Random Sampling - How do you make sure your sample is random?

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Objective

Students will be able to conduct a random sampling simuation.

Big Idea

How many tomatoes are on your tree? Students will do a simulation of random sampling to calculate the average number of tomatoes on each tree in a crop.

Launch

10 minutes

Opener:  As students enter the room, they will immediately begin working on the opener.  The opener is a mixture of previously learned questions, and students should work individually, and then as table groups to discuss the methods for solving the questions.  After approximately 5 minutes, I will call on students to go to the board and solve the opener questions.  As with all openers, I will take volunteers to go to the board – the volunteer is expected to explain their reasoning, and other students are expected to follow along with the work and ask questions/make suggestions as necessary.  By having a student explain their reasoning while others listen and provide feedback, mathematical practice 3 – construct viable arguments and critique the reasoning of others – becomes a natural part of class.

Learning Target: After completion of the opener, I will address the day’s learning targets to the students.  In today’s lesson, the intended target is, “I can make a conclusion based on a simulation of random sampling.” Students will jot the learning targets down in their agendas (our version of a student planner, there is a place to write the learning target for every day).   

Explore

40 minutes

Random Sampling Inquiry and Notes:  In this lesson, students will begin to explore the concept of random sampling through inquiry.  Using a map of a gardener’s tomato crop (I make a poster out of the Tomato Crop Map), students will drop paperclips onto the map to develop a random sample.  Students will then calculate the average of the tomatoes on the ten plants that they chose.  Next, students will use an alternate method of sampling, which is to just choose the first ten plants on the map – and the students are asked to calculate the mean of the tomatoes on those plants.  Then, they compare the two means.  After finding the means of the two different samples, students are asked to calculate the mean of the entire crop – there are 100 numbers, so I ask that each person at the table calculates the mean because it would be easy to make a mistake, perhaps they can sum each row, then find the sum of all rows, then find the mean.  After finding the mean of the entire population, students are asked to make comparisons from the means of their samples to the mean of the actual population.  Finally, students are asked to discuss why they think the mean of their first sample was closer to the actual mean than the second sample.  Points to discuss are that the first sample was truly random, but the second sample only looked at the first row – what if that row got more sun, more rain, was attacked by an animal, etc. 

By conducting an actual sampling experiment that could be used to solve a real world problem, students are truly modeling the mathematics, and then abstracting information from their model to draw conclusions and make predictions, mathematical practices 2 and 4.  Given that it would not be feasible to use an actual 10 by 10 crop of tomato plants, the students use the tools that they do have, map of crop, paperclips, calculator strategically in an effort to model a real crop - mathematical practice 5.

We will discuss the questions as a class, and then move onto to notes – introducing the idea of biased and unbiased samples.  This is the topic of tomorrow’s class, so today will just be a brief intro to the idea. 

Summarize

10 minutes

To summarize this lesson, after we discuss the concept and relate it to the experiment, students will be given three scenarios and asked as a table to determine which of the three would be considered a true random (unbiased) sample.  I will have students stand for 1, 2, or 3 depending on which scenario they believe is truly random.  This activity will help me to gauge their understanding of the day’s lesson.