geometry teaching guide - activities
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8/2/2019 Geometry Teaching Guide - Activities
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g 68 P 1 2
Poster/teaching guide
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GEOMETRY
Lessons & Worksheets to Build Skills in
m P, a, s a, Vl
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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
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OBJECTIVES: Students will understand ormulas used to measure the
volume o these basic three-dimensional shapes: a rectangular prism, acylinder, and a square pyramid.
Time Required: 20 minutes, plus additional time or worksheets
Materials: Student Worksheet 3
Extensions: Bonus Worksheet 3, Take-Home Activity 3
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.
Time Required: 20 minutes, plus additional time or worksheets
Materials: Student Worksheet 2
Extensions: Bonus Worksheet 2, Take-Home Activity 2
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
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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
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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 '
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8/2/2019 Geometry Teaching Guide - Activities
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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'
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8/2/2019 Geometry Teaching Guide - Activities
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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
y av.
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'
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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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