ALEX Resources

   View Standards     Standard(s): [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) 26 :

26. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify that a dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

b. Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.

In this video from KCPT, watch an animated demonstration of rotating and dilating a triangle on the coordinate plane. In the accompanying classroom activity, students watch the video; draw rotations and dilations of a triangle; and identify center of rotation, angle of rotation, and scale factors in classmates drawings. To get the most from the lesson, students should be comfortable graphing points on the coordinate plane and reproducing a drawing of a geometric shape at a different scale. Prior exposure to rotation and dilation is helpful.

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

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] ACC-7 (7) 35 :

35. Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections. [Grade 7, 19]

[MA2019] ACC-7 (7) 41 :

41. Use formulas to calculate the volumes of three-dimensional figures to solve real-world problems. [Grade 8, 30]

[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] 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] ACC-7 (7) 41 :

41. Use formulas to calculate the volumes of three-dimensional figures to solve real-world problems. [Grade 8, 30]

[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.

[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] GEO (9-12) 35 :

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]

[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.

Students study the basic properties of two-dimensional and three-dimensional space, noting how ideas shift between the dimensions. They learn that general cylinders are the parent category for prisms, circular cylinders, right cylinders, and oblique cylinders, and study why the cross-section of a cylinder is congruent to its base. Next students study the explicit definition of a cone and learn what distinguishes pyramids from general cones, and see how dilations explain why a cross-section taken parallel to the base of a cone is similar to the base.  Students revisit the scaling principle as it applies to volume and then learn Cavalieri’s principle, which describes the relationship between cross-sections of two solids and their respective volumes. This knowledge is all applied to derive the volume formula for cones, and then extended to derive the volume formula for spheres. Module 3 is a natural place to see geometric concepts in modeling situations. Modeling-based problems are found throughout Topic B and include the modeling of real-world objects, the application of density, the occurrence of physical constraints, and issues regarding cost and profit.