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Lesson plan Grade 6 Math

Corsley's Pumping Log: A Dot Plot of Fifteen Days

A grade 6 lesson on a small town's fifteen-day pumping log: students build a dot plot, then test how one main-break day pulls the mean away from the median.

Objectives

  1. Students will build a dot plot from fifteen days of daily pumping totals.
  2. Students will compute the mean, median, and range of a data set and interpret each one in context.
  3. Students will explain why a single unusual day moves the mean more than it moves the median.

Vocabulary

Key words: dot plot, outlier, mean, median, range, distribution

Teacher overview

In the fictional town of Corsley, the operator on duty writes one number in a logbook at the end of every shift: the total gallons pumped from the wells into the storage tank that day. Fifteen of those numbers make a data set that students can plot by hand in a few minutes, and one of the fifteen days is unusual, because a water main broke while the fire department was also drawing from a hydrant. The misconception this lesson addresses is the belief that the mean is always the best summary of a data set. Here the mean sits above every ordinary day in the log, so a student who reports it as the typical day is describing a level the town reached only once.

Procedure

Hook (5 min): Write five numbers on the board as five days from a smaller system in the town of Corsley: 12, 13, 12, 14, and 59 thousand gallons. Ask students for the typical day. Take two or three answers and write them down without ruling on them, then say that the class will settle the question with a plot and two statistics.

Direct instruction (12 min): Model a dot plot with those five numbers on a number line marked from 10 to 60 by fives, stacking one dot per value. Name the parts aloud: the cluster from 12 to 14, the gap, and the single point far to the right. Compute the mean (110 divided by 5, which is 22) and the median (13), and mark both on the same number line so students see the mean sitting in the empty gap where no day falls. Define outlier using this picture rather than a formal rule.

Guided practice (8 min): Distribute the handout and have pairs build the dot plot for the fifteen Corsley days on their own number line, marked from 40 to 90 by fives. Circulate and check that stacked dots line up vertically and that the day recorded as 88 is plotted rather than skipped. Do not let pairs begin the computations until their plot is complete.

Independent practice (15 min): Students answer handout questions 1 through 7 on their own. Watch for students who divide by 14 instead of 15 in question 2, and for students who report the middle value of the unsorted list in question 3.

Closure (5 min): Poll the class: hold up one finger for the mean and two for the median as the better description of a typical day in Corsley. Look for at least 80 percent of hands showing two fingers, and ask one student holding two fingers to justify the choice using the dot plot. If fewer than 80 percent choose the median, reteach the gap on the number line before moving on.

Corsley's Pumping Log: A Dot Plot of Fifteen Days — Grade 6 Math
Name: ____________________________ Date: ______________

Student Handout

The town of Corsley pumps water from two wells into one storage tank. The operator records the total pumped each day. Here are fifteen days from the log, in thousands of gallons. On day 11 a water main broke while the fire department was drawing water from a hydrant.

Day Gallons pumped (thousands) Day Gallons pumped (thousands)
1 42 9 44
2 44 10 43
3 41 11 88
4 43 12 41
5 42 13 42
6 45 14 45
7 45 15 43
8 42

Build a dot plot of the fifteen values on a number line before you answer. Show your work.

  1. How many days of data are in the log?
  2. What is the mean of the fifteen daily totals, in thousands of gallons?
  3. What is the median of the fifteen daily totals, in thousands of gallons?
  4. What is the range of the fifteen daily totals, in thousands of gallons?
  5. Which single value in the log is the outlier?
  6. Leave out the outlier and find the mean of the other fourteen days, in thousands of gallons.
  7. Which measure of center better describes a typical day in Corsley, the mean or the median? Explain your choice in one or two sentences.

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: 15 days.
    Students count the rows in the table, or count the dots on their plot as a check.
  2. Answer: 46 thousand gallons.
    The fifteen values total 690 thousand gallons, and 690 divided by 15 is 46.
  3. Answer: 43 thousand gallons.
    Sorted, the values are 41, 41, 42, 42, 42, 42, 43, 43, 43, 44, 44, 45, 45, 45, 88. With fifteen values the median is the eighth one, which is 43.
  4. Answer: 47 thousand gallons.
    The largest value is 88 and the smallest is 41, and 88 minus 41 is 47.
  5. Answer: 88 thousand gallons.
    It sits far to the right of the cluster from 41 to 45, with a wide gap and no dots between.
  6. Answer: 43 thousand gallons.
    Removing 88 leaves a total of 602 thousand gallons over 14 days, and 602 divided by 14 is 43.
  7. Answer: The median.
    The median of 43 falls inside the cluster where fourteen of the fifteen days sit. The mean of 46 is pulled up by the single 88, and no day in the log is close to 46. Accept any answer choosing the median that refers to the outlier or to the gap on the plot.

Differentiation

Support

Give these students the sorted list of the fifteen Corsley values already written out, so the work of question 3 is locating the eighth value rather than ordering fifteen numbers. Provide a pre-drawn number line from 40 to 90 with tick marks every unit, and allow a calculator for question 2 and question 6.

Extension

Ask these students to compute the mean absolute deviation of the fourteen ordinary days in the Corsley log and compare it with the mean absolute deviation of all fifteen. They should be able to state in writing how much one day widens a measure of spread.

ELL supports

Pre-teach outlier, cluster, and typical with the hook data from the town of Corsley still on the board, pointing at the plot as each word is said. Provide sentence frames for question 7: "The better measure is ___ because most days are near ___, and the mean is pulled toward ___."

Extensions

Have the class collect a real fifteen-day data set from the school, such as the number of water bottles filled at a filling station or the daily lunch count, and repeat the analysis. Comparing the shape of that plot with the shape of the Corsley plot gives students a second case where they must decide which measure of center to report.