STANDARD 8 - NUMERICAL OPERATIONS
All students will understand, select, and apply various methods of
performing numerical operations.
Standard 8 - Numerical Operations - Grades 9-12
In the ninth through twelfth grades, the themes described in the
K-12 Overview - estimation, mental computation, and
appropriate calculator and computer use-become the focus
of this standard. What is different about this standard at this level
when compared to the traditional curriculum is its mere presence. In
the traditional academic mathematics curriculum, work on numerical
operations was basically finished by eighth grade and focus then
shifted exclusively to the more abstract work in algebra and geometry.
But, in the highly technological and data-driven world in which
today's students will live and work, strong skills in numerical
operations have perhaps even more importance than they once did. By
giving older students a variety of approaches and strategies for the
computation that they encounter in everyday life, approaches with
which they can confidently approach numerical problems, they will be
better prepared for their future.
The major work in this area, then, that will take place in the high
school grades, is continued opportunity for real-world
applications of operations, wise choices of appropriate
computational strategies, and integration of the numerical
operations with other components of the mathematics
The new topics to be introduced in this standard for these grade
levels involve factorials, matrices, operations with polynomials, and
operations with irrational numbers as useful tools in problem solving
Estimation, mental math, and technology use
should fully mature in the high school years 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 paper-and-pencil techniques. Students need to continue
to develop alternatives to paper-and-pencil as they learn about
operations with matrices and other types of number, but the work here
is almost exclusively on the continuing use of all of the strategies
in rich, real-world, problem solving settings.
The topics that should comprise the numerical operations focus of
the ninth through twelfth grade mathematics program are:
- operations on real numbers
- translation of arithmetic skills to algebraic operations
- operations with factorials, exponents, and matrices
Standard 8 - Numerical Operations - Grades 9-12
Indicators and Activities
The cumulative progress indicators for grade 12 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 9, 10,
11 and 12.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 9-12 will be such that all students:
6*. Select and use appropriate
computational methods from mental math, estimation,
paper-and-pencil, and calculator methods, and check the
reasonableness of results.
- Students frequently use all of these computational
strategies in their ongoing mathematics work. Inclinations to
over-use the calculator, in situations where other strategies would be
more appropriate, are overcome with five minute "contests,"
speed drills, and warm-up exercises that keep the other skills sharp
and point out their superiority in given situations.
- Numerical problems in class are almost always worked out
in "rough" form before any precise calculation takes place
so that everyone understands the "ballpark" in which the
computed answer should lie and which answers would be considered
- Students use estimation in their work with
irrational numbers, approximating the results of operations such as
sqrt(15) + sqrt(17) or sqrt(32)
sqrt(8), and developing general rules.
- Students discuss the advantages and disadvantages
of using graphing calculators or computers to perform computations
- Students demonstrate their ability to select and
use appropriate computational methods by generating examples of
situations in which they would choose to use a calculator, to
estimate, or to use mental math.
- Students solve given computational problems using
an assigned strategy and discuss the advantages and disadvantages of
using that particular strategy with that particular
13. Extend their understanding and use of operations
to real numbers and algebraic procedures.
- Students work on the painted cube problem to enhance their
skill in writing algebraic expressions: A 3-inch cube is painted
red. It is then cut into 1-inch cubes. How many of them have
3 red faces? 2 red faces? 1-red face? No red faces? Repeat the
problem using an original 4-inch cube, then a five-inch cube,
then an n-inch cube.
- Students develop a procedure for binomial multiplication
as an extension of their work with 2-digit whole number multiplication
arrays. Using algebra tiles, they uncover the parallels
between 23 x 14 (which can be thought of as
(20+3)(10+4)) and (2x+3)(x+4).
- Students work through the Ice Cones lesson
that is described in the First Four Standards ofthis Framework.
They discover that in order to graph the equation to determine the
maximum volume of the cones, they need to use algebraic procedures to
solve for h in terms of r.
- Students devise their own procedures and
"rules" for operations on variables with exponents by
performing trials of equivalent computations on whole numbers.
- Students use algebra tiles to develop
procedures for adding and subtracting polynomials.
- Students use compasses and straightedges to
construct a Golden Rectangle and find the ratio of the length to the
width (1 + sqrt(5))/2.
- Students consider the ratios of successive terms
of the Fibonacci sequence (1, 1, 2, 3, 5, 8, ...), where
each term after the first two is the sum of the two preceding terms,
finding that the ratios get closer and closer to the Golden Ratio
(1 + û5)/2.
14. Develop, apply, and explain methods for solving
problems involving factorials, exponents, and matrices.
- Students work through the Breaking the Mold lesson
that is described in the Introduction to this Framework. It
uses a science experiment with growing mold to involve students in
discussions and explorations of exponential growth.
- Students use their graphing calculators to find a curve
that best fits the data from the population growth in the state of New
Jersey over the past 200 years.
- Students discover the need for a factorial notation and
later incorporate it into their problem solving strategies when
solving simple combinatorics problems like: How many
different five card poker hands are there? In how many
different orders can four students make their class
presentations? In how many different orders can six packages be
delivered by the letter carrier?
- Students compare
(250) 2, and
3x2100 to decide which is largest.
They explain their reasoning.
- Students read "John Jones's
Dollar" by Harry Keeler and discuss how it demonstrates
exponential growth. They check the computations in the story,
determining their accuracy.
Keeler, Harry Stephen. "John Jones's Dollar," in
Clifton Fadiman, Ed. Fantasia Mathematica. New York:
Simon and Schuster, 1958.
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
* Activities are included here for Indicator 6,
which is also listed for grade 8, since the Standards specify that
students demonstrate continued progress in this indicator.