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

Standard 10  Estimation  Grades 56
Overview
In grades 5 and 6, students extend estimation to new types of
numbers, including fractions and decimals. As indicated in the K12
Overview, they continue to work on determining the reasonableness
of results, using estimation strategies, and applying
estimation to measurement, quantities, and computation.
In fifth and sixth grade, estimation and number sense are even more
important skills than algorithmic paperandpencil computation with
multidigit whole numbers. Students should become masters at applying
estimation strategies so that answers displayed on a calculator are
instinctively compared to a reasonable range in which the correct
answer lies.
The new estimation skills that are also important in fifth and
sixth grade are skills in estimating the results of fraction and
decimal computations. Even though the study of the concepts and
arithmetic operations involving fractions and decimals begins before
fifth grade, a great deal of time will be spent on them here. A sample
unit on fractions for the sixthgrade level can be found in Chapter 17
of this Framework. As students develop an understanding of
fractions and decimals and perform operations with them, estimation
ought always to be present. Estimation of quantities in fraction or
decimal terms and of the results of operations on those numbers is
just as important for the mathematically literate adult as the same
skills with whole numbers.
Children should understand that, sometimes, an estimate will be an
accurate enough number to serve as an answer. At other times, an
exact computation will need to be done, either mentally, with
paperandpencil, or with a calculator to arrive at a more precise
answer. Which procedure should be used is dependent on the setting
and the problem. Even in cases where exact answers are to be
calculated, however, students must understand that it is always a good
idea to have an estimate in mind before the actual exact computation
is done so that the computed answer can be checked against the
estimated one.
Standard 10  Estimation  Grades 56
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 5 and
6.
Building upon knowledge and skill gained in the preceding grades,
experiences in grades 56 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.
 Given pairs of reallife situations, students determine
which is the one in which estimation is the best approach and which is
the one needing an exact answer. For example, one such pair, might
be: planning how long it would take to drive from Boston to New
York and submitting a bill for mileage to your boss.
 Students collect data for a week on various situations
when they and their families had to do some computation and describe
when an exact answer was necessary (and why) and when an estimate was
sufficient (and why).
 When doing routine problems, the students are always
reminded to consider whether their answers make sense. For instance,
in the following problem, an estimate makes much more sense than an
exact computation. Molly Gilbert is the owner of a small apple
orchard in South Jersey. She has 19 rows of trees with 12
trees in each row. Last year the average production per tree
was 761.3 apples. At that rate, what can she expect the total
yield to be this year? For this problem, an exact computation is
certain to be wrong and will also be a number that is very hard to
remember or use in further planning.
 Students look for examples of estimation language
in their reading and/or in the newspapers.
 Students investigate what decisions at the school
are based on estimates (e.g., quantity of food for lunch, ordering of
textbooks or supplies, making up class schedules, organizing bus
routes) by interviewing school employees and making written and/or
oral reports.
6. Determine the reasonableness of an answer by
estimating the result of operations.
 Students estimate the size of a crowd at a rock concert
from a picture. They share all of their various strategies with the
rest of the class.
 Students demonstrate their understanding by estimating
whether or not they can buy a set of items with a given amount of
money. For example: You have only $10. Explain how you
can tell if you have enough money to buy: 4 cans of tuna @
79 cents each, 2 heads of lettuce @ 89 cents
each, and 2 lbs. of cheese @ $2.11 per lb.
 Students place decimal points in multidigit numbers to
make absurd statements more reasonable. Some typical
statements could be Mr. Brown averages 2383 miles per gallon
when he drives to work, My dog weighs 5876 pounds, or
Big Burger charges $99 for a large order of french
fries.
8. Develop, apply, and explain a variety of different
estimation strategies in problem situations involving
quantities and measurement.
 Students develop the concept of a billion by estimating
its size relative to one hundred thousand, one million, and so on.
For instance, they explore questions like If a calculator
is programmed to repeatedly add 1 to the previous display and
it takes about forty hours to reach a million, how long would
it take to reach a billion? Students might relate this to the
size of the national debt (now $5 trillion).
 Students estimate the area inside a closed curve in square
centimeters and then check the estimate with centimeter graph or grid
paper.
 After reading Shel Silverstein's poem
"How Many, How Much" and Tom Parker's Rules
of Thumb, students write their own rules of thumb (e.g.,
You should never have homework higher than one
inch.).
 Students make estimates about things that happen
in one day at school after reading about some of the data in In
One Day, by Tom Parker. For example, they estimate: How
much pizza is eaten? How much milk is drunk? How many
students go home sick? or How many students forget something?
They then interview people around the school or conduct a survey
to check their estimates.
 Students develop strategies for estimating sums and
differences of fractions as they work with them. For example, a fifth
grade class is asked to determine which of the following computation
problems have answers greater than 1 without actually performing the
calculation; many students' strategies will hinge on comparisons
of the given fractions to 1/2.
 Students make use of strategies like
clustering and compatible numbers in estimating the
results of computations. They recognize that a sum of numbers that
are approximately the same, such as 37, 39, and 42, can be replaced in
an estimate by the product 3 x 40 (clustering). They also know
that other computations can be performed easily by changing the
numbers to numbers that are closely related to each other, such as
changing 468 divided by 9 to 450 divided by 9 (compatible
numbers).
 Students work through the Mathematics at
Work lesson that is described in the Introduction to this
Framework. A parent discusses a problem which her company
faces regularly: to determine how large an air conditioner is needed
for a particular room. To solve this problem, the company has to
estimate the size of the room.
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 a number makes estimation easier than
another form. For example, rather than estimating the product 3/5 x
4, students consider 0.6 x 4 which yields a much quicker estimate.
 In their beginning work with percent, students master the
common fraction equivalents for familiar percentages and use fractions
for estimation in appropriate situations. For example, an estimate of
65% of 63 can be easily obtained by considering 2/3 of 60.
10. Determine whether a given estimate is an overestimate
or an underestimate.
 Students decide, as they discuss each new estimation
strategy they learn, whether the strategy is likely to give an
overestimate, an underestimate, or neither. For instance, using
frontend digits will always give an underestimate; rounding
everything up (as one might do to make sure she has enough money to
pay for items selected in a grocery store) always gives an
overestimate; and ordinary rounding may give either an overestimate or
an underestimate.
 Students frequently use guess, check, and revise as
a problem solving strategy. With this strategy, the answer to a
problem is estimated, then calculations are made using the estimate to
see whether this estimate meets the conditions of the problem, and the
estimate is then revised upwards or downwards as a result. For
example, students are asked to find two consecutive pages in a book
the product of whose page numbers is 1260. An initial guess
might be 30 and 31, the check might involve concluding that
this product is a little over 900, and, as a result, they might
revise their estimate upwards.
References

Parker, Tom. In One Day. Boston: Houghton Mifflin,
1984.
Parker, Tom. Rules of Thumb. Boston: Houghton Mifflin,
1983.
Silverstein, Shel. "How Many, How Much," in A
Light in the Attic. New York: Harper and Row, 1981.
OnLine 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 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.
