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PA-GM-2-2

Page history last edited by Brenda Butz 6 years, 3 months ago

PA.GM.2.2 Calculate the surface area of a cylinder, in terms of 𝜋 and using approximations for 𝜋, using decomposition or nets. Use appropriate measurements such as cm².

In a Nutshell

By laying out a cylinder into a net, students will discover that the surface area of a cylinder is equal to the total area of two circles and a rectangle. When using the different area formulas (for a rectangle and a circle), students will work with the concept of leaving an answer in terms of 𝞹, calculating approximate answers using a 𝞹 button on a calculator and find answers using the traditional approximation of 𝞹 = 3.14. Students should be aware of any unit assigned to their measures and practice multiplying the units as well as the numerical measures, leading to the understanding of squared units as they apply to area of a figure.

Student Actions	Teacher Actions
Develop Mathematical Reasoning by decomposing a cylinder into equivalent circles and a rectangle to explore the net and surfaces of the figure. Develop Ability to Make Conjectures, Model and Generalize when deriving a formula for calculating surface area of a cylinder by writing an area formula to calculate each face and then using distributive and number properties to simplify the formula. Develop Mathematical Reasoning by making the connection that the rectangle's width is equivalent to the circumference of the circle. Develop Mathematical Reasoning by discovering that the surface area of a 3-D shape is equivalent to the total area of its composite figures and that because the formula is derived from the area calculation of each 2-D surface that the measurement is in squared units. Develop Ability to Communicate Mathematically when comparing student created formula to the standard formula for calculating the surface area of a cylinder and use your decomposed shape to justify the formula and discuss results with the class. Develop Accurate and Appropriate Procedural Fluency when analyzing the pros and cons of leaving 𝜋 in your answer vs using an approximation of 𝜋 to 3.14 vs using the 𝜋 button on your calculator.	Implement tasks that promote reasoning and problem solving by supplying students with a variety of 3-D cylinders that students can decompose to form nets to explore individual faces of the shape. Support productive struggle by allowing time and structural support for students to develop the formula for calculating the surface area of cylinders by finding the area of each face and then simplifying. Implement tasks that promote reasoning and problem solving, such as wrapping a paper towel tube, for students to test their created formula in a real world context. Facilitate meaningful mathematical discourse to allow students to analyze the pros and cons of leaving 𝜋 in your answer vs using an approximation of 𝜋 to 3.14 vs using the 𝜋 button on your calculator, so they can build an understanding of when to use the different representations of pi.
Key Understandings	Misconceptions
Knowledge and use of area of circle Calculate the surface area of a cylinder using both in terms of 𝜋 and as an approximation using 3.14. Justify the formula for the surface area of a cylinder by decomposing the surface into two circles and a rectangle.	Confusing "lateral surface area" with "total surface area." It is helpful for students to draw nets of cylinders to calculate surface areas to reinforce the differences. If the cylinder is lying on its side, students sometimes misread the height as the vertical distance, which would actually be the diameter in this orientation, when height is actually the distance between the two circular bases.