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Students connect algebra to geometric concepts with polygons as they explore the distance formula, slope criteria for parallel and perpendicular lines, and learn to calculate and justify the area and perimeter of polygons.
In Unit 5, Polygons & Algebraic Relationships, students connect algebra to geometric concepts with polygons through distance on the coordinate plane, partitioning line segments, slope criteria for perpendicular and parallel lines, area (with composition and decomposition), and perimeter. Students then use this knowledge, in part, to describe properties and prove theorems of triangles and parallelograms.
In this unit, students will draw on previous understanding of elementary geometry standards as well as many middle school standards. The primary foundational content students will need to have prior to beginning this unit are application of the Pythagorean Theorem from eighth grade; areas of polygons from sixth grade; and algebraic skills with square roots and factoring from eighth grade, Algebra 1, and Unit 4.
The Unit begins with Topic A, Distance on the Coordinate Plane. Students develop the distance formula using the Pythagorean Theorem to partition line segments proportionally. In Topic B, Classify Polygons using Slope Criteria and Proportional Line Segments, students describe and apply the slope criteria for parallel and perpendicular lines in order to algebraically identify characteristics of triangles and quadrilaterals, with a particular focus on midsegments, medians, and diagonals. Extending these skills, Topic C, Area and Perimeter On and Off the Coordinate Plane, focuses on calculating and justifying the area and perimeter of polygons. In addition, students identify scale factors of dilated polygons and use quadratic equations and systems of inequalities to describe polygons on and off the coordinate plane.
The material from this unit is foundational to the next unit, Three-Dimensional Measurement and Application, where students will need to use the composite area concepts and the Pythagorean Theorem, and focus on measurement units to solve application problems.
This assessment accompanies Unit 5 and should be given on the suggested assessment day or after completing the unit.
Internalization of Standards via the Unit Assessment
Internalization of Trajectory of Unit
|Distance formula||Pythagorean Theorem|
|Ratio||Directed line segment|
|Median||Point of concurrency|
|Composite shape||Irregular shape|
|Scale factor||Quadratic equation|
|System of inequalities|
Use the Pythagorean Theorem to calculate distance on a coordinate plane. Develop a formula for calculating distances.
Partition horizontal and vertical line segments into equal proportions on a number line.
Divide a line segment on a coordinate plane proportionally and identify the point that divides the segment proportionally.
Algebraically and using the Pythagorean Theorem, determine the slope criteria for perpendicular lines.
Describe and apply the slope criteria for parallel lines.
Identify and create parallelograms, rectangles, rhombuses, and squares on the coordinate plane.
Algebraically verify midsegment, median, and parallel line relationships in triangles.
Algebraically verify diagonal relationships in quadrilaterals and parallelograms.
Calculate and justify the area and perimeter of polygons and composite shapes off the coordinate plane.
Calculate and justify the area and perimeter of parallelograms and triangles on the coordinate plane.
Calculate and justify composite and irregular areas on the coordinate plane.
Describe how the area changes when a figure is scaled.
Solve area applications by creating and solving quadratic equations.
Describe a polygon on the coordinate plane using a system of inequalities.
Calculate and justify the area and perimeter of polygons on the coordinate plane given a system of inequalities.
Key: Major Cluster Supporting Cluster Additional Cluster