by Louisa Johnson
Dr. Johnson has written a number of articles for this newsletter. A retired mathematics educator and former university dean, she has served as a volunteer on several occasions at International CHED in Dallas.
Because of its practical use in finding solutions to real-world problems and contributing to mathematical reasoning, estimation is a topic that should never be overlooked in curriculum.
By the time children enter school, they have already had experience with estimation. Statements such as “I’m almost six years old,” “My sister is about as tall as you are,” “You gave her more M and Ms than you gave Nancy,” “My brother has a little more money than I do” are examples of sentences used by kindergarten students that exemplify their use of estimation.
If children are encouraged to continue to use estimation and are helped to develop specific strategies to assist them in doing so, they will come to accept estimation as an appropriate part of mathematics and to appreciate it as a forceful mathematical idea that can be used in solving problems and in checking the reasonableness of results. It is important for a child to learn that mathematics involves more than exactness.
What Kind of Answer?
In the process of solving real-world problems, we usually have to decide what kind of answer is desired. When determining the amount of paint needed to redo our kitchen, the answer will be an approximate number—an estimate. It will usually be an amount that is slightly more than what is needed for the job.
If Johnny is figuring how much money he must save to buy the bicycle he saw in the store, the answer will be the exact amount needed. When deciding how much money to give a child being sent to the store to get some groceries, a parent will frequently conclude the child should have slightly more than the estimated need.
Bill wishes to hire a child to mow his lawn, and he wants an idea of how much it will cost. The child charges $6 an hour. Bill asks how long it will take to mow his yard. Since it would be impossible to tell exactly how long it would take, Bill should expect the answer to be an estimate.
These are just a few examples showing that part of the problem-solving process is determining the most desirable type of answer. In order to become an efficient problem solver, a child should have experiences that will help him or her come to know what is meant by an estimate, when an exact answer is desired, when it is appropriate to estimate, and how close an estimate is required in a given situation.
Several specific strategies can assist children in estimating while they are doing computation. Front-end estimation, rounding, using comparisons, and selecting easy numbers with which to work are four of the most commonly used strategies.
Using front-end estimation, the estimate of the sum 42 + 11 + 36 would be 40 + 10 + 30 or 80. The estimate could be improved by observing that the sum of the second digits (2 + 1 + 6) is about 10 so the sum is about 90.
Using rounding for the same sum, 42 would round down to 40, 11 would round down to 10 and 36 would round up to 40. 90 would be the estimate since the sum 40 + 10 + 40 equals 90. (All numbers have been rounded to the nearest 10.)
If one wishes to find out if 57 x 2 is greater than 100, the product could be compared with a known product. Since 57 is greater than 50, and 2 x 50 is 100, 57 x 2 must be greater than 100.
When estimating the sum of 44 3/5, 89 1/4, and 56 1/2, add 44 and 56 first because their sum is 100. It is easy to find the sum 100 + 89, so an estimate would be 189. (I used 44, 56 and 89 because they were easy numbers to work with.)
When multiplying 49 by .21, 48 and 1/4 would be nice numbers to work with because 48 is divisible by four, and .21 is close to .25 (i.e. 1/4) . An estimate of 12 results from 48 x 1/4.
When computing mixed numbers containing decimals or fractions, rounding numbers off to the nearest whole number often creates simpler numbers to work with.
A practice exercise for rounding to the nearest whole number involves using a number line marked at fractional points (e.g. 0, 1/8, 2/8, 3/8, ..., 7/8, 1, 11/8, 12/8, ..., 17/8, 2, 21/8,..., 47/8, 5). Ask the child to locate a point on the number line and tell which whole number it rounds to. For example, ask a child to locate the number 34/8 and tell what whole number it rounds to.
Continue with other numbers. Instead of fractions, mark the points on the number line for decimals (i.e. tenths). The number line can also be helpful when a child is first learning how to round numbers to the nearest tens or hundreds.
Children need help to recognize that estimation strategies such as those above are helpful when one wishes to obtain an amount that is close enough for the purpose at hand. Estimation should be looked on as as simplification, not as just another cumbersome process to be mastered. If done correctly, an estimation should be obtained quickly and with less work than computing an exact amount.
It is often prudent to allow a child to invent strategies based on what he/she knows about numbers and operations. This can be done by beginning with a problem that can be solved by estimation and encouraging the child to solve it in any reasonable way and then share with you the strategy used.
Estimation and the Calculator
Frequently the calculator is a useful tool for computation. This is especially true when working with two- or three-digit numbers which would require tedious and time-consuming computations using paper and pencil algorithms (one method of computing). A child may also wish to solve a real-world problem before he/she has learned the necessary algorithms.
Although the calculator can be helpful in obtaining desired answers, mistakes can be made when entering numbers into the calculator. It is, therefore, advisable to provide children with ways to check the reasonableness of computations. Estimation skills, as well as number and algorithmic sense, are valuable assets for doing the checking.
If the calculator showed the product of 8.24 and 4.89 to be 4029.36, a quick estimation (observing that 8.24 is close to 10, 4.89 is close to 5, and 5 x 10 = 50) would be helpful to confirm that an error had occurred. A more careful analysis would reveal that the decimal point in either 8.24 or 4.89 had been omitted.
Quantities and Measurement
Estimation can be applied in working with quantities and measurements as well as computation and problem solving. Children should have practice in estimating large as well as small quantities. One good exercise is to have them estimate the number of beans in a jar. Encourage them to count the number of a small amount and use this information to assist when making their estimate. After they have made the estimate, have them find the exact amount and compare the numbers. With a little practice, many can become surprisingly accurate in their estimations.
A useful activity in lessons on estimation for middle school students is to ask them to estimate the length of objects in either yards or meters and to compare their answers with the actual measured length. This not only helps in developing estimation skills but also enhances their concept of length. In a similar manner, as they are developing concepts in volume, they could be asked to estimate the number of cubes it would take to fill a box, prior to determining the actual number required.
Estimation is an important mathematical skill. Mathematics lessons should encourage students to recognize that estimation is not simply guessing but informed reasoning. They should come to appreciate it as a way of obtaining appropriate answers to many real-world problems in a timely manner and as an efficient way to check the accuracy of problem solutions involving computations.
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