Building Geometric
Understanding
Hands-on Exploration in
Geometric
Measurement
Grades 3-5
WALT:
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We are learning to:
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Understand the concepts of area and volume as
they are sequenced in the CCSS for 3-5th grades
and incorporate the Math Practice Standards in
our teaching
Describe relationships between perimeter and
area
Describe relationships between surface area and
volume
Success Criteria:

We know we are successful when we can
describe how explorations in geometric
measurement meet the criteria of the CCSS
in both content and practice standards.
Effective Classroom Practices
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Manipulatives
Cooperative groups
Goal setting - WALT
Effective questioning
Student thinking explained
Connections to prior knowledge
Multiple exposures
CCSS Practice Standards
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#2 Reason abstractly and quantitatively
#3 Construct viable arguments and critique
the reasoning of others
#4 Model with mathematics
#5 Use appropriate tools strategically
#6 Attend to precision
CCSS Content Standards
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Grade 3
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Grade 4
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Geometric measurement: Understand concepts of area and
relate area to multiplication and to addition
Geometric measurement: recognize perimeter as an
attribute of plane figures and distinguish between linear and
area measures
4.MD.3 Apply the area and perimeter formulas for
rectangles in real world and mathematical problems
Grade 5
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Geometric measurement: understand concepts of volume
and relate volume to multiplication and addition
Battista
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Powerful mathematics learning can occur in problem-centered
inquiry-based teaching
To develop powerful mathematical thinking, instruction must
carefully guide and support students’ personal construction of
concepts and ways of reasoning while the students intentionally
try to make sense of situations.
Pay careful attention to classroom talk
Battista, M.T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquirybased classroom. Journal for Research in Mathematics Education, 30, 417-448. In Lessons Learned from
Research, NCTM, 2002, pg. 75-83. In Adding It Up, National Research Council, 2001, p 284-288.
Why Examine Perimeter and Area
Relationships?
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Woodward and Byrd (1983) found that
almost two-thirds of 8th graders studied
believed that rectangles with the same
perimeter occupy the same area.*
This is a 3rd grade content piece in the new
CCSS.
*Stone, Michael E. (1994). Teaching relationships between area and perimeter using geometer’s sketchpad.
Mathematics Teacher, Nov. 590-594.
Erika was wondering how to arrange 20 pieces of
fencing to make a rectangular dog run.
Table Task 1:
 Build a rectangle with 20 toothpicks (fencing
pieces)
 Sketch, label dimensions and find area.
 Display all rectangles on chart paper.
 Label which arrangement has the largest
area and which has the smallest.
 Post
Wait a minute…
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We have the same number of toothpicks for
the perimeter but different areas. How can
this happen?
Discuss with your table group how students
in 3rd – 5th may respond to the above
question.
Area
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Although students can recall standard formulas for
areas and perimeters, other aspects of area
measure remain problematic.
Rectangular area is treated as simply multiplying
length times width; research suggests many
elementary students do not see this product as a
measurement.
A Research Companion to Principles and Standards for School Mathematics. Reston: NCTM,
2003,p.185
Erika has 20 square pieces of sod (grass) for the
dog run. Which rectangular arrangement of sod
would take the most fencing? The least fencing?
Table Task 2:
 Build a rectangle with 20 tiles
 Sketch, label and find the perimeter
 Display all rectangles on chart paper
 Label which requires the most fencing and
which requires the least fencing
 Post
Wait a minute….
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We always have the same number of tiles
but the number for our perimeter changes.
How can this happen?
Discuss with your table group how students
in 3rd – 5th may respond to the above
question.
From Perimeter to Area to Volume
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As students progress in their understanding
of geometric measurement, underlying
concepts build upon one another.
Fourth grade focuses on angle measurement
but perimeter and area should be reinforced.
Fifth grade introduces the measurement of
volume.
Why explore understanding of
volume?
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In one study, Lehrer and Schauble found that fifth
graders who had a wide range of experience with
representations of volume and its measurement
typically organized space into three-dimensional
arrays.*
Three dimensional thinking is vital in the fields of
engineering and science
Lehrer and Schauble.( 2000). Inventing data structures for representational purposes:
Elementary grade students’ classification models. Mathematical Thinking and Learning, 2,
pg.49-72. In Adding It Up, Helping Children Learn Mathematics, National Research
Council. Washington, DC: National Academy Press, 2001.
Patrick Thompson,
Vanderbilt University
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Students in a 5th grade teaching experiment
on area and volume alerted us to the
distinction between understanding a formula
numerically and understanding it
quantitatively.
Assessment Item
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What is the volume of this box?
17 in2
6 in
Thompson, Patrick W. and Saldanha, Luis. Fraction and multiplicative reasoning. In A Research
Companion to Principles and Standards for School Mathematics, NCTM, 2003.
Student Interview A
Discussion about how to find volume of the figure:
Student: “There’s not enough information”
Interviewer: “What information do you need?”
“I need to know how long the other sides are.”
“What would you do if you knew those numbers?”
“Multiply them.”
“Any idea what you would get when you multiply them?”
“No, it would depend on the numbers.”
“Does 17 have anything to do with these numbers?”
“No, it’s just the area of that face.”
Student Interview B
Discussion about how to find volume of the figure:
Student: “Somebody’s already done part of it for us.”
Interviewer: “What do you mean?”
“All we have to do now is multiply 17 and 6.”
“Some children think that you have to know the other two
dimensions before you can answer this question. Do
you need to know them?”
“No, not really.”
“What would you do if you knew them?”
“I’d just multiply them.”
“What would you get when you multiplied them?”
“17”
Difficult for students: 3D
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Students have considerable difficulty determining #
of cubes in 3-D rectangular buildings
Students told to first predict, then check with cubes,
then reflect and refine mental models
Student Reflection: discrepancies between predicted
and actual number of cubes
Battista, M.T. (1999). Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquirybased classroom. Journal for Research in Mathematics Education, 30, 417-448. In Lessons Learned from
Research, NCTM, 2002, pg. 75-83. In Adding It Up, National Research Council, 2001, p 284-288.
How do you find the volume and
surface area of a cube?
Table Task 3
 Build cubes with various side lengths
 Sketch, label dimensions and find volume
 Use “Examining Cubes Record Sheet” to
gather information
Examining Cubes
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Look at the Examining Cubes record sheet
Look for patterns. What is the relationship
between surface area and volume in a cube?
How might students be led to discover how to
generalize finding Volume, Area of each face
and Total Surface Area for a cube with side
length n?
Discuss with your table group
Kelly wants to wrap 20 golf balls, each in a cubeshaped box, together in one larger box. Which
arrangement will use the least wrapping paper?
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Build a box with 20 cubes
Sketch each box, label dimensions, find area
of each face and the total surface area
Display all boxes on chart paper
Label which arrangement has the largest
surface area and which has the smallest.
Post
Wait a minute…
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How can the boxes have the same volume of
20 cubes and have different surface areas?
Discuss with your table group how students
in 5th grade may respond to the above
question.
CCSS Practice Standards
Reread these practice standards and answer: How do
the exercises and the discussion questions help
students experience the richness of these Practice
Standards?
 #2 Reason abstractly and quantitatively
 #3 Construct viable arguments and critique the
reasoning of others
 #4 Model with mathematics
 #5 Use appropriate tools strategically
 #6 Attend to precision
WALT:

We are learning to:
–
–
–
Understand the concepts of area and volume as
they are sequenced in the CCSS for 3-5th grades
and incorporate the Math Practice Standards in
our teaching
Describe relationships between perimeter and
area
Describe relationships between surface area and
volume
Success Criteria:

We know we are successful when we can
describe how explorations in geometric
measurement meet the criteria of the CCSS
in both content and practice standards.