STANDARD 10  ESTIMATION
All students will use a variety of estimation strategies and
recognize situations in which estimation is appropriate.

Standard 10  Estimation  Grades 912
Overview
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 paperandpencil 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 userdefined 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.
Standard 10  Estimation  Grades 912
Indicators and Activities
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 912 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 realworld 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?
 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 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
0ring: 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 ONEHALF 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.
OnLine Resources

http://dimacs.rutgers.edu/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 gradespecific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
^{*} Activities are included here for Indicator 6,
which are also listed for grade 8, since the Standards specify that
students demonstrate continued progress in this indicator.
