723A Chapter 12
LESSON AT A GLANCE
Ch t 12
About the MathProfessional Development
LESSON 12.5
Interactive Student Edition
Personal Math Trainer
Math on the Spot Video
Animated Math Models
iTools: Geometry
HMH Mega Math
Professional Development Videos
Why Teach This Students learned how to describe sides of polygons and about types of angles earlier in this chapter. Those topics provide students with a foundation to describe, classify, and compare quadrilaterals.Quadrilaterals can be described by the types of sides and by the lengths of the sides that they have. A rectangle has two pairs of opposite sides that are parallel, two pairs of sides of equal length, and four right angles. A rhombus has two pairs of opposite sides that are parallel, four sides of equal length, and could have right angles. A square has two pairs of opposite sides that are parallel, four sides of equal length, and four right angles. The term “trapezoid” may have two different meanings. One meaning is that a trapezoid is a quadrilateral with exactly one pair of parallel sides. Another meaning is that it is a quadrilateral with at least one pair of parallel sides. In this book, the latter definition is used.These different meanings will result in different classifications. For example, according to the first meaning, a rectangle is not a trapezoid and according to the second meaning, a rectangle is a trapezoid.
Classify Quadrilaterals
Learning ObjectiveDescribe, classify, and compare quadrilaterals based on their sides and angles.
Language ObjectiveStudents demonstrate and explain how to use the sides and angles to describe quadrilaterals.
MaterialsMathBoard
F C R Focus:Common Core State Standards
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.
MATHEMATICAL PRACTICES (See Mathematical Practices in GO Math! in the Planning Guide for full text.)MP2 Reason abstractly and quantitatively. MP4 Model with mathematics. MP6 Attend to precision.
F C R Coherence:Standards Across the GradesBefore2.G.A.1
Grade 3 3.G.A.1
After4.G.A.2
F C R Rigor:Level 1: Understand Concepts....................Share and Show ( Checked Items)Level 2: Procedural Skills and Fluency.......On Your Own, Practice and HomeworkLevel 3: Applications..................................Think Smarter and Go Deeper
F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 695J.
FOCUS COHERENCE RIGOR
Lesson 12.5 723B
ENGAGE1Daily Routines
Common Core
Lesson 12.5 723B
How can you use sides and angles to help you
describe quadrilaterals?
with the Interactive Student Edition
Essential QuestionHow can you use sides and angles to help you describe quadrilaterals?
Making ConnectionsInvite students to tell you what they know about figures.
What figures do you remember learning about? Triangle, square, circle, etc.
What do squares, rectangles, rhombuses, and trapezoids have in common? They have 4 sides and 4 angles.
Learning ActivityWhat is the problem the students are trying to solve? Connect the story to the problem.
• What do you know about the shape of the sign? It has 4 sides and 4 angles.
• What are other characteristics about the figure that you notice? Answers may vary.
Literacy and MathematicsChoose one or more of the following activities.
• Have students use dot paper to draw 5 to 10 four-sided figures. Have them write about the characteristics that they notice about their figures.
• Have students draw as many road signs as they can. Have students describe the characteristics of each sign.
Vocabulary Builder• Write each of the vocabulary words on
the board.
• Draw a picture of a square, a rectangle, a rhombus, a trapezoid, and a quadrilateral that is none of these.
• Have students come up to the board and write an appropriate vocabulary word under each of the pictures.
• Note that more than one of the vocabulary words may be used to describe a square.
Problem of the Day 12.5The area of a county is 578 square miles. What is 578 rounded to the nearest hundred?
______
Vocabulary rectangle, rhombus, square, trapezoid
Interactive Student EditionMultimedia Glossary e
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Unlock the ProblemUnlock the Problem
Some quadrilaterals cannot be classified as a trapezoid, rectangle, square, or rhombus.
Name
Chapter 12 723
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Classify QuadrilateralsEssential Question How can you use sides and angles to help you describe quadrilaterals?
Quadrilaterals are named by their sides and
their angles.
quadrilateral
__ sides
__ angles
trapezoid
at least __ pair of opposite sides that are parallel lengths of sides could be the same
rectangle
_ pairs of opposite sides that are parallel
_ pairs of sides that are of equal length
_ right angles
square
_ pairs of opposite sides that are parallel
_ sides that are of equal length
_ right angles
rhombus
_ pairs of opposite sides that are parallel
_ sides that are of equal length
Describe quadrilaterals.
Geometry—3.G.A.1
MATHEMATICAL PRACTICESMP1, MP6, MP8
MathTalk MATHEMATICAL PRACTICES 8
Generalize Why can a square also be named a rectangle or a rhombus?
4
2
2
Possible explanation: a square is a rectangle because it
has 2 pairs of opposite sides that are parallel, 2 pairs of
sides of equal length, and 4 right angles. A square is a rhombus
because it has 2 pairs of opposite sides that are parallel and
4 sides of equal length.
4
4
2
4
4
2
4
1
EXPLORE2
rhombustrapezoid
rectanglesquare
Name
2 pairs of opposite sides that are parallel
4 sides that are of equal length
at least 1 pair of opposite sides that are parallel
lengths of sides could be the same
2 pairs of opposite sides that are parallel
2 pairs of sides that are of equal length
4 right angles
2 pairs of opposite sides that are parallel
4 sides that are of equal length
4 right angles
Circle all the words that describe the quadrilateral.
1. 2. 3.
square
rhombus
trapezoid
square
rectangle
quadrilateral
square
rectangle
rhombus
You can classify quadrilaterals by their sides and by their angles.
How can you classify the quadrilateral?
It has only 1 pair of opposite sides that are parallel.
The lengths of all 4 sides are not equal.
So, the quadrilateral is a trapezoid.
Classify Quadrilaterals
Lesson 12.5Reteach
12-13 ReteachChapter Resources© Houghton Mifflin Harcourt Publishing Company
Name
Quadrilateral Riddles
Read the riddles and name the shape that is
being described.
1. I am a quadrilateral with exactly 1 pair of opposite sides that are parallel. What shape am I?
2. I am a quadrilateral that always has 4 sides that are of equal length and 4 right angles. What shape am I?
3. I am a quadrilateral with 2 pairs of opposite sides that are parallel, 2 pairs of sides that are of equal length, and 4 right angles. I am not a square. What shape am I?
4. I am a polygon with 4 sides and 4 angles. I do not have any pairs of opposite sides that are parallel. What shape am I?
5. Jerome drew a shape and described it as a square. Kayla described it as a rectangle. Luis described it as a rhombus. Can they all be correct? Explain.
Lesson 12.5Enrich
squaretrapezoid
quadrilateral
Yes; possible answer: a square is also
a rhombus because it has 4 sides of equal
length. A square is also a rectangle because
it has 2 pairs of opposite sides that are
parallel and 2 pairs of sides that are of
equal length.
Possible answers: rectangle, trapezoid
12-14 EnrichChapter Resources© Houghton Mifflin Harcourt Publishing Company
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3333
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2
3 DifferentiatedInstruction
723 Chapter 12
LESSON 12.5
Enrich 12.5Reteach 12.5
Unlock the ProblemMATHEMATICAL PRACTICES
MP6 Attend to precision. Have students make connections to what they already know about quadrilaterals.
• What is true about all quadrilaterals? They all have four sides and four angles.
• How might quadrilaterals be different? Possible answer: they might have different kinds of angles, and not every quadrilateral will have the same number of sides of equal length.
Have students look at each quadrilateral.
• Which quadrilaterals can have two pairs of opposite sides that are parallel? square, rectangle, trapezoid, and rhombus
• Are all quadrilaterals also rectangles? Explain. No; possible explanation: a trapezoid is a quadrilateral, but it may not a rectangle.
• Are all rectangles also trapezoids? Explain. Yes; possible explanation: trapezoids have at least 1 pair of opposite sides that are parallel and rectangles have 2 pairs of opposite sides that are parallel.
MathTalk Use Math Talk to focus on students’
understanding of why a square is a special type of rectangle and rhombus.
In Grade 4, students will learn that a rectangle, a square, and a rhombus can also be classified as a parallelogram.
ELL Strategy: Understand Context
Students use prior knowledge and context to understand the concept of a pair.
• Explain that a pair is two of something, not a fruit (like pear). Point out things that come in pairs: a pair of shoes, a pair of socks, or a pair of gloves.
• Show two gloves. Here is one pair of gloves. How many gloves are there? 2
• Add two more gloves. Here are two pairs of gloves. How many gloves are there now? 4
• Repeat the activity with other pairs of objects they suggest.
MP2 Reason abstractly and quantitatively. • How can you check if two sides of a
quadrilateral are parallel? Possible answer: The sides are parallel if they never cross.
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.
On Your OwnOn Your Own
Share and ShShare and ShShare and Show MATHBOARDMATHBOARD
724
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4.
rectangle
rhombus
square
trapezoid
5.
rhombus
quadrilateral
square
rectangle
6.
rectangle
rhombus
trapezoid
quadrilateral
Circle all the words that describe the quadrilateral.
7.
rectangle
trapezoid
quadrilateral
rhombus
8.
rectangle
rhombus
trapezoid
square
9.
quadrilateral
square
rectangle
rhombus
Look at the quadrilateral at the right.
1. Outline each pair of opposite sides that are parallel
with a different color. How many pairs of opposite sides
appear to be parallel? __
2. Look at the parallel sides you colored.
The sides in each pair are of __ length.
3. Name the quadrilateral in as many ways as you can.
____
Circle all the words that describe the quadrilateral.
Think: All the angles are right angles.
MathTalk MATHEMATICAL PRACTICES 1
Analyze How can you have a rhombus that is not a square?
2 pairs
equal
rectangle, trapezoid
See below.
Check students’ drawings.
Math Talk: Possible explanation: a rhombus has 4 sides of
equal length, like a square. But a rhombus may not have 4 right angles, like a square.
COMMON ERRORS
Quick Check
If
Rt I RR1
2
3
Then
EXPLAIN3
Advanced Learners
Lesson 12.5 724
Error Students may not identify a square as a rectangle or a rhombus.
Example Students may not circle rectangle or rhombus in Exercise 9.Springboard to Learning Review the properties of a square, a rectangle, and a rhombus. Point out that a square is a special kind of rectangle and a special kind of rhombus.
Share and Show MATHBOARDMATHBOARDMBMMMBBBMATHABOARDMMMAAATHATHTHHAAAAAAAAATTAAAABOARDBOARDBOARD
The first three problems connect to the learning model. Have students use the MathBoard to explain their thinking.Use the checked exercises for Quick Check.
MathTalk Use Math Talk to focus on
comparing properties of quadrilaterals to classify them.
Suggest students use a ruler to determine if sides are or are not equal in length. Students may also use the corner of a piece of paper to help decide if an angle is a right angle.
On Your OwnWhen students have finished Exercises 7–9, ask:
• How can you prove that the quadrilateral in Exercise 7 is NOT a rhombus or a square? All 4 sides are not equal in length.
• How can you prove that the shape in Exercise 8 is a trapezoid? It has at least one pair of parallel sides.
a student misses the checked exercises
Differentiate Instruction with • Reteach 12.5
• Personal Math Trainer 3.G.A.1
• RtI Tier 1 Activity (online)
Visual / SpatialIndividual
Materials ruler, Dot Paper (see eTeacher Resources)
• Write the following directions on the board and have students draw each shape on square dot paper.
1. Draw a quadrilateral with 2 right angles and exactly 1 pair of opposite sides that are parallel.
2. Draw a quadrilateral with 2 pairs of opposite sides that are parallel and 2 angles that are greater than a right angle.
• Then have students draw another quadrilateral and write directions for how to draw it. Possible directions: Draw a quadrilateral with 4 right angles and all sides of equal length.
AB
D
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Chapter 12 • Lesson 5 725
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10. Which quadrilaterals appear to have
4 right angles?
11. Which quadrilaterals appear to have
2 pairs of opposite sides that are parallel?
12. Which quadrilaterals appear to have no right
angles?
Use the quadrilaterals at the right for 10–12.
13. The opposite sides of _ rectangles
are parallel.
15. _ squares are rectangles.
17. _ quadrilaterals are polygons.
14. _ sides of a rhombus are the
same length.
16. _ rhombuses are squares.
18. _ polygons are quadrilaterals.
Write all or some to complete the sentence for 13–18.
19. MATHEMATICALPRACTICE 6 Circle the shape at the
right that is not a quadrilateral. Explain
your choice.
20. SMARTER I am a polygon that has 4 sides and 4 angles. At
least one of my angles is less than a right angle. Circle all
the shapes that I could be.
quadrilateral rectangle square rhombus trapezoid
B, C
B, C, D
D, E, F
Possible explanation: I circled the pentagon
because it has 5 sides and 5 angles. All
quadrilaterals have 4 sides and 4 angles.
all All
All Some
SomeAll
ELABORATE4
Math on the Spot videos are in the Interactive Student Edition and at www.thinkcentral.com.
725 Chapter 12
Explain to students that in Exercises 10–12, they should write the letter names of the quadrilaterals that answer each question. Point out that each exercise has more than one answer.MP6 Attend to precision. Discuss the definition of quadrilaterals with students. Help students understand that a quadrilateral has 4 sides and 4 angles. Guide students to count the number of sides and angles in each shape. Then, ask:
• If the shape in Exercise 19 is not a quadrilateral, what shape is it? It is a pentagon.
• Can a trapezoid have exactly two right angles? Explain. Yes; it can have exactly two right angles and still have one pair of opposite sides that are parallel. For example, quadrilateral A is a trapezoid with two right angles.
SMARTER
Exercise 20 requires students to visualize a polygon that is drawn based on the properties of quadrilaterals.
To extend students’ thinking, have students write the name of a quadrilateral and then write a description of it in their own words. Tell them to be sure to write enough information so that if anyone draws it using the description, they will draw the correct shape. Then have volunteers read their descriptions while other volunteers identify the shapes.
Math on the Spot Video TutorUse this video to help students model and solve this type of Think Smarter problem.
Problem Solving • Applications
Differentiated Centers Kit
DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIES
WRITE Math Show Your Work
MATHEMATICAL PRACTICES ANALYZE
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Compare and ContrastWhen you compare, you look for
ways that things are alike. When
you contrast, you look for ways
that things are different.
Mr. Briggs drew some shapes on
the board. He asked the class to
tell how the shapes are alike and
how they are different.
DEEPER Complete the sentences.
• Shapes _ , _ , _ , _ , _ , _ , and _ are polygons.
• Shapes _ , _ , and _ are not polygons.
• Shapes _ , _ , _ , and _ are quadrilaterals.
• Shapes _ , _ , and _ appear to have only 1 pair of opposite sides that are parallel.
• Shapes _ , _ , and _ appear to have 2 pairs of opposite sides that are parallel.
• All 4 sides of shapes _ and _ appear to be the same length.
• In these polygons, all sides do not appear to be the same length. ____
• These shapes can be called rhombuses. ____
• Shapes _ and _ are quadrilaterals but cannot be called rhombuses.
• Shape _ is a rhombus and can be called a square.
21. SMARTER Identify the quadrilateral that can have two pairs
of parallel sides and no right angles.
A rhombus B square C rectangle
A, B, G, H, I
E, F
A
C
A B E F G H I
E
A
F
B
G
I
E
D
F
J
G
E F
F
A G
EVALUATE5 Formative Assessment
Differentiated Centers Kit
DIFFERENTIATED INSTRUCTION INDEPENDENT ACTIVITIESD
Lesson 12.5 726
Essential QuestionUsing the Language ObjectiveReflect Have students demonstrate and explain to a partner to answer the Essential Question. How can you use sides and angles to help you describe quadrilaterals? All quadrilaterals have 4 sides and have special names if the sides are parallel or perpendicular, or equal in length.
Math Journal WRITE MathExplain how a trapezoid and rectangle are different.
Students complete purple Activity Card 18 by identifying
and defining two-dimensional shapes by playing a game.
Students complete orange Activity Card 18 by
classifying two-dimensional shapes based on their attributes.
ActivitiesClassification Act
ActivitiesWhat Figure?
MATHEMATICAL PRACTICES
SMARTER
This item assesses a student’s ability to classify a quadrilateral based on its sides and angles. Students should recognize that both a square and a rectangle have 4 right angles, and therefore know that the rhombus is the only shape that could fulfill both conditions.
Connect to ReadingStudents will learn the reading skill compare and contrast by considering the attributes of the polygons shown.
Practice and HomeworkUse 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.
727 Chapter 12
A B C ED
Problem SolvingProblem Solving
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Lesson 12.5Practice and Homework
Classify Quadrilaterals
Circle all the words that describe the quadrilateral.
Use the quadrilaterals below for 4–6.
7. A picture on the wall in Jeremy’s
classroom has 4 right angles,
4 sides of equal length, and 2 pairs
of opposite sides that are parallel.
What quadrilateral best describes
the picture?
8. WRITE Math Explain how a
trapezoid and a rectangle are different.
Chapter 12 727
1.
square
rectangle
rhombus
trapezoid
2.
square
rectangle
rhombus
trapezoid
3.
square
rectangle
rhombus
trapezoid
4. Which quadrilaterals
appear to have no right
angles?
5. Which quadrilaterals
appear to have 4 right
angles?
6. Which quadrilaterals
appear to have 4 sides
of equal length?
COMMON CORE STANDARD—3.G.A.1 Reason with shapes and their attributes.
square
B, D, and E A and C B, C, and D
Possible explanation: a rectangle
has 2 pairs of parallel sides, where a
trapezoid has at least 1 pair of parallel
sides.
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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.
Lesson 12.5 728
Personal Math Trainer
FOR MORE PRACTICE GO TO THE
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Lesson Check (3.G.A.1)
1. What word describes the
quadrilateral?
2. Which quadrilaterals appear to
have 2 pairs of opposite sides that
are parallel?
A B C
Spiral Review (3.G.A.1)
5. What word describes the dashed
sides of the shape shown? Write
intersecting, perpendicular, or
parallel.
6. How many right angles does this
shape have?
3. Aiden drew the the polygon shown.
What is the name of the polygon
he drew?
4. How many pairs of parallel sides
does this shape appear to have?
trapezoid
decagon
parallel
A and B
2 pairs
0 right angles
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