**Instructional Implications**Remind the student that the answer reported communicates to the reader a level of accuracy. A report that includes decimal digits that are not reliable can be misleading and one that excludes reliable digits might be omitting important information. Review with the student that the accuracy of a product or quotient is constrained by the least accurate of the original measures being multiplied or divided. Explain that the key for reporting products and ratios is to report a result that has as many digits as the data value with the fewest digits. Ask the student how many digits should be reported in the product if the diameter is reported as 48 cm andÂ Â is approximated by 3.14.
Ask the student to indicate the maximum and minimum values of the tree circumference andÂ Â to three decimal places given that the circumference is accurate to the indicated value. If necessary, guide the student to conclude that the circumference must be between 47.5 and 48.5 andÂ Â is between 3.135 and 3.145. Prompt the student to observe that the largest ratio occurs when the numerator is the largest possible and the denominator is the smallest possible (resulting in ). Â Likewise, the smallest ratio occurs when the quotient contains the smallest possible numerator and the largest possible denominator (). Based upon these results, ask the student to describe the diameter using a number of decimal places that is meaningful considering the accuracy of the data.
Have the student consider the product . IfÂ Â is approximated by 3.14 andÂ Â is approximated by 1.41, the product of the approximations is 4.4274. Ask the student to consider if this result can be accurate to four decimal places. Have the student consider the following possible representations: 4, 4.43, and 4.42740. Explain that since the factors are accurate to three digits, a result to three digits is appropriate. Explain why four is not as accurate as it could be and why 4.42740 could be misleading. Discuss with the student that the way one represents the answer communicates to the reader an estimate for the level of confidence that the writer has in the answer. Ask the student what a reader might interpret from 4.42740. Then ask the student to find the product whenÂ Â is approximated by 3.141593 and Â is approximated by 1.414214. Provide the student with the problem of selecting a representation for the product Â whenÂ Â is approximated by 3.14 and Â is approximated by 3.33333333.
Provide the student additional problems in which a level of accuracy appropriate to the described limitations on measurement must be chosen. Ask the student to identify the accuracy of each of the original measures in the problem and to state an appropriate level of accuracy of the solution before attempting to find the answer.
Consider using the NCTM activity *Dinosaur Bones* (http://www.illustrativemathematics.org/illustrations/16) or the MFAS task *Density* (N-Q.1.3), if not previously used. |