As indicated in the K-12 Overview, students' ability to use estimation appropriately in their daily lives develops as they focus on the reasonableness of answers, explore and construct estimation strategies, and estimate measurements, quantities, and the results of computation.

For this type of development to occur, the atmosphere established in the classroom ought to assure everyone that their estimates are important and valued. Children should feel comfortable taking risks, and should understand that an explanation and justification of estimation strategies is a regular part of the process. Third- and fourth-graders, for the most part, should be beyond just "guessing." As children communicate with each other about how their estimates are formulated, they further develop their personal bank of strategies for estimation.

Students should already feel comfortable with estimation of sums and differences from their work in earlier grades. Nonetheless, they should regularly be asked About how many do you think there will be in all? or About what do you think the difference is? or About how many do you think will be left? in the standard addition and subtraction settings. These questions are appropriate whether or not exact computations will be done. As the concepts and the related facts of multiplication and division are introduced through experiences that are relevant to the child's world, estimation with computation must again be integrated into the development and practice activities.

One of the most useful computational estimation strategies in these grade levels also reinforces an important place-value idea. Students should understand that in multi-digit whole numbers the larger the place value, the more meaningful the digit in that position is in contributing to the overall value of the number. A reasonable approximation, then, of a multi-digit sum or difference can always be made by considering only the leftmost places and ignoring the others. This strategy is referred to as front end estimation and is the main estimation strategy that many adults use. In third and fourth grades, it should accompany the traditional rounding strategies.

Children should understand that, sometimes, the estimate will be accurate enough to serve as an answer. At other times, an exact computation will need to be done, either mentally, with paper-and-pencil, or with a calculator to arrive at a more precise answer. The particular procedure to be used is dependent on the setting and the problem. Also at this level, estimation must be an integral part of the development of concrete, algorithmic, or calculator approaches to multi-digit computation. Students must be given experiences which clearly indicate the importance of formulating an estimate before the exact answer is calculated.

In third and fourth grades, students are developing the concepts of a thousand and then of a million. Many opportunities arise where estimation of quantity is easily integrated into the curriculum. Many what if questions can be posed so that students continue to use estimation skills to determine practical answers.

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 3 and 4.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 3-4 will be such that all students:

1. Judge without counting whether a set of objects has less than, more than, or the same number of objects as a reference set.

• Students estimate the numbers represented by groups of base ten blocks or bundles of popsicle sticks. For example, one set might consist of 1 hundred, 6 tens, and 3 ones, and the other 0 hundreds, 17 tens, and 7 ones. Students first estimate which is more without arranging the blocks or counting them and then they determine the correct answer. These kinds of proportional models allow the "quantity of wood" to be proportional to the actual size of the number.

• Students read The Popcorn Book by Tomie dePaola and make estimates with popcorn. For example, they might consider two quantities of popcorn, one popped and one unpopped. Which contains the largest number of kernels? They might also predict how many measuring cups the unpopped popcorn will fill once it is popped and find out the result after popping the popcorn. (The total number of cups made may be surprising.)

2. Use personal referents, such as the width of a finger as one centimeter, for estimations with measurement.

• Students estimate the height of a classmate in inches or centimeters by standing next to him or her and using their own known height for comparison.

• As standard units like yard and half-inch are introduced, students are challenged to find some part of their body or some personal action that is about that size at this point in their growth. For instance, they may decide that the width of their little finger is almost exactly one half-inch or the length of two giant steps is one yard.

• Students measure the width of their handspan in centimeters (from thumb tip to little finger tip with the hand spread as far as possible) and then use the knowledge of its width to estimate the metric measures of various classroom objects by counting the number of handspans across and multiplying by the number of centimeters.

3. Visually estimate length, area, volume, or angle measure.

• Students estimate the number of 3" x 5" cards it would take to cover their desktops, a floor tile, and the blackboard. They describe the process they used in writing, which is then read by the teacher to determine the students' progress.

• Students work through the Tiling a Floor lesson that is described in the First FourStandards of this Framework. They estimate how many of their tiles it would take to cover one sheet of paper, and compare their answers to the actual number needed.

• Students estimate the capacities of containers of a variety of shapes and sizes, paying careful attention to the equal contributions of width, length, and height to the volume. They sort a series of containers from smallest to largest and then check their arrangements by filling the smallest with uncooked rice, pouring that into the second, verifying the fact that it all fits and that more could be added, pouring all of that into the third, and so on.

• Students estimate the angle formed by the hands of the clock. They can also be challenged to find a time when the hands of the clock will make a angles of 90 degrees, 120 degrees angle, and 180 degrees.

4. Explore, construct, and use a variety of estimation strategies.

• Students develop and use the front end estimation strategy to obtain an initial estimate of the exact answer. For example, to estimate the total mileage in a driving trip where 354 miles were driven the first day, 412 the second day, and 632 the third, simply add all digits in the hundreds' place: 3 + 4 + 6 = 13 hundred or 1300.

• Students learn to adjust the front end estimation strategy to give a more accurate answer. To do so, the second number from the left (in the problem above, the tens) is examined. Here, the estimate would be adjusted up one hundred because the 5 + 1 + 3 tens are almost another hundred. This would give a better estimate of 1400.

• Students use rounding to create estimates, especially in multi-digit addition and subtraction. They do so flexibly, however, rather than according to out-of-context rules. In a grocery store, for example, when a person wants to be sure there is enough money to pay for items that cost $1.89, $2.95, and $4.45, the best strategy may be to round each price up to the next dollar. In this case then, the actual sum of the prices is definitely less than $10.00 (2 + 3 + 5). On the other hand, to be sure that the total requested by the cashier is approximately correct, the best strategy may be to round each price to the nearest dollar and get 2 + 3 + 4 which is $9.

• Students are shown a glass jar filled with about two hundred marbles and are asked to estimate the number in the jar. In small groups, they discuss various approaches to the problem and the strategies they can use. They settle on a strategy to share with the class along with the estimate that resulted.

• Students write about how they might find an estimate for a specific problem in their journals.

5. Recognize when estimation is appropriate, and understand the usefulness of an estimate as distinct from an exact answer.

• Given pairs of real-life situations, students determine which situation in the pair is the one for which estimation is the best approach and which is the one for which an exact answer is probably needed. One such pair, for example, might be: deciding how much fertilizer is needed for a lawn and filling the bags marked "20 pounds" at the fertilizer company.

• Given a set of cartoons with home-made mathematical captions, third graders decide which of the cartoon characters arrived at exact answers and which got estimates. One ofthe cartoons might show an adult standing in the checkout line at a supermarket and another might show the checkout clerk. The captions would read: Mr. Harris wondered if he had enough money to pay for the groceries he had put in the cart and Harry used the cash register to total the bill. Students make up their own similar cartoon.

• Students share with each other various situations within the past week when they and their families had to do some computation and describe when an exact answer was necessary (and why) and when an estimate was sufficient (and why).

6. Determine the reasonableness of an answer by estimating the result of operations.

• Students are regularly asked if their answer makes sense in the context of the problem they were solving. They respond with full sentences explaining what they were asked to find and why the numerical answer they found fits the context reasonably, that is, why it could be the answer.

• Fourth graders might be asked to decide if their estimated answer to the following problem is reasonable. The band has 103 students in it. They line up in 9 rows. How many students are there in each row? The students' responses might indicate, for example, that there should be about 10 students in each row, since 103 is close to 100 and 9 is close to 10.

• Students estimate reasonable numbers of times that particular physical feats can be performed in one minute. For example: How many times can you skip rope in a minute? How many times can you hit the = button on the calculator in a minute? How many times can you blink in a minute? How many times can you write your full name in a minute? and so on. Other students judge whether the estimates are reasonable or unreasonable and then the tasks are performed and actual counts made. (To determine the number of times the = button is hit in a minute, press +1= so that each time the = button is pressed, the display increases by 1.)

• Third-grade students are given a set of thirty cards with three-digit subtraction problems on them. In one minute, they must sort the cards into two piles: those problems whose answers are greater than 300 and those whose answers are less than 300. The correct answers can be on the backs of the cards to allow self-checking after the task is completed.

• For assessment, fourth-grade students might be given a page of one-digit by multi-digit multiplication problems in a multiple choice format with four possible answers for each problem. Within some time period which is much too short for them to perform the actual computations, students are asked to choose the most reasonable estimate from each set of four answers.

7. Apply estimation in working with quantities, measurement, time, computation, and problem solving.

• Students work through the Product and Process lesson that is described in the Introduction to this Framework. It challenges the students to form two three-digit numbers using 3, 4, 5, 6, 7, 8 which have the largest product; estimation is used to determine the most reasonable possible choices.

• Students learn about different strategies for estimating by reading The Jellybean Contest by Kathy Darling or Counting on Frank by Rod Clement.

• Students regularly try to predict the numerical facts presented in books like In an Average Lifetime . . . by Tom Heymann. Using knowledge they have and a whole variety of estimation skills, they predict answers to: What is the number of times the average American eats in a restaurant in a lifetime? (14,411) What is the total length each human fingernail grows in a lifetime? (77.9 inches) and What is the average number of major league baseball games an American attends in a lifetime? (16)

• Students regularly estimate in situations involving classroom routines. For example, they may estimate the total amount of money that will be collected from the students who are buying lunch on Pizza Day or the number of buses that will be needed to take the whole third and fourth grade on the class trip.

• Students investigate environmental issues using estimation. One possible activity is for them to estimate how many gallons of water are used for various activities each week in their home. (See Healthy Environment - Healthy Me.)

            Activity            Total #           Each time  Total # of gallons used
                               in 1 week 
  
       Take shower or bath     _______     x     18 gallons             =
       Flush toilet            _______     x      7 gallons             =
       Wash dishes             _______     x     10 gallons             =
       Wash clothes            _______     x     40 gallons             =
                                                                   _________________
                                 Total # of gallons used in 1 week      =
  

  Students discuss possible reasons for differences among their estimates, and they compute the class total for the number of gallons consumed during that week.

References

Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth Stevens Children's Books, 1991.

Darling, Kathy. The Jellybean Contest. Champaign, IL: Garrard, 1972.

de Paola, Tomie. The Popcorn Book. New York: Holiday House, 1978.

Environmental and Occupational Health Sciences Institute. Healthy Environment - Healthy Me. Exploring Water Pollution Issues: Fourth Grade. New Jersey: Rutgers University, 1991.

Heymann, Tom. In an Average Lifetime ... New York: Random House, 1991.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.