Standard #: MAFS.912.G-GMD.1.1 (Archived Standard)

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.

This task is the first in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are shown a regular n-gon inscribed in a circle. They are asked to use the formula for the area of the n-gon to derive an equation that describes the relationship between the area and circumference of the circle.

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.

This task is the third in a series of three tasks that assess the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students are given the definition of pi as the area of the unit circle, A(1), and are asked to use this representation of pi along with the results from the two previous tasks to generate formulas for the area and circumference of a circle.

This task is the second in a series of three tasks that assesses the students’ understanding of informal derivations of the formulas for the area and circumference of a circle. In this task, students show that the area of the circle of radius r, A(r), can be found in terms of the area of the unit circle, A(1) [i.e., A(r) = r² · A(1)].

Students are guided through the process of writing an informal argument for the volume of a pyramid formula using Cavalieri’s Principle.

This is a lesson where students investigate, compare, dissect, and use the relationship between volume of a cone and cylinder with equal corresponding dimensions.

Students create a cone and a cylinder with the same height and base. At the conclusion of the lesson, the students will know that the volume ratio between the cone and cylinder is 1:3.

Using reasoning skills, students will understand how the formulas for circumference and area of a circle are derived. Students will use a wide array of skills such as deductive reasoning, finding patterns, using algebra, modeling and transformation of an object. The teacher ensures student success through direct instruction, investigation and collaborative group work.

Using volume in the real world

An architect discusses how he uses circumference and area calculations to accurately create designs and plans.

A science teacher demonstrates stepwise calculations involving multiple variables for designing robots with desired characteristics.