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GEOMETRY Lessons & Worksheets to Build Skills in
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Lesson 1
Perimeter and area of 2d shaPes
Dear T eacher: W elcome t o Set ti n g t he St a ge w it h , a new ma t h program al igned w i t h Geomet r y
hel p s t uden t s NC TM s t andards t ha t is des igned t o ic geome t r y in grades 6–8 bu il d and re in f orce bas . sk il ls f or measur ing 2D and 3D shapes ion, De vel oped b y The Ac t uar ial Founda t -bu il d ing t h is program seeks t o pro v ide sk il l t s become ma t h ac t i v i t ies t o hel p your s t uden real -worl d success f ul in t he cl assroom and in e you s i t ua t ions ou t s ide o f school . We hop en jo y t h is new program!
Geometry Works! The Stage Takes Shape OBJECTIVES: Students will understand ormulas used to measure the perimeter and area o these basic two-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. Perimeter is the total distance around the outside o a polygon (a closed gure made up o line segments).
2. On the board, draw a rectangle labeled with a length o 4 eet and width o 3 eet. Then draw a right triangle with a base o 4 eet, height o 3 eet, and hypotenuse (the side opposite the right angle) o 5 eet. Explain that to measure the perimeter o any polygon, you add together the lengths o each side.
S incerel y, The Ac t uar ial Founda t ion
getting started 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 Geometrics is planning a big concert at a school, but they need help to build a stage and promote the show. Some students volunteer to become the Geometrics Stage Crew and use their geometry knowledge to help.
Three lesson 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 in the 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 reference sheet 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.
3. Ask students what the perimeter o the rectangle is. Show students the ormula or the rectangle’s perimeter on the poster and ask why it’s correct. The ormula o P (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. Ask them what the diameter is. (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) = p • d (diameter). Tell students that p 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 o area on the poster: the measure o a bounded 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) = l • w . Ask them to calculate the area o the 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 • r 2, where r 2 means radius squared, or r • r . The answer is p (3.14) 2 • r (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 complete all 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).
Lesson 2
Lesson 3
surface area of 3d shaPes That Should Cover It! OBJECTIVES: Students will understand ormulas used 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 the ormulas 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 on the 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 object’s 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 3 eet and the height at 4 eet.
6. Show students the surace area ormula or cylinders on the poster: SA = (2 • p • r 2) + (p • d • h) 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 the dimensions 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 l • w like any square or rectangle.
8. Show students the surace area ormula or square pyramids on the poster, SA = (BA) + 1/2 • P • slant h 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 P in 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 should complete 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 = (p • r 2) + (p • r • slant )]. Go over all correct answers as a class, reerring to the Worksheet Answer Key (see back cover).
VoLume of 3d shaPes Pack It Up! What Will Fit? OBJECTIVES: Students will understand ormulas used to measure the volume 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 3 Extensions: Bonus Worksheet 3, Take-Home Activity 3 DIRECTIONS: 1. Explain to your students that now that they’ve mastered measuring the 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 the Lesson 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 the poster: V (volume) = l • w • h. 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 = p • r 2 • h. 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 measurement can 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 dimensions as 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 • BA • h. 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 complete all 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 • r 2 • h]. Go over all correct answers as a class reerring to the Worksheet Answer Key (see back cover).
Look for more math resources at The Actuarial Foundation Web site at:
www.actuarialfoundation.org/programs/for_teachers.shtml n n
Advancing Student Achievement® Grants “Expect the Unexpected With Math®” Series: Shake, Rattle, & Roll (probability) Bars, Lines, & Pies (graphing) Conversions Rock (converting decimals, fractions, and percents)
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The Math Academy Series: Using Math in the Real World
al l pbl p p : www.scholastic.com/unexpectedmath
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Pt, a, su a, Vu: rvw T, b sp, u Terminology : the measure o a bounded region o a two-dimensional shape expressed in square units
t: the diagonal distance rom the top o a cone to its base t t: the height o one o the triangular aces o a
u: the distance around the edge o a circle
pyramid
t: the distance across a circle through its
u : the sum o all the areas o all suraces o a
center point
three-dimensional object, measured in square units
ptu: the side opposite the 90° angle in a right
vu: the amount o space inside a three-dimensional
triangle, also the longest side o a right triangle
shape, measured in cubic units
pt: the total distance around the outside o a polygon
p p: the circumerence o any circle divided by its diameter, rounded to the number 3.14
u: the measure rom the center o a circle to a point on the circle
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
basic shaPes and ormUlas 2D ShAPeS: PeRimeteR AnD AReA
3D ShAPeS: SuRfACe AReA AnD Volume
rt
rtu P
P = 2 • (l + w) A = l • w
SA = 2 • (l • w + l • h + w • h) V = l • w • h
w
w
T
squ P
P = side a + side b + side c A = 1/2 • (b • h)
SA = (BA) + 1/2 • P • slant h V = 1/3 • BA • h h y p o t e n u s e
obtuSe tRiAngle
ACute tRiAngle
Right tRiAngle
Note: base area (BA) of a square or rectangular pyramid is l • w of the base, and P is perimeter of the base.
s l a n t h
w
r
c d
C =p•d A = p • r2
c 2
a
2
SA = (2 • p • r2) + (p • d • h) V = p • r2 • h
1
Tpz P = side a + b1 + b2 + side c A = 1/2 • (b1 + b2) • h
c
r
c
SA = (p • r ) + (p • r • slant) V = p • 1/3 • r2 • h
s l a n t
r
stut Wkt 1
Name:___________________________________________________ Date:_____________
gt Wk! T st Tk sp
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.
1
First, they want a main stage that is rectangular-shaped, measuring a length o 24 eet and a width o 16 eet. What are the perimeter and area o that stage?
Prr:
2
Second, the band’s lead guitarist wants the Geometrics Stage Crew to build a smaller circular stage in ront o the main stage 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:
3
Ara:
Ara:
The bass player has a thing or triangles and sees hersel on a triangular platorm o to the let o the stage. When viewed 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:
T u wt t tpz-p pt. T qu t gt st cw t w u t tpz [ a = 1/2 • (b 1 + b 2) • h ]. T tpz w t wt t ut t. b 1 ( b 1) = 8 t. b 2 (b 2) = 5 t. T t u 6 t. Wt t ?
bonUs:
5'
6'
8'
stut Wkt 2
Name: ___________________________________________________ Date:_____________
Tt su cv it! 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.
1
The dimensions o the rectangular prism or the lead guitarist are height = 7 eet, width = 4 eet, and length = 3 eet.
3
Srac Ara =
The dimensions o the square pyramid or the bass player are base length = 4, base width = 4, and a slant height o 7.28 eet. Solve with a decimal, then also round to the nearest hal oot. Srac Ara =
2
The dimensions o the cylinder or the drummer are radius = 3.5 eet and height = 7 eet. Solve with a decimal, then also round to the nearest hal oot.
4
Can the stage crew paint the surace area o all three shapes with just one can o paint?
Srac Ara =
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 = (p • r 2) + (p • r • t).
bonUs:
. Wt t u ? exp t u t t t t.
7.8' 7'
3.5'
. i pt v 350 qu t, ut w u t pt t ?
stut Wkt 3
Name:___________________________________________________ Date:_____________
Pk it Up!
Wt W t?
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.
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
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?
b vu, wu t t t t u 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 • r 2 • h .
bonUs:
7.8' 7'
3.5'
worksheet answer key Student Worksheet 1: Geometry Works! The Stage Takes Shape 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
3.
1.
4.
Bonus Worksheet 3: Turn Up the Volume! 1. 2.
Student Worksheet 2: That Should Cover It! 1. 2.
2 • (3 • 4 + 3 • 7 + 4 • 7) = 2 • (12 + 21 + 28) = 2 • 61 = 122 square eet (2 • 3.14 • 12.25) + (3.14 • 7 • 7) = (76.93) + (153.86) = 230.79 square eet , rounded = 231 square eet 3. (4 • 4) + 1/2 • 16 • 7.28 = 16 + 58.24 = 74.24 square eet, rounded = 74 square eet 4. No. The total surace area is 427.03 square eet and one can o paint will 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 = 124.187 square 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, leaving extra room volume-wise. Bonus: The cone has a volume o 3.14 • 1/3 • 12.25 • 7 = 89.75 cubic eet or 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: What’s the Angle?
Using the surace area ormula or a rectangular prism, each CD has a 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 paper needed to wrap 100 CDs: 65.5 • 100 = 6,550 square inches o paper. The answer uses the surace area ormula or a square pyramid but without the base area: 1/2 • 40 eet • 10 eet = 200 square eet.
3.
4.
The volume o the room is 180,000 cubic eet, so the band can turn up their ampliers 10 notches in this gym. 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 5 square eet (6,000 ÷ 1,200 = 5). For 1,500 people: 1,500 • 5 = 7,500 square eet. 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 = foor area • height. Rearrange the ormula to: Height = Volume ÷ foor area. H = 280,000 ÷ 8,000 = 35 eet. Using the ormulas or the volume o a square pyramid and the area o a rectangle, students can nd the length o one o the hologram’s base sides: 6,250 = 1/3 BA • 30, so BA = 625. Because BA = l • w and 25 • 25 = 625, one side o the pyramid’s 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. 2.
Answers will vary Area o rectangular posters: 8.5 • 11 = 93.5. 93.5 • 10 = 935 square inches, 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 or circumerence and area.
For a wood foor, the wall angles are: [180 – (90 + 80)] = 10° or a maximum sae 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 sae foor 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
Take-Home Activity 2: Covering Up
Bonus Worksheet 2: That’s a Wrap!
Take-Home Activity 3: The Perfect Fit
1.
1. 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 • r 2 • 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 nea rest inch the height is 7 inches.
2.
The length o the poster is the same as the trash can’s height and the width o the poster is equal to the trash can’s circumerence. The circumerence is 3.14 •3 = 9.42 eet. The poster’s dimensions are 4 eet long and 9.42 eet wide, or 4 eet by 9.5 eet rounded to the nearest hal oot. The surace o the trash cans without the top and bottom can be derived using part o the ormula or a cylinder’s surace area: SA = p • d • h. 3.14 •3 • 4 = 37.68 square eet or 38 square eet when rounded to the nearest square oot.
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 square inches + 648 square inches = 5,598 square inches. 144 square inches = 1 square oot , so 5,598/144 = 38.875 square eet, or 38 square eet and 126 square inches. Now Try This: To gure out the area o the hat, use the ormula p • r • slant h. 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.
aLignment with standards: nl cl t m (nctm)
t p l nctm g s g 6–8: p://..//p6/.
