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

[MA2015] PRE (9-12) 38 :

[MA2019] REG-8 (8) 30 :

5 ) Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [G-CO5]

[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 : 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) 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) 12 :

[MA2019] GEO-19 (9-12) 36 :

12 ) Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. *Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.* [G-CO12]

[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.

[MA2019] GEO-19 (9-12) 38 : 38. Use the mathematical modeling cycle involving geometric methods to solve design problems.

*Examples: Design an object or structure to satisfy physical constraints or minimize cost; work with typographic grid systems based on ratios; apply concepts of density based on area and volume.*

Discover parallels between the natural world and origami—the Japanese art of paper folding—and witness how artists, mathematicians, and scientists are pioneering complex new designs, in this video from *NOVA: The Origami Revolution*. While origami is steeped in tradition, computer programs are being used today to develop new, more complex designs so that almost any three-dimensional object can be translated into a two-dimensional pattern that can be folded to model the object. This resource is part of the NOVA Collection.

View Standards
**Standard(s): **
[MA2015] GEO (9-12) 16 :

[MA2015] GEO (9-12) 17 :

[MA2019] GEO-19 (9-12) 27 :

16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

[MA2015] GEO (9-12) 17 :

17 ) Prove theorems about triangles. *Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity.* [G-SRT4]

[MA2019] GEO-19 (9-12) 27 :

27. Given two figures, determine whether they are similar by identifying a similarity transformation (sequence of rigid motions and dilations) that maps one figure to the other.

[MA2019] GEO-19 (9-12) 34 : 34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.

[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.

Students learn what it means for two figures to be similar in general, and then focus on triangles and what criteria predict that two triangles will be similar. Length relationships within and between figures are studied closely and foreshadows work in Module 2, Topic D. The topic closes with a look at how similarity has been used in real-world application.