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 K-2
Students can develop a clear sense of number from consistent
ongoing experiences in classroom activities where a variety of
manipulatives and technology are used. 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 numbers.
Kindergarten, first, and second graders are just beginning to
develop their concepts of number. They have most likely come to
school with some ability to count, but with differing notions of what
that activity means. It is in these grades that they begin to attach
meaning to the numbers that they hear about and see all around them.
One useful activity that can be repeated many times throughout this
age range is the keeping of a scrapbook reflecting all the uses of
numbers that the children can identify. It would probably include
telephone numbers, addresses, ages, page numbers, clothing sizes, room
numbers, and many others. Discussions of the similarities and
differences in all of these uses can provide some interesting
In terms of numeration, students in these grades start by
constructing meaning for one-digit numbers and build up to formal work
with three-digit numbers. The regular and consistent use of concrete
models for that development is essential. Kindergartners need a
variety of things to count, from poker chips to marbles to beans.
Both concrete and rote counting are critically important in developing
a sense of number. Adequate attention to counting activities
throughout these grades will help to assure both a good sense of
magnitude (size) of numbers and a real readiness for all
four basic operations. (See Standard 8.) Counting by ones should be
followed by counting back; skip counting by twos, fives, and tens;
counting from a given starting number to a given target number by ones
and by other numbers; counting on by tens from non-multiples of ten
like 43; and so on.
As students are able to handle larger numbers, place value and
base-ten ideas are introduced through grouping activities. Many of
the models with which they are comfortable for single units can
translate nicely into beginning base-ten models; poker chips can be
put in groups of ten into small paper cups; beans can be pasted in
tens onto tongue depressors, and so on. These newly enhanced models,
along with the single digit units, are then used to represent
two-digit numbers. As the next step, of course, groups of ten tens
can be made to create hundreds. These first models of base-ten number
are the best ones to use with young children who are first
encountering these notions because they can actually build larger
units from smaller units. Such models are called bundle-able.
Another property these have is proportionality, because the
model for a ten is actually ten times as large as the model for a one.
A widely used model which is both bundle-able and proportional
involves popsicle sticks which are wrapped into tens and hundreds with
The next type of model to be used would be one which is still
proportional, but no longer bundle-able. The best examples of this
type are the standard base-ten blocks. They require the child to
trade ten ones for a ten rather than directly constructing a ten from
the ones, and, as a result, are slightly more sophisticated. The last
level of sophistication in this sequence of models includes those that
are neither proportional nor bundle-able. Two models of this type
which are regularly used are chip trading materials and play money.
With chip trading materials, there is no inherent concrete ten-to-one
relationship between the red chips and the green chips; the red chips
are not ten times as large as the green ones. The relationship holds
only because of an external rule that is made up and followed.
Similarly, there is no inherent concrete ten-to-one relationship that
exists between dimes and pennies. The relationship only exists
because of a rule that is external to the coins themselves. As a
result, these most sophisticated models should be used after
the underlying concepts are developed with the earlier models.
Children at these grade levels also begin to learn about
equivalence. When youngsters find as many "names" as
they can for the number 7 (such as 2 + 5, 9 -
2, and one more than 6), they are creating equivalent forms
of the same number. Slightly older students should be using similar
activities to generate equivalent forms of multi-digit numbers, partly
in preparation for operations involving them: 67 = 6 tens and 7
ones = 5 tens and 17 ones = 4 tens and 27 ones.
Estimation should be a routine part, not only of daily
mathematics lessons, but also 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 out here in the playground? About how
many pieces of construction paper will we need for this project
if everyone needs three different colors? and How many of
your great graphs do you think will fit on the bulletin board
without overlapping? 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 to
develop good number sense.
Technology plays an important role in number sense at these
grade levels. Calculators can be wonderful teaching tools when
programmed to count forward and backward by some constant. Children
can do the programming easily themselves and try to anticipate the
calculator display. Appropriate computer software provides
environments in which students can first develop a sense of small
whole numbers and then build an understanding of place-value and
The topics that should comprise the number sense focus of the
kindergarten through second grade mathematics program are:
- whole number meanings through three digits
- place value and number base
- counting and grouping
Standard 6 - Number Sense - 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 grades 1 and 2.
Experiences will be such that all students in grades K-2:
1. Use real-life experiences, physical materials,
and technology to construct meanings for whole numbers,
commonly used fractions, and decimals.
- Young students make and use a variety of models for
"number" ranging from poker chips to dot patterns on a paper
plate, to Cuisenaire Rods, to tally marks, to domino and dice
combinations. A large component of their early work with number
focuses on the various parts into which any given number can be
- Students play the Broken Key game on their
calculators. Kindergartners try to get the calculator display to show
7 while pretending that the 7 key is broken and cannot
be pressed. Second graders might try to get the display to show
45 without pressing the 4 or the 5 key.
- Students use 5-frames and 10-frames to help
develop initial ideas of small numbers. By filling up a 5-cell grid
with counters first and then putting out 2 more while trying to show
"7 in all," the child not only learns about "7"
but also about its relationship to "5."
- Students use numbers throughout the school day as they
discuss the date, attendance, time, snacks, money, etc.
- Students investigate fractions by listening to the story
Gator Pie by Louise Mathews and by discussing how Alvin and
Alice can share their pie with more and more alligators.
- Second-graders record prices as decimals ($0.39) and use
this notation to find totals over $1 on a calculator.
- Students find half of a sheet of paper by folding
horizontally, by folding vertically, and by folding diagonally. They
compare the results and discuss how they are alike and how they are
- Students use Balancing Bear software to find
combinations of numbered weights that will balance a seesaw or that
will be greater or less than a given weight.
2. Develop an understanding of place value
concepts and numeration in relationship to counting and
- Calendar activities at the beginning of the school day
incorporate a Daily Count feature where each day another
popsicle stick is added to a collection representing all of the days
of school to date. Whenever 10 single sticks are available, they are
bundled with a rubber band and are thereafter counted as a ten.
On the hundredth day of school, the ten tens are wrapped
together to make a hundred, and the class celebrates the event
with a party.
- Students progress from a proportional and
bundle-able base ten model like popsicle sticks to a
proportional but not bundle-able model like base-ten
blocks to a model that is neither proportional nor
bundle-able like pennies and dimes. (See K-2 Overview)
- Pairs of students play Race to One Hundred with base
ten blocks. Each, in turn, rolls one or two dice and takes that many
unit cubes. Whenever there are ten unit cubes in a player's
collection, the player must trade for a ten block. The first
player able to trade ten ten blocks for a hundred block is the
- Students have 3 dimes and 4 pennies to spend on a variety of
items that are displayed in a classroom store. The items have tags
ranging from 3 cents to 56 cents and the children are asked: Which
of these items can be bought for exactly the amount of money that you
have (requiring no change)? Which items can you buy and have
some money left over? Which of these items cannot be bought
because you do not have enough money? What items are left?
- Student understanding of place value for two-digit numbers
is assessed by asking each student to represent a different number
using popsicle sticks or base 10 blocks.
3. See patterns in number sequences, and use
pattern-based thinking to understand extensions of the number
- Students find patterns in a hundred number chart. When
asked to describe patterns that they see, some children see a counting
by ones pattern horizontally, others see the tens digit increasing and
the ones digit staying the same as they move down the chart
vertically, and still others see in the last column the numbers that
they use to count by tens.
- Students use the constant function feature of their
calculators to program a skip count. They press + 2 ===
to watch the display count by twos, try to anticipate what number
comes next and make predictions to each other. Any number can replace
the "2" to add difficulty to the activity.
- Students play Find the Number on a hundred number
chart located at the front of the room, with each of the numbers
covered by a Post-it or a small tag. One child calls out a number,
like 45, and a volunteer tries to identify where it is on the
chart. The indicated Post-it is then lifted to check the guess.
4. Develop a sense of the magnitudes of whole
numbers, commonly used fractions, and decimals.
- Children are presented with four jars of jelly beans -
one with 3 beans in it, one with 19 beans in it, one with 52 beans in
it, and one with 156. The teacher then asks Which of these
jars do you think has about 50 beans in it? The students
discuss their reasons for believing as they do.
- Second graders are challenged to guess how many sheets of
paper are in the ream of paper on the front table. After everyone has
made a guess, one student counts out 25 sheets from the top of the
pile and places them next to the rest of the pile. Everyone is
offered a chance to change their estimates and to discuss the reason
for their change. Then students agree on a way to verify their
guesses before trying to guess how many such reams it would take to
reach the ceiling!
- Students work through the Will a Dinosaur
Fit? lesson that is described in the First Four Standards of this
Framework. They discuss how many dinosaurs of different types
might fit into the classroom.
- Students fold paper circles into halves, fourths, and
eighths and are asked questions like: Which would you rather have,
a half of a cherry pie or a fourth of the pie? How about
three-eighths of a pizza or one-fourth?
- Students read or listen to a piece of children's
literature that has fractions as its theme, such as Eating
Fractions by Bruce McMillan.
5. Understand the various uses of numbers
including counting, measuring, labeling, and indicating
- A kindergarten teacher announces to her class: Boys and
Girls! Great News! The principal told me that our class has
just won FIVE! A discussion then ensues regarding the need for
that number to exist in some context, to have some unit or label
before it makes sense.
- Second graders are given a stack of old magazines. They cut
out any information which uses numbers and sort them according to how
they are used: as page numbers, as prices, as dates, as addresses, and
- The class takes a walk around the school or neighborhood
pointing out to each other the numbers they see, and discuss how they
6. Count and perform simple computations with
- Students use play money to show different combinations of
coins that can be used to "buy" an object. For example, an
11 cents pencil can be bought with 11 pennies, a dime and a penny, one
nickel and six pennies, or two nickels and a penny.
- Students earn 2 cents each day for attendance and 1 cent
for good behavior. They keep their play money in a bank, count it
regularly and use it to buy objects from a treasure chest.
- Students play Spend a Dollar. They each start with
$1 (either as a bill or in change) and then roll one or two dice to
find out how much they "spend" on that turn. They trade
coins as needed. The student who spends all of her money first
- Students play a shopping board game. They each begin with a
given amount of money in coins. They roll two dice to determine how
far they move each turn. As they land on a space, they must buy
whatever is shown. Some spaces may provide refunds. The winner is
the first person to go around the board and still have money left.
- Students' abilities to recognize coins and find the
value of a group of coins are assessed by having each student select
three objects to "buy," identify and name the coins needed
to purchase each object, and find the total amount of money required
to purchase all three.
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 2-digit numbers with base-ten models such as
popsicle sticks, base-ten blocks, or pennies and dimes, students are
frequently asked to show all the ways they can make a given number.
Children then begin to see that 3 tens and 7 ones, 2 tens and 17
ones, 1 ten and 27 ones, and 37 ones all represent
the same number 37.
- Students each develop questions whose answers are all
equivalent to some target number. For example, if the target is
8, students may ask the following questions: What is 4+4?
What is 9-1? What is 8+0? How many hands do four children
have? How many days is one more than a week? or How
much is a nickel and three pennies?
- Students use geoboards, pattern blocks, Cuisenaire Rods,
paper folding, and tangrams to explore simple common fractions like
halves, thirds, and fourths. For instance, they may be challenged to
model 1/2 with all of the different models.
8. Compare and order whole numbers, commonly used
fractions, and decimals.
- Young students use dot pattern cards or dominoes to practice
more, less, and same. For example, given a card with 6
dots on it, students use counters to make a set that is more, another
that is less, and one that is the same. They can then label the sets
with cards that show the appropriate words. With dominoes, students
work in pairs to compare the dots on the two halves and state which is
more and by how much.
- Students play the old favorite card game war with
either dot cards or with a deck of regular playing cards minus the
face cards. Every now and then, the rule changes so that the student
with the card that is less wins the play.
- Students play Guess the Point. A long number line
with endpoints of 20 and 75, for example, is drawn on the board where
all of the intermediary points are labeled above the line. The labels
are then covered by a long piece of 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.
9. Explore real-life settings which give rise to
- Primary classrooms are equipped with Celsius thermometers,
in addition to Fahrenheit ones, so that "below zero" outdoor
temperatures can be recorded. Temperature reports, possibly in both
scales, become a part of the everyday calendar routine.
Baratta-Lorton, Mary. Mathematics Their Way. Menlo Park,
CA: Addison-Wesley, 1995.
Mathews, Louise. Gator Pie. New York: Dodd, Mead, and
McMillan, Bruce. Eating Fractions. New York: Scholastic,
National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School
Mathematics. Reston, VA, 1989.
Balancing Bear. Sunburst Communication.
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