STANDARD 9  MEASUREMENT
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.

Standard 9  Measurement  Grades 78
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 onehalf 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
realworld settings.
Standard 9  Measurement  Grades 78
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 7 and
8.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 78 will be such that all students:
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 longterm assessment project, students make a
threedimensional 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 threedimensional
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 onethird 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.
OneLine 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 gradespecific activities submitted
by New Jersey teachers, and to provide a forum to discuss the
Mathematics Standards.
