Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Tab 8: Introduction to Measurement
Table of Contents
Master Materials List
8-ii
Measurement Overview
8-1
Of Course, We Have High Standards (and Nonstandards)!
Handout 1-Recording Sheet #1
Handout 2- Recording Sheet #1 Sample Responses *
8-18
8-26
8-27
* This document was developed as a resource for trainers, but it may be used with participants at the
trainer's discretion.
Tab 8: Introduction to Measurement: Table of Contents
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Tab 8: Introduction to Measurement
Master Materials List
4 identical blank transparencies
Tongue depressor
Blank paper
Tray or lid in which to set up
Pencils
demonstration
Tape
Unpopped popcorn kernels
Large Ziploc bag or other container with an assortment of items such as:
Paper clips (assorted sizes)
1-inch cubes
1-inch square tiles
Pattern blocks
Base ten blocks
Pencils (sharpened to various
lengths and unsharpened)
Bathroom cups
Pompoms
Centimeter cubes
Post-it notes
Cotton balls
Crayons (new and used)
Quarter
Rulers
Eyedroppers
Small jars
Gram stackers
Small yogurt cups
Marbles
Tape measure
Measuring cups
Water bottles
Measuring spoons
Yarn
Milk lids
Nickels
Of Course, We Have High Standards (and Nonstandards)! Handouts
The following materials are not in the notebook. They can be accessed on the MTR
website until the K-5 MTR CDs are available.
Introduction to Measurement PowerPoint
K-5 Mathematics TEKS
Tab 8: Introduction to Measurement: Master Materials List
8-ii
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Activity:
Measurement Overview
TEKS:
This lesson is designed for teachers.
Overview:
This activity is designed to remind participants of some key ideas that are
important for measurement.
Realizing that trainers will often need to provide professional development
for teachers from different grade level configurations, the MTR Trainer of
Trainers (TOT) materials have been developed to be easily adapted for
sessions involving only K-2 teachers, only 3-5 teachers, or combined
groups of K-5 teachers. In addition to teachers’ grade level, trainers
should consider teachers’ experiences and their understanding of
concepts addressed when making decisions on whether to include various
measurement activities in the training they deliver. Suggestions related to
appropriate audience are included with each activity. This activity,
Measurement Overview, is suggested for all trainings for grades K-5
teachers.
Trainers should keep in mind that the Measurement activities use a
hands-on approach which facilitates learning for all students and is
especially appropriate for ELL and other students with special needs.
Building a strong conceptual foundation for measurement is important for
all students’ success and is especially critical for ELL and other students
with special needs.
Materials:
Introduction to Measurement PowerPoint
Blank paper, 1 sheet per participant
Pencils
4 identical blank transparencies: 2 for demonstration and 2 for verification
Tape
Unpopped popcorn kernels
Tongue depressor
Tray or lid in which to set up demonstration
K-5 Mathematics TEKS, 1 copy per participant
Grouping:
Groups of 4
Time:
30 – 45 minutes
Measurement Overview
8-1
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Lesson:
Procedures
1.
Introduction to
Measurement
Notes
Ask participants to reflect back on
the beginning activity from Day 1,
What Are the Changes? and to
recall which strand had the most
numerous refinements and/or
significant changes.
(measurement strand)
The content pieces in this section
will address that strand.
2.
Engage
How have you used measurement in the
last week?
ƒ Jot down 3 or 4 ways you have used
measurement in your daily life.
ƒ Then, label each example with a “T” (used a
tool) or an “E” (made an estimate).
ƒ Finally, note how you arrived at each
measurement: perceptually, using a
benchmark, or unit (standard or nonstandard).
Provide blank paper and pencils at
each table.
If necessary, relate a short story to
participants before they proceed
with the assignment in order to
exemplify the task.
Example: In testing this
introduction on my husband, he
remarked that it might be difficult
for people to come up with
measurement tasks from the last
week. I reminded him that he had
just been to the store to buy
groceries, helped me cook dinner,
wrapped several Christmas
presents, and related to me how
he was going to have to get off
earlier in the morning because of
the icy road conditions.
Likewise, you may have to clarify
what is meant by the terms,
“perceptually,” and “benchmark.”
When speaking of perception, one
may have heard the phrases,
“eyeball it,” or “ocular estimate.”
Generally, it means we are using
our senses or perceptions of
distance, weight, time, etc. to
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
make an estimate. Benchmarks
are personal models or referents
that are about the same size as a
given unit of measure. For
example, the width of your little
finger is about the same size as a
centimeter. A nickel has a mass of
about 5 grams.
Remember that the tool can be
standard or nonstandard.
3.
Sharing Time
ƒ For the next few minutes, share one or two of
your measurement experiences with your
tablemates.
ƒ As a group, discuss whether most of your daily
experiences were the result of using an estimate
or using a tool.
ƒ And, share any benchmarks or strategies that
you used in arriving at your measurements.
ƒ Be prepared to share your table’s observations.
4.
Measurement in the Real World
ƒ Measurement is integral to our everyday
lives.
ƒ We learn measurement concepts and
skills through experience.
ƒ Most of our daily measurements are
estimates.
ƒ The real world is interdisciplinary.
Measurement Overview
Allow a few minutes for groups to
share. Monitor the room and
make mental notes of some of the
measurements, estimates,
strategies, etc. for possible use
during the next discussion.
If time permits, allow several
tables to share some interesting
experiences.
Here are some possible
comments to highlight each bullet
– please personalize to your
audience and style.
Measurement is integral to our
everyday lives. It is an essential
life skill. We use measurement
and spatial skills to make sense of
the physical world in which we
live. As adults, most of us utilize
measurement concepts and skills
automatically. A goal of
mathematics education should be
to help our students acquire
competence in this type of
measurement. You might remind
participants that in the “Engage”
activity, they readily identified
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
“mathematics in everyday
situations,” which is a K-5 SE.
We learn measurement
concepts and skills through
experience. If we want our
students to become good
estimators and measurers, we
can’t tell them or show them how
to measure. They must have
extensive concrete, hands-on
experiences and frequent practice
with the appropriate grade level
concepts and skills. Hands-on
experiences with measurement
tools and manipulatives address
another K-5 SE, “use tools such
as real objects, manipulatives, and
technology to solve problems.”
Most of our daily measurements
are estimates. Our students
need to develop personal
benchmarks for frequently used
units of measure. They can use
these benchmarks to not only
estimate measures, but also to
judge the reasonableness and
accuracy of their actual
measurements. Knowing the
purpose for a particular
measurement will determine
whether an accurate measure is
needed or an estimate will do.
The real world is
interdisciplinary. Measurement
relates to other content areas
(science, art, physical education,
music, social studies, etc.) as well
as to other mathematical strands.
(This is a good segue into the next
demonstration.)
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Procedures
5.
Explore
ƒ Two identical transparencies have been rolled into
cylinders - one, tall and slender and the other, short and
wide.
ƒ They have been taped along the edges to avoid any
overlap.
Tarleton State University
Notes
You may want to have the
demonstration already set up. If
so, just show participants two
identical transparencies and how
they were rolled, so they
understand how the cylinders
were formed. Be sure to set the
entire demonstration inside a tray
or lid so the popcorn can be
contained and collected after the
demonstration is over.
Emphasize the fact that the two
transparencies are the same type
and size, and that you have taped
them very carefully to avoid any
overlap of edges.
6.
Explore
The tall, slender cylinder is then placed inside the short,
wide one and filled to the top with popcorn.
7.
Prediction
When the tall cylinder is lifted,
will the popcorn…
1. Fill the shorter cylinder to the top,
2. Overflow, or,
3. Not completely fill the shorter cylinder?
In place of popcorn kernels, you
can use rice, beans, centimeter
cubes, etc….
You may want to have a tongue
depressor handy to verify that the
popcorn is level with the top of the
tall cylinder.
Have participants either predict to
themselves silently or share their
prediction with a partner or table
group. You may not want them to
predict in the large group. It could
be embarrassing if their initial
prediction doesn’t match the
results.
Don’t give them much time to think
or analyze the situation. This is
not the time for discussion or
debate.
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
8.
Procedures
After the participants have made their
predictions, carefully pull the tall cylinder up
to let the popcorn kernels fall into the short
cylinder BEFORE going to the next slide.
Tarleton State University
Notes
9.
When the tall slender cylinder is
lifted, the popcorn kernels do not
completely fill the shorter cylinder,
so the shorter cylinder “holds
more than” the taller cylinder.
10.
Important to say upfront before
discussing bullets:
I purposefully misled you by
emphasizing that my two
transparencies were the same
size.
Explain
ƒ What attribute are we actually measuring
when we fill the cylinders with popcorn?
ƒ What attribute are we measuring when we
look at the lateral sides made by the two
transparencies?
What attribute are we actually
measuring when we fill the
cylinders with popcorn?
We are measuring volume or
capacity when we fill the cylinders
with popcorn; that is, how much
space does the popcorn take up?
What attribute are we measuring
when we look at the lateral sides
made by the two transparencies?
We are measuring surface area
when we consider the lateral sides
of the cylinders made by the
identical transparencies.
Emphasize the importance of
knowing what attribute is being
measured.
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
Sometimes a participant may ask
why the shorter cylinder holds
more than the taller cylinder. In
our example with the two
cylinders, we have the constraint
of the dimensions of the
transparencies (8.5 inches by 11
inches). Let’s consider the
shorter, wider cylinder first. When
the transparency is rolled, the
resulting radius for the cylinder will
11
be
or approximately 1.75
2π
inches as shown in the diagrams
below.
h = 8.5 in.
length = 11 in.
length of rectangle = circumference
of base of
cylinder
length of rectangle = 2 π r
11 in. = 2 π r
so r =
Measurement Overview
11
in. or r ≈ 1.75 in.
2π
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Procedures
Notes
h = 8.5 in
r
r =
11
in. or r ≈ 1.75 in.
2π
The lateral surface area of the
cylinder is 11 inches x 8.5 inches,
which gives 93.5 square inches.
The volume of the cylinder is the
area of the base, π r 2 , times the
height of the cylinder. The volume
of the cylinder is calculated below:
V =π r2 h
⎛ 112 ⎞
⎟ (8.5 )
V = π ⎜⎜
2 ⎟
(
)
π
2
⎝
⎠
⎛ 121 ⎞
⎟⎟ (8.5 ) cu. in.
V = ⎜⎜
⎝ 4π ⎠
V ≈ 81.85 cu. in.
Now let’s consider the taller,
narrower cylinder. When the
transparency is rolled, the
resulting radius for the cylinder will
8 .5
be
or approximately 1.35
(2 π )
inches as shown in the diagrams
below.
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Procedures
Notes
h = 11 in.
length = 8.5 in.
length of rectangle = circumference
of base of
cylinder
length of rectangle = 2 π r
8.5 in. = 2 π r
so r =
8 .5
in. or r ≈ 1.35 in.
2π
h = 11 in.
r
Measurement Overview
r =
8.5
in. or r ≈ 1.35 in.
2π
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
The lateral surface area of the
cylinder is still 11 inches x 8.5
inches, or 93.5 square inches.
The volume of the cylinder is the
area of the base, π r 2 , times the
height of the cylinder. The volume
of the cylinder is calculated below:
V =π r2 h
⎛ 8 .5 2 ⎞
⎟ (11)
V = π ⎜⎜
2 ⎟
(
)
2
π
⎠
⎝
⎛ 72.25 ⎞
⎟⎟ (11) cu. in.
V = ⎜⎜
⎝ 4π ⎠
V ≈ 63.24 cu. in.
The participants may be able to
relate this problem to an
analogous 2-dimensional situation
of finding the maximum area of a
rectangle with a given perimeter.
Finding maximum (or minimum)
values of a quantity within certain
constraints is an application in
algebra and in calculus courses as
well.
11.
Elaborate
ƒ How would you build a cylinder if you
wanted to contain the maximum volume of
corn using the same amount of lateral
surface materials?
ƒ Can you think of a real world context for
this activity?
Real world contexts might include
grain silos. A short, fat silo may
be more cost effective than a tall,
skinny one. You could store more
grain using the same amount of
lateral surface materials.
Also, our water towers are
constructed more like the short,
wide cylinders.
On the other hand, some
companies might use tall, slender
cylinders for packaging for their
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Procedures
12.
Elaborate
ƒ What are the various attributes of a
cylinder that can be measured?
ƒ Take a few minutes to discuss this with
your tablemates.
13.
Important Measurement Ideas
The process of measurement includes:
ƒ Selecting/identifying the attribute or property
to be measured
ƒ Choosing an appropriate unit of measure that
has the same attribute
ƒ Making an estimate
ƒ Comparing the chosen unit to the object to be
measured
ƒ Determining the number of those units by
counting, using an instrument, and/or a
formula
Measurement Overview
Tarleton State University
Notes
products in an effort to create the
impression that there is more.
Now that we’ve talked about the
importance of identifying the
attribute we are measuring, let’s
brainstorm other attributes of
cylinders that can be measured.
Possible responses include:
•
Capacity
•
Volume
•
Mass
•
Weight
•
Lateral Surface Area
•
Total Surface Area
•
Height of the cylinder
•
Radius of the cylinder
•
Diameter of the cylinder
•
Area of the bases
•
Circumference of the bases
There are many variations on this
process.
Reinforce the important first step
of identifying the attribute to be
measured.
Making an estimate should always
precede measuring, although this
is not always included in the
process steps.
Measuring tools/instruments make
measurement easier and more
efficient. It is easier to use a
measuring tape than it is to lay
centimeter lengths end-to-end
(iteration). Having students
construct simple measurement
tools/instruments will help them
understand how actual
measurement instruments work.
8-11
Mathematics TEKS Refinement 2006 – K-5
Procedures
14.
Important Measurement Ideas
Tarleton State University
Notes
In elaborating on this slide, you
could use the example:
Numbers are adjectives.
Adjectives modify nouns.
Remember to express the measurement in
terms of a number and a unit.
15.
Family Circus Cartoon Connecting the appropriate tool
and unit with the attribute or
property being measured is
often very difficult for
elementary students.
© Bil Keane, Inc. King Features
Syndicate
© Bil Keane, Inc. King Features Syndicate
Measurement Overview
MTR has permission to use this
cartoon in the training for up to
one year. The permission expires
on August 14, 2007. If you are
leading MTR training after that
date, contact the following:
King Features Syndicate
North America Syndicate
Permissions
A Unit of the Hearst Corporation
P.O. Box 536463
Orlando, Florida 32853-6463
(800) 708-7311 Ext. 246
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Mathematics TEKS Refinement 2006 – K-5
Procedures
16.
Used with permission; http://www.cartoonstock.com
17.
Important Measurement Ideas
Definition of Measurement:
Measurement is the process of quantifying
the attribute or property of an object by
comparison with some unit (nonstandard
or standard).
18.
Important Measurement Ideas
ƒ Measurements are approximations!
ƒ There is always a margin of error. Why?
ƒ The smaller the unit, the more precise the
measurement.
ƒ What is the smallest unit that our students
are required to use for each attribute of
measurement at each grade level in the
TEKS?
Measurement Overview
Tarleton State University
Notes
Second Cartoon –
Then, observing how students
utilize an instrument or tool
often points to other
misconceptions.
The activities and lessons in this
training are designed to provide
students with many opportunities
to correct these misconceptions.
This cartoon is used with
permission from
http://www.cartoonstock.com. The
permission is valid for the “first
edition” of the MTR training. The
cartoonist’s name is Vahan
Scirvanian.
This slide is optional, and there
are many different ways to say the
same thing.
Measurements are
approximations! Measurements
are inherently imprecise. We can
refine our measurements by using
smaller units or fractional parts of
units, but mathematically there is
no such thing as the smallest unit.
We cannot say that this paper clip
is exactly 3 centimeters long.
There is always a margin of
error. Why? In addition, human
error, the limitations of our
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Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
instruments, and influences of the
environment influence the
accuracy (or inaccuracy) of our
measurements.
The smaller the unit, the more
precise the measurement. How
precise a measurement needs to
be depends on the context or
purpose of the measurement.
Also, the development of the
students’ understanding of
measurement determines the
reasonable unit. *See Notes on
Page 8-15.
What is the smallest unit that
our students are required to use
for each attribute of
measurement at each grade
level in the TEKS? The TEKS do
not identify the smallest unit that
our students are required to use
for each attribute of measurement
at each grade level. We want the
participants to look for it to see
that it doesn’t exist… don’t spoil
the surprise! Let them see for
themselves that it isn’t there.
Discussions related to this issue
need to be held at the district level
during curriculum alignment and
professional development.
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Procedures
19.
Let’s Measure!
Notes:
Tarleton State University
Notes
We’ve highlighted some of the big
ideas in the measurement
process. So, let’s measure!
A question may arise during your training concerning the use of the terms
“precise (or precision)” and “accurate (or accuracy).” Some background
material regarding these terms is provided here for the trainer so that you
can be prepared for this question, should it arise.
Below are some definitions and clarifications that the MTR writers used
when determining which terms to use within this context. (See Page 8-13,
Procedure 18, Bullet 3.) The material is summarized here, and references
are cited so that you may read more about the topic if you wish.
The following information comes from the science position paper on
accuracy and precision. This paper is available electronically on the Texas
Education Agency website at
http://www.tea.state.tx.us/curriculum/science/Accuracy_and_Precision.html.
“For the purposes of the Grades 10 and 11 Science TAKS:
Precision will be viewed as the –
1. repeatability of a measurement or measurements made in the
same way and which consistently return a similar value.
2. degree of refinement of a measurement as limited by the design
of the instrument.
Accuracy will be viewed as the –
1. reading of an instrument against a true value as set by a
calibration standard.
2. ability to determine if a measurement is reasonable or to
experimentally calculate certain constants and standards.”
To clarify these definitions for elementary teachers, the following sources
may be helpful.
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
First, the Ask Dr. Math website contains several items in its archives on this
issue. This next information comes from a question/answer that was
posted on the Ask Dr. Math website on 01/07/2003 at
http://mathforum.org/library/drmath/view/61947.html.
The following question was submitted to Dr. Math:
“Using a metric ruler student A measured the length of an object to
the nearest tenth (0.1) of a centimeter, while student D measured its
length to the nearest centimeter. Which measurement is more
accurate?”
Here is an excerpt from the response:
“We have to distinguish between “accuracy,” which means the
closeness of a measurement to the exact value, and “precision,”
which means the claimed or implied closeness. For example, if I
said my desk was 2 meters wide, and you said it was 2.345 meters
wide, your answer would be more precise (claiming that you know it
down to the millimeter); but if the desk is really 2.123 meters wide,
then my answer is more accurate!”
Another helpful source for clarifying the precision vs. accuracy issue for
elementary teachers is Math Matters: Understanding the Math You Teach,
Grades K-6 by Suzanne H. Chapin and Art Johnson. On page 180, Chapin
and Johnson state the following:
“The precision of a measuring device tells us how finely a
measurement is made. Measurements made using small units (e.g.,
centimeters) are more precise than measurements made using
larger units (e.g., meters). The accuracy of a measure is how
correctly a measurement has been made. Accuracy can be affected
either by the person doing the measuring or by the measurement
tool. In addition, two measurements can both be accurate (e.g., a
pumpkin weighs 7 pounds on one scale and 6 pounds 15 ounces on
another scale), but one is more precise than another. Older
students need to discuss precision and accuracy and become
comfortable with using language that features the approximate
nature of measurement – The stick is about two meters long; to be
more precise, the stick is 197 centimeters long.”
Dr. Chapin clarified these ideas further for the MTR writers through a series
of email correspondence. In summary, the precision of a measurement
depends on the size of the smallest measuring unit used. Measuring the
1
cm is more precise than measuring the
length of a book to the nearest
100
Measurement Overview
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
1
cm. Accuracy on the other
10
hand, refers to how “correctly” the measurements have been made and to
how the measurements compare to the true measure. For example, can a
1
student measure the length of the book to the nearest
inch correctly? If
2
1
inches, can the student determine
the book’s length is closer to 10 and
2
this or does he or she say that the book is 10 inches in length?
length of the book to the nearest cm or
Measurement Overview
8-17
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Activity:
Of Course, We Have High Standards (and Nonstandards)!
TEKS:
This investigation is designed for teachers.
Overview:
This lesson will allow the participants an opportunity to consider the use of
standard units and nonstandard units in the measurement process. A key
point in this lesson is that one must pay careful attention to the attribute to
be measured when selecting a tool to use in measuring. In addition, the
notion of uniformity is important when selecting and using nonstandard
measurement tools. Participants will explore reasons for using standard
units and nonstandard units in the mathematics classroom, as well as
investigate the TEKS to see the vertical articulation in the use of
nonstandard units and standard units from Kindergarten through 5th
Grade.
Trainers should include this activity in all trainings for grades K-2 teachers.
For trainings with only grades 3-5 teachers, trainers should make a
decision about whether to include this activity based on the experiences
and understanding of their audience.
Materials:
Handout 1 – Recording Sheet # 1, one per participant (page 8-26)
K-5 Mathematics TEKS, one copy per participant
Large Ziploc bag or other container (one per group) with an assortment of
items such as the following:
Tape measure
Gram stackers
Rulers
Measuring cups
Measuring spoons
Crayons (new and used)
Paper clips (assorted
Pencils (unsharpened and
sizes)
sharpened to various lengths)
Post-it notes
Marbles
1-inch cubes
1-inch square tiles
Centimeter cubes
Base ten blocks
Pattern blocks
Cotton balls
Pompoms
Yarn
Bathroom cups
Small yogurt cups
Eyedroppers
Water bottles
Nickels
Quarters
Milk lids
Small jars
Grouping:
Groups of 4
Of Course, We Have High Standards (and Nonstandards)!
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Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
30 – 45 minutes
Time:
Lesson:
1.
Procedures
In advance, prepare bags of assorted items
that could be used in the measurement
process.
Notes
The large 2.5 gallon Ziploc bags
work well for assembling this
collection of items.
See the Materials List above for suggestions
for the contents of the bags.
The bags for each of the groups
should contain the same types of
items to aid in discussion during
the debrief for this lesson.
2.
Distribute one bag to each group of
participants.
3.
Tell the participants: You have been given a
collection of items at each of your tables that
could be used in the measurement process.
Sort the items into the following categories:
• Tools for measuring with standard units
• Tools for measuring with nonstandard
units
Of Course, We Have High Standards (and Nonstandards)!
Tools for measuring with
standard units:
Tape measure
Gram stackers
Rulers
Measuring cups
Measuring spoons
Tools for measuring with
nonstandard units:
Pencils (sharpened and
unsharpened)
Crayons (new and used)
Post-it notes
Marbles
Paper clips (assorted sizes)
1-inch cubes*
1-inch square tiles*
Centimeter cubes*
Base ten blocks*
Pattern blocks
Cotton balls
Pompoms
Yarn
Bathroom cups
Small Yogurt cups
Eyedroppers
Water bottles
Nickels
Quarter
8-19
Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Milk lids
Small jars
Notes
* Note: Participants may include
items such as 1-inch tiles, 1-inch
cubes, centimeter cubes, base-ten
blocks, etc. in the set of tools for
measuring with standard units.
The classification depends upon
what you call the item when you
use it. For example, if you call the
cubes “1-inch cubes,” then the
item might fit better with the tools
for measuring with standard units.
However, if you call the
manipulative “colored cubes,” then
the item might fit better with the
tools for measuring with
nonstandard units. The 1-inch
tiles, 1-inch cubes, centimeter
cubes, etc., are wonderful
manipulatives for approximating
standard units. Since the items
are not labeled with “inch”
markings or “centimeter”
markings, they work well to help
students bridge from nonstandard
units to standard units.
4.
Ask participants to share with the large group
the items identified as tools for measuring
with standard units.
5.
As emphasized in the Measurement
Overview activity (pages 8-1 – 8-17), remind
participants that it is important to identify the
attribute we wish to measure, then to select
a unit that possesses that same attribute.
Keeping this statement in mind, ask the
participants to turn their attention to the set
of items identified as “tools for measuring
with nonstandard units.”
Ask the participants to identify which of the
Of Course, We Have High Standards (and Nonstandards)!
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Mathematics TEKS Refinement 2006 – K-5
Procedures
items from the “tools for measuring with
nonstandard units” would be most
appropriate when measuring the attribute of
length.
6.
Ask participants to share responses with the
large group by asking each group to share
one item from their list and briefly justify why
their group chose the item.
See Notes in the right column for examples
of things that might come out during the
discussion.
The main points to bring out in this
discussion are the following:
1) one must focus on the attribute of
length (for this example) of the item
when choosing the item as a tool for
measuring the length of some other
object
2) uniformity is important when using a
tool in the measurement process
Tarleton State University
Notes
Answers will vary depending on
the contents of the bag.
When groups are justifying their
choices, make sure to emphasize
(if the group doesn’t emphasize it)
that one must focus on the
attribute of length of the item when
choosing the item as a tool for
measuring length of some other
object. For example, a square tile
could be used to measure length
IF you focus on the length of one
of the sides of the square tile. A
quarter could be used to measure
length IF you focus on the length
of the diameter of the quarter.
It is also important to emphasize
that the chosen item(s) must be
uniform in order to be used
appropriately. For example, paper
clips would be a good choice to
use for measuring the attribute of
length, as long as you used paper
clips that were all the same size.
You could measure the length of
an object in small paper clips or
measure the length of an object in
large paper clips, but you would
not want to mix the sizes of paper
clips for measuring.
Another example could involve
unsharpened pencils vs.
sharpened pencils. Unsharpened
pencils would be a good choice for
measuring the attribute of length
because they are uniform in
length. You could lay several
Of Course, We Have High Standards (and Nonstandards)!
8-21
Mathematics TEKS Refinement 2006 – K-5
Procedures
Tarleton State University
Notes
unsharpened pencils end-to-end
to see how many pencil lengths
will fit along the length of the
object you wish to measure.
Sharpened pencils would be
appropriate if you iterated a single
sharpened pencil along the length
of the object you wish to measure.
7.
Let’s consider measuring other attributes
besides length.
If we wanted to measure area using
nonstandard units, which items would be
most appropriate?
8.
If we wanted to measure volume/capacity
using nonstandard units, which items would
be most appropriate?
Possible responses include:
Post-it notes
1-inch square tiles
Pattern blocks
Possible responses include:
Marbles
1-inch cubes
Centimeter cubes
Base ten blocks
Cotton balls
Pompoms
Bathroom cups
Small Yogurt cups
Eyedroppers
Water bottles
Milk lids
Jars
You might choose to bring out the
point that cotton balls and
pompoms, even when uniform in
size, are not the best choice for
measuring volume/capacity
because the cotton balls and
pompoms can be compressed.
We will explore volume and
capacity in more detail during
many of the subsequent lessons
and activities included in the
training.
Of Course, We Have High Standards (and Nonstandards)!
8-22
Mathematics TEKS Refinement 2006 – K-5
9.
Procedures
If we wanted to measure mass/weight using
nonstandard units, which items would be
most appropriate?
Tarleton State University
Notes
Possible responses include:
Pencils
Crayons
Marbles
Paper clips
1-inch cubes
1-inch square tiles
Centimeter cubes
Base ten blocks
Pattern blocks
Cotton balls
Pompoms
Nickels (about 5 grams)
Quarter
Milk lids (about 2 grams)
Paper clips
We will explore mass and weight
in more detail later in the training.
10.
Now let’s look at the K-5 Mathematics TEKS
(see Materials List for link) to see what grade
level expectations are present in the TEKS
concerning standard and nonstandard units.
Have the participants examine the K-5
Mathematics TEKS with a focus on standard
units and nonstandard units. Ask
participants to record both the TEKS
number/letter for the Knowledge and Skills
Statement and the Student Expectation,
along with a brief summary of their findings
on Recording Sheet #1 (page 8-26).
11.
A sample of a completed chart is
included at the end of this lesson.
Lead participants in a short discussion of
their findings concerning standard units
and/or nonstandard units for the attribute of
length by asking the participants what is
expected and appropriate in Kindergarten,
1st Grade, 2nd Grade, 3rd Grade, and so on.
Repeat the discussion for the attributes of
area, then for capacity/volume, then for
mass/weight.
Of Course, We Have High Standards (and Nonstandards)!
Remind participants that we will
deal with capacity/volume and
mass/weight in more detail later in
this training.
8-23
Mathematics TEKS Refinement 2006 – K-5
12.
Tarleton State University
Procedures
By doing so, the participants will get a feel
for the progression of measurement
development for each attribute throughout
grades K-5.
Notes
As we round out this discussion, let’s
consider for a few moments some reasons
for using nonstandard units and standard
units in the mathematics classroom.
According to Van de Walle (2004,
p. 319), some key ideas that might
come out of the brainstorming
session include the following:
Have each group appoint a recorder for the
brainstorming session. Ask participants to
brainstorm in their groups for a few minutes
about reasons for using nonstandard units
and reasons for using standard units.
Reasons for using nonstandard
units:
• Allows the student to focus
directly on the attribute being
measured: For example, when
measuring area of an irregular
shape, covering the region with
lima beans as the unit will
produce a different
measurement for the area than
covering the region with
square tiles as the unit. Each
unit covers area however, and
the resulting discussion with
the students can highlight what
it means to measure area.
• Allows the magnitude of the
numbers to be kept
reasonable: For example, the
measures of length can be
kept smaller even when
measuring longer distances by
choosing a larger unit.
• Prevents conflicting objectives
during introductory lessons:
For example, is the lesson
about measuring volume or is
the lesson about
understanding cubic
centimeters?
• Allows students to see the
need for standard units.
Quickly go around the room and allow each
group to share one reason for using either
nonstandard or standard units.
Reasons for using standard units:
• Convention of society: The
Of Course, We Have High Standards (and Nonstandards)!
8-24
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
Procedures
•
13.
Notes
student must develop
familiarity with standard units
and understand appropriate
relationships between them.
Ease of use after the
measurement concept has
been developed.
As participants reflect back on the lesson,
remind them that the use of standard and
nonstandard units is very important.
Reporting the unit used is a crucial part of
the measuring process in order that we might
communicate clearly our measurement
results.
In addition, remind participants that beyond
being developmentally appropriate in grades
K-2, the use of nonstandard units in grades
K-2 is justified by the TEKS.
Finally, encourage the participants to
develop a working relationship with the
science faculty in the schools. An open line
of communication between the mathematics
teacher and the science teacher concerning
the introduction of measurement units and
tools can reap great benefits in terms of
student success.
Resources:
Leedy, L. (1997). Measuring penny. New York: Henry Holt and
Company.
Murphy, S.J. (1999). Super sand castle saturday. New York:
HarperTrophy.
Van de Walle, J. A. (2004). Elementary and middle school
mathematics: Teaching developmentally. Boston: Pearson
Education, Inc.
Of Course, We Have High Standards (and Nonstandards)!
8-25
Tarleton State University
Kindergarten
1st Grade
Of Course, We Have High Standards (and Nonstandards)!
Mass/Weight
Capacity/Volume
Area
Attribute
Length
2nd Grade
3rd Grade
4th Grade
Handout 1
8-26
5th Grade
Examine the Mathematics TEKS for Kindergarten through 5th Grade to see what the TEKS say about the use of standard
units and nonstandard units. Summarize your findings in the chart below. Include the number/letter of the Knowledge
and Skills Statement and/or Student Expectation along with a brief summary of your findings.
Of Course, We Have Standards (and Nonstandards)!
Recording Sheet #1
Mathematics TEKS Refinement 2006 – K-5
Tarleton State University
K.10B
•
Direct
comparisons
K.10C
•
Direct
comparisons
K.10D
•
Direct
comparisons
Area
Capacity/Volume
Mass/Weight
1.7F
•
Direct
comparisons
1.7E
•
Direct
comparisons
1.7D
•
Direct
comparisons
1.7ABC
•
Direct
comparisons
•
Nonstandard
units
1st Grade
Of Course, We Have High Standards (and Nonstandards)!
K.10A
•
Direct
comparisons
Kindergarten
Length
Attribute
2.9D
•
Direct
comparisons
•
Nonstandard
units
3.11D
•
Direct
comparisons
•
Standard Units
(concrete
models)
3.11AB
•
Direct
comparisons
•
Standard units
(linear
measurement
tools and
perimeter)
3.11C
•
Direct
comparisons
•
Standard units
(concrete and
pictorial
models)
3.11EF
•
Direct
comparisons
•
Standard units
(concrete
models)
2.9A
•
Direct
comparisons
•
Nonstandard
units
•
Standard units
(concrete
models)
2.9B
•
Direct
comparisons
•
Nonstandard
units
2.9C
•
Direct
comparisons
•
Nonstandard
units
3rd Grade
2nd Grade
5.10AB
•
Standard units
(conversions;
connect models
to formulas)
5.10AB
•
Standard units
(conversions;
connect models
to volume
formula for
rectangular
prism)
4.11A
•
Standard units
(measurement
tools)
4.11ABC
•
Standard units
(measurement
tools for
capacity;
concrete models
for volume)
•
Standard units
(conversions)
4.11AB
•
Standard units
(measurement
tools)
•
Standard units
(conversions)
Handout 2
8-27
5.10A
•
Standard units
(conversions)
5.10AB
•
Standard units
(conversions;
connect models
to formulas)
5th Grade
4.11AB
•
Standard units
(measurement
tools)
•
Standard units
(conversions)
4th Grade
Examine the Mathematics TEKS for Kindergarten through 5th Grade to see what the TEKS say about the use of standard
units and nonstandard units. Summarize your findings in the chart below. Include the number/letter of the Knowledge
and Skills Statement and/or Student Expectation along with a brief summary of your findings.
Of Course, We Have Standards (and Nonstandards)!
Recording Sheet #1 – Sample Responses
Mathematics TEKS Refinement 2006 – K-5