Geometry Teaching Guide - Activities

8/2/2019 Geometry Teaching Guide - Activities

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Poster/teaching guide

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GEOMETRY

Lessons & Worksheets to Build Skills in

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2/8

OBJECTIVES: Students will understand

ormulas used to measure the perimeter and area o these basictwo-dimensional shapes: rectangles, circles, and triangles.

Time Required: 20 minutes, plus additional time or worksheets

Materials: Student Worksheet 1

Extensions: Bonus Worksheet 1, Take-Home Activity 1

DIRECTIONS:

1. Review with students the concept o perimeter. Perimeteris the totaldistance around the outside o a polygon (a closed gure made up oline segments).

2. On the board, draw a rectangle labeled with a length o 4 eet andwidth o 3 eet. Then draw a right triangle with a base o 4 eet, heighto 3 eet, and hypotenuse (the side opposite the right angle) o 5

eet. Explain that to measure the perimeter o any polygon, you addtogether the lengths o each side.

3. Ask students what the perimeter o the rectangle is. Show studentsthe ormula or the rectangles perimeter on the poster and ask why

its correct. The ormula oP(perimeter) = 2 (l+ w) is correct because

a rectangle has two sets o sides that are each o equal length. The

perimeter o this rectangle is 2 (4 + 3) = 14 eet.

4. Ask what the perimeter o the triangle is. Show them the ormula:P= side a + side b + side c. The perimeter is 3 + 4 + 5, or 12 eet.

5. Tell students that triangles can be classied by angles in three ways: 1)right triangles with one 90 angle where the base and height meet; 2)acute triangles with all angles less than 90; and 3) obtuse triangles

with one angle greater than 90. The angles o any triangle equal 180.6. Draw a circle on the board. Draw a line rom the center o the circle to

the edge and mark it as 3 eet. Tell students that this is the radius. Askthem what the diameteris. (The answer is 6 eet.) Then explain that

the length o the line that orms the circle is called the circumference.There is a unique ormula or calculating the circumerence: C(circumerence) = pd(diameter). Tell students thatp is the

circumerence o any circle divided by its diameter and equals a

number with an innite decimal: 3.14159.... The decimal continues

on innitely, but to solve most math problems, people use a rounded

ratio o 3.14. Ask students to gure out the circumerence o the circle

you have drawn. Ask them to provide the answer to the nearest hal

oot. As 3.14 6 = 18.84 eet, the answer is 19 eet.

7. Now go over the denition oarea on the poster: the measure o abounded region o a two-dimensional shape expressed in square units,

e.g., square inches or square eet. Show your students the ormula or

area o a rectangle:A (area) = lw. Ask them to calculate the area othe rectangle you had drawn earlier (4 3 = 12 square eet).

8. Now point out the ormula or the area o a triangle on the poster:A = 1/2 [b (base) h (height)]. Reer back to your drawing o a right

triangle with a base o 4 eet and height o 3 eet. Ask students to

calculate the area. The answer is 1/2 (4 3) = 6 square eet.

9. Finally, go over the area ormula or circles. Again, reer to the poster:A = p r2, where r2 means radius squared, or rr. The answer is p (3.14)

r2 (3 3) = 28.26 square eet. Ask students to provide the answer to

the nearest hal oot. The answer is 28.5 eet or 28 eet and 6 inches.

10. Distribute Student Worksheet 1. Tell students they should completeall the questions. Explain that the bonus question introduces a new

ormula or the area o trapezoids. Go over correct answers as a class

using the Worksheet Answer Key(see back cover).

In the lessons and worksheets or this program, students will learn

and reinorce these geometry skills:

1. measuring perimeter and area of 2D shapes;

2. measuring surface area of 3D shapes; and

3. measuring volume of 3D shapes.

The materials are taught through this story line: A popular

band called The Geometricsis planning a big concert at aschool, but they need help to build a stage and promote the

show. Some students volunteer to becomethe Geometrics

Stage Crewand use their geometry knowledge to help.

Threelesson plans teach basic measuring skills; each lesson

eatures a worksheet, and is also supplemented by a bonus

worksheet and a take-home activity.

Beore launching the lessons, you can engage students in a

discussion about real-world geometry with the classroom

poster. Show your class how geometric shapes can be ound inthe concert setting on the poster. Ask students where they have

seen these shapes in their daily lives.

The poster includes undamental ormulas you can display in the

classroom year-round. In addition, there is a handy referencesheet o ormulas and denitions or teachers and students. The

reerence sheet also eatures drawings o all the shapes included

in these lessons.

Note: All program pages appear in full color, yet are designed to easily

reproduce in black and white.

getting started

Perimeter and areaof 2d shaPesGeometry Works! The

Stage Takes Shape

Lesson 1

DearTeacher:

Welcome toSettingtheStagewith

Geometry,anewmathprogramalignedwith

NCTMstandardsthatisdesignedtohelpstudents

ingrades68buildandreinforcebasicgeometry

skillsformeasuring2Dand3Dshapes.

DevelopedbyTheActuarialFoundation,

thisprogramseekstoprovideskill-building

mathactivitiestohelpyourstudentsbecome

successfulintheclassroomandinreal-world

situationsoutsideofschool.Wehopeyou

enjoythisnewprogram!

Sincerely,

TheActuarialFoundation


3/8

OBJECTIVES: Students will understand ormulas used to measure the

volume o these basic three-dimensional shapes: a rectangular prism, acylinder, and a square pyramid.




DIRECTIONS:

1. Explain to your students that now that theyve mastered measuringthe surace area o 3D shapes, they can move on to measuring volume,

which is the amount o space inside a 3D shape, measured in cubic

units. Reer to the poster, which provides essential ormulas.

2. Again, draw a rectangular prism on the board like the one rom theLesson 2 surace area unit with these measurements: height = 3 eet,

length = 4 eet, and width = 5 eet.

3. Show students the volume ormula or rectangular prisms on theposter: V(volume) = lwh. Ask students what the volume o the

prism is. The answer is 4 5 3 = 60 cubic eet.

4. Now draw a cylinder again with the same dimensions as in Lesson 2:The radius is 3 eet and the height is 4 eet.

5. Show students the volume ormula or cylinders on the poster: V=pr2h. Ask students what the volume o the cylinder is when rounded to

the nearest hal oot. As 3.14 9 4 = 113.04 cubic eet, the answer is

113 cubic eet.

6. You might add that a cylinder is like a barrel, and volume measurementcan help determine how much liquid will t in a container this size.

One cubic oot = 7.48 gallons. Ask students how much water this

cylinder would hold. The answer is 7.48 113.04, or 845.54 gallons(when rounded to the nearest hundredth). Students may need a

calculator to solve this problem.

7. Finally, draw a square pyramid on the board with the same dimensionsas in Lesson 2: The square pyramid has a base length o 6 eet and a

base width o 6 eet. The height o the pyramid is 4 eet.

8. Show students the volume ormula or square pyramids on the poster:V=1/3 BAh. Ask students or the volume o the square pyramid.

The answer is 1/3 36 4 = 48 cubic eet.

9. Distribute Student Worksheet 3. Tell students they should completeall the questions. You may want to take some extra time to go over the

bonus question, which introduces the ormula or the volume o a cone

[V=p 1/3 r2h]. Go over all correct answers as a class reerring to

the Worksheet Answer Key(see back cover).

OBJECTIVES: Students will understand ormulasused to measure the surace area o these basic

three-dimensional shapes: a rectangular prism,

a cylinder, and a square pyramid.




DIRECTIONS:

1. Ater mastering the area o 2D shapes, students can now learn theormulas to measure 3D shapes.

2. Draw a rectangular prism on the board with these measurements:height = 3 eet, length = 4 eet, and width = 5 eet.

3. Show students the surace area ormula or rectangular prisms onthe poster:SA = 2 (l w+ l h + w h). Explain to them that the

surface area o 3D objects is measured in square units, just like the

area o 2D objects, and is the sum o all o the 3D objects suraces.4. Ask students what the surace area is o the shape you have drawn.

The answer is 2 (20 + 12 + 15) = 94 square eet.

5. Now draw a cylinder and mark the dimensions with the radius at 3eet and the height at 4 eet.

6. Show students the surace area ormula or cylinders on the poster:SA = (2 pr2) + (pdh) and ask them to calculate the answer

to the nearest hal oot. As (2 3.14 9) + (3.14 6 4) = 131.88

square eet, the answer is 132 square eet.

7. Finally, draw a square pyramid on the board and mark thedimensions with a base length o 6 eet and a base width o 6 eet.

Show the slant height as 5 eet by drawing a perpendicular line

rom the center o one o the base sides to the top o the pyramid.The square pyramid has a base area (BA) measurable by lwlike

any square or rectangle.

8. Show students the surace area ormula or square pyramids on theposter,SA = (BA) + 1/2 Pslanth and ask students to calculate

the answer. This ormula adds together the area o the base with

the area o the our triangular sides o the square pyramid. The Pin

the ormula reers to the perimeter o the base. The answer is 36 +

1/2 24 5 = 96 square eet.

9. Distribute Student Worksheet 2. Tell students they shouldcomplete all the questions. You may want to take some extra

time in class to go over the bonus question, which introduces the

ormula or measuring the surace area o a cone [SA = (pr2) + (p

rslant)]. Go over all correct answers as a class, reerring to theWorksheet Answer Key(see back cover).

VoLume of3d shaPesPack It Up!

What Will Fit?

surface area of 3d shaPesThat Should Cover It!

Lesson 3Lesson 2

n Advancing Student Achievement Grants

n Expect the Unexpected With Math Series:Shake, Rattle, & Roll (probability)

Bars, Lines, & Pies (graphing)

Conversions Rock(converting decimals, fractions, and percents)

n

The Math Academy Series: Using Math in the Real World

Look formore math resourcesat The Actuarial Foundation Web site at:www.actuarialfoundation.org/programs/for_teachers.shtml

al l pbl p p : www.scholastic.com/unexpectedmath


4/8

r st

: the measure o a bounded region o a two-dimensionalshape expressed in square units

u:the distance around the edge o a circle

t:the distance across a circle through itscenter point

ptu: the side opposite the 90 angle in a righttriangle, also the longest side o a right triangle

pt: the total distance around the outside

o a polygonp p: the circumerence o any circle divided by itsdiameter, rounded to the number 3.14

u: the measure rom the center o a circle to a pointon the circle

t: the diagonal distance rom the top o a cone to its base

t t: the height o one o the triangular aces o apyramid

u : the sum o all the areas o all suraces o athree-dimensional object, measured in square units

vu: the amount o space inside a three-dimensionalshape, measured in cubic units

rtP = 2 (l + w)A = l w

rtu PSA = 2 (l w + l h + w h)V = l w h

TP = side a + side b + side c

A = 1/2 (b h)

Right tRiAngleACute tRiAngleobtuSe tRiAngle

squ PSA = (BA) + 1/2 P slant h

V = 1/3 BA h

Note: base area (BA) of a square or

rectangular pyramid is lw of the base,

andP is perimeter of the base.

cSA = (2 p r2) + (p d h)

V = p r2 h

cSA = (p r2) + (p r slant)

V = p 1/3 r2 h

Pt, a, su a, Vu:rvw T, b sp, u

basic shaPes and ormUlas

Terminology

2D ShAPeS: PeRimeteR AnD AReA 3D ShAPeS: SuRfACe AReA AnD Volume

hypotenuse

w

w

TpzP = side a + b1 + b2 + side c

A = 1/2 (b1 + b2) h

1

2

a c

cC = p d

A = p r2

r

slant

abbreViaTions: d = dar r = rads

A = ara = SA = srac ara

= as = sa = sa

bA = as ara P = prr V = v

C = crcrc p= p = 3.14 w = wd

r

w

rd

slanth


5/8

1 First, they want a main stage that is rectangular-shaped, measuring a length o 24 eet and a width o 16 eet. What arethe perimeter and area o that stage?

Prr: Ara:

2 Second, the bands lead guitarist wants the Geometrics Stage Crew to build a smaller circular stage in ront o the mainstage that he can step onto and play a solo. The diameter has to be one-third o the length o the main stage. What is

the circumerence and area? Round your answer to the nearest oot.

Crcrc: Ara:

3 The bass player has a thing or triangles and sees hersel on a triangular platorm o to the let o the stage. Whenviewed rom above, the right triangle has a height o 8 eet, a base o 6 eet, and a third side (called the hypotenuse) o

10 eet. What is the perimeter and area?

Prr: Ara:

stut Wkt 1 Name:___________________________________________________ Date:_____________

The popular band The Geometrics

wants to play a special concert at

your school, but they need a stage

crew to help. The frst step or the

Geometrics Stage Crew is d

a ara sa ar

dry sapd scs.

gtWk!T stTk sp

bonUs: T u wt t tpz-p pt.T qu t gt st cw t w

u t tpz [a = 1/2 (b1 + b2) h]. T tpz w t wt t ut t. b 1 (b1) = 8 t.b 2 (b2) = 5 t. T t u 6 t. Wt t ?

8 '

5 '

6 '


6/8

1 The dimensions o the rectangular prism or thelead guitarist are height = 7 eet, width = 4 eet, andlength = 3 eet.

Srac Ara =

2 The dimensions o the cylinder or the drummer areradius = 3.5 eet and height = 7 eet. Solve with a decimal,then also round to the nearest hal oot.

Srac Ara =

3 The dimensions o the square pyramid or the bass playerare base length = 4, base width = 4, and a slant heighto 7.28 eet. Solve with a decimal, then also round to thenearest hal oot.

Srac Ara =

4 Can the stage crew paint the surace area o all threeshapes with just one can o paint?

stut Wkt 2 Name: ___________________________________________________ Date:_____________

The Geometrics love shapes. For the upcoming

concert, the three main players each want to

emerge rom human-size shapes o a rectangular

prism, a cylinder, and a square pyramid. While

they already have these props built, the band

asks the Geometrics Stage Crew to paint over

them completely (even the bottom o each

object). The stage crew knows that 1 gallon o

paint covers 350 square eet. To buy the right

amount o paint, sa crw as caca srac ara ac sap.

Tt sucv it!

T u tt k t v kup wt t t u t t 7.8 t.

T t w w u t fu ut t u :

sa = (pr2) + (pr t).

. Wt t u ? exp t u t t t t.

. i pt v 350 qu t, ut w u t pt t ?

bonUs:

7'

7.8'

3.5'


7/8

1 First, get the complete volume o all the objects combined.a. What is the volume o the rectangular prism?

________________________________________________________

b. What is the volume o the cylinder? (Give the decimal answer and then round it to the nearest cubic oot.) ________

c. What is the volume o the square pyramid? _________________________________________________________

d. What is the total volume o all the objects combined, rounded to the nearest oot? __________________________

2 The van cargo space measures 8 eet tall by 5 eet wide by 13 eet deep. What is the volume o the cargo space?

3 Based on the volume measurements, can you estimate i the objects will ft in the van?

The dimensions o The objecTs again are:

Rcaar Prs: height = 7 eet, width = 4 eet, and length = 3 eet

Cydr: radius = 3.5 eet and height = 7 eet

Sqar Pyrad: base length = 4, base width = 4, a slant height o 7.28 eet, and a height o 7 eet

The Geometrics Stage Crew now has to

transport the painted props o a rectangular

prism, a cylinder, and a square pyramid to

the concert. They have to make sure the van

they have is big enough to carry the props.

To do this, they are going to asr

v car spac ad cpar

a v r jcs

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Pk it Up!Wt W t?

b vu, wu t t t tu t Wkt 2? T u t = 3.5 t

t = 7 t t = 7.8 t. T ut t vu (u ut) t , u t w u: V= p 1/3 r2h.

bonUs:

stut Wkt 3 Name:___________________________________________________ Date:_____________

7'

7.8'

3.5'


8/8

Student Worksheet 1: Geometry Works! The Stage Takes Shape

1. Perimeter: 2 (24 + 16) = 80 eet

Area: 24 16 = 384 square eet

2. 8 eet is one-third the length o the main stage.

Circumerence: 3.14 8 = 25.12 eet, rounded = 25 eet

Area: 3.14 42 = 50.24 square eet, rounded = 50 square eet

3. Perimeter: 8 + 6 + 10 = 24 eet

Area: 1/2 (8 6) = 24 square eet

Bonus: Trapezoid Area = 1/2 (8 + 5) 6 = 39 square eet

Student Worksheet 2:That Should Cover It!

1. 2 (3 4 + 3 7 + 4 7) = 2 (12 + 21 + 28) = 2 61 = 122 square eet

2. (2 3.14 12.25) + (3.14 7 7) = (76.93) + (153.86) = 230.79 squareeet , rounded = 231 square eet

3. (4 4) + 1/2 16 7.28 = 16 + 58.24 = 74.24 square eet, rounded = 74square eet

4. No. The total surace area is 427.03 square eet and one can o paintwill cover only 350 square eet, so the stage crew needs more paint.

Bonus: a. (3.14 12.25) + (3.14 3.5 7.8) = 38.465 + 85.722 = 124.187square eet, rounded = 124 square eet; b. 124.187/350 = .35482 or about

.35, or 35% o the gallon

Student Worksheet 3: Pack It Up! What Will Fit?

1. a. 7 4 3 = 84 cubic eet

b. 3.14 12.25 7 = 269.255 cubic eet, rounded = 269 cubic eet

c. 1/3 (4 4) 7 = 37.33 cubic eet, rounded = 37cubic eet

d. 84 + 269 + 37 = 390 cubic eet

2. 13 5 8 = 520 cubic eet

3. Yes, because the total volume o the objects is 390 cubic eet, leavingextra room volume-wise.

Bonus: The cone has a volume o 3.14 1/3 12.25 7 = 89.75 cubic eetor 89 cubic eet and 1,296 cubic inches (1,728 cubic inches = 1 cubic oot).There was approximately 130 cubic eet let in the van. So based on volume

alone, there should still be enough room in the van to t the cone.

Bonus Worksheet 1: Whats the Angle?

For a wood foor, the wall angles are: [180 (90 + 80)] = 10 or a maximumsae foor angle, and [180 (90 + 68)] = 22 or a minimum sae foor angle.

1. Since 22 is LARGER than 10, then 22 is the MAXIMUM sae wall angle.

2. Since 10 is SMALLER than 22, then 10 is the MINIMUM sae wall angle.

For a carpet, the wall angles are: [180 (90 + 85)] = 5 or a maximum saefoor angle, and [180 (90 + 30)] = 60 or a minimum sae foor angle.

3. Since 60 is LARGER than 5, then 60 is the MAXIMUM sae wall angle.

4. Since 5 is SMALLER than 60, then 5 is the MINIMUM sae wall angle.

5. 1/2 (8 14) = 56 square inches

Bonus Worksheet 2: Thats a Wrap!1. The length o the poster is the same as the trash cans height and

the width o the poster is equal to the trash cans circumerence.The circumerence is 3.14 3 = 9.42 eet. The posters dimensionsare 4 eet long and 9.42 eet wide, or 4 eet by 9.5 eet rounded tothe nearest hal oot.

2. The surace o the trash cans without the top and bottom can be derivedusing part o the ormula or a cylinders surace area: SA = pdh.3.14 3 4 = 37.68 square eet or 38 square eet when rounded to thenearest square oot.

3. Using the surace area ormula or a rectangular prism, each CD hasa surace area o: 2 (6 .25 + 6 5 + .25 5) = 65.5 square inches.Multiply the surace area o 1 CD 100 to nd the total amount o paperneeded to wrap 100 CDs: 65.5 100 = 6,550 square inches o paper.

4. The answer uses the surace area ormula or a square pyramid butwithout the base area: 1/2 40 eet 10 eet = 200 square eet.

Bonus Worksheet 3: Turn Up the Volume!

1. The volume o the room is 180,000 cubic eet, so the band can turn uptheir ampliers 10 notches in this gym.

2. First calculate the area o the gym foor: 60 100 = 6,000 square eet.I 1,200 people t into 6,000 square eet, then one person occupies 5square eet (6,000 1,200 = 5). For 1,500 people: 1,500 5 = 7,500square eet.

3. In the ormula or the volume o a rectangular prism (V = l w h),the l w is actually the area o the foor, so you can say V = foorarea height. Rearrange the ormula to: Height = Volume foor area.H = 280,000 8,000 = 35 eet.

4. Using the ormulas or the volume o a square pyramid and the area oa rectangle, students can nd the length o one o the holograms basesides: 6,250 = 1/3 BA 30, so BA = 625. Because BA = l w and 25 25

= 625, one side o the pyramids base is 25 eet.Take-Home Worksheet Front Cover: Warm-Up

1. area; 2. surace area; 3. volume; 4. cylinder; 5. cone

Take-Home Activity 1: Poster-Crazy

1. Answers will vary

2. Area o rectangular posters: 8.5 11 = 93.5. 93.5 10 = 935 squareinches, or 6.5 square eet

Area o circular posters: p 12 = 3.14 square eet 5 = 15.7 square eet

Area o triangular poster: 1/2 (3 3) = 4.5 square eet

Total area o posters: 26.7 square eet

Now Try This: Answers will vary depending on the width o the doorway,which will determine the diameter o the welcome mats. Students need

to measure the diameter and put their numbers in the ormulas orcircumerence and area.

Take-Home Activity 2: Covering Up

1. The student is painting 2 sides and a top each measuring 3 square eet,and the back measuring 9 square eet. 9 + 3 + 3 + 3 = 18 square eet

2. (75 54) + [2(75 6)] + [2(54 6)] = 4,050 square inches + 900 squareinches + 648 square inches = 5,598 square inches. 144 square inches= 1 square oot , so 5,598/144 = 38.875 square eet, or 38 square eetand 126 square inches.

Now Try This: To gure out the area o the hat, use the ormula prslanth.3.14 2 7 = 43.96 square inches. I a 1-ounce jar covers 33 square inches,the student does not have enough paint or his or her hat.

Take-Home Activity 3: The Perfect Fit1. 6 5 .25 = 7.5 cubic inches; 7.5 80 = 600 cubic inches

2. Answers will vary.

3. 152 3.14 5 = 3,532.5 cubic eet, or 26,423.1 gallons

Now Try This: V= p r2 h, so 22 = 3.14 1 h. To get the answer or h,divide 22 by 3.14. 22 3.14 = 7.006. Rounded to the nearest inch theheight is 7 inches.

worksheet answer key

aLignment with standards: nl cl t m (nctm)

t p l nctm g s g 68:

p://..//p6/.

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Geometry Teaching Guide - Activities