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Lesson plan Grade 7 Math

Tollbury's Beach Closure Record: A Grade 7 Sampling Lesson

A 50-minute grade 7 math lesson where students estimate a summer's beach closure rate from five random samples of a fictional lake's 60-day record.

Objectives

  1. Students will estimate a population percent from a random sample of days.
  2. Students will compare five sample estimates to describe how much estimates vary.
  3. Students will explain why a sample must be random to represent a whole season.

Vocabulary

Key words: population, sample, random sample, estimate, variation, range

Teacher overview

Health departments test swimming water for bacteria and close a beach when a reading rises above the safe-swimming limit. This lesson uses a fictional record from Wistermere Lake in the fictional town of Tollbury, where the beach was tested on each of 60 summer days and either stayed open or closed. Students estimate the closure rate from five random samples of ten days, then compare those estimates to the rate in the full record. The misconception this lesson targets is the belief that one sample gives the exact answer: students who get 30 percent from their own sample often decide that a classmate who got 10 percent made an arithmetic error, rather than recognizing that two honest random samples of the same population routinely disagree.

Procedure

Hook (5 min): Tell students that the fictional town of Tollbury wants to know how often its lake beach was closed last summer, but that the health department has time to look up only ten of the 60 daily records. Ask what could go wrong if the town looks up the first ten days, or the ten days it remembers as rainy. Collect two or three answers and name the idea of a representative sample.

Direct instruction (12 min): Distribute the handout and read the 60-day record aloud, explaining that O marks a day the beach was open and C marks a day it was closed. Model one sample that is not on the handout: days 8, 12, 19, 24, 30, 33, 41, 47, 55, 60. Look each day up in the record, count the closed days, and show the estimate as a fraction of ten converted to a percent. Say plainly that this estimate belongs to the sample, not to the summer.

Guided practice (10 min): Work handout items 1 and 2 with the class, looking up each of Sample A's and Sample B's ten days in the record. Have students check their counts against a partner before moving on, since a miscounted day changes the percent by ten points.

Independent practice (15 min): Students complete handout items 3 through 9 on their own. Circulate and watch for students who divide by 60 instead of by 10 when working with a sample, and for students who report the number of closed days instead of a percent.

Closure and exit ticket (8 min): Write the five sample estimates on the board in the order students found them and ask what the spread tells the town about trusting a single sample. Then collect an exit ticket on an index card: "A sixth random sample of ten days contained 5 closed days. What percent of that sample was closed, and is that estimate higher or lower than the summer rate of 30 percent?" If more than a quarter of the class answers 5 percent, reteach the step that converts a count out of ten into a percent.

Tollbury's Beach Closure Record: A Grade 7 Sampling Lesson — Grade 7 Math
Name: ____________________________ Date: ______________

Student Handout

Wistermere Lake is a fictional lake in the fictional town of Tollbury. The health department tested the swimming beach on each of the 60 days of last summer. O means the beach stayed open that day. C means the beach was closed that day.

Days 1 to 10: O O C C O O O O C O
Days 11 to 20: O O O C O O C C O O
Days 21 to 30: O C O O O O C O O O
Days 31 to 40: C O O O C C O O O O
Days 41 to 50: O C O O C O O C O O
Days 51 to 60: C O C O O O C O C O

Five random samples of ten days were drawn from this record.

Sample A: days 2, 9, 15, 21, 27, 33, 38, 44, 52, 58
Sample B: days 4, 8, 13, 18, 24, 30, 35, 41, 47, 55
Sample C: days 3, 12, 17, 23, 29, 36, 40, 45, 50, 56
Sample D: days 5, 11, 16, 22, 26, 34, 39, 46, 54, 60
Sample E: days 1, 7, 14, 19, 25, 31, 37, 43, 51, 58

  1. What percent of Sample A's ten days were closed days?
  2. What percent of Sample B's ten days were closed days?
  3. What percent of Sample C's ten days were closed days?
  4. What percent of Sample D's ten days were closed days?
  5. What percent of Sample E's ten days were closed days?
  6. Find the mean of the five sample estimates.
  7. Using the full 60-day record, what percent of the summer's days were closed days?
  8. What is the range of the five sample estimates?
  9. How many percentage points below the summer rate is Sample D's estimate?

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: 20 percent.
    Sample A contains days 9 and 27, and both are marked C. Two closed days out of ten is 2 divided by 10, which is 0.20, or 20 percent.
  2. Answer: 30 percent.
    Sample B contains days 4, 18, and 35, all marked C. Three out of ten is 0.30, or 30 percent.
  3. Answer: 40 percent.
    Sample C contains days 3, 17, 36, and 45, all marked C. Four out of ten is 0.40, or 40 percent.
  4. Answer: 10 percent.
    Sample D contains only day 22 as a closed day. One out of ten is 0.10, or 10 percent.
  5. Answer: 30 percent.
    Sample E contains days 14, 31, and 51, all marked C. Three out of ten is 0.30, or 30 percent.
  6. Answer: 26 percent.
    Add the five estimates: 20 plus 30 plus 40 plus 10 plus 30 equals 130. Divide by 5 to get 26 percent.
  7. Answer: 30 percent.
    The record holds 18 closed days: three in days 1 to 10, three in days 11 to 20, two in days 21 to 30, three in days 31 to 40, three in days 41 to 50, and four in days 51 to 60. Then 18 divided by 60 equals 0.30, or 30 percent.
  8. Answer: 30 percentage points.
    The largest estimate is 40 percent and the smallest is 10 percent, so the range is 40 minus 10.
  9. Answer: 20 percentage points.
    Sample D estimated 10 percent while the summer rate is 30 percent, and 30 minus 10 equals 20.

Exit ticket answer: 50 percent, which is higher than the summer rate of 30 percent. Five closed days out of ten is 0.50.

Differentiation

Support

Give these students a copy of the record with the closed days already circled, so the task becomes counting rather than searching. Let them complete items 1 through 4 and item 7 only, and provide a filled-in example showing that 3 closed days out of 10 equals 30 percent.

Extension

Ask these students to draw two more random samples of ten days using a ten-sided die or a random number generator, compute both estimates, and then write a short paragraph on what happens to the spread of estimates when each sample contains twenty days instead of ten.

ELL supports

Pre-teach "open," "closed," "sample," and "estimate" with a labeled strip of ten squares in which some squares are shaded. Provide the sentence frame "Sample ___ had ___ closed days out of 10, so the estimate is ___ percent" for items 1 through 5.

Extensions

Have students look at the record's last ten days, which contain four closed days, and explain why a sample taken only from the end of the summer would give a poor estimate of the whole season. Ask them to name one real reason closures might cluster late in a summer, such as warmer water or a run of storms, and to say how a random sample protects against that clustering.