Estimation is a combination of content and process. Students' abilities to use estimation appropriately in their daily lives develop 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. At the high school level, estimation includes focusing on the reasonableness of answers and using various estimation strategies for measurement, quantity, and computations.

In the high school grades, estimation and number sense are much more important skills than algorithmic paper-and-pencil computation. Students need to be able to judge whether answers displayed on a calculator are within an acceptable range. They need to understand the displays that occur on the screen and the effects of calculator rounding either because of the calculator's own operational system or because of user-defined constraints. Issues of the number of significant digits 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. Appropriate issues 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 experience in computing them.

The cumulative progress indicators for grade 12 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 9, 10, 11, and 12.

Building upon knowledge and skills gained in the preceding grades, experience in grades 9-12 will be such that all students:

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

• Students are routinely asked if the answers they've 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?

• On a test, students are asked the following question: Jim used the zoom feature of his graphics calculator and found the solution to the system y = 2x + 3 and y = -2x - 1 to be (-.9997062,1.0005875). When he got his test back his teacher had taken points off. What was wrong with Jim's answer?

11. Estimate probabilities and predict outcomes from real-world data.

• Students use tables of data from an almanac to make estimates of the means and medians of a variety of measures such as the average state population or the 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 for the following week. They provide as detailed and statistical a predictionas possible. At the end of the week, predictions are compared to the actual performance. Written reports are evaluated by the teacher.

12. Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.

• Students in high school learn methods for estimating the magnitude of error in their estimations at the same time as they learn the actual computational procedures. Discussions regarding the acceptability of a given magnitude of error are a regular part of classroom activities when estimation is being used.

• Students work in small groups to carefully measure the linear dimensions of a rectangular box and determine its volume using measures to the nearest 1/8 inch or 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 exact answer lies by first using a measure for each dimension which is clearly short of the actual measure and multiplying them, and then by finding a measure for each dimension which is clearly longer than the actual measure and multiplying those. The exact answer then lies between those two products. Each group prepares a written report outlining their procedures and results.

• 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 of an inch. The volume of this "sphere" is 0.09314 cubic inches. Therefore (1728/0.09314) = 18552.71634 or 18,000 pieces of popcorn.

• Students write a computer program to round any number to the nearest hundredth.

• Students analyze the error involved in rounding to any value. For example, a number rounded to the nearest ten, say 840, falls into this range: 835 < X < 845. The error involved could be as large as 5. Similarly, a number rounded to the nearest hundredth, say .84, falls into the range: 0.835 < X < 0.845. The error could be as large as .005.

• Students discuss what is meant by the following specifications for the diameter of an 0-ring: 2.34 + 0.005 centimeters.

• 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 to see whether they accurately reflect the data that is provided. For example, did the editor make appropriate use of the data given below? (Based on an example in Exploring Surveys and Information from Samples by James Landwehr.)

  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.

References

Landwehr, James. Exploring Surveys and Information from Samples. Palo Alto, CA: Dale Seymour, 1987.

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.