Classroom Resources (6) |

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 35 :

[MA2015] PRE (9-12) 38 :

[MA2019] REG-7 (7) 19 :

35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. *Use dissection arguments, Cavalieri's principle, and informal limit arguments.* [G-GMD1]

[MA2015] PRE (9-12) 38 :

38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

[MA2019] REG-7 (7) 19 :

19. Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections.

[MA2019] REG-8 (8) 30 : 30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.

[MA2019] GEO-19 (9-12) 16 : 16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

[MA2019] GEO-19 (9-12) 17 : 17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

[MA2019] GEO-19 (9-12) 23 : 23. Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Use the Pythagorean theorem to unroll a cone and find its surface area with this interactive video from the School Yourself Geometry series.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 35 :

[MA2015] PRE (9-12) 38 :

[MA2019] REG-8 (8) 30 :

35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. *Use dissection arguments, Cavalieri's principle, and informal limit arguments.* [G-GMD1]

[MA2015] PRE (9-12) 38 :

38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

[MA2019] REG-8 (8) 30 :

30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.

[MA2019] GEO-19 (9-12) 17 : 17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

Apply Cavalieri's principle to determine when two solids must definitely have the same volume with this interactive video from the School Yourself Geometry series.

35 ) Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. *Use dissection arguments, Cavalieri's principle, and informal limit arguments.* [G-GMD1]

[MA2015] PRE (9-12) 38 :

38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

[MA2019] REG-8 (8) 30 :

30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems.

[MA2019] GEO-19 (9-12) 17 : 17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

How can you find the volume of prisms that aren't rectangular? Learn how with this interactive video from the School Yourself Geometry series.

View Standards
**Standard(s): **
[MA2015] PRE (9-12) 38 : 38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

[MA2019] REG-8 (8) 30 :30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems. [MA2019] GEO-19 (9-12) 17 : 17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

[MA2019] REG-8 (8) 30 :

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

Use the Pythagorean theorem and apply Cavalieri's principle to cones and cylinders to find the volume of a sphere with this interactive video from the School Yourself Geometry series.

View Standards
**Standard(s): **
[MA2015] PRE (9-12) 38 : 38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

[MA2019] REG-8 (8) 30 :30. Use formulas to calculate the volumes of three-dimensional figures (cylinders, cones, and spheres) to solve real-world problems. [MA2019] GEO-19 (9-12) 16 : 17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

[MA2019] REG-8 (8) 30 :

16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

[MA2019] GEO-19 (9-12) 17 : b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.

[MA2019] GEO-19 (9-12) 23 : 23. Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

[MA2019] GEO-19 (9-12) 36 : 36. Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.

In school, you learn about shapes with sides and edges, but there are weird shapes out there (beyond our 3 dimensions) that defy our normal idea of geometry. QuanQuan and Jenny explain, knit, and 3D print their way through these strange shapes.

View Standards
**Standard(s): **
[MA2015] AL2 (9-12) 5 :

[MA2015] AL2 (9-12) 6 :

[MA2015] PRE (9-12) 13 :

[MA2015] ALT (9-12) 5 :

[MA2015] ALT (9-12) 6 :

[MA2015] PRE (9-12) 37 :

[MA2015] PRE (9-12) 38 :38 ) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [G-GMD2]

5 ) (+) Extend polynomial identities to the complex numbers.

Example: Rewrite *x*^{2} + 4 as (*x* + 2*i*)(*x* - 2*i*). [N-CN8]

[MA2015] AL2 (9-12) 6 :

6 ) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9]

[MA2015] PRE (9-12) 13 :

13 ) (+) Know and apply the Binomial Theorem for the expansion of (*x* + *y*)^{n} in powers of *x* and *y* for a positive integer *n*, where *x* and *y* are any numbers, with coefficients determined, for example, by Pascal's Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) [A-APR5]

[MA2015] ALT (9-12) 5 :

5 ) (+) Extend polynomial identities to the complex numbers. [N-CN8]

Example: Rewrite *x*^{2} + 4 as (*x* + 2*i*)(*x* - 2*i*).

[MA2015] ALT (9-12) 6 :

6 ) (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. [N-CN9]

[MA2015] PRE (9-12) 37 :

37 ) (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. [G-GPE3]

[MA2015] PRE (9-12) 38 :

Module 3, Topic A brings students back to the study of complex roots of polynomial functions. Students briefly review quadratic and cubic functions and then extend familiar polynomial identities to both complex numbers and to general polynomial functions. Students use polynomial identities to find square roots of complex numbers. The binomial theorem and its relationship to Pascal’s Triangle are explored using roots of unity. The topic concludes with students’ use of Cavalieri’s principle to derive formulas for the volume of the sphere and other geometric solids.

**Note: This module is identified as Precalculus and Advanced Topics in the EngageNY curriculum. It also corresponds to the Algebra II and Algebra II with Trigonometry Alabama Courses of Study.**