Under Construction:

A Different Perspective

Steve Roeder

El Camino High School

Oceanside, CA

stephen.roeder@oside.us

EVALUATION POLL CODE: 3109

PSCC-Mesquite D

Session 171

Under Construction: A Different Perspective

Our objective:

1. Use the properties of quadrilaterals as the basis for all types of geometric constructions. Approach constructions from a different perspective.

2. Gain ideas to promote conceptual understanding rather than memorizing steps.

For more information, visit:

Mathematics Vision Project

www.mathematicsvisitproject.org

Secondary Mathematics One: Integrated Pathway

Authors: Scott Hendrickson, Joleigh Honey, Barbara Kuehl,

Travis Lemon, Janet Sutorius

Past/current practices of

instruction for Constructions

look something like this…

How constructions have been taught:

Constructing Angle Bisectors

Given: ABC

Construct an angle bisector.

1. Using B as center, choose any radius and draw an arc

intersecting BA and BC. Label X & Y.

2. Using X as center, choose a radius greater than ½ XY.

Draw an arc in the interior of ABC.

3. Repeat using Y as center & the same radius. Label the

point of intersection Z.

4. Draw BZ. Then BZ bisects ABC.

B

A

C

Bisect the given angle.

B

A

C

1. 2. L

B

K

Constructing Congruent Angles

Given: XYZ

Construct an angle congruent to XYZ.

1. Draw a ray & label it YZ.

2. Using Y as center, choose any radius. Draw an arc

that intersects YX & YZ. Label points S & T.

3. Using Y as center & the same radius, draw an arc

intersecting YZ. Label the point of intersection Q.

4. Using T as center, find radius equal to TS. Draw

arc through point S.

5. Using Q as the center, draw arc using radius equal

to TS. Label point of intersection P.

6. Draw YP. Then . ZYPXYZ

Y

X

Z

Construct a congruent angle to XYZ.

Y

X

Z

Constructing Parallel Lines.

Given: Point X and line l. Construct the line parallel to l containing X.

1. Place point A anywhere on line l . Draw AX.

2. At point X, construct 1 so that it is

congruent to XAB. Let m be the line drawn

for 1.

3. Then m is parallel to l.

X

l

1. Construct a line parallel to AB through X.

A B

2. Construct a line parallel to line g through X.

X

X

g

Why did we teach this way? • We spent a few days before the California

Standards Test going over constructions, and

we hoped our students would guess the correct

answers based on recognition.

Do you remember these?

Do you remember these?

Why did we teach this way?

• If our students had great scores, then we

looked good, and we avoided the wrath from

our principal and No Child Left Behind.

• 4 out of 65 questions on the Geometry CST

were about constructions. 6% of the test!

Common Core: A Fresh Start

Now that Common Core rolled on through, and our

incoming freshmen of the 2014-15 were not going

to take SBAC until their junior year, then we had a

chance to breathe and be creative.

We adopted integrated curriculum from

Mathematics Vision Project, and we found a new

perspective how constructions can be a tool to

reinforce understanding of triangles and

quadrilaterals.

Integrated 1/Geometry CC Standards

Make geometric constructions. [Formalize and explain processes.]

12. Make formal geometric constructions with a variety of tools

and methods (compass and straightedge, string, reflective

devices, paper folding, dynamic geometric software, etc.).

Copying a segment; copying an angle; bisecting a segment;

bisecting an angle; constructing perpendicular lines, including

the perpendicular bisector of a line segment; and constructing

a line parallel to a given line through a point not on the line.

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Under Construction: A Different Perspective • Depending on the flow of older curricula, constructions were

dispersed throughout textbooks (but ignored) or thrown at

the end in its own chapter.

• Many books teach constructions procedurally with no

justifications.

• But, some textbooks may justify steps using congruent

triangles.

• Today, we will understand constructions from a

perspective of the properties of quadrilaterals.

• If you are familiar with constructions, I challenge you today,

as we go through these activities, to try to deprogram the

procedural methods you know and reflect on why steps

naturally follow one after the other.

Building a Foundation From Quadrilaterals

• Let’s start with 4 specific quadrilaterals:

Parallelogram

Rhombus

Square

Rectangle

Definitions & Symmetries: Parallelogram: quadrilateral with 2 pairs of parallel sides

Rhombus: quadrilaterals with 4 congruent sides

Rectangle: quadrilaterals with 4 right angles

Square: quadrilaterals with 4 congruent sides and 4 right angles

• 180º rotational symmetry at the intersection

of the diagonals

• 180º rotational symmetry at the intersection

of the diagonals

• Diagonals are the lines of symmetry

• 180º rotational symmetry at the intersection of the

diagonals

• 2 lines of symmetry (thru midpoints of opposite sides)

• 90º, 180º, 270º rotational symmetry at the intersection

of the diagonals

• 4 lines of symmetry (both diagonals and 2 lines thru

midpoints of opposite sides)

Your first task:

On the handout, brainstorm and

record the properties (additional

features) that describe each of the 4

quadrilaterals.

Definitions, Symmetries, & Properties:

Parallelogram: quadrilateral with 2 pairs of parallel sides • 180º rotational symmetry at the intersection of the

diagonals

Opposite sides are congruent

Opposite angles are congruent

Diagonals bisect each other

Rhombus: quadrilaterals with 4 congruent sides

• 180º rotational symmetry at the intersection

of the diagonals

• Diagonals are the lines of symmetry

Opposite sides are parallel

Opposite sides are congruent

Opposite angles are congruent

Diagonals bisect each other

Diagonals are perpendicular

Diagonals bisect interior

angles

Opposite sides are parallel

Opposite sides are congruent

Opposite angles are congruent

Diagonals bisect each other

Opposite sides are parallel

Diagonals bisect each other

Diagonals are congruent

Diagonals are perpendicular

Diagonals bisect interior angles

Rectangle: quadrilaterals with 4 right angles

Square: quadrilaterals with 4 congruent sides and 4 right angles

• 180º rotational symmetry at the intersection of the

diagonals

• 2 lines of symmetry (thru midpoints of opposite sides)

• 90º, 180º, 270º rotational symmetry at the intersection

of the diagonals

• 4 lines of symmetry (both diagonals and 2 lines thru

midpoints of opposite sides)

Definitions, Symmetries, & Properties:

Parallelogram “Family” Venn Diagram

Parallelograms

Rectangles

Coming into high school math, students will know circles:

One last thing… Circles

Circle: Set of all points in a plane that the

are equidistant from a fixed point.

Radius: Distance from the center to a

point on the circle.

Diameter: segment that goes through the center

with both endpoints on the circle.

arc: part of the circumference of a circle

radius

All radii are congruent in a circle

Basic Constructions:

1. Use a compass to draw a circle with the given radius. Use

the labeled point as the center.

A

2. Construct line segment congruent to 𝐹𝐺 using only a straight

edge and a compass.

F G

3. Using your compass, draw at least three concentric circles

that have point A as the center and then draw the same sized

circles that have point B as the center.

A B

What do you notice about where all the circles with center A

intersect all the corresponding circles with center B?

Construct a Rhombus A

B

1. Knowing what you know about circles and line segments, how

might you locate point C on the ray in the diagram above so the

distance from B to C is the same as the distance from B to A.

Describe how you will locate point C and how you know

𝐵𝐶 ≅ 𝐵𝐴 , then construct point C on the diagram above.

2. Now that we have three of the four vertices of the rhombus, we

need to locate point D, the fourth vertex. Describe how you will

locate point D and how you know 𝐶𝐷 ≅ 𝐷𝐴 ≅ 𝐴𝐵, then

construct point D on the diagram above.

Construct a Rhombus… try again!

Construct a rhombus on the segment 𝐴𝐵 that is given below

and that has point A as a vertex. Be sure to check that your final

figure is a rhombus.

A

B

• All sides are congruent (definition).

As you complete the next several constructions, think about how

the properties of a rhombus help us make these constructions.

• Opposite sides are parallel.

• Opposite angles are congruent.

• The diagonals bisect each other.

• The diagonals are perpendicular to each other.

• The diagonals bisect the interior angles of the rhombus.

Properties of a Rhombus

2. Construct a rhombus using segment 𝐴𝐵 and the angle given.

A

B

Bisect an Angle

Bisect each angle with a compass and straight edge.

D

F

E

C

A

B

Copy an Angle

Copy each angle below with a compass and straight edge.

D

F

E

C

A

B

Construct Perpendicular Bisector

A B

Construct the perpendicular bisector of 𝐴𝐵.

Justifications

Let’s go back and justify the constructions we have completed.

On the next handout you are receiving, the steps for each

construction were developed by my students on their own, in

groups, and during whole-group class discussions.

Let’s justify why we completed these steps.

Construct a Square The only difference between constructing a rhombus and constructing a square is that a square is a rhombus with right angles. Therefore, we need to construct perpendicular lines using only a compass and straightedge.

S R

1. Construct a perpendicular line to RS. The right angle formed with help us construct a square.

2. Label the midpoint of 𝑅𝑆 on the diagram above as point M. Using segment 𝑅𝑀 as one side of the square, and the right angle formed by segment 𝑅𝑀 and the perpendicular line drawn through point M as the beginning of a square. Finish constructing this square on the diagram above. (Hint: Remember that a square is also a rhombus.)

Construct a perpendicular to a line through a

given point

P

Construct a line parallel to a given line

through a given point

Q

Construct Parallelogram

J

K

Construct a parallelogram with sides 𝐽𝐾 and 𝐾𝐿 and the given angle K.

L

Construct Rectangle

O

R

Construct a rectangle using side lengths 𝑂𝑃 and 𝑅𝑆.

S

P

Construct an Equilateral Triangle

Construct an equilateral with side length 𝐺𝐻.

H

G

Construct Hexagon Inscribed in a Circle

Center of Rotation of an Equilateral Triangle

Given the equilateral triangle below, find the center of rotation of the triangle using compass and straight edge.

A

B

C

Neon Brown

Circle Master Compass

http://www.eaieducation.com/Product/531646/Circle_Master_Compass_Neon.aspx

$2.65 each

Circle Perfect Compass $2.19 each

$62.95 for set of 30