Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16
Standards for Mathematical Practice
Critical Area(s): Area
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:
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.MD.D Reason with shapes and their attributes.
3.MD.D.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 perimeters.
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3.MD.C.7 Relate area to the operations of multiplication and addition.
b. Multiply side lengths to find areas of rectangles with whole number
side lengths in the context of solving real world and mathematical
problems, and represent whole-number products as rectangular
areas in mathematical reasoning.
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Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16
Foundational Learning
2.MD.A
2.MD.B.5
3.MD.C
Future Learning
4.MD.A.3
Key Student Understandings
 Students understand that perimeter is a measurable attribute of polygons; they understand that perimeter
is the measure of the distance around a polygon.

Students connect their understanding of addition and multiplication to determine different methods for
finding perimeters of polygons.

Students understand the difference between the measures of perimeter and area; students explore various
rectangles to determine if there is a relationship between area and perimeter.
Assessments

Formative Assessment Strategies

Evidence for Standards-Based Grading
Common Misconceptions/Challenges
3.MD.D Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
 When presented with a drawing of a rectangle with only two of the side lengths shown or a problem situation with only two of the side lengths provided,
students may think these are the only dimensions needed to find the perimeter. Encourage students to fill in the appropriate dimensions on the other
side(s) of the rectangle. With problem situations, encourage students to use a drawing to represent situations in order to find the perimeter.

Students may confuse area and perimeter. At this point in the year, students should have a solid sense of area through
their work with multiplication. Be sure to spend time distinguishing between these two measurements. After
conceptual understanding is explored and established, use of a visual (examples, see right) may be helpful.

When exploring irregular polygons or polygons with more than 4 sides, students may skip a segment or end the count
incorrectly. Model and encourage students to develop a method for keeping track of each segment as they work.
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 in 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 35-36—May/June
enVisionmath2.0—Topic 16
Instructional Practices
Domain: 3.MD
Cluster: 3.MD.D Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.
3.MD.D.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 perimeters.

A perimeter is the boundary of a two-dimensional shape. For a polygon, the length of the perimeter is the sum of the lengths of the sides. Initially, it is
useful to have sides marked with unit length marks, allowing students to count the unit lengths. Later, the lengths of the sides can be labeled with
numerals. As with all length tasks, students need to count the length-units and not the end-points. Next, students learn to mark off unit lengths with a
ruler and label the length of each side of the polygon.

Students can develop an understanding of the concept of perimeter by walking around the perimeter of a room, using rubber bands to represent the
perimeter of a plane figure on a geoboard, or tracing around a shape on an interactive whiteboard. They find the perimeter of shapes and objects; use
addition to find perimeters; and recognize the patterns that exist when finding the sum of the lengths and widths of rectangles.

Students should also use tools, such as geoboards, tiles, and graph paper to find all the possible rectangles that have a given perimeter (e.g., find all
possible rectangles with a perimeter of 14 cm.) They should record these possibilities using dot or grid paper, compile the possibilities into an organized
list or a table, and determine whether they have all the possible rectangles. Following this experience, students can reason about connections between
their representations, side lengths, and the perimeter of the rectangles.

Given a perimeter and a length or width, students use objects or pictures to find the missing length or width. They justify and communicate their
solutions using words, diagrams, pictures, numbers, and an interactive whiteboard. Students use geoboards, tiles, graph paper, or technology to find all
the possible rectangles with a given area (e.g. find the rectangles that have an area of 12 square units.) They record all the possibilities using dot or graph
paper, compile the possibilities into an organized list or a table, and determine whether they have all the possible rectangles. Students then investigate
the perimeter of the rectangles with an area of 12.
The patterns in the chart allow for opportunities for students to identify the factors of 12, connect the results to the commutative property, and discuss
the differences in perimeter within the same area. This chart can also be used to investigate rectangles with the same perimeter. It is important to
include squares tiles in the investigation.
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Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16

For rectangles, parallelograms, and regular polygons, students can discuss and justify faster ways to find the perimeter length than just adding all of the
lengths. Rectangles and parallelograms have opposite sides of equal length, so students can double the lengths of adjacent sides and add those numbers
or add lengths of two adjacent sides and double that number. A regular polygon has all sides of equal length, so its perimeter length is the product of one
side length and the number of sides. Provide opportunities for students to explore and discuss and generalize these strategies, rather than simply telling
them to students.

Students need to explore how measurements are affected when one attribute to be measured is held constant and the other is changed. Using square
tiles, students can discover that the area of rectangles may be the same, but the perimeter of the rectangles varies. Geoboards can also be used to
explore this same concept.

Perimeter problems for rectangles and parallelograms often give only the lengths of two adjacent sides or only show numbers for these sides in a
drawing of the shape. The common error is to add just those two numbers. Having students first label the lengths of the other two sides as a reminder is
helpful. Students then find unknown side lengths in more difficult “missing measurements” problems and other types of perimeter problems.
(Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 16)
(Progressions for the CCSSM, K-5 Geometric Measurement, CCSS Writing Team, June 2012, page 16)
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Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16
Domain: 3.MD
Cluster: 3.MD.C Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
3.MD.C.7 Relate area to the operations of multiplication and addition.
a. Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the
side lengths.
b. Multiply side lengths to find areas of rectangles with whole number side lengths in the context of solving real world and mathematical problems,
and represent whole-number products as rectangular areas in mathematical reasoning.
c. Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area
models to represent the distributive property in mathematical reasoning.
d. Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the
non-overlapping parts, applying this technique to solve real world problems.

Students can learn how to multiply length measurements to find the area of a rectangular region. But, in order that they make sense of these quantities,
they must first learn to interpret measurement of rectangular regions as a multiplicative relationship of the number of square units in a row and the
number of rows. This relies on the development of spatial structuring. To build from spatial structuring to understanding the number of area-units as the
product of number of units in a row and number of rows, students might draw rectangular arrays of squares and learn to determine the number of
squares in each row with increasingly sophisticated strategies, such as skip-counting the number in each row and eventually multiplying the number in
each row by the number of rows.

They learn to partition a rectangle into identical squares by anticipating the final
structure and forming the array by drawing line segments to form rows and columns.
They use skip counting and multiplication to determine the number of squares in the
array.

Many activities that involve seeing and making arrays of squares to form a rectangle
might be needed to build robust conceptions of a rectangular area structured into
squares.
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Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16

Students should understand and explain why multiplying the side lengths of a rectangle yields the same measurement of area as counting the number of
tiles (with the same unit length) that fill the rectangle’s interior. For example, students might explain that one length tells how many unit squares in a row
and the other length tells how many rows there are. (Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 17)

Students should tile rectangles and then multiply the side lengths to show the measurements are the same.
To find the area one
could count the squares
OR multiply 3 x 4 = 12.

Students should solve real world and mathematical problems.
o Example: Drew wants to tile the bathroom floor using 1 foot tiles. How many square foot tiles will he need?

Students might solve problems such as finding all the rectangular regions with whole-number side lengths that have an area of 12 unit squares, doing this
for larger rectangles (e.g., enclosing 24, 48, 72 unit squares), making sketches rather than drawing each square. Students learn to justify their belief they
have found all possible solutions. (Progressions for the CCSSM, Geometric Measurement, CCSS Writing Team, June 2012, page 18)

Using concrete objects or drawings students build competence with composition and decomposition of shapes, spatial structuring, and addition of area
measurements. Students learn to investigate arithmetic properties using area models. For example, they learn to rotate rectangular arrays physically and
mentally, understanding that their areas are preserved under rotation, and thus, for example, 4 x 7 = 7 x 4, illustrating the commutative property of
multiplication. Students also learn to understand and explain that the area of a rectangular region of, for example, 12 length-units by 5 length-units can
be found either by multiplying 12 x 5, or by adding two products, e.g., 10 x 5 and 2 x 5, illustrating the distributive property. (Progressions for the CCSSM,
Geometric Measurement, CCSS Writing Team, June 2012, page 18)
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Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16

This standard uses the word rectilinear. A rectilinear figure is a polygon that has all right angles.
o Example: A storage shed is pictured below. What is the total area? How could the figure be decomposed to help find the area?
o
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Example: Students can decompose a rectilinear figure into different rectangles. They find the area of the figure by adding the areas of each of the
rectangles together.
Property of MPS
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Mathematics 2016-17—Grade 3
Weeks 35-36—May/June
enVisionmath2.0—Topic 16
Differentiation
3.MD.D Geometric measurement: recognize perimeter as an attribute of plane figures and distinguish between linear and
area measures.
Struggling/On-Level:
 Initially, have struggling students mark graph paper with unit length marks, allowing students to count the unit lengths.
Later, the lengths of the sides can be labeled with numerals.
 Have tools available for students to use as needed; some students may need to work with concrete tools longer than others.
To transition students from concrete tools to pictorial models/drawings, allow them to model with concrete tools first, and
then encourage them to explain their model to a classmate or teacher. Then have students draw/represent the models they
created on paper.
Literacy Connections

Academic Vocabulary
Terms

Vocabulary Strategies

Literacy Strategies
Possible Enrichment Tasks:
 Illustrative Mathematics: Shapes and their Insides https://www.illustrativemathematics.org/content-standards/tasks/1514
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/draw all the possible figures that 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 16 Pacing Guide
Grade 3 Games to Build Fluency
Multi-Digit Addition & Subtraction Resources
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Property of MPS
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