Students should be comfortable with the definitions of and distinctions between a
ratio and a
proportion. Review of these topics may be a valuable warm-up for students.
Students should already have a method for solving a proportion. For example:
- Solve: 11/15 = x/75.
- James knows that 2/3 of the class is going on the field trip. If 24 students go on the trip, how many students are in the class?
Before Class
Prepare the cups of beans. Each group/pair of students should receive their own cup of unmarked beans. The cups should have more beans than can be counted visually (around 200). The cups may have the same number of beans, or they may be different, depending on your preference.
In Class
Introduce the lesson with a discussion about the following situation:
| Scientists often study the health of a habitat by gathering data about the number of animals that live in the area. Suppose you wanted to know how many robins lived in a particular forest. |
 |
Ask students how they think the number of robins in a forest could be counted.
Elicit student responses to questions such as:
- If you tried to gather all of the robins and count them, how would you know if you had indeed counted every single one?
- Do you need to know the exact number of robins?
To begin the activity, distribute the Capture-Recapture activity sheet, a cup full of white beans, and a marker to each pair or group of students.
Emphasize for students that:
- An initial handful of beans must be marked, counted, recorded on the activity sheet for Question 1 and returned to the cup at the beginning of the activity.
- After each sampling and counting (for trials 1 through 6), all the beans are returned to the cup before the next trial.
Although the steps to be taken are outlined on the activity sheet, you may want to walk students through filling out one of the rows. For instance, if a handful of 33 beans is pulled from the cup and 6 beans are marked, then the first row of the table in Question 5 would be filled out like this:
When students complete the activity with the beans, they may need some guidance for setting up their first proportion (in Question 7). Model for them what is being compared, and develop a model to estimate the total population when the sample tagged, sample size, and total tagged are known.
For example, suppose that they initially marked 42 beans. Then, the proportion would be 6/33 = 42/x for the first trial. In Question 8, the activity sheet would then be filled in as follows:
Students should write and solve a proportion representing each trial with the beans. Their results won't be perfect, but most answers should fall within a reasonable range of the actual number of beans in the cup.
As students finish their calculations, have them write their answers to the remaining questions on the activity sheet. This will enrich the responses that students might give to the Questions For Students that you may wish to ask to the entire class.
After Guessing the Total Number of Beans
In Question 14 of the activity sheet, students are asked to examine the process and think about what would make an acceptable estimate. Students should take additional samples and then determine what percent of their population estimates fall within the interval they say is acceptable. By doing this, students have a frame of reference to determine if an estimate is good or poor. They can look at the distribution of their estimates and be fairly certain that the estimates that are at either end of the range of distributions are likely to be poor. As a class, choose one group's work, build a bar graph on the board, and discuss the group’s results.
If there's time before the end of the period, ask students to count the total number of beans in their cup to check their answers. Students should be discouraged from counting the beans instead of using proportions during the activity, but once they've committed to an answer, they can count the actual number of beans in the cup to know how reasonable their calculated estimate is.
