Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Common Core and Research from the CCSS Progression Documents
Geometry
Students learn to analyze and relate categories of two-dimensional and three-dimensional shapes explicitly
based on their properties.5.G.4
Based on analysis of properties, they classify two-dimensional figures in hierarchies. For example, they
conclude that all rectangles are parallelograms, because they are all quadrilaterals with two pairs of opposite,
parallel, equal-length sides (MP3). In this way, they relate certain categories of shapes as subclasses of other
categories.5.G.3 This leads to understanding propagation of properties; for example, students understand that
squares possess all properties of rhombuses and of rectangles. Therefore, if they then show that rhombuses’
diagonals are perpendicular bisectors of one another, they infer that squares’ diagonals are perpendicular
bisectors of one another as well.
• Note that in the U.S., that the term “trapezoid” has two different meanings: T(E)exclusive and T(I) inclusive
 T(E): a trapezoid is a quadrilateral with exactly one pair of parallel sides
 T(I): a trapezoid is a quadrilateral with at least one pair of parallel sides.
These different meanings result in different classifications. According to T(E), a parallelogram is not a
trapezoid; according to T(I), a parallelogram is a trapezoid.
th
th
Both definitions are legitimate; however; when teaching trapezoids to 4 and 5 graders, we will use the
inclusive definition since most college bound textbooks and PARCC use the inclusive definition.
• A note about research The ability to describe, use, and visualize the effects of composing and decomposing
geometric regions is significant in that the concepts and actions of creating and then iterating units and higher-order units in the context of constructing patterns, measurin
g, and computing are established bases for mathematical understanding and analysis. Additionally, there is suggestive evidence that this type of composition corresponds
with, and may support, children’s ability to compose and decompose numbers.
Coordinate Planes
Students extend their Grade 4 pattern work by working briefly with two numerical patterns that can be related and examining these relationships within sequences of
5.OA.3
ordered pairs and in the graphs in the first quadrant of the coordinate plane.
This work prepares students for studying proportional relationships and functions in middle
school.
Fifth grade students extend their knowledge of the coordinate plane, understanding the continuous nature of two-dimensional space and the role of fractions in specifying
locations in that space. Thus, spatial structuring underlies coordinates for the plane as well, and students learn both to apply it and to distinguish the objects that are
structured. For example, they learn to interpret the components of a rectangular grid structure as line segments or lines (rather than regions) and understand the precision of
location that these lines require, rather than treating them as fuzzy boundaries or indicators of intervals. Students learn to reconstruct the levels of counting and
quantification that they had already constructed in the domain of discrete objects to the coordination of (at first) two continuous linear measures. That is, they learn to apply
5.G.1
their knowledge of number and length to the order and distance relationships of a coordinate grid and to coordinate this across two dimensions.
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Although students can often “locate a point,” these understandings are beyond simple skills. For example, initially, students often fail to distinguish between two different
ways of viewing the point (2, 3), say, as instructions: “right 2, up 3”; and as the point defined by being a distance 2 from the x-axis and a distance 3 from the y-axis. In these
two descriptions the 2 is first associated with the x-axis, then with the y-axis.
They connect ordered pairs of (whole number) coordinates to points on the grid, so that these coordinate pairs constitute numerical objects and ultimately can be operated
upon as single mathematical entities. Students solve mathematical and real-world problems using coordinates. For example, they plan to draw a symmetric figure using
5.G.2
computer software in which students’ input coordinates that are then connected by line segments.
Progressions for the Common Core State Standards in Mathematics (draft)
Commoncoretools.wordpress.com
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
The chart below highlights the key understandings of this cluster 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
Two-Dimensional figures are classified by their
properties.
Essential Questions
How can plane figures be categorized and classified?
What is a quadrilateral?
Two-Dimensional figures can fit into more than one
category.
Identify and describe properties of two-dimensional
figures more precisely.
What are the properties of quadrilaterals?
How can you classify different types of quadrilaterals?
How are quadrilaterals alike and different?
Graphical representations can be used to make
predications and interpretations about real world
situations.
How can angle and side measures help us to create and
classify triangles?
Why are some quadrilaterals classified as parallelograms?
Why is a square always a rectangle?
What are ways to classify triangles?
Where is geometry found in your everyday world?
How does the coordinate system work?
How do coordinate grids help you organize information?
Key Vocabulary
The terms below are for teacher reference only and
are not to be memorized by the students. Teachers
should present these concepts to students with models
and real life examples. Students should understand the
concepts involved and be able to recognize and/or
demonstrate them with words, models, pictures, or
numbers.
Geometry
acute angle
acute triangle
circle
congruence/congruent
equilateral triangle
half circle/semicircle
hexagon
irregular polygon
isosceles triangle
kite
obtuse angle
parallel lines
parallelogram
pentagon
perpendicular lines
plane figure
polygon
quadrilateral
rectangle
regular polygon
right angle
right triangle
rhombus/rhombi
scalene triangle
square
triangle
trapezoid
two-dimensional
vertex
Coordinate Planes
Coordinate system
Coordinate plane
First quadrant
Points
Lines
Axis/Axes
x-axis
y-axis
Horizontal
Vertical
Intersection of lines Origin
Ordered Pairs
Coordinates
x-coordinate
y-coordinate
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Throughout this cluster, students will develop their use of the 8 Mathematical Practices while learning the instructional standards.
Specific connections to this cluster and instructional strategies are provided in the following chart.
Standards for Mathematical Practice
Cluster Connections and Instructional Strategies
1. Make sense of problems and persevere in solving
them
Students make sense of problems involving two-dimensional figures based on their geometric properties.
Students make sense of solving real world problems involving points on the coordinate plane.
2. Reason abstractly and quantitatively
Students demonstrate abstract reasoning about rational relationships among geometric properties. Students go
beyond simple recognition to an analysis of the properties and how they interrelate.
Students demonstrate abstract reasoning about ordered pairs with their visual representations.
Students construct and critiques arguments regarding their knowledge of triangles and the ability to belong to one
or more of the categories and sub-categories.
Students construct and critique arguments regarding patterns and relationship of ordered pairs as they are plotted
on a coordinate plane to represent real-world contexts.
Students use tables to identify and draw all three types of triangles comparing their attributes in mathematical and
real-world contexts.
Students use the coordinate plane to compare two numbers in mathematical and real-world contexts.
Students select and use tools such as tables and the quadrilateral hierarchy to represent situations involving the
categories and sub-categories of two-dimensional figures.
Students select and use tools such as number line models and the coordinate plane to represent situations involving
positive numbers.
Students attend to the geometric precision when classifying two-dimensional figures in the hierarchy.
Students attend to the language of real-world situations to determine how far to travel from the origin and the given
direction of the coordinates being represented.
Students relate the attributes belonging to a category of two-dimensional figures also belong to all subcategories of
that category.
Students relate the structure of number lines to values of positive integers as they use the coordinate plane.
Students relate new experiences to experiences with similar contexts when studying the hierarchy of polygons based
on properties.
Students relate new experiences to experiences with similar contexts when studying positive representations of
distance and quantity.
3. Construct viable arguments and critique the
reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated
reasoning
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Classify two-dimensional figures into categories based on their
properties.
Maryland College and CareerReady Standards
5.G.3 Understand that
attributes belonging to
a category of twodimensional figures
also belong to all
subcategories of that
category. For example,
all rectangles have four
right angles and
squares are rectangles,
so all squares have four
right angles.
SMP.3 Construct viable
arguments and critique
the reasoning of others
Instructional Strategies and Resource Support
This standard calls for students to reason about the attributes (properties) of shapes. Student should have
experiences discussing the property of shapes and reasoning.
Example: Examine whether all quadrilaterals have right angles. Give examples and non-examples.
Example: If the opposite sides on a parallelogram are parallel and congruent, then rectangles are parallelograms
A sample of questions that might be posed to students include:
A parallelogram has 4 sides with both sets of opposite sides parallel. What types of quadrilaterals are
parallelograms?
Regular polygons have all of their sides and angles congruent. Name or draw some regular polygons.
All rectangles have 4 right angles. Squares have 4 right angles so they are also rectangles. True or False?
A trapezoid has 2 sides parallel so it must be a parallelogram. True or False?
http://illuminations.nctm.org/ActivityDetail.aspx?ID=70
Use tangrams to have students make shapes using a designated number of tangram pieces. For example: make a
square, rectangle, trapezoid and a parallelogram by using 3 tangram pieces for each. Which of the figures can you
make using only 2 tangram pieces?
Make quadrilaterals using Geoboards. Record several on dot paper. Have students bring the dot paper quadrilaterals
to small group instruction and use them to sort the quadrilaterals by different attributes or properties.
SMP.7 Look for and
make use of structure
SMP.5 Use appropriate
tools strategically
Teaching Student-Centered Mathematics (Grades 3-5)
pg. 212 Shape Sorts - Activity 8.1 (BLMs 20-26)
pg. 213 What's My Shape - Activity 8.2
pg. 214 Constructing & Dissecting Shapes (BLM 27)
pg. 217 *Can You Make It - Activity 8.5
pg. 221 Categories of Two-Dimensional Shapes - Table 8.1
Sample Formatives
What property do
they have in
common?
Name each shape.
Explain how the
two shapes are
alike and how they
are different.
Use the Geoboards
to draw two
different
quadrilaterals.
Name each of the
quadrilaterals.
Compare the
properties of the
quadrilaterals you
drew.
Complete the table
by placing Xs in
each column to
indicate properties
of each figure.
Students make use of structure to build a logical progression of statements and explore hierarchical relationships
among 2-dimensional shapes (SMP.3, SMP.7)
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Classify two-dimensional figures into categories based on
their properties.
Maryland College and CareerReady Standards
5.G.4. Classify twodimensional figures
in a hierarchy based
on properties.
Instructional Strategies and Resource Support
Figures from previous grades: polygon, rhombus/rhombi, rectangle, square, triangle, quadrilateral, pentagon,
hexagon, cube, trapezoid, half/quarter circle, circle
SMP.3 Construct
viable arguments
and critique the
reasoning of others
Example: Create a Hierarchy Diagram using the following terms:
 polygons – a closed plane figure formed from line segments that meet only at their endpoints.
 quadrilaterals - a four-sided polygon.
 parallelogram – a quadrilateral with two pairs of parallel and congruent sides.
 rectangle – a quadrilateral with two pairs of congruent, parallel sides and four right angles.
 rhombus – a parallelogram with all four sides equal in length.
 square – a parallelogram with four congruent sides and four right angles.
 trapezoid-a quadrilateral with at least one pair of parallel sides.
SMP.7 Look for and
make use of
structure
Student should be able to reason about the attribute/properties of shapes by examining: What are ways to classify
triangles? Which quadrilaterals have opposite angles congruent and why is this true of certain quadrilaterals? How
many lines of symmetry does a regular polygon have? Why can’t all rectangles be classified as squares?
SMP.5 Use
appropriate tools
strategically
Teaching Student-Centered Mathematics (Grades 3-5)
pg. 222 Figure 8.11 (Bottom figure)
pg. 225 Triangle Sort - Activity 8.7 (BLM 29)
pg. 226 Property Lists for Quadrilaterals -Activity 8.8 (BLM 30-33)
pg. 227 Diagonal Strips - Activity 8.9 , *Figure 8.16
pg. 230 Minimal Defining Lists - Activity 8.11
pg. 231 True or False - Activity 8.12
Sample Formatives
Put the labels from the
word box in the
appropriate box to
make the diagram true.
Students make use of structure to build a logical progression of statements and explore hierarchical relationships
among 2-dimensional shapes (MP.3, MP.7)
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Graph points on the coordinate plane to solve real-world and
mathematical problems.
Maryland College and Career-Ready Standards
5.G.1. Use a pair of perpendicular number
lines, called axes, to define a coordinate
system, with the intersection of the lines
(the origin) arranged to coincide with the 0
on each line and a given point in the plane
located by using an ordered pair of
numbers, called its coordinates.
Understand that the first number indicates
how far to travel from the origin in the
direction of one axis, and the second
number indicates how far to travel in the
direction of the second axis, with the
convention that the names of the two axes
and the coordinates correspond (e.g., xaxis and x-coordinate, y-axis and ycoordinate).
5.G.2. Represent real world and
mathematical problems by graphing points
in the first quadrant of the coordinate
plane, and interpret coordinate values of
points in the context of the situation.
SMP.6 Attend to precision
SMP.4 Model with mathematics
SMP.2 Reason abstractly and
quantitatively
SMP.5 Use appropriate tools strategically
Instructional Strategies and Resource Support
TSCM-page 239
5.G.1 and 5.G.2 These standards deal with only the first quadrant (positive numbers) in the
coordinate plane. Although students can often “locate a point,” these understandings are
beyond simple skills. For example, initially, students often fail to distinguish between two
different ways of viewing the point (2, 3), say, as instructions: “right 2, up 3”; and as the point
defined by being a distance 2 from the y-axis and a distance 3 from the x-axis. In these two
descriptions the 2 is first associated with the x-axis, then with the y-axis.
Sample Formatives
Plot the ordered
pairs on the
coordinate plane.
Label the points as
indicated.
Students precisely describe the coordinate of points and the relationship of the coordinate
plane to the number line (SMP.6). Students both generate and identify relationship in
numerical patterns, using the coordinate plane as a way of representing these relationship and
patterns (SMP.4)
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Carroll County Public Schools Elementary Mathematics Instructional Guide (5th Grade)
August-September (12 days)
Unit #1 : Geometry
Graph points on the coordinate plane to solve real-world
and mathematical problems.
Maryland College and Career-Ready Standards
5.G.2. Represent real world and
mathematical problems by graphing points in
the first quadrant of the coordinate plane,
and interpret coordinate values of points in
the context of the situation.
Instructional Strategies and Resource Support
This standard references real-world and mathematical problems, including the traveling from
one point to another and identifying the coordinates of missing points in geometric figures,
such as squares, rectangles, and parallelograms.
Example 1:
SMP.6 Attend to precision
SMP.4 Model with mathematics
SMP.5 Use appropriate tools strategically
SMP.1 Make sense of problems and
persevere in solving them
Sample
Formatives
Plot the
following points
on the
coordinate grid.
Plot a fourth
point on the grid
to form a
rectangle with all
4 point are
connected.
Write the
coordinates for
the point you
plotted.
Students precisely describe the coordinate of points and the relationship of the coordinate
plane to the number line (SMP.6). Students both generate and identify relationship in
numerical patterns, using the coordinate plane as a way of representing these relationship and
patterns (SMP.4)
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