Download Transforming Quadrilaterals Philadelphia 2012

Transforming Quadrilaterals and Their Changing Diagonals
NCTM Annual Meeting and Exposition
Philadelphia, Pennsylvania
April 25 – 28, 2012
Charlene Keen, Dauphin County Technical School, Harrisburg, PA
Please email for an electronic handout, please email [email protected].
Thanks to Bill Hornung, Hornung’s Hardware, Linglestown, PA, for his help with
materials and ideas. Dauphin County Technical School always gives me support.
Materials:
8 dowels
8 tubes
4 elastic grommet tabs
4 yarn strands,
2 long and 2 short
2 colors for each model
Towel with soap
Two magnet clips on each top dowel will hold the models to a magnetic board
and make them easier to use.
When the yarn strand is pulled, the quadrilateral will change from rectangle to
parallelogram OR rhombus to square.
As the shifts take place the diagonals will always bisect each other.
When the quadrilateral angles are right angles, the diagonals are congruent.
You can also discuss what the diagonals do to the quadrilateral angles.
I bought the tubing from McMaster-Carr #5234K72 Super soft latex rubber tubing 3/16”ID, 5/16” OD, 1/16” Wall
Each kit needs about 7” cut into 8 pieces, each ¾” long. I did make tubes from double knit material cut on bias, hot glued.
Two 48” – ¼” dowels for each kit. Each cut 9”, 9”, 11”, and congruent remainder.
I made the tabs from leather grommet tape and elastic. I cut the grommet tape halfway between the grommets. I folded the
elastic over both sides of the tape and sewed it in place by hand.
Directions
Parallelogram/ Rectangle
Choose 2 pairs of dowels that are different lengths.
Slip tabs over both ends of a short dowel.
Wipe the end of each dowel with a small amount of soap.
Use tubes to connect the dowels to make a parallelogram.
Dowels should just touch inside the tubing when it bends.
Slip the tabs over the tubes.
Center each grommet at the vertex.
Lay the parallelogram so that the tabs are closest to you.
Use the two longer lengths of yarn. Use two different colors.
Tie a length of yarn (different colors) over tubing at each top corner.
Make a knot in such a way that you do not have a lot of wasted yarn.
To form diagonals, slip the yarn through the grommet in the opposite
corner.
Hold the free ends together and tie at the bottom.
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Tie
Tie
Knot
Rhombus/Square
Use the 4 congruent dowel lengths.
Use the two shorter yarn lengths.
Follow the previous directions. All four sides are the same length.
Illuminations http://illuminations.nctm.org Lessons Geometry Key Word: Diagonals to Quadrilaterals
Thinkfinity
Commercial vendor: GeoLegs and AngLegs
Vocational applications www.pde.state.pa.us Go: Math Council BP Math Council Carpentry; HVA
Directions to Construct Quadrilaterals using Diagonals and SketchPad
Rectangle: The diagonals of a rectangle bisect each other and are the same length.
Construct the
midpoint of a
segment.
Construct a circle whose
center is the midpoint. The
circle contains the segment
endpoints.
Construct a point on the circle.
Construct a LINE through the
point and the center.
Construct segments through the four
consecutive intersection points on the
circle. This is the rectangle.
This diameter will be the same
length as the first segment.
Translate by moving either endpoint or
the circle point. Show that a square is a
special rectangle.
Square: The diagonals of a square are perpendicular, congruent and bisect each other.
Construct the
midpoint of a
segment.
Construct the perpendicular
through the midpoint.
Construct a circle whose center is
the midpoint. The segment
endpoints are on the circle.
Construct segments through the four
consecutive intersection points on
the circle.
Rotate using
either
endpoint to
obtain
different
orientations.
Rhombus: The diagonals are perpendicular and bisect each other; may be different lengths.
Construct the
midpoint of a
segment.
Construct the
perpendicular bisector.
Construct a circle such that the
center is the midpoint AND
the segment endpoints are
NOT on the circle.
Connect consecutive-- Endpoint, Intersection
point, Endpoint, Intersection point--to form a
rhombus.
Translate by moving
either endpoint.
Move the indicated
circle point to the
segment endpoint
(making congruent
diagonals) to show
that a square is a
special case of the
rhombus.
This is the first
diagonal.
Thus the two diagonals will
not be congruent.
Parallelogram: The diagonals bisect each other. They may be different lengths.
Construct the
midpoint of a
segment.
This is the first
diagonal.
Construct a circle such
that the center is the
midpoint AND the
segment endpoints are
NOT on the circle
Construct a point on the
circle. Construct the LINE
through the circle point and
the center.
Connect consecutive--Endpoint, Intersection
point, Endpoint, Intersection point--to form a
parallelogram.
Translate by moving
the endpoints and the
indicated circle point
to change the
parallogram
into a rectangle,
rhombus, and square.
Isosceles Trapezoid: The diagonals are the same length and the intersection point makes congruent
segments above and below.
Construct a ray.
There are two points
shown—the
endpoint and a “ray
point”.
Construct a circle
with the ray point
as the center. The
circle contains the
endpoint.
Construct a point on the
circle. Construct a line
through the point and the
center.
Highlight the RAY and
construct a point.
Construct a second circle
with the same center but a
larger diameter. This new
larger circle contains the new
point on the ray.
To form the isosceles
trapezoid, connect
consecutive
intersections of the two
small circles and the
two large circles.
The diagonals are congruent
and the center point creates a
1:1 ratio.
The best translations use either point on the small circle and the ray-large circle intersection.
Rectangles and squares are formed when the first circle overlaps the second.
Trapezoid: In general the diagonals are different lengths and the intersection point makes proportional
sections. I am using a 1 : 2 ratio. Recently I was able to simplify this just using circles and the lines.
Construct a ray. There are two points
shown—the endpoint and a “ray point”.
Make a new initial
point off the ray
but close to it.
Construct a
line
through the
initial point
and the ray
point.
Hide the
initial point
Construct a circle using the “ray point” as a center; the
circle contains the endpoint of the ray. Click where the
circle intersects the LINE.
This intersection point will be the center of a new 2nd circle.
Construct the second circle.
Use the “line-circle
intersection” point as the
center. The second circle
contains the “ray point”
(1st circle center).
Click where the 2nd circle
intersects the line to form an
important point.
For a smaller circle, carefully move the ray point closer to
the ray’s endpoint or start again
Highlight the LINE and construct a
point on the lower line somewhat
close to the ray point.
Construct a circle on the lower LINE
using the “ray point” as the center; the
circle contains the new “line point”.
Connect the proper points—
Ray endpoint, 2nd top point on
line, 1st circle- lower ray
intersection, 2nd circle-line
intersection. Be careful.
The circles have the
same radii. You have
marked off two
congruent lengths on the
upper LINE, compared
to one length on the
upper RAY.
Hide the two circles.
Check parallelism by copy and
pasting the right base and dragging
it over the left base OR check that
same-side interior angles are
supplementary
Make a second circle-- the center is
the new “line point”; the second circle
contains the “ray point”.
Again there is a 2:1 ratio on the LINE compared to the
RAY. The trapezoid diagonals have 2:1 ratio.
The best translations occur by moving the ray endpoint and the first lower line point.
When the ray endpoint is moved to the first large circle you have a parallelogram. To form a rhombus, move the ray
endpoint around, but on, the circle, until the line and ray are perpendicular.
Or onsider displaying the two upper circles with a thin rather than hiding them. Move the endpoint or the first lower
line point so that first large circle overlaps first small circle to form a 1 : 1 ratio that again produces the same figures.
Move the first lower line point to the “ray point” to make a triangle -- a special trapezoid with top base length of zero.
AT THE CONFERENCE I MADE BOTH TRAPEZOIDS USING INTERSECTING LINES.
ISOSCELES TRAPEZOIDS: FROM THE INTERSECTION I MADE A SMALL CIRCLE AND A LARGE
CIRCLE. THE ISO. TRAP. WAS MADE BY CONNECTING THE INTERSECTIONS THAT THE
CIRCLES MADE WITH THE LINES. SMALL ABOVE, SMALL ABOVE, LARGE BELOW, LARGE
BELOW.
REGULAR TRAPEZOID: FROM LINE INTERSECTION I MADE ONE LARGE AND ONE SMALL
CIRCLE. WHERE THE CIRCLES INTERSECTED THE SECOND LINE I MADE AN ADDITIONAL
PAIR OF CIRCLES, LARGE AND SMALL THAT WERE CONGRUENT TO THE FIRST PAIR.
CONNECT OUTERMOST INTERSECTION POINTS.
IF THESE ARE CONFUSING PLEASE EMAIL ME AT [email protected]. Thanks
This has thrown off my spacing. There is more below, including the question pages. You could want to
separate the page into two as it represents two topics.
Vocational Applications
Masonry, Carpentry, Building Construction:
Lay out a Foundation
Determine if Walls are Square
Ornamental Horticulture:
Lay out a garden
Carpentry, Ornamental Horticulture:
Determine the Center of a Circle
www.pde.state.pa.us
Magazine that Lowe’s distributes.
Auto Body:
Bumper Alignment
Auto Mechanics:
Tire Alignment
Blue Prints, Drafting: Marking the center of a ceiling
Go: Math Council
BP Math Council Carpentry
CHRYSLER 1984 LEBARON
UNDERHOOD VIEW
All Underhood View Dimensions are Point-to-Point.
Holes are Measured to the Closest Edge.
Bolts are Measured to Center.
Draw diagonals with a straight edge.
Quadrilateral
Questions
This quadrilateral is a ___________________
Use patty paper to compare
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides Same Different
Are the opposite sides parallel? Yes No
Questions
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
This quadrilateral is a ___________________
Use patty paper to compare:
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides Same Different
Are the opposite sides parallel? Yes No
List another observation:
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
This quadrilateral is a ____________________
Use patty paper to compare:
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides
Same Different
Are the opposite sides parallel? Yes No
List another observation:
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
This quadrilateral is a ____________________
Use patty paper to compare:
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides
Same Different
Are the opposite sides parallel? Yes No
List other observations:
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
This quadrilateral is a ____________________
Use patty paper to compare:
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides
Same Different
Are the opposite sides parallel? Yes No
List other observations:
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
This quadrilateral is a ____________________
Use patty paper to compare:
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides
Same Different
Are the opposite sides parallel? Yes No
List other observations:
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
This quadrilateral is a ____________________
Use patty paper to compare:
Sizes of the Opposite angle Same Different
Lengths of Opposite Sides
Same Different
Are the opposite sides parallel? Yes No
List other observations:
Diagonals are the same length Yes No
Diagonals bisect each other
Yes No
Diagonals are perpendicular
Yes No
Diagonals bisect Quadrilateral angles Y / N
Other Observations?
Use the Quadrilateral kits to further confirm
you answers.
Use the Quadrilateral kits to further confirm you answers.
Diagonals of Quadrilaterals Lab:
Materials: Rope: 2 congruent lengths and one different—mark the midpoint of each,
Long piece of rope, Tape, Direction/Answer sheet
Directions: Students are in groups of 2 or 3.
Students lay out the diagonals as described on paper. They tape the ends to the floor. Then they connect the
endpoints with the long rope. They write the name of the shape on the handout, by the proper description.
DIAGONALS
NAME THE SHAPE
Same Length
Bisect each other
Perpendicular
Same Length
Bisect each other
Not Perpendicular
Different Length
Bisect each other
Perpendicular
Different Length
Bisect each other
Not Perpendicular
Same Length
Top part of each
diagonal is same
length; they don't
bisect
Does not matter
Different lengths
(could be same)
Only one is bisected
Perpendicular
B
C
Illuminations
Diagonals and Quadrilaterals
Illuminations, from NCTM, has a
worksheet similar to this.
There is an interactive applet.
A
D
Diagonals congruent. Top small portion
of each diagonal congruent.