### Clarifications

*Clarification 1*: Instruction includes using compass and straightedge, string, reflective devices, paper folding or dynamic geometric software.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**912

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- 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, 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, Δ*APG*≅ Δ*AQG*by SSS and since corresponding parts of congruent triangles are congruent (CPCTC), ∠*PAG ≅ ∠QAG*and*AG*is the angle bisector of ∠*BAC*.

- 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

- 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.

- 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

- 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.

- For example, when constructing the angle bisector, if the compass width is not
changed throughout the process, then quadrilateral
- 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*.

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

## 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