Teacher Resources

## Collection Contents

### 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.

by Allen Wolmer

Exercises from Illustrative MathematicsA set of 4 exercises, commentary, and solutions
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### Students review the area formula for rectangles and use a square grid to estimate the area of a curved region

by Allen Wolmer

Students review the area formula for rectangles with rational side lengths and prove the area formula for anarbitrary rectangle.Students use a square grid to estimate the area of a curved region using lower approximations, upperapproximations, and average approximations.Teacher and Student versions of full lesson from engageNY
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### Students use inscribed and circumscribed polygons for a circle (or disk) of radius r and circumference C to show that the area of a circle is (1/2)Cr or as it is usually written, ?r^2.

by Allen Wolmer

Students use inscribed and circumscribed polygons for a circle (or disk) of radius r and circumference C to show that the area of a circle is (1/2)Cr or as it is usually written, ?r^2.Teacher and Student versions of full lesson from engageNY
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### Students understand the precise language that describes the properties of volume.

by Allen Wolmer

Students understand the precise language that describes the properties of volume.Students understand that the volume of any right cylinder is given by the formula area of base×height.Teacher and Student versions of full lesson from engageNY
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### Students understand the principle of parallel slices in the plane, and understand Cavalieri’s principle

by Allen Wolmer

Students understand the principle of parallel slices in the plane, and understand Cavalieri’s principle as ageneralization of the principle of parallel slices.Students use Cavalieri’s principle to reason that the volume formula for a general cylinder is area of base×height.Teacher and Student versions of full lesson from engageNY
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### Students use Cavalieri’s principle

by Allen Wolmer

Students use Cavalieri’s principle and the cone cross section theorem to show that a general pyramid or cone has volume (1/3)Bh where B is the area of the base and h is the height by comparing it with a right rectangular pyramid with base area B and height h.Teacher and Student versions of full lesson from engageNY
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### Area of a Circle

by Allen Wolmer

Area of a CircleWhat if you wanted to figure out the area of a circle with a radius of 5 inches? After completing this Concept, you'll be able to answer questions like this.Lessons, videos, exercises, and text from CK-12. Additional resources available at this site.
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### Area and Circumference of Circles

by Allen Wolmer

Area and Circumference of CirclesYou can use regular polygons with an increasing number of sides to help explain why a circle of radius 1 unit has an area of pi un^2 . Where does the “  r^2 ” come from in the formula for the area of a circle?Lessons, videos, exercises, and text from CK-12. Additional resources available at this site.
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