Reflection: Advanced Students Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections - Section 3: You Choose: Practice Thinking About Volume or Take it Further


I wanted to design a lesson that would allow all my students to see how area and volume relate to one another—specifically how the area of a 2D cross-section relates to the volume of the 3D solid from which it was taken.  However, since making sense of Cavalieri’s Principle is a “plus” standard in the Common Core, I knew I had to teach this lesson at a level appropriate for each student.


Student engagement was very high in the group that wanted to make sense of the volume formula for a sphere.  By differentiating this segment of the lesson, I gave my students high cognitive tasks that targeted them at the “right” level.  In particular, the group that took on the sphere volume task grappled with really important mathematical ideas.  They had to decipher what exactly each variable represented in the cross-sections taken from two different 3D solids, figure out how the variables related to one another, and consider how they would write a proof about the areas of these cross-sections to then derive the volume formula for a sphere.  Overall, I was really proud of my students who engaged in this work because they used the kind of thinking that is often used in Calculus BC and later math classes.


The groups that worked on the prism volume task were able to see that they could quadruple a prism’s volume by quadrupling either the prism’s height or base area.  Because one of the groups had actually built four rectangular prisms, they physically saw that doubling (not quadrupling) the dimensions of the base would quadruple the prism’s volume, an important concept I wanted students to take away from the task.

  Why Differentiate?
  Advanced Students: Why Differentiate?
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Volume Formulas, Cavalieri's Principle, and 2-D Cross-Sections

Unit 10: Geometric Measurement and Dimension
Lesson 11 of 14

Objective: Students will be able to identify the shapes of 2D cross-sections of 3D objects and use geometric shapes, their measures, and properties, to describe objects.

Big Idea: In a differentiated lesson based on student choice, students will be able to practice finding the volume of prisms or apply Cavalieri's Principle to determine the volume formula for a sphere.

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4 teachers like this lesson
Math, Geometry, Measurement, modeling, space, shapes, Cavalieri's Principle, Differentiated Instruction
  80 minutes
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