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

Standard 8  Numerical Operations  Grades 78
Overview
Traditionally, tremendous amounts of time were spent at these grade
levels helping students to finish their development of complex
paperandpencil procedures for the four basic operations with whole
numbers, fractions, and decimals. While some competency with
paperandpencil computation is necessary, 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.
As indicated in the K12 Overview, then, the major shift in the
curriculum that will take place at these grade levels 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.
Seventh and eighthgraders are relatively comfortable with the
unit shift discussed in this standard's Grades 56 Overview.
Operations on fractions and decimals, as well as whole numbers, should
be relatively well developed by this point, allowing the focus to
shift to a more holistic look at operations. "Numerical
operations" becomes less a specific object of study and more a
process, a set of tools for problem setting. It is critical that
teachers spend less time focused on numerical operations, per se, so
that the other areas of the Standardsbased curriculum receive
adequate attention.
One important set of related topics that needs to receive some
significant attention here, however, is ratio, proportion,
and percent. Seventh and eighth graders are cognitively
ready for a serious study of these topics and to begin to incorporate
proportional reasoning into their set of problem solving tools. Work
in this area should start out informally, progressing to the student
formulation of procedures that make proportions and percents the
powerful tools they are.
Two other topics that receive greater attention here, even though
they have been informally introduced earlier, are integer
operations and powers and roots. Both of these types of
operations further expand the students' knowledge of the types of
numbers that are used and the ways in which they are used.
Estimation, mental math, and technology use
begin to mature in seventh and eighth grades as students use these
strategies in much the same way that they will as adults. If earlier
instruction in these skills has been successful, students will be able
to make appropriate choices about which computational strategies to
use in given situations and will feel confident in using any of these
in addition to paperandpencil procedures. For example, students
should evaluate simple problems involving fractions, such as
what's twothirds of 5 tablespoons? using mental
math. Students need to continue to develop alternatives to
paperandpencil as they learn more about operations on other types of
numbers, but the work here is primarily on the continuing use of all
of the strategies in rich realworld problem solving settings.
The topics that should comprise the numerical operations focus of
the seventh and eighth grade mathematics program are:

rational number operations 

powers and roots 
integer operations 

proportion and percent 
Standard 8  Numerical Operations  Grades 78
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 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 78 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.
 Students choose a stock from the New York Stock Exchange
and estimate and then compute the net gain or loss each week for a
$1,000 investment in the company.
 Students use spreadsheets to "program" a set of
regular, repeated, calculations. They might, for example, create a
prototype online order blank for a school supply company that lists
each of the ten items available, the individual price, a cell for each
item in which to place the quantity ordered, the total computed price
for each item, and the total price for the order.
 Students regularly have human vs. calculator races.
Given a list of specially selected computation exercises (e.g., 53
x 20, 40 x 10, 95 + 17 +5 ), one student uses mental
math strategies and another uses a calculator. They quickly come to
realize that the human has the advantage in many situations.
8. Extend their understanding and use of arithmetic
operations to fractions, decimals, integers, and rational
numbers.
 Given a decimal or a fractional value for a piece of a
tangram puzzle, the students determine a value for each of the other
pieces and a value for the whole puzzle.
 Students use fraction squares to show why the
multiplication of two fractions less than one results in a product
that is less than either.
 Students demonstrate their understanding of operations on
rational numbers by formulating their own reasonable word problems to
accompany given number sentences such as
3/4 divided by 1/2 = 1 1/2.
9. Extend their understanding of basic arithmetic
operations on whole numbers to include powers and
roots.
 Students play powers max out. Each student
has a set of 5 blanks, into each of which will be written a digit.
They are in the form VW^{X} + YZ. One student rolls a die and
everyone must write the number showing into one of their
blanks. Once written, anumber can not be moved. Another roll 
another number written, and so on. The object is to be the player
with the largestvalued expression when all five digits have been
written. If a player has the largest possible value that can be made
from the five digits rolled, there is a bonus for maxing
out.
 Students develop their own "rules" for
operations on numbers raised to powers by rewriting the expressions
without exponents. For example, 7^{2} x
7^{4} = (7 x 7) x
(7 x 7 x 7 x 7) = 7 x 7 x 7
x 7 x 7 x 7 = 7^{6}. You
just add the exponents!
 Students read The
King's Chessboard, The Rajah's Rice: A
Mathematical Folktale from India, or A Grain of
Rice. All of these stories involve a situation in which a
quantity is doubled each day. Students use the story to discuss
powers of 2 and to look for patterns in the sums of the powers
of 2.
 Students use the relationship between the area
of a square and the length of one of its sides to begin their study of
roots. Starting with squares on a geoboard with areas of 1,
4, 9, and 16, they then are asked to find squares
whose areas are 2, 5, and 13.
 Students work through the Rod Dogs lesson
that is described in the First Four Standards of this
Framework. They investigate how the surface area and volume of
an object changes as it is enlarged by various scale
factors.
10. Develop, apply, and explain procedures for
computation and estimation with whole numbers, fractions,
decimals, integers, and rational numbers.
 Students use a videotape of a youngster walking forward
and backward as a model for multiplication of integers. The
"product" of running the tape forward (+) with the student
walking forward (+) is walking forward (+). The "product"
of running it backward () with the student walking forward (+)
is walking backwards (). The other two combinations also work
out correctly.
 Students use base ten blocks laid out in an array to show
decimal multiplication. How could the values of the blocks
be changed to allow it to work? What new insights do we gain
from the use of this model?
 Students judge the reasonableness of the results of
fraction addition and subtraction by "rounding off" the
fractions involved to 0, 1/2, or 1.
 Students explore the equivalence between
fractions and repeated decimals by finding the decimal representations
of various fractions and using the resulting patterns to find the
fractional equivalents of some repeated decimals.
11. Develop, apply, and explain methods for solving
problems involving proportions and percents.
 Students use The Geometer's Sketchpad
software to draw a geometric figure on a computer screen, scale it
larger or smaller, and then compare the lengths of the sides of the
original with those of the scaled image. They also compare the areas
of the two.
 Students are comfortable using a variety of approaches to
the solution of proportion problems. Example: If 8 pencils
cost 40 cents, how much do 10 pencils cost? This problem
can be solved by:

unitrate method 
8 pencils for 40 cents means 5 cents per pencil or 10 x 5=50 cents for 10 
factorofchange method 
10 pencils is 10/8 of 8 pencils, so cost is (10/8)x40=50 cents 
cross multiplication method 
8/40 = 10/x, 8x = 400, so x=50 cents. 
 Students set up a part/whole proportion as one method of
solving percent problems.
 Students spend $100 by selecting items from a
catalog. They must compute sales tax and consider it in deciding what
they will buy.
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 "3 + 15
÷ 3" and then compare their calculator
displays. Some of the displays show 6 and others show 8.
Why? Which is right? Are the other calculators broken?
Students decide what key sequence would work for the calculators
that do not use order of operations.
 Students play with the software How the West was One +
Three x Four, which requires them to construct numerical
expressions that use the standard order of operations.
 Students use the digits 1, 2, 3, and 4 to
find expressions for each of the numbers between 0 and
50. For example, 7 = (3x4)/2 + 1.
References

Barry, David. The Rajah's Rice: A
Mathematical Folktale from India. San Francisco, CA:
W. J. Freeman, 1995.
Birch, David. The King's Chessboard. Puffin
Pied Piper Books, 1988.
Pittman, Helena Clare. A Grain of Rice. Bantam
Skylark, 1986.
Software

Geometer's Sketchpad. Key
Curriculum Press.
How the West Was One + Three x Four.
Sunburst Communications.
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.
