Teaching the Lesson materials

Key ActivitiesStudents construct convex and concave polygons. They identify the characteristics that various polygons have in common and develop definitions of a polygon and a regular polygon.

Key Concepts and Skills• Construct convex and nonconvex (concave) polygons. [Geometry Goal 2]• Develop definitions for convex and nonconvex (concave) polygons. [Geometry Goal 2]• Describe properties of polygons and regular polygons. [Geometry Goal 2]• Identify types of polygons according to the number of sides. [Geometry Goal 2]

Key Vocabularyside • angle • pentagon • polygon • vertex (vertices) • convex • nonconvex or concave • hexagon • heptagon • octagon • nonagon • n -gon • interior • regular polygon • equilateral triangle

Ongoing Assessment: Recognizing Student Achievement Use journal page 12. [Geometry Goal 2]Ongoing Assessment: Informing Instruction See page 45.

Ongoing Learning & Practice materials

Students practice and maintain skills through Math Boxes and Study Link activities.

Differentiation Options materials

Students use rubber bandsto construct polygons on ageoboard.

Students use straws to create polygons found inThe Greedy Triangle.

Students identify propertiesof kites and rhombuses.

� Teaching Masters (Math Masters,pp. 19–22)

� geoboards and rubber bands� straightedge; straws and twist-ties� The Greedy Triangle

See Advance Preparation

ENRICHMENTREADINESSREADINESS

3

� Math Journal 1, p. 13� Study Link Master (Math Masters,

p. 18)

2

� Math Journal 1, p. 12� Student Reference Book, p. 97� Study Link 1� 4� +, – Fact Triangles� straws and twist-ties

See Advance Preparation

1

Lesson 1�5 41

Objectives To provide opportunities to identify properties of polygons and distinguish between convex and nonconvex (concave) polygons; and to explore geometric definitions and classification.

Technology Assessment Management SystemJournal page 12, Problem 1See the iTLG.

Additional InformationAdvance Preparation

For Part 1, place twist-ties and full-length, half-length, and �34�-length straws in 4 separate

boxes. Each student will need 8 twist-ties and 3 of each size straw. Have extras on hand.

For the optional Readiness activity in Part 3, obtain the book The Greedy Triangle by Marilyn Burns (Scholastic, 1994).

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42 Unit 1 Naming and Constructing Geometric Figures

� Math Message Follow-UpNOTE Although this lesson’s vocabulary list is extensive, many of the wordshave been used in Everyday Mathematics prior to fourth grade and should befamiliar to many students. Do not insist that students memorize these words.They will become part of their vocabulary through repeated use. As geometricterms are discussed in this lesson, write them on the board, and include drawings,to support English language learners.

Ask students to look at the shape they made with the straws andrecall the name of a shape that has five sides and five angles.pentagon Students may remember that triangles, quadrangles,and pentagons are examples of polygons.

Language Arts Link The word polygon comes from theGreek words polu, meaning “many,” and gonia, meaning

“angle.” Many geometric terms are derived from Greek and Latinwords. Breaking these terms into parts can make learning themeasier and promotes useful word analysis skills.

Ask students to push one vertex or several vertices of theirpentagons toward the inside. This new shape is also a pentagon,but it has a special name: It is called a nonconvex or concavepentagon.

WHOLE-CLASS ACTIVITY

1 Teaching the Lesson

Getting Started

Study Link 1� 4Follow-Up Ask volunteers to sharetheir answers for Problem 2. Have students indicate thumbs-up if they agreewith the answers and explanations.

Math MessageTake 3 of each size strawand 8 twist-ties. Make a shape with 5 sides.

Mental Math andReflexes Students take out their �, � FactTriangles and practice the facts in theirTry Again piles. When time is up, studentstransfer appropriate triangles to the OKpile, fasten their new piles with paperclips, and store them.

convex nonconvex, or concave

NOTE Some students may benefit fromdoing the Readiness activity before you beginPart 1 of the lesson. See the Readiness activity in Part 3 for details.

NOTE A line segment connecting any two points on a convex polygon lies entirelyinside or on the polygon. That is not always the case for concave polygons.

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Lesson 1�5 43

Adjusting the Activity

Point out the following:

� A polygon in which all vertices are pushed outward is called aconvex polygon.

� A polygon in which at least one vertex is pushed inward (“cavesin”) is called a nonconvex, or concave, polygon.

To support English language learners, explain the meaning of cave in so that the hint is understood.

Ask students if it is possible to create a concave triangle no or a concave quadrangle. yes Have them explain their reasoning.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

� Constructing Convex and Concave PolygonsHave students work in groups of four. Ask each group to makepolygons with 6 straws, 7 straws, 8 straws, and 9 straws. (That is,each student in a group makes one of the polygons. Some shouldhave all sides equal, some not. Students will need to share strawsand twist-ties.) Bring the class together and review the names ofthe polygons: A polygon with 6 sides is a hexagon, with 7 sides aheptagon, with 8 sides an octagon, and with 9 sides a nonagon.

Ask students who made hexagons to hold up their constructions forall to see. Ask them to distinguish between convex and nonconvex(concave) polygons. Repeat with the other kinds of polygons.● Is it possible to make a polygon out of 25 straws? yes Out of

100 straws? yes

A nonconvex (concave) 25-gon

Polygons are sometimes called n -gons. For example, a 25-sided polygon can be called a 25-gon.

Language Arts Link Remind students of the prefixes in thenames of polygons: tri- in triangle means “three”; quad- in

quadrilateral and quadrangle means “four”; penta- means “five”;hexa- means “six”; hepta- means “seven”; octa- means “eight”;nona- means “nine”; deca- means “ten”; and dodeca- means“twelve” (two and ten).

SMALL-GROUP ACTIVITY NOTE The interior (inside) of a polygon is

not part of the polygon. A polygon with its interior is known as a polygonal region.

interiorside

vertex

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44 Unit 1 Naming and Constructing Geometric Figures

12

What Is a Polygon?LESSON

1�5

Date Time

These are polygons.

These are NOT polygons.

1. If you had to explain what a polygon is, what would you say? (Think: What doPolygons 1–4 have in common? How are Shapes 5–9 different from Polygons 1–4?)

2. Choose one of the shapes from above. Explain why the shape is not a polygon.

5 6 7 8 9

1 2 3 4

Sample answer: A polygon is a figure madeup of line segments called sides. The sidesare connected end to end and make oneclosed path. The sides do not cross (intersect).

Sample answers: Shape 5 has more than oneinterior and more sides than vertices; Shape 6 has two sides that are curved; Shape 7 hasonly two sides; Shapes 8 and 9 have sidesthat are not connected end to end.

96

�

Math Journal 1, p. 12

Student Page

Geometry and Constructions

Convex PolygonsA convex polygon is a polygon in which all the sides are pushedoutward. The polygons below are convex.

Nonconvex (Concave) PolygonsA nonconvex, or concave, polygon is a polygon in which at least two sides are pushed in. The polygons at the right are nonconvex.

Regular PolygonsA polygon is a regular polygon if (1) all the sides have the samelength; and (2) all the angles inside the figure are the same size. A regular polygon is always convex. The polygons below are regular.

Check Your UnderstandingCheck Your Understanding

1. What is the name of a polygon having a. 4 sides? b. 6 sides? c. 8 sides?2. a. Draw a convex hexagon. b. Draw a concave octagon.3. Explain why the cover of this book is not a regular polygon.

Check your answers on page 342.

triangle quadrangle (or quadrilateral)

pentagon hexagon

quadrangle (or quadrilateral)

pentagon

hexagon

octagon

equilateral triangle square regular pentagon

regular octagon

decagonheptagon octagon nonagon

regular nonagonregular hexagon

Student Reference Book, p. 97

Student Page

Adjusting the Activity

� Defining Properties of Polygons(Math Journal 1, p. 12)

Working in pairs, have students examine the two sets of shapes atthe top of journal page 12. Suggest that they use these shapes tohelp them discuss the properties of polygons. Then ask students tocomplete journal page 12 on their own.

Ongoing Assessment:Recognizing Student Achievement

Use journal page 12, Problem 1 to assess students’ understanding of the properties of polygons. Students are making adequate progress if responsesinclude the following:� A polygon is made up of line segments; curved lines are not line segments.� The sides of a polygon must form one closed path.� The sides of a polygon must not cross.� A polygon can have only one interior (inside).Some students may include additional information such as the following:� The number of sides is the same as the number of angles and vertices.� A polygon must have at least three sides.

[Geometry Goal 2]

� Defining the Properties of Regular Polygons(Student Reference Book, p. 97)

Ask pairs of students to look at the regular polygons on page 97of the Student Reference Book.

ELL

Discuss the everyday and mathematical meanings of the word regular.Write regular polygons on the board. Label and draw pictures of the regular andnonregular polygons you discuss in class. Display these labeled illustrationsthroughout the unit.

A U D I T O R Y � K I N E S T H E T I C � T A C T I L E � V I S U A L

PARTNER ACTIVITY

PARTNER ACTIVITY

Journal page 12 �Problem 1

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Links to the Future

Ask students to make a list of things that regular polygons have in common. The list should include the following:

� All sides of a regular polygon have the same length.

� The angles inside the figure are the same size.

� Regular polygons are convex.

� Regular polygons are symmetric.

Ongoing Assessment: Informing InstructionWatch for students who list properties of all polygons (for example, sides are line segments, there is only one interior, and so on). Ask them to focus on what makes regular polygons different from other polygons.

Call attention to the two regular polygons with special names: An equilateral triangle is a regular triangle, and a square is a regular quadrangle. Other regular polygon names are alwayspreceded by the adjective regular: regular pentagon, regular 13-gon, and so on.

In Unit 11 of Fourth Grade Everyday Mathematics, students apply their knowledgeof polygons as they identify and construct polyhedrons—geometric solids whosesurfaces are all flat and formed by polygons.

� Math Boxes 1� 5(Math Journal 1, p. 13)

Mixed Practice Math Boxes in this lesson are paired with Math Boxes in Lesson 1-7. The skill in Problem 6 previews Unit 2 content.

� Study Link 1� 5(Math Masters, p. 18)

Home Connection Students solve polygon riddles andwrite one of their own.

INDEPENDENTACTIVITY

INDEPENDENTACTIVITY

2 Ongoing Learning & Practice

13

Math Boxes LESSON

1�5

Date Time

3. Draw and label line segment AB.

What is another name for AB�?

A BSample answer:

4. Name as many rays as you can in thefigure below.

Write their names.

M N O

1. Subtract mentally.

a. 7 � 0 �

b. 10 � 7 �

c. � 9 � 4

d. � 14 � 6

e. 13 � 7 �

f. 16 � 9 � 76

85

37

2. Draw �MRT.

What is another name for �MRT?

�TRM

5. Which polygons have 2 pairs of parallelsides? Circle the best answer.

A. square and trapezoid

B. rectangle and rhombus

C. triangle and parallelogram

D. pentagon and square

6. Put these numbers in order from least to greatest.

10,005 51,000

5,100 10,500

51,00010,50010,005

5,100

92

91

499 100

90

MN�� (or MO��),BA� ON�� (or OM��),

NM��, NO��

T

M

R

Math Journal 1, p. 13

Student Page

Lesson 1�5 45

STUDY LINK

1� 5 Polygon Riddles

Name Date Time

Try This

Answer each riddle. Then use a straightedge to draw a pictureof the shape in the space to the right.

1. I am a quadrangle. I have 2 pairs of parallel sides.All of my angles are right angles.I am not a square.

What am I?

2. I am a polygon.All of my sides have the same measure.All of my angles have the same measure.I have 3 sides.

What am I?

3. I am a polygon.I am a quadrangle.All of my sides are the same length.None of my angles are right angles.

What am I? rhombus

Equilateral triangle

rectangle

4. On the back of this page, make up your own polygon riddle using 4 clues.Make 2 of the clues hard and 2 of the clues easy. Check your riddle by using astraightedge to draw a picture of the polygon. Ask a friend or someone athome to solve your polygon riddle. Answers vary.

96–100

5. 8 � 9 � 6. 7 � 8 � 7. 90 � 70 �

8. � 60 � 50 9. � 54 � 59 10. 185 � 366 � 5511131101601517

Practice

Math Master, p. 18

Study Link Master

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46 Unit 1 Naming and Constructing Geometric Figures

LESSON

1�5

Name Date Time

Polygons on a Geoboard

Practice using rubber bands to make polygons on a geoboard, thenfollow the directions below. Use a straightedge to record your work.

1. Make a triangle in which each sidetouches at least 4 pins.

3. Make a trapezoid.

2. Make a square in which each sidetouches at least 3 pins.

4. Make a hexagon that only touches 8 pins.

5. Compare your polygons with those of a partner. In the space below,make a list of how the polygons are alike and how they are different.

Answers vary.

96

Sample answers:

Math Masters, p. 19

Teaching Master

� Constructing Polygons on a Geoboard(Math Masters, p. 19)

To explore the properties of polygons using a concrete model, have students use rubber bands to construct polygons on ageoboard. Partners look for similarities and differences among thepolygons they made.

� Exploring Side Angle Properties(Math Masters, p. 20)

Literature Link To explore classifying polygons according totheir number of sides, have students read or listen to the

story The Greedy Triangle by Marilyn Burns (Scholastic, 1994)and build the polygons from the story using straws and twist-ties.Ask students to record their work on Math Masters, page 20.

� Identifying Properties of Kites and Rhombuses(Math Masters, pp. 21 and 22)

To apply students’ understanding of the properties ofkites and rhombuses, have them compare examples andnonexamples of each. They use this information todescribe the properties of both kinds of polygons.

Planning Ahead

Students play Polygon Pair-Up in Part 2 of Lesson 1-6. They willneed to cut apart the Polygon Deck and Property Deck found onMath Masters, pages 496 and 497. Consider copying the cards on cardstock.

15–30 Min

INDEPENDENTACTIVITYENRICHMENT

15–30 Min

PARTNER ACTIVITYREADINESS

5–15 Min

PARTNER ACTIVITY

READINESS

3 Differentiation Options

4 sides cannot have the same length.of equal length are next to each other. All 2 pairs of sides of equal length. The sides Sample answer: A kite is a quadrangle with

LESSON

1�5

Name Date Time

What Is a Kite?

These are NOT kites.

If you had to explain what a kite is, what would you say?

These are kites.

Math Masters, p. 21

Teaching Master

equilateraltrianglequadrilateral

ShapeDrawing of

ShapeNumber of

Sides

3

4

Number ofAngles

3

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