04-CROSS-CURRICULA+LEARNING+CENTER+DESIGN

===** Overall Goal of the Lesson: ** The overall goal of this lesson is to extend each student’s conceptual understandings of surface area. By doing so, students will learn how surface area can be increased without increasing the volume of an object. ===

===** This lesson plan belongs to group presentations. Different groups will present different topics. Each center will be guided by a member of the group. The same procedure will be followed by the other group presenters. **===

** NCTM Grades 6 – 8 Expectations Addressed: ** Students should –

 * === precisely describe, classify, and understand relationships among types of two- and three- dimensional object using their defining properties ===
 * === describe sizes, positions, and orientations of shapes under informal transformations ===
 * === use two-dimensional representations of three dimensional object to visualize and solve problems such as those involving surface area and volume ===
 * === use geometric models to represent and explain numerical and algebraic relationships ===
 * === recognize and apply geometric ideas and relationships in areas outside the mathematics classroom, such as art, science, and everyday life ===

** PA Standards and Common Core Standards Addressed: **
=== 7.G.6 – Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects comprised of triangles, quadrilaterals, polygons, cubes, and right prisms. === ||
 * === PA Standards: === || === Common Core Standards: === ||
 * === 2.3.8.A – Use formulas and procedures for determining measurements (e.g., area, surface area, volume, distance). 2.3.8.D – Estimate, use, and describe measures of distance, rate, perimeter, surface area, area, volume, weight, mass, and angles. 2.3.8.E – Describe how a change in linear dimension of an object affects its perimeter, area, and volume. === || === 6.G.4 – Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. ===

** Materials / Technology Needed: **

 * === Teacher Materials / Technology: === || === Student Materials / Technology: === ||
 * * === Balloons ===
 * === Soda Cans (Empty) ===
 * === Heatsink (Computer Part, when available, to show application) ===
 * === Presentation in Power Point. ===


 * === Computer / Projector ===
 * === Internet Access ===
 * === Polydrons (Manipulative) ===
 * === Calculators === ||  ||

** Assessment Strategies: **

 * * === Informal assessment of understandings during lesson – evaluation based upon participation, discussion, and content understandings during class ===
 * === Informal assessment of student solutions to application problem === ||

** Lesson Procedures: **

 * ===** Introduction / Anticipatory Set: **===


 * === The Essential Question is introduced to all students ** “How to increase surface Area without increasing the volume?” **===

** The student in charge of this center will **

 * === Begin the activity by having students, in their groups, to calculate the surface area and volume of the rectangular prisms (polydrons – one rectangular prism/one cube) on their table. Notes: The long side of rectangle is 9 “parts” and the short side is 6 “parts.” The cube is 6 “parts” per side. ===
 * === Hint (may help struggling students): To calculate, you can unfold the polydron to a 2d surface to measure area. There will be an animation that shows how to do this. The animation is available at the following URL: [] When the group are finished, Teacher will ask the following questions: ===

** The student in charge of this center will ask the following questions. Each member of the group will have to answer correctly this questions before moving to the Center 2 **

 * ===** Is the answer correct? **===
 * ===** What are the formulas used to find the solution? **===
 * ===** How do you find the solution? **===
 * ===** Anyone used a different method? **===
 * ===** If yes, what are the advantages of each one? **===
 * ===__ Answers __ : Rectangular prism. Surface Area = 288sq.p or 288 p2. Volume = 324 p3 Cube. Surface Area = 216 p2. Volume = 216 p3 === ||
 * ===** Center 2 Exploration / Demonstration / Explanation of Content: **===
 * ===** Center 2 Exploration / Demonstration / Explanation of Content: **===


 * === Restate main topic of discussion: ** “How can you increase surface area without increasing the volume?” ** ===
 * === Begin the //balloon and can example//, relating volume and surface area. ===
 * === Tell students that we will be relating volume and surface areas of both the //balloon and soda can// before and after a transformation has occurred to each. ===
 * === Crush the can and blow up the balloon. ===
 * === Try to have students point out the following and why this is by asking the questions to the groups and use //think, pair, share//: ===

Answer:

 * === For the can, the can has the same surface area both before and after being crushed. Therefore, the surface area remains constant, even though the volume is decreased. ===
 * === For the balloon, as it is blown up, the volume increases as does the surface area. ===

||
 * * === Explain the concept of thermodynamics applied to the class (Real word Problem) “Heat conduction, also called diffusion, is the direct microscopic exchange of kinetic energy of particles through the boundary between two systems. When an object is at a different [|temperature] from another body or its surroundings, [|heat] flows so that the body and the surroundings reach the same temperature, at which point they are in [|thermal equilibrium].” In other words, heat goes from the warmer object to the cooler object, through the areas of contact (Boundaries), until they reach the same temperature. ===
 * === Restate main topic of discussion: “How can you increase surface area without increasing the volume?” ===
 * === Leader Student will ask students if they can come up with an answer. ===
 * === Leader Student will model one possible solution. ===
 * === Tell students they will be using the polydrons already on their table. ===
 * === Explain how to assemble object. ===
 * === Have students calculate the surface area and the volume of the assembled object. __Answer__: Surface Area = 3784 p2 ===
 * === Volume = 7260 p3 ===
 * === Have groups brainstorm and create a model of a way of increasing the surface area of the object without changing the relative size/volume of the object. ===
 * === Have students then calculate the new surface area of their object and determine how many square parts it increased by. ===

** How much did the surface area increased? **

 * === Leader Student will point out that the new surface area is 51% more, without increasing the original volume. ===
 * === Leader Student will ask the following questions: ===


 * === Show parts of a youtube video on a heatsink to show how increasing surface area without increasing volume can be applied to computer and physics. Also, show actual heatsink for a computer. ===
 * === The video is available at the following URL: ===
 * === [|http://www.youtube.com/watch?v=9VSDkB3st4&list=PLB38FA4D49509699D&feature=player_embedded] ===
 * === Display the following summarizing questions on the board. ===
 * === Have students discussing the question in their groups and then review as a whole class. **1.** How can the surface area of an object be increased without increasing the volume of the object? **Answer:** Making holes on the prism, or making the surface wrinkle, like the crushed soda can and the picture of the motorcycle engine. ===

||

** Possible Student Misconceptions: **

 * * === Students may have difficulty recalling and/or using the concepts of surface area and other vocabulary terms that are prerequisites for this lesson. ===
 * === Students may have misunderstandings about the order of operations or make calculation errors when calculating surface area. ===
 * === Students may have difficulty understanding questions due to reading or comprehension difficulties. ===
 * === Students may have trouble handing the responsibility of using manipulatives. === ||

** Adaptations for Diverse Learners: **

 * === This lesson appeals to learners that are diverse in their methods of learning. ===
 * === Appeals to visual learners by using a video that relates the concept to the heatsink cooling system for a computer. ===
 * === Appeals to auditory learners by verbally discussing processes, understandings, and procedures with their group and as a whole class. ===
 * === Appeals to hands-on learners by utilizing manipulatives that allow learners to use the concept (polydrons). === ||