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Worksheet Grade 8 Math

Ilmenwood Pipe Age and Main Breaks: Judging a Line of Fit

A grade 8 worksheet where students plot pipe age against main breaks for a fictional district, use a given line of fit to predict, and judge how well the line fits.

Objectives

  1. Students will construct a scatter plot from a table of pipe ages and break rates.
  2. Students will use a given line of fit to predict a break rate for an age not in the table.
  3. Students will judge how well a line fits by measuring how far the points sit from it.

Vocabulary

Key words: line of fit, slope, predict, scatter plot

Teacher overview

Utilities decide which pipes to replace first by tracking how often each group of mains breaks, and older iron pipe usually breaks more often than newer pipe. The fictional Ilmenwood Water District sorted its mains into eight age groups and counted the breaks in each group over one year, reporting the count for every 100 miles of pipe so that groups of different sizes could be compared. A common error on this worksheet is treating the line of fit as if it must pass through every point; the line is a model of the trend, and students are asked to measure how far the points sit from it rather than to force it through them.

Ilmenwood Pipe Age and Main Breaks: Judging a Line of Fit — Grade 8 Math
Name: ____________________________ Date: ______________

Student Worksheet

The table shows eight groups of mains in the fictional Ilmenwood Water District. Age is the age of the pipe in years. Breaks is the number of breaks in one year for every 100 miles of that pipe.

Age of pipe (years) Breaks per 100 miles
30 6
40 12
50 20
60 31
70 36
80 46
90 52
100 62

The district's engineer drew this line of fit through the points, where a is the age in years and b is the breaks per 100 miles:

b = 0.8a - 19

  1. Plot the eight points on graph paper, with age across the bottom and breaks up the side, then draw the engineer's line on the same axes.
  2. As the pipe gets older, does the break count rise or fall? Name the type of association.
  3. Use the line b = 0.8a - 19 to predict the breaks per 100 miles for pipe that is 55 years old.
  4. Use the line b = 0.8a - 19 to predict the breaks per 100 miles for pipe that is 75 years old.
  5. The line predicts 29 breaks for 60-year-old pipe. The table shows 31 breaks. How far is that point from the line?
  6. Is the point for 60-year-old pipe above the line or below it?
  7. The line predicts 53 breaks for 90-year-old pipe. The table shows 52 breaks. How far is that point from the line?
  8. The slope of the line is 0.8. Explain what that slope means for pipe in this district.
  9. Solve 0.8a - 19 = 45 to find the age at which the line predicts 45 breaks per 100 miles.
  10. For pipe that is 10 years old, the line gives a negative number of breaks. Explain why the line should not be used for pipe that young.
  11. No point in the table sits more than 2 breaks away from the line. Is the line a good fit for this data? Explain.

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: A completed scatter plot with the line drawn on it. Accept any plot with age from about 20 to 110 across the bottom, breaks from 0 to about 70 up the side, the eight points placed correctly, and a straight line passing near them from about (30, 5) to about (100, 61).
  2. Answer: The break count rises, so the association is positive. Each older group in the table has more breaks than the group before it.
  3. Answer: 25 breaks per 100 miles. Multiply 0.8 by 55 to get 44, then subtract 19 to get 25.
  4. Answer: 41 breaks per 100 miles. Multiply 0.8 by 75 to get 60, then subtract 19 to get 41.
  5. Answer: 2 breaks per 100 miles. Subtract the predicted value from the table value: 31 minus 29 equals 2.
  6. Answer: Above the line. The table value of 31 is larger than the predicted value of 29, so the point sits above the line by 2.
  7. Answer: 1 break per 100 miles. Subtract the table value from the predicted value: 53 minus 52 equals 1, so the point sits 1 below the line.
  8. Answer: For each extra year of pipe age, the line predicts about 0.8 more breaks per 100 miles in a year. Across a ten-year difference in age, that comes to about 8 more breaks per 100 miles.
  9. Answer: 80 years. Add 19 to both sides to get 0.8a = 64, then divide both sides by 0.8 to get a = 80.
  10. Answer: The line gives 0.8 times 10 minus 19, which is negative 11, and a pipe group cannot have a negative number of breaks. The line was drawn to match pipe from 30 to 100 years old, so it should not be stretched to ages far outside that range.
  11. Answer: Yes, the line fits well. Every point lies within 2 breaks of the line, which is small compared with the range from 6 to 62 breaks across the table, so the line describes the trend closely.

Differentiation

Support

Provide graph paper with the axes drawn, labeled, and scaled, and with the first two points already plotted. Give these students a filled-in row showing the substitution for one prediction, and mark items 1 through 7 as the required set.

Extension

Ask students to write a different line of fit for the same eight points, using two points of their own choosing to find a slope and an intercept, then compare how far the points sit from their line and from the engineer's line. Have them state which line they would give the district and why.

ELL supports

Post a labeled diagram showing a scatter plot with a line through it, with arrows marked "above the line" and "below the line." Provide the sentence frames "When the age goes up by 1 year, the breaks go up by ___" and "This point is ___ breaks above the line."

Extensions

Have students look up the total miles of water main in their own community, then use the engineer's line to estimate how many breaks a system of that size would see in a year if all of its pipe were 70 years old, and explain why the real number would differ.