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Geometry Module 4, Topic D: Partitioning and Extending Segments and Parameterization of Lines

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Geometry Module 4, Topic D: Partitioning and Extending Segments and Parameterization of Lines


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Type: Lesson/Unit Plan


Topic D concludes the work of Module 4. In Lesson 12, students find midpoints of segments and points that divide segments into more equal and proportional parts. Students also find locations on a directed line segment between two given points that partition the segment in given ratios (G-GPE.B.6). Lesson 13 requires students to connect this work to proving classical results in geometry (G-GPE.B.4). For instance, the diagonals of a parallelogram bisect one another, and the medians of a triangle meet at the point of the way from the vertex for each. Lesson 14 is an optional lesson that allows students to explore parametric equations and compare them with more familiar linear equations (G-GPE.B.6, G-MG.A.1). Parametric equations make both the variables in an equation dependent on a third variable, usually time. In this lesson, parametric equations model the robot’s horizontal and vertical motion over a period of time. Introducing parametric equations in the Geometry course prepares students for higher-level courses and also represents an opportunity to show coherence between functions, algebra, and coordinate geometry. Students extend their knowledge of parallel and perpendicular lines to lines given parametrically. Students complete the work of this module in Lesson 15 by deriving and applying the distance formula (G-GPE.B.4) and with the challenge of locating the point along a line closest to a given point, again given as a robot challenge.

Content Standard(s):
MA2015 (2016)
Grade: 9-12
32 ) Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [G-GPE6]

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.HS.32- Identify the midpoint between two points on a vertical or horizontal line.

MA2019 (2019)
Grade: 9-12
Geometry with Data Analysis
32. Use coordinates to prove simple geometric theorems algebraically.
Unpacked Content
Evidence Of Student Attainment:
  • 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
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.
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.
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.

Tags: algebra, coordinates, geometric theorems, line segment, partition, point, ratio
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There are four lessons on this topic.

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Author: Hannah Bradley