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 78
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.
Seventh and eighthgraders can use a variety of manipulatives
to enhance their mathematical understanding and problem solving
ability. For example, new approaches to the teaching of elementary
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 best understand issues of projection,
perspective, and shadow by actually building concrete constructions
out of blocks or cubes and viewing them from a variety of directions
and in different ways. Slicing clay models of various
threedimensional figures convinces students of the resulting planar
shape that is the crosssection. Using pipe cleaners and straws,
students can build their own version of a Sierpinski tetrahedron.
Seventh and eighthgraders 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.
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 the 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, Using Calculators in
the Middle Grades, in The New Jersey Calculator Handbook,
David Glatzer suggests that there are three major categories of
calculator use in the middle grades:
To explore, develop, and extend concepts  for example, when
the students use the square and square root keys to try to understand
these functions and their relationship to each other.
As a problem solving tool  for example, to see how increasing
each number in a set by 15 increases the mean of the set.
To learn and apply calculatorspecific skills  for example,
to learn how to use the memory function of a calculator to efficiently
solve a multistep problem.
These three categories provide a good framework for thinking about
calculators in the seventh and eighth grade. For another powerful
example of the first category, consider the question of compounding
interest. When asked how much a bank account might accumulate after 10
years with an initial balance of $1000 and a simple annual interest
rate of 6 percent, most students would first calculate the interest
for the first year, add it to the initial balance to get the new
balance, multiply that by 0.06 to get the interest for the second
year, add that to the previous year's balance, and so forth.
After discussion of that iteration, most seventh and eighthgraders
are able to understand that each year's balance is the product of
the previous year's balance times 1.06, so to find the
balance after three years, one could simply use the formula:
$1000 x 1.06 x 1.06 x 1.06. After
still more discussion, most students will transform this into the
standard formula, which is easy to apply with a calculator: $1000
x 1.06^{n}. These concepts develop
nicely in a classroom where all of the students have calculators and
can do the computations easily and quickly. In a traditional
classroom without calculators, the progression takes much longer and
the resulting formula is much less believable to students.
Students at this level should also have some experience with
graphing calculators. Although these tools will be most useful in the
high school curriculum, middle school students should be exploring
graphs of linear functions and other simple graphs and should be
making use of the statistical capabilities of most graphing
calculators. They should also be exploring the use of Calculator
Based Laboratories (CBL) which enables them to gather data and display
the data graphically in the viewing window.
Computers are also an essential resource for students in
seventh and eighth grade, and the software tools available for them
are more like adult tools than those available for younger children.
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 solutions to
problems.
In terms of specific mathematics education software, there are also
many good choices. Logo, of course, can be used effectively by
students at these levels to explore computer programming and geometry
concepts at the same time. It is an ideal tool to learn about one of
the critical cumulative progress indicators for Standard 14 (Discrete
Mathematics) for grades 58: the use of iterative and recursive
processes. Oregon Trail II is a very popular CDROM program
that effectively integrates mathematics applications with social
studies. How the West Was One + Three x Four also uses an old
west theme to work on arithmetic operations.
A variety of computer golf games allow students to play a
competitive game while sharpening their estimation ability with angle
and length measure. The Geometric Supposer and PreSupposer
series is one of the most popular geometry construction tools for
students of this age. With it, 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.
Graph Power, the Graphing and Probability Workshop,
AppleWorks, TableTop, Graphers, and MacStat
are some of the many tools available that include database,
spreadsheet, or graphing facilities written for students at this age.
Many other valuable pieces of software are available.
The World Wide Web can be an exciting and eyeopening tool for
seventh and eighthgraders as they retrieve and share information.
Specifically, in these grades, they might look for good math problems
from the Web bulletin boards, biographical data about famous
mathematicians, and census data for local towns.
Standard 5  Tools and Technology  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:
1. Select and use calculators,
software, manipulatives, and other tools based on their utility and
limitations and on the problem situation.
 In problem solving situations, students are no
longer provided with instructions concerning which tool to use, but
rather are expected to select the appropriate tool from the array of
manipulatives, calculators, computers and other tools that are always
available to them.
 Students continue with activities used in
previous grades and use problems like those of Target Games:
Estimation is Essential! in The New Jersey Calculator
Handbook, in which students learn to identify reasonable and
unreasonable answers in the calculator display.
 Students use a variety of materials to
demonstrate their understandings of basic mathematical properties and
relationships. For instance, they are able to use geoboards, dot
paper, and Geometer's Sketchpad to demonstrate the
Pythagorean Theorem.
 Students work through the Rod Dogs lesson
that is described in The First Four Standards of this
Framework. They use Cuisenaire rods to model the increase of
the dimensions of an object by various scale factors, but when they
realize that there are not enough rods to simulate the situation, they
find other models which can be used.
2^{*}. Use physical
objects and manipulatives to model problem situations, and to develop
and explain mathematical concepts involving
number, space, and data.
 Students use twocolored counters to model
signed numbers and integer operations. On the red side, a counter
represents +1, on the white side, 1. As sets of counters are
combined or separated to model the operations, students look for
patterns in the answers so that they can write rules for completing
the operations without counters.
 Students use the same counters and also red and
white cubes, representing +x and x, to model and
solve equations. By setting up counters and cubes to represent the
initial equation and then removing equal sets from both sides,
students model the essential elements of solving linear equations and
develop the appropriate language with which to discuss those elements.
Conversion to symbolic processes comes soon after mastery is achieved
with the concrete objects.
 Students make threeinch cubes of clay and then
experiment to see in how many different ways they can slice the cube
with a plane to produce different crosssections. Drawings of the
crosssections and a description of the cuts that created them are
displayed on a poster in the classroom.
3^{*}. Use a variety of
technologies to discover number patterns, demonstrate number sense,
and visualize geometric objects and
concepts.
 Students work on this seemingly simple problem
from the New Jersey Department of Education's Mathematics
Instructional Guide: A copy machine makes 40 copies per
minute. How long will it take to make 20,000 copies? A) 5
hours B) 8 hours 20 minutes C) 8 hours 33 minutes D) 10
hours. Students immediately decide to use their calculators to
solve the problem, but then have an interesting discussion regarding
the calculator display of the answer. When 20,000 is divided by 40,
the display shows 8.33333. Which of the answer choices is
that? Why?
 Students use the Geometer's
Sketchpad to create a triangle on the computer screen and
simultaneously place on the screen the measures of each of the angles
as well as the sum of the three angles. They notice that the sum of
the angles is 180 degrees. They then click and drag one of the
vertices around the screen to make a whole variety of other triangles.
They notice that even though the measure of each of the angles changes
in this process, the sum of 180 degrees never changes, thus
intuitively demonstrating the triangle sum theorem.
 Students use videotape of a person walking to model integer
multiplication. The videotape shows a person walking forward with a
sign that says "forward" and then walking backward with a
sign that says "backward." When run forward, the video
shows the "forward" walker walking forward (+
× + = +). When run backward, the video shows the
"forward" walker walking backward ( × +
= ). The other two possibilities also work out
correctly to show all of the forms of integer multiplication.
4. Use a variety of tools to
measure mathematical and physical objects in the world around
them.
 Groups of students build toothpick bridges in a
competition to see whose bridge can hold the most weight in the center
of the span. Each group has the same materials with which to work and
the bridges must all span the same distance. In the process of
building the bridges, the students conduct a good deal of research
into bridge designs and about factors that contribute to structural
strength.
 Students work through the Sketching
Similarities lesson that is described in the First Four Standard
of this Framework. They use the Geometer's
Sketchpad to measure the length of sides and the angles of similar
figures to discover the geometric relationships between corresponding
parts.
 Students explore the relationship between the
height of a ramp and the length of time ittakes a matchbox car to roll
down it. The teacher provides stopwatches, long wooden boards, and
meter sticks. The students use a spreadsheet program to enter their
data relating height and time for several different heights, and use
the spreadsheet's integrated graphing program to plot the ordered
pairs. They then look for the relationship between the height and the
time.
 Students are challenged to answer the question:
In how many ways can you measure a ball? After the
obvious spatial characteristics are named (volume, diameter,
circumference, and so on), students get more creative and suggest
bounceability, density, fraction of its height that it loses if a five
pound book is placed on it, weight, number of times it bounces when
dropped from one meter, and so on. When the list is complete,
different groups of students select several of the possible
characteristics and develop ways to measure them.
5. Use technology to gather,
analyze, and display mathematical data and information.
 Students use a temperature probe connected to a
graphing calculator to collect data about the rate of cooling of a cup
of boiling water. The data is displayed in chart form and in a graph
by the calculator after the experiment is performed.
 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 gather data from their fellow students
regarding the number of people in their households. They then enter
the data into a graphing calculator and learn how to produce a
histogram, showing the number of students with each size household,
from the data on the calculator.
 Students decide to resolve the debate that two
of them were having about which of their favorite baseball players was
the better hitter. They find a great deal of numerical data on their
respective teams' World Wide Web homepages regarding the number
of at bats, singles, doubles, triples, homeruns, and walks each batter
had accumulated in his career. The class decides what weight to
attribute to each type of hit and then computes a weighted score for
each player to decide who the winner is.
 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."1150
6. Use a variety of technologies to
evaluate and validate problem solutions, and to investigate the
properties of functions and their graphs.
 Students use Geometer's
Sketchpad to work out a solution to this problem from the New
Jersey Department of Education's Mathematics Instructional
Guide: Two of the opposite sides of a square are
increased by 20% and the other two sides are decreased by
10%. What is the percent of change in the area of the original
square to the area of the newly formed rectangle? Explain the
process you used to solve the problem. In their solution
attempts, they construct a square on the screen with known sides, and
then a rectangle with the sides indicated in the parameters of the
problem. The program calculates the areas of the two figures and the
students are close to a solution.
 Students use their calculators to solve this
problem from the New Jersey Department of Education's
Mathematics Instructional Guide:
A set of test scores in Mrs. Ditkof's class of 20
students is shown below.

62 
77 
82 
88 
73 
64 
82 
85 
90 
75 
74 
81 
85 
89 
96 
69 
74 
98 
91 
85 
Determine the mean, median, mode, and range for the
data.
Suppose each student completes an extracredit assignment
worth 5 points, which is then
added to his/her score. What is the mean of the set of scores
now if each student received
the extra five points? Explain how you calculated your
answer.
 Students play Green Globs and Graphing
Equations, a computer game in which they score points for writing
the equations of lines that will pass through several green globs
splattered on the xy plane.
7. Use computer spreadsheets and graphing
programs to organize and display quantitative information and
to investigate properties of functions.
 Students use a simple spreadsheet/graphing
program to solve this problem from the New Jersey Department of
Education's Mathematics Instructional Guide:
VOTING RESULTS
Class Colors 
Number of Votes 
red and white 
10 
green and gold 
12 
blue and orange 
5 
black and yellow 
9 
Rather than using the tools the problem suggests (protractor,
compass, and straight edge), the students enter the data into a
spreadsheet and construct a circle graph from the spreadsheet.
 Students measure the temperatures of a variety
of differently heated and cooled liquids in both Fahrenheit and
Celsius. They then enter the collected data into a spreadsheet as
ordered pairs in two adjacent columns, measurements in Fahrenheit
followed by measurements in Celsius. They have the spreadsheet
program graph the pairs on an xyplane. After they discover that all
of the points lie on a line, they draw the line and use it to
determine the Fahrenheit temperature for a given Celsius temperature
and vice versa.
 Students configure a spreadsheet to act as an
orderprocessing form for a stationery store (or some other retail
operation). They decide on the five items they'd like to sell,
enter the prices they'll charge, and then program all of the
surrounding cells to compute the prices for the quantities of items
ordered, add the tax, and compute the final charge.
Items 
Price 
Quantity Ordered 
Cost 
Pencils 
.05 


Pens 
.29 


Paper Pads 
.59 


Tape 
.49 


Scissors 
1.39 




Tax: 



Total: 

References

Association of Mathematics Teachers of New Jersey. The New
Jersey Calculator Handbook. 1993.
Glatzer, D. "Using Calculators in the Middle Grades,"
in The New Jersey Calculator Handbook. New Jersey:
Association of Mathematics Teachers of New Jersey, 1993.
New Jersey Department of Education. Mathematics Instructional
Guide. D. Varygiannes, Coord. Trenton, NJ: 1996.
Software

AppleWorks. Apple Computer Corp.
Geometer's Sketchpad. Key Curriculum
Press.
Geometric PreSupposer. Sunburst
Communications.
Geometric Supposer. Sunburst Communications.
Geometry Workshop. Scott Foresman.
Graph Power. Ventura Educational Systems.
Graphers. Sunburst Communications.
Graphing and Probability Workshop. Scott
Foresman.
Green Globs and Graphing Equations.
Sunburst Communications.
How the West Was One + Three x Four.
Sunburst Communications.
HyperStudio. Roger Wagner.
Logo. Many versions of Logo are commercially
available.
MacStat. Minnesota Educational Computing
Consortium (MECC).
Oregon Trail II. Minnesota Educational
Computing Consortium (MECC).
TableTop. TERC.
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,
4, and 5 which are also listed for grade 4, since the Standards
specify that students demonstrate continual progress in these
indicators.
