A Close Look at Measurement
Grades 3-5
Cynthia Santosuosso
Objectives
1. Understand the progression of measurement concepts
in relation to the Common Core
2. Develop content knowledge about measurement
3. Collaborate with colleagues on teaching of
measurement
4. Learn effective strategies to promote student learning
of measurement
Why Measurement?
• “It bridges the two main areas of school
mathematics-geometry and number.”
▫ NCTM, 2000
• “Measurement is justifiably seen as important,
because it provides the main route to the
application of mathematics to quantities in all
daily life, science, and technology practice.”
▫ Ryan and Williams, 2007
Reflecting on Measurement
Performance
Only 24.36%
answered
correctly…
NAEP, Grade 4, 2011
The square has a perimeter of 12 units. What is the area of the square?
a.
b.
c.
d.
6 square units
8 square units
9 square units
12 square units
The Components of the Standard
Measurement
and Data
Geometric
Measurement
Measuring
Data
Common Core Critical Area
Grade 3: Students recognize area as an attribute of
two-dimensional regions. They measure the area of a
shape by finding the total number of same-size units of
area required to cover the shape without gaps or
overlaps, a square with sides of unit length being the
standard unit for measuring area. Students understand
that rectangular arrays can be decomposed into
identical rows or into identical columns. By decomposing
rectangles into rectangular arrays of squares, students
connect area to multiplication, and justify using
multiplication to determine the area of a rectangle.
Common Core Critical Area
Grade 5: Students recognize volume as an attribute of threedimensional space. They understand that volume can be
measured by finding the total number of same-size units of
volume required to fill the space without gaps or overlaps. They
understand that a 1-unit by 1-unit by 1-unit cube is the standard
unit for measuring volume. They select appropriate units,
strategies, and tools for solving problems that involve estimating
and measuring volume. They decompose three-dimensional
shapes and find volumes of right rectangular prisms by viewing
them as decomposed into layers of arrays of cubes. They measure
necessary attributes of shapes in order to determine volumes to
solve real world and mathematical problems.
The Progressions Documents
• Project designed by the University of Arizona
• CCSS were originally built on the narrative progression of a topic across
grade levels
• University of Arizona is recreating these, because of the following benefits:
• Explain why standards are sequenced in order
• Identify cognitive difficulties for students
• Provide pedagogical solutions
• Explain difficult areas of mathematics in depth
K-2 Progression of Measurement
Kindergarten
Describe and
compare
measurable
attributes
First
Critical Area:
Measure lengths
indirectly and by
iterating length
units
Second
Critical Area:
Measure and
estimate lengths in
standard units
Relate addition
and subtraction to
length
Task: Close Reading of Progressions
1. Closely read your section of the Progressions
Document
2. Summarize the key points on chart paper
3. Share out
Progressions Reflections…
Complete the reflection sheet.
Carousel Activity: Connections to the
Mathematical Practices
What opportunities do students
have to engage in each
mathematical practice while
teaching your grade level
measurement standards?
What is measurement?
Measurement is the process of quantifying the properties
of an object by expressing them in terms of a standard
unit.
Measurements are made to answer such questions as,
How heavy is this box? How tall is my daughter? How
much chlorine is in this water?
The Measurement Process
Select an attribute of the item you wish to
measure.
Choose an appropriate unit of measurement for
measuring that attribute.
Determine the magnitude, or number of units
needed.
Turn and Talk: How can you measure
this bucket?
• Height
• Depth
• Diameter
• Volume
• Surface area
• Weight
Fundamental Components of
Measurement
Partitioning
Conservation
Transitivity
Unit iteration
Accumulation of distance
Relation between number and measurement
Partitioning
• Dividing an object into same-size units. That this
is not obvious to children becomes apparent
when they are asked what the hash marks on a
ruler mean. A child who does not understand
partitioning of length sees the 3 on a standard
ruler as a hash mark rather than as the end of a
space that is divided into 3 equal-size units.
Conservation
The principle that an object maintains the same
size and shape even if it is repositioned or divided
in certain ways. The understanding that as an
object moves, its length does not change.
Transitivity
When you can't compare two objects directly, you must compare
them by means of a third object.
The understanding that if the length of one object is equal to the
length of a second object, which is equal to the length of a third
object that cannot be directly compared with the first, the first
and third objects are also the same length.
Example:
For example, to compare the length of a bookshelf in one room
with the length of a desk in another room, you might cut a
string that is the same length as the bookshelf. You can then
compare the piece of string with the desk. If the string is the
same length as the desk, then you know that the desk is the
same length as the bookshelf.
Unit Iteration
Unit iteration is the repetition of a single unit along the length
of an object. Children with little understanding of this concept
often leave gaps between subsequent units or even overlap the
units.
Example:
If you are measuring the length of a desk with straws, it is easy
enough to lay out straws across the desk and then count them.
But if only one straw is available, then you must iterate
(repeat) the unit (straw). You first have to visualize the total
length in terms of the single unit and then reposition the unit
repeatedly.
Accumulation of distance
• When you iterate a unit along an object’s length
and count the iterations, the number words
convey the space covered by all units counted up
to that point.
five inches
Relation between number and
measurement
• Many children fall back on their earlier counting
experiences to interpret measuring tasks (for
example, beginning with 1 rather than 0).
Students who simply read a ruler procedurally
have not related the meaning of the number to
its measurement.
Units Versus Numbers
• “When measuring with a ruler of this type,
children must deal directly with the fact that the
length of an object is the number of units
(spaces, not marks, on the ruler) between the
beginning point and the end point”
Why are these important?
These are the building blocks
of measurement! Without a
solid foundation, students will
not master area, angles,
conversions, or volume.
General Measurement Misconceptions
• Measurement and measurement units are
abstract concepts for children.
Misconceptions About Length
• Students have difficulty in determining the length of
a line when the ruler is not aligned with the object
starting at zero. This is because they count the
marks/hashes on the ruler rather than the units.
• Measurement is difficult for students because they
confuse counting and measuring.
• Children rarely understand that numbers are
represented on the number line by lengths; instead,
numbers are thought to be represented by the points
they label.
Misconceptions About Area
• Children see area as calculation—a number
separate from the real-world situation—rather
than a measurement.
• Children have difficulty in tiling a region with a
unit of different shape than the region and they
will not violate any boundary of the region while
tiling.
• Formula replaces the measurement with an
instrument and impedes the understanding of
measurement units.
Misconceptions About Volume and
Angle
• When students have to find the volume of a
space filled with unit cubes, they usually count
only the cubes that are visible from a given
angle.
• Students think that the length of the rays
constructing the angle has an effect on the
measure of the angle.
• They rarely understand that angle is a measure
of rotation.
Analysis of Student Misconceptions
• Review the student misconceptions about
measurement.
• Identify the specific misconception.
• Is there a specific area of measurement that the
child is struggling with?
Common Instructional Goals
Familiarity
with the units
Ability to select an
appropriate unit
Knowledge of relationships
between units
Estimation v. Precision
Both are
extremely
important in
measurement!
Estimation
• Helps students focus on the attribute being
measured and the measuring process.
• When standard units are being measured,
estimation helps develop familiarity with the
unit. Ex: If you are measuring the height of a
door in meters, you must first think about the
size of a meter.
• Forces students to check for reasonability!
Strategies for teaching estimation in
measurement
• Develop and use benchmark measurements
▫ Ex: A paperclip is one inch; a sack of flour is 5 lbs
• Use “chunking”
Strategies for teaching estimation in
measurement
• Iterate a unit mentally or physically. Use your
hands, feet, fingers, folds in paper to keep track
of your measurement.
Before you start measuring….
Precision
• Measurement, by its very nature, is
approximate.
• The precision of the measuring device tells us
how finely a particular measurement was made.
• Measurements made using small units, such as
square millimeters, are more precise than
measurements made using larger units, such as
square centimeters.
Strategies for Teaching Precision
• Have students specify unit of measure
• Encourage students to make precise
measurements
• The accuracy of a measure is determined by how
correctly a measurement has been made.
Accuracy can be affected by the person making
the measurement and/or by the measurement
tool.
Strategies for Teaching Length
• Have students look at spaces, rather than hash
marks. Students should count the units of
measure.
• Allow students to measure real-world objects,
rather than only pictures on paper.
• Create strong benchmark measurements for
students to use as references for estimate and to
check reasonability. Example: A small
paperclip is about one inch long; I am about
five feet tall.
Strategies for Teaching Perimeter
• “Unroll” the sides of a shape by drawing a line
segment by tracing the length of each side of the
figure.
• If necessary, add the lengths on a ruler.
• Provide hands-on experiences beyond
calculating perimeter on paper.
Strategies for Teaching Area
• Arrays!!!
• Unit covering/Tiling
• Determining the area of a
shape that is superimposed
on a grid
Strategies for Teaching Area/Perimeter
• Pentominoes
Strategies for Teaching Angles
• Asking students to estimate the angle prior to
measuring may help prevent them from
misreading the protractor. Students can identify
whether the angle is acute, obtuse or more or
less than a right angle prior to measuring.
Strategies for Teaching Angles
• Provide students with opportunities to compare
angles without measuring them.
Strategies for Teaching Volume
• Use models (cubes, boxes, stacks of books) to explain the formula
• Make box models
Transfer to EnVision
• Review the following items:
• Progressions reflection sheet
• Common Misconceptions
• Teaching strategies
• Use sticky notes to mark up your upcoming
EnVision measurement topics with important
points that you want to remember when teaching
these topics to students.
It’s your turn! Try these measurement tasks and see what
concepts and strategies you apply while problem solving.
Measurement Task
Problem of the Month: Surrounded and Covered
Level D
Level E
Turn and Talk
• What key ideas in measurement do these tasks
promote?
Sources
• Oberdorf, Christine; Schultz-Ferrell, Karren. Math
Misconceptions, PreK-Grade 5 Heinemann. Kindle Edition.
2010.
• Bamberger, Honi and Christine Oberdorf. Activities to Undo
Math Misconceptions. Portsmouth: Heinemann, 2010.
• El Paso Collaborative for Academic Excellence.
“Measurement: Student Misconceptions
and Strategies for Teaching.”
http://www.epcae.org/uploads/documents/Measuremen
t_pck_SEP21.pdf
• Annenberg Learner. “Fundamentals of Measurement.”
http://www.learner.org/courses/learningmath/measurement
/session2/index.html