G-GMD.1.1

Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

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Formative Assessments
Formative Assessments
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Students are asked to derive and explain a formula for the volume of a cone given a pyramid with the same height and the same cross-sectional area at every height.
Volume of a Cone
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Students are asked to derive and explain a formula for the volume of a cylinder given a prism with the same height and the same cross-sectional area at every height.
Volume of a Cylinder
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Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.
Volume of a Pyramid
Virtual/Hands-on Manipulatives
Virtual/Hands-on Manipulatives
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Assessment Questions
Assessment Questions
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Videos of Expert Instruction
Videos of Expert Instruction
Story of Pi
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A science teacher demonstrates stepwise calculations involving multiple variables for designing robots with desired characteristics.
Robot Mathematics: Gearing Ratio Calculations for Performance
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Math Literature
Math Literature
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High school students will know that the circumference of a circle of radius r is 2πr and therefore the goal of this task is to help them understand this formula from the point of view of similarity.
Circumference of a Circle
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The goal of this task is to calculate the volume of a particular pyramid with square base, which is easily reasoned by viewing it as one sixth of a cube.
Volume of a Special Pyramid
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This purpose of this task is to develop an understanding of the formula for the area of the circle. The argument presented is essentially due to Archimedes.
Area of a Circle
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The goal of this task is to establish formulas for volumes of right prisms and cylinders. Students encounter these formulas in eighth grade (8.G.9) but arguments based on dissections are introduced in high school.
Volume Formulas for Cylinders and Prisms