## Geometry Scope and Sequence

Students will be able to:

• Derive and use the distance, slope and midpoint formulas in one- and two-dimensional coordinate systems.
• Use distance, slope and midpoint formulas to verify geometric relationships, including congruence of segments and parallelism or perpendicularity of pairs of lines.
• Determine an equation of a line parallel or perpendicular to a given line that passes through a given point.
• Distinguish between undefined terms, definitions, postulates, conjectures and theorems.

Activities

• Exploring Linear Functions
• Points, Lines and Angles—Always,Sometimes,Never
• Slopes of Parallel and Perpendicular Lines Inquiry
• Understanding the Distance Formula

Students will be able to:

• Identify the converse, inverse and contrapositive of a conditional statement, and determine its validity.
• Recognize a biconditional statement as a true conditional statement with a true converse.
• Use counterexamples to verify that a conjecture is false.

Activities

• Conditional Statements
• Conditionals in Mathematics
• OnOneCondition

Students will be able to:

• Use a compass and straightedge to construct segments, congruent angles, segment and angle bisectors, perpendicular lines and perpendicular bisectors, and lines parallel to a given line through a point not on the line; use the constructions to make conjectures about geometric relationships.
• Analyze patterns of angle measures to make conjectures about angles formed by intersecting lines and parallel lines cut by a transversal.
• Verify theorems and solve problems about angles formed by parallel lines and transversals and use these relationships to solve problems.
• Prove equidistance between the endpoints of a segment and points on its perpendicular bisector and use this relationship to solve problems.
• Derive formulas for the measures of interior and exterior angles of polygons.
• Explore relationships created by special segments of triangles and diagonals of quadrilaterals.

Activities

• Basic Geometric Constructions
• Angles in a Triangle
• Angles and Lines at a Point
• Angles of Polygons
• Transversals

Students will be able to:

• Verify the Triangle Inequality Theorem and apply it to solve problems.
• Determine the conditions required for triangle congruence.
• Prove two triangles are congruent by applying the triangle congruence conditions.
• Identify congruent figures and their corresponding sides and angles.
• Prove segments or angles congruent by applying the corresponding parts theorem.
• Verify theorems about relationships in triangles, including the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and use these relationships to solve problems.

Activities

• Triangle Sides and Angles
• Interior and Exterior Angles of a Triangle
• Congruent or Not

Students will be able to:

• Describe and perform transformations of figures in a plane using coordinate notation.
• Determine the image or pre-image of a two-dimensional figure under a composition of transformations.
• Identify the sequence of transformations that will carry a  given pre-image onto an image.
• Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding parts.
• Distinguish between reflectional and rotational symmetry in a plane figure.

Activities

• Reflections
• Rotations
• Scale Factor
• Transformations

Students will be able to:

• Apply the definition of similarity in terms of a dilation to identify similar figures and their proportional sides and the congruent corresponding angles.
• Prove the Angle-Angle Theorem and the Triangle Proportionality Theorem and apply them to solve problems.
• Identify and apply the relationships that exist when an altitude is drawn to the hypotenuse of a right triangle, including the geometric mean, to solve problems.

Activities

• Midsegments of Triangles
• Sidesplitter Theorem
• Geometric Mean

Students will be able to:

• Explore proofs of the Pythagorean Theorem.
• Use Pythagorean triples, the Pythagorean Theorem, and the converse of Pythagorean Theorem to solve problems involving the measures of the sides and angles of a right triangle.
• Apply the relationships in special right triangles (30-60-90 and 45-45-90) to solve problems.
• Determine the measures of the sides and angles of a right triangle by applying the trigonometric ratios (sine, cosine, tangent).

Activities

• Pythagorean Theorem
• Special Right Triangles
• Trig Ratios

Students will be able to:

• Prove a quadrilateral is a parallelogram, rectangle, square or rhombus using opposite sides, opposite angles, or diagonals.

Activities

• Parallelogram Properties Interactive Worksheet
• Properties of Parallelograms
• Rhombi, Kites and Trapezoids
• Working with Quadrilaterals Circuit Training

Students will be able to:

• Investigate patterns in special segments and angle measures of circles.
• Apply theorems about circles, including relationships among angles, radii, chords, tangents, and secants, to solve problems.
• Apply the proportional relationship between the measure of an arc length and the circumference of a circle to solve problems.
• Describe radian measure of an angle as the ratio of the length of an arc intercepted by a central angle and the radius of the circle.
• Relate the equation of a circle to the distance formula.
• Determine the equation for the graph of a circle.

Activities

• Circle Vocabulary Activity
• Angles and Arcs
• Circle Product Theorems
• Putting It All Together (Circles)

Students will be able to:

• Derive and apply the formulas for the area of regular polygons to solve problems using appropriate units of measure.
• Determine the area of composite two-dimensional figures comprised of a combination of polygons and/or sectors of circles to solve problems, using appropriate units of measure.

Activities

• Area of a Regular Polygon
• Boardroom Table
• Arcs, Sector Areas and Segment Areas Grudgeball

Students will be able to:

• Identify the shapes of two-dimensional cross-sections of prisms, pyramids, cylinders, cones, and spheres.
• Identify three-dimensional objects generated by rotations of two-dimensional shapes.
• Derive and apply the formulas for the total and lateral surface area of three-dimensional prisms, pyramids, cones, cylinders, spheres, and composite figures to solve problems, using appropriate units of measure.
• Derive and apply the formulas for the volume of three-dimensional figures, including prisms, pyramids, cones, cylinders, spheres, and composite figures, to solve problems, using appropriate units of measure.
• Determine how changes in the linear dimensions of a shape affect its perimeter, area or volume, including proportional and non-proportional dimensional change.

Activities

• Surface Area of a Cylinder
• Family Tent Quandary
• Volume of Solids Chain Activity
• Wax Museum
• Making a Splash

Students will be able to:

• Develop strategies to use permutations and combinations to solve problems.
• Determine probabilities based on area to solve contextual problems.
• Identify whether two events are independent and compute the probability of two events occurring together with or without replacement.
• Apply conditional probability to solve contextual problems.
• Apply independence in contextual problems.

Activities

• Permutations and Combinations Formula Discovery
• Coin Toss Games
• Permutations and Combinations Calculator Activity