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 5-6
Fifth and sixth graders should have a good sense of whole numbers
and their orders of magnitude and should be focusing mostly on
developing number sense with decimals, fractions, and rational
numbers, which require substantially different ways of thinking about
numbers. They also should be exploring two relatively new topics:
ratios and integers. The key components of number sense, as explained
in the K-12 Overview, are an awareness of the uses of numbers
in the world around us, a good sense of approximation,
estimation, and magnitude, the concept of
numeration, and an understanding of comparisons and the
equivalence of different representations and forms of
Students at this age are capable of categorizing all of the ways in
which numbers are used in our society. An excellent activity is to
have them collect ways in which they see numbers being used during a
twenty-four hour period. Their uses of numbers would probably
include telephone numbers, addresses, ages, page numbers, clothing
sizes, library book numbers, room numbers, and many others.
Discussions of the similarities and differences among these uses
should resolve themselves into some of the standard categorizations:
counts, measures, labels, and indicators of location. The
students' data can then be graphed according to these
Their numeration work in earlier grades, having focused on
models of whole numbers, has taken these students to the point where
they are able 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. The
focus now shifts to a real sense of the meanings of decimals and
fractions and to providing models which adequately serve that purpose.
Money continues to provide a superb setting for the learning of
decimal concepts (at least up to two decimal places) because of the
students' increasing familiarity with it, because of the vast
array of real-world applications that it makes available, and because
of its inherent motivational quality. Base-ten blocks are also useful
as a slightly more abstract model. They have the added advantage of
being able to represent any number with four digits in
the place value system. A block model of 3274 with which the students
are familiar can become a model of the decimal 3.274 if, instead of
thinking of the smallest block as one unit, they think of the largest
block as one unit.
In addition, models are essential for the continued exploration of
fraction meaning and fraction operations. Fraction Circles or Fraction
Bars help children establish rudimentary meaning for fractions, but
have the drawback of using the same size unit for all the pieces.
This is a fairly serious drawback leading to the misconception, for
instance, that 1/3 is always less than 1/2 without regard to
the units in which those fractions are expressed; students need
to be aware, for example, that 1/3 of a large pizza is frequently
larger than 1/2 of a small one. Cuisenaire Rods or paper folding can
also be used to accomplish many of the same goals without the same
drawback. A sample unit on fractions for the sixth-grade level can be
found in Chapter 17 of this Framework.
Work with ratios, percents, and integers in grades 5 and 6 should
be limited to informal exploration, with no use of more formal,
symbolic procedures. Students should use models as they explore these
topics. Theymight use 2 red tiles and 1 yellow tile to illustrate
mixing paint in the ratio 2:1 and extend this pattern in order to make
larger (and smaller) quantities. By using base ten blocks and 10 x 10
grids, they can visualize percent more easily. Two-color counters or
the number line might be used to model positive and negative numbers
Students in grades 5 and 6 should begin to understand the ways in
which different types of numbers are related. For example, they
should understand that every whole number is a rational number, since
it can be written as a fraction. Similarly, every decimal is a
rational number. By the end of sixth grade, they should have had
sufficient experiences with integers to realize that the integers
consist of the whole numbers and their opposites (additive
Students at these grade levels continue their learning about
equivalence, but there is a significant shift in what that
means. As third and fourth graders, they have explored simple
fractions and decimals, and their work with equivalence has focused
primarily on the multiple ways to represent whole numbers (8 = 2 +
6 = 9 - 1, 23 = 2 tens + 3 ones = 1 ten + 13 ones, and
so on). Now, as fifth- and sixth-graders, they should begin to focus
on the representation of the same quantity with different types
of numbers. Their work with equivalent fractions (1/2 = 2/4) and
equivalent decimals (3 tenths = 30 hundredths), for example, should
lead to exploration of the basic equivalences of fractions and
decimals (1/2 = 0.5). They should be engaged in activities using
concrete models to generate equivalent forms of many different kinds
of numbers. They also begin to explore the role of ratios and
percents in this mix. Ten-by-ten grid paper helps enormously with
these activities, since all forms of a quantity can frequently be
represented on it.
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 what fraction (percentage) of the
kids in the playground do you think are wearing gloves? About
one-third of our students stay for the after-school program in
the afternoon; if there are 500 students in the school about how many
of them stay? After children have had several 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 good number sense.
Technology plays an important role in number sense at these
grade levels. Calculators can be wonderful exploration tools when
examining new relationships. Many insights about the relationships
between fractions and decimals, for instance, can be achieved by
simply dividing the numerators of fractions by their denominators.
Generalizations about what kinds of fractions produce what kinds of
decimals start to flow very freely in such open-ended explorations.
Computer software also creates environments in which students
manipulate decimal models on-screen and explore fraction and decimal
The topics that should comprise the number sense focus of the fifth
and sixth grade mathematics program are:
- ratio and percent
Standard 6 - Number Sense - Grades 5-6
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
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 will be such that all students:
10. Understand money notations, count and compute money,
and recognize the decimal nature of United States currency.
11. Extend their understanding of the number system by
constructing meanings for integers, rational numbers, percents,
exponents, roots, absolute values, and numbers represented in
- Students each develop questions, the answers to all of which
are equivalent to some target number. For example, if the target
number is 24, students may ask the following questions: What
is 8 x 3? What is (-25) - (-49)?
What is 52 -1? What is 3 more
than the sixth triangular number? What is 1 less than
one-fourth of 100? or What is the smallest positive number
with 8 factors?
- Students continue to refine their concepts of fractions
using all available models to answer questions like: Is 1/4 always
larger than 1/8? Is 1/4 of every pizza larger than 1/8 of
every other pizza? Issues that point out the importance of
defining the unit are special topics for discussion.
- Students use two-color counters to construct models of the
set/subset meaning of fraction. You might ask: Given 3 red counters
in a set of 12, what are the equivalent fractions thatrepresent
the reds as a part of the set?
- Students also use two-color counters to model and begin to
make sense of positive and negative integers. In this system, a
positive 1 is represented by one color and a negative 1 by the other.
Students determine the value of a pile of counters by pairing up
counters, one of each color, setting aside all pairs, and counting the
- Students read Shel Silverstein's poem A Giraffe and
a Half and discuss how to describe an amount that is more than one
whole but less than two.
- Students read The Phantom Tollbooth and discuss the
relationships between decimals and fractions in the book. For
example, Milo meets half a child (actually, .58 of a child since the
average family has 2.58 children).
- Students construct a time line to scale to show the history
of the earth. Significant periods and events are shown along the line
with numbers reflecting the number of years since the earth's
12. Develop number sense necessary for
- Students imagine collecting a million of something. They
discuss what objects would be reasonable to collect (such as
toothpicks, punched holes from fan-folded computer paper or pages in
telephone books to be recycled), how much space this collection would
take up, and how much it would weigh.
- Students make estimates to answer the question: How much
drinking water do you think Columbus' ships carried
with them on their trip across the ocean? Then they gather the
data they need to make more informed estimates. (Addenda Series
Grade 4 Book.)
- Students determine the number of decimal places in a simple
decimal multiplication product, not by mechanically adding the number
of places in the factors, but by estimating a reasonable range for the
product and placing the decimal point so that their computed product
falls within that range.
- Students investigate the question: What size room would
be needed to hold one million ping-pong balls?
- Students read Counting on Frank and estimate how many
dogs would fill their classroom.
- Students use estimates to compare fractions. For example,
3/7 < 9/16 since 3/7 is less than half and 9/16 is more than
13. Expand the sense of magnitudes of different number
types to include integers, rational numbers, and roots.
- Students challenge each other to find target numbers on a
number line. First one student asks another to find 3.2. The second
target number then must be between 0 and 3.2, say 1.74. The third
must then be between 0 and 1.74 and so on.
- Students use their calculators to explore the types of
decimal expansions for common fractions. They discover that some such
decimals terminate, some repeat, and some appear to do neither.
(Actually, if the calculators could exhibit more digits, each such
decimal would either terminate or repeat.)
- Students use calculators to come as close as they can to an
answer to: What number multiplied by itself gives an answer
14. Understand and apply ratios, proportions, and percents in
a variety of situations.
- Students begin to see a ratio as both
the comparison of two quantities and as a number in its own right.
They are challenged to find ratios that they frequently use like
$0.65 per pound, 55 miles per hour, and so on.
- In a social studies unit, students use population and area
data for countries in South America to compute population densities,
and then compare their results to those for other areas of the
- Students use two different sizes of grid paper to copy a
simple drawing of a house from the smaller grid to the larger grid,
investigating and discussing the change from one to the other and
exploring ways to represent it numerically. They then copy the same
drawing onto a third grid, smaller than the second but larger than the
- Students are challenged to use any combination of the digits
3, 4, 5, and 9 to make a ratio as close as possible to 90%. As
follow-up, they invent other closeness problems for each
- Students search for as many uses of percent as they can find
over the course of a week. The sources for the uses, however, are to
be exclusively within the school setting. Likely entries in the
resulting list are: grades on tests, foul shot success of the
basketball team, a measure of how close the PTA is to their
fund-raising goal for the new playground equipment, and so
on. For each use found, the students explain what 100% would
represent and whether percentages above 100% would make any sense in
the given context.
15. Develop and use order relations for integers and rational
- Students use a deck of fraction cards for a variety
of tasks. A deck consists of one card containing each of a number of
fractions, for example, 1/4, 9/10, 2/19, 4/7, 7/9, 1/3, 12/15, 2/5,
and 5/8. They are asked to: Find the smallest fraction in the
set. Sort into two groups more than 1/2 and less than 1/2.
Determine which pairs have a value close to 1.
- Students use a number line, including both positive and
negative integers, to graph inequalities stated verbally. For
example: Show all of the numbers larger than
- Students gain understanding of the order relationships among
fractions and integers by comparing them with similar ones for whole
numbers. How is the comparison of 4/7 to 5/7 like the
comparison of 4 to 5? How is comparing 4/7 to 4/8 like comparing 7 to
8? How is comparing -4 to -7 like
comparing 4 to 7? They answer similar questions on their test.
- Students use 4 digits, say 2, 3, 4, and 5 to
construct as many true fraction sentences as they can. For example,
2/3 < 5/4.
16. Recognize and describe patterns in both finite and
infinite number sequences involving wholenumbers, rational
numbers, and integers.
- Given the first four rows, students formulate a rule for
generating succeeding rows of Pascal's triangle. They look for
other patterns in the triangle.
- Students explore the well-known problem of taking a long
walk by first doing half of it, then half of what remains, half again
of what remains, and so on. They write the series as 1/2 + 1/4 + 1/8
+ ... . What happens to the walker?
- Students solve this classic problem: Which would you
choose as the method for getting your allowance next month:
$1.00 every day; or 1 cent the first day, 2 cents the second, 4
cents the third, 8 cents the fourth, and so on?
17. Develop and apply number theory concepts, such as,
primes, factors, and multiples, in real-world and mathematical
- Students build rectangular arrays with square tiles to
determine which of the first fifty counting numbers are rectangular
(composite) and which are non-rectangular (prime).
- Students use the Sieve of Eratosthenes to generate a list of
all the primes in the first 100 counting numbers.
- Students use common multiples to solve problems like this:
Hot dog buns come in packages of 8. Hot dogs come in
packages of 6. What is the smallest number of packages of each
that can be bought so that there are no extra buns or hot
18. Investigate the relationships among fractions,
decimals, and percents, and use all of them
- Students address the questions How are 0.50 and 40/100
alike? and How are they different? Answers can be
written in their math journals.
- Students use shadings on a ten by ten grid to discuss all of
the different equivalences. For example, the same shading can be
named 3/10, 30/100, 0.3, 0.30, .3, .30, and 30%. Thinking of the grid
as $1.00 leads to some interesting insights about two-place
- Students explore the density property of numbers by
addressing problems like: Find 4 decimals between 0.456 and
0.457. Find 3 fractions between 3/5 and 4/5.
19. Identify, derive, and compare properties of
- Students use Venn diagrams to explore the multiple sets to
which particular number belong. For example, a Venn diagram is created
for these three sets of numbers less than 25: multiples of 3, factors
of 24, primes; the Venn diagram is used to answer questions like:
How many numbers are in exactly two of these sets? A similar
question is used on their test.
- Students explore the property of closure for a
variety of sets of numbers under various operations. For example:
Using subtraction, is there always an answer within the set of
positive whole numbers for any member of the set minus any
other? (no); Is there always an answer within the set of
integers? (yes); Is there always an answer within the set of
even integers? (yes); within the set of odd
- Students explore the properties of odd and even numbers
under various operations. For instance: What can always be said
about the sum of two even numbers? of two odd numbers? of an
even and an odd number?
- Students create a book about zero for
- Students explore the concepts of place value and zero by
learning about other number systems. For example, they might use the
computer program Maya Math to learn about the Mayan number
Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth
Stevens Publishing, 1991.
Juster, Norton. The Phantom Tollbooth. New York: Random
National Council of Teachers of Mathematics. Addenda Series
Grade 4 Book. Reston, VA, 1993.
Silverstein, Shel. A Giraffe and a Half. New York: Harper
& Row, 1964.
Maya Math. Sunburst Communications.
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