Erin Murphy
INVESTIGATIONS
A Mathematical
Measurement Mystery
his department features children’s hands-on
and minds-on explorations in mathematics
and presents teachers with open-ended
investigations to enhance mathematics instruction. The tasks are designed to invoke problem
solving and reasoning, require communication
skills, and connect various mathematical concepts and principles. The ideas presented here
have been tested in classroom settings.
A mathematical investigation—
T
• has multidimensional content;
• is open-ended, with several acceptable solutions;
• is an exploration requiring a full period or
longer to complete;
• is centered on a theme or event; and
• is often embedded in a focus or driving question.
In addition, a mathematical investigation involves
processes that include—
•
•
•
•
researching outside sources;
collecting data;
collaborating with peers; and
using multiple strategies to reach conclusions.
Although this department presents a scripted
sequence and set of directions for an investigation, NCTM’s Standards (NCTM 2000) encourage teachers and students to explore multiple
approaches and representations while engaging
in mathematical activities. This month’s investigation explores the concepts of nonstandard and
Erin Murphy, [email protected], teaches first grade at Westlawn Elementary School in
Falls Church, Virginia. She began teaching at Westlawn through a PDS partnership with
George Mason University.
Edited by Patricia S. Moyer-Packenham, [email protected], George Mason University,
Fairfax, VA 22030. This section is designed for teachers who wish to give students new
insights into familiar topics in grades K–6. This material can be reproduced by classroom
teachers for use with their own students without requesting permission from the National
Council of Teachers of Mathematics. Readers are encouraged to send manuscripts appropriate for this section to “Investigations,” Teaching Children Mathematics, 1906 Association Dr.,
Reston, VA 20191-1502; or send electronic submissions to [email protected].
54
standard measurement, specifically length. The
purpose of this investigation is for children to
recognize the attributes of length, use nonstandard units to compare objects by length, discover
the need for standard units of measurement for
length, and represent lengths of objects in standard units of measurement. These objectives
encompass several expectations of the Measurement Standard for grades pre-K–2 (NCTM
2000).
The concept of measurement can be difficult for
primary-grade children to grasp. It is an area of
mathematics that requires understanding that various attributes of objects can be measured (such as
weight, length, or capacity), and numerous standard units of measure exist from which to choose
(such as grams, inches, or gallons). Measurement
easily lends itself to hands-on investigations
because students can explore measurement concepts by experimenting with objects and discussing
their findings.
In the following activities, students explore
the concept of length measurement using nonstandard units. These explorations enhance students’ conceptual knowledge of the processes
required to measure an object’s length and of
how using different units results in different measurements. During the investigations, students
work in pairs and small groups. This gives each
child the opportunity to measure the objects,
record the measurements, and reach conclusions
with their group. The lessons described in this
article can be taught over the course of one week,
allowing for connectivity and the development of
conceptual knowledge. Although these lessons
provide a rich introduction to the concepts of
measurement, teaching the lessons to students
who have a foundational understanding of these
concepts is also possible.
As a framework for the measurement activities,
the classroom teacher created a scenario in which
the elementary school had the opportunity to
receive some new furniture to replace the furniture
that was currently in the first-grade classroom and
Teaching Children Mathematics / September 2004
Copyright © 2004 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
Photograph by Patricia S. Moyer; all rights reserved
some new playground equipment to replace the
school’s existing equipment. The teacher placed
the students in charge of deciding whether the new
furniture and playground equipment would be the
right size to replace the existing classroom furniture and playground equipment. By creating this
real-life scenario, the teacher contextualized the
problem and made it a meaningful and motivating
student investigation.
Exploring the Concept of
Measurement Using
Nonstandard Units
The following are required materials:
•
•
•
•
•
•
•
Links
Unifix cubes
Popsicle sticks
Tens rods
Colored counting bears
Student Recording Sheet #1
Large chart-sized record sheet for class results
Teaching Children Mathematics / September 2004
• Measurement Words Chart for recording important vocabulary
The teacher began the investigation by presenting the students with the following scenario
to spark enthusiasm in the problem-solving
process:
The school needs our help with a measurement problem! The school has the opportunity to get some free furniture and playground equipment from another school that
has closed. The principal is not sure if the
furniture and equipment will be the right size
to replace the things we have in our classroom and on the playground. Some of the
items that the other school has to give away
are on this list. We need to help the principal
figure out if the new items are the same size
as the items we already have and if they will
fit in our classroom and on our playground.
We need to know if there is room for everything or not.
55
Figure 1
Photograph by Patricia S. Moyer; all rights reserved
Students use counting bears to measure the TV.
The first graders were very excited about this
dilemma. The teacher asked the students to work
together in pairs and think about what they could
do to help the principal with this mathematical
problem. She asked them to decide how they
would know if the new furniture could replace the
old furniture and if some items would fit in the
classroom and some items would not. The students sat in pairs and shared ideas for completing
the task.
When the students returned to the whole
group, many of them made suggestions that
involved moving the furniture and trying to fit the
new items into the classroom by trial and error.
Some groups suggested measuring the old items
before bringing the new ones into the classroom.
The teacher then asked the students what they
already knew about measuring. Activating prior
knowledge provides insight into what the students
already know and what misconceptions they bring
to the lesson. Students were asked to demonstrate
how to measure something and tell the length of
the object.
56
The teacher clarified the meanings of the
words measuring and length and wrote these on a
chart that she titled “Measurement Words Chart.”
These words, along with their definitions, were
used as resources in preparation for the rest of the
investigation. The word chart helps build students’ mathematical vocabulary and increase their
understanding of how to use mathematical terms
in context. For these first graders, the chart also
helped with word recognition and the spelling of
mathematical terms associated with the concept
of measurement. The chart can be used on a daily
basis to reinforce the words and their meanings.
In this class, the students referred to the chart several times throughout the measurement unit to
complete various tasks and writing activities.
In order to provide additional reasoning opportunities in this and subsequent lessons, the teacher
explained that the other school had created a chart
that listed the measurements of the furniture and
playground equipment and sent the measurements
to their principal. This chart did not include exact
units of measure such as inches or feet; instead, it
Teaching Children Mathematics / September 2004
Teaching Children Mathematics / September 2004
Figure 2
The teacher’s chart to record information
Photograph by Patricia S. Moyer; all rights reserved
used the term unit. The word unit was added to
the Measurement Words Chart and defined as
“something used to measure.” The use of the word
unit is important during these early activities
because it familiarizes students with the terminology of standard measurement.
The teacher modeled how to measure the
length of the classroom easel using colored counting bears, identifying the bears as a unit of measure. Counting bears were lined up along the bottom of the easel and the class counted the number
of bears across the easel from left to right.
Emphasis was placed on making sure the bears
were positioned end to end with no spaces
between them in order to make sure they accurately measured the entire length of the easel. The
teacher explained that the students would need to
choose a unit that they could use for measuring
the items on the list. The list of items to measure
in the classroom included a bookcase, TV, desk,
writing table, magnetic letter board, and easel.
Each student would measure the items on the list
so that they could later be compared with the list
from the other school.
Working in pairs, students used different tools to
measure like objects in their classroom. Figure 1
shows students using different tools for measuring. Student groups can be larger; however, in this
class the students worked in pairs to maximize
their opportunities for measuring each of the
objects and to increase the likelihood that they
would engage in discourse about the measurement process with their partners. This also
allowed for differentiation with students of varied
abilities as well as second-language learners. Students used Recording Sheet #1 to measure each
object and record the number of units needed to
measure the length of the objects.
After collecting the measurements, the students gathered together to report their findings and
make comparisons. Students’ findings were
recorded on a large chart that the teacher had prepared. The teacher listed each piece of furniture on
the vertical axis and the units that students used
for measuring on the horizontal axis. Figure 2
shows how the teacher set up the chart to record
student information. During this time, the teacher
asked questions that were open-ended to allow
students to make predictions and discoveries on
their own. One of the first things that students recognized was that their papers had many different
numbers for the measurements of the same
objects. As each measurement was recorded, the
teacher encouraged the students to brainstorm
about why so many differences were present in
the numbers. The students ultimately reached the
conclusion that their numbers were different
because they were using different units of measurement. They discovered that the bigger the
measurement tool (or unit), the smaller the number showing the length of the object. They also
found that the smaller the measurement tool, the
more times they needed to use it to measure the
object.
Next, the students made comparisons between
their measurements and the measurements of the
classroom furniture sent by the principal from the
other school. Soon it became apparent that the
measurements the students had recorded from
their classroom and the measurements from the
furniture in the other school were very different.
Some of the units the students used to measure
the classroom items showed that the new items
57
Figure 3
Photograph by Patricia S. Moyer; all rights reserved
Students use an inchworm tape to measure.
would fit in the classroom, and some of the units
showed that the new items would be too big or
too small. The students and teacher worked
together to make a list of strategies they would
use the next day to solve their mathematical mystery. To close the lesson, the teacher read the
book How Big Is a Foot? by Rolf Myller (1991)
to illustrate the concept of nonstandard versus
standard measurement and evoke interest for the
subsequent lessons.
Exploring Standard Units
of Measurement: The Inch
The following are required materials:
•
•
•
•
Inchworm paper
Scissors
Student Recording Sheet #2
Large chart-sized record sheet
The second lesson opened with a review of the
chart that the students created during the first lesson. The teacher asked students to revisit and comment on their findings. This discussion led students
back to the problem they had discovered in the different measurements during the previous lesson.
The teacher asked, “How can we solve our problem
of having so many different measurements?” Stu58
dents sat in pairs and decided how to solve the new
problem. When they came together as a class, they
concluded that they would all need to measure the
classroom furniture with the same unit (or measuring tool). The teacher wrote the word standard on
the word chart and the students defined it as “the
same.”
The class discussed and compared the difference in their measurements with the measurements that the other school had provided. The students realized that they would have to use the
same units that the other school had used to measure the furniture in their own classroom. The
principal arrived just in time to let the class know
that the other school had used inches to measure
the furniture. The class continued the conversation, focusing on suggestions for solving the
mathematical mystery. After a bit of discussion,
the students determined that they would need to
use inches as well.
To introduce the concept of an inch, the teacher
read Leo Lionni’s picture book Inch by Inch (1995)
to provide a link to literature and spark interest.
The word inch was added to the Measurement
Words Chart and defined as “a unit of length.”
After the teacher read the book, she gave each child
a paper inchworm tape—twelve inchworms on a
tape that mimics a ruler. The teacher also showed
the students a standard ruler and compared the
Teaching Children Mathematics / September 2004
inchworm units with the inches on the ruler. This
point could be made at the end of the lesson as
well. The teacher modeled how to use the inchworm tapes by measuring the easel, providing a
connection to the previous lesson. The students
worked in pairs to re-measure all the classroom
furniture using the inchworm tapes. Figure 3
shows students using the inchworm tape to measure. Students recorded the number of inchworms
they used to measure each furniture item (see
Recording Sheet #2). When some of the furniture
did not measure to an exact inch, students counted
only the “whole” inchworms and not the additional
parts.
After students finished measuring, they gathered together as a whole group and helped the
teacher record their measurements on the large
chart paper. Although some of the measurements
varied by one or two inchworms, the students discovered that their new measurements were much
closer to one another than the numbers they
recorded during the previous lesson. Students concluded that this difference existed because they had
all used the same unit, or a “standard,” for measuring. In addition, students began to understand that
human error plays a part in measurement and
sometimes variations occur because people measure differently. The inchworm tape can be used to
illustrate this point by placing the tape in different
places on an object and showing how this human
error would change the measurement.
Next, the teacher displayed the measurement
chart showing measurements for the new furniture.
The students discovered that their measurements
were much smaller than the measurements for the
furniture from the other school. The teacher asked
the students to conjecture the reasons why the
number of inches for the furniture in their classroom was so much smaller than the number of
inches for the furniture in the other school. She
wondered aloud, “Will this new furniture fit in our
classroom?” Discussion ensued as the students
compared the number of inchworms for their classroom furniture with the greater number of inches
used for the furniture from the other school. The
students found that the furniture from the other
school was much bigger than their classroom furniture. Some of the students thought that it was
“super-size” furniture or “giant” furniture because
it was so much bigger than the furniture from their
classroom. They hoped that the playground equipment would not have this mysterious problem
when they measured it.
Teaching Children Mathematics / September 2004
Nonstandard Units of
Measurement for Larger
Objects
The following are required materials:
•
•
•
•
Student Recording Sheet #3
Large chart-sized class record sheet
Mathematics journals
Measurement Words Chart
On the third day of the unit, the teacher
reminded students about the second part of the scenario—that the other school also had some playground equipment to give away. The class discussed ways to measure objects that were as large
as the playground equipment. Students worked in
pairs to brainstorm ideas for measuring larger
objects. To illustrate the difference between measuring a smaller object, such as an easel, and a
larger object, such as a swing set, the teacher used
Unifix cubes. She asked the students to predict how
many Unifix cubes would be needed to measure
the length of the classroom carpet. The discussion
focused on using a larger unit to measure larger
things. This discovery was linked to the previous
lesson in which students used different units to
measure the same objects. Because the characters
in the book How Big Is a Foot? (Myller 1991) from
the previous lesson used their feet to measure, the
teacher modeled this strategy by counting off her
own steps across the carpet.
The teacher asked students to select something
larger to use as their unit for measuring the playground equipment. Reminding children to place
units end to end with no spaces in between is important. The reasoning component of these lessons is
essential for students to begin to think about measurement conceptually and build knowledge on their
own. Throughout the lessons, the students made predictions frequently in order to guide them in building knowledge as they learned and explored.
At this point in the unit, many of the students
may come to the conclusion that they all need to
use the same unit when measuring the playground
equipment. In this case, the teacher should modify
the lesson to accommodate the new learning and
reasoning abilities of the students. If students assert
the need for the same unit of measurement, the lesson should proceed with the same unit (nonstandard or standard) that the class has chosen.
The students realized that they should use the
same units of measurement that the other school
59
Figure 4
Photograph by Patricia S. Moyer; all rights reserved
Students measure the playground equipment in “feet.”
used to measure the playground equipment. The
teacher showed the students a chart similar to the
classroom furniture chart. She explained that the
other school had used feet to measure the equipment. The students determined that they would use
their own feet as the standard unit. They measured
the length of items on the school playground,
including the benches, the swing set, and other
equipment using their feet. Figure 4 shows students using their feet to measure the length of the
playground equipment. Students worked in pairs to
measure equipment on the playground and
recorded this information on Recording Sheet #3.
Students mimicked the heel-to-toe motion that the
teacher used when measuring the carpet.
After the measuring was complete, students
gathered together and recorded their measurements on a chart-sized record sheet. Students
speculated about the differences in their measurements. The teacher facilitated the discussion and
students were able to conclude that their measure60
ments were slightly different because the sizes of
their feet were different. The discussion related
again to the book How Big Is a Foot? and how the
problem for the characters in the book was similar
to that of the class. The teacher provided an additional connection by showing the ruler and pointing out that all the inches on the ruler together
make one foot. She explained that the unit of feet
that the other school had used was not really a person’s feet but was a standard unit of measurement
just like the inch.
In addition to the ruler, the teacher showed the
students a yardstick and the class used several
rulers to discover that three rulers placed end to
end were the same length as the yardstick. The
class discussed how the yardstick was comprised
of feet and compared it with the ruler that was comprised of inches. Students decided that they wanted
to measure the playground equipment using yardsticks, and they completed this additional portion
of the unit later that week. To end the lesson, students wrote about their discoveries and drew pictures of the measuring units in their mathematics
journals. This activity was open-ended and allowed
differentiation among the children in the class
because they could draw and write according to
their own abilities. Having students write about
mathematical concepts gives the teacher insight
into two major areas of the children’s thinking:
How are the concepts building in students’ minds,
and what misconceptions are still apparent that
need additional focus? In addition to assessment,
mathematics journals provide an opportunity for
students to explore their own thinking using writing. This part of the lesson can facilitate reasoning
and clarify concepts in the minds of the students.
The Giant Playground
Equipment
The following are required materials:
• Large chart-sized record sheets from previous
lessons
• Record sheets from the “school” mentioned in
the problem scenario
• Large banner paper
• Crayons
• Markers
• Measurement Words Chart
On the final day of the unit, students compared
the measurements of the playground equipment.
Teaching Children Mathematics / September 2004
The interpretation and comparison of the measurements from the playground at the other school and
from their own school playground made for a
lively discussion. Students found that all the playground equipment from the other school was
much too large to fit on their playground. These
discussions with students were important to illustrate how to interpret measurements. Although the
students were told that another school was closed
and would no longer need the playground equipment, the scenario was fabricated to provide a
framework and purpose for the children’s measuring activities. The teacher purposefully created
measurements for the furniture and playground
equipment that were much too large for the classroom and playground.
The Measurement Words Chart and the data
that the students gathered were used to assist the
class in writing a letter to the other school. Students explained why the furniture and playground
equipment would not fit at their school. After writing the letter, students worked together on a large
mural for the hallway that presented and illustrated the important things the class had learned
about measurement. Students chose a concept or a
word from the chart to focus on. The mural
showed what measurement is and how to measure
the length of objects. It also displayed information
about standard units and the specific units of
inches and feet.
Reflections
Throughout the unit, students constructed their
own knowledge about measurement in a problem-solving context. This required reasoning
skills, problem solving through trial and error,
and effective communication. By beginning with
nonstandard units of measurement, the students
explored the concept of length with familiar
manipulatives, which created a safe environment
for discovery and knowledge construction. As
they made comparisons, students were able to
reach conclusions about why they found differences among the measured amounts. By the final
lesson, students had determined that they needed
to bypass the nonstandard measurement activity
for the playground and move right to the use of a
standard unit. Perhaps the most significant aspect
of this series of lessons is that the students were
engaged in problem solving that involved the use
of mathematics for a purpose. Creating the scenario of “giant-sized” furniture and equipment
from another school gave students an intriguing
mystery to solve. This made the process of learning about measurement meaningful for them. It
also left the class with some “wonderings,” such
as “I wonder what kind of school would have
such large furniture and such oversized playground equipment.” For this first-grade class,
these questions remain a mathematical measurement mystery!
References
Lionni, Leo. Inch by Inch. New York: Harper Trophy,
1995.
Myller, Rolf. How Big Is a Foot? New York: Yearling
Books, 1991.
National Council of Teachers of Mathematics (NCTM).
Principles and Standards for School Mathematics.
Reston, Va.: NCTM, 2000.
The author wishes to thank her first-grade mathematicians and her principal, Kim Dockery, at Westlawn Elementary School in the Fairfax County
Public Schools in Falls Church, Virginia. ▲
Editor’s Note
Looking for additional resources? NCTM offers a variety of resources related to measurement:
Learning and Teaching Measurement. 2003 Yearbook of the National Council of Teachers of Mathematics. Reston, Va.: NCTM, 2003.
Navigating through Measurement in Prekindergarten–Grade 2. Reston, Va.: NCTM, 2003.
Visit www.nctm.org for more information on all of NCTM’s resources, including professional development offerings and publications available in the online catalog.—Ed.
Teaching Children Mathematics / September 2004
61
MEASUREMENT Using Nonstandard Units:
Recording Sheet #1
Names ___________________________ and ___________________________
The school needs our help! We need to know if the new furniture will fit
in the classroom. Measure the length of each object on this list and write
the measurement in the blank.
What units are you using to measure?
We are using ________________ to measure.
What We Measured
Length in ______________
1. Easel
2. Writing Table
3. Letter Board
4. Bookcase
5. TV
6. Desk
Think
Did you remember to write the units after each measurement?
62
Teaching Children Mathematics / September 2004
MEASUREMENT Using the Inch:
Recording Sheet #2
Names ___________________________ and ___________________________
Let’s measure using inches! Measure the classroom furniture. This time,
use the inchworm tapes. Count the number of worms it takes to measure
the furniture.
What We Measured
Length in Inches
1. Easel
2. Writing Table
3. Letter Board
4. Bookcase
5. TV
6. Desk
Think
Did you remember to write inches after each measurement?
Teaching Children Mathematics / September 2004
63
MEASUREMENT Using Feet:
Recording Sheet #3
Names ___________________________ and ___________________________
Now let’s measure the equipment on the playground using feet! Can we
replace the old equipment with the new? Measure the length of each
object on this list and write the measurement in the blank.
What We Measured
Length in Feet
1. Bench
2. Bouncy Tube
3. Swing Set
4. Fire Truck
5. Straight Slide
6. Bridge
Think
Did you remember to write feet after each measurement?
64
Teaching Children Mathematics / September 2004