geometry and measurement of plane figures activity set 3 · geometry and measurement of plane...

Geometry and Measurement of Plane Figures

Activity Set 3

Trainer Guide

geometry and measurement of Plane figures—activity set 3 Mid_PGe_03_TGCopyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

GEOMETRY AND MEASUREMENT OF PlANE FIGURESACTIvITY SET 3

NGSSS 6.A.2.1 NGSSS 6.A.2.2 NGSSS 7.A.1.1

geometry and measurement of Plane figures—activity set 3 Mid_PGe_03_TG

Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development 1

It’s like This

In this activity, participants will distinguish between similar and congruent plane figures using ratios.

Materials

• Transparency/Page:It’sLikeThis• Transparency/Page:It’sLikeThisAnswerKey• Transparency/Page:ImageDilation• Transparency/Page:SimilarSides• Transparency/Page:SimilarSidesAnswerKey• blank transparencies• ruler with centimeters (1 for each participant)• protractor (1 for each participant)

Vocabulary

• congruentfigures• similarfigures• dilation• scalefactor

tiMe: 20 minutes

INTRODUCE

•Explain to participants that they will investigate the relationships between similar figures and congruent figures.

•Display Transparency:It’sLikeThis and have participants take out their matching pages.

•Instructparticipantstousetheirprotractorsandrulerstomeasure and compare the three triangles on the page.

•Direct participants to write the measurements on the figures and then describe the relationships between Figures 1 and 2 and Figures 1 and 3.

geometry and measurement of Plane figures—activity set 3 Trans_MS_PG_03Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development 2

Fig. 3

Fig. 1 Fig. 2

A C

B

D

E

F

H

IG

it’s like this

1. How are Figures 1 and 2 the same?

How are Figures 1 and 2 different?



Measure the angles and sides of each figure andlabel the diagrams.

Transparency: It’s Like This




•Ask participants to work individually.

•Allowparticipants5–7minutestocomplete the activity.

DISCUSS AND DO

•Callthegrouptogether.

•Askvolunteerstosharetheirangleandlengthmeasurements for each figure.

•Recordthemeasurementsonthetransparency.

•Ask volunteers to share their findings to the following questions:

◆ How are Figures 1 and 2 the same? (They are the same size and shape; all corresponding angles and all corresponding sides are equal.)

◆ How are they different? (They are exactly the same.)

•Explaintoparticipantsthatfiguresthatareexactlythesame size and same shape are congruent.

•Write,onthetransparency,ABC DEF, and point out that this means that the two triangles are congruent.

•Ask volunteers to share their findings to the following questions:

◆ How are Figures 1 and 3 the same? (They are the same shape; the corresponding angles are the same size.)

◆ How are they different? (The corresponding sides are of different lengths.)

•Tellparticipantsthatfiguresthatarethesameshapeand have congruent angles, but that have corresponding side lengths that are proportional rather than equal, are similar.

•Write,onthetransparency,ABC GHI and point out that this means the two triangles are similar.




•Draw,onablanktransparency,twosimilartrianglesas illustrated below.

•Tellparticipantsthatthesetrianglesaresimilar.

•Askparticipantswhatobservationscanbemade about these triangles. (Their corresponding angles are congruent and their corresponding sides are proportional.)

•Drawanotherpairofsimilartrianglesas illustrated below.

Note: The angles are labeled for these scalene triangles. Side measures are rounded to the nearest tenth and angle measures are rounded to the nearest degree.

•Askparticipantsifthesetrianglesaresimilar.(yes)

•Askparticipantshowtheycantellthatthesetrianglesare similar. (Their corresponding angles are congruent and their corresponding sides are proportional.)

3cm

3cm

3cm

2cm

2cm

2cm

3 m

6 m55° 29°

96° 4.7 m 4 m

8 m

55° 29°

96° 6.3 m




teaching tip: Ask participants if all rectangles are similar, as they all have congruent corresponding angles. (No, their sides may not be in proportion to each other. Different rectangles may have different shapes, even though all angles are congruent. Squares, however, are always similar because their sides are always proportional.)

•DisplayTransparency:ImageDilation

•Explaintoparticipantsthatyoucanenlargethe image by creating a larger space that is proportional to this one.

•Explaintoparticipantsthatthisprocessisoftenusedby artists and craftspeople to enlarge a sketch to its full size when producing a painting or a mural.

•Demonstratehowtocreateaproportionalgridbydrawing a new box below the existing image. Make it with a scale factor of 2 : 1. The new box will be twice as wide and twice as tall.

•Pointoutasmallsectionoftheimageandsketchthepart contained in the section into the corresponding section in larger grid by using the 2 : 1 ratio for each line.

•Havestudentscompletetheprocessontheirpages.

•Allow5–7minutesforthisactivity.

•Getthegroup’sattention.

•Explaintoparticipantsthatwhenproportionsareused to create images larger or smaller than their originals, the proportions are referred to as scalefactors.Whenacopyisfivetimesthesizeoftheoriginal,ithasbeenincreasedbyascalefactorof5.

geometry and measurement of Plane figures—activity set 3 Trans_MS_PG_03Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

image Dilation

Transparency: Image Dilation




geometry and measurement of Plane figures—activity set 3 Trans_MS_PG_03Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

M

N

D F

5050

E

xC

150

A

150

120

B

Similar Sides

1. ABC ~ DEF. Find the length of DF.

2. Rectangle M is similar torectangle N. The ratio ofrectangle N’s width torectangle M’s width is 3:2.Rectangle M has a length of24 cm and a width of 16 cm.What is the perimeter ofrectangle N?

Transparency: Similar Sides

•Askparticipantstodescribetherelationshipbetweenthe height of the original and that of the copy. (The copy is twice as long. In terms of a ratio, the height of copy: height of original is 2 : 1.

•Askparticipantswhatthescalefactoris.(2)

•Askvolunteerparticipantshowtheareaofthecopycompares to the area of the original. (The copy is 4 times the area of the original. In terms of a ratio, area of copy : area of original is 4 : 1.

•Explainthatthisscalefactorisoftenexpressedas22 to indicate that it refers to area or square units.

CONClUDE

•DisplayTransparency:SimilarSides and have participants take out their matching pages.

•Haveparticipantsworkinpairs.

•Tellparticipantstousewhattheyhavelearnedaboutratios to find the missing dimensions of the figures on their pages.

•Allow5–7minutesforthisactivity.

•Callthegrouptogether.

•Askforvolunteerstosharetheirprocessesandtheirfindings.Writethestepsforeachsolutiononthedisplayed transparency.

•RefertoTransparency:SimilarSidesAnswer Key to resolve any questions.

End of It’s like This




Race to Place

In this activity, participants will use geometric knowledge that they remember to match pictures of angles and shapes with their definitions.

Materials

• Transparency/Page:RacetoPlaceDirections• Transparency/Page:TriangleFactsAnswerKey• Transparency/Page:AngleFactsAnswerKey• Transparency/Page:AnglesinShapesAnswerKey• Transparency/Page:LineFactsAnswerKey• Transparency/Page:CircleFactsAnswerKey• RacetoPlaceCards• 5pocketcharts• bell

tiMe:15minutes

teaching tip: Post the pocket charts with their titles and the definitions before the beginning of the activity. Use the Facts transparencies as a guide for the definitions that go with each title. Space the charts around the room with a lot of room between them.




INTRODUCE

•Suggesttoparticipantsthatovertimetheyhaveaccumulated a lot of knowledge about the way lines, shapes, and angles work.

•Pointoutthefivechartsandtheirdefinitions.

•Explaintoparticipantsthattheywillcompeteasteams to match geometric definitions with pictures that illustrate the concepts defined.

teaching tip: If you have a large group, assign pairs instead of single people to each card.

DISCUSS AND DO

•DisplayTransparency:RacetoPlaceDirections.

•Gooverthestepsofthegame.

•Haveparticipantsmoveinto4or5equal-sizedgroups.

•Distributethecards—allofonecoloredshapetoeachgroup, one card per person.

•Call,“Go.”

•Havethefirstgrouptofinishsendonemembertothefront of the room to ring the bell.

teaching tip: If a team member cannot place his or her card, he or she should go to the end of the line and wait to place the card after other team members have placed their cards.

teaching tip: If the group is inexperienced, permit them a few moments to look at the definition sheets (Answer Keys) before the game.

geometry and measurement of Plane figures—activity set 8 TRANS_MS_PG_08Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

Directions • Distribute your team cards evenly among the

members of your team.

• Have team members play their cards in relay fashion.

• Have a player:

• race to the chart that holds the definition of the picture on his or her card

• place the card next to the definition

• race back to the team and sit down

• Have the next person race to the chart and place hisor her card.

• Have one team member race to the front and ring thebell when all the team’s cards are correctly placed.

race to place

Transparency: Race to Place Directions




CONClUDE

•Congratulateparticipantsforbeingabletorememberso many geometry concepts and definitions.

•Displaytheanswerkeytransparenciesinturn,quicklyreviewing the definitions.

•Emphasizethefollowingdefinitionsforeachkey(these will be of use in subsequent activities):

◆ Transparency:TriangleFactsAnswerKey –equilateraltriangle –righttriangle(esp.hypotenuse)

◆ Transparency:AngleFactsAnswerKey –straightangle –verticalangles

◆ Transparency:AnglesinShapesAnswerKey –triangle –equilateraltriangle

◆ Transparency:LineFactsAnswerKey –alternateinteriorangles

◆ Transparency:CircleFactsAnswerKey –circumference

End of Race to Place

geometry and measurement of Plane figures—activity set 3 TRANS_MS_PG_03Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

• A scalene triangle has no congruent sides and no congruent angles.

• An isosceles triangle has two congruent sides and two congruent angles.

• An equilateral triangle has three congruent sides and three congruent angles.

• The angles of an acute triangleare all less than 90˚.

• One angle in an obtuse triangleis greater than 90˚.

• A right triangle has one angle equal to 90˚. The side opposite the 90˚ angle is called the hypotenuse.

triangle Facts Answer Key

Transparencies: Triangle Facts, Angle Facts, Angles in Shapes, Line Facts, and Circle Facts Answer Keys

GEOMETRY AND MEASUREMENT OF PlANE FIGURES—AcTIvITY SET 3 Mid_PGe_03_PM

Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development

Fig. 3

Fig. 1 Fig. 2

A C

B

D

E

F

H

IG

It’s Like This





Measure the angles and sides of each figure and label the diagrams.



It’s Like ThisAnswer Key

Fig. 3

Fig. 1 Fig. 2

A C

B

D

E

F

H

IG90º

53º

37º

90º

53º

37º 90º

8 cm

4 cm 4 cm

6 cm

3 cm

3 cm

53º

37º





Measure the angles and sides of each figure and label the diagrams.

The angle and side measurements are exactly the same.

They are not different. They are exactly the same.

The angle measurements are exactly the same.

The side measurements are different, but they are in proportion to each other.



Image Dilation



M

N

D F

5050

E

xC

150

A

150

120

B

Similar Sides


2. Rectangle M is similar to rectangle N. The ratio of rectangle N’s width to rectangle M’s width is 3:2. Rectangle M has a length of 24 cm and a width of 16 cm. What is the perimeter of rectangle N?



M

N

D F

5050

E

xC

150

A

150

120

B

6,000 = 150x

120 • 50 = 150x

6,000 = x

40 = x

150

120 =

= 2l + 2wP

= 2(36) + 2(24)P

= 72 + 48P

= 120 cmP

x15050

AC =DFABDE

= 3length of N = l length of M = 24

= 3 • 242l

= 722l

= 36l

2

= 3width of N = w width of M = 16

= 3 • 162w

= 482w

= 24w

2

Similar SidesAnswer Key


2. Rectangle M is similar to rectangle N. The ratio of rectangle N’s width to rectangle M’s width is 3:2. Rectangle M has a length of 24 cm and a width of 16 cm. What is the perimeter of rectangle N?



Directions •Distributeyourteamcardsevenlyamongthemembersofyourteam.

•Haveteammembersplaytheircardsinrelayfashion.

•Haveaplayer:

• racetothechartthatholdsthedefinitionofthe picture on his or her card

• placethecardnexttothedefinition

• racebacktotheteamandsitdown

•Havethenextpersonracetothechartandplacehisor her card.

•Haveoneteammemberracetothefrontandringthebellwhenalltheteam’scardsarecorrectlyplaced.

Race to Place



• Ascalene triangle has no congruent sides and no congruent angles.

• Anisosceles triangle has two congruent sides and two congruent angles.

• Anequilateral triangle has three congruent sides and three congruent angles.

• Theanglesofanacute triangle are all less than 90˚.

• Oneangleinanobtuse triangle is greater than 90˚.

• Aright triangle has one angle equal to 90˚. The side opposite the 90˚ angleiscalledthehypotenuse.

Triangle Facts Answer Key



• Themeasureofanacute angle is less than 90˚.

• Themeasureofanobtuse angle is greater than 90˚ and less than 180˚.

• Themeasureofastraight angle is equal to 180˚.

• Themeasureofaright angle is equal to 90˚.

• Anglesthatshareacommonsidebetweenthemareadjacent.

• Twoangleswithmeasuresthatsumto 180˚ are called supplementary.

• Nonadjacentanglesformedby two intersecting lines are called vertical angles.Theyhavethe same measure.

Angle FactsAnswer Key



• Atriangle has angles that sum to 180˚.

• Arectangle has angles that sum to 360˚.

• Anglesinsideashapeare interior angles.

•Exterior angles are angles outside a shapethatareformedbyextendinga side of the shape.

• Thebaseanglesandopposite sides of an isosceles triangle are congruent.

• Thesidesandanglesofanequilateral triangle are congruent.

Angles in Shapes Answer Key



• Asetofpoints,astraightpath, thatextendsindefinitelyin two opposite directions is a line.

• Aline segment is two endpoints and the straight pathbetweenthem.

• Perpendicular lines form right angles.

• Ifalineintersectstwoparallel lines,thealternate interior angles are equal.

• Parallel lines are equidistant from each other.

Line FactsAnswer Key

6 cm6 cm



• Acompleterevolutionaroundthecenter of a circle has 360º.

• Achord is a line segment that connects two points on the circumference of a circle.

• Thelinesegmentjoiningthecenterofthecircle and a point on its circumference is called a radius.

• Adiameter is a chord that passes through thecenterofacircle.Itslengthistwicethatof the radius of the circle.

• Acircle is the set of all points in a plane that are equidistant from a specified point.

• Thedistancearoundacircleiscalled its circumference.

Circle Facts Answer Key


Copyright© by the McGraw-Hill Companies—McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF PlANE FIGURES—AcTIvITY SET 1 BLM_MS_PG_01Copyright© 2002 by the McGraw-Hill Companies—McGraw-Hill Professional Development

GlossaryGeometry and Measurement of Plane Figures

acute angle An angle with a measure less than 90 degrees (°).

angle A geometric figure composed of two raysor line segments that sharethe same endpoint,called a vertex.

area The numberof square units in a region.

circle The set of all points in a plane that are the same distance from afixedpoint (the center of the circle).

circumference The perimeter of (distance around) a circle. Thecircumference can befound using the formula C = 2πr,where C is the circumference of the circle and r is the radius of the circle.

congruent figures Two figures that haveidentical size and shape so thatwhen one is placed overthe other,theycoincide exactly.

coordinate pair An ordered pair of numbersthat indicates the position ofa point on a plane. The first numberof a coordinate pair givesthepoint’s location in relation to the x-axis.The second numberin acoordinate pair givesthe point’s location in relation to the y-axis.

coordinate plane A plane containing an x-axisand a y-axis.Everypoint on the plane can bedescribedusing a coordinate pair.

degree (°) A unit of measure for angles. 1° is 1360 of a complete revolutionaround a point.

equilateral The propertyof havingequal,or congruent,sides.

equilateral triangle A three-sided polygonwith all sides and with all angles congruent.

hexagon A six-sidedpolygon.

irregular polygon A polygonin which not all the sides are congruent and not all the angles havethe same measure.



isosceles triangle A triangle that has two congruent sides and two congruent angles.

line The set of all contiguous (touching) points that form a straight pathextendingindefinitelyin two directions opposite each other.

line segment A part of a straight line that has two end points and a fixedlength; a straight line segment marksthe shortest distancebetweentwo points.

linear unit A unit of measure for elements of a single dimension—length.

obtuse angle An angle with a measure greater than 90° and less than 180°.

parallel lines Lines that do not intersect and that are everywhereequidistant from each other.

parallelogram A quadrilateral in which bothpairs of opposite sides are parallel.

pentagon A five-sidedpolygon.

perimeter The distance around the outside of a plane shape or figure.

perpendicular At right angles to. Two lines are perpendicular if theirintersection creates right angles.

pi (π) The ratio of the circumference of anycircle to its diameter(3.141592653 . . .). Pi is usuallyrepresented bythe Greekletter,π.

plane A flat surface that extendsforeverin all directions.

plane figure A figure that lies entirelyin one plane.

point A location in space.

polygon A simple,closed plane shape composed of a minimum of threestraight-line segments.

Glossary (continued)



Glossary (continued)

quadrilateral A four-sided polygon.

radius A segment connecting the center of a circle to anypoint on thecircle; the length of the radius.

ray A subsetof a line that includes one endpoint and that extendsinfinitelyfrom that endpoint in one direction.

rectangle A quadrilateral that includes four interior right angles.

regular polygon A polygonin which all the sides are congruent and all theangles havethe same measure.

rhombus A parallelogram in which all sides are congruent.

right angle An angle with a measure of 90°.

right triangle A triangle with one right angle.

scalene triangle A triangle in which no sides are congruent and no angleshavethe same measure.

similar figures Figures that havecongruent corresponding angles and inwhich corresponding sides are proportional.

square A quadrilateral in which all sides and all angles are congruent.

square unit A unit of measure used to describethe surface (area) of figuresof two dimensions—length and width.

straight angle An angle with a measure of 180°.

trapezoid A quadrilateral in which onlyone pair of sides is parallel.

triangle A three-sided polygon.

vertex (pl. vertices) The intersection point shared bytwo sides of apolygonor the two sides (rays)of an angle. Also the intersection pointshared bythree or more edges of a polyhedron.



Ra

ce t

o P

lace

Ca

rds

(1 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(2 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(3 o

f 2

0)6 cm

6 cm



Ra

ce t

o P

lace

Ca

rds

(4 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(5 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(6 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(7 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(8 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(9 o

f 2

0)



Ra

ce t

o P

lace

Ca

rds

(10

of

20

)



Ra

ce t

o P

lace

Ca

rds

(11

of

20

)

A complete revolution around the center of a

circle has 360º.

A chord is a line segment that connects two points

on the circumference of a circle.

The line segment joining the center of the circle and

a point on its circumference is called a radius.



Ra

ce t

o P

lace

Ca

rds

(12

of

20

)

Ifa line intersects two parallel lines,the

alternate interior angles are equal.

Parallel lines are equidistant from

each other.

The measure of a right angle is equal to 90°.



Ra

ce t

o P

lace

Ca

rds

(13

of

20

)

A set of points that extendindefinitelyin two opposite

directions is a line.

A line segment has two endpoints.

Perpendicular lines form right angles.



Ra

ce t

o P

lace

Ca

rds

(14

of

20

)

Exterior angles are angles outside a shape that are formed by extending a

side of the shape.

The base angles and opposite sides of an

isosceles triangle are congruent.

The sides and angles of an equilateral triangle

are congruent.



Ra

ce t

o P

lace

Ca

rds

(15

of

20

)A triangle has angles that sum to 180˚.

A rectangle has angles that sum to 360˚.

Angles inside a shape are interior angles.



Ra

ce t

o P

lace

Ca

rds

(16

of

20

)

Nonadjacent angles formed by two intersecting

lines are called vertical angles. They have the

same measure.

Angles that share a common side between

them are adjacent.

Two angles that sum to 180˚ are called

supplementary.



Ra

ce t

o P

lace

Ca

rds

(17

of

20

)The measure of an acute angle is less than 90˚.

The measure of an obtuse angle is greater

than 90˚ and less than 180˚.

The measure of a straight angle is

equal to 180˚.



Ra

ce t

o P

lace

Ca

rds

(18

of

20

)

The angles of an acute triangle are all

less than 90˚.

Oneangle in an obtuse triangle is

greater than 90˚.

A right triangle has one angle equal to 90˚.



Ra

ce t

o P

lace

Ca

rds

(19

of

20

)

The diameter is a chord that passes through the

center of a circle.

A circle is the set of all points in a plane that are

equidistant from a specified point.

The distance around a circle is called its circumference.



Ra

ce t

o P

lace

Ca

rds

(20

of

20

)

A scalene triangle has no congruent sides and

no congruent angles.

An isosceles triangle has two congruent sides and

two congruent angles.

An equilateral triangle has three congruent sides

and three congruent angles.

geometry and measurement of plane figures activity set 3 · geometry and measurement of plane...

Documents