Construct figures, using technology and other tools, in order to make and test conjectures about their properties.
Identify different sets of properties necessary to define and construct figures.
Develop and use precise definitions of figures such as angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats.
Investigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment’s endpoints.
Investigate, prove, and apply theorems about triangles, including but not limited to: the \sum of the measures of the interior angles of a triangle is $180^{\circ}$; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.
Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.
Example: Prove that rectangles are parallelograms with congruent diagonals.
Use coordinates to prove simple geometric theorems algebraically.
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.
Example: Find the equation of a line parallel or perpendicular to a given line that passes through a given point.
Use congruence and similarity criteria for triangles to solve problems in real-world contexts.
Discover and apply relationships in similar right triangles.
Derive and apply the constant ratios of the sides in special right triangles ($45^{\circ} -45^{\circ} -90^{\circ}$ and $30^{\circ} -60^{\circ} -90^{\circ}$).
Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.
Explain and use the relationship between the sine and cosine of complementary angles.
Demonstrate the converse of the Pythagorean Theorem.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.
Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: 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.
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.
Explain how the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for an additional notation for radicals using rational exponents.
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Define the imaginary number i such that $i^2 = -1$.
Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity.
Example: Interpret the accrued amount of investment $P(1 + r)^t$, where $P$ is the principal and $r$ is the interest rate, as the product of $P$ and a factor depending on time $t$.
Use the structure of an expression to identify ways to rewrite it.
Example: See $x^4 - y^4$ as $(x^2)^2 - (y^2)^2$, thus recognizing it as a difference of squares that can be factored as $(x^2 - y^2)(x^2 + y^2)$.
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines.
Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one.
Use the properties of exponents to transform expressions for exponential functions.
Example: Identify percent rate of change in functions such as $y = (1.02)^t$, $y = (0.97)^t$, $y = (1.01)^{12t}$, $y = (1.2)^{t/10}$, and classify them as representing exponential growth or decay.
Add, subtract, and multiply polynomials, showing that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication.
Explain why extraneous solutions to an equation involving absolute values may arise and how to check to be sure that a candidate solution satisfies an equation.
Select an appropriate method to solve a quadratic equation in one variable.
Use the method of completing the square to transform any quadratic equation in $x$ into an equation of the form $(x - p)^2 = q$ that has the same solutions. Explain how the quadratic formula is derived from this form.
Solve quadratic equations by inspection (such as $x^2 = 49$), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real.
Select an appropriate method to solve a system of two linear equations in two variables.
Solve a system of two equations in two variables by using linear combinations; contrast situations in which use of linear combinations is more efficient with those in which substitution is more efficient.
