STANDARD 5  TOOLS AND TECHNOLOGY
All students will regularly and routinely use calculators,
computers, manipulatives, and other mathematical tools to enhance
mathematical thinking, understanding, and power.

Standard 5  Tools and Technology  Grades 912
Overview
This standard addresses the use of calculators, computers and
manipulatives in the teaching and learning of mathematics. These
tools of mathematics can and should play a vital role in the
development of mathematical thought in students of all ages.
High school students can use a variety of manipulatives to
enhance their mathematical understanding and problem solving ability.
For example, new approaches to the teaching of concepts of algebra
incorporate concrete materials at many levels. Twocolored counters
are used to represent positive and negative integers as students build
a sense of operations with integers. Algebra tiles are used to
represent variables and polynomials in operations involving literal
expressions. Concrete approaches to equation solving are becoming
more and more popular as students deal meaningfully with such
mathematical constructs as equivalence, inequality, and balance.
In geometry, students can create solids of revolution by cutting
plane figures out of cardboard, attaching a rubber band along an axis
of rotation, winding it up, and then letting it unwind by itself,
creating a vision of the solid as it does so. Students can use Miras
to do reflective geometry  to find the center of a circle or a
perpendicular bisector of a line segment. They might build models of
a pyramid with a square base and a cube with the same size base to use
in an investigation of the relationship of their volumes. Using pipe
cleaners and straws, students can build their own version of a
Sierpinski tetrahedron.
High school students should also be in the habit of using a variety
of materials to help them model problem situations in other areas of
the mathematics curriculum. They might use spinners or dice to
simulate a variety of reallife events in a probability experiment.
Suppose, for instance, they had statistics about the frequency of
occurrence of a particular genetic trait in fruit flies and were
interested in the probability that they would see it in a given
population. A simulation using dice or a spinner might be a useful
approach to the problem. They should be able to use a variety of
measurement tools to measure and record the data in a science
experiment. They might use counters to represent rabbits as they
simulate Fibonacci's famous question about rabbit
populations.
This list is, of course, not intended to be exhaustive. Many more
suggestions for materials to use and ways to use them are given in the
other sections of this Framework. The message in this section
is a very simple one  concrete materials help students to
construct mathematics that is meaningful to them.
There are many appropriate uses for calculators in these
grade levels as well. In his article, "Technology and
Mathematics Education: Trojan Horse or White Knight?" in The
New Jersey Calculator Handbook, Ken Wolff suggests that the
availability of calculators, especially graphing calculators, has
presented a unique opportunity for secondary mathematics educators.
He asserts: "Tedium can be replaced with excitement and wonder.
Memorization and mimicry can be replaced with opportunities to explore
and discover." Wolff offers some challenging problems for
students to try with their calculators to illustrate the scope of what
is now possible in secondary classrooms:
What happens when we continually square a number close to the
value 1? Try continually taking the square root of a number.
Does it matter what number you start with?
Enter the radian measure of an angle and continually take the
sine of the resulting values. What happens? Can you explain
why it happens? Replace the sine operator with the tangent and repeat
the experiment.
Where does the graph of y = 2 sin (3x) intersect the graph of
y =  4x + 3?
He suggests that these are but a few of the problems that students
will gladly try if they have a calculator, but would be very reluctant
to do without one. Many secondary teachers have had similar
experiences with graphing calculators. As graphing calculators
pervade the world of secondary mathematics, what we teach and how we
teach it will dramatically change. Nowhere more than in these
classrooms will the educational impact of this technology be felt. A
sample unit on finding regression lines using graphing calculators can
be found in Chapter 17 of this Framework.
High school students should be using Calculator Based Laboratories
(CBL) in conjunction with their graphing calculators, to generate,
analyze, and display data obtained using a variety of probes;
discussions of these activities should be coordinated with activities
in their science classrooms.
Computers are also an essential resource for students in
high school, and the software tools available for them are very much
like adult tools. The standard computer productivity tools 
word processors, spreadsheets, graphing utilities, and databases
 can all be used as powerful tools in problem solving
situations, and students should begin to rely on them to help in
finding and conveying problem solutions.
In terms of specific mathematics education software, there are also
many good choices. The Geometric Supposer and PreSupposer
series, Geometer's Sketchpad, and Cabri
Geometry are all popular geometry construction tools for students
of this age. With them, students construct geometric figures on the
screen, measure them, transform them, and identify a variety of
geometric properties of their creations. Discoveryoriented lessons
using these types of software are easy to create and very engaging and
useful for students.
Algebra tools include Derive, Maple, and
Mathematica. These tools all can manipulate algebraic symbols
and equations, solve a variety of equations, do two and
threedimensional plotting, and much more. The programs offer
significantly more power than the graphing calculators, but are also
more expensive. They can be used very effectively for classroom
presentations with a projection viewing device.
There is also a good variety of algebra learning programs. The
Function Supposer, Green Globs and Graphing
Equations, and The Algebra Sketchbook are all popular
pieces of software that deal with functions and their graphs. Many
other valuable pieces of software are available.
The World Wide Web can be an exciting and eyeopening tool for
ninth through twelfthgraders as they retrieve and share information.
Specifically, in these grades, they might look for information about
colleges in which they might be interested, the history of
mathematics, or ecology experiments in which students are gathering
and contributing local data.
Standard 5  Tools and Technology  Grades 912
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 912 will be such that all students:
1^{*}. Select and use calculators,
software, manipulatives, and other tools based on their utility and
limitations and on the problem situation.
 Students have a variety of tools available to
them in the wellequipped mathematics classroom: a bank of computers
loaded with algebraic symbol manipulation and functionplotting
programs, spreadsheet and graphing programs, and geometry construction
programs; a set of graphing calculators for relevant explorations and
computations; and manipulative materials related to the content
studied. The students easily move from one type of tool to another,
understanding both their strengths and limitations.
 Students work through the Making Rectangles
lesson that is described in The First Four Standards of this
Framework. They use algebra tiles organized in rectangular
form to help them develop procedures for rewriting binomial
expressions as multiplication problems (factoring).
 Students can use algebra tiles, HandsOn
Equations materials, and a variety of equationmanipulating
software to simplify and solve equations. They understand and can
demonstrate the relationship between various manipulations of tiles or
pawns and the corresponding symbolic actions in the software solution
procedure.
 Students use both graphing calculator techniques and
paperandpencil techniques for solving systems of equations.
Depending on the complexity of the system and on the degree of
accuracy needed in the answer, they may try to locate the intersection
of two graphs by tracing and zooming on the calculator screen, by
calculating the solution with matrices, or by using a simple addition
or substitution paperandpencil method.
2^{*}. Use physical objects and
manipulatives to model problem situations, and to develop and
explain mathematical concepts involving number, space, and
data.
 Students use a process described in Algebra in a
Technological World to construct cones from a circular piece of
paper by cutting a wedgeshaped sector from it and then taping
together the edges. They then try to find the cone constructed in this
manner that has the largest volume. A similar activity is described
in The Ice Cones lesson in The First Four Standards of this
Framework.
 Students use molds to make cones of clay and then
experiment to see in how many different ways they can slice the cone
with a plane to produce different crosssections. Drawings of the
cross sections and a description of the cuts that created them are
displayed on a poster in the classroom.
 In one of the units of the Interactive Mathematics
Program, students read The Pit and the Pendulum by
Edgar Allan Poe and then work in groups to investigate the properties
and behavior of pendulums. The ultimate goal, after a good deal of
measurement and statistical manipulations of their data, is to
determine how much time the prisoner in the story has to escape from
the 30foot, razorsharp, descending pendulum.
3. Use a variety of technologies to
discover number patterns, demonstrate number sense, and
visualize geometric objects and concepts.
 Students play Green Globs and Graphing Equations, a
computer game in which they score points for writing the equations of
functions that will pass through several green globs splattered on the
xy plane. As they gain experience with the game, their ability to
hit the targets with more and more creative functions improves.
 Students use a set of spherical materials like the
Lenart Sphere to study a nonEuclidean geometry. With these
materials, students make geometric constructions on the surface of a
sphere to realize that, in some geometries, a triangle can have
three right angles, and to find the spherical equivalent of the line
that is the shortest distance between two points.
 Students use calculators to investigate interesting number
patterns. For example, they try to determine why this old trick
always works: Enter any threedigit number into the
calculator. Without clearing the display, enter the same three
digit number again so that you have a sixdigit number. Divide
the number by 7. Then divide the result by 11. Then divide
that result by 13. What is in the display?
5^{*}. Use technology to gather,
analyze, and display mathematical data and information.
 Students use a simulation program to check their
predictions regarding the answer to this problem from the New Jersey
Department of Education's Mathematics Instructional
Guide: Two standard dice are rolled. What is the
probability that the sum of the two numbers rolled will be less
than 5? A) 1/3 B) 1/6 C) 1/9 D) 1/12. After
determining the probability theoretically, they use a simulation
program for 1000 rolls of two dice and check the outcome data to see
if their predicted probability was in the right ballpark.
 Students use HyperStudio to create the reports
they write about biographies of mathematicians, about how mathematics
is used in real life, or about solutions to problems they've
solved. The software allows them to create true multimedia
presentations.
 Students explore the great wealth of
mathematical information available at the University of
St. Andrews' History of Mathematics World Wide Web site
(http://www.groups.dcs.stand.ac.uk/~history/).
 Students use Algebra Animator software to
simulate and manipulate the motion of a variety of objects such as
cars, projectiles, and even planets. They gather data about the
motion and directly visualize both the functions that describe the
motion and their graphs.
 Working in small groups, students use a distance
probe connected to a graphing calculator to collect data about the
rate of approach of a classmate walking toward the calculator. After
the walk is finished, the calculator plots the student's position
relative to the calculator as a function of time. The group then
presents the finished graph to the rest of the class and challenges
them to describe the walk that was taken: What rate of progress
was made? Was it steady progress? Where did the student stop?
Was there ever any backward walking?
 Students explore the rich links suggested on the
Cornell University Math and Science Gateway World Wide Web Site
(http://www.tc.cornell.edu/Edu/MathSciGateway).
 There is always math help available at the
Dr. Math World Wide Web site (dr.math@forum.swarthmore.edu). In
Dr. Math's words, "Tell us what you know about your problem,
and where you're stuck and think we might be able to help you.
Dr. Math will reply to you via email, so please be sure to send us
the right address. K12 questions usually include what people learn
in the U.S. from the time they're five years old
through when they're about eighteen."
7. Use computer spreadsheets and graphing
programs to organize and display quantitative information and
to investigate properties of functions.
 Students work through
the Building Parabolas lesson that is described in The First
Four Standards of this Framework. They use both the Green
Globs software and their graphing calculators to investigate how
the various coefficients affect the graph of parabolas.
 Students use calculators, a spreadsheet, and an
integrated plotter to work on this problem from Algebra in a
Technological World:
A new professional team is in the process of determining the
optimal price for a special ticket package for its first season. A
survey of potential fans reveals how much they are willing to pay for
a fourgame package. The data from the survey are displayed
below.
Price of the FourGame Package 
Number of Packages That Could Be Sold at That Price 
$96.25 
5,000 
90.00 
10,000 
81.25 
15,000 
56.25 
25,000 
50.00 
27,016 
40.00 
30,000 
21.25 
35,000 
On the basis of the foregoing data, find a relationship that
describes the price of a package as a function of the number sold (in
thousands). Then determine the selling price which will maximize the
revenue, and its number of packages likely to be sold at this
price.
 Students investigate the growth of the
world's population by researching estimates of the level of
population at various times in history and plotting the corresponding
ordered pairs in a piece of software called Data Models. They
then use the software tool to find a line or curve of best fit and use
the resulting graph to predict the population in the year 2100. As a
last step, they find the predictions made by several social scientists
and compare them to their own.
 Students work on a lesson from The New Jersey
Calculator Handbook which uses graphing calculators to focus on
the linear functional relationship between circumference and diameter.
The students measure everyday circular objects to collect a sample of
diameters and circumferences. They then enter their data into a
calculator which plots a scattergram for them and finds a line of best
fit. The slope of the line is, of course, an approximation of
pi.
 Students use the Geometry Inventor for
constructions which illustrate a proof of the Pythagorean Theorem.
With the construction tool, they create a right triangle in the center
of the computer screen, and a square on each of the legs. They then
make a table of the areas of the three squares. As they manipulate
the triangle to adjust the relationship among the lengths of the legs,
they notice that the basic additive relationship of the areas of the
three squares remains the same.
 Students use The Geometer's
Sketchpad to create an initial polygon and then apply a series of
complex transformations to it resulting in a whole sequence of
transformed polygons spread out across the screen. The results are
often striking colorful images that the students can preserve as
evidence of the connections between geometry and modern artistic
design.
8. Use calculators and computers effectively and
efficiently in applying mathematical concepts and principles to
various types of problems.
 Students quickly determine the appropriate
window for finding the intersection of two functions by playing with
the zoom and range functions on a graphing calculator.
 Students solve a variety of online trigonometry
problems posted on the Trigonometry Explorer World Wide
Web site (http://www.cogtech.com/EXPLORE).
 Having just conducted a science experiment where
they collected data about the rates of cooling of a liquid in three
different containers, the students quickly and efficiently enter the
data into a computer spreadsheet and generate brokenline graphs to
represent the three different settings.
 Students solve the following problem by writing
a function that describes the volume of the box, plotting the function
on a graphing calculator, and searching visually for the peak of the
graph. An opentopped box is made from a sixinch square piece of
paper by cutting a square out of each corner, folding up the
sides and taping them together. What size square should be cut
out of the corners to maximize the volume of the box that is
formed?1150
References

Association of Mathematics Teachers of New Jersey. The New
Jersey Calculator Handbook. 1993.
Fendel, D., D. Resek, L. Alper, and S. Fraser. Interactive
Mathematics Program. Key Curriculum Press.
Heid, M.K., et al. Algebra in a Technological
World. Reston, VA: National Council of Teachers of
Mathematics, 1995.
Lenart Sphere. Key Curriculum Press.
New Jersey Department of Education. Mathematics Instructional
Guide. D. Varygiannes, Coord. Trenton, NJ, 1996.
Wolff, K. "Technology and Mathematics Education: Trojan
Horse or White Knight?" in The New
Jersey Calculator Handbook. Association of Mathematics
Teachers of New Jersey, 1993.
Software

Algebra Animator. Logal.
The Algebra Sketchbook. Sunburst
Communications.
Cabri Geometry. IBM.
Data Models. Sunburst Communications.
Derive. Soft Warehouse.
The Function Supposer. Sunburst
Communications.
Geometer's Sketchpad. Key
Curriculum Press.
Geometric PreSupposer. Sunburst Communications.
Geometric Supposer. Sunburst
Communications.
Geometry Inventor. Logal.
Green Globs and Graphing Equations.
Sunburst Communications.
HyperStudio. Roger Wagner.
Maple. Brooks/Cole Publishing Co.
Mathematica. Wolfran Research.
OnLine Resources

http://dimacs.rutgers.edu/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 Indicators 1, 2, 3,
5, and 7 which are also listed for grade 8, since the Standards
specify that students demonstrate continual progress in these
indicators.
