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. Estimations of measurements as well as of quantities should pervade the classroom activity. As students 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 with whole number computation 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 would each of us get if we divided these equally in the standard settings. These questions are appropriate whether or not actual exact computations will be done.

In fifth and sixth grade, estimation and number sense are even more important skills than algorithmic pencil-and-paper computation with multi-digit whole numbers. Students should become masters at applying estimation strategies so that answers displayed on a calculator can be instinctively compared to a sense of the range in which the correct answer lies.

The new estimation skills that are also important in fifth and sixth grade are skills in estimating the results of fraction and decimal computation. Even though study of the concepts and arithmetic operations involving fractions and decimals begins before fifth grade, a great deal of time will be spent on them here as well. As students develop understandings of the numbers and operations on them, estimation ought to be always present. Estimation of quantities in fraction or decimal terms and of the results of operations on those numbers is just as important for the mathematically literate adult as the same skills with whole numbers.

Children should understand that sometimes, an 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. Even in cases where exact answers are to be calculated, however, students must understand that it is almost always a good idea to have an estimate in mind before the actual exact computation is done so that the computed answer can be checked against the estimated one.

Building upon K-4 expectations, experiences in grades 5-6 will be such that all students:

H. develop, apply, and explain a variety of different estimation strategies in problem situations involving quantities and measurement.

• Students develop the concept of a billion by estimating its size relative to one hundred thousand, one million, and so on. For instance, they explore questions like If a calculator is programmed to repeatedly add 1 to the previous display and it takes about forty hours to reach a million, how long would it take to reach a billion?
• Students estimate the area inside a closed curve in square centimeters and then check the estimate with centimeter graph or grid paper.
• Students develop strategies for estimating sums and differences of fractions as they work with them. For example, a fifth grade class is asked to determine which of these computation problems have answers greater than 1 without actually performing the calculation:

  	1/2 + 3/4 =	1/3 + 5/6 =	1-5/16 - 1/2 =
  

  Most of the strategies the students develop hinge on whether each of the fractions are larger or smaller than 1/2.

• Students use fractions, decimals, or mixed numbers interchangeably when one form or another of the number eases estimation. For example, rather than estimating the product 3/5 x 4 directly, students consider 0.6 x 4 which yields a much quicker estimate.
• Similarly, in their beginning work with percent, students master the common fraction equivalents for familiar percentages and use fractions for estimation in appropriate situations. For example, an estimate of 65% of 63 can be easily obtained by considering 2/3 of 60.

• Students estimate the size of a crowd at a rock concert from a picture. They then share all of their various strategies with the rest of the class.
• Students estimate whether or not they can buy a set of items with a given amount of money. For example, You have only $10. Do you have enough money to buy:
  4 cans of tuna @ 79 cents each
  2 heads of lettuce @ 89 cents each
  2 lbs. of cheese @ $2.11 per lb.

• Students place decimal points in multi-place numbers to make absurd statements more reasonable. Some typical statements could be Mr. Brown averages 2383 miles per gallon when he drives to work, My dog weighs 5876 pounds, or Big Burger charges $99 for a large size of french fries.

• 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: Planning how long it would take to drive from Boston to New York and Submitting a bill for mileage to your boss.
• 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).
• When doing routine problems, the students are always reminded to consider whether their answers make sense. For instance, in the following problem, an estimate makes much better real world sense than an exact computation. Molly Gilbert is the owner of a small apple orchard in South Jersey. She has 19 rows of trees with 12 trees in each row. Last year the average production per tree was 761.3 apples. What can she expect to be the total yield this year? For this problem, an exact computation is certain to be wrong and will also be a number that is very hard to remember and to use in further planning.

• Students decide, as they discuss each new estimation strategy they learn, whether the strategy is likely to give an overestimate, an underestimate, or neither. For instance, using front-end digits will always give an underestimate; rounding everything up (as one might do to make sure she has enough money to pay for items selected in a grocery store) always gives an overestimate; and ordinary rounding may give either an overestimate or an underestimate.
• Students frequently use Guess, Check, and Revise as a problem solving strategy. With this strategy, an approximate answer is chosen which comes close to the parameters set in the problem. It is used computationally to check to see if it could be the right answer and, if not, whether it gives a result that is too large or too small. It is then revised up or down as a consequence. This success of this series of successive approximations is dependent upon understanding whether the original estimate was too large or too small.