# MA.912.GR.5.1

Construct a copy of a segment or an angle.

### Clarifications

Clarification 1: Instruction includes using compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.
General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Angle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In elementary grades, students drew lines and angles using a variety of tools, including rulers and protractors. In Geometry, students are introduced to constructions for the first time, specifically copying a segment or an angle. These two procedures are embedded in other basic constructions, and the concept of constructing and identifying copies of segments and angles is closely connected to visualizing and understanding congruence.
• 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, students can use tracing/folding paper (e.g., patty paper) to trace the copy of an angle, or the copy of a segment, and verify that the angle and its copy are congruent, or that the segment and its copy are congruent. Additionally, using several folds, it is possible to verify the congruency of two angles or two segments drawn on the same piece of paper.
• Instruction includes the connection to logical reasoning and visual proofs when verifying that a construction works.
• Instruction includes discussing the role of the compass in a geometric construction, beyond drawing circles, and how a string can replace a compass. Most of the time in this course, compasses will be used to draw arcs. Discuss how no matter the point chosen on the arc, the distance to the given point is the same.
• For example, students can place the compass at P and draw an arc. Choosing two points on the arc, A and B, the distance to P is the same, AP = BP and AP ≅ BP are radii of the circle containing the drawn arc centered at P
• Instruction includes the student understanding that in a geometric construction, one does not use the markings on a ruler or on a protractor to copy 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.
• 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
• 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 understand that the size of the angle, “the opening,” is what is being measured when copying an angle.
• Students may not understand why they are not using the marking on rulers and protractors to copy segments and angles.

• Create a construction of quadrilateral JKLM so that it is congruent to quadrilateral ABCD.

• Given angle EFG below, create a copy so that it creates parallelogram EFGH.

### Instructional Items

Instructional Item 1
• Construct the necessary segments and angles to construct quadrilateral EFGH so that it is congruent to quadrilateral EFGH. Assume ∠DAB ≅ ∠EFG,  DA ≅ EF and AB ≅ FG.

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

## Related Courses

This benchmark is part of these courses.
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206310: Geometry (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206315: Geometry for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7912065: Access Geometry (Specifically in versions: 2015 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.912.GR.5.AP.1: Construct a copy of a segment.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Constructing a Congruent Angle:

Students are asked to construct an angle congruent to a given angle.

Type: Formative Assessment

Constructing a Congruent Segment:

Students are asked to construct a line segment congruent to a given line segment.

Type: Formative Assessment

## Lesson Plans

Geometric Construction Site:

This lesson takes students from simple construction of line segments and angles to an optional extension worksheet for creating triangles.

Type: Lesson Plan

Construction of Inscribed Regular Hexagon:

A GeoGebra lesson for students to become familiar with computer based construction tools. Students work together to construct a regular hexagon inscribed in a circle using rotations. Directions for both a beginner and advanced approach are provided.

Type: Lesson Plan

Copying and Bisecting an Angle:

This lesson is a gradual release model for constructing congruent angles and bisecting angles.

Type: Lesson Plan

Inscribe Those Rims:

This lesson will engage students with an interactive and interesting way to learn how to inscribe polygons in circles.

Type: Lesson Plan

Construction Junction:

Students will learn how to construct an equilateral triangle and a regular hexagon inscribed in a circle using a compass and a straightedge.

Type: Lesson Plan

Inscribe it:

This activity allows students to practice the construction process inscribing a regular hexagon and an equilateral triangle in a circle using GeoGebra software.

Type: Lesson Plan

Construct Regular Polygons Inside Circles:

Students will be able to demonstrate that they can construct, using the central angle method, an equilateral triangle, a square, and a regular hexagon, inscribed inside a circle, using a compass, straightedge, and protractor. They will use worksheets to master the construction of each polygon, one inside each of three different circles. As an extension to this lesson, if computers with GeoGebra are available, the students should be able to perform these constructions on computers as well.

Type: Lesson Plan

Construct This:

In this lesson, students will construct a square inscribed in a circle using the properties of a square and determine if there is more than one way to complete the construction.

Type: Lesson Plan

I Am Still Me Transformed.:

Students explore ways of applying, identifying, and describing reflection and rotation symmetry for both geometric and real-world objects, for them to develop a better understanding of symmetries in transformational geometry.

Type: Lesson Plan

Constructing an Angle Bisector:

Students construct an angle bisector given a straightedge and compass then verify their process. The Guided Practice is done in stations. One that is teacher-led and one that is student-led. In order to complete the student-led Guided Practice, access to a teacher computer and projector is needed. Then the students independently create their own angle and its bisector and verify their work for a grade. Students use patty paper and protractors to confirm the accuracy of the construction.

Type: Lesson Plan

Back to the Basics: Constructions:

Students will use a compass and straightedge to develop methods for constructions. GeoGebra directions are also provided.

Type: Lesson Plan

Sage and Scribe - Points, Lines, and Planes:

Students will practice using precise definitions while they draw images of Points, Lines, and Planes. Students will work in pairs taking turns describing an image while their partner attempts to accurately draw the image.

Type: Lesson Plan

Fundamental Property of Reflections:

This lesson helps students discover that in a reflection, the line of reflection is the perpendicular bisector of any segment connecting any pre-image point with its reflected image.

Type: Lesson Plan

Crafty Circumference Challenge:

Students learn about geometric construction tools and how to use them. Students will partition the circumference of a circle into three, four, and six congruent arcs which determine the vertices of regular polygons inscribed in the circle. An optional project is included where students identify, find, and use recycled, repurposed, or reclaimed objects to create "crafty" construction tools.

Type: Lesson Plan

## Original Student Tutorials

Angle UP: Player 1:

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

Meet Me Half Way:

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

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.

## MFAS Formative Assessments

Constructing a Congruent Angle:

Students are asked to construct an angle congruent to a given angle.

Constructing a Congruent Segment:

Students are asked to construct a line segment congruent to a given line segment.

## Original Student Tutorials Mathematics - Grades 9-12

Angle UP: Player 1:

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.

Meet Me Half Way:

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.

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Angle UP: Player 1:

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

Meet Me Half Way:

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

Reflected Triangles:

This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.