# Mathematics (Grades 5–12) Subtest 2

## Subarea I.1. Measurement and Geometry

0008
Apply principles of measurement and geometry to solve problems.

- solving problems involving conversions within and between different systems of measurement, including dimensional analysis
- solving mathematical and real-world problems involving measurable attributes of simple or composite shapes, regions, and solids (e.g., length, area, capacity, degree and radian angle measures)
- solving mathematical and real-world problems involving indirect measurement (e.g., proportional reasoning, Pythagorean theorem, trigonometric ratios, unit circle trigonometry)
- solving problems involving the effects of changing linear dimensions of a two-dimensional or three-dimensional object on its perimeter, area, and volume

0009
Analyze figures and shapes in two and three dimensions.

- analyzing the relationships between points, lines (e.g., parallel, perpendicular, skew), rays, angles (e.g., supplementary, vertical), and planes
- applying the properties of triangles (e.g., congruence conditions, triangle inequality, special triangles) and other polygons (e.g., properties of quadrilaterals, sum of interior angles) to solve mathematical and real-world problems
- applying the properties of circles (e.g., tangents, arc length, chords, inscribed angles) to solve mathematical and real-world problems
- applying the properties of three-dimensional shapes (e.g., pyramids, cones, spheres) to solve mathematical and real-world problems
- analyzing and evaluating formal and informal arguments, reasoning processes, proofs, and the validity of conclusions
- demonstrating knowledge of the axioms and key features (e.g., parallelism, angle sums, great circles, congruence) of Euclidean and non-Euclidean (e.g., spherical, hyperbolic, fractal) geometries

0010
Analyze figures and shapes using coordinate and transformational geometry.

- classifying, representing, and analyzing two-dimensional shapes in the coordinate plane, including translating between geometric descriptions and equations of conic sections
- applying geometric properties and the concepts of distance, midpoint, parallelism, and slope in two-dimensional and three-dimensional coordinate systems to solve mathematical and real-world problems
- applying transformations (e.g., rotations, reflections, translations) to figures in the coordinate plane and analyzing topological properties and transformations
- translating between two-dimensional and three-dimensional representations of geometric figures (e.g., cross sections, nets) and generating three-dimensional figures from two-dimensional shapes
- analyzing the use of the techniques of coordinate geometry in geometric proofs