STANDARD 6 - NUMBER SENSE
All students will develop number sense and an ability to represent
numbers in a variety of forms and use numbers in diverse
Standard 6 - Number Sense - Grades 3-4
In third and fourth grade, students continue to develop their
number sense by using manipulatives and technology. The key
components of number sense, as identified in the K-12 Overview,
include an awareness of numbers and their uses in the world
around us, a good sense of place value concepts,
approximation, estimation, and magnitude, the concept of
numeration, and an understanding of comparisons and the
equivalence of different representations and forms of
Third and fourth graders are refining their understanding of whole
numbers and are just beginning to develop understanding of numbers
like decimals and fractions that require substantially different ways
of thinking about numbers. An excellent activity that can be used to
impress upon these students the omnipresence of numbers around them is
the keeping of a journal reflecting all of the uses of numbers
that they can find in magazines and books. In fourth grade, they
focus particularly on uses of fractions and decimals. They may
include decimal prices in advertisements, fraction-off sales, and
decimal or fraction measurements. Discussions of the uses found and
the meanings of the numbers involved can provide interesting
Their numeration work in earlier grades, having focused on
models of number, has enabled students to use relatively sophisticated
models like play money or chip trading to represent whole numbers up
to three digits. The regular and consistent use of concrete models is
essential for the continuing development of their understanding of
numeration. They should use base-ten models not only to extend their
experiences with whole numbers to four places, and then symbolically
beyond that, but also to create meaning for decimals.
In addition, models are essential for the initial explorations of
the meaning of fractions. Fraction Circles or Fraction Bars help
children establish rudimentary meaning for fractions, but have the
drawback of using the same size unit for all of the pieces.
Cuisenaire Rods or paper folding can be used to accomplish the same
goals without this drawback.
Children at these grade levels also continue their learning about
equivalence. They should be engaged in activities using
concrete models to generate equivalent forms of many different kinds
of numbers. For multi-digit numbers, equivalences such as: 367 = 3
hundreds, 6 tens and 7 ones = 3 hundreds, 5 tens and 17 ones =
2 hundreds, 14 tens and 27 ones are useful in promoting a
confident feeling about place value and will help in understanding
multi-digit computation. Early explorations of equivalent fractions
(1/2 = 2/4) and equivalent decimals (3 tenths = 30 hundredths) can
accompany the exploration of the basic equivalences between fractions
and decimals (1/2 = 0.5).
Estimation should be a routine part not only of mathematics
lessons, but of the entire school day. Children should be regularly
engaged in estimating both quantities and the results of operations.
They should respond to questions that arise naturally during the
course of the day like: About how many kids do you think there
are in the auditorium? About how many paper cranes will each
student have to fold if the class needs to make 200 altogether?
and How many floor tiles do you think are on the floor?
After several children have had chances to make estimates about
numbers like these, they should defend their estimates by giving some
rationale for thinking they are close to the actual number. These
discussions can be invaluable in helping them develop number
Technology plays an important role in number sense at these
grade levels. Calculators can be wonderful exploration tools when
examining new numbers. Students will themselves raise questions about
decimals when someone divides 30 by 60 inadvertently instead of 60 by
30 and wonder what the 0.5 in the display means. Computers provide
software that creates environments in which students manipulate
base-ten models on-screen and explore initial fraction and decimal
The topics that should comprise the number sense focus of the third
and fourth grade mathematics program are:
- whole number meanings through many digits
- place value and number base
- initial meanings for fractions and decimals
Standard 6 - Number Sense - Grades 3-4
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
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 3-4 will be such that all students:
1. Use real-life experiences, physical materials,
and technology to construct meanings for whole numbers,
commonly used fractions, and decimals.
- Students are comfortable using a full array of base-ten
models including money, base-ten blocks, and chip trading materials to
represent both whole numbers and decimals.
- Students use computer software that provides easy pictorial
representation of large whole numbers, decimals, and fractions like
the MECC Math Tools, the Silver Burdett Math
Workshop, and the Wasatch Math Construction
- Students use geoboards to model common fractions. For
example, they search for multiple ways to show 1/4 on the
- Students use Cuisenaire Rods to model fractions, frequently
switching the rod or length used as the whole to avoid the
misconception that, for instance, the yellow rod is always
2. Develop an understanding of place value
concepts and numeration in relationship to counting and
- Pairs of students play Race to Five Hundred and its
opposite, Race to Zero, with base ten blocks. In the first
game, each student, in turn, rolls a red die and a green die and makes
a two digit number from the faces showing (using the red die as the
tens digit). He or she then takes that many tens and ones from the
bank. Whenever there are ten tens or ten ones in a player's
collection, the player must trade for a larger block. The first
player to collect 5 hundreds is the winner. In Race to Zero,
the players start with 5 hundreds and give back blocks
according to their dice rolls.
- Students use a die to generate random digits from 1 to 6.
After each roll, they decide where to place the digit in a 4-digit
whole number. The goal is to produce as large a number as possible.
If a ten-sided die or spinner with ten equal sectors is available,
students should use it to generate random digits from 0 to 9 and
repeat the activity.
- Students work in groups to decide what the next base-ten
block after the thousands block would look like.
- Students read and listen to children's literature that
is related to a numeration theme like Millions of Cats
and The 500 Hats of Bartholomew Cubbins.
3. See patterns in number sequences, and use
pattern-based thinking to understand extensions of the number
- Students use the constant function feature of their
calculators to program a skip count. They press + 12 ===
to watch the calculator display count by twelves, trying to
anticipate what number will come next and making predictions to each
other. Any number can replace the 12 to change the difficulty
level of the activity.
- Students also use their calculators to play Guess My Rule
games. One student secretly programs the calculator by typing
something like x 2 =. Thereafter, every time a number
is pressed followed by an equals sign, the original number will be
multiplied by two. A second student must guess the rule that was
programmed. Rules like + 3 =, ÷ 4 =, and - 2
= also work.
- Students create and solve arrow puzzles on a hundred
number chart. By naming a number and then giving directions for
movement on the chart, instructions are given to arrive at some other
number. For example: 72, down, down, right, right, up leaves
the student at the number 84. After examples of different
patterns are demonstrated on the chart, students point out patterns
and try to solve puzzles mentally.
- Students make a table to reflect how many handshakes there
would be if everyone shook hands in groups of different sizes. For
example, for 2 people, 1 handshake; for 3 people, 3 handshakes; for 4
people, 6 handshakes. As they extend the table for larger groups, the
students look for a pattern in the emerging numbers.
- Students search for patterns in the addition of even and odd
numbers by using unifix cubes to represent the numbers and trying to
arrange the sum into two stacks of equal height. (This will work for
even numbers, but not for odd ones.)
4. Develop a sense of the magnitudes of whole
numbers, commonly used fractions, and decimals.
- Students imagine collecting 10,000 of something. They
discuss what objects would be reasonable to collect (such as bottle
caps, pennies for charity, or pebbles from the beach), how much space
this collection might take up, and how much it would weigh?
- Students cut and paste sheets of base-ten graph paper to
make models of the different powers of ten: 1, 10, 100, 1000,
- Students locate numbers as points on a number line strung
across the room, continuing to attach labels as they learn more about
numbers. Paper clips or tape are used to fasten equivalent forms of a
number to the same point.
- Students estimate and investigate how long a million seconds
is using calculators.
- Students write a Logo or BASIC computer program which will
count to 100, printing the numbers to the screen as it runs, and
timing how long it takes. Then they predict how long it would take
the program to count to one thousand, to one hundred thousand, and to
one million. They make the required changes to the program and check
5. Understand the various uses of numbers
including counting, measuring, labeling, andindicating
- Students keep a 24-hour diary recording all of the ways they
use or see others use numbers. They pool all of these uses of numbers
and classify them into categories that they design.
- As part of a geography unit, students make a map of a
fantasy island, using a Cartesian coordinate system to help describe
the location of various places on the island. They use other numbers
to describe geographical properties of the sites: elevation, longitude
and latitude, population, and the like.
- The students brainstorm ways to describe their math book in
terms of numbers: its width, the number of pages, the publication
date, a student-generated "quality rating" of the book, the
area of the cover, and so on.
6. Count and perform simple computations with
- Students establish a school store and make transactions on a
regular basis, with different students assigned as clerk each day.
- Students read Dollars and Cents for Harriet and then
decide how they would spend five dollars.
- Students practice making change with coins by counting
up to the amount given. For example, if the bill is $1.73, and
$2.00 is the amount given, the students would count up to $2.00 by
starting with two pennies and saying, "$1.74" and
"$1.75"; then they add one quarter to bring the total to
$2.00. They would then count this change to find its value of
- Students play Treasure Math Storm on the computer or
use IBM's Exploring Measurement, Time and
7. Use models to relate whole numbers, commonly
used fractions, and decimals to each other, and to represent
equivalent forms of the same number.
- When modeling 3- and 4-digit numbers with a base-ten model
like base-ten blocks or place value chips, the students are frequently
asked questions like: Show all the ways you can make
327. Children thus begin to see that 3 hundreds, 2 tens, and 7
ones; 2 hundreds, 12 tens, and 7 ones; 2 hundreds, 11 tens, and
17 ones; and 32 tens and 7 ones all represent the same
number. Students are assessed by asking them to show 327 in two
- Students use shadings on ten-by-ten grids to represent
fractions and decimals that are equivalent. For example, the
representation for 0.4 is the same as that for 4/10.
- Students develop their own questions, the answers to which
are equivalent to some target number. For example, if the target
number is 24, students may ask the following questions: What
is 20 + 4? What is 2 x 12? What is 2 x 2 x 2
x 3? How much is 2 dozen? How many is 3 less than the
number of children in our class? or How much would something
cost if you paid a quarter and got back a penny in change?
- Students use geoboards, pattern blocks, Cuisenaire Rods,
paper folding, and tangrams to explore common fractions. They may be
challenged to model 3/4, for instance, with all of the different
- Students use money to represent decimals. For example, 8
dimes = $0.80 = .8. They also represent fractional parts of a dollar
as a decimal (a quarter = 1/4 = 25 cents = .25).
- Students use graham crackers, candy bars, pizzas, and other
food to illustrate fractions.
- Students work through the Sharing Cookies
lesson that is described in the First Four Standards of this
Framework. They realize that 8 is not readily divisible by 5
and try to find ways to solve that sharing problem using real
- Students play Bowl a Fact by rolling three dice and
using the numbers shown to make number sentences whose answers equal
numbers from 1 to 10. For each different answer, they
knock down the bowling pin labeled with that number. For example, if
they roll 2, 5, and 3, they can make these number sentences: 2 + 5
+ 3 = 10, 5 + 3 - 2 = 6, 5 x 2 - 3
= 7, 5 - 3 + 2 = 4, and 3 x 2
- 5 = 1, and therefore knock down the 10, 6, 7, 4
and 1 pins. If they cannot knock down all ten pins on the
first roll, they roll the dice again and try to get the remaining
pins. The students are assessed by giving all of them the same
outcomes of two rolls of the three dice to play the game.
8. Compare and order whole numbers, commonly used
fractions, and decimals.
- Students use base-ten materials such as, blocks, sticks or
money to make models of pairs of 3- or 4-digit numbers like 405
and 450 and compare them to see which is larger. Responses
and reasons can be written in a journal.
- Students play Guess the Point. A long number line
with endpoints of 130 and 470, for example, is drawn on the board with
the intermediary points labeled as multiples of ten above the line.
The labels are then covered by a long piece of butcher paper that can
be lifted to reveal them. A student places a finger somewhere on the
line and others must estimate the numerical label of the point chosen.
The paper is then lifted to check the accuracy of their responses.
- Pairs of students play Hi-Lo with whole numbers and
decimals. One student thinks of a number within a given range such as
1 to 1000. The other student tries to guess the number, receiving
feedback after each guess as to whether the guess was too high or too
low, and keeping a written record of the guesses and the feedback.
The goal is to find the number using as few guesses as possible.
- When using Cuisenaire Rods, students choose a base
rod to represent one whole, and then determine the values of all
of the rest of the rods. They then use the rods to model the
comparison of the relative sizes of two fractions with different
9. Explore real-life settings which give rise to
- Students record daily low Celsius temperatures throughout
the winter and draw a line graph of those temperatures. Students
discuss changes in the graph and the meaning of the line dipping below
the zero degree line.
- Students examine a videotape of a section of a football game
and record the results of a series of plays as a series of integers
- gains as positive integers and losses as negative integers (for
example: -3, +5, +9, first down; -5, -4, +6, punt).
They use their record to determine the total yardage gained during the
- Students use an almanac to find the altitudes of selected
cities around the country, and discuss what it means for a city to be
below sea level.
Gàg, Wanda. Millions of Cats. New York: Coward,
McCann, & Geoghegan, 1928.
Maestro, Betsy and Maestro, Giulio. Dollars and Cents for
Harriet. New York: Crown Publishers, Inc., 1988.
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School
Mathematics. Reston, VA, 1989.
Seuss, Dr. The 500 Hats of Bartholomew Cubbins. New York:
Exploring Measurement, Time, and Money. IBM.
Math Construction Tools. Wasatch.
Math Tools. Minnesota Educational Computing Consortium
Math Workshop. Silver Burdett.
Treasure Math Storm. The Learning Company.
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 Mathematics