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Standard #: MAFS.912.G-GPE.2.5 This document was generated on CPALMS - www.cpalms.org 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). Subject Area: Mathematics Grade: 912 Domain: Geometry: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically - Geometry - Major Cluster Date Adopted or Revised: 02/14 Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters. Content Complexity Rating: Level 2: Basic Application of Skills & Concepts - More Information Date of Last Rating: 02/14 Status: State Board Approved Remarks/Examples Geometry - Fluency Recommendations Fluency with the use of coordinates to establish geometric results, calculate length and angle, and use geometric representations as a modeling tool are some of the most valuable tools in mathematics and related fields. Related Courses Course Number 1206310: 1206315: 1206320: 7912065: 1200400: 1207310: 1200410: 1200700: Course Title Geometry (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Geometry Honors (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Access Geometry (Specifically in versions: 2015 and beyond (current)) Intensive Mathematics (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Mathematics for College Readiness (Specifically in versions: 2014 - 2015, 2015 and beyond (current)) Related Access Points Access Point Access Points Number MAFS.912.G-GPE.2.AP.5a: MAFS.912.G-GPE.2.AP.5b: Access Points Title Using slope, prove lines are parallel or perpendicular. Find equations of lines based on certain slope criteria such as; finding the equation of a line parallel or perpendicular to a given line that passes through a given point. Related Resources Lesson Plan Name Description Students will be guided through the investigation of y = mx+b. Through this lesson, students will be able to determine "When will we ever meet?": whether lines are parallel, perpendicular, or neither by looking at the graph and the equation. Starting with a set of three points, students will practice finding equations of lines and the lines that are perpendicular to Forget Waldo - Where is 'the them. The students will repeat this process three times - using different colors for differentiating one line from the next. orthocenter'?: The big finale brings all the work together and the students realize this activity leads to finding the orthocenter of a triangle. The lesson teaches students about an important characteristic of lines: their slope. Slope can be determined either in page 1 of 3 Graphing Equations on the Cartesian Plane: Slope: Investigating Lines With Our Minds!: graphical or algebraic form. Slope can also be described as positive, negative, zero, or undefined. Students get an explanation of when and how these different types of slope occur. Finally, students learn how slope relates to parallel and perpendicular lines. When two lines are parallel, they have the same slope and when they are perpendicular their slopes are negative reciprocals of one another. Prerequisite knowledge: Students must know how to graph points on the Cartesian plane. They must be familiar with the x- and y- axes on the plane in both the positive and negative directions. Students will investigate lines to discover the relationships of the slopes of parallel and perpendicular lines using X-Y Coordinate Pegboards. Problem-Solving Task Name A Midpoint Miracle: Description This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not. This problem solving task challenges students to use ideas about linear functions in order to determine when certain Triangles inscribed in a circle: angles are right angles. Unit Squares and Triangles: This problem solving task asks students to find the area of a triangle by using unit squares and line segments. Formative Assessment Name Finding Equations of Parallel and Perpendicular Lines: Description This lesson is intended to help you assess how well students are able to understand the relationship between the slopes of parallel and perpendicular lines and, in particular, to help identify students who find it difficult to: Find, from their equations, lines that are parallel and perpendicular. Identify and use intercepts. It also aims to encourage discussion on some common misconceptions about equations of lines. Proving Slope Criterion for Parallel Lines - One: Proving Slope Criterion for Parallel Lines - Two: Proving Slope Criterion for Perpendicular Lines - 1: Students are asked to prove that two parallel lines have equal slopes. Proving Slope Criterion for Perpendicular Lines - 2: Students are asked to prove that if the slopes of two lines are both opposite and reciprocal, then the lines are perpendicular. Writing Equations for Parallel Lines: Students are asked to identify the slope of a line parallel to a given line and write an equation for the line given a point. Writing Equations for Perpendicular Lines: Students are asked to identify the slope of a line perpendicular to a given line and write an equation for the line given a point. Students are asked to prove that two lines with equal slopes are parallel. Students are asked to prove that the slopes of two perpendicular lines are both opposite and reciprocal. Worksheet Name Midpoints of the Sides of a Quadrilateral: Description The students will construct a quadrilateral on graph paper, determine the midpoints of each of the four sides, then connect the midpoints of adjacent sides. The question then is the following: what are the properties of the resulting quadrilateral? Students need to justify their conclusions. Tutorial Name Parallel Lines: Description Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line. Perpendicular Lines: Perpendicular lines have slopes which are negative reciprocals of each other, but why? Video/Audio/Animation Name Parallel Lines: Description This video illustrates how to determine if the graphs of a given set of equations are parallel. Parallel Lines 2: This video shows how to determine which lines are parallel from a set of three different equations. Perpendicular Lines 2: This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the equation of a line and an ordered pair from the other line. Assessment Name Sample 1 - High School Geometry State Interim Assessment: Sample 2 - High School Geometry State Interim Assessment: Description This is a State Interim Assessment for 9th-12th grade. This is a State Interim Assessment for 9th-12th grade. page 2 of 3 Sample 3 - High School Geometry State Interim Assessment: Sample 4 - High School Geometry State Interim Assessment: This is a State Interim Assessment for 9th-12th grade. This is a State Interim Assessment for 9th-12th grades. Student Resources Name A Midpoint Miracle: Description This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not. Parallel Lines: Parallel lines have the same slope and no points in common. However, it is not always obvious whether two equations describe parallel lines or the same line. Parallel Lines: This video illustrates how to determine if the graphs of a given set of equations are parallel. Parallel Lines 2: This video shows how to determine which lines are parallel from a set of three different equations. Perpendicular Lines: Perpendicular lines have slopes which are negative reciprocals of each other, but why? This video describes how to determine the equation of a line that is perpendicular to another line. All that is given initially the Perpendicular Lines 2: equation of a line and an ordered pair from the other line. Triangles inscribed in a This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles. circle: Unit Squares and Triangles: This problem solving task asks students to find the area of a triangle by using unit squares and line segments. Parent Resources Name A Midpoint Miracle: Description This problem solving task gives students the opportunity to prove a fact about quadrilaterals: that if we join the midpoints of an arbitrary quadrilateral to form a new quadrilateral, then the new quadrilateral is a parallelogram, even if the original quadrilateral was not. Triangles inscribed in a This problem solving task challenges students to use ideas about linear functions in order to determine when certain angles are right angles. circle: Unit Squares and Triangles: This problem solving task asks students to find the area of a triangle by using unit squares and line segments. page 3 of 3