Estimating Unknown Quantities

Estimation is an important aspect of quantitative thinking -- and a critical life skill in a world in which we often need to make decisions on the basis of inexact or undefined information.
Teaching Strategies:
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Updated on: March 15, 2007
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Why Is It Important?

Estimation serves as an important companion to computation. It provides a tool for judging the reasonableness of calculator, mental, and paper-and-pencil computations. Most importantly, the ability to make a reasonable estimate of an unknown quantity is an empowering skill—it frees us from the need to be precise when precision is impossible and lets a number that is reasonably accurate be good enough.

Unfortunately, mathematics instruction too often focuses on getting precise answers. For example, we frequently ask students to solve problems such as, "How long will it take a car to travel 5 miles if it is traveling 60 miles per hour?" We seldom ask students to calculate how fast cars are traveling along the street outside the school. Both involve the same sort of mathematics. The first problem is relatively easy to solve, has a single correct answer, and is not very interesting. The second is somewhat harder to solve because it involves taking some actual measurements; there is no single correct answer. This type of question might motivate students because it involves a part of their everyday lives.

How Can You Make It Happen?

Estimation is not a skill that you can teach on a given morning; it is a "habit of mind" that should be cultivated over a lifetime. Experienced teachers are always on the lookout for problems that allow students to apply their skills in mathematics, science, and communication.

  1. Fermi Problems

    One way of helping students develop estimation skills is to ask them to solve Fermi Problems. Fermi Problems, named after the Italian physicist Enrico Fermi, are open-ended problems with no easy solution. Rather, they require students to make rough calculations based on a reasonable set of assumptions. Here are some examples:

    • How many leaves are on the two trees standing outside the school?

    • How many gallons of water does our class drink every day?

    • How many pencils are there in our school? What is the average length?

    • How much money could we save by turning the lights off when we leave this room?

    • Are Spanish words longer than English words? Do they use more vowels?

    To solve problems such as these successfully, students need to have good computation skills, their own benchmarks, an understanding of the concept of an assumption, and comfort with a range of possible values as opposed to an exact value.

  2. Benchmarks

    A benchmark is a point of reference for making estimations and is developed through past experiences. An important aspect of estimation is making sure students have developed a variety of benchmarks in all areas of measurement. Students should have plenty of experiences in estimating heights, lengths, weights, volume, and time to develop personal benchmarks that they can apply to future estimations. A benchmark can be anything that allows students to relate the relative size or magnitude of something to a known size or magnitude. For example, if students have internalized the benchmark of 1 foot, they can estimate the length of a car to be about 10 feet.

    A simple activity is to time the length of a minute, and ask students to raise their hands when they think a minute has passed. After repeating this activity several times, students will have a much better sense of the length of a minute. (You might teach them the trick of counting seconds by saying "one Mississippi, two Mississippi, three Mississippi...")

    Practice estimating using benchmarks, and have students share their strategies for making estimations.

  3. Assumptions

    Assumptions are the building blocks of an estimate. To estimate how much punch you need for your party, you make assumptions about the number of guests who actually will come (which may be different from the number that you invite) and the amount of punch that each will drink. The assumptions that you make will be different depending on a your past experiences. For example, an experienced painter may estimate that a house will take four days to paint, based on the assumption that the house is similar to another house she painted previously. Someone who has never painted a house may estimate that it will take seven days to paint it. Once you've made your assumptions, the estimate itself is simply a matter of calculation.

    It takes a certain amount of courage to make an assumption. Making an assumption is risky because you could be wrong. Some students may balk at making assumptions for fear of not getting it right. Therefore, it is important to create an environment in which it is safe to make, and challenge, assumptions.

  4. Range of Acceptable Values

    It is often useful to consider a range of values for a particular estimate. For example, in estimating how many people speak English as a primary language, an estimate would not be 301,025,601 people. An acceptable estimate may be a range, such as between 250 and 400 million people.

    One way to introduce the idea of a range of possible values is to have students, working individually or in groups, come up with estimates for the same problem. For example, divide the class into groups, and have each group estimate the area of the classroom. Poll the groups, and write the estimates on the board, possibly in the form of a bar graph. Ask the "outliers" (groups whose estimates are far above or below the average) to explain their assumptions, which may lead them to change their estimates. Then suggest that the true answer is probably somewhere between the highest and the lowest estimates. Students can then measure the area of the classroom and compare their estimates to the actual measurement.

  5. Teaching Estimation Every Day

    Students should understand that estimation is a skill that is used every day in many different situations and is not only a topic in math class. Explain to students the different types of estimates and when each would be used. You may want to start by having students brainstorm situations in which they have made estimates. Then, categorize the examples to show the wide use of estimation in real-world situations. For example, students might use estimates to predict the amount of money they need to go to the mall for a day or how long their homework will take them on a given night. They might make eyeball estimates of the length of their hair or the weight of objects in the classroom. Ask students to describe how they make estimates, and be sure that they understand all types of estimation. This will allow you to determine what assistance they may need in developing appropriate estimates.

To introduce a class to making estimates, give students a simple Fermi problem, such as, "If we lined up all of our shoes in a row, end to end, how long would the row of shoes be?"

As students think about the problem, ask them to record their ideas. Here are some questions to get them thinking.

  1. Define the problem. What sort of answer are we looking for? What would be an example of a solution?

  2. What are the appropriate units of measure?

  3. What do we need to know to arrive at an estimate?

  4. What assumptions can we make?

  5. What calculations can we make? What mathematical operations can we use?

  6. How can we check our estimate to see if it makes sense?

For Step 1, be sure students understand that they are trying to estimate the length of the row of shoes, not how many shoes there would be. They should decide if their row consists of shoes lined up end to end, or side by side.

For Step 2, have students choose the unit they will use: yards, inches, feet, meters, or centimeters. Remind students of the benchmarks they know from past experiences. Students may know that an inch is about the length of the top section of their thumb, a foot is the length of a ruler or about the length of a standard piece of notebook paper, or a meter is about the length of a baseball bat.

In Step 3, students need to find out how many shoes will be in the row and how the length of each shoe will be calculated. For Step 4, guide students to make assumptions about how wide or long the shoes are and how many shoes will be in the row. Students can measure the length or width of five shoes and find the average, or measure a few shoes and take the most frequently occurring measurement.

During Step 5, students should calculate the length of the row by multiplying the number of shoes by the shoe length or width they arrived at in Step 4 by using their assumptions. Younger students can repeatedly add the length of shoes to reach an estimate. If students are working in groups, discuss the range of reasonable estimates based on the assumptions students made. Ask them to make new assumptions if their estimates are outside of that range. Once all students have estimates that are within the acceptable range of values, the class can measure the actual length of all the lined-up shoes. Have students compare the actual length to their estimates.

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