STANDARD 9 - MEASUREMENT
All students will develop an understanding of and will use
measurement to describe and analyze phenomena.
Standard 9 - Measurement - Grades 5-6
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
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
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
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
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
- 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
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
- 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
- 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
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
- 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
- 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
- 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
- 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
- Students compare the measurements of an object to those of
its shadow on a wall as the distance between the object and the wall
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
- 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
Anno, Mitsumasa. Anno's Sundial. New York:
Philomel Books, 1987.
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.
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