Surface Area of a Cylinder

Surface Area of a Cylinder Tim Carson Mary Ann Murphy Rebecca Mersereau Kate Peart

Teacher Reference • Grade Level: 8 • Unit: Geometry • SCO: Students should be able to estimate and calculate volumes and surface areas of right prisms and cylinders. • Prior Knowledge: Students should know how to calculate the circumference and surface area of a circle. They will also know how to calculate the surface area of a rectangle and a rectangular prism.

Setting the Stage • Review the properties (circumference, radius, etc.) of a circle • Review the area of a rectangle and circle • Introduce and prepare the students for the surface area of a cylinder • Define surface area

First Activity Use Transparency to review the properties of circles and rectangles. Review known formula for area with students.

Experimenting with Shapes and Area. • Have students create pictures with shapes they cut out from the handout. Then calculate the area

Area and Surface Area • Definitions: • Area: The measurement of the space inside the perimeter of a 2-d object • Surface Area: the sum of the area of each individual surface of a 3-d object

Providing Context • Problem: Oh Henry has decided to create a new chocolate bar in the shape of a cylinder. They have made the chocolate bar to be 15 cm long and 3 cm in diameter. They now need to know how much packaging they need to cover their first order of 100 bars.

Assessment Opportunity • During the last 5 minutes of class have the students write in their journals the answer to the following question: How would you go about finding the answer to Oh Henry’s problem?

“Homework” • Have the students each bring in an example of a cylinder from home. (ideally cans with the volume marked on them) • However all cylinders are suitable. • Provides context for a class discussion • These will be used in the enrichment phase.

Core Discovery • See Jodi MacInnis’ manip presentation • http://people.stu.ca/%7Epheeney/5873Manip_SurfaceAreaCylinder.pdf

Core Discovery Alternative Option. • The link below is to a movie on You Tube demonstrating the same activity. Students who need to see the demonstration again can use it for follow up, or absent students can be directed to this resource. • http://youtube.com/watch?v=O5_7ymhemi0

Youtube video

Post Discovery. Reinforcement and Application: • Reinforce the formula for surface of the cylinder • Apply the formula to various problems • Determine the effect of changing the dimensions of a cylinder on its surface area

Reinforcement • Review the key principles from the manipulation (teacher facilitates discussion) • Surface area depends on the radius of the base and height of the cylinder • The width of the rectangle is the same as the circumference of the circle. • Total surface area is the area of the top, bottom and side.

Assessment Opportunity • Demonstrate a couple sample questions from the ‘Find the Surface Area’ Overhead and have the students complete selected questions from the sheet before continuing. • See where common mistakes occur; gives students the opportunity to review or self reinforce the concept of finding the surface area of a cylinder.

Word Problem • Suppose you work for a company that makes water tanks. A customer wants a tank that measures 10 meters in height and 3 meters in diameter. • How much metal should you order to make this tank?

Revisit Context Question: • Oh Henry has decided to create a new chocolate bar in the shape of a cylinder. They have made the chocolate bar to be 15 cm long and 3 cm in diameter. They now need to know how much packaging they need to cover their first order of 100 bars. • After they figure out this out, ask if they have any idea how to find out how much chocolate they will need to make the chocolate bars.

Assessment Opportunity • This asks students to apply their understanding of the surface area of a cylinder and transform the cylinder; it also allows them to question the surface area at different stages, make and justify a hypothesis.

Post Discovery: Enrichment • Use the formula to derive the surface areas of different geometric shapes • Compare the relationship between surface area and volume

Brain Teaser? • A cylinder 105 cm high has a circumference of 20 cm. A string makes exactly 7 complete turns round the cylinder while its two ends touch the cylinder's top and bottom. How long is the string in cm?

Brain Teaser • The figure alongside illustrates how the cylinder can be unrolled into a rectangle for the case where the string makes exactly 4 complete turns. • Since the students have not learned Pythagoreans theorum formula they would be expected to measure the length of the string with a ruler

Working with Cylinders • Have the students get the cylinders that they brought in and calculate the surface area of each one. Having the students produce table similar to the table provided below • Graph the total surface area against the volume of the cylinder. Is there any relation between the surface area and the volume

Confirming Knowledge • As a reinforcement, this website allows students to manipulate the dimensions of rectangular and triangular prism to obtain the same relationships

Assessment Opportunity • After completing the graph of the relationship between surface area and volume of a cylinder students are given the opportunity to look at squares and Triangular prisms. • This can provide a discussion opportunity for students to share what they have learned.

Show You Know • For 5-7 minutes, have them in their groups discuss what might be a possible solution to the problem and then work it out. After the groups have finished, have students present their solution to the class and discuss why each solution is either right or wrong. Have them do this using only the symbols not with numbers.

Assessment Opportunity • Within the groups students will be discussing the model of the square with a cylindrical hole. They will be using the formula they know for rectangular prisms also what they have been learning for cylinders. The formulas the groups come up will help inform the teacher of their understanding of the concept.

Thank You • Questions/ Comments?

References: • http://id.mind.net/~zona/mmts/geometrySection/surfaceAreasAndVolumes/areaVolumeBoxCylindricalHole1.html • http://www.syvum.com/cgi/online/serve.cgi/contrib/teasers/string1.tdf?0 • http://illuminations.nctm.org/Lessons/Scaling/Scaling-AS-ScalingAway.pdf • http://www.purplemath.com/modules/perimetr.htm • http://www2.whidbey.net/ohmsmath/webwork/javascript/prismcyl.htm • http://youtube.com/watch?v=O5_7ymhemi0 • http://www.math.wichita.edu/history/activities/geometryact.html#areavolume-act • http://www.math.wichita.edu/history/activities/geometryact.html#areavolume-act

Surface Area of a Cylinder