Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15
Standards for Mathematical Practice
Critical Area(s): Area, 2-Dimensional Geometry
FOCUS for Grade 3
Supporting Work
20% of time
3.MD.B.3-4
3.G.A.1-2
1. Make sense of problems and persevere in solving them.
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.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
Major Work
Additional Work
70% of time
10% of time
3.OA.A.1-2-3-4
3.NBT.A.1-2-3
3.OA.B.5-6
3.MD.D.8
3.OA.C.7
3.OA.D.8-9
3.NF.A.1-2-3
3.MD.A.1-2
3.MD.C.5-6-7
Fluency standards: 3.OA.C.7 and 3.NBT.A.2
Standards in bold are specifically targeted within instructional materials.
Domains:
Geometry
Measurement and Data
Clusters:
Clusters outlined in bold should drive the learning for this period of instruction.
3.MD.C Understand concepts of area and relate area to multiplication and
division.
Standards:
3.G.A Reason with shapes and their attributes.
3.G.A.1 Understand that shapes in different categories (e.g., rhombuses,
rectangles, and others) may share attributes (e.g., having four sides), and
that the 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.
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3.MD.C.5 Recognize area as an attribute of plane figures and understand
concepts of area measurement.
b. A plane figure which can be covered without gaps or overlaps by n
unit squares is said to have an area of n square units.
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Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15
Foundational Learning
Future Learning
2.G.A.1
3.MD.D.8
3.MD.C.5
4.G.A.2
3.G.A.2
3.NF.A
Key Student Understandings
Assessments
 Students understand that two-dimensional figures can be described, classified, and analyzed based on their
 Formative Assessment Strategies
attributes.


Students identify shapes based on their attributes, and they begin to explore the relationships between
shape categories.

Evidence for Standards-Based Grading
Students understand that shapes in different categories may share attributes that place them in a larger
category.
Common Misconceptions/Challenges
3.G.A Reason with shapes and their attributes.
 Students may not realize that an attribute can describe some but not all of the shapes in a group. Point to each shape and ask if that attribute applies to
the individual shape and if it does not apply to all, you cannot use this attribute to describe all of the shapes in that group.
 Students do not recognize that a square is a special kind of rectangle (“square rectangle”) because it has all of the properties of a rectangle; a square is
also a special kind of rhombus because it has all the properties of a rhombus. Students may not look at the properties of a square that are characteristic
of other figures as well. Using straws to make four congruent figures have students change the angles to see the relationships between a rhombus and a
square. As students develop definitions for these shapes, relationships between the properties will be understood.
 In initial explorations, students might not use precise mathematical language to identify shapes and their attributes, e.g., they may refer to a rhombus as
a diamond or use corner instead of angle or vertex. Model and encourage precision in language throughout lessons, and give students opportunities to
rephrase responses with the correct formal terminology.
3.MD.C Understand concepts of area and relate area to multiplication and division.
 Students don’t connect the idea of area to their study of multiplication. Make explicit connections between arrays and area models (both concretely and
pictorially), to help students see that these models, and the related concepts of multiplication and area, are related.
 Students may lose track while counting unit squares for a given figure. Encourage students to generate strategies they might use to help them keep
track.
 Students don’t recognize that unit squares must be the same size. As students create and draw models exploring the area of rectilinear figures,
reinforce precision in the use of unit squares.
 Students may confuse the concepts of area and perimeter when they measure the sides of a rectangle and then multiply. Because they measured the
length of the sides, they think the attribute they are calculating is also length (perimeter). Pose and discuss problem situations that require students to
explain whether the situation is focused on area or perimeter.
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Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15
Instructional Practices
Domain: 3.G
Cluster: 3.G.A Reason with shapes and their attributes.
3.G.A.1 Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that
the 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.

In Grade 2, students identify and draw triangles, quadrilaterals, pentagons,
and hexagons. Grade 3 students build on this experience and further
investigate quadrilaterals (technology may be used during this exploration).
Students recognize shapes that are and are not quadrilaterals by examining
the properties of the geometric figures. They conceptualize that a
quadrilateral must be a closed figure with four straight sides and begin to
notice characteristics of the angles and the relationship between opposite
sides. Students should be encouraged to provide details and use proper
vocabulary when describing the properties of quadrilaterals.
(Progressions for the CCSSM: K-6, Geometry, The Common Core Standards Writing Team, June 2012)
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Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15

The broad category quadrilaterals include all types of parallelograms, trapezoids and other four-sided figures. Explore quadrilaterals by having students
use properties of geometric figures to compare shapes that are and are not quadrilaterals. Parallelograms include: squares, rectangles, rhombi, and/or
other shapes that have two pairs of parallel sides. Summaries should include discussions around characteristics of the angles and the relationship
between opposite sides. Students should be encouraged to provide details and use proper vocabulary when describing the properties of quadrilaterals.
(Progressions for the CCSSM; K-6, Geometry, CCSS Writing Team, June 2012, page 13)
o
Examples for exploration of quadrilaterals:
 Draw a picture of a quadrilateral. Draw a picture of a rhombus. How are they alike? How are they different?
 Is a quadrilateral a rhombus? Is a rhombus a quadrilateral? Justify your thinking.

Grade 4 students should have a firm foundation of several shape categories; these categories can be the basis for thinking about the relationships
between classes. Students should classify shapes by attributes, and draw shapes that fit specific categories. Students can form larger, categories, such as
the class of all shapes with four sides, or quadrilaterals, and recognize that it includes other categories, such as squares, rectangles, rhombuses,
parallelograms, and trapezoids. They also recognize that there are quadrilaterals that are not in any of those subcategories.

The notion of congruence (“same size and same shape”) may be part of classroom conversation but the concepts of congruence and similarity do not
appear until middle school.
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Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15

In the U.S., the term “trapezoid” may have two different meanings; research identifies these as inclusive and exclusive definitions. (Progressions for the
CCSSM: Geometry, The Common Core Standards Writing Team, June 2012)
o
o
The inclusive definition states: A trapezoid is a quadrilateral with at least one pair of parallel sides.
The exclusive definition states: A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Domain: 3.MD
Cluster: 3.MD.C Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.C.5 Recognize area as an attribute of plane figures and understand concepts of area measurement.
a. A square with side length 1 unit, called “a unit square,” is said to have “one square unit” of area,
and can be used to measure area.
b. A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an
area of n square units.

This standard calls for students to explore the concept of covering a region with “unit squares,” which
should include square tiles and then shading on grid or graph paper. Based on students’ development,
they should have ample experiences filling a region with square tiles before transitioning to pictorial
representations on graph paper.
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Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15

The concept of multiplication can be related to the area of rectangles using arrays. Students need to discover that the length of one dimension of a
rectangle tells how many rows (groups) of unit squares are in a rectangular array, and the length of the other dimension of the rectangle tells how many
unit squares are in each row (items in each group). Ask questions about the dimensions if students do not make these discoveries. For example:
o How do the squares covering a rectangle compare to an array?
o How is multiplication used to count the number of objects in an array?

Students need to develop the meaning for computing the area of a rectangle. The area of a rectangle can be determined by having students lay out unit
squares and count how many square units it takes to completely cover the rectangle completely without overlaps or gaps. A connection needs to be
made between the number of squares it takes to cover the rectangle and the dimensions of the rectangle. Ask questions such as:
o What does the length of a rectangle describe about the squares covering it?
o What does the width of a rectangle describe about the squares covering it?

Provide students with the area of a rectangle (i.e., 42 square inches) and have them determine possible lengths and widths of the rectangle. Expect
different lengths and widths such as, 6 inches by 7 inches or 3 inches by 14 inches.
Differentiation
3.G.A Reason with shapes and their attributes.
Struggling/On-Level:
 Consider exploring and classifying triangles of all shapes and sizes, as a foundation for exploring quadrilaterals.
Have students verbalize attributes of triangles that identify if a shape is or is not a triangle.
 Have students sort shapes. Allow students to play “Guess my rule” when sorting. Students can work in small
groups to pick an attribute for a “rule”. Others can then try to find shapes that “fit the rule” or “don’t fit the rule”.
Discussion should focus on increasing vocabulary using the attributes of the different shapes.
 Provide students with ample opportunities to develop Geometric Thinking in Level 0: Visualization and Level 1:
Analysis.
Literacy Connections

Academic Vocabulary Terms

Vocabulary Strategies

Literacy Strategies
Source:
https://www.researchgate.net/publication/228460004_Students_Movement_Through_Van_Hiele_Levels_In_A_Dynamic_Geometry_Guided_Reinvention_Process
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Mathematics 2016-17—Grade 3
Weeks 33-34—May
enVisionmath2.0—Topic 15
3.MD.C Understand concepts of area and relate area to multiplication and division.
Struggling/On-Level:
 Determine a specific area to be covered and identify the boundaries of the area. Cover the area with square tiles or
pieces of paper. After several experiences, increase the size of the area so students are pushed to use a
multiplicative counting strategy.
 Give students a specific area measurement, and have them model or draw all the possible figures that could have
the given area.
Above Level:
 Have students compare objects/figures with different areas, first visually and then by using a unit of measurement.
Students can share and justify their thinking with concrete models/representations and verbal/written
explanations.
The Common Core Approach to Differentiating Instruction (engageny How to Implement a Story of Units, p. 14-20)
Linked document includes scaffolds for English Language Learners, Students with Disabilities, Below Level Students, and
Above Level Students.
Resources
enVisionmath2.0
Developing Fluency
Multiplication Fact Thinking Strategies
Topic 15 Pacing Guide
Grade 3 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
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