Students create scale drawings of polygonal figures by the Ratio Method.
Given a figure and a scale drawing from the Ratio Method, students answer questions about the scale factor and the center.

Students create scale drawings of polygonal figures by the Parallel Method.
Students explain why angles are preserved in scale drawings created by the Parallel Method using the theorem on parallel lines cut by a transversal.

Students understand that the Ratio and Parallel Methods produce the same scale drawing and understand the proof of this fact.
Students relate the equivalence of the Ratio and Parallel Methods to the Triangle Side Splitter Theorem: A line segment splits two sides of a triangle proportionally if and only if it is parallel to the third side.

Students divide a line segment into equal pieces by the Side Splitter and Dilation Methods.
Students know how to locate fractions on the number line.

Students understand that similarity is reflexive, symmetric, and transitive.
Students recognize that if two triangles are similar, there is a correspondence such that corresponding pairs of angles have the same measure and corresponding sides are proportional. Conversely, they know that if there is a correspondence satisfying these conditions, then there is a similarity transformation taking one triangle to the other respecting the correspondence.

Students prove the side-angle-side criterion for two triangles to be similar and use it to solve triangle problems.
Students prove the side-side-side criterion for two triangles to be similar and use it to solve triangle problems.

Students state, understand, and prove the Angle Bisector Theorem.
Students use the Angle Bisector Theorem to solve problems.

Students understand that parallel lines cut transversals into proportional segments. They use ratios between corresponding line segments in different transversals and ratios within line segments on the same transversal.
Students understand Eratosthenes’ method for measuring the earth and solve related problems.

Students understand how the Greeks measured the distance from the earth to the moon and solve related problems.

Students multiply and divide expressions that contain radicals to simplify their answers.
Students rationalize the denominator of a number expressed as a fraction.

It is convenient, as adults, to use the notation “ ” to refer to the value of the square of the sine function. However, rushing too fast to this abbreviated notation for trigonometric functions leads to incorrect understandings of how functions are manipulated, which can lead students to think that is short for “ ” and to incorrectly divide out the variable, “ .”

To reduce these types of common notation-driven errors later, this curriculum is very deliberate about how and when we use abbreviated function notation for sine, cosine, and tangent:

In geometry, sine, cosine, and tangent are thought of as the value of ratios of triangles, not as functions. No attempt is made to describe the trigonometric ratios as functions of the real number line. Therefore, the notation is just an abbreviation for the “sine of an angle” ( ) or “sine of an angle measure” ( ). Parentheses are used more for grouping and clarity reasons than as symbols used to represent a function.

In Algebra II, to distinguish between the ratio version of sine in geometry, all sine functions are notated as functions: is the value of the sine function for the real number , just like is the value of the function for the real number . In this grade, we maintain function notation integrity and strictly maintain parentheses as part of function notation, writing, for example, , instead of .

By pre-calculus, students have had two full years of working with sine, cosine, and tangent as both ratios and functions. It is finally in this year that we begin to blur the distinction between ratio and function notations and write, for example, as the value of the square of the sine function for the real number , which is how most calculus textbooks notate these functions.

Students prove that the area of a triangle is one-half times the product of two side lengths times the sine of the included angle and solve problems using this formula.
Students find the area of an isosceles triangle given the base length and the measure of one angle.

Students find missing side lengths of an acute triangle given one side length and the measures of two angles.
Students find the missing side length of an acute triangle given two side lengths and the measure of the included angle.

Students understand that the Law of Sines can be used to find missing side lengths in a triangle when you know the measures of the angles and one side length.
Students understand that the Law of Cosines can be used to find a missing side length in a triangle when you know the angle opposite the side and the other two side lengths.
Students solve triangle problems using the Laws of Sines and Cosines.

Students develop an understanding of how to determine a missing angle in a right triangle diagram and applythis to real world situations.

Given a physical situation (e.g., a room of a certain shape and dimensions, with objects at certain positions and a robot moving across the room), students impose a coordinate system and describe the given in terms of polygonal regions, line segments, and points in the coordinate system.

Students describe rectangles (with edges parallel to the axes) and triangles in the coordinate plane by means of inequalities. For example, the rectangle in the coordinate plane with lower left vertex (1,2) and upper right vertex (10,15) is {(x,y) l 1 < x < 10 & 2 < y < 15} , the triangle with vertices at (0,0), (1,3), and (2,1) is {(x,y) l x/2 < y < 3x & y < -2x + 5}.

Given two points in the coordinate plane and a rectangular or triangular region, students determine whether the line through those points meets the region, and if it does, they describe the intersections as a segment and name the coordinates of the endpoints.