STANDARD 9 - MEASUREMENT
K-12 Overview
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
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Descriptive Statement
Measurement helps describe our world using numbers. We use numbers
to describe simple things like length, weight, and temperature, but
also complex things such as pressure, speed, and brightness. An
understanding of how we attach numbers to those phenomena, familiarity
with common measurement units like inches, liters, and miles per hour,
and a practical knowledge of measurement tools and techniques are
critical for students' understanding of the world around
them.
Meaning and Importance
Measurement is important because it helps us to quantify the world
around us. Although it is perfectly natural to think about length,
area, volume, temperature, and weight as attributes of objects that we
measure, a little reflection will produce many other measurable
quantities: speed, loudness, pressure, and brightness, to name just a
few. An understanding of the processes of measurement, the concept of
a unit, and a familiarity with the tools and common units of
measurement, are all critical for students to develop an understanding
of the world around them.
This standard is also, in many ways, the prototypical
"integrated" standard because of its strong and essential
ties to almost every one of the other content standards. Measurement
is an ideal context for dealing with numbers and with numerical
operations of all sorts and at all levels. Fractions and decimals
appear very naturally in real-world measurement settings. Metric
measures provide perhaps the most useful real-world model of a
base-ten numeration system we can offer to children. Similarly
geometry and measurement are almost impossible to consider separately.
For instance, treatments of area and perimeter are called
"measurement" topics in some curricula and
"geometry" topics in others because they are, quite simply,
measurements of geometric figures. Another of the content standards
which is inextricably linked to measurement is estimation. Estimation
of measures should be a focus of any work that students do with
measurement. Indeed, the very concept that any continuous measurement
is inexact - that it is at best an "estimate" - is
a concept that must be developed throughout the grades.
Think about how many different content standards are incorporated
into one simple measurement experience for middle school students:
the measurement of a variety of circular objects in an attempt to
explore the relationship between the diameter and circumference of a
circle. Clearly involved are the measurement and
geometry of the situation itself, but also evident are
opportunities to deal with patterns in
the search for regularity of the relationship, estimation in
the context of error in the measurements, and number sense and
numerical operations in the meaning of the ratio that
ultimately emerges.
K-12 Development and Emphases
Throughout their study and use of measurement, students should be
confronted explicitly with the important concept of a measurement
unit. Its understanding demands the active involvement of the
learner; it is simply not possible to learn about measurement units
without measuring things. The process of measurement can be thought
of as matching or lining up a given unit, as many times
as possible, with the object being measured. For instance, in its
easiest form, think about lining up a series of popsicle sticks, end
to end, to see how many it takes to cover the width of the
teacher's desk; or consider how many pennies it takes to balance
the weight of a small box of crayons on a pan balance. At a slightly
more sophisticated level, multiple units and more standard units might
be used to add precision to the answers. The desk might be measured
with as many orange Cuisenaire Rods as will fit completely and then
with as many centimeter cubes as will fit in the space remaining; the
crayons, with as many ten-gram weights as can be used and then
one-gram weights to get a better estimate of its weight. These types
of activities - this active iteration of units - make the
act of measurement and the relative sizes of units
significantly more meaningful to children than simply reading a number
from a measurement instrument like a meter stick or a postal scale.
Of course, as the measures themselves become the focus of study,
rather than the act of measurement or the use of actual physical
units, students should become knowledgeable in the use of a variety of
instruments and processes to quickly and accurately determine
measurements.
Much research has dealt with the development of children's
understanding of measurement concepts, and the general agreement in
the findings points to a need for coherent sequencing of curriculum.
Young children start by learning to identify the attributes of objects
that are measurable and then progress to direct comparisons of
those attributes among a collection of objects. They would suggest,
for instance, that this stick is longer than that one or that
the apple is heavier than the orange. Once direct comparisons
can consistently be made, informal, non-standard units like pennies or
"my foot" can be used to quantify how heavy or how long an
object is. Following some experiences which illustrate the necessity
of being able to replicate the measurements regardless of the measurer
or the size of the measurer's foot, these non-standard units
quickly give way to standard, well-defined units like inches
and grams.
Older students should continue to develop their notions of
measurement by delving more deeply into the process itself and by
measuring more complex things. Dealing with various measurement
instruments, they should consider questions concerning the inexact
nature of their measures, and to adjust for, or account for, the
inherent measurement error in their answers. Issues of the
degree of precision should become more important in their
activities and discussion. They need to appreciate that no matter how
accurately they measure, more precision is always possible with
smaller units and better instrumentation. Decisions about what level
of precision is necessary for a given task should be discussed and
made before the task is begun, and revised as the task unfolds.
Older students should also begin to develop procedures and formulas
for determining the measures of attributes like area and volume that
are not easily measured directly, and to develop indirect
measurement techniques such as the use of similar triangles
to determine the height of a flagpole. Their universe of measurable
attributes should expand to include measures of a whole variety of
physical phenomena (sound, light, pressure) and a consideration of
rates as measures (pulse, speed, radioactivity).
Connections are another strong focus of students' work
with measurement. The growth of technology in schools opens up a wide
range of new possibilities for students of all ages. Inexpensive
instruments attached to appropriately programmed graphing calculators
and computers are capable of making and recording measurements of
temperature, distance, sound and light intensity, and many other
physical phenomena. The calculators and computers may then be used to
graph those measurements with respect to time or any other measure, to
present them in tabular form, or to manipulate them in other ways.
These opportunities for scientific data collection and analysis using
this technology are unlike any that have been available to mathematics
and science teachers in the past and hold great promise for real-life
investigations and for the integration of these two disciplines.
In summary, measurement offers us the challenge to
actively and physically involve students in their learning as well as
the opportunity to tie together seemingly diverse components of their
mathematics curriculum like fractions and geometry. It is also one of
the major vehicles by which we can bring the real worlds of other
disciplines such as the natural and social sciences, health, and
physical education into the mathematics classroom.
Note: Although each content
standard is discussed in a separate chapter, it is not the intention
that each be treated separately in the classroom. Indeed, as
noted in the Introduction to this Framework, an effective
curriculum is one that successfully integrates these areas to
present students with rich and meaningful cross-strand
experiences.
Standard 9 - Measurement - Grades K-2
Overview
Students can develop a strong understanding of measurement and
measurement systems from consistent experiences in classroom
activities where a variety of manipulatives and technology are used.
The key components of this understanding, as identified in the K12
Overview, are: the concept of a measurement unit; standard
measurement units; connections to other mathematical areas and to
other disciplines; indirect measurement; and, for older
students, measurement error and degree of
precision.
Students in the early grades encounter measurement in many
situations, from their daily work with the calendar, to situations in
stories that they are reading, to describing how quickly they are
growing. Hands-on science activities often require students to
measure objects or compare them directly. Daily calendar activities
frequently offer work with temperature, time, and money, in addition
to number. Thus, many opportunities for connections present
themselves in a natural way.
The study of measurement also encourages students to develop their
number sense and to practice their counting skills. By using
measures, students can recognize that numbers are often used to
describe and compare properties of physical objects. Students in the
early grades should make estimates not only of discrete objects like
marbles or seeds but also of continuous properties like the length of
a jumprope or the number of children's feet which might fit in a
dinosaur's footprint.
Students need to focus on identifying the property that they wish
to measure. They should understand what is meant by the length of an
object or its weight or its capacity. Concrete experiences in
describing the properties of objects, in sorting objects, and in
comparing and contrasting objects provide them with opportunities to
develop these concepts.
Students begin by making direct comparisons. Which string is
longer? Which child is taller? Which rock is heavier? Which
glass holds more? Making comparisons will help children better to
understand the properties which they are discussing. They also begin
to make some indirect measurements. For example, in order to
compare the height of the blackboard with the height of a window, they
might measure both objects using links and then compare the number of
links used for each. In this way, they start to see a need for a
measurement unit, a unit that they can use over and over to
compare to a variety of objects.
In grades K-2, students should use a variety of non-standard units
to measure objects. How many links long is a desk? How
many erasers high are you? How many pennies balance a Unifix cube?
In each case, students should first be asked to make an estimate
and then proceed to actually measure the object. Students should also
use different units to measure the same object. They should begin to
understand that when the size of a measuring unit increases, the
number of units needed to measure the object decreases.
In these grades, students also begin to use standard measurement
units and standard measurement devices such as rulers and scales.
It is important that the students see the use of the standard devices
as simply an extension of their earlier activities. For example, the
use of an inch ruler is just a more efficientprocedure than lining up
a series of cubic inch blocks. Students should explore length using
inches, feet, centimeters and meters; liquid capacity using quarts,
pints, cups, and liters; mass/weight using pounds, ounces, grams, and
kilograms; time using days, weeks, months, years, seconds, minutes,
and hours; and temperature using degrees Fahrenheit and Celsius.
Whether making direct comparisons, using non-standard units, or
using standard measurement units, students in the early grades should
always estimate a measure first and then perform the measurement. In
this way, their estimation and number sense skills will be
reinforced.
Standard 9 - Measurement - 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. Use and describe measures of length, distance,
capacity, weight, area, volume, time, and temperature.
- Students find out how many cubes long their hand is. The
class can then generate a graph showing the results.
- Using a large map of the school community, students
estimate and then use paper clips or links to measure who lives
farthest from school. This type of activity might be related to a
specific story that was used in the Social Studies unit on
community.
- Students name objects big enough to hold a football or too
small to hold a soccer ball.
- Students lay out a model zoo with several toy animals,
using boxes of different sizes for their cages or yards. They also
cut doors of appropriate sizes in the boxes for the animals.
- Students listen to and look at the book
Let's Find Out about What's Light and
What's Heavy by Martha and Charles Shapp. The
simple text and humorous illustrations lead to the conclusion that
weighing things using a standard unit of weight helps answer the
question in the title.
- Students name objects they can lift and ones that they
cannot lift.
- Students estimate and then use balances to find out how
many pennies balance a small familiar object.
- Students cut strips of paper to fit around a pumpkin or to
make Santa's belt.
- Students fill a large bottle with water using first a 4
ounce cup and then an 8 ounce cup. They then compare the results.
- Students make their own measuring jug using a large plastic
jar. They pour in one cupful of water and mark the water level on the
jar with a marker; they repeat this procedure with one cupful after
another until no further cupfuls will fit inside the jar.
- Students read The Little Gingerbread Man and make a
gingerbread village. In doing so, they measure lengths and
capacities.
- Students make their own paper clip ruler. First they make
a paper clip chain and then paste it down on a long cardboard strip.
They draw a small vertical line where each paper clip ends.
- Students estimate and measure the distance around an object
using Unifix cubes or a paper clip chain.
- Students conduct experiments using timers: How many
times can you bounce a ball before all the water runs out of
the can? How many times can you clap your hands before the
sand runs out of the timer? How many times can you blink your
eyes before the second hand goes all the way around the
clock?
- Students read or listen to The Very Hungry
Caterpillar by Eric Carle which shows the time of day at which
various activities occur.
- Students make a book describing their day at school. On
each page, they stamp a clock face and write underneath a time that
the teacher has written on the board. They then draw the hands on the
face to show the time. When the actual time of day on the classroom
clock matches a time in their book, students draw a picture of what
they are doing next to the correct clock face.
- Students line up Cuisenaire Rods in different combinations
to measure the width of a sheet of paper.
2. Compare and order objects according to some
measurable attribute.
- Kindergarten students listen to and look
at the book Big Friend, Little Friend by Eloise Greenfield. In
it a young boy and his two friends explore situations that clearly
demonstrate what it means to be big and what it means to be little.
As a nice followup assessment activity to this and other manipulative
activities, the teacher has the students draw pictures which
illustrate "big and little."
- Students compare the lengths of pencils to find out which
is longest. They arrange a set of pencils in order from longest to
shortest.
- Students use water, rice, or sand to fill different
objects, pouring from one object into another to find out which object
holds more. They explain how the shape of each object plays a role in
the amount it holds.
- Students line up in order, from tallest to shortest.
- Children make stick drawings of a family: father, mother,
school-aged child, and baby. They discuss which stick drawings are
taller or shorter than others, and relate these to the relative size
of the individuals in the family.
- Students collect a large variety of cardboard boxes and
arrange them in order from smallest to largest.
- Each group of students is given a cup and several
containers of different sizes, plain white paper, some uncooked rice,
and 1 inch graph paper. They find out how many cups of rice will fit
in each container and show the number of cups as a bar on the graph
paper under a picture of the container. After completing the graph,
they arrange the containers in order from largest to smallest.
- Students work through the Will a Dinosaur
Fit? lesson that is described in the First Four Standards of
this Framework. They arrange the dinosaurs in order from
tallest to smallest according to their height, and from longest to
shortest according to their length.
3. Recognize the need for a uniform unit of
measure.
- Students measure the width of their desks by counting how
many widths of their hands it would take to go from one end of the
desk to the other. They compare their results and discuss what would
happen to the number of hands if the teacher's hand were used
instead.
- Students read and discuss How Big Is a Foot? by
Rolf Myllar. The king wishes to give the queen a special bed for her
birthday and measures the size using his own foot. He gives the
measurements to the carpenter, who gives them to the little
apprentice. The bed that he makes is too small, but the apprentice
solves the problem and everyone lives happily ever after. The
students use their own feet to measure the width or length of the
hallway and compare their results. Finally, they measure the hallway
using meter sticks.
- As an assessment of the students'
understanding of units, the teacher has the students measure the
length of their math book using paper clips, unifix cubes, and yellow
Cuisenaire Rods. They write about their results and explain why they
are different.
4. Develop and use personal referents for standard
units of measure (such as the width of a finger to approximate
a centimeter).
- Students identify parts of their body that are the same
length as the unit cube from a base tens block set (1 centimeter).
- Students make a list of foods or drinks that come in quarts
and others that come in liters.
- Students find out that ten pennies weigh about an
ounce.
- Students find that first-graders are a little taller or a
little shorter than a meter.
5. Select and use appropriate standard and
non-standard units of measurement to solve real-life
problems.
- Students decide whether they should use paper clips or
pennies to measure the weight of a pencil.
- Students discuss whether they should use links or meter
sticks to measure the length of the gym.
- Students write about how they might measure the distance
from the cafeteria to their classroom.
6. Understand and incorporate estimation and
repeated measures in measurement activities.
- Students estimate how many of their shoes will fit in a
giant's footprint (left conveniently on the classroom
blackboard!) and write their estimates. They trace around their shoes
and cut out the tracings. After the teacher has pasted a few shoes
onto the giant's footprint, the students revise their estimate.
They then check the accuracy of their estimates by pasting as many
shoes as will fit into the footprint.
- Students estimate the weight of various objects in beans
and then use a balance scale to check the accuracy of their
measurements.
References
-
Carle, Eric. The Very Hungry Caterpillar. New York:
Philomel Books, 1987.
Greenfield, Eloise. Big Friend, Little Friend. New York:
Black Butterfly Children's Books, 1991.
Myllar, Rolf. How Big is a Foot? New York: Dell
Publishing, 1962.
Shapp, Martha, and Charles Shapp. Let's Find Out
about What's Light and What's Heavy. New
York: Franklin Watts, 1975.
General reference
-
Burton, G., and T. Coburn. Curriculum and Evaluation
Standards for School Mathematics:
Addenda Series: Kindergarten Book. Reston, VA: National
Council of Teachers of Mathematics, 1991.
One-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 9 - Measurement - Grades 3-4
Overview
Students can develop a strong understanding of measurement and
measurement systems from consistent experiences in classroom
activities where a variety of manipulatives and technology are used.
The key components of this understanding, as identified in the K12
Overview, are: the concept of a measurement unit; standard
measurement units; connections to other mathematical areas and to
other disciplines; indirect measurement; and, for older
students, measurement error and degree of
precision.
Students in grades 3 and 4 continue to encounter measurement
situations in their daily lives and in their schoolwork. They
investigate how much weight different structures will support or make
a model of the solar system in science class, they make maps in social
studies, and they read and discuss stories in which people measure
objects. Measurement continues to provide opportunities for making
mathematical connections among subject areas.
Measurement also helps students make connections within
mathematics. For example, as students begin to develop their
understanding of fraction concepts, they extend their understanding of
measurement to include fractions of units as well. Measurement is
interwoven with developing understanding of the geometric concepts of
perimeter, area, and volume. Furthermore, students develop their
estimation skills as they develop their understanding of
measurement.
Students also continue to learn about more attributes of objects
that can be measured. In addition to length, distance, capacity,
weight, area, volume, time, and temperature, they now are able to
discuss the size of angles and the speed of a car or a bike. Students
begin to make more indirect measurements as well. For example,
they will measure a desk to find out whether it will fit through a
door, or measure how far a toy car goes in a minute and divide to find
its speed in feet per second.
The emphasis in these grades is on building on the students'
earlier experience with non-standard units and their developing
concept of measurement unit to the use of more sophisticated
standard units of measurement. They solidify their
understanding of the basic units introduced in the earlier grades and
begin to use fractional units. Students use half-inches,
quarter-inches, and eighths of an inch, for example, in measuring the
lengths of objects. Students also begin to use some of the larger
units: miles, kilometers, and tons.
Some students may begin to discover formulas to help count units.
For example, students may use shortcuts to find out how many squares
cover a rectangle, multiplying the number of rows times the number of
squares in each row. Or they may find the distance around an object
by measuring each side and then adding the measures.
In summary, in grades 3 and 4, it is important that all students
get extensive hands-on experience with measuring the properties of a
variety of physical objects. They will learn to measure by actually
doing so with an appropriate measuring instrument.
Standard 9 - Measurement - 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. Use and describe measures of length, distance,
capacity, weight, area, volume, time, and temperature.
- Students find out how many inches long their hand is. The
class then generates a graph showing the results.
- Students use rulers to measure the length of the
room in feet and inches and then in metric units.
- Students move thermometers to different parts of the
school, recording the temperature at each location. For example, it
may be hot in the cafeteria and cold in the gym. They learn to
identify appropriate reference points on both Celsius and Fahrenheit
scales (e.g., 30 degrees Centigrade is a hot day).
- Students investigate truth-in-packaging by reading labels,
estimating weights, and then using balances to weigh foods.
- Students investigate how many cups in a pint, how many
pints in a quart, and how many quarts in a gallon by making lemonade
and filling various sizes of containers.
- Students make their own rulers, marking off intervals equal
in length to one centimeter.
- Students estimate and measure the distance around an object
using a length of string which they then measure with centimeter
cubes.
- Students conduct experiments using timers: how many
times can you bounce a ball, clap your hands, or blink your
eyes in one minute? They discuss how many times each would occur
in 10 minutes, in an hour, or in a day, if they continued at the same
rate, and why their answers might be different.
- Students measure all sorts of performances in
their physical education class: the time it takes to run 100 meters,
the length of a long jump in inches, and the length of a softball
throw in meters.
- Students read Time to . . . by Bruce
McMillan. In this book about a farm boy and his daily activities,
clock faces are always there to remind the reader what time it
is.
- Students use calculators to help them find out how many
days old they are.
- When going on a field trip, students determine how much
time they will have available at a museum by considering when they
will arrive and when they must leave.
- Students use cubes to fill rectangular boxes of various
sizes as they explore the concept of volume.
2. Compare and order objects according to some measurable
attribute.
- Students compare the areas of different leaves and order
them from smallest to largest. They use a variety of strategies; some
students cover the leaves with centimeter cubes, others make a copy of
the leaf on grid paper, and still others just "eyeball" it.
They discuss the different strategies used, comparing their advantages
and disadvantages.
- Students bring in a variety of cereal boxes from home and
estimate their order from smallest volume to largest volume. They
then check their accuracy by filling the boxes with cubic inch blocks,
with cubic centimeter blocks, and with sand, and discuss the reasons
for the differences in their results.
- Students build bridges using straws and pipe cleaners,
estimate how many round metal washers their bridge will hold, and then
place the washers on their bridge until it buckles or breaks. They
compare different types of bridges to determine what type is
strongest.
- Students estimate and then weigh objects, putting them in
order from heaviest to lightest.
3. Recognize the need for a uniform unit of
measure.
- Students measure the length of their classroom using their
paces and compare their results. They discuss what would happen if the
teacher measured the room with his or her pace.
- Students read and study the illustrations in the
book Long, Short, High, Low, Thin, Wide by James Fey. There
are many activities in this attractive book that take the students
through an historical account of the development of standard
units.
4. Develop and use personal referents for standard units
of measure (such as the width of a finger to approximate a
centimeter).
- Students identify parts of their body that are the same
length as ten centimeters and use them to measure the length of their
pencil.
- Students find things in their environment that weigh about
an ounce.
- Students use a meter stick to identify a personal
referent which is approximately a meter. For example, for one child it
may be an armspan, for another it might be the distance between a
kneecap and the top of the head.
- Students measure the length of their pace in inches and use
that information, along with a measurement of the length of the room
in paces, to find the length of the room in inches.
5. Select and use appropriate standard and
non-standard units of measurement to solve real-life
problems.
- Students decide what units they should use to measure the
weight of a textbook.
- Students discuss what units they should use to measure the
length of the hallway outside their classroom.
- Students write about how they might measure the distance
from the cafeteria to their classroom or the area of the gym.
- A nice approach to assessment of students'
skills with this topic is to make a list of items that they can
measure, such as the length of a piece of notebook paper, the weight
of a teacher, the amount of water a bucket can hold, and the distance
between Trenton and Newark, and ask them to name a measurement unit
with which it would be appropriate to measure the given item. They
discuss their choice of unit, estimate the measure of each item, and
then actually measure it and compare it to their estimate and to the
results of other students.
6. Understand and incorporate estimation and
repeated measures in measurement activities.
- Students read and laugh
about the pictures in Counting on Frank by Rod Clement. Frank
is a dog whose young owner, challenged by his father to use his brain,
estimates and imagines all sorts of measurements leading to some
pretty silly situations. Possible extensions are plentiful and easy
to devise. As an openended assessment followup to the story, the
teacher asks each of the students to make a
"CountingonFranklike" estimate of how many of something
(television, little sister, car, dog) would fit into their bedroom and
to draw a picture showing all of them there.
- Students estimate the weight of various objects in grams
and then use a balance scale to check the accuracy of their
measurements.
- Students estimate the weight of, and then weigh the
contents of a box of animal crackers, graphing their results and
comparing them to the weights indicated on the packages.
References
-
Clement, Rod. Counting on Frank. Milwaukee, WI: Gareth
Stevens Publishing, 1991.
Fey, James. Long, Short, High, Low, Thin, Wide. New
York, NY: Thomas Y. Crowell Publishers, 1971.
McMillan, Bruce. Time to . . . Lothrop, Lee and
Shepard, 1986.
General reference
-
Burton, G., et al. Curriculum and Evaluation Standards for
School Mathematics: Addenda Series:
ThirdGrade Book. Reston, VA: National Council of
Teachers of Mathematics, 1992.
One-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 9 - Measurement - Grades 5-6
Overview
Students can develop a strong understanding of measurement and
measurement systems from consistent experiences in classroom
activities where a variety of manipulatives and technology are used.
The key components of this understanding, as identified in the K12
Overview, are: the concept of a measurement unit; standard
measurement units; connections to other mathematical areas and to
other disciplines; indirect measurement; and, for older
students, measurement error and degree of
precision.
Why teach measurement? The ability to measure enables
students to connect mathematics to the environment and offers
opportunities for interdisciplinary learning in social studies,
geography, science, music, art, and other disciplines. In addition,
measurement tools and skills have a variety of uses in everyday adult
life.
However, in the most recent international assessment of mathematics
achievement, 13-year-olds in the United States performed very poorly
in comparison to other nations. The results of this study indicated
that, while students are given instruction on measurement, they do not
learn the concepts well. For example, some students have difficulty
recognizing two fundamental ideas of measurement, the concept of a
unit and the iteration of units. A common error is counting number
marks on a ruler rather than counting the intervals between the marks.
Another difficult concept is that the size of the unit and the number
of units needed to measure an object are inversely related; as one
increases the other decreases. In the fifth and sixth grades, students
begin to encounter both very small and very large standard
measurement units (such as milligrams or tons), and these
ideas become increasingly critical to understanding measurement.
Students must be involved in the act of measurement; they must have
opportunities to use measurement skills to solve real problems if they
are to develop understanding. Textbooks by themselves can only
provide symbolic activities. Middle grade teachers must take
responsibility for furnishing hands-on opportunities that reinforce
measurement concepts with all common measures.
Using measurement formulas as a more efficient approach to some
types of direct measurement is an important part of fifth and sixth
grade mathematics. It represents the first formal introduction to
indirect measurement. Multiplying the length by the
width of a rectangle is certainly an easier way to find its area than
laying out square units to cover its surface. The formulas should
develop, however, as a result of the students' exploration and
discovery; they should be seen as efficient ways to count iterated
units. Having students memorize formulas that, for them, have no
relation to reality or past direct measurement experiences will be
unsuccessful. Fewer than half of the U.S. seventh grade students
tested in the international competition could figure out the area of a
rectangle drawn on a sheet of graph paper, and only slightly more than
half could compute the area given the dimensions of length and height.
Often length and width are taught separately and how the two
measurements combine to form the square units of area is not
emphasized in instruction. In addition, area and perimeter are often
confused with each other by middle grade students. Limiting
students' experiences with measurement to the printed pages of
textbooks restricts flexibility so that their understanding cannot be
developed or generalized.
In order to further strengthen students' understanding of
measurement concepts, it is important to provide connections of
measurement to other ideas in mathematics and to other areas of
learning. Students should measure objects, represent the information
gathered visually (e.g., in a graph), model the situation with symbols
(e.g., with formulas), and apply what they have learned to real-world
events. For example, they might collect information about waste in
the school lunchroom and present their results to the principal with
suggestions for reducing waste. Integrating across mathematical
topics helps to organize instruction and generates useful ideas for
teaching the important content of measurement.
In summary, measurement activities should require a dynamic
interaction between students and their environment, as students
encounter measurement outside of school as well as inside school.
Students should use each measuring instrument until its use becomes
second nature. The curriculum should focus on the development of
understanding of measurement rather than on the rote memorization of
formulas. This approach can be reinforced by teaching students to
estimate and to be aware of the context whenever they make an
estimate. For example, when buying carpeting it is advisable that the
estimate be too high rather than too low. Students must be given the
opportunity to extend their learning to new situations and new
applications.
Standard 9 - Measurement - 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:
7. Use estimated and actual measurements to
describe and compare phenomena.
- Students estimate the number of square centimeters in a
triangle. Then they enclose the triangle in a rectangle and use
centimeter cubes or a transparent square centimeter grid to find the
area of the rectangle. They also count squares to find the area of
the triangle and that of any other triangles formed by the rectangle.
They look for a pattern in their results and compare their results to
their estimates.
- Students explain why the following is or is not reasonable:
An average person can run one kilometer in one minute.
- Students measure how long it takes to go 10 meters, first
using "baby steps," then using normal steps, and finally
using "giant steps." They then compare their rates.
- Students measure the area of their foot by tracing around
it on centimeter graph paper and counting the number of squares
covered. To ease the counting task, students can color the squares
completely inside the outline blue, those that are onehalf inside
green, those that are onethird inside yellow, and those that are
onefourth inside orange. Then all of the likecolored squares can be
counted more easily and the various totals added to each other.
- An interesting openended group assessment project
to use after the previous foottracing activity has been completed is
to tell the students that Will Perdue (of the Chicago Bulls) wears a
size 18 « shoe which measures 21 « inches long. Students
are asked to use what they know about the areas of their own feet to
estimate the area of his foot. Students who make a good estimate will
deal with several issues: the fact that they have information about
the length of their own feet, but only about the length of Will's
shoe; the fact that as the foot gets longer, it also gets wider; and
the issue of how to set up a proportion between appropriate
quantities.
8. Read and interpret various scales, including
those based on number lines and maps.
- Students use a given scale to compute the actual length of
a variety of illustrated dinosaurs.
- Students make a scale drawing of their classroom and use
two-dimensional scale models of its furniture in order to propose new
ways of arranging the classroom so that they can work more efficiently
in cooperative groups.
- Students use a map to find the distance between two
cities.
- Students work through the Short-Circuiting
Trenton lesson that is described in the Introduction to this
Framework. Using a map of Trenton and a ruler, students
determine the distances between various sites, and then find the most
efficient walking tour for their class trip.
9. Determine the degree of accuracy needed in a
given situation and choose units accordingly.
- Students plan a vegetable garden, determining the unit of
measure appropriate for the garden, estimating its size, and then
computing the perimeter (for fencing) and area (for fertilizer).
- Groups of students use a scale drawing of an apartment (1
cm = 1 foot) to find out how many square yards of carpeting are needed
for the rectangular (9' x 12') living room and other rooms.
- Statements work through the Mathematics at
Work lesson that is described in the Introduction to this
Framework. A parent discusses a problem which her company
faces regularly: to determine how large an air conditioner will be
needed for a particular room. To solve this problem, the company has
to estimate the size of the room, in terms of its volume and the areas
of any windows, after determining the appropriate units.
10. Understand that all measurements of continuous
quantities are approximate.
- Students measure a specific distance in the room and
compare their results, focusing on the idea that any measurement is
approximate.
- Before a cotton ball toss competition, students discuss
what units should be used to measure the tosses. They decide that
measuring to the nearest centimeter should be close enough, even
though the actual tosses will probably be slightly more or slightly
less.
11. Develop formulas and procedures for solving
problems related to measurement.
- Students complete a worksheet showing several rectangles on
grids that are partially obscured by inkblots. In order to find the
area of each rectangle, they must use a systematic procedure involving
multiplying the length of the rectangle by its width.
- Students develop the formula for finding the volume of a
rectangular prism by constructing and filling boxes of various sizes
with centimeter or inch cubes and looking for patterns in their
results. As a journal entry, they describe the "shortcut"
way to find the volume.
12. Explore situations involving quantities
which cannot be measured directly or conveniently.
- Students work in groups to estimate the number of bricks
needed to build the school building. They explain their results in a
class presentation, describing the strategies they used.
- Students are asked to estimate how many heads tall they
are. Then they work in groups to develop a procedure for finding out
how many heads tall each student is.
- Students construct a measuring tool that they can use to
find the height of trees, flagpoles, and buildings when they are
standing a fixed distance from the object to be measured.
13. Convert measurement units from one form to
another, and carry out calculations that involve various units
of measurement.
- Students are asked to find how many pumpkin seeds there are
in a kilogram. They decide to measure how much 50 seeds weigh and use
this result to help them find the answer.
- Students use approximate "rules of thumb" to help
them convert units. For example:
- 1 km is about 6/10 of a mile
- 1 liter is a little bigger than a quart
- 1 meter is a little bigger than a yard
- 1 kg is about 2 pounds
- 20 degrees Centigrade is about 70 degrees Fahrenheit (room temperature)
- 1000 ml of water normally weighs about 1 kg
14. Understand and apply measurement in their own lives and
in other subject areas.
- Students measure the heights of bean plants at regular
intervals under different conditions. Some are in sunlight and some
are not. The students discuss their results and make a graph of their
findings.
- Students estimate and weigh cups filled with jellybeans,
raisins, dried beans, peanuts, and sand to find out that equal volumes
of different objects do not always weigh the same.
- Students learn how much water is in different foods by
first trimming pieces of 5 different foods to a standard 15 grams,
then measuring their weights again the next day. Where did
the water go?
- Students estimate what fraction of an orange is edible,
then weigh oranges, peel them and separate the edible parts. They
weigh the edible part and then compute what fraction is actually
edible and compare that fraction to their estimate.
- Students create their own food
recipes.
- An ice cube is placed on a plastic tray in five
different parts of the classroom. One group of students is assigned
to each ice cube and tray. The students are asked to estimate how
long it will take each ice cube to melt. They then observe the ice
cube at five-minute intervals, recording their observations. After
the ice cubes have melted, the groups share their observations and
compare the length of time it took for the ice cubes to melt. They
make a conjecture about the warmest spots in the classroom and then
measure the temperature in each location to confirm their
conjecture.
- Students read Anno's Sundial
by Mitsumasa Anno. This sophisticated threedimensional popup book
presents an extraordinary amount of information about the movement of
the earth and the sun, the relationship between those movements, and
how people began to tell time. It is an ideal kickoff for an
integrated, multidisciplinary unit with upperelementary students,
incorporating reading, social studies, science, and mathematics.
- Students estimate and then develop a plan to find out how
many pieces of popped popcorn will fit in their locker.
- Students work in pairs to design a birdhouse that can be
made from a single sheet of wood (posterboard) that is 22" x
28". The students use butcher paper to lay out their plans so
that the birdhouse is as large as possible. Each pair of students
must show how the pieces can be laid out on the posterboard before
cutting.
- Students compare the measurements of an object to those of
its shadow on a wall as the distance between the object and the wall
increases.
15. Understand and explain the impact of the change of
an object's linear dimensions on its perimeter,
area, or volume.
- Students use pattern blocks to see how the area of a square
changes when the length of its side is doubled. They repeat the
experiment using equilateral triangles.
- Students use cubes to explore how the volume of a cube
changes when the length of one side is doubled, then when the lengths
of two sides are doubled, and, finally, when the lengths of all three
sides are doubled.
- Students use graph paper to draw as many rectangles as they
can that have a perimeter of 16 units. They find the area of each
rectangle, look for patterns, and summarize their results.
16. Apply their knowledge of measurement to the
construction of a variety of two- and three-dimensional
figures.
- Students use paper fasteners and tagboard strips
with a hole punched in each to investigate the rigidity of various
polygon shapes. For shapes that are not rigid, they determine how
they can be made so.
- Students design and carry out an experiment to
see how much water is wasted by a leaky faucet in an hour, a day, a
week, a month, a year.
- Students use straw and string to construct models
of the two simplest regular polyhedra, the cube and the
tetrahedron.
References
-
Anno, Mitsumasa. Anno's Sundial. New York:
Philomel Books, 1987.
General references
-
Geddes, D. Curriculum and Evaluation Standards for School
Mathematics: Addenda Series:
Measurement in the Middle Grades. Reston, VA: National
Council of Teachers of Mathematics, 1991.
Bright, G. W., and K. Hoeffner. "Measurement, Probability,
Statistics, and Graphing," in D. T. Owens, Ed. Research
Ideas for the Classroom: Middle Grades Mathematics. New York:
Macmillan Publishing Company, 1993.
LaPoint, A. E., N. A. Mead, and G. W. Phillips. A World of
Differences: An International
Assessment of Mathematics and Science. Princeton, NJ:
Educational Testing Service, 1989.
One-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 9 - Measurement - Grades 7-8
Overview
Students can develop a strong understanding of measurement and
measurement systems from consistent experiences in classroom
activities where a variety of manipulatives and technology are used.
The key components of this understanding, as identified in the K12
Overview, are: the concept of a measurement unit; standard
measurement units; connections to other mathematical areas and to
other disciplines; indirect measurement; and, for older
students, measurement error and degree of
precision.
In grades seven and eight, students begin to look at the
measurement process more abstractly while continuing to develop their
actual measurement skills and using measurement in connection with
other subjects and other topics in mathematics.
All measurement activities should involve both estimation and
actual measurement at these grade levels. Estimation strategies should
include (1) having a model or referent (e.g., a doorknob is about one
meter from the floor), (2) breaking an object to be estimated into
parts that are easier to measure (chunking), and (3) dividing
the object up into a number of equal parts (unitizing).
Students should also discuss when an estimate is appropriate and when
an actual measurement is needed and they should have opportunities to
select appropriate measuring tools and units.
Especially in the context of making measurements in connection
with other disciplines, the approximate nature of measure is an
aspect of number that needs particular attention. Because of
students' prior experience with counting and with using numerical
operations to obtain exact answers, it is often difficult for them to
develop the concept of the approximate nature of measuring. Only
after considerable experience do they recognize that when they
correctly measure to the nearest "unit," the maximum
possible error would be one-half of that unit. Teachers must help
students to understand that the error of a measurement
is not a mistake but rather a result of the limitations of the
measuring device being used. Only through measurement activities can
students discover and discuss how certain acts, such as the selection
and use of measuring tools, can affect the degree of precision
and accuracy of their measurements.
Students in grades seven and eight should expand their
understanding of measurement to include new types of measures,
especially those involving indirect measurement. For example,
they learn about density and force and how these characteristics are
measured in science class. Middle school students also should
develop a deeper understanding of the concept of rate, by experiencing
and discussing different rates. Constructing scale drawings and scale
models or relating biological growth and form provide excellent
opportunities for students to use proportions to solve problems, as
does using a variety of measuring tools to find the measures of
inaccessible objects. Such personal experiences help students
recognize and appreciate the use of measurement concepts in other
real-world settings.
Standard 9 - Measurement - 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:
7. Use estimated and actual measurements to
describe and compare phenomena.
- Students estimate the number of square centimeters in a
trapezoid. Then they use a transparent grid and count squares to find
the area. They compare that result to the area of a rectangle whose
base is the average of the two bases of the trapezoid and whose height
is the same as that of the trapezoid. They look for a pattern in
their results and compare their results to their estimates.
- Students build a bridge out of paper to connect two bricks
and place weights on the bridge until it breaks, noting how much
weight it held.
- Students read and discuss sections of This
Book Is About Time by Marilyn Burns. Its many engaging activities
and experiments are interspersed with an historical treatment of time
and the instruments designed to measure it.
8. Read and interpret various scales, including those
based on number lines and maps.
- Students use objects shown in a movie poster for King
Kong to determine how tall the ape is.
- As a long-term assessment project, students make a
three-dimensional scale model of their classroom.
- Students use a map to plan an auto trip across the United
States, finding the distance traveled each day and the amount of time
required to drive each day's route.
9. Determine the degree of accuracy needed in a
given situation and choose units accordingly.
- Students plan a school garden, determining the unit of
measure appropriate for the garden, estimating its size, and then
computing the perimeter (for fencing) and area (for fertilizer).
- Groups of students make and use a scale drawing (1/4 inch =
1 foot) of an apartment and use scale models of the furniture to
furnish the living room and dining room.
- Students make a floor plan for a small restaurant
furnished with round tables.
10. Understand that all measurements of continuous
quantities are approximate.
- Students measure a given hallway in school and compare
their results, noting that their results are different because any
measurement is approximate. They discuss how accuratetheir individual
measurements are (degree of precision) and, after reviewing all of
their measurements, determine the likely errors in their individual
measurements. They also discuss how more precise measures may be
obtained and what degree of precision is needed in this situation.
- Each student in a group measures the circumferences and
diameters of several round objects using a tape measure or ruler and
string. They compare their measurements and decide what the most
accurate set of measurements is for each of the objects. They use a
calculator to find the ratio of the circumference to the diameter for
each object.
11. Develop formulas and procedures for solving
problems related to measurement.
- Students develop a formula for finding the surface area of
a rectangular prism by constructing boxes of various sizes using graph
paper, finding the area of each side and adding them, and looking for
patterns in their results. They describe their findings in their
journals.
- Students construct different parallelograms whose base and
height have the same length on their geoboards. They sketch each
parallelogram and record its area (found by counting squares). They
discuss their results.
- Students work through the Sketching
Similarities lesson that is described in the First Four Standards
of this Framework. They use a computer program and measure the
sizes of the corresponding sides and corresponding angles of similar
figures. They conclude that similar figures have equal corresponding
angles and their corresponding sides have the same ratio.
- Students use plastic models of a pyramid and a
prism, each having the same height and polygonal base, to investigate
the relationship between their volumes.
12. Explore situations involving quantities which
cannot be measured directly or conveniently.
- Students and parents in the McKinley School have been
dropping pennies into the very large plastic cylinder in the school
lobby in an effort to raise money for new playground equipment. The
students are challenged to devise a method to estimate how much money
is in the cylinder as a function of the height of the pennies at any
given time.
- Students construct a measuring tool that they can use to
find the height of trees, flagpoles, and buildings, using cardboard,
graph paper, straws, string, and washers.
- Students use proportions to find the height of the flagpole
in front of the school.
13. Convert measurement units from one form to
another, and carry out calculations that involve various units
of measurement.
- Students are given a ring and asked to find the height of
the person who lost the ring. They measure their own fingers and their
heights, plotting the data on a coordinate graph. They use a piece of
spaghetti to fit a straight line to the plotted points and make a
prediction about the height of the person who owns the ring, based on
the data they have collected.
- Students plan the weird name Olympics by renaming
standard events in different measurement units. For example,
"the hundred meter dash" becomes the "100,000
millimeter marathon" and the "ten meter dive" could
become the "onehundredth of a kilometer splash."
- Students continue to use approximate "rules of
thumb" learned in earlier grades to help them convert units. For
example:
- 1 km is about 6/10 of a mile
- 1 liter is a little bigger than a quart
- 1 meter is a little bigger than a yard
- 1 kg is about 2 pounds
- 1 inch is about 2.5 centimeters
- 20 degrees Centigrade is about 70 degrees Fahrenheit (room temperature)
- 1000 ml of water normally weighs about 1 kg
14. Understand and apply measurement in their own
lives and in other subject areas.
- Students design and carry out an experiment to
see how much water is wasted by a leaky faucet in an hour, a day, a
week, a month, and a year.
- Students are told that when people get out of a bath, a
film of water about 0.05 cm thick clings to their skin. They are then
challenged to find what volume of water clings to the skin of an
average eighth grader. In order to make the estimate, of course, they
need to estimate the surface area of the body. They can do so by
considering a collection of cylinders and spheres that approximates a
human.
- Students investigate the concept of density by
finding objects for which they can find both the volume and the
weight, measuring both, and dividing the latter by the former.
Interesting objects to use include an orange, a block of wood, a
textbook, a rubber sponge ball, and an airfilled rubber ball.
Discussions of the results should lead to interesting conjectures
about density which can then be confirmed with additional
experimentation.
15. Understand and explain the impact of the change of
an object's linear dimensions on its perimeter,
area, or volume.
- Students use the computer program The Geometric
SuperSupposer to explore the relationship in similar triangles
between corresponding sides and the perimeters of the triangles. They
also analyze the relationship between corresponding sides and the
areas of the triangles.
- Students build a "staircase" using wooden cubes.
Then they double all of the dimensions and compare the number of cubes
used in the second staircase to the number used in the original
staircase.
- Students work through the Rod Dogs lesson
that is described in the First Four Standards of this
Framework. They discover that if an object is enlarged by a
scale factor, its new surface area is the scale factor squared times
the original area, and its new volume is the scale factor cubed times
the original volume.
16. Apply their knowledge of measurement to the
construction of a variety of two- and three-dimensional
figures.
- Students use straw and
string to construct models of the five regular polyhedra: the cube,
the tetrahedron, the octahedron, the icosahedron, and the
dodecahedron.
- Students use cardboard and tape to construct a
model that demonstrates that the volume of a pyramid is one-third that
of a prism with the same base and height.
- Students build scale models of the classroom, the
school, or a monument.
References
-
Burns, Marilyn. This Book is About Time. Boston, MA:
Little, Brown & Co., 1987.
Software
-
The Geometric SuperSupposer. Sunburst Communications.
General references
-
Geddes, D. Curriculum and Evaluation Standards for School
Mathematics: Addenda Series:
Measurement in the Middle Grades. Reston, VA: National
Council of Teachers of Mathematics, 1991.
Owens, D. T., Ed. Research Ideas for the Classroom: Middle
Grades Mathematics. New York, NY: MacMillan, 1993.
One-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 9 - Measurement - Grades 9-12
Overview
Students can develop a strong understanding of measurement and
measurement systems from consistent experiences in classroom
activities where a variety of manipulatives and technology are used.
The key components of this understanding, as identified in the K12
Overview, are: the concept of a measurement unit; standard
measurement units; connections to other mathematical areas and to
other disciplines; indirect measurement; and, for older
students, measurement error and degree of
precision.
Building upon the measurement skills and understandings developed
in grades K-8, high school students move to a more routine use of
measurement. They examine measurement as a more abstract process,
focusing on measurement error and degree of precision.
They spend much more time on indirect measurement
techniques than they did in earlier grades, expanding their
repertoire to include not only the use of proportions and similarity
but also the use of the Pythagorean Theorem and basic right triangle
trigonometric relationships.
Students at the high school level will frequently use measurement
to help develop connections to other mathematical concepts.
For example, students may use a computer program that measures angles
to help them discover the relationship between the measures of two
vertical angles formed by intersecting lines or the measures of
inscribed angles intercepting the same arc of a circle. They may also
develop algebraic techniques to help them find measures, as, for
example, when they develop a formula for finding the distance between
two points in the coordinate plane.
High school students also use measurement frequently in
connection with other subject areas. Science experiments require
a precise use of measurement. Social studies activities often require
students to read and interpret maps and/or scale drawings. In
technology classes, woodshop, drafting, sewing, and cooking, students
must also use a variety of measuring tools and techniques; and in
physical education, students frequently will need to measure distances
and rates.
Standard 9 - Measurement - 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:
17. Use techniques of algebra, geometry, and
trigonometry to measure quantities indirectly.
- Students use coordinate geometry techniques to determine
the distance between two points.
- Students use similar figures and proportions to measure the
height of a tree or a flagpole.
- Students use the Pythagorean Theorem to determine how long
a ladder is needed to climb a wall, including a determination of a
safe angle at which to place the ladder.
- Students use right-triangle trigonometry to measure the
width of a canyon or the height of a waterfall.
- Students work through the Ice Cones lesson
that is described in the First Four Standards of this
Framework. They use the formula for the volume of a cone and a
graphing calculator to determine the maximum volume of a cone made
from a paper circle of radius 10 cm which is cut along a
radius.
18. Use measurement appropriately in other subject areas
and career-based contexts.
- Students investigate how the volume of a cereal box changes
with its area by finding the volume and surface area of a box of their
favorite kind of cereal. They also discuss how the shape of the box
affects its volume and surface area and why the volume of the box is
so large for the amount of cereal it contains.
- As a question on their takehome final exam in Algebra,
students are asked to measure their own and gather data about the
length of other people's femurs and overall height in an effort
to determine whether there is a relationship between the two lengths.
They plot the resulting ordered pairs on a coordinate plane and find a
line of best fit for their data. For extra credit, they make a
prediction of the height of a male with a 22 inch femur.
- Students discuss the possible meaning of
"light year" as a unit of measure and find units equivalent
to it.
19. Choose appropriate techniques and tools to measure
quantities in order to achieve specified degrees of precision,
accuracy, and error (or tolerance) of measurements.
- Students use significant digits appropriately in measuring
large distances, such as the distance from one school to another, from
one city to another, and from one planet to another.
- Students find the distance between two cities by adding the
numbers given on a road map for the segments of the trip, by measuring
the segments and using the mileage scale, and by referring to a
published mileage table. They explain the different results by
referring to the degrees of prevision of the different
measurements.
- In making a scale drawing of a house, students discuss the
degree of accuracy of their measurements.
- Students read and discuss the photographs in
Powers of Ten by Phillip and Phylis Morrison and the office of
Charles and Ray Eames, and view the associated videotape. This
wellknown book takes the reader on a trip through perspectives
representing fortytwo powers of ten, from the broadest view of the
universe to the closest view of the nucleus of an atom. The
measurement units used and the progression from one to another
highlight the range and power of our system of
measurement.
- Students use computer drawing and measuring utilities to
discover geometric concepts. They also discuss the limitations of such
a program. For example, a program may give 14.7 for the length of the
base of a triangle and 7.3 for its midline (the segment joining the
midpoints of the other two sides); however, because of the
program's measurement limitations, its answer for the length of
the midline may not be exactly half the length of the base, as is the
case in reality.
- Students determine what kind of measuring instrument needs
to be used to measure ingredients for pain-relievers, for cough syrup,
for a cake, and for a stew. They bring to class a variety of empty
bottles and packages and note how the ingredients are measured: What
does 325 mg (of acetaminophen) mean, or one fluid ounce (of cough
syrup) as opposed to 1/4 cup of oil (for a cake). They discuss the
accuracy, error, and tolerance of each measurement.
References
-
Morrison, Philip, Phylis Morrison, and the office of Charles and
Ray Eames. Powers of Ten. Scientific American, 1988.
(Revised, 1994. See also reference under Video.)
General reference
-
Froelich, G. Curriculum and Evaluation Standards for School
Mathematics: Addenda Series:
Connecting Mathematics. Reston, VA: National Council of
Teachers of Mathematics, 1991.
Video
-
Powers of Ten. Philip Morrison, Phylis Morrison, and the
office of Charles and Roy Eames. New York: Scientific American
Library, 1991.
One-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.
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