Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
Maryland College and Career Readiness
Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles,
and connect these with definitions of shapes. The emphasis is NOT on learning the terminology of geometry. Students recognize area and
perimeter as an attribute of two dimensional regions. They distinguish the difference between linear and area measures and focus on the
relationship them.
Research
Students pass through five sequential levels of development, called the Van Hiele levels, as they make sense of geometric concepts. The levels
describe how we think and what types of geometric ideas we think about. It is not based on age/grade or how much knowledge we have, but
on a student’s level of thinking. A student is likely to be at different levels at any one time. Geometric experience is the greatest single factor
influencing advancement through the levels. It is important to be aware of these levels, so that instruction can be designed to help students
move through this hierarchy or thinking. “When instruction or language is at a level higher than that of the students, there will be a lack of
communication. Students required to wrestle with objects of thought that have not been constructed at the earlier level may be forced into rote
learning and achieve only temporary and superficial success.” (Van de Walle page 208)
The first three levels (0-2) of thinking will most likely be developed during elementary school and are briefly described below. See TSCM pages
206-210 for more detailed information.
Level 0: Visualization: Students identify shapes and figures only on the bases of appearance.
Classroom instruction needs to include examples of a variety of each type of shape shown in various orientations. This encourages students not to
focus on irrelevant properties such as side length, angle measure, or orientation, but instead make generalizations about common features.
Level 1: Analysis: Students begin to analyze shapes and figures based on properties and attributes.
In order for students to be able to make generalizations about how different shapes relate to one another, students need to spend time
comparing and classifying properties of shapes.
Level 2: Informal Deduction: Students can generalize relationships between various shapes.
Students are not expected to understand or create proofs, but should be able to justify their generalizations (“if-then” reasoning) by providing
examples and non-examples.
Level 3: Formal Deduction: This is the level of the traditional high school geometry course.
Level 4: Rigor: This is the level of college mathematics.
1
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
The chart below highlights the key understandings of this unit along with important questions that teachers should pose to
promote these understandings. The chart also includes key vocabulary that should be modeled by teachers and used by
students to show precision of language when communicating mathematically.
Enduring Understandings
2

Geometric shapes can be
identified, described, and
classified by their attributes.

Area is an attribute of twodimensional regions.

Perimeter is an attribute of plane
figures.

Problems can be solved using
drawings and equations.
Essential Questions

How can geometric shapes be described
and classified?

How does what we measure determine the
units we use to measure?

How does a change in the area of an
object affect its perimeter?

How does a change in the perimeter of an
object affect its area?

What drawings and equations can be used
to solve this problem and why?
Key Vocabulary
adjacent
area
congruent
perimeter
unit
height
inch
length
measure
parallel sides
parallelogram
polygon
quadrilateral
rectangle
rhombus
square
square corner/right angle
trapezoid
vertex
vertices
width
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
Throughout this unit, students will develop their use of the 8 Mathematical Practices while learning the instructional standards. Specific
connections to this unit and instructional strategies are provided in the following chart.
Standards for Mathematical Practice
Unit Connections and Instructional Strategies
1. Make sense of problems and
persevere in solving them.
Make sense of problems involving perimeter, area and their relationship with one another.
2. Reason abstractly and quantitatively
Create an equation for a problem involving area and perimeter and be able to explain how the equation relates to the
original problem. .
Solve the equation outside of the context of the problem and reconnect the solution to the problem.

Ask questions such as these to make meaning of numbers in equations:
What do the numbers in this equation represent? How do you know that?
Is your solution reasonable? How do you know?
Explain why a statement about a shape is true or false, and provide examples and non-examples to support thinking.
Justify statements about shapes, and analyze/critique the justifications of others.
Create and test conjectures about the attributes and measures of shapes.
Create and test conjectures about the relationship between perimeter and area.
Defend:
- choice of measurement unit
- the solution
- the strategy used to solve measurement word problems

Create a risk taking environment so students are comfortable freely sharing their mathematical ideas by accepting
multiple solution paths.

Provide multiple opportunities for students to explain, explore, and record thinking.

Model how effective labeling communicates math reasoning.

Record conjectures on a chart and display them in the classroom.
Recognize shapes in the world around them.
Create drawings of figures to represent specific attributes of the figure.
Use models and drawings to represent situations involving perimeter and area.
Represent word problems using drawings and equations.

Students solve “real world” problems in which they have to determine the perimeter and area.
Use an index card to identify a right angle or “square corner”.

Use concrete materials and drawings efficiently and accurately.

Allow students to choose the manipulative/tool that is “best fit” to solve problems presented.

Discuss why one tool may be more appropriate than another.

Explain how tools can help them see patterns and relationships with shapes.

Provide students with examples of measurements that are flawed due to incorrect use of a tool. Ask students to
analyze the errors.
Create accurate drawings and representations of mathematical situations involving area and perimeter.
Use specific math vocabulary to communicate mathematical ideas.
Use labels to communicate the appropriate unit sizes (square unit, square inch, etc.)
Compute accurately.
3. Construct viable arguments and
critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
3
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
7. Look for and make use of structure
Determine categories and subcategories of shapes by identifying and reasoning about their attributes.
Test conjectures about the relationship between area and perimeter.
Explain the relationship between area and perimeter.

Compose and decompose shapes and reason about the relationship between the whole and its parts.

Recognize shapes organized into repeated units and use repeated units to create original designs.

Recognize that shapes are the same regardless of orientation
8. Look for and express regularity in
repeated reasoning
Use the formulas for area and perimeter to solve problems.
4
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
Maryland College and
Career Readiness Standards
Reason with shapes and their attributes.
Standard(s)
3.G.1 Understand that
shapes in different
categories (e.g.
rhombuses, rectangles,
and others) may share
attributes (e.g., having four
sides) AND shared
attributes can define a
larger category (e.g.,
quadrilaterals).
- Recognize rhombuses,
rectangles, and squares as
examples of quadrilaterals,
AND draw examples of
quadrilaterals that do not
belong to any of these
subcategories.
SMP.1 Make sense of
problems and persevere in
solving them
SMP.3 Construct viable
arguments and critique the
reasoning of others
SMP.6 Attend to precision
SMP.7 Look for and make
use of structure
Instructional Strategies and Resource Support
Can You Make It? Activity 8.5 (TSCM, 3-5 p. 217) (SMP.1)
- have students use multiple geoboards to compare shapes
- have students copy designs on geoboard paper for display and further discussion
Online geoboard:
http://nlvm.usu.edu/en/nav/frames_asid_277_g_1_t_3.html?open=activities&from=topic_t_3.html
-Make a shape on a geoboard that touches ___pegs. Share and compare shapes. Record shapes on
geopaper. Cut out shapes and create an anchor chart labeled with the shape name/attributes.
- A group of 10 students create quadrilaterals on their own geoboard. Students place the geoboard so all
students can view them. The teacher/ student sorts them. Students discuss in small groups how they have
been sorted.
Provide a description of a quadrilateral.
- Students create it using anglegs . (Quadrilaterals can be traced onto a piece of chart paper and labelled.
Students can also measure the quadrilaterals and find the perimeter.)
-How many different looking quadrilaterals share the described attributes?
-Trace and display the figures on anchor charts.
-Create a shape that does not have the described attribute (non-example)
Given an attribute, the students complete the Frayer Model organizer naming the attributes, drawing an
example(s), non- example(s), and a real world connection. (SMP.6- precision of vocabulary)
Mystery Definition Activity 8.6 (TSCM, 3-5 p. 224) BLM 29 (SMP.1)
Property Lists for Quadrilaterals Activity 8.8 (TSCM, 3-5 p. 226)
Sort quadrilaterals according to attributes using Quadrilateral Sorting Cards
Sort according to given rule http://illuminations.nctm.org/Activity.aspx?id=3581
Record commonalities and differences in a Venn Diagram/Table (SMP.7)
Prove if a statement is true or false. (See TSCM,3-5 p. 231 for some examples.) (SMP.3)
- If it is a _____, then it is also a ______.
- All _____ are _____.
- Some ____ are _____.
Classify angles in comparison to a right angle or “square corner”-right angle, less than a right angle, or greater
than a right angle.
Compare shapes in a category with those that look similar, but are missing a defining attribute.
Rectangles vs Parallelograms http://illuminations.nctm.org/Lesson.aspx?id=1323
5
Sample Formative
Assessments
Use a geoboard(s) to create
two different quadrilaterals
that share at least two
attributes. Record each
quadrilateral. List the shared
attributes.
Sort the quadrilaterals by the
number of square
corners/right angles. Use the
sort to answer the questions.
Sort the quadrilaterals by the
number of parallel sides. Use
the sort to answer the
questions.
Sort the quadrilaterals by the
number of congruent sides.
Use the sort to answer the
questions.
Additional sorts:
- Opposite Sides Are
Congruent vs. Opposite Sides
Are NOT Congruent
- No Adjacent Sides are
Congruent, One Pair of
Adjacent Sides are
Congruent, Two Pairs of
Adjacent Sides are Congruent
Use the chart to compare the
square and rhombus.
Molly said that the figure
below has two pairs of parallel
sides. Is she correct? Explain
your thinking.
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
Recognize perimeter as an attribute of plane figures and distinguish between
linear and area measures.
Maryland College and Career
Readiness Standards
6
Standard(s)
3.MD.8 Solve real world
and mathematical
problems involving
perimeters of polygons,
including finding the
perimeter given the side
lengths, finding an
unknown side length, and
exhibiting rectangles with
the same perimeter and
different areas or with the
same area and different
perimeter.
(3.MD.8 is addressed in
full in this unit and focuses
on distinguishing
between linear and area
measures and examining
their relationship).
MP.2 Reason abstractly
and quantitatively
MP.4 Model with
mathematics
MP.5 Use appropriate
tools strategically
Instructional Strategies and Resource Support
Same perimeter and different area:

Activity - Dog Yards, Sizing Up Measurement Grades 3-5, pgs. 73-77 (SMP.4)

Activity- Perimeters of 30, Sizing Up Measurement Grades 3-5, pgs. 78-81 (SMP.5)

Activity- Changing Garden, Navigating Through Measurement 3-5, pgs. 62 -65. (SMP.4)



Activity- Foot Stuff, Math By All Means Area and Perimeter Grades 5-6, pgs. 41-53.(SMP.
4&5)
Activity- Perimeter Stays the Same Part 1, Math By All Means Area and Perimeter Grades
5-6, pgs. 78-85.(SMP.2 &5)
Activity- Perimeter Stays the Same Part 2, Math By All Means Area and Perimeter Grades
5-6, pgs. 134-149. (SMP.7)
Same area and different perimeter:

Activity- The Area Stays the Same, Math By All Means Area and Perimeter Grades 5-6, pgs.
24-35.(SMP.2 & 5)

Activity- The Banquet Problem, Math By All Means Area and Perimeter Grades 5-6, pgs.
100 – 112. (SMP.2 &5)

Activity- Perimeter with Cuisenaire Rods, Math By All Means Area and Perimeter Grades 56, pgs. 149-162 (SMP.5)
.
Students can explore the relationship between area and perimeter at the websites below:
http://www.shodor.org/interactivate/activities/AreaExplorer/
http://www.mathplayground.com/area_perimeter.html
http://www.mathplayground.com/geoboard.html
http://www.mathplayground.com/PartyDesigner/PartyDesigner.html
Students can independently work through the lesson “area and perimeter” to understand the
relationship of area and perimeter in rectangles at this website:
http://www.learnalberta.ca/content/me5l/html/Math5.html
- Click on the activity button to be able to print a series of activities the students can work
through to develop this relationship.
- Click on the assessment button to access the following activity:
Students will draw a rectangular dog pen on grid paper and then explain in words the effect
of decreasing the length of the dog pen by 1 unit, 2 units, and 3 units on its perimeter and on
its area. Students then generalize the relationship between changing one dimension of the
rectangle to its perimeter and its area.
Activity- Spaghetti and Meatballs for All, Math By All Means Area and Perimeter Grades 5-6,
pgs. 54-74. (SMP.2, 4 & 5)
Activity- The Biggest Garden, Teaching Arithmetic Lessons for Introducing Multiplication Grade
3, by Marilyn Burns pgs. 139-141. (SMP.2 & 4)
Sample Formative Assessments
Create three rectangles that have a perimeter
of 24 yds., but have different areas.
Create three rectangles that have an area of
12 square units, but have different perimeters.
Nancy said that the area of her garden is 28
square feet and that the perimeter is 22 feet.
Sam said that the area of his garden is 28
square feet and that the perimeter is 58 feet.
Bryan says they are both correct. Do you
agree? Defend your thinking using words,
numbers, and/or drawings.
The perimeter of the rectangular garden
measures 42 ft. What might the dimensions of
the garden be?
What might the area of a garden with a
perimeter of 24 yards be?
Sam’s yard measures 27 feet by 15 feet. He
wants to place a fence around the perimeter
of his yard. How many feet of fencing will he
need to buy?
Jen has a square piece of wood with the area
of 36 square feet. What is the perimeter of the
square piece of wood?
Assessment- The Blob, Math By All Means Area
and Perimeter Grades 5-6, pgs. 36-40. You
want to find the area of the “blob” below.
Your friend has a suggestion. Do you agree
with his method? Explain your thinking.
Assessment- The Garden Problem, Math By All
Means Area and Perimeter Grades 5-6, pgs.
126-133. Greg, Jenna, and Bill are working
together to make the garden plot larger. The
garden plot has a fence around the perimeter.
Each has a different idea. Who is correct?
Explain your thinking.
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
Maryland College and Career Readiness
Standards
Instructional Strategies and Resource Support
Sample Formative Assessments
Recognize perimeter as an attribute of plane figures and distinguish
between linear and area measures.
Standard(s)
7
3.MD.8 Solve real world and mathematical
problems involving perimeters of polygons,
including finding the perimeter given the side
lengths, finding an unknown side length, and
exhibiting rectangles with the same perimeter
and different areas or with the same area and
different perimeter.
(3.MD.8 is addressed in full in this unit and
focuses on distinguishing between linear and
area measures and examining their
relationship).
MP.2 Reason abstractly and quantitatively
MP.4 Model with mathematics
MP.5 Use appropriate tools strategically
3.OA.7 Fluently multiply and divide within 100,
using strategies such as the relationship
between multiplication and division (e.g.
knowing that 8 x 5 = 40 ,one knows 40/8=5) or
properties of operations.
-By the end of Grade 3, know from memory all
products of two one-digit numbers.
Activity- Design a House, Sizing Up Measurement Grades 3-5, pgs. 87-92
(SMP.4)
Activity- My Dream Bedroom, Sizing Up Measurement Grades 3-5, pgs. 92-95
(SMP.4)
Good Questions, by Sullivan and Liburn
- A rectangle has an area of 36 cm squared. What might its perimeter be?
- The area of a 4 by 4 ft. square table top is 16 sq. ft. By putting four tables
together what shaped tables can I make and what is the perimeter of those
shapes?
Great Ways to Differentiate Math Instruction, by Marian Small
- The areas of two shapes are almost the same, but the perimeters are very
different. What might the shapes be?
- One square has a greater area than another. Does its perimeter also have
to be greater? -One rectangle has a greater area then another. Does its
perimeter also have to be greater?
Refer to the Fact Fluency chart (page 8) for the instructional sequence of the
multiplication and division facts.
Lynn is making a display board for the
school talent show. The display board
is a 5 ft. by 8 ft. rectangle. If ribbon
costs $2 per foot, how much will it cost
to add a ribbon border around the
entire display board?
Ingrid has a rectangular flower garden
that measures 8 ft. by 5 ft. One weed
mat can cover 20 square ft. How
many weed mats will she need to
cover the entire garden?
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade Three)
Geometry and Measurement- Unit 7
Third Grade Fact Fluency
3.OA.7
As students show mastery of a set of multiplication and division facts, they should progress to the next
set in order to know from memory all products of two one-digit numbers by the end of Grade 3.
Set
Number
Multiplication and Division
Facts
1
Foundation Facts
x 2, x 10, x 5, x 1, x 0
÷ 2, ÷ 10, ÷ 5, ÷1, ÷ 0
Possible Strategies




Skip counting
Doubling
Base ten (10’s)
Properties
Foundation Facts

AND



Multiply by two and then add
another group (3’s)
Double double (4’s)
Double a multiple of 3 (6’s)
Multiply by 5 and then add
another set (6’s)
2
x 3, x 4, x 6
÷ 3, ÷ 4, ÷ 6
Mastering the Basic
Facts in
Multiplication and
Division by Susan
O’Connell and John
SanGiovanni
Pages 27-90
Pages 91-128
Assessment
- Observation of
classroom
performance
- Individual
Formative
Assessments
- Focused
strategy-based
assessments
- Time
assessments
Foundation Facts, x 3,
x 4, x 6, ÷ 3, ÷ 4, ÷ 6


AND

3
x 9, x 8, x 7
÷ 9, ÷ 8, ÷ 7
8
Double double double (8’s)
Multiply by 10 and then take
away the extra group (9’s)
Commutative property (leaves
only 7 x7 to learn)
Pages 129-159
*Students should
have
opportunities to
assess orally
and/or in
writing.