ALEX Classroom Resource

  

Geometry, Module 4, Topic B: Perpendicular and Parallel Lines in the Cartesian Plane

  Classroom Resource Information  

Title:

 Geometry, Module 4, Topic B: Perpendicular and Parallel Lines in the Cartesian Plane

URL:

 https://www.engageny.org/resource/geometry-module-4-topic-b-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

The challenge of programming robot motion along segments parallel or perpendicular to a given segment leads to an analysis of slopes of parallel and perpendicular lines and the need to prove results about these quantities (G-GPE.B.5). MP.3 is highlighted in this topic as students engage in proving the criterion for perpendicularity and then extending that knowledge to reason about lines and segments. This work highlights the role of the converse of the Pythagorean theorem in the identification of perpendicular directions of motion (G-GPE.B.4). In Lesson 5, students explain the connection between the Pythagorean theorem and the criterion for perpendicularity (G-GPE.B.4). Lesson 6 extends that study by generalizing the criterion for perpendicularity to any two segments and applying this criterion to determine if segments are perpendicular.

In Lesson 7, students will recognize when a line and a normal segment intersect at the origin. Lesson 8 concludes Topic B when students recognize parallel and perpendicular lines from their slopes and create equations for parallel and perpendicular lines. The criterion for parallel and perpendicular lines and the work from this topic with the distance formula is extended in the last two topics of this module as students use these foundations to determine the perimeter and area of polygonal regions in the coordinate plane defined by systems of inequalities. Additionally, students study the proportionality of segments formed by diagonals of polygons.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
32. Use coordinates to prove simple geometric theorems algebraically.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Given coordinates and geometric theorems and statements defined on a coordinate system, use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others.
Teacher Vocabulary:
  • Simple geometric theorems
  • Simple geometric figures
Knowledge:
Students know:
  • Relationships (e.g. distance, slope of line) between sets of points.
  • Properties of geometric shapes.
  • Coordinate graphing rules and techniques.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Accurately determine what information is needed to prove or disprove a statement or theorem.
  • Accurately find the needed information and explain and justify conclusions.
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
Understanding:
Students understand that:
  • Modeling geometric figures or relationships on a coordinate graph assists in determining truth of a statement or theorem.
  • Geometric theorems may be proven or disproven by examining the properties of the geometric shapes in the theorem through the use of appropriate algebraic techniques.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.32.1: Apply formulas, and properties of polygons, angles, and lines to draw conclusions from the given information.
GEO.32.2: Identify properties of perpendicular and parallel lines, properties of polygons.
GEO.32.3: Illustrate polygons created by given coordinates on a coordinate plane.
GEO.32.4: Identify distance formula, circle formula, Pythagorean Theorem, midpoint.

Prior Knowledge Skills:
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate an understanding of an extended coordinate plane.
  • Draw and label a 4 quadrant coordinate plane.
  • Draw and extend vertical and horizontal number lines.
  • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
  • Recall how to graph points in all four quadrants of the coordinate plane.
  • Define ordered pairs.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Identify which signs indicate the location of a point in a coordinate plane.
  • Recall how to plot ordered pairs on a coordinate plane.
  • Identify the length between vertices on a coordinate plane.
  • Calculate the perimeter and area using the distance between the vertices.
  • Define a right angle, Pythagorean Theorem, converse, and proof.
  • Recognize examples of right triangles.
  • Demonstrate how to find square roots.
  • Solve problems with exponents.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Mathematics
MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
33. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

Example: Find the equation of a line parallel or perpendicular to a given line that passes through a given point.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given a line,
  • Create lines parallel to the given line and compare the slopes of parallel lines by examining the rise/run ratio of each line.
  • Create lines perpendicular to the given line by rotating the line 90 degrees and compare the slopes by examining the rise/run ratio of each line.
  • Use understandings of similar triangles and logical reasoning to prove that parallel lines have equal slopes and the slopes of perpendicular lines are negative reciprocals.Given a geometric problem involving parallel or perpendicular lines.
  • Apply the appropriate slope criteria to solve the problem and justify the solution including finding equations of lines parallel or perpendicular to a given line.
Teacher Vocabulary:
  • Parallel lines
  • Perpendicular lines
  • Slope
  • Slope triangle
Knowledge:
Students know:
  • Techniques to find the slope of a line.
  • Key features needed to solve geometric problems.
  • Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
  • Explain and justify conclusions reached regarding the slopes of parallel and perpendicular lines.
  • Apply slope criteria for parallel and perpendicular lines to accurately find the solutions of geometric problems and justify the solutions.
  • Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
Understanding:
Students understand that:
  • Relationships exist between the slope of a line and any line parallel or perpendicular to that line.
  • Slope criteria for parallel and perpendicular lines may be useful in solving geometric problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.33.1: Define slope, point slope formula, slope-intercept formula, standard form of a line, parallel lines, and perpendicular lines.
GEO.33.2: Demonstrate and explain algebraically how perpendicular lines have only one common point.
GEO.33.3: Demonstrate and explain algebraically how parallel lines have no common points.
GEO.33.4: Write and solve equations of parallel and perpendicular lines.
GEO.33.5: Illustrate graphically how perpendicular lines have only one common point.
GEO.33.6: Illustrate graphically how parallel lines have no common points.
GEO.33.7: Write an equation of a line in slope intercept form.
GEO.33.8: Find the slope of a given line.

Prior Knowledge Skills:
  • Define slope, intercept, linear, equation, and bivariate.
  • Recall how to determine the rate of change (slope) from a graph.
  • Identify the parts of the slope-intercept form of an equation.
  • Recognize how to read a graph.
  • Recall how to write an equation in slope-intercept form.
  • Apply the identification of the slope and the y-intercept to a real-world situation.
  • Create a graph to model a real-word situation.
  • Define proportional relationships, unit rate, and slope.
  • Demonstrate how to graph on a Cartesian plane.
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Define linear functions, nonlinear functions, slope, and y-intercept.
  • Recognize linear equations.
  • Identify ordered pairs.
  • Recognize ordered pairs.
  • Generate the slope of a line using given ordered pairs.
  • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
  • Graph a function given the slope-intercept form of an equation.
  • Recognize that two sets of points with the same slope may have different y-intercepts.
  • Graph a linear equation given the slope-intercept form of an equation.
Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Tags: algebra, coordinates, geometric theorems, parallel, perpendicular, slope
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There are four lessons on this topic.

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Author: Hannah Bradley
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