Standard 4.1C: Math

Content Area: Math

Index: 4.1C Grade 12 CPI 1

Standard: 4.1 - Number and Numerical Operations

Strand: C - Estimation

Cumulative Progress Indicator: 1 - The student will recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.

Grade: 12

Sample Activities:

· Numerical problems in class are almost always worked out in "rough" form before any precise calculation takes place so that everyone understands the "ballpark" in which the computed answer should lie and which answers would be considered unreasonable.

· Students use estimation in their work with irrational numbers, approximating the results of operations such as sqrt(15) + sqrt(17) or sqrt(32) sqrt(8), and developing general rules.

· 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? 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] 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 discuss what is meant by the following specifications for the diameter of an 0-ring: 2.34 ± 0.005 centimeters.

· 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.

· Students act as quality assurance officers for mythical companies and devise procedures to keep errors within acceptable ranges. One possible scenario:

o 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?

· The World Series of Mechanisms