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Lesson plan Grade 8 Math

Ledworth Filter Run: Reading a Graph That Is Not a Line

A grade 8 lesson where students describe a fictional sand filter's headloss graph, decide where it is linear or curved, and sketch one from words.

Objectives

  1. Students will describe where a graph is increasing and where it is decreasing using intervals of the input.
  2. Students will distinguish the linear part of a graph from the nonlinear part and justify the choice with the data.
  3. Students will sketch a graph that shows the qualitative features of a relationship given only in words.

Vocabulary

Backwash
Backwash is when a treatment plant reverses the flow of water through a filter to flush out the dirt and particles it has trapped. It is like rinsing out a dirty sponge so the filter can keep working well.

Also: headloss, filter run, linear, nonlinear, interval, steepness

Teacher overview

Operators at the fictional Ledworth Water Treatment Plant track a measurement called headloss, which is the pressure the water gives up as it passes through a bed of filter sand. Headloss rises slowly while the sand is clean, rises faster as particles pack into the bed, and falls back to its starting value once the filter is washed. Students plot one filter's record from the fictional Ledworth plant and describe it in words, then sketch a second filter's graph from a verbal description. A misconception worth naming aloud is that students read the tallest part of a graph as the place where the quantity is changing fastest; steepness, not height, shows the rate of change, and on a curved graph the highest point and the steepest point are not always the same point.

Procedure

Hook (5 min): Project the headloss table from the Student Handout without any commentary and ask what students think the graph will look like. Record two or three predictions on the board. Ask whether they expect a straight line, and keep both answers visible for the rest of the period.

Direct instruction (12 min): Plot the first five rows together on the board, using hours on the horizontal axis and feet of headloss on the vertical axis. Show that the readings rise by the same amount every six hours, so that stretch of the graph is linear. Then plot the last four rows and show that the rises grow larger each time, so the graph curves upward. Introduce the language the handout expects: increasing, decreasing, linear, nonlinear, and steepest. Model one full sentence of description, such as "from hour 0 to hour 24 the graph increases at a constant rate."

Guided practice (10 min): As a class, describe the interval from hour 42 to hour 48 in the same sentence frame, then describe what the graph does at hour 48, when the operator at the fictional Ledworth plant washes the filter. Ask a check-for-understanding question with a thumbs poll: "The graph is steepest near hour 0, where it starts." Students point up for true and down for false. If fewer than three quarters of the class points down, work one more interval together and repeat the poll before moving on.

Independent practice (13 min): Students complete the Student Handout individually, plotting the full record, answering the description items, and sketching the graph for Filter 5 from the written description in item 7. Circulate and check that axes are labeled with units.

Closure and exit ticket (5 min): On an index card, students write one sentence describing a graph that rises slowly, then rises steeply, then drops straight down. Collect the cards and sort them into two piles, those that mention the change in steepness and those that do not, as the entry point for the next lesson.

Ledworth Filter Run: Reading a Graph That Is Not a Line — Grade 8 Math
Name: ____________________________ Date: ______________

Student Handout

A filter at the fictional Ledworth Water Treatment Plant was watched for one filter run. The table gives the headloss in feet at six-hour intervals. At hour 48 the operator washed the filter, and the headloss went back to 0.2 feet.

Hours into the run Headloss (feet)
0 0.2
6 0.5
12 0.8
18 1.1
24 1.4
30 1.9
36 2.6
42 3.6
48 5.0

Plot the nine points on graph paper. Put hours on the horizontal axis and feet of headloss on the vertical axis. Label both axes with their units. Then answer the items below.

  1. Between hour 0 and hour 48, is the headloss increasing or decreasing? Give the interval in your answer.
  2. Look only at hour 0 through hour 24. Is that part of the graph linear or nonlinear? Use the table values to support your answer.
  3. Look only at hour 24 through hour 48. Is that part of the graph linear or nonlinear? Use the table values to support your answer.
  4. Over which six-hour interval is the graph steepest?
  5. At hour 48 the filter is washed. Describe what the graph does at that moment.
  6. A student says the headloss is climbing fastest at hour 0 because that is where the graph starts. Is the student correct? Explain what part of the graph actually shows how fast headloss is climbing.
  7. Filter 5 at the same plant behaves this way: headloss rises slowly and steadily for two days, then rises faster and faster for one more day, then drops straight back to where it began. Sketch a graph of headloss against time that shows those features. Label the axes.
  8. Describe in words a headloss graph that would tell an operator to wash a filter earlier than planned.

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: Increasing over the whole interval from hour 0 to hour 48. Every reading in the table is larger than the one before it, so the graph rises from left to right across that interval.
  2. Answer: Linear. The headloss rises 0.3 feet in each six-hour step, from 0.2 to 0.5 to 0.8 to 1.1 to 1.4 feet, so the rate of change is constant and the points fall on a straight line.
  3. Answer: Nonlinear. The rises are 0.5, 0.7, 1.0, and 1.4 feet, so each step is larger than the one before it. A constant rise is required for a line, and this part of the graph curves upward.
  4. Answer: From hour 42 to hour 48. The headloss rises 1.4 feet over that interval, which is more than any other six-hour step in the table.
  5. Answer: The graph drops sharply, from 5.0 feet back down to 0.2 feet, and then begins rising again. Washing the filter returns the headloss to about where the run started.
  6. Answer: The student is not correct. Height on the graph shows how much headloss there is, not how fast it is changing. Steepness shows the rate of change, and the graph is flattest near hour 0 and steepest near hour 48, so headloss is climbing slowest at the start.
  7. Answer: A sketch that rises gently and in a straight line for the first two days, then bends upward and grows steeper through the third day, then falls almost vertically to its starting height. Accept any sketch with labeled axes that shows those three features in that order; exact values are not required.
  8. Answer: A graph that bends upward much sooner than expected, so the steep part arrives early in the run. That shape means headloss is building quickly and the filter will reach its limit before the scheduled wash.

Differentiation

Support

Give these students the plotted graph already drawn for the fictional Ledworth record, so the work begins at the description stage rather than at plotting. Provide the sentence frame "From hour ___ to hour ___ the graph is ___ because ___" and ask for items 1 through 5 only. Two highlighters help: one color for the straight stretch and one for the curved stretch.

Extension

Ask students to add a second curve to their graph for a filter that starts a run with headloss already at 1.0 feet and reaches 5.0 feet in only 30 hours, then to write two sentences comparing the two runs in terms of steepness rather than height.

ELL supports

Pre-teach headloss, filter run, and backwash with a simple cross-section drawing of a sand bed, showing clean sand and clogged sand side by side. Post a word bank with increasing, decreasing, linear, nonlinear, steep, and level, and pair each word with a small arrow sketch. Allow students to answer items 1 through 5 orally before writing.

Extensions

Have students collect a real record of something that fills or clogs over time, such as the time a classroom sink takes to drain with a partly blocked strainer measured once a week, and describe the resulting graph using the same vocabulary. Ask them to state which part of their graph is linear and which part, if any, is not, and to explain what event would make their graph drop suddenly the way the fictional Ledworth graph drops at hour 48.