tells them when to run extra pumps. This lesson has students read the shape of one day at the fictional Wickerstone plant and then sketch graphs from written descriptions. Teaching note: students commonly read a falling section of a graph as "the plant received no water," confusing a decreasing amount with none at all. Ask repeatedly whether a section shows no water at all or a smaller amount than before.

Procedure

  1. Opening, 6 minutes. Ask students to predict the hour of the day when the most water would arrive at a plant serving their town, and record three predictions on the board without comment.
  2. Set the context, 6 minutes. Explain that the town of Wickerstone and its plant are fictional, but the daily pattern is typical. Define inflow as the water arriving at the plant, measured here in thousands of gallons each hour.
  3. Build the graph, 12 minutes. Distribute the handout. Have students plot the twelve readings on graph paper with time on the horizontal axis, then connect the points with a smooth curve rather than straight segments.
  4. Guided reading of the shape, 10 minutes. Work items 1 through 3 as a class. Mark the steepest rise and the steepest fall on a projected copy, and name each one aloud as increasing or decreasing.
  5. Independent work, 14 minutes. Students complete items 4 through 9. Item 6 asks for a sketch from words, so remind students that no numbers are needed on the axes for that item.
  6. Share and compare, 5 minutes. Post two or three student sketches for item 6 and ask the class which features every correct sketch must show, regardless of scale.
  7. Exit check, 2 minutes. Students write one sentence naming the two-hour interval with the sharpest rise and stating why an operator would care.
Wickerstone Plant Inflow: The Shape of a 24-Hour Day — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

The town of Wickerstone is fictional. Its water reclamation plant recorded the inflow every two hours for one day. Inflow is measured in thousands of gallons each hour.

Time Inflow
12 a.m. 180
2 a.m. 140
4 a.m. 110
6 a.m. 160
8 a.m. 300
10 a.m. 320
12 p.m. 280
2 p.m. 250
4 p.m. 260
6 p.m. 310
8 p.m. 290
10 p.m. 230
  1. Plot the readings and connect them with a smooth curve. During which two-hour interval does the inflow rise the most?
  2. During which two-hour interval does the inflow fall the most?
  3. At what time is the inflow lowest?
  4. Is the relationship between time and inflow linear or nonlinear? Explain what your graph shows.
  5. Which peak is higher, the morning peak or the evening peak?
  6. Sketch a graph for this description of a storage tank. The tank fills at a steady rate for 3 hours, holds the same level for 2 hours, then drains faster than it filled. Describe the three parts of your sketch in words.
  7. Each clock time in the table is paired with one inflow reading. Explain why that makes the day's record a function of time.
  8. Explain why an operator would care about the shape of the day rather than only the total water for the day.
  9. Suppose a storm sends a sudden surge of water into the plant at 3 p.m., and the surge fades over the next four hours. Describe how your graph from item 1 would change.

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: 6 a.m. to 8 a.m.
    The inflow climbs from 160 to 300, a rise of 140, which is larger than any other two-hour change in the table.
  2. Answer: 8 p.m. to 10 p.m.
    The inflow drops from 290 to 230, a fall of 60, which is the largest decrease in the table.
  3. Answer: 4 a.m.
    The reading of 110 at 4 a.m. is the smallest value in the table, and the curve reaches its low point there.
  4. Answer: Nonlinear.
    The curve rises, falls, rises again, and falls again, so the rate of change is not constant and the points do not lie on one straight line.
  5. Answer: The morning peak.
    The morning peak is 320 at 10 a.m., while the evening peak is 310 at 6 p.m., so the morning peak is higher by 10.
  6. Answer: Rise, flat, steeper fall.
    Accept any sketch that shows three parts in order: a straight segment rising with constant slope for 3 hours, a horizontal segment for 2 hours, and then a falling segment steeper than the rising one. Axis numbers are not required.
  7. Answer: Each time has exactly one reading.
    A function pairs each input with exactly one output. No clock time in the table appears twice with two different inflow readings, so the record is a function of time.
  8. Answer: The peaks decide the staffing and pumping.
    A plant sized only for the daily average would be overwhelmed at the morning and evening peaks. Knowing when the peaks arrive tells the operator when to bring extra pumps or basins online.
  9. Answer: A spike at 3 p.m. that fades by about 7 p.m.
    Accept any description showing a sharp rise starting at 3 p.m., a new high point above the ordinary afternoon values, and a gradual return toward the original curve over the following four hours.

Differentiation

Support

Give students graph paper with the axes labeled and the first four points already plotted, so the plotting task begins mid-table. For item 6, provide three cut-out graph segments, one rising, one flat, and one falling, and let students arrange them in order before drawing.

Extension

Ask students to sketch what the fictional plant's graph would look like on a day when a large factory in town closes for a holiday, and to mark which part of the day changes the most. Have them justify the change with one sentence about who sends water to the plant at each hour.

ELL supports

Pre-teach inflow, peak, steady, and drain with arrows drawn directly on a sample graph, labeling one rising section, one flat section, and one falling section. Provide the sentence frame for items 1 and 2: "The inflow ___ the most between ___ and ___."

Extensions

Have students track one household water use pattern for a day, such as the times a faucet or shower runs, and sketch a graph of that pattern with no numbers on the axes. Ask them to compare the shape of their household graph with the shape of the fictional Wickerstone plant graph and explain why the two look similar.