Suggested Unit Length and Description:
Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola. (Mathematics Appendix A, p.33)
· Use coordinates to prove simple geometric theorems algebraically.
· Practice with distance formula; relate to Pythagorean Theorem.
· Relate work on parallel lines to systems of equations having no solution or infinitely many solutions.
G-GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1,√3) lies on the circle centered at the origin and containing the point (0,2).
G-GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
G-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★
Focus Standards of Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
· I can recall previous understandings of coordinate geometry (including, but not limited to: distance, midpoint and slope formula, equations of a line, definitions of parallel and perpendicular lines, etc.)
· I can use coordinate to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0,2)
· I can recognize that slopes of parallel lines are equal
· I can recognize that slopes of perpendicular lines are opposite reciprocals (i.e., the slopes of perpendicular lines have a product of -1)
· I can fins the equation of a line parallel to a given line that passes through a given point
· I can find the equation of a line perpendicular to a given line that passes through a given point
· I can prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems
· I can recall the definition of ratio
· I can recall previous understandings of coordinate geometry
· I can, given a line segment (including those with positive and negative slopes) and ratio, find the point on the segment that partitio9ns the segment into the given ratio
· I can use the coordinates of the vertices of a polygon to find the necessary dimensions for finding the perimeter (i.e., the distance between vertices)
· I can use the coordinates of the vertices of a triangle to find the necessary dimensions (base, height) for finding the area (i.e., the distance between vertices by counting, distance, formula, Pythagorean Theorem, etc.)
· I can use coordinates of the vertices of a rectangle to find the necessary dimensions (base, height) for finding the area (i.e., the distance between vertices by counting, distance, formula)
· I can formulate a model of figures in contextual problems to compute area and/or perimeter
· Representation of geometric ideas and relationships allow multiple approaches to geometric problems and connect geometric interpretations to other contexts.
· Communicating mathematically appropriate arguments are central to the study of mathematics.
· Geometric representations in the coordinate plane are a useful way to model various problem situations and physical phenomena.
· Use of appropriate properties and measurement leads to efficient design.
· Logical deduction and clear reasoning are essential tools in creating and understanding arguments inside and outside the field of mathematics.
· Characteristics, properties, and mathematical arguments about geometric relationships can be analyzed and developed using logical and spatial reasoning.
· How does my understanding of algebraic principles help me solve geometric problems?
· How does geometry explain or describe the structure of our world?
· How can reasoning be used to establish or refute conjectures?
· What are the relationships that exist between angles, sides, and diagonals of parallelograms?
· What is the relationship between the slopes of parallel lines and perpendicular lines?
· How can geometric relationships be proven through the application of algebraic properties to geometric figures represented in the coordinate plane?