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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 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.
Instructional Tasks
Instructional Task 1 (MTR.2.1, MTR.3.1)- Create a construction of quadrilateral JKLM so that it is congruent to quadrilateral ABCD.
Instructional Task 2 (MTR.2.1, MTR.5.1)
- 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.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorials
Perspectives Video: Teaching Ideas
Problem-Solving Task
MFAS Formative Assessments
Students are asked to construct an angle congruent to a given angle.
Students are asked to construct a line segment congruent to a given line segment.
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.
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
Perspectives Video: Teaching Idea
Students will learn to use basic geometric construction tools to copy a segment and an angle, then apply these skills repeatedly to construct a rhombus. The definition of a parallelogram is verified.
Type: Perspectives Video: Teaching Idea
Problem-Solving Task
This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Task
This task asks students to use a straightedge and compass to construct the line across which a triangle is reflected.
Type: Problem-Solving Task
