High School Geometry

Student Learnings: What students should know and be able to do

Mathematical Reasoning

Apply skills of mathematical representation, communication and reasoning throughout the remaining three content strands.

• Assess the reasonableness of a solution by comparing the solution to appropriate graphical or numerical estimates or by recognizing the feasibility of solutions in a given context and rejecting extraneous solutions.
• Appropriately use examples and counterexamples to make and test conjectures, justify solutions, and explain results.
• Support mathematical results by explaining why the steps in a solution are valid and why a particular solution method is appropriate.
• Determine whether or not relevant information is missing from a problem and if so, decide how to best express the results that can be obtained without that information.
• Know and use the relationship that exists among a logical implication of the form “if A, then B,” its converse “if B, then A,” its inverse “if not A, then not B,” and its contrapositive “if not B, then not A.

Number Sense, Computation and Operations

Number Sense
Use real numbers, represented in a variety of ways, to quantify information and to solve real-world and mathematical problems.

• Use real numbers, represented in a variety of ways, to quantify information and to solve real-world and mathematical problems.

Computation and Operation
Appropriately use calculators and other technologies to solve algebraic, geometric, probabilistic and statistical problems.

• Apply the correct order of operations and grouping symbols when using calculators and other technologies.
• Know, use and translate calculator notational conventions to mathematical notation.
• Recognize the impact of units such as degrees and radians on calculations.
• Understand that use of a calculator requires appropriate mathematical reasoning and does not replace the need for mental computation.

Patterns, Functions, and Algebra

Algebra (Algebraic Thinking)
Solve simple equations and inequalities numerically, graphically and symbolically. Use recursion to model and solve real-world and mathematical problems.

• Translate among equivalent forms of expressions, such as, simplify algebraic expressions involving nested pairs of parentheses and brackets, simplify rational expressions, factor a common term from an expression and apply associative, commutative and distributive laws.
• Understand the relationship between absolute value and distance on the number line and graph simple expressions involving absolute value such as, [x-3] = 6 or [x÷2] < 5.
• Find equations of a line given two points on the line, a point and the slope of the line or the slope and the y-intercept of the line.
• Translate among equivalent forms of linear equations and inequalities.
• Use a variety of models such as equations, inequalities, algebraic formulas, written statements, tables and graphs or spreadsheets to represent functions and patterns in real-world and mathematical problems.
• Apply the laws of exponents to perform operations on expressions with integer exponents.
• Solve linear equations and inequalities in one variable with numeric, graphic and symbolic methods.
• Find real solutions to quadratic equations in one variable with numeric, graphic and symbolic methods.
• Understand how slopes can be used to determine whether lines are parallel or perpendicular. Given a line and a point not on the line, find the equations for the lines passing through that point and parallel or perpendicular to the given line.

Spatial Sense, Geometry and Measurement

Spatial Sense
Use models to represent and understand two- and three-dimensional shapes and how various motions affect them. Recognize the relationship between different representations of the same shape.

• Use models the visualization to understand and represent three-dimensional objects and their cross sections from different perspectives.

Geometry
Apply basic theorems of plane geometry, right triangle trigonometry, coordinate geometry and a variety of visualization tools to solve real-world and mathematical problems.

• Know and use theorems about triangles and parallel lines in elementary geometry to justify facts about various geometrical figures and solve real-world and mathematical problems. These theorems include criteria for two triangles to be congruent or similar and facts about parallel lines cut by a transversal.
• Know and use theorems about circles to justify geometrical facts and solve real-world and mathematical problems. These theorems include the relationships involving tangent lines and radii, the relationship between inscribed and central angles and the relationship between the measure of a central angle and arc length.
• Know and use properties of two- and three-dimensional figures to solve real-world and mathematical problems such as finding area, perimeter, volume and surface area; applying direct or indirect methods of measurement; the Pythagorean theorem and its converse; and properties of 45º-45º-90º triangles.
• Apply the basic concepts of right triangle trigonometry including sine, cosine and tangent to solve real-world and mathematical problems.
• Use coordinate geometry to represent and examine geometric concepts such as the distance between two points, the midpoint of a line segment, the slope of a line and the slopes of parallel and perpendicular lines.
• Use numeric, graphic and symbolic representations of transformations such as reflections, translations and change of scale in one-, two- and three-dimensions to solve real-world and mathematical problems.
• Perform basic constructions with a straightedge and compass.
• Draw accurate representations of planar figures using a variety of tools.

Measurement
Use the interconnectedness of geometry, algebra and measurement to explore real-world mathematical problems.

• Use the interconnectedness of geometry, algebra and measurement to explore real-world mathematical problems.

Trigonometry and Geometry

Understand the properties of the standard trigonometric functions and apply them to real-world and mathematical problems, especially geometrical problems. Develop increased mastery of geometric proof methodology.

• Know the six trigonometric functions defined for an angle in a right triangle.
• Solve applied problems about triangles using the law of sines including the ambiguous case.
• Solve applied problems about triangles using the law of cosines.

Copyright 2004, Holt Rinehart Winston Publishing
Geometry