STANDARD 5 - TOOLS AND TECHNOLOGY
K-12 Overview
All students will regularly and routinely use calculators,
computers, manipulatives, and other mathematical tools to enhance
mathematical thinking, understanding, and power.
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Descriptive Statement
Calculators, computers, manipulatives, and other mathematical tools
need to be used by students in both instructional and assessment
activities. These tools should be used, not to replace mental math
and paper-and-pencil computational skills, but to enhance
understanding of mathematics and the power to use mathematics.
Historically, people have developed and used manipulatives (such as
fingers, base ten blocks, geoboards, and algebra tiles) and
mathematical devices (such as protractors, coordinate systems, and
calculators) to help them understand and develop mathematics.
Students should explore both new and familiar concepts with
calculators and computers, but should also become proficient in using
technology as it is used by adults, that is, for assistance in solving
real-world problems.
Meaning and Importance
Both mathematics and the way we do mathematics have changed
dramatically in the last few decades. The presence of technology has
made more mathematics accessible to us, has allowed us to solve
mathematical problems never before solved, and has brought a new and
extraordinarily higher level of proficiency with mathematical
operations to all members of our society. At the same time, our
resulting increased need to truly understand that mathematics has
brought on a renewed interest in developing materials and approaches
that provide concrete models and that can engage learners. Both of
these new directions combine in this one standard which calls for the
appropriate and effective use of tools and technology.
Manipulatives are concrete materials that are used for
"modeling" or representing mathematical operations or
concepts. In much the same way that children make and use models of
race cars or the human skeleton so that they can study and learn about
them, students can make and learn from models of two-digit numbers or
multiplication. The difference between the two situations is that
while the car and skeleton models are smaller versions of actual
concrete things, the number models or multiplication models are
concrete models of abstract concepts and operations.
When students use bundles of sticks and single sticks to represent
tens and ones, or algebra tiles to represent polynomials, they are
using manipulatives to "model" mathematical ideas.
Technically, even when young children count on their fingers, they are
concretely modeling numbers.
The mechanism by which concrete modeling aids children in
constructing mathematical knowledge is still not completely
understood. There is little doubt, however, that it does. There is a
good deal of research which shows that the optimal presentation
sequence for new mathematical content is
concrete-pictorial-abstract. Activities with concrete
materials should precede those which show pictured relationships and
those should, in turn, precede formal work with symbols. Ultimately,
students need to reach that final level of symbolic proficiency with
many of the mathematical skills they master, but the meanings of those
symbols must be firmly rooted in experiences with real objects.
Otherwise, their performance of the symbolic operations will simply be
rote repetitions of meaningless memorized procedures.
Calculators and computers provide still other
benefits for students. In the 1994 Position Statement on The
Use of Technology in the Learning and Teaching of
Mathematics, the National Council of Teachers of Mathematics
states:
Students are to learn how to use technology as a tool for
processing information, visualizing and solving problems,
exploring and testing conjectures, accessing data, and verifying their
solutions.
. . . In a mathematics setting, technology must be an
instructional tool that is integrated into daily
teaching practices, including the assessment of what students
know and are able to do.
The availability of technology requires that we re-evaluate our
mathematics curricula. What we teach and how we teach it are now
inextricably linked to these new tools. Their presence makes some
traditional mathematics topics obsolete - we certainly do not
still teach the square root extraction algorithm, but what about hours
and hours of paper-and-pencil practice with the long division
algorithm? How much is necessary? How much is adequate? How much
will our students need that skill when they become adults? The
presence of the technology also makes some traditional topics more
important than ever - efficient calculator use requires a high
level of estimation, place value, and mental math skills so that
calculations can be quickly checked for reasonableness and accuracy.
And the technology allows some topics to be dealt with that were never
accessible to students previously - graphing calculators allow
students to instantly see the graphs of complex functions that would
have taken a whole class period to graph by hand, and computer-based
geometry construction tools allow students to experiment with
animation to see the effects of transformations of figures in
three-dimensional space.
Stephen Willoughby, a former president of the National Council of
Teachers of Mathematics, offers this analogy to those who might be
reluctant to change the traditional curriculum:
When automobiles first appeared, there were undoubtedly many
people who kept a spare horse in the garage lest the automobile
fail, but very few of us still do today. Calculators, with and
without batteries, have become so inexpensive and reliable that
it is more efficient to keep an extra calculator handy than it
is to learn to do well everything a calculator does better. Most of
us no longer find the ability to shoe a horse or cinch a saddle
to be essential skills. Is it not reasonable that in the near
future we may feel the same way about multi-digit long
division? (Willoughby, 1990, p. 62)
The New Jersey State Board of Education, recognizing the need for
students to develop appropriate and integrated technology skills,
requires the use of calculators in the state's mathematics
assessment program at the eighth and eleventh grade levels. To be
adequately prepared to use calculators on those assessments, it is
critical that the students have ample opportunity to use them as a
regular and natural part of their mathematics classes and other
testing.
We are only just beginning to explore the ways in which technology
can be helpful to mathematics learners, but already there are
tremendous opportunities at all levels. Two additional tools that
have not yet been mentioned in this Overview possibly provide the most
futuristic vision of what mathematicsclassrooms might soon look like.
Calculator-Based Laboratories (CBLs) allow measurement probes to be
connected to hand-held calculators to collect and analyze scientific
data such as light, distance, voltage, temperature, and so on. This
direct observation, collection, and display of data allows the student
to focus more on the hypothesis, interpretation, and analysis phases
of experiments. And the World-Wide Web, or graphical Internet, holds
untold amounts of data and information. From geographic and census
data, to current information about almost any mathematical or
scientific subject, to rich sources of mathematical problems for K-12
students, the Internet will greatly expand the research capability of
both teachers and students.
Strategies and teaching approaches which utilize technology have
been shown to improve student attitudes, problem solving ability,
ability to visualize mathematics, and overall performance. Here,
possibly more than anywhere else in this Framework, we need to
be open to new ideas and receptive to new approaches.
K-12 Development and Emphases
Many specific suggestions for appropriate uses of calculators,
computers, and manipulatives are given in the following
pages and, indeed, throughout this Framework. The point to be
made here, though, is that the frequent, well-integrated use of these
tools at all levels is essential. Young children find the use of
concrete materials to model problem situations very natural. Indeed
they find such modeling more natural than the formal work they do with
number sentences and equations. Older students will realize that the
adults around them use calculators and computers all the time to solve
mathematical problems and will be prepared to do the same. Perhaps
more challenging, though, is the task of getting the
"reverse" to happen as well, so that technology is also used
with young children, and the older students'
learning is enhanced through the use of concrete models. Such
opportunities do exist, however, and new approaches and tools are
being created all the time.
In summary, mathematical tools play an ever-more important
role in today's mathematics. Students who will be expected to be
knowledgeable users of such tools when they leave school must see
those tools as a regular and routine part of "doing
mathematics" in school.
References
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National Council of Teachers of Mathematics. Position
Statement on The Use of Technology in
the Learning and Teaching of Mathematics. Reston, VA,
1994.
Willoughby, S. Mathematics Education for a Changing World.
Alexandria, VA: Association for Supervision and Curriculum
Development, 1990.
General References
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Barnes, B., et al. Tales from the Electronic Frontier.
San Francisco: WestEd Eisenhower Regional Consortium for
Science and Mathematics Education (WERC), 1996.
Kinslow, J. Internet Jones. Philadelphia, PA: Research
for Better Schools, 1996.
Standard 5 - Tools and Technology - Grades K-2
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.
In the primary grades, manipulatives are the most natural of
the three types of tools to use. Primary grade teachers have
traditionally used many manipulative materials in their teaching of
mathematics because they correctly perceived them to be of great value
for young children. Typically, concrete materials are used to model
mathematical concepts such as number or shape when those concepts are
first introduced to the students.
Young children counting with lima beans, colored chips, linking
cubes, smooth stones, or their fingers is a familiar sight in many New
Jersey classrooms as they begin to master early counting skills and
are introduced to addition and subtraction concepts. More
sophisticated models should then be used, though, to begin to explore
more sophisticated number concepts. Colored rods in graduated lengths
give students a different sense of number than a set of discrete
objects. Students should be able to see both a yellow rod and
five colored chips as representative of the number five - the
first being more of a measure model and the second a count model. Ice
cream sticks and base ten blocks as well as chip trading activities
help students begin to understand the very abstract concepts involved
with place value and number base.
Attribute blocks, blocks with different shapes and colors, help
students begin to classify and categorize objects and recognize their
specific characteristics. Pattern blocks allow them to make patterns
and geometric designs as they become familiar with the geometric
properties of the shapes themselves. Geoboards allow students to
explore the great variety of shapes that can be made and also to deal
with issues of properties, attribute, and classification.
A great variety of different materials should be used to explore
measurement. Paper clips, shoes, centimeter and decimeter rods, paper
cutouts of handspans, and building blocks can all be used as
non-standard units of length (even though some of them are really
standard). Students place them down one after another to see how many
paper clips long the desk is or how many handspans wide the doorway
is. The transition can then be made to more standard measures and,
following that, to rulers.
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.
Calculators have not been used traditionally in primary
classrooms, but there are several appropriate uses for them. It is
never too early for students to be introduced to the tool that most of
the adults around them use whenever they deal with mathematics. In
fact, many students now come to kindergarten having already played
with a calculator at home or somewhere else. To ignore calculators
completely at this level is to send the harmful message that the
mathematics being done at school is different from the mathematics
being done at home or at the grocery store.
The use of calculators at this level does not imply that students
don't need to develop the arithmetic skills traditionally
introduced at the primary level. They certainly do need to develop
these skills. This Standard does not suggest that all traditional
learning be replaced by calculator use; rather, it calls for the
appropriate and effective use of calculators.
One of the most effective uses of the calculator with young
children is the use of the constant feature of most calculators to
count, forward or backward, or to skip count, forward or
backward, by twos or threes or other numbers. This process allows
children to anticipate what number will come next and then get
confirmation of their guess when they see it appear in the display.
Students can also greatly enhance their estimation ability through
calculator use. Range-finding games ask students, for
instance, to add a number to 34 that will give them an answer between
80 and 90. After the estimate is made, it is punched into the
calculator to see whether or not it did the job. Calculators will
prompt young students to be curious about mathematical topics that are
not typically taught at their level. For example, when counting back
by threes by entering 15 - 3 = = = . . . into the calculator,
after the expected sequence of 12, 9, 6, 3, 0, the child will see
-3, -6, -9, . . . A curious child will begin to ask
questions about what those numbers are, but will also begin to develop
an intuitive notion about negative numbers.
Computers are a valuable tool for primary children. As more
and more computers find their way into primary classrooms, the
software available for them will dramatically improve; however, there
are already many good programs that can be used with kindergartners
and first and second graders. MathKeys links on-screen
manipulative materials to standard symbolic representations and to a
writing tool for children to use. A number of different counting
programs match objects on the screen to a standard symbolic
representation of the number and the number is said aloud so that a
young student can count along with the program. Many other new
programs focus on money skills and help children recognize different
coins and determine the values of sets of coins through simulated
purchases.
Standard 5 - Tools and Technology - Grades K-2
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in kindergarten
and grades 1 and 2.
Experiences will be such that all students in grades K-2:
1. Select and use calculators, software,
manipulatives, and other tools based on their utility and
limitations and on the problem situation.
- Students participate in races to complete a set
of computation problems between some students who use calculators and
others who use mental math. They try to determine what makes the
calculator a useful tool in some circumstances (large numbers, harder
operations) and not terribly useful in others (basic facts, easy
numbers).
- Students are regularly asked to make their own
decisions about what is the right type of linear measuring device for
a particular situation: mental estimation, colored rods, ruler, yard
or meter stick, or tape measure. Different decisions are made in
different circumstances: Estimation is fine when you are deciding
whether you will fit through a small doorway, but accurate ruler
measurement is important if you are cutting out a frame for a
picture.
- In problem solving situations, students are
regularly provided with calculators, manipulatives, and other tools so
that they may choose for themselves what will be useful to help solve
the problem.
2. Use physical objects and manipulatives to model
problem situations, and to develop and explain mathematical
concepts involving number, space, and data.
- Students use popsicle sticks to model
multi-digit base-ten numbers and then use them to further model
operations with the numbers.
- Students use pipe cleaners and straws to make
models of two-dimensional geometric shapes. They then compare,
contrast, and sort all the shapes using whatever criteria they think
are important, including number of corners, straight or curvy sides,
number of sides, and so on.
- Students work through the Shapetown
lesson that is described in the First Four Standards of this
Framework. Students in kindergarten are challenged to build towns
with attribute blocks and loops based on a rule or pattern they make
up.
- Kindergarten students each use a cubic inch
block to represent himself or herself in a bar graph that describes
the favorite flavors of ice cream of all the students in the class.
On a table in the front of the room, the teacher has placed mats that
say Vanilla, Chocolate, and Strawberry. One by one, the
students walk past the table, dropping their blocks on one of the
piles that build up on the mats. When this concrete "bar
graph" is complete, the children ask questions that can be
answered with the data displayed: What's the
mostfavorite flavor in the class? What's the least
favorite? Are there more people who like vanilla than
chocolate?1150
3. Use a variety of technologies to discover
number patterns, demonstrate number sense, and visualize
geometric objects and concepts.
- Students use the constant function on a
calculator to count by ones, twos, tens, fourteens, and other numbers,
both forward and backward. As they do so, they try to keep up with
the calculator by saying the numbers orally as they come up in the
display, and even trying to say them before they come
up.
- Students use a beginner's Logo to explore
movement in two-dimensional space. They move the turtle on the
computer screen forward and backward with simple commands and also
turn the turtle through predetermined angles to the right and to the
left with other commands. The turtle leaves a trail of where
it's been on the screen so that its movements actually create a
drawing of a figure. The students try to have the turtle draw a
square, a different rectangle, and a triangle before progressing to
harder tasks.
- Students use a geoboard to make shapes that are
composed of unit squares. One challenge they are given is to find as
many shapes as possible that are made up of 10 unit
squares.
4. Use a variety of tools to measure mathematical
and physical objects in the world around them.
- Young students develop meaning for rulers by
first measuring with individual paper clips, then a paper clip chain,
then taping the clip chain to a paper strip, then marking and
numbering the ends of the clips on the strip, and last, removing the
clip chain from the paper strip leaving just the marks and the
numbers. This leaves the students with paper clip
rulers with which they can measure the lengths of a variety of
objects. The unit of measurement is, of course, a paper
clip.
- Students use a balance scale to determine the
weights of a variety of classroom objects in terms of units that are
other classroom objects; for example, How many pennies does a
math book weigh? How many paper clips does a pencil
weigh?
- Students work through the Will a Dinosaur
Fit? lesson that is described in the First Four Standards of this
Framework. Second grade students measure the size of their
classroom and other places in a variety of ways to determine whether
dinosaurs they are studying would fit into them.
- As part of the morning calendar routine, second
graders check each of two thermometers - one Fahrenheit and one
Celsius - and make daily recordings of the outside temperature.
They record the temperatures in a chart and look for interesting
patterns. They notice that, as the school year progresses and the
temperatures change, whenever one of the temperatures goes up or down,
so does the other.
- Students regularly use both analog and digital
stopwatches to practice timing events that are usually measured in
seconds such as: the amount of time it takes a classmate to say the
alphabet, how long a classmate goes without blinking, or how long the
morning announcements take.
5. Use technology to gather, analyze, and
display mathematical data and information.
- Students take a survey to determine every
child's birth month and then use the Graph Club or
Primary Graphing and Probability Workshop software to display
the resulting data in graphs.
- Using a World Wide Web page that reports
meteorological data (possibly
http://www.rainorshine.com/weather/index/sites/njo/), students find
the predicted high temperatures for a variety of cities in different
regions around the country, write those numbers on a map of the United
States, and then look for patterns and trends in different
regions.
- Students use Table Top software to make a Venn
diagram to show which of them have brothers, which have sisters and
which have both (the intersection of the two sets). Students who have
no siblings are shown outside the rings. Other attributes of the
children are also used to make Venn diagrams.
References
Software
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Graph Club. Tom Snyder Productions.
Logo. Many versions of Logo are
commercially available.
Primary Graphing and Probability Workshop.
Scott Foresman.
TableTop. TERC.
MathKeys. Minnesota Educational Computing
Consortium (MECC).
On-Line Resources
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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 grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 5 - Tools and Technology - Grades 3-4
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.
In grades 3 and 4, manipulatives have traditionally not been
used as much as they have been in the primary grades. 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, the 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 at these grade levels as it is for younger
students. The content being modeled is, of course, different and so
the models are different - but no less important.
Third- and fourth-graders can use square tiles to model one-digit
multiplication arrays in a manner that makes the operation very
meaningful for them, and later use base-ten blocks to model two-digit
multiplication arrays. The added advantage to this kind of a model is
the degree to which students who have used it can visualize
what's happening with the factors in the problem and so can
develop much better estimation and mental math skills than students
who have simply learned the standard paper-and-pencil algorithms. The
relationship between multi-digit multiplication and division is also
clearly shown by such models.
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 should use geoboards to explore area
and perimeter and to begin to develop procedures for finding the areas
of irregular shapes. They can also use construction materials like
pipe cleaners and straws to make three-dimensional geometric shapes
like cubes and pyramids so that they can study them directly. Such
models make it much easier to determine the number of faces or edges
in a figure than two-dimensional drawings.
Third- and fourth-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 different colored
unifix cubes to represent all of the different double-decker ice cream
cones that can be made with three different flavors of ice cream.
They should be able to use a variety of measurement tools to measure
and record the data in a science experiment. They might use coin
tosses or dice throws to simulate real-world events that have a
one-in-two chance or a one-in-six chance of happening.
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 construct
mathematics that is meaningful to them.
There are several appropriate uses for calculators at these
grade levels. It is never too early for students to be introduced to
the tool that most of the adults around them will use whenever they
deal with mathematics.
The use of calculators at this level does not imply that students
don't need to develop arithmetic skills traditionally introduced
at the primary level. They certainly do need to develop these skills.
This Standard does not suggest that all traditional learning be
replaced by calculator use; rather, it calls for the appropriate and
effective use of calculators.
One of the most effective uses of the calculator with young
children which can be continued in grade three is the use of the
constant feature of most calculators to count, forward or backward, or
to skip count. This process allows children to anticipate what number
will come next and then get confirmation of their guess when they see
it come up in the display. Students can greatly enhance their
estimation ability through calculator use. Range-finding games
ask students, for instance, to add a number to 342 that will give them
an answer between 800 and 830. After the estimate is made, it is
punched into the calculator to see whether or not it did the job.
Calculators will also prompt students to be curious about
mathematical topics to which they are about to be introduced. For
example, while routinely using calculators in problem solving
activities, some students may notice that whenever they add, subtract,
or multiply two whole numbers, they get a whole number for an answer.
Sometimes that happens for division, too, but sometimes when they
divide they get an answer like 3.5. What does that mean?
These kinds of questions offer a great opportunity for some further
exploration and investigation; for example, Which problems give you
answers like those? What happens when you solve those problems
using pencil-and-paper?
Computers are a valuable tool for students in third and
fourth grade. As more and more computers find their way into these
classrooms, the software available for them will dramatically improve;
however, there are already many good programs that can be used with
students of this age. MathKeys links on-screen manipulative
materials to standard symbolic representations and to a writing tool
for children. Logo can be used by students to explore computer
programming and geometry concepts at the same time.
Tesselmania! and other programs offer an opportunity to
play with geometric transformations on the screen and produce striking
designs. The King's Rule is a program that asks
students to determine the rules that distinguish one set of numbers
from another, fostering creative and inductive thinking. The World
Wide Web can be an exciting and eye-opening tool for third-and
fourth-graders as they retrieve and share information. Specifically,
in these grades, they might look for state populations, meteorological
data, and updates on current events.
Standard 5-Tools and Technology-Grades 3-4
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 3-4 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 participate in races between some
students who use calculators and others who use mental math, each
working to complete a set of computation problems involving newly
learned arithmetic skills. They try to determine what makes the
calculator a useful tool in some circumstances (large numbers, harder
operations) and not terribly useful in others (basic facts, easy
numbers).
- Students work through the Tiling a Floor
lesson that is described in the First Four Standards of this
Framework. Third grade students test various shapes made of a
variety of materials to determine which can be used to tessellate an
area.
- Students choose to use a computer spreadsheet on
their classroom computer as a neat way to organize tables and charts,
but they also use a full-function word processor when there is a good
deal of text involved or when using different fonts and text
formatting.
- Students use base ten blocks rather than
popsicle sticks when performing operations with large numbers because
they can create models more efficiently and more quickly with
them.
2. Use physical objects and manipulatives to model
problem situations, and to develop and explain mathematical
concepts involving number, space, and data.
- Students use base ten blocks to demonstrate the
operations of multiplication and division with multi-digit numbers
using both repeated subtraction and partition methods.
- Students work through the Sharing Cookies
lesson that is described in the First Four Standards of this
Framework. Fourth grade students use manipulatives to
determine how to divide 8 cookies equally among 5
people.
- Students use a variety of devices such as dice, coin flips,
spinners, and decks of cards for generating random numbers and
understand the essential equivalence of these devices.
- Students use pipe cleaners and straws to build
and study three-dimensional objects, finding it easier to discuss
things like numbers of edges, faces, and vertices and the
relationships among them if they have a physical model with which to
work.
- Students use geoboards to solve Farmer
Brown's problem. She has 16 meters of fencing and wants to fence
in the largest rectangular area possible for her dog to romp around
in.
- Students use colored rods or pattern blocks to
develop early notions of fractions, using different rods or blocks as
the unit and discovering by trial-and-error the resulting fractional
values of all of the other pieces.
3. Use a variety of technologies to discover number
patterns, demonstrate number sense, and visualize geometric
objects and concepts.
- Students play the game target practice in
the New Jersey Calculator Handbook. In it, one student enters
a number into a calculator to be used as an operand, enters an
operation (addition, subtraction, multiplication, or division) into
the calculator by pressing the appropriate sign, and then specifies a
"target range" for the answer. For instance, the student
may enter: 82 x and specify the range as 2000-3000. A
second student must then enter a second operand into the calculator
and press the equals key. If the answer is within the specified
target range, the shot was a bull's eye.
- Students play The Biggest Product, also
from The New Jersey Calculator Handbook. In it, four
cards are dealt face up from a shuffled deck of cards containing only
the cards from ace to nine. The students who are playing then use
their calculators to try to compose the multiplication problem that
uses only the digits on the cards, each only once, that has the
largest possible product. After several rounds, the students begin to
notice a pattern in their answers and become much more efficient at
finding the correct problems.
- Students begin to use Logo to create geometric
figures on the computer screen. They write routines that have the
turtle's path describe a square, a rectangle, a triangle, and
other standard polygons. As a challenge, they write a routine to have
the turtle draw a simple house with windows and a roof.
- 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
specified tasks.
- After reading Counting on Frank by Rod
Clement, students practice their estimation skills by using software
of the same title.
4. Use a variety of tools to measure mathematical and
physical objects in the world around them.
- Students regularly use both analog and digital
stopwatches to practice timing events that happen in short time
periods such as: the amount of time it takes a classmate to recite
the Pledge of Allegiance or count to 60, how long a classmate takes to
run a 50 meter dash, or how long the morning announcements take. They
begin to record the elapsed time in decimals that include tenths or
hundredths of a second.
- Students first estimate and then use a metric
trundle wheel to measure long distances such as the distance from the
cafeteria doors to the sandbox, the distance from the classroom door
to the principal's office door, or the distance all the way
around the school on the sidewalk.
- Students read Counting on Frank by Rod Clement and
repeat some of the estimates made by the boy in the book. How many
peas would it take to fill up the room? How long a line can a
pen write? They make up their own silly things to estimate, and
devise ways to make the appropriate measures and estimates.
5. Use technology to gather, analyze, and display
mathematical data and information.
- Students use the New Jersey State homepage
http://www.state.nj.us on the World Wide Web to gather data about the
latest reported populations for each of the municipalities in their
county. They then enter the collected data into a simple spreadsheet
and use its graphing function to produce a bar graph of all of the
populations of the towns and cities. They highlight their own town to
show where it stands in relationship to the others.
- Students use the Graphing and Probability
Workshop or similar software to generate large amounts of random
data. This software simulates a variety of probability experiments
including up to 300 coin tosses, spinner spins, and dice rolls.
Discussions focus on whether the simulated outcomes were as expected
or were different from what was expected.
- 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 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."
References
-
Association of Mathematics Teachers of New Jersey.
The New Jersey Calculator Handbook. 1993.
Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth
Stevens Children's Books, 1991.
Software
-
Counting on Frank. EA Kids Software.
Graphing and Probability Workshop. Scott
Foresman.
Logical Journey of the Zoombinis. Broderbund.
Logo. Many versions of Logo are commercially
available.
MathKeys. Minnesota Educational Computing
Consortium (MECC).
Tesselmania! Minnesota Educational
Computing Consortium (MECC).
The King's Rule. Sunburst Communications.
On-Line 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 grade-specific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
Standard 5 - Tools and Technology - Grades 5-6
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.
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
proportion.
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
business dealing.
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
three times.
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
6.
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
display.
- 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
display.
- Students explore the rich source of problems at
the Math Forum World Wide Web site at Swarthmore College
(http://forum.swarthmore.edu).
2*. Use physical objects and
manipulatives to model problem situations, and to develop and
explain mathematical concepts involving number, space, and
data.
- 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
concepts.
- 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
image.
- 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
them.
- 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
information.
- 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
presentations.
- 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
school.
- 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 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
specified tasks.
- 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
necessary.
- 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
puzzles.
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. Association of
Mathematics Teachers of New Jersey, 1993.
Schwartz, D. How Much is a Million? New York: A
Mulberry Paperback Book, 1985.
Software
-
AppleWorks. Apple Computer Corp.
Elastic Lines. Sunburst Communications.
Geometry Workshop. Scott Foresman.
Graph Power. Ventura Educational
Systems.
Graphers. Sunburst Communications.
Graphing and Probability Workshop. Scott
Foresman.
How the West Was One + Three x Four.
Sunburst Communications.
HyperStudio. Roger Wagner.
Logical Journey of the Zoombinis.
Broderbund.
MacStat. Minnesota Educational Computing
Consortium (MECC).
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).
On-Line 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 grade-specific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
Standard 5 - Tools and Technology - Grades 7-8
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 eighth-graders 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.
Two-colored 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
three-dimensional figures convinces students of the resulting planar
shape that is the cross-section. Using pipe cleaners and straws,
students can build their own version of a Sierpinski tetrahedron.
Seventh- and eighth-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 spinners or dice
to simulate a variety of real-life 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 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 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 eighth-graders
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 1.06n. 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 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.
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 Pre-Supposer
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. Discovery-oriented 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 eye-opening tool for
seventh- and eighth-graders 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 7-8
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 7-8 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 two-colored 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 three-inch cubes of clay and then
experiment to see in how many different ways they can slice the cube
with a plane to produce different cross-sections. Drawings of the
cross-sections 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.st-and.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 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."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 extra-credit 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 x-y 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 x-yplane. 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
order-processing 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 Pre-Supposer. 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.
On-Line 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
grade-specific activities submitted by
New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
Standard 5 - Tools and Technology - Grades 9-12
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. Two-colored 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 real-life 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 Pre-Supposer
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. Discovery-oriented 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
three-dimensional 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 eye-opening tool for
ninth- through twelfth-graders 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 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:
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 well-equipped mathematics classroom: a bank of computers
loaded with algebraic symbol manipulation and function-plotting
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 re-writing binomial
expressions as multiplication problems (factoring).
- Students can use algebra tiles, Hands-On
Equations materials, and a variety of equation-manipulating
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
paper-and-pencil 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 paper-and-pencil 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 wedge-shaped 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 cross-sections. 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 30-foot, razor-sharp, 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
x-y 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 non-Euclidean 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 three-digit number into the
calculator. Without clearing the display, enter the same three
digit number again so that you have a six-digit 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.st-and.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 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."
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 four-game package. The data from the survey are displayed
below.
Price of the Four-Game 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 on-line 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 broken-line 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 open-topped box is made from a six-inch 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 Pre-Supposer. 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.
On-Line 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
grade-specific 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.
* 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.
|