STANDARD 9 - MEASUREMENT
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
Standard 9 - Measurement - Grades 9-12
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
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
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
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
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
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
- 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
- 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.
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.)
Froelich, G. Curriculum and Evaluation Standards for School
Mathematics: Addenda Series:
Connecting Mathematics. Reston, VA: National Council of
Teachers of Mathematics, 1991.
Powers of Ten. Philip Morrison, Phylis Morrison, and the
office of Charles and Roy Eames. New York: Scientific American
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