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Lesson plan Grade 6 Math

Ivensdale Plant's Basin Log: Center, Spread, and Shape

A 45-minute grade 6 lesson using fifteen days of basin readings from a water reclamation plant to compare mean and median and describe a skewed distribution.

Objectives

  1. Students will compute the mean, median, and range of a fifteen-day data set.
  2. Students will explain how one unusually high reading changes the mean more than the median.
  3. Students will describe the overall shape of a distribution as symmetric or as having a tail.

Vocabulary

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

Teacher overview

Treatment plants test the water leaving each basin every day, so an operator reads a data set rather than a single number when asked how the plant is running. This lesson uses fifteen days of readings from the fictional Ivensdale Water Reclamation Plant, where bacteria in an aeration basin break down the material a laboratory test reports in milligrams per liter. Fourteen of the days sit close together and one storm day runs far higher, which is what makes the set worth summarizing two ways. The misconception this lesson addresses is that the mean is always the best description of a typical day, when a single extreme reading pulls the mean away from the value most days actually show.

Procedure

Hook (5 min): Write two numbers on the board, a mean of 16 and a median of 14, and tell students both describe the same fifteen days at the fictional Ivensdale plant. Ask which number they would put in a report titled "a typical day," and record the split without settling it.

Direct instruction (10 min): Explain what the log measures. The laboratory test reports how much material is still left in the water for bacteria to break down, so a lower number means cleaner water leaving the basin, and the plant's permit is written as a monthly average rather than as a single day. Review mean as the total divided by the number of readings, median as the middle value of the ordered list, and range as the largest value minus the smallest. Order five sample values on the board together so students see that the median depends on position rather than on size alone.

Guided practice (12 min): Hand out the log and have students build a class dot plot on the board, one sticky note per reading placed above a number line from 10 to 45. Ask what the picture shows that a list of numbers does not, and name the long tail on the right. Pause for a formative check: ask each student to write on the back of a sticky note whether removing the day 15 reading would change the mean, the median, both, or neither, and re-teach with the dot plot if fewer than three quarters answer that only the mean changes.

Independent practice (13 min): Students complete the handout, showing the ordered list they used for the median and the total they used for the mean.

Closure (5 min): Return to the two numbers from the hook and ask three students to defend the median using their item 9 answer. Close by noting that operators report both numbers, because the mean carries the storm day that the plant still has to account for.

Ivensdale Plant's Basin Log: Center, Spread, and Shape — Grade 6 Math
Name: ____________________________ Date: ______________

Student Handout

The fictional Ivensdale Water Reclamation Plant tested the water leaving one aeration basin on fifteen days in a row. Lower numbers mean cleaner water. A storm hit the town on day 15. The plant's permit allows a monthly average of 30 mg/L.

Day Reading (mg/L) Day Reading (mg/L)
1 15 9 12
2 13 10 14
3 14 11 16
4 12 12 13
5 16 13 15
6 14 14 14
7 13 15 44
8 15
  1. How many readings are in the log?
  2. What is the range of the fifteen readings?
  3. What is the median of the fifteen readings?
  4. What is the mean of the fifteen readings?
  5. Which reading sits far away from all the others?
  6. Leave out the day 15 reading. What is the mean of the other fourteen readings?
  7. By how much does the day 15 reading raise the mean of the whole set?
  8. Leave out the day 15 reading again. What is the median of the other fourteen readings?
  9. Which measure of center better describes a typical day at this basin, the mean or the median? Use your answers to items 4, 6, and 8 to explain.
  10. Is the shape of this data set symmetric, or does it have a tail on one side? Name the side.

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 readings.
    The log lists one reading for each of days 1 through 15.
  2. Answer: 32 mg/L.
    The largest reading is 44 and the smallest is 12, and 44 minus 12 is 32.
  3. Answer: 14 mg/L.
    In order the readings are 12, 12, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 44. With fifteen values the median is the eighth one, which is 14.
  4. Answer: 16 mg/L.
    The readings total 240, and 240 divided by 15 is 16.
  5. Answer: 44 mg/L.
    The day 15 reading of 44 is 28 above the next highest reading of 16, while every other day falls between 12 and 16.
  6. Answer: 14 mg/L.
    Without day 15 the total is 240 minus 44, which is 196, and 196 divided by 14 is 14.
  7. Answer: 2 mg/L.
    The mean with day 15 is 16 and the mean without it is 14, and 16 minus 14 is 2.
  8. Answer: 14 mg/L.
    With fourteen values the median is the mean of the seventh and eighth values, which are both 14, so the median stays 14.
  9. Answer: The median.
    The median is 14 whether or not the storm day is counted, while the mean jumps from 14 to 16 because of that one day, so the median describes what happens on most days at this basin.
  10. Answer: A tail on the high side.
    Fourteen readings cluster between 12 and 16 and one reading reaches 44, so the distribution is not symmetric, and its tail stretches toward the high end.

Differentiation

Support

Give these students the fifteen readings from the fictional plant already written in order on a number strip, so the work begins with locating the middle value rather than with sorting. Assign items 1, 2, 3, 4, and 5, and provide the total of 240 for item 4 so the division stands alone.

Extension

Ask these students to find what the mean of the fictional plant's fifteen days would be if the storm day had read 24 instead of 44, and to state how far the median moves under that change. They should then write two sentences explaining why the mean is called sensitive to extreme values while the median is not.

ELL supports

Pre-teach mean, median, range, and tail with a labeled dot plot of five values on the board, pointing to the middle dot for median and to the distance between the end dots for range. Provide these sentence frames: "There are ___ readings, and the middle one is ___." and "The mean is ___ because the total is ___ divided by ___."

Extensions

Have students collect fifteen days of a measurement from their own lives, such as minutes spent walking to school or cups of water drunk each day, and summarize it with the same four numbers they found for the fictional Ivensdale log. Comparing which of their own data sets has a long tail, and which does not, gives the class a second reason to report both a mean and a median rather than choosing one by habit.