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 students 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 student 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 whole number computation assumes less importance at these grade levels than previously as students mastery over the various strategies and approaches should be somewhat established. Formal strategies should still be compared with informal strategies which students develop, however, as they estimate answers to various problems under varying conditions. Students should also continue to 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 seventh and eighth grade, estimation and number sense are much more important skills than algorithmic pencil-and-paper computation with 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. With calculators and computers being used on a consistent basis in these grade levels, it is critical that students understand the displays that occur on the screen and the effects of calculator rounding either because of the calculators own operational system or because of user-defined constraints. Issues of number of significant figures and what kinds of answers make sense in a given problem setting create new reasons to a focus on reasonableness of answers.

The new estimation skills that were begun in fifth and sixth grade are also still being developed in seventh and eighth grades. These include skills in estimating the results of fraction and decimal computation. As students deepen their understandings of these 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.

In addition, the seventh and eighth grades present students with opportunities to develop strategies for estimation with ratios, proportions, and percents. Estimation and number sense must play an important role in the lessons dealing with these concepts so that students feel comfortable with the relative effects of operations on them. Another new opportunity here is estimation of roots. It should be well within every eighth graders ability, for example, to estimate the square root of 40.

Students 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-6 expectations, experiences in grades 7-8 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 regularly have opportunities to estimate answers to straight-forward computation problems and to discuss the strategies they use in making the estimations. Problems as mundane as these cause interesting discussions and a greater shared sense of number in the class:

  	23% of 123,   5 x 38,   28 x 42y,   486 x 2004,   423/71
  

• Given a ream of paper, students work in small groups to estimate the thickness of one sheet of paper. Answers and strategies are compared across groups and explanations for differences in the estimates are sought.
• Students develop strategies for estimating the results of operations on fractions as they work with them. For example, a seventh grade class is asked to determine which of these computation problems have answers greater than 1 without actually performing the calculation:

  	1/2 x 3/4 = 	1/3 + 5/6 = 	1 5/16 / 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 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 most easily be obtained by considering 2/3 of 60.
• Students collect and bring to class sales circulars from local papers which express the discounts on sale items in a variety of ways including percent off, fraction off, and dollar amount off. For items chosen from the circular, the students discuss which form is the easiest form of expression of the discount, which is most understandable to the consumer, and which makes the sale seem the biggest bargain.

• Students evaluate various statements made by public figures to decide whether they are reasonable. For example, The Phillies center fielder announced that he expected to get 225 hits this season. Do you think he will? In order to determine what confidence to have in the prediction, a variety of factors need to be estimated: number of at-bats, lifetime batting average, likelihood of injury, whether a baseball strike will occur, and so on.
• Students estimate whether or not they can buy a set of items with a given amount of money. For example, I have only $50. Can I buy a reel, a rod, and a tackle box during the sale advertised below?
  ALL ITEMS I/3 OFF AT JAKES FISHING WORLD!

  ITEM	REGULAR PRICE
  Daiwa Reels	$29.95 each
  Ugly Stick Rods	$20.00 each
  Tackle Boxes	$17.99 each

• Students discuss events in their lives that might have the following likelihoods occurring:

  	100%,     0.5 %,    3/4,     95%
  

• Students regularly tackle problems for which estimation is the only possible approach. For example, How many hairs are on your head? or How many grains of rice are in this ten-pound bag? Solution strategies are always discussed with the whole class.
• 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 create a plan to "win a contract" for by bidding on projects. For example, Your class has been given one day to sell peanuts at Shea Stadium. Prepare a presentation that includes the amount of peanuts to order, the costs of selling the peanuts, the profits that will be made, and the other logistics of selling the peanuts. Organize a schedule with estimated times for completion for the entire project.

• Using calculators, but without using the square root key, students try to find approximations for a few square roots. For example, Find a good approximation for the square root of 40. Through a series of approximations, they make a guess, perform the multiplication on the calculator, determine whether the approximation was too large or too small, adjust it, and begin again. This series of approximations, in itself a very useful strategy, continues until an approximation is reached that is satisfactory.
• 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.