STANDARD 10 - ESTIMATION
All students will use a variety of estimation strategies and
recognize situations in which estimation is appropriate.
Standard 10 - Estimation - Grades K-2
As indicated in the K-12 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.
One of the estimation emphases for very young children is the
development of the idea that guessing is an important and exciting
part of mathematics. The teacher must employ sound management
practices which ensure that everyone's guess is important and which
encourage student risk-taking and sharing of ideas about how their
guesses were determined. When first asked to guess an answer, many
students will give nonsense responses until they establish appropriate
experiences, build their sense of numbers, and develop informal
strategies for creating a guess. Children begin to make reasonable
estimates when the situations involved are relevant to their immediate
world. Building on comparisons of common objects and using personal
items to build a sense of lengths, weights, or quantities helps
children gain confidence in their guessing. As children communicate
with each other about how guesses are formulated they begin to develop
informal strategies for estimation.
Estimation with computation is as important at these early
grade levels as it is at all the other grade levels. Estimation of
sums and differences should be a part of the computational process
from the very first activity with any sort of computation. Children
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. Children
should understand that, sometimes, the 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
paper-and-pencil, 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.
One of the most useful computational estimation strategies
at these grade levels also reinforces an important place value
idea. Students should understand that in two-digit numbers the tens
digit is much more meaningful than the ones digit in contributing to
the overall value of the number. A reasonable approximation, then, of
a two-digit sum or difference can always be made by considering only
the tens digits and ignoring the ones. This strategy is referred to
as front end estimation and is used with larger numbers as
well, although then the first two digits may be used. It is the main
estimation strategy that many adults use.
Standard 10 - Estimation - Grades K-2
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 kindergarten and in grades 1 and 2.
Experiences will be such that all students in grades K-2:
1. Judge without counting whether a set of objects
has less than, more than, or the same number of objects as a
- Students place various amounts of counters or other small
objects in individual plastic bags. Working in groups of four
students, the children choose one bag to be the reference set and
judge whether each of the other bags has more than, less than, or the
same as the reference set. Initially, they should try to make the
judgments without counting. The teacher observes the groups as they
work, making notes about the students' progress.
- Young children benefit from frequently comparing sets of
objects to some given number. For example, given sets of colored chips
arranged on a table, they should name which sets have more than five
and which have less than five.
- Students play the card game War with a set of cards
without numerals (i.e., cards which only show sets of hearts, clubs,
diamonds, or spades). Students will easily distinguish between the
7's and the 3's, but will be reluctant to make judgments
about closer numbers like the 4's and 5's without counting.
As they play more often, however, their ability to distinguish will
visibly improve. They may also begin to notice patterns involving
even and odd numbers on their own.
- Students learn to recognize certain arrangements
of dots or stars as representing certain numbers. Using flashcards,
they estimate the number of dots or stars, and then count to check
- After reading Ten Black Dots by Donald
Crews, students make up their own uses for 1-10 black dots. They use
adhesive dots to create their own books that include uses for each of
the numbers from 1 through 10. They then estimate and count the total
number of dots they actually use for their book. (The total might
surprise them: 55.)
- As an assessment of students' ability to
judge without counting, the teacher puts some counters (more than
five) on the overhead projector, turns it on for a few seconds, and
then asks the students to write whether the number of counters shown
is closer to 10 or to 20.
2. Use personal referents, such as the width of a
finger as one centimeter, for estimations with
- Students estimate lengths of pieces of spaghetti, yarn,
paper, pencils, paper clips, etc., using suggested non-standard
personal units such as width of thumb, length of a foot, and so on.
They note that different students get different "right"
- As standard units like foot and centimeter are introduced,
students are challenged to findsome 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 centimeter or the length of one baby step is one
- Students use their self-discovered personal body referents
to estimate the measures of various classroom objects like the length
of the blackboard or the width of a piece of paper. They compare
their answers, noting that when larger units are used, the estimated
answer is a smaller number; and when smaller units are used, the
estimated answer is a larger number.
3. Visually estimate length, area, volume,
or angle measure.
- Students look at a quantity of sand, salt, flour, water,
macaroni, corn, or popcorn and estimate how many times it could fill
up a specified container.
- Students estimate how many pieces of notebook paper it
would take to cover a given area such as the blackboard, or a portion
of the classroom floor.
- Students regularly estimate lengths using a variety of
non-standard units such as my feet, Unifix cubes, paper
clips, and orange Cuisenaire Rods. They then measure to
verify or revise their estimates.
- Students begin to develop an understanding of
angle measure by making right-hand (or left-hand) turns repeatedly to
turn completely around. They also compare angles to right-angle
"corners," and decide whether an angle is more than or less
than a "corner."
- Students note that there are 12 numbers on a
clock face and discuss how far each hand moves in an hour. They note
that each hand moves in a circle, but that the hour hand moves much
- Students work through the Will a Dinosaur
Fit? lesson that is described in the First Four Standards of this
Framework. They determine the size of the room, hear their
classmates' presentations about the dinosaurs, and then as a
whole class activity estimate which dinosaurs, and how many of them,
might fit into the room.
4. Explore, construct, and use a variety of
- Students are asked if a sixty-seat bus will be adequate to
take the two first grade classes on their field trip. After it is
known that there are 23 children in one class and 27 in the other,
individuals volunteer their answers and give a rationale to support
their thinking; front end estimation should lead to the
conclusion that the total number of students is between 40 and 60. A
discussion might be directed to the question of whether an exact
answer to the computation was needed for the problem.
- Students are shown a glass jar filled with about eighty
marbles and asked to estimate the number in the jar. In small groups,
they discuss various approaches to the problem and strategies they can
use. Each group shares one strategy with the class, and the estimate
that resulted. The teacher makes notes about students' work
throughout the activity.
- Second grade students can be challenged to estimate the
total number of students in the school. They will need to talk
informally about the average number of students in each class, the
number of classes in a grade level, and the number of grade levels in
the school. They might then use calculators to get an answer, but the
result, even though the exact answer to a computation, is still an
estimate to the original problem. They discuss why that is so.
- Primary-grade students explore the meanings of
comparison words by listening to How Many is Many? by
Margaret Tuten. They compare big and small, long and
short, a lot and a few. They list how many pieces
of candy would be a few and how many pieces would be many, eventually
reaching general agreement, perhaps on 5 as a few. Then they consider
whether 5 teaspoons of medicine would be a few.
5. Recognize when estimation is appropriate, and
understand the usefulness of an estimate as distinct from an
- Given a pair of real-life situations, students determine
which situation in the pair is the one for which estimation is a good
approach and which is the one that probably requires an exact answer.
One such pair, for example, might be: sharing a bag of peanuts
among 3 friends and paying for 3 tickets at the movie
- Given a set of cartoons with home-made mathematical
captions, first graders decide which of the cartoon characters arrived
at exact answers and which got estimates. Two of the cartoons might
show an adult and a child looking at a jar of jellybeans and the
captions might read: Susie guessed that there were 18 jellybeans
left in the jar and Susie's mom counted the 14
jellybeans left in the jar.
- Students read or listen to newspaper headlines
and discuss which involve exact numbers and which might be
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. For example, first graders
might be asked to decide if their answer to the following problem
makes sense: Mary made 27 cookies and Jose made 15. How
many cookies did they make in all? Some responses might
indicate that the answer should be more than 20 + 10 = 30 and
less than 30 + 20 = 50. Other students might say that they
know that 25 and 15 is 40, so the answer should
be a little more.
- Students estimate reasonable numbers of times that
particular physical feats can be performed in one minute. For
example: How many times can you bounce a basketball in a
minute? How many times can you hop on one foot in a minute? How many
times can you say the alphabet in a minute? and so on.
Other students judge whether the estimates are reasonable or
unreasonable and then the tasks are performed and the actual counts
- Second-grade students are given a set of thirty cards with
two-digit addition problems on them. In one minute, they must sort
the cards into two piles: those problems whose answers are greater
than 100 and those less than 100. The correct answers can be on the
backs of the cards to allow self-checking after the task is
- Second-grade students are given a page of addition or
subtraction problems in a multiple choice format with 4 possible
answers for each problem. Within some time period whichis much too
short for them to do the computations, students are asked to choose
the most reasonable answer from each set of four.
7. Apply estimation in working with quantities,
measurement, time, computation, and problem solving.
- Students have small pieces of yarn of slightly different
lengths ranging from 2 to 6 inches. Each student first estimates the
number of his or her pieces it would take to match a much longer piece
- about 30 inches long - and then actually counts how many.
Then they use their individual pieces to measure other objects in the
room. Each child is responsible for estimating the lengths in terms
of his or her own yarn, but they can use evidence from other
children's measuring to help make their own estimates.
- Students regularly estimate in situations involving
classroom routines. For example, at snack time, they may guess how
many cups can be filled by each can of juice or how many crackers each
student will get if all of the crackers in the box are given out.
- Kindergartners always have fun deciding which color is
best represented in a group of multi-colored objects. Good examples
of such an activity would be choosing the color that shows up most
often (or least often) in bags of M&M's, in handfuls of small
squares of colored paper, or in a jar full of marbles. After everyone
has committed to a guess, the children can sort the objects and count
each color. They can then make bar graphs to show the distribution
of the different colors.
- Students use Tana Hoban's photographs in
Is It Larger? Is It Smaller? as a starting point for
investigating and comparing quantities and measures in their
classroom. For example, on one page, three vases are shown filled
with three different kinds of flowers. The reader must decide which
objects to compare, such as the vases, before ordering them -
from tallest to shortest and/or by volume.
Crews, Donald. Ten Black Dots. New York: Greenwillow,
Hoban, Tana. Is It Larger? Is It Smaller? New York:
Tuten, Margaret. How Many is Many? Chicago:
Children's Press, 1970.
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 grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the