Meaning and Importance

People who use mathematics in their lives and careers find estimation to be preferable to the use of exact numbers in many circumstances. Frequently, it is either impossible to obtain exact answers or too expensive to do so. An air conditioning salesperson preparing a bid would be wasting time and money by measuring rooms exactly. Astronomers attempting to determine movements of celestial objects cannot obtain precise measurements. Many people use approximations because it is easier than using exact numbers. Shoppers, for example, use approximations to determine whether they have sufficient funds to purchase items. Travelers use rough estimates of time, distance, and cost when planning trips. Commonly reported data often use levels of precision which have been accepted as appropriate, even though they may not be considered "exact". Astronomers always report information to two significant digits, and baseball players always report their batting averages as three place decimals.

K-12 Development and Emphases

Instruction in estimation has traditionally focused on the use of rounding. There are times when rounding is an appropriate process for finding an estimate, but this standard emphasizes that it is only one of a variety of processes. Computational estimation strategies are a new and important component in the curriculum. Clustering, front-end digits, compatible numbers, and other strategies are all helpful to the skillful user of mathematics, and can all be mastered by young students. Selection of the appropriate strategy to use depends on the setting and the numbers and operations involved.

The foregoing discussion describes a new emphasis on the use of estimation in computational settings, but students should also be thoroughly comfortable with the use of estimation in measurement. Students should develop the ability to estimate measures such as length, area, volume, and angle size visually as well as through the use of personal referents such as the width of a finger being about one centimeter. Measurement is also rich with opportunities to develop an understanding that estimates are often used to determine approximate values which are then used in computations and that results so obtained are not exact but fall within a range of tolerance.

Estimation should be an emphasis in many other areas of the mathematics curriculum in addition to the obvious uses in numerical operations and measurement. Within statistics, for example, it is often useful to estimate measures of central tendency for a set of data; estimating probabilities can help a student determine when a particular course of action would be advisable; problem situations related to algebraic concepts provide opportunities to estimate rates such as slopes of lines and average speed; and working with sequences in algebra and increasing the number of sides of a regular polygon in geometry yield opportunities to estimate limits.

In summary, estimation is a combination of content and process. Students ability to use estimation appropriately in their daily lives develops as they have regular opportunities to explore and construct estimation strategies and as they acquire an appreciation of its usefulness through using estimation in the solution of problems.

One of the estimation emphases for very young children is the development of the idea that guessing is an important and exciting part of mathematics. The teacher must employ sound management practices which ensure that everyone's guess is important and which encourage risk taking and sharing of ideas about the how guesses were determined. Estimations of measurements as well as of quantities should pervade the classroom activity. When first asked to guess an answer, many students will give nonsense responses until they establish appropriate experiences, build their sense of numbers, and develop informal strategies for creating a guess. Children begin to make reasonable estimates when the situations involved are relevant to their immediate world. Building on comparisons of common objects and using personal items to build a sense of lengths, weights, or quantities helps children to gain confidence in their guessing. As children communicate with each other about how guesses are formulated they begin to develop informal 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. Estimation of sums and differences should be a part of the computational process from the very first activity with any sort of computation. Children 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. 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.

One of the most useful computational estimation strategies in these grade levels also reinforces an important place value idea. Students should understand that in two-digit numbers the tens digit is much more meaningful than the ones digit in contributing to the overall value of the number. A reasonable approximation, then, of a two digit sum or difference can always be made by considering only the tens digits and ignoring the ones. This strategy is referred to as front end estimation and is used with larger numbers as well. It is the main estimation strategy that many adults use.

Experiences will be such that all students in grades K-2:

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

• Students place various amounts of counters, candy, or other small objects in individual plastic bags. Working in groups of four students, the children choose one bag to be the reference set and judge whether each of the other bags has more than, less than, or the same as the reference set. Initially, they should try to make the judgments without counting.
• Young children benefit from comparing sets of objects to some given number. For example, given five sets of poker chips arranged on a table, they should often practice going through the sets and name which ones are more than five and which are less than five.
• Students play the card game War with a set of cards without numerals. All that is on the cards are the sets of hearts, clubs, diamonds, or spades. Students will initially easily distinguish between the 7s and the 3s, but will be reluctant to make judgments about closer numbers like the 4s and 5s without counting. As they play more, though, distinctions will become finer.

• Students estimate lengths of pieces of spaghetti, yarn, paper, pencils, paper clips etc. using suggested non-standard personal units such as width of thumb, length of a foot, and so on.
• 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 use their self-discovered personal body referents to estimate the measures of various classroom objects like the length of the blackboard or the width of a piece of paper in the associated units.

• Students look at a quantity of sand, salt, flour, water, or popcorn and estimate how many times it could fill up a specified container.
• Students estimate how many pieces of notebook paper it would take to cover a given area such as the blackboard, or a portion of the classroom floor.
• Students regularly estimate lengths in a whole variety of non-standard units such as my feet, unifix cubes, paper clips, and orange Cuisenaire Rods. They then measure to verify or correct their estimates.

• Students are asked if a sixty-seat bus will be adequate to take the two first grade classes on their field trip. After it is known that there are 23 children in one class and 27 children in the other, individual children volunteer their answers and give a rationale to support their thinking. A discussion might be directed to the question of whether an exact answer to the computation was needed for the problem.
• Students are confronted with a glass jar filled with about eighty 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.
• Second grade students can be challenged to estimate the total number of students in the school. They will need to talk about the average number of students in each class, the number of classes in a grade level, and the number of grade levels in the school. They might then want to use calculators to solve the problem, but the result, even though the exact answer to a computation, is still an estimate. They can then discuss why that is so.

• 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: Sharing a bag of peanuts among 3 friends and Paying for tickets at the movie theater.
• Given a set of cartoons with home-made mathematical captions, first graders decide which of the cartoon characters arrived at exact answers and which got estimates. Two of the cartoons might show an adult and a child looking at a jar of jelly beans and the captions might read: Susie guessed that there were 18 jelly beans left in the jar and Susie's mom counted the 14 jelly beans left in the jar.

• 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.
• Second-grade students are given a set of thirty cards with two-digit addition problems on them. In one minute, they must sort the cards into two piles: those problems whose answers are greater than 100 and those whose answers are less than 100. The correct answers can be on the backs of the cards to allow self-checking after the task is completed.
• Second-grade students might also be given a page of addition or subtraction 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 have small pieces of yarn of slightly different lengths ranging from 2 to 6 inches. Each student first estimates the number of his or her pieces it would take to match a much longer piece - about 30 inches long - and then actually matches the two pieces up and counts how many. Then they use their individual pieces to measure other objects in the room. Each child is responsible for first estimating the lengths in terms of his or her own yarn, but they can use evidence from other childrens measuring to help make their own estimates.
• Students regularly estimate before appropriate classroom procedures. For example, at snack time, they may guess how many cups can be filled by each can of juice or how many crackers each student will get if all of the crackers in the box are given out.
• Kindergartners always have fun estimating which color is best represented in a group of multi-colored objects. Good examples of such activity would be choosing the m&m color that shows up most in a bag of m&ms, choosing the Fruit Loops color that shows up the most in a bowl of Fruit Loops, or choosing the color that shows up most in a bowl full of marbles. After everyone has committed to a guess, the children can sort the objects and count each color. They can even make a bar graph to show the distribution of the different colors.

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.

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.

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.

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.

In the high school grades, estimation and number sense are much more important skills than algorithmic pencil-and-paper computation. 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.

Measurement settings are rich with opportunities to develop an understanding that estimates are often used to determine approximate values which are then used in computations and that results so obtained are not exact but fall within a range of tolerance. Issues also appropriate for discussion at this level include acceptable limits of tolerance, and assessments of the degree of error of any particular measurement or computation.

Another topic appropriate at these grade levels is the estimation of probabilities and of statistical phenomena like measures of central tendency or variance. When statisticians talk about "eyeballing" the data, they are explicitly referring to the process where these kinds of measures are estimated from a set of data. The skill to be able to do that is partly the result of knowledge of the measures themselves and partly the result of years of experience in computing them. Students can begin to develop these skills as well.

Students should, by this point in their educations, 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-8 expectations, experiences in grades 9-12 will be such that all students:

M. determine the reasonableness of answers to problems solved using pencil-and-paper techniques, mental math, algebraic formulas and equations, computers or calculators.

• Students are constantly asked if the answers theyve computed make sense. Latisha's calculator displayed 17.5 after she entered 3 times the square root of 5. Is this a reasonable answer?
• Students are sometimes presented with hypothetical scenarios that challenge both their estimation and technology skills: During a test, Paul entered

  	y = .516x - 2 	and 	y = .536x + 5
  

  in his graphics calculator . After analyzing the two lines displayed on the "standard" screen window setting [-10,10, -10,10], he decided to indicate that the lines were parallel and that there was no point of intersection. Was Paul's answer reasonable?

• Another example: Jim used the zoom feature of his graphics calculator and found the solution to y = 2x + 3 and y = -2x - 1 to be (-.9997062,1.0005875). When he got his paper back his teacher had taken points off. What was wrong with Jim's response?

• Students use tables of data from an almanac to make estimates of the means and medians of a variety of measures from average state population to average percentage of voters in presidential elections. Any table where a list of figures, but no mean, is given can be used for this kind of activity. After estimates are given, actual means and medians can be computed and compared to the estimates. Reasons for large differences between the means and medians ought to also be explored.
• Students collect data about themselves and their families for a statistics unit on standard deviation. After everyone has entered data in a large class chart regarding number of siblings, distance lived away from school, oldest sibling, and many other pieces of numerical data, the students work in groups to first estimate and then compute means and medians as a first step toward a discussion of variation.
• Students each track the performance of a particular local athlete over a period of a few weeks and use whatever knowledge they have about past performance to predict his or her performance over the next week. They provide as detailed and statistical a prediction as possible. At the end of the week, predictions are compared to the actual performance.

• Students in the high school grades learn methods for estimating the magnitude of error in their estimations at the same time as they learn the main procedures. Discussions regarding the acceptability of a given magnitude of error are a regular part of classroom activity when estimation is being used.
• Students work in small groups to measure the linear dimensions of a rectangular box and determine its volume using measures to the nearest 1/8 inch since that is the smallest unit on their rulers. After their best measurements and computation, the groups share their estimates of the volume and discuss differences. Each group then constructs a range in which they are sure the actual exact answer lies by using a measure for each dimension which is clearly short of the actual measure and multiplying them and finding a measure for each dimension which is clearly longer than the actual measure and multiplying them. The range is specified as between those two products.
• Students are presented with these two solutions to the following problem and discuss the error associated with each approach:

  How many kernels of popcorn are in a cubic foot of popcorn?

  1. There are between 3 and 4 kernels of popcorn in 1 cubic inch. There are 1728 cubic inches in a cubic foot. Therefore there are 6048 kernels of popcorn in a cubic foot. [3.5 (the average number of kernels in a cubic inch) x 1728 = 6048]
  2. The diameter of a kernel of popcorn is approximately 9/16". The volume of this "sphere" is 0.09314 cubic inches. Therefore (1728/0.09314) = 18552.71634 or 18,000 pieces of popcorn.

• Students act as Quality Assurance officers for mythical companies and devise procedures to keep errors within acceptable ranges. One possible scenario:

  In order to control the quality of their product, Paco's Perfect Potato Chip Company guarantees that there will never be more than 1 burned potato chip for every thousand that are produced. The company packages the potato chips in bags that hold about 333 chips. Each hour 9 bags are randomly taken from the production line and checked for burnt chips. If more than 15 burnt chips are found within a four hour shift, steps are taken to reduce the number of burnt chips in each batch of chips produced. Will this plan ensure the company's guarantee?

• Students regularly review statistical claims reported in the media as to their accuracy given the data that is available. One possible case study (the idea for which was taken from: Landwehr, J. Exploring Surveys and Information from Samples. Palo Alto CA: Dale Seymour, 1987, page 2.):

  The March, 1985 Gallup survey asked 1,571 American adults "Do you approve or disapprove of the way Ronald Reagan is handling his job as president."

  56% said that they approved. For results based on samples of this size, one can say with 95% confidence that the error attributable to sampling and other random effects could be as much as 3 percentage points in either direction.

  A newspaper editor read the Gallup survey report and created the following headline:
  BARELY ONE-HALF OF AMERICA APPROVES OF THE JOB REAGAN IS DOING AS PRESIDENT.

  Did this editor make appropriate use of the data for the headline?