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

Standard 10  Estimation  Grades 34
Overview
As indicated in the K12 Overview, students' ability to use
estimation appropriately in their daily lives develops as they focus
on the reasonableness of answers, explore and construct
estimation strategies, and estimate measurements,
quantities, and the results of computation.
For this type of development to occur, the atmosphere established
in the classroom ought to assure everyone that their estimates are
important and valued. Children should feel comfortable taking risks,
and should understand that an explanation and justification of
estimation strategies is a regular part of the process. Third and
fourthgraders, for the most part, should be beyond just
"guessing." As children communicate with each other about
how their estimates are formulated, they further develop their
personal bank of strategies for estimation.
Students should already feel comfortable with estimation of sums
and differences from their work in earlier grades. Nonetheless, they
should regularly be asked About how many do you think there will
be in all? or About what do you think the difference
is? or About how many do you think will be left? in the
standard addition and subtraction settings. These questions are
appropriate whether or not exact computations will be done. As the
concepts and the related facts of multiplication and division are
introduced through experiences that are relevant to the child's
world, estimation with computation must again be integrated
into the development and practice activities.
One of the most useful computational estimation strategies
in these grade levels also reinforces an important placevalue
idea. Students should understand that in multidigit whole numbers
the larger the place value, the more meaningful the digit in that
position is in contributing to the overall value of the number. A
reasonable approximation, then, of a multidigit sum or difference can
always be made by considering only the leftmost places and ignoring
the others. This strategy is referred to as front end
estimation and is the main estimation strategy that many
adults use. In third and fourth grades, it should accompany the
traditional rounding strategies.
Children should understand that, sometimes, the 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 to arrive at a more precise
answer. The particular procedure to be used is dependent on the
setting and the problem. Also at this level, estimation must be an
integral part of the development of concrete, algorithmic, or
calculator approaches to multidigit computation. Students must be
given experiences which clearly indicate the importance of formulating
an estimate before the exact answer is calculated.
In third and fourth grades, students are developing the concepts of
a thousand and then of a million. Many opportunities arise where
estimation of quantity is easily integrated into the curriculum. Many
what if questions can be posed so that students continue to use
estimation skills to determine practical answers.
Standard 10  Estimation  Grades 34
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 34 will be such that all students:
1. Judge without counting whether a set of objects
has less than, more than, or the same number of objects as a
reference set.
 Students estimate the numbers represented by groups of
base ten blocks or bundles of popsicle sticks. For example, one set
might consist of 1 hundred, 6 tens, and 3 ones, and the other 0
hundreds, 17 tens, and 7 ones. Students first estimate which is more
without arranging the blocks or counting them and then they determine
the correct answer. These kinds of proportional models allow the
"quantity of wood" to be proportional to the actual size of
the number.
 Students read The Popcorn Book by Tomie
dePaola and make estimates with popcorn. For example, they might
consider two quantities of popcorn, one popped and one unpopped.
Which contains the largest number of kernels?
They might also predict how many measuring cups the unpopped popcorn
will fill once it is popped and find out the result after popping the
popcorn. (The total number of cups made may be
surprising.)
2. Use personal referents, such as the width of a
finger as one centimeter, for estimations with
measurement.
 Students estimate the height of a classmate in inches or
centimeters by standing next to him or her and using their own known
height for comparison.
 As standard units like yard and halfinch are introduced,
students are challenged to find some part of their body or some
personal action that is about that size at this point in their growth.
For instance, they may decide that the width of their little finger is
almost exactly one halfinch or the length of two giant steps
is one yard.
 Students measure the width of their handspan in
centimeters (from thumb tip to little finger tip with the hand spread
as far as possible) and then use the knowledge of its width to
estimate the metric measures of various classroom objects by counting
the number of handspans across and multiplying by the number of
centimeters.
3. Visually estimate length, area, volume, or
angle measure.
 Students estimate the number of 3" x 5" cards it
would take to cover their desktops, a floor tile, and the blackboard.
They describe the process they used in writing, which is then read by
the teacher to determine the students' progress.
 Students work through the Tiling a Floor
lesson that is described in the First FourStandards of this
Framework. They estimate how many of their tiles it would take
to cover one sheet of paper, and compare their answers to the actual
number needed.
 Students estimate the capacities of containers of a
variety of shapes and sizes, paying careful attention to the equal
contributions of width, length, and height to the volume. They sort a
series of containers from smallest to largest and then check their
arrangements by filling the smallest with uncooked rice, pouring that
into the second, verifying the fact that it all fits and that more
could be added, pouring all of that into the third, and so on.
 Students estimate the angle formed by the hands of the
clock. They can also be challenged to find a time when the hands of
the clock will make a angles of 90 degrees, 120 degrees angle, and
180 degrees.
4. Explore, construct, and use a variety of
estimation strategies.
 Students develop and use the front end estimation
strategy to obtain an initial estimate of the exact answer. For
example, to estimate the total mileage in a driving trip where 354
miles were driven the first day, 412 the second day, and 632 the
third, simply add all digits in the hundreds' place: 3 + 4 +
6 = 13 hundred or 1300.
 Students learn to adjust the front end
estimation strategy to give a more accurate answer. To do so, the
second number from the left (in the problem above, the tens) is
examined. Here, the estimate would be adjusted up one hundred because
the 5 + 1 + 3 tens are almost another hundred. This would give
a better estimate of 1400.
 Students use rounding to create estimates,
especially in multidigit addition and subtraction. They do so
flexibly, however, rather than according to outofcontext rules. In a
grocery store, for example, when a person wants to be sure there is
enough money to pay for items that cost $1.89, $2.95, and $4.45, the
best strategy may be to round each price up to the next
dollar. In this case then, the actual sum of the prices is
definitely less than $10.00 (2 + 3 + 5). On the other hand, to
be sure that the total requested by the cashier is approximately
correct, the best strategy may be to round each price to the
nearest dollar and get 2 + 3 + 4 which is $9.
 Students are shown a glass jar filled with about two
hundred marbles and are asked to estimate the number in the jar. In
small groups, they discuss various approaches to the problem and the
strategies they can use. They settle on a strategy to share with the
class along with the estimate that resulted.
 Students write about how they might find an
estimate for a specific problem in their journals.
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 situation in the pair is the one for which estimation is the
best approach and which is the one for which an exact answer is
probably needed. One such pair, for example, might be: deciding
how much fertilizer is needed for a lawn and filling the
bags marked "20 pounds" at the fertilizer
company.
 Given a set of cartoons with homemade mathematical
captions, third graders decide which of the cartoon characters arrived
at exact answers and which got estimates. One ofthe cartoons might
show an adult standing in the checkout line at a supermarket and
another might show the checkout clerk. The captions would read:
Mr. Harris wondered if he had enough money to pay for the
groceries he had put in the cart and Harry used the
cash register to total the bill. Students make up their own
similar cartoon.
 Students share with each other various situations within
the past week 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).
6. Determine the reasonableness of an
answer by estimating the result of operations.
 Students are regularly asked if their answer makes sense
in the context of the problem they were solving. They respond with
full sentences explaining what they were asked to find and why the
numerical answer they found fits the context reasonably, that is, why
it could be the answer.
 Fourth graders might be asked to decide if their
estimated answer to the following problem is reasonable. The band
has 103 students in it. They line up in 9 rows. How many
students are there in each row? The students' responses
might indicate, for example, that there should be about 10 students in
each row, since 103 is close to 100 and 9 is close to 10.
 Students estimate reasonable numbers of times that
particular physical feats can be performed in one minute. For
example: How many times can you skip rope in a minute? How
many times can you hit the = button on the calculator in a minute?
How many times can you blink in a minute? How many times can
you write your full name in a minute? and so on. Other students
judge whether the estimates are reasonable or unreasonable and then
the tasks are performed and actual counts made. (To determine the
number of times the = button is hit in a minute, press +1= so that
each time the = button is pressed, the display increases by 1.)
 Thirdgrade students are given a set of thirty cards with
threedigit subtraction problems on them. In one minute, they must
sort the cards into two piles: those problems whose answers are
greater than 300 and those whose answers are less than 300. The
correct answers can be on the backs of the cards to allow
selfchecking after the task is completed.
 For assessment, fourthgrade students might be given a
page of onedigit by multidigit multiplication problems in a multiple
choice format with four possible answers for each problem. Within
some time period which is much too short for them to perform the
actual computations, students are asked to choose the most reasonable
estimate from each set of four answers.
7. Apply estimation in working with quantities,
measurement, time, computation, and problem solving.
 Students work through the Product and
Process lesson that is described in the Introduction to this
Framework. It challenges the students to form two threedigit
numbers using 3, 4, 5, 6, 7, 8 which have the largest product;
estimation is used to determine the most reasonable possible
choices.
 Students learn about different strategies for
estimating by reading The Jellybean Contest by Kathy Darling
or Counting on Frank by Rod Clement.
 Students regularly try to predict the numerical facts
presented in books like In an Average Lifetime . . .
by Tom Heymann. Using knowledge they have and a whole variety of
estimation skills, they predict answers to: What is the number of
times the average American eats in a restaurant in a lifetime?
(14,411) What is the total length each human fingernail grows
in a lifetime? (77.9 inches) and What is the average number
of major league baseball games an American attends in a
lifetime? (16)
 Students regularly estimate in situations involving
classroom routines. For example, they may estimate the total amount
of money that will be collected from the students who are buying lunch
on Pizza Day or the number of buses that will be needed to take the
whole third and fourth grade on the class trip.
 Students investigate environmental issues using
estimation. One possible activity is for them to estimate how many
gallons of water are used for various activities each week in their
home. (See Healthy Environment  Healthy
Me.)
Activity Total # Each time Total # of gallons used
in 1 week
Take shower or bath _______ x 18 gallons =
Flush toilet _______ x 7 gallons =
Wash dishes _______ x 10 gallons =
Wash clothes _______ x 40 gallons =
_________________
Total # of gallons used in 1 week =
Students discuss possible reasons for differences among their
estimates, and they compute the class total for the number of gallons
consumed during that week.
References

Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth
Stevens Children's Books, 1991.
Darling, Kathy. The Jellybean Contest. Champaign, IL:
Garrard, 1972.
de Paola, Tomie. The Popcorn Book. New York: Holiday
House, 1978.
Environmental and Occupational Health Sciences Institute.
Healthy Environment  Healthy Me.
Exploring Water Pollution Issues: Fourth Grade.
New Jersey: Rutgers University, 1991.
Heymann, Tom. In an Average Lifetime ... New York:
Random House, 1991.
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.
