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Surface Area, Volume & Heat Transfer Aimee Frame 1 , Gabrea Bender 2 1 Department of Engineering, University of Cincinnati, Cincinnati OH; 2 Newport High school, Newport, KY Abstract Pop-up Box Activity Prerequisite Knowledge Goals Temperature Loss Activity Reflections & Modifications Objectives State Standards Pre/Post Assesment References & Acknowledgments Conclusions Project STEP is funded through NSF Grant # DGE058532. Students will be able to: Determine the surface area of a scaled figure. Determine the volume of a scaled figure. Describe the relationship between scale factor, surface area and volume. Kentucky Core Content: MA-HS-2.1.1: Students will determine the surface area and volume of right rectangular prisms, pyramids, cylinders, cones and spheres in real-world and mathematical problems. MA-HS-2.1.2: Students will describe how a change in one or more dimensions of a geometric figure affects the perimeter, area and volume of the figure. The purpose of this lesson is to help the students understand how changes in dimensions of a figure affect other geometric properties. The activities in this lesson specifically address the relationship between scale factor, surface area and volume. The concept is first introduced using a pop-up box activity that creates a visual aid for the students. The engineering concept of heat transfer is then used as a real-world example that is dependent on the relationship between surface area and volume. Examples include gloves, heating of buildings, and the use of heat sinks in computers. http://www.eng.uc.edu/STEP/activities/ Finding area of rectangular prisms Finding surface area of rectangular prisms This hands-on activity is meant to create a visual that students can use to see the relationship between scale factor and volume/surface area. The students are given a worksheet that contains a diagram for the creation of the pop-up boxes along with data tables that are to be completed relating to the boxes that they just created. The result of the activity is the development of the general relationships between scale factor and volume/surface area. This activity is meant to create interest in the engineering concept of heat transfer and to provide a practical, real-world application where surface area and volume are used. The students are asked to complete a worksheet describing five different containers (as shown below). After a discussion on which container will have the hottest water after ten minutes, the actual temperature data is shown to the students. This lesson was created to help the students gain a better understanding of the relationships between a figure’s dimensions and its surface area and volume since it is a part of the core content with which the students typically struggle. The first day’s lesson is centered around the pop-up box activity. This activity allows the students to create a visual aid that demonstrates the differences in the surface area and volume of various cubes and rectangular prisms as their lengths are doubled or tripled. By completing the worksheet that accompanies the activity, the students can develop a general relationship between scale factor and volume or surface area. The second day’s lesson provides a practical, real-world application where surface area and volume are used. The students are asked to apply the general relationship developed during the previous day’s lesson to complete a worksheet containing data about five different containers. Based on their answers, the students are then asked to choose which container will be the best at retaining heat if they are all filled to the top with hot water. After the students have decided on a container, an explanation of how surface area and volume affect heat loss is given through the use of a power point presentation that includes discussion questions. The lesson then concludes with actual temperature data taken using the containers described in the worksheet. Although the lesson is intended to demonstrate how surface area and volume change when an object’s dimensions are multiplied by a scaling factor, it is also a good lesson to reinforce the prerequisite knowledge listed above. The pop-up boxes created in the first day’s activity give the students a manipulative to work with when trying to calculate surface area and volume in addition to comparing rectangular prisms of double or triple dimensions. These concepts are then extended to the second day’s activity which requires the students to apply their knowledge to a logic problem (temperature loss worksheet). a) What happens to the surface area of the cube when the length of the side is doubled? b) What happens to the volume of the cube when the length of the side is doubled? Length of Side Scale Factor Surface Area SA ratio Volume V ratio 2 cm X X X 4 cm 8 cm 16 cm This Fill in the missing values in the table below using the following information: a)Container #2 is double the size of container #1. b)Container #3 is triple the size of container #1. c)Container #5 is half the size of container #4. Each container is filled with hot water and the temperature is measured five minutes later. Predict which container will have the hottest water (highest temperature reading) after the five minutes are up. Explain your answer. # Lengt h Width Heigh t Surf Area Volume 1 2 2 160 3 12 4 4 6 5 54 h w l V ) ( * 2 h w h l w l SA length width height Visualizing concepts Volume Amount of plastic needed to make a toy Amount of rice needed to fill up a container Surface Area Amount of paint needed to paint the toy Amount of wrapping paper needed to wrap a gift Relationship between scale factor and surface area/volume Use idea of toy manufacturing (want to double size of toy) Will need four times as much paint Will need eight times as much plastic A set of connectable cubes can be used to illustrate the relationship between surface area, volume and scale factor. Pop-up Box Activity To save time, copy the cutting and folding lines onto cardstock instead of having the students measure them Viewing the boxes from the side better illustrates the idea that the volume more than doubles when the dimensions are doubled Temperature Loss Activity Conducting the experiment in class can be difficult Need a lab classroom or ability to have hot water in the classroom Takes time to get enough temperature data Would be good to do in conjunction with a science class May need to ask leading questions to help students fill out worksheet What happens to the volume when the dimensions are doubled? How do you calculate surface area? Etc. Do not show the students the containers until after the worksheet has been completed Students must rely on calculations to make their decision Leads to better discussion on how they made their decision Presentation on Heat Transfer Initial slides engage students and introduce the relationship between surface area and heat transfer Bringing in heat sinks to pass around also helped engage the students Volume/surface area ratio Higher number – better heat retention (can relate back to temperature loss activity) Does not stay constant when object is scaled 1. A box has the dimensions shown. a) Find the volume of the box. b) Find the surface area of the box. 2. Use the given information about the boxes to find their scale factor. a) b) 3. The Mattox Aquarium has a 90 ft tall fish tank that contains 210,000 gallons of water when full. a) If the tank has a square bottom, find the length of the side of the tank. Show your work. (1 gallon = 0.1137 ft 3 ) b) How tall would the tank in part (a) have to be in order to double the volume? Show 1 in 2 in 3 in V=27 ft 3 V=216 ft 3 SA=24 in 2 SA=384 in 2
