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 5-6
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.
Traditionally, by grades 5 and 6, teachers are devoting relatively
little time to student modeling with manipulatives since they
have begun to concentrate on symbolic and abstract approaches to
content. It is fairly common for teachers at this level to think that
once initial notions of number and shape have been established with
concrete materials in the lower grades, such materials are no longer
necessary and a more symbolic approach is preferable. Research shows,
however, that concrete materials and the modeling of mathematical
operations and concepts is just as useful in these grades as it is for
younger students. The content being modeled is, of course, different
and so the models are different - but they are no less important.
Fifth- and sixth-graders can use a great variety of materials,
including colored rods, base-ten blocks, pattern blocks, fraction
strips and circles, and tangrams, to develop very rich notions of
rational numbers and the operations associated with them. Initial
fraction notions are well-modeled with colored rods. Students use
different length rods to represent different units and then decide on
the fractional or mixed number value of the other rods. Base-ten
blocks, with the values 1, 0.1, 0.01, and 0.001 assigned to the
different sizes, model operations with decimals just as well as the
values normally associated with base-ten blocks model whole number
computation. Pattern blocks and tangram pieces are comfortable and
familiar tools with which to begin to explore notions of ratio and
Geometry models, both two- and three-dimensional, are an important
part of learning about geometry and development of spatial sense in
students of this age. Students can use geoboards to develop
procedures for finding the areas of polygons or irregular shapes.
They can also use construction materials like pipe cleaners and straws
or cut-out cardboard faces to make complex three-dimensional geometric
figures which can be studied directly. It is much easier to determine
the number of faces or edges in a figure from such a model than from a
two-dimensional drawing of the figure.
Fifth- and sixth-graders 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 two-colored
counters to represent positive and negative integers in initial
explorations of addition and subtraction of signed numbers. They
should be able to use a variety of measurement tools to measure and
record the data in a science experiment. They might use play money to
concretely construct solutions to coin problems or to riddles that ask
how much an individual actually profited or lost in some complex
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 children to
construct mathematics that is meaningful to them.
There are many appropriate uses for calculators in these
grade levels. In his article, Using Calculators inthe
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 calculator-specific skills - for example,
to learn how to use the memory function of a calculator to efficiently
solve a multi-step problem.
These three categories provide a good framework for thinking about
calculators at these grade levels. For another powerful example of
the first category, consider using an ordinary four-function
calculator to explore and begin to describe the relationships between
common fractions and decimals. Entering 2/3 into the calculator by
pressing 2, then the division key, and then 3 gives a
result of 0.6666667. Discussion of this result, attempts to create
other similar results, and working out some of the problems by hand
lead to discoveries about terminating and non-terminating decimals,
repeating decimals, and fraction-decimal equivalence. Such
explorations also should be used to highlight the limitations of the
calculator, which does not always give the answer 1 when 1/3 is added
Computers are a valuable resource for students in fifth and
sixth 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 problem solutions.
In terms of specific mathematics education software, there are many
good choices. Logo, of course, can be used effectively by students at
these grade 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 5-8: the use of iterative and recursive
processes. Oregon Trail II is a very popular CD-ROM 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. Tesselmania!, the
Teaching and Learning with Computers series, Elastic
Lines, and the Geometry Workshop all allow students
to make geometric constructions on the computer screen and then
transform them in a variety of ways in order to experiment with the
effects of the transformations.
Graph Power, 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 eye-opening tool for
fifth- and sixth-graders as they retrieve and share information.
Specifically, in these grades, they might look for demographic data
about geographic locations in which they are interested, summaries of
the vote totals for different precincts in local elections, and home
pages from other schools in this country and abroad.
Standard 5 - Tools and Technology - Grades 5-6
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
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 5-6 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 work through the Pizza Possibilities
lesson that is described in the First Four Standards of this
Framework. They use a variety of manipulatives to help them
visualize and solve the problem.
- Having used paper-and-pencil to develop an interesting
geometric shape which tessellates the plane, the students go to the
computer to use Tesselmania! to reproduce it and then to tile
the computer screen with it. They color the printout from the program
to produce a unique piece of artwork which is then posted in a class
- Students engage in the four activities of
Target Games: Estimation is Essential! in The New
Jersey Calculator Handbook. In these activities, students learn
the role that estimation plays in effective calculator use and learn
to identify reasonable and unreasonable answers in the calculator
- Students explore the rich source of problems at
the Math Forum World Wide Web site at Swarthmore College
2*. Use physical objects and
manipulatives to model problem situations, and to develop and
explain mathematical concepts involving number, space, and
- Using standard base ten blocks as a model, where
the four blocks, as usual, represent one, ten, one hundred, and one
thousand, students demonstrate their understanding of place value on
an assessment by writing in their journals a verbal description
(accompanied by drawings) of what a ten-thousand block and a
one-hundredth block might look like.
- Students use gumdrops and toothpicks to build a
variety of polyhedra. Using these models, they try to generalize a
relationship among the faces, edges, and vertices that works for all
solids. (There is one! It's called Euler's formula after
its discoverer and is: F + V = E + 2.)
- Students read How Much is a Million by
David Schwartz. The book describes how tall a stack of a million
children standing on each other's shoulders would be, how long it
would take to count to a million, and so on. The students pick some
object of their own and try to determine how big a space would be
needed to contain a million of them. Typical objects to inquire about
include blades of grass, pennies, and dollar bills.
3. Use a variety of
technologies to discover number patterns, demonstrate number sense,
and visualize geometric objects and
- Students are asked to enter the number 6561 into
their calculators and then to keep pressing the square root key to try
to discover what it does. After more experimentation with the key,
the students are asked to predict what would have to be entered into
the calculator's display if they wanted to press the square root
key 6 times and wind up with the number 4.
- Students work through the Two-Toned Towers
lesson that is described in the First Four Standards of this
Framework. Students use manipulatives to determine how many
towers can be built which are 4 cubes tall and use no more than 2
colors, and then discuss the pattern that results when the length of
the towers can be 4,5,6, or a larger number of cubes. They also relate
their answers to the solution of the Pizza Possibilities lesson
on the following page in the First Four Standards.
- Students use Elastic Lines or another version of an
electronic geoboard to construct geometric figures and then transform
them through rotations, reflections, and translations. Students create
a figure and its transformed image on the screen and challenge each
other to describe the specific transformation that created the
- Students play a computer golf game where they
must hit a ball into a hole. The ball and the hole are both visible
on the screen, but at opposite sides. Players specify an angular
orientation (where 0o is straight up) and a number of units
of length which will describe the path of the ball once it is struck.
The object, just like in real golf, is to get the ball in the hole in
as few strokes as possible. Good estimation of both angle measure and
length are critical to success.
4*. Use a variety of
tools to measure mathematical and physical objects in the world around
- Students divide into groups to make a scale
model of their classroom by accurately measuring critical elements of
the room, using a standard proportional relationship to convert the
actual measurements to the model's measurements, and then
measuring again to cut the modeling material (cardboard, balsa wood,
or manila paper) to the correct size. Their model of the room should
also contain models of the blackboard, the teacher's desk, some
student desks, the shelves in the room, and so on. Each student group
is responsible for different elements of the room.
- Students measure the volumes of several
rectangular boxes by filling them with cubic inch blocks or cubic
centimeter blocks. After some thought and discussion, they
devise formulas to compute the volumes from direct measurement of the
three appropriate dimensions.
- Students use ratios and proportions to determine the
heights of objects that are too tall to easily measure directly.
Measuring the heights of some known objects and the lengths of the
shadows they cast, students determine the heights of the school
building, the flagpole, and the tallest tree outside the school by
measuring their shadows.
5*. Use technology
to gather, analyze, and display mathematical data and
- Students use the New Jersey State homepage
http://www.state.nj.us on the World Wide Web to gather data about the
latest reported population for each county in the state and about the
area of the counties. They enter the collected data into two adjacent
columns in a spreadsheet and configure a third column to calculate the
population density for each county (population / area). They
highlight their own county in the printout of the spreadsheet to show
where it stands in relationship to the others.
- 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
- Students measure various body parts such as
height, length of forearm, length of thigh, length of hands, and arm
span. They enter the data into a spreadsheet and produce various
graphs as well as a statistical analysis of the class. They update
their data every month and discuss the change, as it relates both to
individuals and to the class.
- Students conduct a survey of the population of
the entire school to determine the most popular of all of the choices
for school lunches. After gathering the data, they enter it into a
spreadsheet and use the program to graph it in a variety of ways
- as a bar graph, a circle graph, and a pictograph. They discuss
which of the graphs best illustrates their data and publish the one
they choose in a report distributed to all of the students in the
- There is always math help available at the
Dr. Math World Wide Web site (firstname.lastname@example.org). 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 e-mail, so please be sure to send us
the right address. K-12 questions usually include what people learn
in the U.S. from the time they're five years old through when
they're about eighteen."
6. Use a variety of technologies to
evaluate and validate problem solutions, and to investigate the
properties of functions and their graphs.
- Students solve the problems posed in Logical
Journey of the Zoombinis by using logic and classification and
categorization skills. In it, they create Zoombinis, little creatures
that have specific characteristics that allow them to accomplish
- Students use their knowledge of theoretical
probability to predict the relative frequency of occurrence of each of
the possible sums when rolling a pair of dice. They use simulation
software like the Graphing and Probability Workshop to simulate
the rolling of 300 pairs of dice. They examine the simulated
frequencies and judge them to either be consistent or inconsistent
with their predictions and reexamine their predictions if
- Students use the data they gathered earlier
concerning the heights of objects and the lengths of the shadows they
cast at the same time on a sunny day. They enter the data as ordered
pairs (height, shadow) into a simple graphing program and notice that
the resulting points all lie on a line. They use the line to predict
the heights of objects whose shadows they can measure.
- Students make a pattern using square tiles to build
increasingly larger squares (a 1x1, a 2x2, a 3x3, and so on). They
count the number of tiles it took to build each successive square and
plot the resulting ordered pairs ((1,1), (2,4), (3,9), (4,16),
. . . ) on an x-y plane. The resulting parabola is a non-linear
function which is easy to discuss.
7. Use computer spreadsheets and graphing
programs to organize and display quantitative information and
to investigate properties of functions.
- Students measure each of a variety of objects in
both inches and centimeters. They enter the collected data into a
spreadsheet as ordered pairs in two adjacent columns, measurements in
inches followed by measurements in centimeters. They have the
spreadsheet program graph the ordered pairs on an x-y plane. After
they discover that all of the points lie on a line, they draw the line
and use it to determine the customary measure of an object whose
metric measure they know and vice versa.
- Students configure a simple spreadsheet to assist them in
finding magic squares by automatically computing all of the sums. For
example, they reserve a three-by-three array of cells for the magic
square somewhere in the middle of the spreadsheet. In the cells that
are at the end of the rows, they enter formulas that show the sums of
the entries in the cells in each row, and enter similar formulas at
the end of each column and diagonal. When proposed entries are placed
in the magic square cells, their various sums are instantly provided
in the adjacent cells, facilitating adjustment of the entries. The
students then use their new tool to solve and create magic square
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. Association of
Mathematics Teachers of New Jersey, 1993.
Schwartz, D. How Much is a Million? New York: A
Mulberry Paperback Book, 1985.
AppleWorks. Apple Computer Corp.
Elastic Lines. Sunburst Communications.
Geometry Workshop. Scott Foresman.
Graph Power. Ventura Educational
Graphers. Sunburst Communications.
Graphing and Probability Workshop. Scott
How the West Was One + Three x Four.
HyperStudio. Roger Wagner.
Logical Journey of the Zoombinis.
MacStat. Minnesota Educational Computing
Oregon Trail II. Minnesota Educational
Computing Consortium (MECC).
Table Top. TERC.
Teaching and Learning with Computers.
International Business Machine, Inc. (IBM).
Tesselmania! Minnesota Educational
Computing Consortium (MECC).
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 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