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

Standard 10  Estimation  Grades 78
Overview
Estimation, as described in the K12 Overview includes three
primary themes: determining the reasonableness of answers,
using a variety of estimation strategies in a variety of
situations, and estimating the results of computations.
In seventh and eighth grade, estimation and number sense are much
more important skills than algorithmic paperandpencil computation
with whole numbers. Students should become masters at applying
estimation strategies so that an answer displayed on a calculator is
instinctively compared to a reasonable range in which the correct
answer lies. It is critical that students 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 figures
and what kinds of answers make sense in a given problem setting create
new reasons to focus on reasonableness of answers.
The new estimation skills begun in fifth and sixth grade are still
being developed in the seventh and eighth grades. These include
skills in estimating the results of fraction and decimal computations.
As students deepen their understanding of these numbers and perform
operations with them, estimation ought always to be present.
Estimation of quantities in fraction or decimal terms as a result 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 grader's ability, for example, to
estimate the square root of 40.
Students should understand that sometimes, an estimate will be
accurate enough to serve as an answer. At other times, an exact
computation will need to be done, either mentally, with
paperandpencil, or with a calculator. 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 so that
the computed answer can be checked against it.
Standard 10  Estimation  Grades 78
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 78 will be such that all students:
5^{*}. Recognize when estimation is
appropriate, and understand the usefulness of an estimate as
distinct from an exact answer.
 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
tenpound bag? Solution strategies are always discussed
with the whole class.
 Students create a plan to "win a contract" 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.
 Students use estimation skills to run a business
using Hot Dog Stand or Survival Math
software.
 Students apply estimation skills to algebraic
situation as they try to guess the equations to hit the most globs in
Green Globs software.
6. Determine the reasonableness of an answer by
estimating the result of operations.
 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 1/3 OFF AT JAKE'S FISHING WORLD!
ITEM 

REGULAR PRICE 
Daiwa Reels 

$29.95 each 
Ugly Stick Rods 

$20.00 each 
Tackle Boxes 

$17.99 each 
To assess students' performance, the teacher asks them to
write about how they can answer this question without doing any exact
computations.
If 6% sales tax is charged, can you tell whether $50 is
enough by estimating? Explain. Calculate the exact price
including tax.
 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 that prediction, a variety of
factors need to be estimated: number of atbats, lifetime batting
average, likelihood of injury, whether a baseball strike will occur,
and so on.
 Students discuss events in their lives that might have the
following likelihoods of occurring:
100%, 0.5 %, 3/4, 95%
 Students simulate estimating the number of fish
in a lake by estimating the number of fish crackers in a box using the
following method. Some of the fish are removed from the box and
"tagged" by marking them with food coloring. They are
"released" to "swim" as they are mixed in with the
other crackers in the box. Another sample is drawn and the number of
tagged fish and the total number of fish are recorded. This data is
used to set up a proportion (# tagged initially: total # fish = #
tagged caught: total # caught) to predict the total number of fish in
the box. The estimate is further improved by taking additional
samples, making predictions based on each sample of the total number
of fish in the box, and finally averaging all the
predictions.
8. 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 straightforward computation problems and to discuss the strategies
they use in making the estimations. Even relatively routine problems
generate interesting discussions and a greater shared number sense
within the class:
23% of 123, 5 x 38, 28 x 425, 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. To assess student understanding, the teacher
asks each student to write about how his or her group solved the
problem.
 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:
 Students make estimates of the number of
answering machines, cellular phones, and fax machines in the United
States and check their results against data from America by the
Numbers by Les Krantz.
 Students estimate the total number of people
attending National Football League games by determining how many teams
there are, how many games each plays, and what the average attendance
at a game might be. They then use these estimates to determine
theoverall answer (17,024,000 according to America by the
Numbers, p. 194).
 Students determine the amount of paper thrown
away at their school each week (or month or year) by collecting the
paper thrown away in their math class for one day and multiplying this
by the number of classes in the school and then by five (or 30 or
50).
9. Use equivalent representations of numbers such as
fractions, decimals, and percents to facilitate
estimation.
 Students use fractions, decimals, or mixed numbers
interchangeably when one form of the number makes estimation easier
than another. For example, rather than estimating 33_% of $120,
students consider 1/3 x 120 which yields a much
quicker estimate.
 Similarly, in their work with percents, students master
the common fraction equivalents for familiar percentages and use
fractions for estimation in appropriate situations. For example, an
estimate of 117% of 50 can most easily be obtained by considering 6/5
of 50.
 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.
10. Determine whether a given estimate is an
overestimate or an underestimate.
References

Krantz, Les. America by the Numbers. New York: Houghton
Mifflin, 1993. (Note: Some of the entries in this book are
unsuitable for seventh and eighthgraders.)
Software

Green Globs and Graphing Equations. Sunburst
Communications.
Hot Dog Stand. Sunburst Communications.
Survival Math. Sunburst Communications.
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 Indicators 5 and 6,
which are also listed for grade 4, since the Standards specify that
students demonstrate continued progress in these indicators.
