*Use dissection arguments, Cavalieri’s principle, and informal limit arguments.*

**Subject Area:**Mathematics

**Grade:**912

**Domain-Subdomain:**Geometry: Geometric Measurement & Dimension

**Cluster:**Level 3: Strategic Thinking & Complex Reasoning

**Cluster:**Explain volume formulas and use them to solve problems. (Geometry - Additional Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

**Date Adopted or Revised:**02/14

**Date of Last Rating:**02/14

**Status:**State Board Approved - Archived

**Assessed:**Yes

**Assessment Limits :**

Informal arguments are limited to dissection arguments, Cavalieri’s

principle, and informal limit arguments.Items may require the student to recall the formula for the

circumference and area of a circle.**Calculator :**Neutral

**Clarification :**

Students will give an informal argument for the formulas for the circumference of a circle; the area of a circle; or the volume of a cylinder, a pyramid, and a cone.**Stimulus Attributes :**

Items may be set in a real-world or mathematical context.Items may ask the student to analyze an informal argument to determine mathematical accuracy.

**Response Attributes :**

Items may require the student to use or choose the correct unit of

measure.

**Test Item #:**Sample Item 1**Question:**Alejandro cut a circle with circumference C and radius r into 8 congruent sectors and used them to make the figure shown.

Alejandro noticed that the figure was very close to the shape of a parallelogram. Select all the statements that apply to the figure..

**Difficulty:**N/A**Type:**MS: Multiselect

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## Lesson Plans

## Perspectives Video: Professional/Enthusiast

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## MFAS Formative Assessments

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.

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*) = *r*^{2} · *A*(1)].

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.

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.

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.

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

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## Video/Audio/Animation

This video dynamically shows how Pi works, and how it is used.

Type: Video/Audio/Animation