STANDARD 8  NUMERICAL OPERATIONS
All students will understand, select, and apply various methods of
performing numerical operations.

Standard 8  Numerical Operations  Grades 56
Overview
As indicated in the K12 Overview, the widespread availability of
computing and calculating technology has given us the opportunity to
significantly reconceive the role of computation and numerical
operations in our fifth and sixth grade mathematics programs. Some
skills are less important while others, such as estimation, mental
computation, and understanding the meanings of the standard
arithmetic operations, all play a more significant role
than ever in the everyday life of a mathematically literate adult.
The major shift in the curriculum that will take place in grades 5
and 6, therefore, is one away from drill and practice of
paperandpencil symbolic procedures and toward realworld
applications of operations, wise choices of appropriate
computational strategies, and integration of the numerical
operations with other components of the mathematics curriculum. At
these grade levels, students are consolidating their understanding of
whole number operations (especially multiplication and division) and
beginning to develop computational skills with fractions and decimals.
A sample unit on fractions for the sixthgrade level can be found in
Chapter 17 of this Framework.
Much research in the past decade has focused on students'
understandings of operations with large whole numbers and work with
fractions and decimals. Each of these areas requires students to
restructure their simple conceptions of number that were adequate for
understanding whole number addition and subtraction.
Multiplication requires students to think about different meanings
for the two factors. The first factor in a multiplication problem is
a "multiplier." It tells how many groups one has of a size
specified by the second factor. Thus, students need different
understandings of the roles of the two numbers in the operation of
multiplication than their earlier understandings of addition, in which
both addends meant the same thing.
A similar restructuring is necessary for dealing appropriately with
operations involving fractions and decimals. This restructuring
revolves around the role of the "unit" in these numbers. In
earlier grades, students thought about 5 or 498 as
numbers that represented that many things. The understood
unit, one, is the number which was used to count a group of objects.
With fractions and decimals, though, the unit, still one and still
understood, is a harder concept to deal with because its essential use
is to help define the fraction or decimal rather than as a counter.
When we speak of 5 poker chips or 35 students, our
message is reasonably clear to elementary students. But when we speak
of 2/3 of the class or 0.45 of the price of the
sweater, the meaning is significantly less clear and we must be
much more explicit about the role being played by the unit.
The topics that should comprise the numerical operations focus of
the fifth and sixth grade mathematics program are:
 multidigit whole number multiplication and division
 decimal multiplication and division
 fraction operations
 integer operations
Standard 8  Numerical Operations  Grades 56
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
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 56 will be such that all students:
6^{*}. Select and use appropriate
computational methods from mental math, estimation,
paperandpencil, and calculator methods, and check the
reasonableness of results.
 Fifth and sixth grade students have calculators available
to them at all times, but frequently engage in competitions to see
whether it is faster to do a given set of computations with the
calculators or with the mental math techniques they've learned.
 Fifth graders make rectangular arrays with baseten blocks
to try to figure out how to predict how many square foot tiles they
will need to tile a 17' by 23' kitchen floor.
 Students are challenged to answer this
question and then discuss the appropriate use of estimation when an
exact answer is almost certain to be wrong: The Florida's Best
Orange Grove has 15 rows of 21 orange trees. Last year's yield
was an average of 208.3 oranges per tree. How many oranges
might they expect to grow this year? What factors might affect
that number?
 Students play multiplication max out. Each student
has a 2 x 2 array of blanks (in the standard form of a
2digit multiplication problem) into each of which a digit will be
written. One student rolls a die and everyone must write the number
showing into one of the blanks. Once a number is written, it cannot
be moved. Another rollanother number written, and so on. The
object is to be the player with the largest product when all four
digits have been written. If a player has the largest possible
product that can be made from the four digits rolled, there is a bonus
for maxing out.
8. Extend their understanding and use of arithmetic
operations to fractions, decimals, integers, and rational
numbers.
9. Extend their understanding of basic arithmetic
operations on whole numbers to include powers and
roots.
 Students explore the exponent key, the x^{2}
key, and the square root key on their calculators. The groups are
challenged to define the function of each key, to tell how each works,
and to create a keypress sequence using these keys, the result of
which they predict before they key it in.
 Students work through The Powers of the Knight
lesson that is described in the Introduction to this Framework.
It introduces a classic problem of geometric growth which engages them
as they encounter notions of exponential notation.
 Students work through the Pizza Possibities
lesson that is described in the First FourStandards of this
Framework. In it, students discover that the number of pizzas
possible doubles every time another choice of topping is added. They
work through the TwoToned Towers lesson that is also
described in the First Four Standards and note the similarities in the
problems and in their solutions.
 Students join the midpoints of the sides of a 2 x
2 square on a geoboard to form a smaller square. They determine the
area of the smaller square and explore the lengths of its four
sides.
10. Develop, apply, and explain procedures for
computation and estimation with whole numbers, fractions,
decimals, integers, and rational numbers.
 Students working in groups develop a method to estimate the
products of twodigit whole numbers and decimals by using the kinds of
baseten block arrays described in Indicator 6 above. Usually just
focusing on the "flats" results in a reasonable
estimate.
 Students follow up a good deal of experience with concrete
models of fraction operations using materials such as fraction
bars or fraction squares by developing and defending their
own paperandpencil procedures for completing those operations.
 Students develop rules for integer operations by using
postman stories, as described in Robert Davis' Discovery in
Mathematics. The teacher plays the role of a postman who delivers
mail to the students. Sometimes the mail delivered contains money
(positive integers) and sometimes bills (negative integers).
Sometimes they are delivered to the students (addition) and sometimes
picked up from them (subtraction).
 Students model subtraction with twocolor chips
by adding pairs of red and yellow chips. (First, they must agree that
an equal number of red and yellow chips has a value of 0.) For
example, to show 4  (2), they lay out
4 red chips, add 2 pairs of red and yellow chips (whose value is
0), and then take away 2 yellow chips. They note that 4
 (2) and 4 + 2 give the
same answer and try to explain why this is so.
11. Develop, apply, and explain methods for solving
problems involving proportions and percents.
 Students develop an estimate of pi by carefully
measuring the diameter and circumference of a variety of circular
objects (cans, bicycle tires, clocks, wooden blocks). They list the
measures in a table and discuss observations and possible
relationships. After the estimate is made, pi is used to solve a
variety of realworld circle problems.
 Students use holiday circulars advertising big sales on
games and toys to comparison shop for specific items between different
stores. Is the new Nintendo game, Action Galore, cheaper at
Sears where it is 20% off their regular price of $49.95 or at
Macy's where it's specially priced at
$41.97?
 One morning, as the students arrive at school, they see a
giant handprint left on the blackboard overnight. They measure it and
find it to be almost exactly one meter long. How big was the person
who left the print? Could she have fit in the room to make the
print, or did she have to reach in through the window? How
could you decide how much she weighs?
 Students read Jim and the Giant Beanstalk
by Raymond Briggs. Jim helps the aging giantby measuring his head and
getting giant eyeglasses, false teeth and a wig. The students use the
measures given in the book to find the size of the giant's hand
and then his height.
 Students develop a sampling strategy and use
proportions to determine the population of Bean City (NCTM
Addenda Booklet, Grade 6), whose inhabitants consist of three
types of beans.
 Students discuss different ways of finding
"easy" percents, such as 50% of
30 or 15% of 25. They then generate percent
exercises that can be solved mentally and share them with their
classmates.
12. Understand and apply the standard algebraic order
of operations.
 Students bring in calculators from home to examine their
differences. Among other activities, they each key in " 6 + 2
x 4 = " and then compare their calculator
displays. Some of the displays show 32 and others show
14. Why? Which is right? Are the other calculators
broken?
 Students play rolling numbers. They use four white
dice and one red one to generate four working numbers and one target
number. They must combine all of the working numbers using any
operations they know to formulate an expression that equals the target
number. For example, for 2, 3, 4, 5 with target number
1, the following expression works: (2+5)/(3+4)=1.
Questions about order of operations and about appropriate use of
parentheses frequently arise.
References

Briggs, Raymond. Jim and the Giant Beanstalk.
CowardMcCann, Inc., 1970.
Ciardi, John. "Little Bits," in You Read to Me,
I'll Read to You. New York: Lippincott, 1962.
Davis, Robert. Discovery in Mathematics. New Rochelle,
NY: Cuisenaire, 1980.
Hiebert, J., and Behr, M. (Eds.) Number Concepts and
Operations in the Middle Grades. Reston, VA: National Council
of Teachers of Mathematics, 1988.
National Council of Teachers of Mathematics.
Addenda Booklet, Grade 6. Reston, VA, 1992.
Schwartz, David. If You Made a Million. New York:
Lothrop, Lee, & Shepard Books, 1989.
Silverstein, Shel. A Giraffe and a Half. New York:
Harper and Row, 1964.
Tahan, Malba. "Beasts of Burden," in The Man Who
Counted: A Collection of Mathematical
Adventures. W. W. Norton, 1993.
OnLine Resources

http://dimacs.rutgers.edu/archive/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.
^{*} Activities are included here for Indicator 6,
which is also listed for grade 4, since the Standards specify that
students demonstrate continued progress in this indicator.
