# Courses of Study : Mathematics

GEOMETRY
Congruence
Experiment with transformations in the plane.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
1 ) Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc. [G-CO1]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.1- Compare properties of perpendicular lines, parallel lines, line segments, angles, and circles.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
2 ) Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [G-CO2]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.2- Given a triangle on a coordinate grid, recognize the image of the triangle after a vertical or horizontal translation.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
3 ) Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO3]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.3- Identify the reflection of a polygon.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 2 Classroom Resources: 2
4 ) Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.4- Given a geometric figure of a reflection or a translation of that figure, identify if the geometric figure is a reflection or translation.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 3 Classroom Resources: 3
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]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.5- Given a figure and that figure after a vertical or horizontal translation, identify the vertical or horizontal translation.

Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for development of concept of geometric proof.)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
6 ) Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. [G-CO6]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.6- Identify corresponding congruent parts of transformed shapes (squares, rectangles, triangles, pentagons).

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
7 ) Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [G-CO7]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.7- Given two congruent triangles and side lengths of one of the triangles, identify the side lengths of the other triangle.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 2 Classroom Resources: 2
8 ) Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.8- Given two congruent triangles and angle measures of one of the triangles, identify the angle measures of the other triangle.

Prove geometric theorems. (Focus on validity of underlying reasoning while using variety of ways of writing proofs.)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
9 ) Prove theorems about lines and angles. Theorems include vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints. [G-CO9]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.9- Given the intersection of two non- perpendicular lines and the measure of one angle, identify the measure of its vertical angle.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 3 Classroom Resources: 3
10 ) Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180o, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.10- Given a measure of a leg or base angle of an isosceles triangle, identify the measure of the other leg or other base angle.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 2 Classroom Resources: 2
11 ) Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. [G-CO11]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.11- Given the measure of one side or one angle of a parallelogram, identify the measure of the opposite side or opposite angle.

Make geometric constructions. (Formalize and explain processes.)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 6 Learning Activities: 3 Classroom Resources: 3
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]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.12- Given a drawing with angles and a protractor overlay, determine which angles are congruent. Sample image below.
Image

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 3 Classroom Resources: 3
13 ) Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.13- Identify an equilateral triangle from a set of triangles or identify a regular hexagon from a set of hexagons. Make sure sides/angles are marked so that students can identify congruence.

Similarity, Right Triangles, and Trigonometry
Understand similarity in terms of similarity transformations.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
14 ) Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT1]

a. 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. [G-SRT1a]

b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. [G-SRT1b]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
15 ) Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. [G-SRT2]

Example 1:

Given the two triangles above, show that they are similar.

4/8 = 6/12

They are similar by SSS. The scale factor is equivalent.

Example 2:

Show that the two triangles are similar.

Two corresponding sides are proportional and the included angle is congruent. (SAS similarity)

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 2 Classroom Resources: 2
16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

Prove theorems involving similarity.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 5 Classroom Resources: 5
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]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
18 ) Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [G-SRT5]

Define trigonometric ratios and solve problems involving right triangles.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
19 ) Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles. [G-SRT6]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
20 ) Explain and use the relationship between the sine and cosine of complementary angles. [G-SRT7]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
21 ) Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
22 ) (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [G-SRT10]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
23 ) (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). [G-SRT11]

Circles
Understand and apply theorems about circles.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
24 ) Prove that all circles are similar. [G-C1]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
25 ) Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. [G-C2]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 3 Classroom Resources: 3
26 ) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
27 ) (+) Construct a tangent line from a point outside a given circle to the circle. [G-C4]

Find arc lengths and areas of sectors of circles. (Radian introduced only as unit of measure.)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
28 ) Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [G-C5]

Expressing Geometric Properties With Equations
Translate between the geometric description and the equation for a conic section.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
29 ) Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. [G-GPE1]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.29- Given a circle graphed on the coordinate plane and a point on the edge of the circle, determine the length of the radius of the circle.

Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
30 ) Use coordinates to prove simple geometric theorems algebraically. [G-GPE4]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.30- Given a rectangle plotted on a coordinate grid with sides parallel to the x- and y-axis, determine the lengths of the sides to verify that opposite sides have the same length.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
31 ) Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). [G-GPE5]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.31- Given a set of parallel lines in a coordinate plane and the slope of one of the lines, identify the slope of the other line.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 1 Classroom Resources: 1
32 ) Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [G-GPE6]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.32- Identify the midpoint between two points on a vertical or horizontal line.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
33 ) Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [G-GPE7]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.33- Given a graph of a square, a rectangle, or a right triangle in the first quadrant, find the area and perimeter of the figure. (limit to squares/rectangles with sides parallel to axes and right triangles with legs parallel to axes)

Use coordinates to prove simple geometric theorems algebraically. (Alabama)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
34 ) Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (Alabama)

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.34- Find the perimeter of an equilateral triangle, square, or regular pentagon given the length of one side.

Geometric Measurement and Dimension
Explain volume formulas and use them to solve problems.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 5 Classroom Resources: 5
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]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.35- Make a prediction about the volume of a container, the area of a figure, or the perimeter of a figure. Ex: how many cubes will go in one figure vs. another. Limit to cylinder, circle.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
36 ) Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [G-GMD3]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.36- Given a cylinder and a cone with the same height and radius, identify that the volume of the cone will be one-third the volume of the cylinder.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
37 ) Determine the relationship between surface areas of similar figures and volumes of similar figures. (Alabama)

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.37- Identify that a rectangle with sides that are two times as large as another rectangle will have an area that is four times as large by using models.

Visualize relationships between two-dimensional and three-dimensional objects.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
38 ) Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. [G-GMD4]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.38- Identify the shapes of two-dimensional cross-sections of three-dimensional objects (limited to sphere, rectangular prism, triangular prism).

Modeling With Geometry
Apply geometric concepts in modeling situations.
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
39 ) Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [G-MG1]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.39- Identify objects that have a similar shape to a cylinder, rectangular prism, and cone.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
40 ) Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).* [G-MG2]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.40- Perform computation operations (addition, subtraction, multiplication, division) in context- based problems about weight, length, or capacity using units.

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
41 ) Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).* [G-MG3]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.41- Solve a context-based problem involving area of rectangles.

Using Probability to Make Decisions
Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules.)
 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 0
42 ) (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6]

 Mathematics (2016) Grade(s): 9 - 12 Geometry All Resources: 2 Classroom Resources: 2
43 ) (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7] (Alabama)

Example:

What is the probability of tossing a penny and having it land in the non-shaded region'

Geometric Probability is the Non-Shaded Area divided by the Total Area.