Students will:

• Change everyday statements or combined ideas into symbols, like using symbols for "not," "and," "or," "if...then," and "if and only if." This includes statements about shapes and their properties.
• Recognize and check the reverse versions of "if...then" statements to see if they still make sense. Understand how "if and only if" relates to true "if...then" statements with true reverses, especially with statements about shapes.
• Use Venn diagrams (overlapping circles) to show how sets (groups of things) are related, including their union, intersection, subsets, and opposites.
• Understand and explain what Venn diagrams mean, including when they represent real-life situations.
• Evaluate the truth value of simple and compound logical statements using truth tables. (Honors)

Identify the conclusion of a series of true conditional statements. (Honors)

Students will:

• Show and explain the relationships between pairs of angles created when two parallel lines are crossed by another line (a transversal), including angles that match, angles inside the lines but on opposite sides, angles outside the lines but on opposite sides, and angles on the same side either inside or outside the lines.
• Prove that two or more lines are parallel by using angle measurements, either with numbers or algebraic expressions.
• Use the relationships between angle pairs formed by the intersection of two parallel lines and a transversal to solve problems.
• Given a diagram with more than three lines, determine the minimum amount of information needed to determine parallelism; and evaluate when statements about parallel lines are always, sometimes, or never true, including cannot be determined. (Honors)
• Construct a line parallel to a given line and explain how the steps of the construction connect to theorems about parallelism. (Honors)

Investigate Euclid’s parallel postulate and compare properties and relationships of geometries of flat surfaces and spherical/elliptical surfaces. (Honors)

Students will:

• Find, count, and draw lines of symmetry in a shape, including shapes in real-world situations.
• Decide if a shape has point symmetry, line symmetry, both, or neither, including shapes in real-world contexts.
• Look at an image or its original shape and figure out which transformation(s) happened, such as:
  • Sliding (translation)
  • Flipping over a horizontal or vertical line, or over the lines y = x or y = -x (reflection)
  • Rotating clockwise or counterclockwise by 90°, 180°, 270°, or 360° around the origin on a grid (rotation)
  • Enlarging or shrinking from a fixed point on a grid (dilation).
• Determine whether a figure has rotational symmetry and identify the angel and order of the rotational symmetry. (Honors)
• Write compound transformations with coordinate notations. (Honors)

Explore self-similar fractals such as Sierpinksi triangles. (Honors)

Students will:

• When given the lengths of three segments, figure out if they can form a triangle.
• If given the lengths of two sides of a triangle, determine the possible range for the length of the third side.
• Arrange the sides of a triangle in order from shortest to longest when you know the angles.
• Arrange the angles of a triangle in order from smallest to largest when you know the lengths of the sides.
• Find the missing interior or exterior angle of a triangle when given the other two angles.
• Determine the midsegments of a triangle and use related properties to solve problems. (Honors)
• Investigate and construct the points of concurrency in a triangle. (Honors)

Construct the inscribed and circumscribed circles of a triangle. (Honors)

Students will:

• Prove that two triangles are congruent using definitions, postulates, and theorems like Side-Side-Side (SSS), Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), and Hypotenuse-Leg (HL).
• Use algebra to prove that two triangles are congruent.
• Use coordinate methods, like the slope and distance formulas, to prove that two triangles are congruent.
• Create a congruent triangle using constructions with congruent segments, angles, and perpendicular lines based on SSS, SAS, ASA, AAS, and HL.
• Prove that triangles are similar using definitions, postulates, and theorems like Side-Angle-Side (SAS), Side-Side-Side (SSS), and Angle-Angle (AA).
• Use algebra to prove that triangles are similar.
• Use coordinate methods, like the slope and distance formulas, to prove that two triangles are similar.
• Describe a series of transformations that can be used to confirm the similarity of triangles in the same plane.
• Solve problems involving the properties of similar triangles, including real-world scenarios.
• Determine minimum amount of information needed to prove a given congruence relationship, and evaluate when statements about congruent triangles are always, sometimes, or never true, including cannot be determined. (Honors)
• Given two figures, use the definition of congruence in terms of rigid transformations to decide if they are congruent. (Honors)

Determine minimum amount of information needed to prove a given similarity relationship; and evaluate when statements about similar triangles are always, sometimes, or never true, including cannot be determined. (Honors) 

Students will:

• Determine if a triangle with three given side lengths is a right triangle.
• Find and confirm trigonometric ratios (sine, cosine, and tangent) using right triangles.
• Solve problems, including real-life situations, using right triangle trigonometry (sine, cosine, and tangent ratios).
• Solve problems using the properties of special right triangles (like 45°-45°-90° triangles).
• Find missing lengths in geometric figures using properties of 45°-45°-90° triangles, including rationalizing denominators when necessary.
• Find missing lengths in geometric figures using properties of 30°-60°-90° triangles, including rationalizing denominators when necessary.
• Solve problems involving right triangles using the Pythagorean Theorem and its converse, including identifying Pythagorean Triples.
•  Connect similar right triangles and trig ratios, and extend to the Law of Sines. (Honors)
• Discover the relationship between the sine and cosine of complementary angles. (Honors)

Investigate vectors and their application, including the connection between vector notation and the magnitude and direction of a vector. (Honors)

Students will:

• Solve problems using the properties of parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids.
• Prove and explain that quadrilaterals have specific properties using methods like the slope formula, distance formula, and midpoint formula.
• Prove and explain theorems and properties of quadrilaterals using logical reasoning.
• Use constructions (like congruent segments, congruent angles, angle bisectors, perpendicular lines, and parallel lines) to verify properties of quadrilaterals.
• Solve problems involving the number of sides of a regular polygon, given the measures of the interior and exterior angles.
• Explain and solve problems related to the relationship between the sum of the interior and exterior angles of a convex polygon.
• Explain and solve problems involving the measure of each interior and exterior angle of a regular polygon.
• Given two points on a particular quadrilateral, describe the possible points to complete the quadrilateral. (Honors)
• Compare quadrilateral congruence relationships with triangle congruence theorems. (Honors)
• Use constructions to generate quadrilaterals and explain how the steps of the construction connect to the properties of the quadrilaterals. (Honors)
• Investigate non-convex polygons and explain how relationships between the sums of angels and the number of sides compare to those relationships in regular and convex polygons. (Honors)

Explain why triangles, squares, and hexagons can tessellate on their own and explore what other combinations of polygons can tessellate. (Honors)

Students will:

• Identify the shape of a flat (two-dimensional) cross section that results from slicing a three-dimensional figure.
• Create models and solve problems involving the surface area of three-dimensional figures, including combined (composite) shapes.
• Solve multistep problems involving the volume of three-dimensional figures, including composite shapes.
• Find unknown measurements of three-dimensional figures using information such as the length of a side, the area of a face, or the volume.
• Explain how changing one or more dimensions of a figure affects other measurements like perimeter, area, total surface area, and volume.
• Describe how changes in surface area and/or volume impact the dimensions of a figure.
• Solve problems, including real-world situations, that involve changing the dimensions or derived measures of a three-dimensional figure.
• Compare the ratios of side lengths, perimeters, areas, and volumes of similar figures.
• Recognize when two- and three-dimensional figures are similar and solve problems involving the attributes of similar geometric figures.
• Solve contextual problems of increasing complexity, including those involving composite figures formed by subtraction or overlapping three-dimensional figures. (Honors)
• Identify the two-dimensional figure formed by the intersection of two three-dimensional figures. (Honors)
• Solve contextual problems of increasing complexity, including those increase complexity of problems by offering different information. (Honors)

Express the volume of surface area of three-dimensional figures as a function of a single variable and analyze and describe the behavior of the function, including finding maximum or minimum as appropriate. (Honors)

Students will:

• Understand the proportional relationship between the arc length or area of a sector and other parts of a circle.
• Solve problems involving arc measures and angles in a circle formed by central angles.
• Solve problems involving arc measures and angles in a circle that involve inscribed angles.
• Calculate the length of an arc of a circle.
• Calculate the area of a sector of a circle.
• Use arc length or sector area to find unknown measurements of the circle, such as the radius, diameter, arc measure, central angle, arc length, or sector area.
• Derive the equation of a circle using the center and radius, applying the Pythagorean Theorem.
• Solve problems in the coordinate plane using equations of circles:
  • Identify the center of the circle from a graph or equation in standard form.
  • Determine the center of a circle using the coordinates of the endpoints of its diameter.
  • Find the length of the radius or diameter from a graph or equation in standard form.
  • Calculate the length of the radius or diameter using the coordinates of the endpoints of the diameter.
  • Determine the length of the radius or diameter using the center and a point on the circle.
  • Identify the coordinates of a point on the circle given the center and radius.
• Find the equation of a circle when given:
  • A graph of a circle with a center at integer coordinates.
  • The center and a point on the circle.
  • The center and the radius or diameter.
  • The endpoints of a diameter.
• Solve contextual problems by determining angle measures formed by intersecting chords, secants, and/or tangents. (Honors)
• Construct polygons inscribed in a circle and use the constructions to verify properties of circles. (Honors)
•  Construct a tangent line to a circle from a point outside a circle and use the construction to verify properties of circles. (Honors)
• Investigate equations of ellipses. Identify characteristics of the ellipse including center, vertices, and axes, and make connections to the equation and characteristics of a circle with the same center. (Honors)

Use algebraic methods such as factoring and completing the square to make connections between the standard form of a circle equation and the standard form of a polynomial in two variables. (Honors)

In this unit, teachers will provide differentiated opportunities for students to review previous content and/or explore content at a deeper level.