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Worksheet Grade 8 Engineering & Design

Sable Reach Brook: Reading a Streambed Slope Profile

A grade 8 worksheet turning a streambed survey table into a linear function, comparing two reach slopes, and placing rock riffles where storm flow runs fastest.

Objectives

  1. Students will calculate the rate of change of a streambed from a table of survey elevations.
  2. Students will write and use a linear function for streambed elevation at a given station.
  3. Students will use two slope values to decide where a restoration crew should place rock riffles.

Vocabulary

Key words: station, streambed, rate of change, riffle, reach

Teacher overview

Sable Reach Brook is a fictional stream that runs behind a middle school and washes out its banks during heavy storms. A survey crew walked the channel and recorded the streambed elevation every 200 feet, and a restoration team now has to decide where to place rock riffles that slow the water. This worksheet turns that survey table into a linear function, compares the slope of the upper reach with the slope of the lower reach, and uses the comparison to place the rock. Students often report the total drop as the rate of change, so items 1 and 2 separate the drop in feet from the drop per foot of channel length.

Sable Reach Brook: Reading a Streambed Slope Profile — Grade 8 Engineering & Design
Name: ____________________________ Date: ______________

Student Worksheet

The town of Sable Reach surveyed 1,000 feet of the brook below the school bridge. Station 0 is at the bridge. Use this survey table for every item, and show your work.

Station (feet below the bridge) Streambed elevation (feet)
0 118.0
200 116.0
400 114.0
600 112.0
800 111.0
1,000 110.0
  1. How many feet does the streambed drop between station 0 and station 600?
  2. Write the rate of change for that upper reach in feet of drop for each foot of length.
  3. Write a linear function for E, the streambed elevation at station x, for stations 0 through 600.
  4. Use your function to find the streambed elevation at station 300.
  5. How many feet does the streambed drop between station 600 and station 1,000?
  6. Write the rate of change for that lower reach in feet of drop for each foot of length.
  7. Which reach is steeper? Use both rates in your answer.
  8. Describe the graph of elevation against station for the whole 1,000 feet. Say what happens at station 600.
  9. Water moves faster where the bed is steeper. In which reach would you expect the fastest water during a storm?
  10. The team places rock riffles where the water moves fastest. Which stations should they target?
  11. The team proposes adding rock so the upper reach drops only 0.005 feet for each foot of length, starting from station 0. Under that plan, what would the elevation be at station 400?
  12. How deep would the new rock have to be at station 400 to reach the elevation you found in item 11?

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 feet.
    118.0 - 112.0 = 6.0 feet of drop between station 0 and station 600.
  2. Answer: 0.01 feet per foot.
    6 feet of drop ÷ 600 feet of length = 0.01 feet per foot. This is the slope of the upper reach.
  3. Answer: E = 118 - 0.01x.
    The elevation starts at 118.0 feet at station 0 and falls 0.01 feet for each foot downstream, so the rate of change is negative.
  4. Answer: 115.0 feet.
    E = 118 - 0.01(300) = 118 - 3 = 115.0 feet. The table supports this, since station 300 sits halfway between 116.0 and 114.0.
  5. Answer: 2 feet.
    112.0 - 110.0 = 2.0 feet of drop between station 600 and station 1,000.
  6. Answer: 0.005 feet per foot.
    2 feet of drop ÷ 400 feet of length = 0.005 feet per foot.
  7. Answer: The upper reach.
    The upper reach drops 0.01 feet per foot and the lower reach drops 0.005 feet per foot. The upper rate is twice the lower rate, so that reach is steeper.
  8. Answer: Two straight segments that both fall, with a bend at station 600.
    From station 0 to station 600 the line falls steeply. At station 600 the line bends and becomes less steep, then continues falling to station 1,000. The graph is not one straight line across the whole survey.
  9. Answer: The upper reach.
    The upper reach falls at twice the rate of the lower reach, and a steeper bed lets gravity pull the water downhill faster, so storm flow speeds up there first.
  10. Answer: Stations 0 through 600.
    That is the steep reach, where the fast water does the most damage to the banks. Riffles placed there break up the flow before it reaches the flatter lower reach.
  11. Answer: 116.0 feet.
    E = 118 - 0.005(400) = 118 - 2 = 116.0 feet under the proposed plan.
  12. Answer: 2.0 feet.
    The current elevation at station 400 is 114.0 feet, and the plan calls for 116.0 feet, so the crew would place 116.0 - 114.0 = 2.0 feet of rock at that station.

Differentiation

Support

Provide a third column in the survey table labeled "drop from the row above," with the first two rows filled in, so the pattern of a 2.0-foot drop every 200 feet is visible before students compute a rate. Give the form y = mx + b printed beside item 3.

Extension

Have students extend the survey a further 400 feet, using sample elevations of 109.6 and 109.4 feet at stations 1,200 and 1,400. Ask them to compute that third reach's rate, explain what the flattening pattern suggests about where sediment settles, and mark the likely deposit zone.

ELL supports

Provide a side-view sketch of the brook with the bridge, the six stations, and the elevations labeled, so the word "station" is anchored to a picture before item 1. Allow students to answer items 7 through 10 with a labeled sketch and the two rate values rather than full sentences.