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  • Lesson
Number and Operations
Zoe Silver
Location: unknown

In this lesson, students experience an application of proportion that scientists actually use to solve real-life problems. Students learn how to estimate the size of a total population by taking samples and using proportions. The ratio of “tagged” items to the number of items in a sample is the same as the ratio of tagged items to the total population.

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:

  1. Solve: \frac{{11}}{{15}} = \frac{x}{{75}}. [x=55.]
  2. James knows that \frac{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? [There are 36 students 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.

2528 robin 

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.

pdficonCapture-Recapture Activity Sheet

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:

2528 cap recap table

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.
        2528 wordprop
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:

2528 cap recap table 2

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.


Assessment Options

  1. Ask students to solve a comparable question:
    Jake wants to know how many students from his school were at the soccer match and saw him score a goal. He knows that there are 850 students at his school. At lunch, he asked 35 students if they had seen the game, and 3 of them said that they had.
    • How can you use this information to estimate how many students from his school were at the game? Estimate how many students were there, and show your work.
    • How confident you are that your estimate is close to the actual number of students who were at the game?
    • What suggestions do you have for Jake to make sure that the estimate is close to the actual number of students that saw the game? [Take a few more samples to see what the range of estimates is or take a larger sample and compare the results.]
  2. Design an experiment to simulate estimating the size of a population by using the capture–recapture method.
1. What if one scientist marks the birds with a blue label, and another scientist uses a red label. What effect does this have on the way they recapture and use the data? Simulate this using a second marker to mark the beans with another colored pen.

[If there were 2 different marking systems used on the robins, then 2 trials and proportions can be generated from one recapture. So half the number of recaptures would yield the same estimates.]

2. Have students write up different scenarios where capture-recapture would be applicable.

[Any situation where finding a population size would be appropriate.]

3. If students have time to count the total number of beans, ask them to calculate the percent of error between their guessed population size and actual population size.

Questions for Students 

1. How do proportions help solve the problem of counting robins in the forest?

[If the scientists capture and tag a specific number of robins, then they can recapture robins later and use a proportion to estimate the total number of robins in the forest.]

2. What might happen if a small number of robins in the population are tagged?

[When a sample is taken it is possible that no tagged robins will be in the sample and the resulting calculation suggests that there are no robins in the forest.

3. What might happen if almost the entire population of robins is tagged?

[When a sample is taken, it is possible that all of the tagged robins will be in the sample. The resulting calculation would then underestimate the number of robins in the entire population.]

4. Develop a plan to estimate the size of the populations of more that one species from one sample.

[Mark each species with a different color. For each species, use a proportion to compare the number of marked to the sample size and the total marked to the population.]

Teacher Reflection 

  • Did you change the lesson as you moved through it?
  • How did you address the definitions of ratio and proportion for students who weren’t comfortable with these words?
  • How did students perform in relation to the stated behavioral objectives?
  • How did student groups/pairs divide up the work?
  • How would you change this lesson if you were to use it again?

Learning Objectives

By the end of this lesson, students will:

  • Recognize equivalent ratios.
  • Determine good and poor estimates.
  • Solve proportions to estimate population size.

NCTM Standards and Expectations

  • Understand and use ratios and proportions to represent quantitative relationship.
  • Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.
  • Develop, analyze, and explain methods for solving problems involving proportions, such as scaling and finding equivalent ratios.
  • Judge the reasonableness of numerical computations and their results.