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Lesson plan Grade 8 Science

Kettleback River Gauge: Forecasting a Flood Crest

A grade 8 science lesson in which students build a linear function from a fictional river gauge table, predict the flood crest, and test where the model fails.

Objectives

  1. Students will determine the rate of change and initial value of a stream gauge record and write the matching linear function.
  2. Students will use a linear model to predict the time a river reaches a stated flood stage.
  3. Students will explain why a straight-line forecast breaks down near the crest of a flood.

Vocabulary

Stream gauge
A stream gauge is a station on a river or stream that continuously measures the water level, which is used to calculate how much water is flowing. Scientists and forecasters use the data to track floods, droughts, and water supply.
Flood stage
Flood stage is the water level in a river at which it starts to overflow its banks and cause flooding problems nearby. Forecasters compare the current water level to flood stage to warn people when flooding is about to happen.
Watershed
A watershed is all the land where rain that falls will drain to the same stream, river, or lake. Everyone lives in a watershed, no matter how far from the water they live.

Also: stage, crest, rate of change, linear function

Teacher overview

The fictional town of Wicklow Ford sits on a bend of the Kettleback River, and a stream gauge at Pettibone Bridge reports the river's height at fixed intervals through a storm. During a long spring rain the readings climb at a steady pace for most of a day, which lets students treat the gauge record as a table of values and build a linear function from it. They then use that function to estimate when the river reaches flood stage and how much warning the town would get. The misconception worth naming out loud is that a river stops rising when the rain stops; runoff from the upper watershed keeps arriving for hours, and the same straight-line model that fits the rise overshoots the crest by more than a foot.

Procedure

Hook (5 min): Project the first three rows of the gauge table from the handout and ask students to predict, without calculating, what the reading will be six hours later. Record two or three predictions on the board and leave them visible for the rest of the lesson.

Direct instruction (12 min): Explain what a stream gauge measures. The gauge reports stage, the height of the water surface above a fixed reference point at the bridge, not the depth of the channel. Model finding the rate of change from the first two rows of the table, stressing that the readings are two hours apart, so the difference must be divided by two to get a rate per hour. Write the function rule on the board using a different scenario than the handout, such as a gauge that starts at 4.0 feet and rises 0.5 feet per hour, and show how the initial value and the rate of change each appear in the rule. Ask students where the flood stage would appear on a graph of that example.

Guided practice (10 min): Students work in pairs on handout items 1 through 3. Circulate and check that pairs divide by two when converting the two-hour differences to an hourly rate, which is the most common error at this step. Ask one pair to explain at the board how they know the relationship is linear rather than simply increasing.

Independent practice (15 min): Students complete handout items 4 through 8 on their own. Item 6 compares the model's prediction with what the river actually did, and it is the item most worth pausing on if many students finish early.

Closure and exit ticket (8 min): Collect the handout and read item 8 aloud. Poll the class with thumbs up or down on the statement "a straight-line model can predict the crest as well as it predicts the rise." Fewer than half the class should agree; if more than half do, reteach the comparison in item 6 the following day before moving on.

Kettleback River Gauge: Forecasting a Flood Crest — Grade 8 Science
Name: ____________________________ Date: ______________

Student Handout

The fictional Wicklow Ford stream gauge at Pettibone Bridge reported these readings during one spring storm. Flood stage at this gauge is 18.0 feet in our scenario.

Time Gauge reading (feet)
6:00 in the morning 7.2
8:00 in the morning 8.8
10:00 in the morning 10.4
12:00 noon 12.0
2:00 in the afternoon 13.6
  1. Using any two rows of the table, find the rate of change of the gauge reading in feet per hour.
  2. The reading at 6:00 in the morning was 7.2 feet. Write a function rule for the gauge reading h, in feet, where t is the number of hours after 6:00 in the morning.
  3. Explain how you know from the table that the relationship between time and gauge reading is linear.
  4. Use your function rule to predict the gauge reading 12 hours after 6:00 in the morning.
  5. Flood stage at this gauge is 18.0 feet. Use your function rule to find how many hours after 6:00 in the morning the river reaches flood stage.
  6. The river actually crested at 16.9 feet, 14 hours after 6:00 in the morning, and then began to fall. By how many feet did your model overpredict the reading at that hour?
  7. The Wicklow Ford emergency manager wants the warning to go out 4 hours before the river reaches 18.0 feet. Using your answer to item 5, how many hours after 6:00 in the morning should the warning go out?
  8. Explain why a straight-line model fits the rising river well for the first several hours yet fails near the crest, and name one thing the town could build or install to reduce flood damage.

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: 0.8 feet per hour.
    The readings are two hours apart and each rises 1.6 feet, so 1.6 divided by 2 equals 0.8 feet per hour. Any pair of rows gives the same rate, which is what makes the record linear.
  2. Answer: h = 0.8t + 7.2.
    The rate of change 0.8 is the coefficient of t, and the 6:00 reading of 7.2 feet is the initial value.
  3. Answer: Every two-hour step adds the same 1.6 feet.
    Accept any response noting the constant difference between consecutive readings, or the equivalent statement that the points would fall on a straight line if graphed.
  4. Answer: 16.8 feet.
    Substituting t equals 12 gives 0.8 times 12 plus 7.2, which is 9.6 plus 7.2, or 16.8 feet.
  5. Answer: 13.5 hours.
    Solving 0.8t plus 7.2 equals 18.0 gives 0.8t equals 10.8, so t equals 13.5 hours after 6:00 in the morning, which is 7:30 in the evening.
  6. Answer: 1.5 feet.
    The model predicts 0.8 times 14 plus 7.2, which is 18.4 feet. The river actually read 16.9 feet, and 18.4 minus 16.9 equals 1.5 feet of overprediction.
  7. Answer: 9.5 hours.
    The model reaches flood stage 13.5 hours after 6:00 in the morning, and 13.5 minus 4 equals 9.5 hours, or 3:30 in the afternoon.
  8. Answer: The rise is steady while runoff keeps arriving, but the crest flattens and turns.
    Accept any explanation that the inflow from the watershed stays roughly constant during the storm, so the stage climbs at a nearly constant rate, while near the crest the inflow falls off and the river levels out and drops, which no straight line can show. For the second part accept any real mitigation measure, such as a levee, a detention basin upstream, a raised road crossing, or a gauge-triggered warning siren.

Differentiation

Support

Give these students a partially completed table with the hourly differences already filled in for the first two rows, and supply the function template h equals ___ times t plus ___ so the work is limited to identifying the two values. Allow a calculator for every item, and pair them with a partner for item 6.

Extension

Ask students to write a second function that models the falling limb of the flood, using the crest of 16.9 feet at hour 14 and a reading of 13.9 feet at hour 20, and then explain why the town needs both functions rather than one. Students should find a rate of change of negative 0.5 feet per hour.

ELL supports

Pre-teach "stage," "crest," and "flood stage" with a labeled side-view sketch of the bridge and the river, marking the gauge zero point and the 18.0-foot line. Provide sentence frames for items 3 and 8: "The relationship is linear because ___" and "The straight-line model stops working near the crest because ___."

Extensions

Students compare the gauge record with a rainfall record for the same storm and describe, in writing, the lag between the heaviest rain and the crest. This sets up a later discussion of why a large watershed crests later than a small one, and why forecasters watch upstream gauges rather than the sky above the town.