For this type of development to occur, the atmosphere established in the classroom ought to assure that everyone's estimate is important and valued, that children feel comfortable taking risks, and that 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." Estimations of measurements as well as of quantities should pervade the classroom activity. As children communicate with each other about how their estimates are formulated, they further develop their personal bank of strategies for estimation.

Activities which provide experiences for the child to determine reasonableness of answers and to establish the difference between estimated answers and exact answers as well as when the use of each is appropriate, should be developed through non-routine problem solving activities that involve measurement, quantities, and computation.

Estimation with computation is important in these grade levels as well as at all grade levels. Formal strategies should be compared with informal strategies which children develop as they estimate answers to various problems under varying conditions.

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 actual 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 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, the more meaningful the digit 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 much more standard and traditional rounding strategies.

Children should understand that sometimes, the estimate will be an accurate enough number 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. Which procedure should 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 multidigit 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 first of a thousand and of a million. Many opportunities arise where estimation of quantity is easily integrated into the curriculum. Many what if type of questions can posed to students so that they continue to use estimation skills to determine practical answers.

Building upon K-2 expectations, experiences in grades 3-4 will be such that all students:

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

• While this expectation is primarily for younger students, third graders can still benefit from estimating the sizes of various multi-digit numbers that are modeled with base ten blocks or bundles of popsicle sticks. For example, one set might be 1 hundred, 6 tens, and 3 ones, and the other be 0 hundreds, 17 tens, and 7 ones. Students first estimate which is more without arranging the blocks or counting them and then the determine the correct answer. These kinds of proportional models allow the "quantity of wood" to be proportional to the actual size of the number and so work well this way.

• 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 centimeter 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 centimeter 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 they are and multiplying by the number of centimeters.

• Students estimate the number of 3"x5" cards it would take to cover their desktops, a floor tile, and the blackboard.
• 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. One way this can be done is by sorting a series of containers from smallest to largest and then checking the predictions by filling the smallest with uncooked rice, pouring that into the second, verifying the fact that it all fits and that more can be added, pouring all of that into the third, and so on.
• Students begin to develop a feel for angle measure by regularly estimating 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 90 degree angle, a 120 degree angle, or a 180 degree angle.

• Students develop and use a front-end strategy when it is not essential that the estimate be either over or under 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 the hundreds: 3+4+6 = 13 hundred or 1300.
• Students also learn to adjust the front-end strategy to give a slightly more accurate answer. To do so, the next place over (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 leave an estimate of 1400.
• Students also use rounding to create estimates, especially in multi-digit addition and subtraction settings. They do so flexibly, however, rather than according to out-of-context rules. In a grocery store, for example, when you want to be sure you have enough money to pay for items that cost $1.89, $2.95, and $4.45, the best strategy is probably to round each price up to the next dollar. In this case, you would be sure that the actual sum of the prices is less than $10.00 (2+3+5)
• Students are confronted with 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 strategies they can use. They settle on one to share with the class along with the estimate that resulted.

• Given pairs of real-life situations, students determine which situation in the pair is the one in which estimation is the best approach and which is the one in 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 of the 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 share with each other various situations in 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 OK (and why).

• 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.
• Students estimate reasonable numbers of times that particular physical feats can be performed in one minute. For example, How many times can you dribble a basketball in a minute, How many times can you hop on one foot in a minute, How many times can you say the alphabet in a minute, and so on. Other students judge whether the estimates are reasonable or unreasonable and then the tasks are performed and the actual counts made.
• 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.
• Fourth-grade students might be given a page of one-digit by multi-digit multiplication problems in a multiple choice format with 4 possible answers for each problem. Within some time period which is much too short for them to do the computation, students are asked to choose the most reasonable answer from each set of four.

• 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 things like the number of times the average American eats in a restaurant in a lifetime (14,411), the total length each human fingernail grows in a lifetime (77.9 inches), and the average number of major league baseball games an American attends in a lifetime (16).
• Students regularly estimate before appropriate classroom procedures. For example, they may estimate the total amount of money that will be collected from the students 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.