Make geometric constructions.
12. Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective
devices, paper folding, dynamic geometric software, etc.). 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.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
2. 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.
5. 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. Convert between
degrees and radians. CA
3.1 Given a quadratic equation of the
form ax2 + by2 + cx + dy + e = 0, use the method for completing the square to put the
equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and
graph the equation. CA
Use coordinates to prove simple geometric theorems algebraically.
4. Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by
four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at
the origin and containing the point (0, 2).
5. 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).
6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.
Translate between the geometric description and the equation for a conic section.
1. 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.
2. Derive the equation of a parabola given a focus and directrix.
3. (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances
from the foci is constant.
Calculate expected values and use them to solve problems.
1. (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space;
graph the corresponding probability distribution using the same graphical displays as for data distributions.
2. (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
3. (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can
be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct
answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find
the expected grade under various grading schemes.
4. (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned
empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in
the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in
100 randomly selected households?
Use probability to evaluate outcomes of decisions.
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
a. Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or
a game at a fast-food restaurant.
b. Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus
a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major
accident.
6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
7. (+) Analyze decisions and strategies using probability concepts (e.g. product testing, medical testing, pulling a hockey
goalie at the end of a game).
Explain volume formulas and use them to solve problems.
1. Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid,
and cone. Use dissection arguments, Cavalieri’s principle, and information limit arguments.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. Visualize relationships between two-dimensional and three-dimensional objects.
4. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects
generated by rotations of two-dimensional objects.
5. Know that the effect of a scale factor k greater than zero on length, area, and volume is to multiply each by k, k2
, and k3, respectively; determine length, area and volume measures using scale factors. CA
6. Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer,
and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve realworld and mathematical problems. CA
Apply geometric concepts in modeling situations.
1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human
torso as a cylinder).
2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per
cubic foot).
3. 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).
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human
torso as a cylinder).