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LESSON
10.4
Classify Quadrilaterals
FOCUS
COHERENCE
RIGOR
LESSON AT A GLANCE
F C R Focus:
Common Core State Standards
4.G.A.2 Classify two-dimensional figures based on the presence or
absence of parallel or perpendicular lines, or the presence or absence of
angles of a specified size. Recognize right triangles as a category, and identify right
triangles.
Mathematical Practices
MP2 Reason abstractly and quantitatively. MP4 Model with mathematics.
MP6 Attend to precision.
F C R Coherence:
Standards Across the Grades
Before
Grade 4 After
3.G.A.1
4.G.A.2
5.G.B.3
5.G.B.4
F C R Rigor:
Level 1: Understand Concepts....................Share and Show ( Checked Items)
Level 2: Procedural Skills and Fluency.......On Your Own
Level 3: Applications..................................Think Smarter and Go Deeper
Learning Objective
Sort and classify quadrilaterals.
Language Objective
Students draw quadrilaterals to demonstrate how
you can sort and classify quadrilaterals.
Materials
MathBoard, color pencils
F C R For more about how GO Math! fosters Coherence
within the Content Standards and Mathematical Progressions
for this chapter, see page 547J.
About the Math
Professional Development
Why Teach This Interactive Student Edition
In this lesson, the definitions of quadrilaterals are
formalized and the relationships among them are
emphasized. First, students identify right angles in
Professional
Development
quadrilaterals and
recognize that
a quadrilateral cannot
have exactly three right angles. Next, students examine
Professional Development
quadrilaterals by the number of pairs of parallel sides and
find that some quadrilaterals have 1 pair, some have
2 pairs, while others have none.
About the Math
About the Math
Personal Math Trainer
Math on the Spot
Animated Math Models
Students then classify quadrilaterals in as many ways as
possible. For example, a figure can be a quadrilateral,
a trapezoid, a parallelogram, and a rectangle.
Through this process, students gain a deeper
understanding of the properties of quadrilaterals and how
they relate to each other.
Professional Development Videos
567A Chapter 10
iTools: Geometry
HMH Mega Math
1 ENGAGE
Daily Routines
Common Core
Problem of the Day 10.4
Simone makes a quilt square using two
triangles as shown. What types of triangles
are they? right
Common Core
Daily Routines
with the Interactive Student Edition
Essential Question
How can you sort and classify quadrilaterals?
Making Connections
Invite students to tell you what they know about toy kites.
What is a toy kite? A kite is a toy that has a light frame with thin material
stretched over it, flown in the wind at the end of a long string. How many
sides does a basic kite have? 4 sides
Learning Activity
What is the problem the students are trying to solve? Connect the
story to the problem.
Vocabulary parallelogram, rectangle,
rhombus, square, trapezoid
• Interactive Student Edition
• Multimedia eGlossary
• What problem are you trying to solve? Find the name of the figure
given in the problem.
• What properties of the given figure do you notice? The figure has
four sides and four angles.
Literacy and Mathematics
Choose one or more of the following activities.
Vocabulary Builder
Materials magazines, scissors, glue, paper, stapler
Identify Quadrilaterals Have students
write the name and definition of each type
of quadrilateral at the top of separate
half-sheets of paper. Then have students
look through old magazines and cut out
examples of each type of quadrilateral.
Students then glue the cutout examples
onto the half-sheet of paper for each
quadrilateral. When done, have students
create a booklet by folding a sheet of paper
in half and inserting the half-sheets inside.
Staple the pages together. Students can title
their booklet “Quadrilaterals” and draw an
example of each type on the front cover.
• Have students write the problem as a short story. Have them
predict what attributes of the figure they could use to determine
the name of the figure.
• Have students create kites using paper and scissors. Have students
compare their kites with each other and note the differences
between the kites.
How can you sor t and
classi fy quadrilaterals?
Lesson 10.4 567B
LESSON
10.4
2 EXPLORE
4.G.A.2 Classify two-dimensional
figures
based“File
on the
presence or absence of parallel or perpendicular lines, or the presence
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through
info”
or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.
CorrectionKey=B
Lesson 10.4
Name
Unlock the Problem
Classify Quadrilaterals
MP6 Attend to precision. Review naming
a quadrilateral by its vertices. Discuss the
quadrilaterals shown. Explain that tick marks
on a figure indicate sides equal in length.
• What are some other possible names for
quadrilateral ABCD? Possible answers:
Unlock
Unlock the
the Problem
Problem
Quadrilateral ABCD is a possible name for the figure shown
at the right. Quadrilateral ACBD is not a possible name,
since points A and C are not endpoints of the same side.
• What can you tell about the sides of a
rhombus? Explain. All four sides have one tick
A
D
The tick marks on the line segments
show that they have the same length.
Sides AD and BC have the same length.
Sides AB and CD have the same length.
Assume that line segments that appear to be parallel are
parallel.
mark each, so all four sides are equal in length.
C
B
A quadrilateral is a polygon with four sides and four angles.
You can name a quadrilateral by the vertices of its angles.
BCDA, CDAB, DABC, ADCB, DCBA, CBAD, BADC
Geometry—
4.G.A.2
MATHEMATICAL PRACTICES
MP1, MP2, MP6
Essential Question How can you sort and classify quadrilaterals?
MATHEMATICAL PRACTICES
Common Quadrilaterals
ELL Strategy:
Illustrate Understanding
Trapezoid
• at least 1 pair of
parallel sides
• On separate index cards, have students draw
and label a quadrilateral, a trapezoid, a
parallelogram, a rhombus, a rectangle, and
a square.
Rectangle
• 2 pairs of parallel
sides
• 2 pairs of sides of
equal length
• 4 right angles
Square
• 2 pairs of parallel sides
• 4 sides of equal length
• 4 right angles
Materials ■ color pencils
Use the Quadrilateral Color Guide to color the quadrilaterals.
Quadrilateral Color Guide
RED:
R
© Houghton Mifflin Harcourt Publishing Company
Remind students to look for the number of
right angles in each quadrilateral.
• How do you know which angles are right
angles? Right angles are marked with square corners.
• How can you describe the two sides of a
quadrilateral that form a right angle? The
Rhombus
• 2 pairs of parallel
sides
• 4 sides of equal
length
Activity 1 Identify right angles in quadrilaterals.
• On the back of the card, have students
draw a line down the center and write the
book’s definition on the left and their own
description on the right.
Activity 1
Parallelogram
• 2 pairs of parallel
sides
• 2 pairs of sides of
equal length
exactly 4 right angles
BLUE:
exactly 2 right angles
ORANGE:
exactly 1 right angle
O
B
No; possible explanation: if a quadrilateral
has 3 right angles, the fourth angle must
also be a right angle.
Math
Talk
MATHEMATICAL PRACTICES 6
Can a quadrilateral have exactly
3 right angles? Explain.
Chapter 10 567
sides are perpendicular.
• Do all parallelograms have right angles?
Explain. No; possible explanation: squares and
rectangles are parallelograms that have right angles, but
other parallelograms do not have right angles.
Math
Talk
Use Math Talk to help students
understand that a quadrilateral
cannot have exactly three right angles.
• Look at the rectangle in Activity 1. Could you
have redrawn it to show a quadrilateral with
only 3 right angles? Explain. Possible answer: no;
4_MNLESE342286_C10L04.indd 567
Reteach 10.4
the acute angle to a right angle, I would have had to
change the obtuse angle to a right angle to still have
4 sides.
567 Chapter 10
2
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1
Lesson 10.4
Reteach
Name Differentiated
Instruction
Enrich 10.4
3
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Lesson 10.4
Enrich
Name Classify Quadrilaterals
Quad Logic
Read each statement carefully. Write true or false.
A quadrilateral is a polygon
with 4 sides and 4 angles.
Some quadrilaterals have
special names:
Quadrilateral
4 sides
4 angles
2. Alltrapezoidsareparallelograms.
3. Allsquaresarerectangles.
4. Somequadrilateralsaretrapezoids.
Parallelogram
2 pairs of
parallel sides
Rectangle
4 right angles
2 pairs of
parallel sides
true
false
true
true
true
false
true
false
1. Someparallelogramsarerectangles.
Trapezoid
At least 1 pair
of parallel sides
if I changed one right angle, I would have had to change
another right angle to still have 4 sides.
• Look at the trapezoid with 2 right angles
in Activity 1. Could you have redrawn it to
show a quadrilateral with 3 right angles?
Explain. Possible answer; no; if I would have changed
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5. Somerectanglesarerhombuses.
6. Allrhombusesaresquares.
7. Someparallelogramsaretrapezoids.
Rhombus
4 sides of
equal length
8. Allrectanglesaresquares.
Make each statement true. Write All, No, or Some.
Square
4 right angles
4 sides of
equal length
All
All
All
Some
9.
10.
11.
Classify each figure as many ways as possible. Write
quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square.
12.
1.
2.
quadrilateral,
trapezoid
quadrilateral,
quadrilateral,
trapezoid,
trapezoid,
parallelogram, parallelogram
rectangle,
rhombus, square
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
4_MNLEAN343153_C10R04.indd 11
3.
10-11
rectanglesareparallelograms.
squaresaretrapezoids.
parallelogramsarequadrilaterals.
quadrilateralsareparallelograms.
13. Stretch Your Thinking Writethreeofyourown
Reteach
quad-logicstatements.Thenexchangethemwithaclassmate
andcompleteeachother’sstatements.
Check students’ work.
Chapter Resources
© Houghton Mifflin Harcourt Publishing Company
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10-12
Enrich
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Activity 2
Direct students to look for pairs of
parallel sides.
• How can you determine which sides are
parallel? Possible answer: I extended the sides, they
Activity 2 Use a Venn diagram to sort quadrilaterals.
Write the names of the quadrilaterals in the Venn diagram.
F
G
E
H
B
C
A
D
R
would never cross.
Q
• What do you notice about the side lengths
of quadrilaterals with two pairs of parallel
sides? Possible answer: the opposite sides are of
P
S
U
K
J
equal length.
N
L
V
W
M
Z
Q
T
• If a quadrilateral has exactly one pair of
parallel sides, do the other sides have to be
of equal length? Explain. No, not necessarily;
R
for example, quadrilateral JKLM has exactly one
pair of parallel sides, but the other sides are not of
equal length.
Quadrilaterals
MP8 Look for and express regularity in
repeated reasoning.
• What names can you use to classify all
of the quadrilaterals that you identified
as having exactly 1 pair of parallel sides?
2 Pairs of
Parallel Sides
At least 1 Pair
of Parallel Sides
No Parallel Sides
quadrilateral PQRS
quadrilateral JKLM
quadrilateral ABCD
quadrilateral QRNZ
quadrilateral TUVW
quadrilateral EFGH
quadrilateral and trapezoid
• What names can you use to classify all of
the quadrilaterals that you identified as
having 2 pairs of parallel sides? quadrilateral,
trapezoid, and parallelogram
b.
c.
quadrilateral
quadrilateral
quadrilateral, trapezoid,
trapezoid
trapezoid
parallelogram, rhombus,
parallelogram
rectangle, square
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Visual
Advanced
Learners
Advanced
Advanced
Learners
Learners
Partners
Materials Venn Diagram (see eTeacher Resources)
•Distribute a Venn diagram to each pair of students.
•Have partners use attributes of quadrilaterals to show
the relationships of the following quadrilaterals in a
Venn diagram: quadrilateral, trapezoid, parallelogram,
rhombus, rectangle, and square.
•Have students explain their Venn diagrams.
Quadrilaterals
Trapezoids
Parallelograms
Rectangles
Rhombuses
Squares
© Houghton Mifflin Harcourt Publishing Company
a.
© Houghton Mifflin Harcourt Publishing Company
Try This! Classify each figure as many ways as possible. Write
quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square.
Try This!
• Is a trapezoid always a quadrilateral?
Explain. Yes; possible explanation: a trapezoid is a
polygon with four sides and four angles, so it is always
a quadrilateral.
• How are a square and a rectangle alike,
and how are they different? Possible answer:
both are quadrilaterals with two pairs of parallel sides
and four right angles. A square always has four sides
of equal length, but a rectangle does not. A rectangle
always has two pairs of sides of equal length.
Teacher Note: Some mathematicians define a
trapezoid as a quadrilateral with exactly one
pair of parallel sides. This book does not use
this definition.
COMMON ERRORS
COMMON ERRORS
Error Students may not identify all possible
ways to classify a figure.
Example quadrilateral,
parallelogram, square,
rhombus
Springboard to Learning Have students
systematically go through the properties for
each type of quadrilateral and determine if
the properties are true for the figure.
Lesson 10.4 568
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Name
3 EXPLAIN
Share
Share and
and Show
Show
MATH
BOARD
1. Tell whether the quadrilateral is also a trapezoid, parallelogram,
Share and Show
rhombus, rectangle, or square.
MATH
BOARD
B
Think:
C
A
3
2
31
2
1
Rt I
Rt I
If
Then
2.
3.
quadrilateral
quadrilateral
quadrilateral, trapezoid
trapezoid, parallelogram
parallelogram
rectangle
rhombus
• How many parallel sides does a trapezoid
have? at least 1 pair
• How many parallel sides do a parallelogram,
a rhombus, a rectangle, and a square have?
• What kind of 4-sided figure has no parallel
sides? a quadrilateral; a quadrilateral can have 1 pair of
Classify each figure as many ways as possible.
Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square.
5.
6.
quadrilateral
trapezoid
parallelogram
In Exercises 5–7, students classify quadrilaterals
in as many ways as possible.
DEEPER
MP7 Look for and make use of structure.
• Is it possible to draw a rhombus that is also
a rectangle? Explain. Yes; possible explanation:
7.
quadrilateral, trapezoid,
parallelogram, rhombus,
quadrilateral
trapezoid
rectangle, square
Chapter 10 • Lesson 4
parallel sides, 2 pairs of parallel sides, or no parallel sides
On Your Own MATHEMATICAL PRACTICES 2
Use Reasoning How would you
classify a figure with 4 sides,
none of which are parallel?
Explain.
On
On Your
Your Own
Own
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Use Math Talk to focus on students’
understanding that a quadrilateral
without any parallel sides can only be named a
quadrilateral.
569 Chapter 10
4.
Math
Talk
Math
Talk
draw a square. A square is a rhombus and a rectangle.
0 right angles
_
Classify each figure as many ways as possible. Write
quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square.
Differentiate Instruction with
• Reteach 10.4
• Personal Math Trainer 4.G.A.2
• RtI Tier 1 Activity (online)
2 pairs
D
parallelogram, rhombus, and
Quadrilateral ABCD is also a _____
a trapezoid
____
.
a student misses the checked
exercises
If
Math Talk: quadrilateral;
possible explanation:
because a parallelogram,
a square, a rectangle, and
a rhombus all have 2 pairs
of parallel sides, and a
trapezoid has at least one
pair of parallel sides
4 sides of equal length
_
Use the checked exercises for Quick Check.
Quick Check
Quick Check
2 pairs of parallel sides
_
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MATHEMATICAL PRACTICES
MODEL • REASON • MAKE SENSE
Problem
Problem Solving
Solving •• Applications
Applications
8.
SMARTER
Explain how
a rhombus and square are alike,
and how they are different.
4 ELABORATE
9.
SMARTER
Classify the figure. Select all
that apply.
Problem Solving • Applications
Possible explanation: a
rhombus and square are both
quadrilaterals, trapezoids, and parallelograms.
Both have 4 sides of equal length and 2 pairs
quadrilateral
rectangle
trapezoid
rhombus
parallelogram
square
MATHEMATICAL PRACTICES
SMARTER
Exercise 8 requires a student to understand
that a rhombus may or may not have 4 right
angles. When it does, it is also a square.
However, a rhombus and a square both
always have 4 sides of equal length.
of parallel sides. A rhombus does not always
The Louvre Museum is located in Paris, France.
Architect I. M. Pei designed the glass and metal
structure at the main entrance of the museum.
This structure is called the Louvre Pyramid.
WRITE
Below is a diagram of part of the entrance to
the Louvre Pyramid.
10.
MATHEMATICAL
PRACTICE
1
Math • Show Your Work
see in the diagram.
DEEPER
How many triangles do you see
in the diagram? Explain.
Possible answer: there are trapezoids
11 triangles; 4 small triangles, 3 triangles
with 1 pair of parallel sides. There are
made of 2 triangles and 1 rhombus,
also trapezoids with 2 pairs of parallel
sides that also look like rhombuses.
If you combine the smaller shapes
in different ways, you can also see
other trapezoids, rhombuses, and
parallelograms.
2 triangles made of 3 triangles and
3 rhombuses, 1 triangle made of
4 triangles and 6 rhombuses, and 1 large
outer triangle.
total triangles 5 4 1 3 1 2 1 1 1 1 = 11
Describe the quadrilaterals you
11.
© Houghton Mifflin Harcourt Publishing Company • Image Credits: (cr) ©Robert Harding Picture Library Ltd/Alamy Images
have 4 right angles, but a square does.
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DIFFERENTIATED INSTRUCTION
INDEPENDENT ACTIVITIES
Differentiated Centers Kit
Students
complete
purple
Activity
Card 13 by
drawing and classifying
polygons.
Use this video to help students model and
solve this type of Think Smarter problem.
Math on the Spot videos are in the Interactive
Student Edition and at www.thinkcentral.com.
SMARTER
Exercise 9 assesses students’ ability to classify
quadrilaterals and identify subgroups based
on their properties. Students who have
difficulty explaining the relationship of a
square and rectangle may not know their
definitions. More likely, however, is that those
students do not understand the heirarchy in
the classification of quadrilaterals.
Connect to Art
570
Activities
Connecting Vertices
Math on the Spot
Video Tutor
Literature
A New Angle on Trains
and Train Stations
Students read
about lines,
angles, and
plane figures
that can be
found in train stations.
MP1 Make sense of problems and
persevere in solving them.
• What type of quadrilateral is formed by
a rhombus and a small triangle in the
diagram? a trapezoid
5 EVALUATE Formative
Assessment
Essential Question
Using the Language Objective
Reflect Have students draw quadrilaterals and
demonstrate to answer the Essential Question.
How can you sort and classify quadrilaterals?
Possible answer: by the number of pairs of parallel sides,
number of equal length sides, and number of right angles
Math Journal
WRITE
Math
Draw and label an example of each type
of quadrilateral: trapezoid, parallelogram,
rhombus, rectangle, and square.
Lesson 10.4 570
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Practice and Homework
Name
Lesson 10.4
Classify Quadrilaterals
COMMON CORE STANDARD—4.G.A.2
Draw and identify lines and angles, and classify
shapes by properties of their lines and angles.
Classify each figure as many ways as possible. Write
quadrilateral, trapezoid, parallelogram, rhombus, rectangle,
or square.
1.
Think: 2 pairs of parallel sides
4 sides of equal length
0 right angles
Practice and Homework
Use the Practice and Homework pages to
provide students with more practice of the
concepts and skills presented in this lesson.
Students master their understanding as they
complete practice items and then challenge
their critical thinking skills with Problem
Solving. Use the Write Math section to
determine student’s understanding of content
for this lesson. Encourage students to use their
Math Journals to record their answers.
quadrilateral, trapezoid, parallelogram, rhombus
2.
3.
quadrilateral, trapezoid,
4.
quadrilateral, trapezoid
quadrilateral
parallelogram, rectangle
Problem
Problem Solving
Solving
5. Alan drew a polygon with four sides and
© Houghton Mifflin Harcourt Publishing Company
four angles. All four sides are equal. None
of the angles are right angles. What figure
did Alan draw?
PROFESSIONAL
DEVELOPMENT
PROFESSIONAL
PROFESSIONAL
DEVELOPMENT
DEVELOPMENT
7.
6. Teresa drew a quadrilateral with 2 pairs
of parallel sides and 4 right angles. What
quadrilateral could she have drawn?
Possible answers: a quadrilateral,
Possible answers: a square,
rhombus, trapezoid
rectangle, trapezoid
Math Draw and label an example of
WRITE
each type of quadrilateral: trapezoid,
parallelogram, rhombus, rectangle, and square.
Check students’ drawings.
Mathematical Practices in Your Classroom
Chapter 10
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Mathematical
Mathematical Practices
Practices in
in Your
Your Classroom
Classroom
CCSS.Math.Practice.MP7 Look for and make use of
structure.
Ask questions such as the following to help students focus on the
structure of geometric figures in order to classify them:
In order to classify quadrilaterals, students must understand the
structure of geometric figures. How many parallel lines does it have?
Does it have right angles? How many sides of equal length does it
have? By analyzing the structure of the figure, students can describe
and classify it. This analysis deepens students’ understanding of
properties of two-dimensional figures and how they are related.
• How can identifying pairs of parallel lines help you classify
quadrilaterals? Possible answer: I can check if the figure has
1/8/15 1:26 PM
2 pairs, exactly 1 pair, or no pair of parallel lines.
• How can identifying the number of right angles help you
classify quadrilaterals? Answers will vary. Possible answer: if the
quadrilateral has 4 right angles, it is a rectangle, a square, a parallelogram, and a trapezoid and sometimes it is a rhombus.
• How can identifying the number of sides of equal length help
you classify quadrilaterals? Answers will vary. Possible answer: if
the quadrilateral has 4 sides of equal length, it is a rhombus,
a square, a rectangle, a parallelogram, and a trapezoid. But not all
rectangles, parallelograms, and trapezoids are rhombuses
or squares.
571 Chapter 10
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TEST
PREP
Lesson Check (4.G.A.2)
1. Joey is asked to name a quadrilateral
that is also a rhombus and has 2 pairs of
parallel sides. What should be his answer?
2. What quadrilateral has at least one pair
of parallel sides, but cannot be called a
parallelogram?
Continue concepts and skills practice with
Lesson Check. Use Spiral Review to engage
students in previously taught concepts and
to promote content retention. Common Core
standards are correlated to each section.
trapezoid
square
Spiral Review (4.OA.B.4, 4.OA.C.5, 4.NF.B.3d, 4.G.A.1)
equal groups. What are all the possible
numbers of eggs that Terence could put
in each group?
4. In a line of students, Jenna is number 8.
The teacher says that a rule for a number
pattern is add 4. The first student in line
says the first term, 7. What number should
Jenna say?
1, 2, 3, 4, 6, 8, 12, 24
6 of a pizza. What fraction of the
5. Lou eats __
8
pizza , in simplest form, is left over?
1
__
4
35
6. Name a capital letter that appears to have
parallel lines.
Possible answer: N
© Houghton Mifflin Harcourt Publishing Company
3. Terrence has 24 eggs to divide into
FOR MORE PRACTICE
GO TO THE
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Personal Math Trainer
1/8/15 1:28 PM
Lesson 10.4 572