Working with the Geoboard © Didax – www.didax.com4

The “Chop Method”

Anothermethodwecanusetofindtheareaofapolygoniscalledthe“chopmethod.”Hereisanexampleofhowthechopmethodworks:

TakethefigureABCD, anirregularpolygon:

Step 1:

Buildarectanglecompletelyaroundtheshape.Notethattheareaofthisrectangle,FABE,is12squareunits.

Step 2:

Beginchopping:

AreaofGAD=12 areaofGAJD=1squareunit.

AreaofHDC=12 areaofHDKC=1squareunit.

AreaofCBE=12 areaofCLBE=1 1

2squareunits.

AreaofFGDH=1squareunit.

Altogether,thischops412 squareunitsoffthe12-square-unitrectangleFABE.

ThismeanstheareaoftheoriginalpolygonABCDis7 12 squareunits.

A

D

B

F

E

C

A

D

C

B

F G

E

H1 1

1

1 12

J

KL

A

D

C

B

1.2

Name: Date:

PRACTICE

© Didax – www.didax.com Working with the Geoboard 5

Find the Area Using the “Rectangle Method”

Make these shapes on your geoboard and find their areas.

1. 2. 3.

Area= sq.units Area= sq.units Area= sq.units

4. 5. 6.

Area= sq.units Area= sq.units Area= sq.units

7. 8. 9.

Area= sq.units Area= sq.units Area= sq.units

PRACTICE

Name: Date:

Working with the Geoboard © Didax – www.didax.com6

Find the Area Using the “Chop Method”

Make these shapes on your geoboard and find their areas.

Example: 1. 2.

Area= sq.units Area= sq.units

3. 4. 5.

Area= sq.units Area= sq.units Area= sq.units

6. 7. 8.

Area= sq.units Area= sq.units Area= sq.units

Areaof ABCD=4

–AreaofAED=1 12

–AreaofEBC = 12

AreaofDEC =2

A E B

CD

Name: Date:

PRACTICE

© Didax – www.didax.com Working with the Geoboard 7

Find the Area

Make these shapes on your geoboard and find their areas using either the rectangle method or the chop method.

1. 2. 3.

Area= sq.units Area= sq.units Area= sq.units

4. 5. 6.

Area= sq.units Area= sq.units Area= sq.units

7. 8. 9.

Area= sq.units Area= sq.units Area= sq.units

PRACTICE

Name: Date:

Working with the Geoboard © Didax – www.didax.com8

What’s the Area?

Makeashapeonyourgeoboard—anyshapeyouwant.Findtheareaoftheshapeyoumade.Sketchyourshapeontheblankgeoboardbelowifitwillhelp.

Askaclassmatetofindtheareaofyourshape.Ifyoudisagreeonwhattheareais,lookatyoursolutionstogethertodeterminethecorrectanswer.

Working with the Geoboard © Didax – www.didax.com10

Properties of Triangles

Atriangleisapolygonwiththreesidesandisclassifiedbythemeasuresofitsanglesorthelengthsofitssides.Ifclassifiedbythelengthsofitssides,atriangleiscalledscalene, isosceles, orequilateral:

Scalene–Notwosidesareequalinlength.Isosceles–Atleasttwosidesareequalinlength.Equilateral–Allthreesidesareequalinlength.

Ifclassifiedbythemeasuresofitsangles,atriangleiscalledacute, obtuse,orright:

Acute–Anacutetrianglehasthreeacuteangles.Acutemeanslessthan90degrees,orlessthanarightangle.Obtuse –Anobtusetrianglehasoneobtuseangle.Obtusemeansgreaterthan90degrees.Obtusemeansgreaterthan90degreesbutlessthan180degrees.Right –Arighttrianglehasonerightangle—thatis,anangleof90degrees.Equiangular–Allanglesareequalinmeasure.

Havestudentsusetheirgeoboardstoexplorethepropertiesoftrianglesintheactivitiesthatfollow.

Main Ideas

2.1

Name: Date:

PRACTICE

© Didax – www.didax.com Working with the Geoboard 11

Scalene, Isosceles, and Equilateral Triangles

Useyourgeoboardtomakethetrianglesonthispage,andcompletethetableonthenextpage.(Hint:Fillinthecorrespondingrowofthetableasyoumakeeachtriangle.)

A

B

C

D

E

F

G

H

I

PRACTICE

Name: Date:

Working with the Geoboard © Didax – www.didax.com12

Scalene, Isosceles, and Equilateral Triangles (cont.)

Refertothetrianglesonthepreviouspageandcompletethefollowingtable.Measurethelengthsofthesideswithyourruler.

Shape # Pegs Inside Scalene Isosceles Equilateral

A

B

C

D

E

F

G

H

I

Name: Date:

PRACTICE

© Didax – www.didax.com Working with the Geoboard 13

Triangles

Useyourgeoboardtomakethetrianglesonthispage,andcompletethetableonthenextpage.(Hint:Fillineachrowofthetableasyoumakeeachtriangle.)

A

B

C

D

E

F

G

H

I

PRACTICE

Name: Date:

Working with the Geoboard © Didax – www.didax.com14

Triangles (cont.)

Refertothetrianglesonthepreviouspageandcompletethethistable.Classifyeachtrianglebyitsanglesandthenbyitssidelengths.

Shap

e#

Pegs

Insi

deA

cute

Obt

use

Rig

htIs

osce

les

Scal

ene

Equi

late

ral

A B C D E F G H I

Name: Date:

CHECK YOUR UNDERSTANDING

© Didax – www.didax.com Working with the Geoboard 15

Triangles

Look back at the tables on pages 12 and 14 and answer the following questions. Circle the correct answer. (You may need to make the triangles on your geoboard again to solve these problems.)

1. Areallisoscelestrianglesacutetriangles? Yes No

2. Areallscalenetrianglesacutetriangles? Yes No

3.Canyoumakearighttrianglethatisisoscelesonyourgeoboard? Yes No

4. Canyoumakearighttrianglethatisobtuse? Yes No

5. Canyoumakearightacutetriangle? Yes No

6. Areallrighttrianglesacute? Yes No

7. Isitpossibletomakeanequilateraltriangleonyour25-pingeoboard? Yes No

8.Triangleswithpegsinsidearealwayslargerinareathantriangleswithoutpegsinside. True False

9.Righttrianglescannothavepegsinside. True False

10. Acutetrianglesarealwayssmallerinareathanobtusetriangles. True False

Name: Date:

PRACTICE

© Didax – www.didax.com Working with the Geoboard 29

How Many Pins?

Belowarethreesetsofexercisesthatwillshowyouanewwaytofindareasonthegeo-board.Eachsetasksyoutomakeashapewithaspecificnumberofpinsinsidetheshape.Foreachshapeyoumake,recordtheareaandnumberofpinstouchedbytherubberbandthatformstheshape.Awonderfulpatternexists.Seewhetheryoucandiscoverit!

1. MakeasmanyshapesasyoucanwithNO pins inside the shape.Recordyourresultsinthetable.

Area (A)Number of Pins Touched

by Rubber Band (N)Number of Pins Inside

the Shape (I)

0

0

0

0

0

2. Makeasmanyshapesasyoucanwithonly 1 pin inside the shape.Recordyourresultsinthetable.

Area (A)Number of Pins Touched

by Rubber Band (N)Number of Pins Inside

the Shape (I)

1

1

1

1

1

PRACTICE

Name: Date:

Working with the Geoboard © Didax – www.didax.com30

How Many Pins? (cont.)

3. Makeasmanyshapesasyoucanwithjust 2 pins inside the shape.Recordyourresultsinthetable.

Area (A)Number of Pins Touched

by Rubber Band (N)Number of Pins Inside

the Shape (I)

2

2

2

2

2

4. Whatpatternsdidyoudiscover?

5. Writetheformulathatdescribesthepattern.

Name: Date:

PRACTICE

© Didax – www.didax.com Working with the Geoboard 35

l = 60 m

w = 40 m

Perimeter Word Problems

Using what you discovered about perimeters with your geoboard, solve the following problems.

Hereisanexample:

Find the perimeter of this rectangle.

Write the formula. P = 2l + 2w

Substitute the data. P = (2 × 60) + (2 × 40)

Solve the problem. P = 120 + 80

P = 200

The perimeter of the rectangle is 200 meters.

1. Ifyouwanttobuildarectangularfencearoundyourgardenthatis12ft.wideand10ft.long,howmuchfencingwillyouneed?

2. Arectangularpicturemeasures39centimetersby50centimeters.Howmuchtrimisneededtogoaroundthepicture?

PRACTICE

Name: Date:

Working with the Geoboard © Didax – www.didax.com36

The Same but Different

Intheshapesshownbelow,noticethatthearea(A)remainsthesamebuttheperimeter(P)changes.

A=6;P=12 A=6;P=10 A=6;P=14

Make some shapes on your geoboard and see whether the perimeter always changes when the shape is different but the area is the same. Sketch your findings on the blank geoboards below. No right triangles allowed!

A = P = A = P = A = P =

A = P = A = P = A = P =

Working with the Geoboard © Didax – www.didax.com74

Answers

Chapter 1: Finding Area on the Geoboard

Page 5:Allanswersareinsquareunits.

1. 5 2.6 3.6 4.8 5.8 6.12 7.4 12 8.10 9.8

Page 6: Allanswersareinsquareunits.

1.4 12 2.34 1

2 3.5 4.4 5.8 6.2 7.4 8.7

Page 7: Allanswersareinsquareunits.

1.4 2.4 3.3 4.8 5.7 6.6 7.9 8.2 9.5

Page 8: Teachercheck.

Page 12:

Shape # Pegs Inside Scalene Isosceles Equilateral

A 1 X

B 3 X

C 0 X

D 1 X

E 1 X

F 0 X

G 3 X

H 3 X

I 0 X

© Didax – www.didax.com Working with the Geoboard 75

Page 14:

Shape # Pegs Inside Acute Obtuse Right Isosceles Scalene Equilateral

A 3 X X

B 0 X X

C 1 X X

D 1 X X

E 1 X X

F 1 X X

G 3 X X

H 0 X X

I 1 X X

Page 15: 1.No 2.No 3.Yes 4.No 5.No 6.No 7.No8.False 9.False 10.False

Page 17:

No,itisnotarhombus.Itdoesnothavefourcongruentsides.Adjacentsidesarenotequalinlength.

Page 18:

Answersmayvary.Possibleanswersinclude:

1.(A)Itisaparallelogram,(B)has4rightangles,and(C)adjacentsidesareequal.

2.(A)Itisaparallelogram,(B)has4rightangles,and(C)adjacentsidesdonothavetobeequal.

3.(A)Itisaquadrilateral,(B)withonlyonepairofparallelsides,and(C)alwayshasanobtuseangle.

4.(A)Itisaparallelogram,(B)withallfoursidesequalinlength,and(C)hastwoobtuseangles.

5.No

6.Yes,butnotonthegeoboard.

7.Yes

© Didax – www.didax.com Working with the Geoboard 77

Page 29–30:

1.Teachercheck.

2.Teachercheck.

3.Teachercheck.

4.Teachercheck.

5.A= 12N +(I–1),ifA=area,P=thenumberofpinstouchedbytherubber

band,andI =thenumberofpinsinsidetheshape.

Page 32:

1.Theareaofthesquareonthehypotenuseequalsthesumoftheareasoftheothertwosquares: 2 .Thesumofthesquaresofthelengthsofthetwo

sidesequalsthesquareofthelengthofthehypotenuse.

2. 13units 3.24units 4. 5units

Page 34:

1.8units 2.16units 3.10units 4.14units 5. 16units

6.14units 7.3 5+ 8.5 17+ 9.4 8+

Page 35:

1.44ft 2.178cm

Page 36: Teachercheck.

Page 37:

1.6 2.Yes 3.Yes:1,2,4,5,8,9;No:3,6,7

4.Yes.Possibleanswers:

5.Yes,a4×4squarehasanareaof16andaperimeterof16.