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PA-GM-2-4

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

PA.GM.2.4 Develop and use the formulas V = 𝜋 r²h and V = Bh to determine the volume of right cylinders, in terms of 𝜋 and using approximations for 𝜋. Justify why base area (B) and height (h) are multiplied to find the volume of a right cylinder. Use appropriate measurements such as cm³.

In a Nutshell

Using area formulas for circles, students will to find the area of the base shape. Introducing the multiplication of the height will have students working with cubic units. It is important for students to understand the difference in right cylinders and oblique cylinders, and how perpendicular height is different in the two. Students should be able to leave exact answers in terms of π and know that it makes a difference when approximating volume after using the π button on a calculator or approximating using 3.14 for 𝜋. Students need to understand the purpose of “B” within the formula.

Student Actions	Teacher Actions
Develop A Deep and Flexible Conceptual Understanding when using technology to construct and deconstruct 3-D cylinders to analyze the number of cubes that each figure is made up. By realizing that your base layer is just repeated by the height layers, a formula to calculate volume can be derived. Develop Ability to Make Conjectures, Model and Generalize when using derived formula to make sense of the capital B in the formula: V=Bh is just the 2-D area of the Base. Develop Strategies for Problem Solving when using real life examples of cylinders to calculate volume and explain your solution with context. Due to the circular base, students can construct the base layer out of graph paper by tracing the cylinder onto graph paper and estimating the area of the circular base. Develop Accurate and Appropriate Procedural Fluency when analyzing and discussing 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.	Support productive struggle and avoid giving students the formula for calculating volume of cylinders, but allow time and structural support for the development of the formula by finding the base area and realizing that the height is just multiple base layers. Implement tasks that promote reasoning and problem solving by providing real world tasks, such as calculating the volume of a tube or canned food for students to test their created formula in a real world context. Facilitate meaningful mathematical discourse when allowing students to discussthe 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 and how each one affects the accuracy of the answer.
Key Understandings	Misconceptions
Justify the formula for the volume of a cylinder. Use an approximation for 𝜋. Students should be able to leave exact answers in terms of 𝜋 and know that it makes a difference when approximating volume after using the 𝜋 button on a calculator or approximating using 3.14 for 𝜋. generalize, algebraically, how to find the volume of a cylinder (Volume = area of the base times height or V = 𝜋r²h).	Students may not understand that the length of the rectangle (the lateral surface of the cylinder) is the circumference of the circle (base). 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. Students may not make the connection that the ‘B’ in the formula is really just the area of the circle.