Math Background

Lesson: Sampling Techniques
Developing the Concept

Your students have had an opportunity to investigate selecting samples. Now they can apply their knowledge to a particular experiment.

Materials: a paper bag and 84 connecting cubes (56 blue and 28 yellow, or whatever colors are available), counters, or other small objects

Preparation: Put the 84 cubes into the paper bag before class begins.

Prerequisite Skills and Concepts: Students should be able to set up a proportion and solve it.

  • Say: Yesterday we introduced the idea of selecting a representative sample from a population. What can you tell me about choosing a sample?
    Students should say that a sample should be representative of a population and should not be biased. If they don't say this, remind them of it.
  • Say: Choosing a sample that is representative of the population is an important step in making good predictions. Biased samples don't represent the population and often lead to incorrect predictions.
    Bring out the paper bag with the 84 cubes in it.
  • Say: This bag contains 84 cubes. Some of them are blue and some are yellow. We want to predict how many there are of each color by selecting a sample. Who would like to come up and select a sample for us?
    Have a student come up and pick four cubes, without looking, from the bag. Place the cubes so the whole class can see them.
  • Ask: What do you think about the sample that was selected?
    Students will probably say that the sample may not be representative because it is too small.
  • Say: Yes, that's right. How could we have picked a better sample so that the results might be more reliable?
    Students should suggest picking more cubes.
  • Say: I will pick a sample of 20 cubes.
    Look inside the bag and pick 20 cubes, mostly all the same color. Say things such as, “Not this one” or “Yes, this one is a good one,” indicating that the sample chosen is not done at random.
  • Ask: What do you think about the sample that I selected?
    Students will probably say that you should not look in the bag when selecting a sample because the sample will be biased.
  • Say: Who would like to come up and select a handful of cubes?
    Have a student come up and pick at least 20 cubes without looking. Assume he or she chose 6 yellow and 14 blue cubes.
  • Say: In our sample, we have 6 yellow and 14 blue. Based on our sample, how many of the 84 cubes do you think are blue and how many are yellow? (Students may make some guesses.) Is there some way we might set up a proportion from our sample to predict the number of each color of cubes?
    Students should suggest setting up the proportion six-twentieths = x over eighty-four. If they don't, explain the use of proportion to figure out the number of yellow cubes in the bag.
  • Say: How do we solve this proportion to find x?
    Students will probably say to cross multiply. (x = 25.2, or about 25.) Have two students count how many of each color are actually in the bag. Discuss the results with the class. Then have them compare the number of blue cubes to the number they obtained based on their sampling of blue cubes. Repeat the experiment with different numbers of cubes.

Wrap-Up and Assessment Hints:
Biologists do a lot of sampling in order to determine the size of a wildlife population or how many specimens of a kind of plant grow in a certain locale. This would be a great opportunity to connect mathematics to science in your classroom. A biologist studying plant life in a forest will take several random samples by staking out a square area called a quadrat. The biologist will then classify all the plants in the area of several quadrats to determine what percent of the forest area is covered by the species of plant. Similarly, a biologist wanting to find out how many black bear live in a national park might capture and tag 30 black bears from different areas in the park. The following year he would go back and capture another group of random samples from all across the park. Based on the ratio of tagged bears to total bears captured in the second sample, the biologist could obtain a good estimate of how many bears there were in the whole park.


Houghton Mifflin Math Grade 6