Geometry Lesson

Compass & Straightedge: Classical Geometric Constructions

Shared by A STEM Educator

High SchoolUniversal Design for LearningCCSS.MATH.CONTENT.HSG.CO.D.12, CCSS.MATH.CONTENT.HSG.CO.D.1350 min

Learning Objective

Students will construct perpendicular bisectors, angle bisectors, parallel lines through a point, and inscribed/circumscribed circles of regular polygons using compass and straightedge, and justify each construction using properties of congruent triangles and circles. Multiple modalities (physical tools, digital tools, verbal explanation, visual annotation) are accepted as valid demonstrations of mastery.

Lesson Overview

Students use compass and straightedge to perform classical constructions — perpendicular bisectors, angle bisectors, parallel lines through a point, and inscribed/circumscribed circles of regular polygons — building both procedural fluency and conceptual understanding of why each construction works.

Materials

Compass and straightedge (one set per student)
Blank white paper and graph paper
Colored pencils or fine-tip markers (for annotating arc intersections)
GeoGebra (browser or app) — digital alternative for compass/straightedge
Printed reference card: 'Construction Steps at a Glance' (visual + text)
Patty paper (wax paper) for folding-based exploration of bisectors
Rulers (for straightedge-only tasks; NOT for measuring in constructions)
Protractors (for verification only, not construction)
Anchor chart: 'Why Arcs Work — Circle Congruence Properties'
Sentence frames for ELL students: 'I set the compass to ___. I place the point at ___. The arc intersects at ___.'
Tablet/laptop with GeoGebra for students needing fine-motor accommodations

Scaffolded Task Progression

1Task 1

Constructing a Perpendicular Bisector — Guided Entry

Below is segment

\overline{A B}

. Your goal: find its midpoint

M

and draw the line perpendicular to

\overline{A B}

through

M

— using only compass and straightedge.

[ Interactive graph / diagram, available in the full lesson ]

Scaffolded Steps (follow in order):
1. Open your compass to MORE than half of

A B

. (It must be more than half — why?)
2. Place the compass point on

A

. Draw an arc above AND below

\overline{A B}

.
3. WITHOUT changing the compass width, place the point on

B

. Draw arcs above and below that cross the first arcs.
4. Label the two intersection points

P

(above) and

Q

(below).
5. Use your straightedge to draw

P Q

. Label the point where

P Q

crosses

\overline{A B}

M

.

Check: Measure

A M

and

M B

with a ruler. Are they equal? Measure the angle at

M

with a protractor — is it

90°

?

Patty-paper alternative: Fold the paper so that

A

lands exactly on

B

. The crease IS the perpendicular bisector. Unfold and compare to your compass construction.

Reflection prompt (choose one): *Why must the compass width be more than half of

A B

?* OR *How do you know

P

and

Q

are both equidistant from

A

and

B

2Task 2

Constructing an Angle Bisector — Guided

Below is

∠ B A C

. Construct the ray from

A

that divides

∠ B A C

into two equal angles.

[ Interactive graph / diagram, available in the full lesson ]

Scaffolded Steps:
1. Place compass point at vertex

A

. Draw an arc that crosses BOTH rays

A B

and

A C

. Label the crossings

D

(on

A B

) and

E

(on

A C

).
2. Place compass point on

D

. Draw an arc in the interior of the angle.
3. Keep the SAME compass width. Place point on

E

. Draw an arc crossing the one from Step 2. Label the intersection

F

.
4. Draw ray

A F

. This is the angle bisector.

Check: Use a protractor to measure

∠ B A F

and

∠ F A C

. Are they each

32°

?

Patty-paper alternative: Fold so ray

A B

lies on top of ray

A C

. The crease from

A

is the bisector.

Why it works (fill in):

A D = A E

because ___.

D F = E F

because ___. So

△ A D F ≅ △ A E F

by ___, which means

∠ D A F = ∠ E A F

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