session 171 under construction: a different perspectiveunder construction: a different perspective...
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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.
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:
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
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.)
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
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