Students discover that a line is tangent to a circle at a given point if it is perpendicular to the radius drawn to that point.
Students construct tangents to a circle through a given point.
Students prove that tangent segments from the same point are equal in length.

Students use tangent segments and radii of circles to conjecture and prove geometric statements, especially those that rely on the congruency of tangent segments to a circle from a given point.
Students recognize and use the fact if a circle is tangent to both rays of an angle, then its center lies on the angle bisector.

Students use the inscribed angle theorem to prove other theorems in its family (different angle and arc configurations and an arc intercepted by an angle at least one of whose rays is tangent).
Students solve a variety of missing angle problems using the inscribed angle theorem.

Students understand that an angle whose vertex lies in the interior of a circle intersects the circle in two points and that the edges of the angles are contained within two secant lines of the circle.
Students discover that the measure of an angle whose vertex lies in the interior of a circle is equal to half the sum of the angle measures of the arcs intercepted by it and its vertical angle.

Students find the measures of angle/arcs and chords in figures that include two secant lines meeting outside a circle, where the measures must be inferred from other data.

Students write the equation for a circle in center-radius form, (x - a)2 (y - b)2 = r2 using the Pythagorean theorem or the distance formula.
Students write the equation of a circle given the center and radius. Students identify the center and radius of a circle given the equation.

Students complete the square in order to write the equation of a circle in center-radius form.
Students recognize when a quadratic in x and y is the equation for a circle.

Given a circle, students find the equations of two lines tangent to the circle with specified slopes.
Given a circle and a point outside the circle, students find the equation of the line tangent to the circle from that point.

Students show that a quadrilateral is cyclic if and only if its opposite angles are supplementary.
Students derive and apply the area of cyclic quadrilateral ABCD as 1/2 AB·CD·sin(w) where w is the measure of the acute angle formed by diagonals AB and CD.

Students determine the area of a cyclic quadrilateral as a function of its side lengths and the acute angle formed by its diagonals.
Students prove Ptolemy’s theorem, which states that for a cyclic quadrilateral ABCD, AC·BD = AB·CD + BC·AD. They explore applications of the result.

Students are familiar with the number line and determining the location of positive fractions, decimals, and whole numbers from previous grades.

Students will build upon reasoning of ratios and rates to formally define proportional relationships and the constant of proportionality. They explore multiple representations of proportional relationships by looking at tables, graphs, equations, and verbal descriptions, extend their understanding about ratios and proportional relationships to compute unit rates for ratios and rates specified by rational numbers.

Students will use an Integer Card Game that creates a conceptual understanding of integer operations and serves as a mental model students can rely on during the module. They will build on their understanding of rational numbers to add, subtract, multiply, and divide signed numbers.

Students will use linear equations to solve unknown angle problems and other problems presented within context to understand that solving algebraic equations is all about the numbers. They use the number line to understand the properties of inequality and recognize when to preserve the inequality and when to reverse the inequality when solving problems leading to inequalities. They also interpret solutions within the context of problems. Additionally, students extend their study of geometric figures and the relationships between them as they apply their work with expressions and equations to solve problems

Students will expand their basic knowledge of positive integer exponents and prove the Laws of Exponents for any integer exponent. Next, students work with numbers in the form of an integer multiplied by a power of 10 to express how many times as much one is than the other. This leads into an explanation of scientific notation and continued work performing operations on numbers written in this form.

Students will learn about translations, reflections, and rotations in the plane and, more importantly, how to use them to precisely define the concept of congruence.

Students will learn about dilation and similarity and apply that knowledge to a proof of the Pythagorean Theorem based on the Angle-Angle criterion for similar triangles.

Students will extend what they already know about unit rates and proportional relationships to linear equations and their graphs. Students understand the connections between proportional relationships, lines, and linear equations in this module.

In this lesson, we focus on what matter is. Matter has a formal science definition as anything that has mass and takes up space (or has volume). This lesson covers matter, mass, and volume in some depth.