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Clarifications
Clarification 1: Instruction includes using compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Angle
Vertical Alignment
Previous BenchmarksNext Benchmarks
Purpose and Instructional Strategies
In elementary grades, students drew lines and angles using a variety of tools, including rulers and protractors, and by making measurements with those tools, they could bisect lines and angles. In Geometry, students are introduced to constructions that do not rely on making measurements, specifically bisecting an angle and bisecting a segment, including perpendicular bisectors, using a compass and straightedge. These two procedures are embedded within constructing an inscribed circle and a circumscribed circle of a triangle as well as in the construction of a square inscribed in a circle.- Instruction includes the use of manipulatives, tools and geometric software. Allowing
students to explore constructions with dynamic software reinforces why the constructions
work.
- For example, the use of paper folding (e.g., patty paper) can be used to determine the angle bisector of a given angle and the midpoint or perpendicular bisector of a given segment.
- Instruction includes the connection to triangle congruence when constructing an angle
bisector.
- For example, have students place the compass at point A and draw an arc intersecting the sides of the angle resulting in the points of intersection P and Q. Students should realize that AP ≅ AQ. Without changing the compass setting, add two arcs intersecting in the interior of the angle at the point G. Students should realize that PG ≅ QG By the Reflexive property of congruence, AG ≅ AG Therefore, ΔAPG ≅ ΔAQG by SSS and since corresponding parts of congruent triangles are congruent (CPCTC), ∠PAG ≅ ∠QAG and AG is the angle bisector of ∠BAC.
- Instruction includes the connection to the converse of the Perpendicular Bisector
Theorem when constructing a perpendicular bisector.
- For example, students can set the compass width more than half the length of AB. Students can draw arcs intersecting above and below the segment at points E and F. Therefore, AE ≅ BE and AF ≅ BF. That is, points E and F are each the same distance to the endpoints of AB and that means they lie on the Perpendicular Bisector.
- Instruction includes the student understanding that when one has constructed the
perpendicular bisector, they have also constructed the midpoint of a segment. (MTR.2.1)
- For example, using the same steps as in the last construction, the midpoint of the segment can be identified as the point where the perpendicular bisector meets the segment.
- Instruction includes the connection to logical reasoning and visual proofs when verifying
that a construction works.
- For example, once the construction of the perpendicular bisector is completed, discuss with students how this construction and a compass can be used to experimentally check the Perpendicular Bisector Theorem. (MA.912.GR.1.1)
- For expectations of this benchmark, constructions should be reasonably accurate and the emphasis is to make connections between the construction steps and the definitions, properties and theorems supporting them.
- While going over the steps of geometric constructions, ensure that students develop vocabulary to describe the steps precisely. (MTR.4.1)
- Instruction includes the connection between constructions and properties of
quadrilaterals, including rhombi.
- For example, when constructing the angle bisector, if the compass width is not changed throughout the process, then quadrilateral AHGF is a rhombus since it has 4 equal sides ( AH HG, GF FA). The diagonals of a rhombus bisect opposite angles. Therefore, ∠HAF is bisected by the diagonal of the rhombus AG and AG is the angle bisector of ∠BAC. Similarly, when constructing the perpendicular bisector, it can be seen that the diagonals of a rhombus are perpendicular.
- Instruction includes the student understanding that in a geometric construction, one does not use the markings on a ruler or on a protractor to bisect a segment or angle. Students should realize that there are limitations on precision that are inherent in the markings on rulers or protractors.
- It is important to build the understanding that formal constructions are valid when the lengths of segments or measures of angles are not known, or have values that do not appear on a ruler or protractor, including irrational values.
- Problem types include identifying the next step of a construction, a missing step in a construction or the order of the steps in a construction.
Common Misconceptions or Errors
- Students may not make the connection that any point on the perpendicular bisector is equidistant from the endpoints of the segment, not just the midpoint of the segment.
- Students may not understand why they are not using rulers and protractors to bisect segments and angles.
Instructional Tasks
Instructional Task 1 (MTR.7.1)- A map of some popular universities is shown below.
- Part A. Prove that Georgia Tech is approximately equidistant from Clemson University and Auburn University.
- Part B. Find one or more universities that are approximately equidistant from Florida State University and Oklahoma State University?
Instructional Items
Instructional Item 1- An image is provided below.
- Part A. Construct the bisector of angle D.
- Part B. Construct the midpoint of segment DB.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Teaching Ideas
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to construct the bisectors of a given segment and a given angle and to justify one of the steps in each construction.
Students are asked to construct a line perpendicular to given line (1) through a point not on the line and (2) through a point on the line.
Original Student Tutorials Mathematics - Grades 9-12
Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.
NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.
Plan a paddle board expedition by learning how to do basic geometric constructions including copying a segment, constructing a segment bisector, constructing a segment's perpendicular bisector and constructing perpendicular segments using a variety of tools in this interactive tutorial.
Learn to construct the perpendicular bisector of a line segment using a straightedge and compass with this interactive tutorial.
Student Resources
Original Student Tutorials
Explore the construction processes for constructing an angle bisector, copying an angle and constructing a line parallel to a given line through a point not on the line using a variety of tools in this interactive, retro video game-themed tutorial.
NOTE: This tutorial uses both the angle bisector construction and the construction to copy an angle as an extension opportunity to also construct a line parallel to a given line through a point not on the line. Students also learn to identify corresponding angles created when a transversal crosses parallel lines, and discover using Geogebra that these angles are congruent.
Type: Original Student Tutorial
Plan a paddle board expedition by learning how to do basic geometric constructions including copying a segment, constructing a segment bisector, constructing a segment's perpendicular bisector and constructing perpendicular segments using a variety of tools in this interactive tutorial.
Type: Original Student Tutorial
Learn to construct the perpendicular bisector of a line segment using a straightedge and compass with this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
This problem solving task challenges students to construct a perpendicular bisector of a given segment.
Type: Problem-Solving Task
This problem solving task challenges students to bisect a given angle.
Type: Problem-Solving Task
This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).
Type: Problem-Solving Task
This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
This problem solving task challenges students to construct a perpendicular bisector of a given segment.
Type: Problem-Solving Task
This problem solving task challenges students to bisect a given angle.
Type: Problem-Solving Task
This problem solving task challenges students to place a warehouse (point) an equal distance from three roads (lines).
Type: Problem-Solving Task
This task provides a construction of the angle bisector of an angle by reducing it to the bisection of an angle to finding the midpoint of a line segment. It is worth observing the symmetry -- for both finding midpoints and bisecting angles, the goal is to cut an object into two equal parts.
Type: Problem-Solving Task
