Download this PDF file - Illinois Mathematics Teacher

Quadrilaterals: Transformations for Developing Student Thinking
Colleen Eddy – University of North Texas, [email protected]
Kevin Hughes – University of North Texas, [email protected]
Vincent Kieftenbeld – Southern Illinois University Edwardsville, [email protected]
Carole Hayata – University of North Texas, [email protected]
Students begin learning about
transformations in the early elementary
grades. They might create simple
tessellations by cutting and pasting
cardboard pieces, or investigate symmetry
by folding figures on patty paper (Common
Core State Standards [CCSS] 4.G.3). Their
knowledge and ease with transformations
continues to grow through middle school,
where the everyday language of turn, flip,
slide, bigger and smaller is connected with
the corresponding mathematical terminology
of rotation, reflection, translation, and
dilation (National Council of Teachers of
Mathematics [NCTM] 2000). By the end of
middle school, students should be able to
describe the effect of transformations on
two-dimensional figures (CCSS 8.G.3).
In high school, however, students
often study transformations in isolation,
establishing few connections with other
parts of geometry. This disconnects the
informal thinking of elementary and middle
school, and the deductive reasoning required
in a high school geometry course. Coxford
and Usiskin (1971, 1975) originally
proposed the use of transformations to build
conceptual understanding based on prior
experiences
before
deriving
proofs.
Transformations also form the basis of
geometric understanding in the vision of the
Common Core. For instance, for high school
geometry “[t]he concepts of congruence,
similarity, and symmetry can be understood
from the perspective of geometric
transformation”
(National
Governors
Association for Best Practices & Council of
Chief State School Officers, 2010,
Geometry: Introduction section, para. 4).
Using transformations to strengthen
your students’ understanding of geometry
sounds like a good idea, but how can you
actually do this in the classroom? This
article describes geometry activities that
incorporate transformations to discover and
justify definitions and properties of
quadrilaterals, and the relationships between
the different quadrilaterals. We provide
examples how you can help students use
transformations in their reasoning. The
lesson described in this article incorporates
CCSS geometry content from across the
grades. Specifically, it addresses the
following standards in the Congruence
domain (G-CO):
Experiment with transformations
in the plane
G-CO.3 Given a rectangle, parallelogram,
trapezoid, or regular polygon,
describe the rotations and
reflections that carry it onto itself.
G-CO.5 Given a geometric figure and a
rotation, reflection, or translation,
draw the transformed figure
using, e.g., graph paper, tracing
paper, or geometry software.
Specify
a
sequence
of
transformations that will carry a
given figure onto another.
Illinois Mathematics Teacher – Fall 2012 .....................................................................................36
Prove geometric theorems
G-CO.11 Prove
theorems
about
parallelograms.
Theorems
include: opposite sides are
congruent, opposite angles are
congruent, the diagonals of a
parallelogram bisect each other,
and conversely, rectangles are
parallelograms with congruent
diagonals.
using transformations of triangles and
discuss how to derive the properties of the
resulting quadrilateral. With teacher
guidance,
students
synthesize
and
summarize their findings and investigate
how these properties relate the different
quadrilaterals to each other. Students then
develop definitions to classify each
quadrilateral shape accordingly (CCSS MP
3). The outline for this activity is as follows:
Completing the activities presented in this
article will help your students develop the
following mathematical practices [CCSS
MP]:
1. Form heterogeneous groups of 3 - 4
students.
2. Assign each group one or two
quadrilaterals to investigate parallelogram, rhombus, rectangle, square,
trapezoid, isosceles trapezoid, or kite.
a. Give each student in the group an
investigation sheet to record their
findings. Blackline masters for these
six investigations can be found at the
end of the article.
b. In addition, give each group a sheet
of graph paper, a ruler, and a sheet of
patty paper, which students can use
to complete the prescribed rotations
in some of the investigations (CCSS
MP 5).
3. During the investigation phase of the
lesson, each group creates a quadrilateral
by completing a set of transformations
on a given triangle. In the process, the
students discuss the various properties of
the quadrilateral shape they create with
respect to the sides, the vertex angles,
the diagonals, and the symmetry of the
quadrilateral.
4. Students are asked to go back and
discuss how they could justify the
various properties using their prior
knowledge of transformations (CCSS
MP 3). In particular, students should
have developed the concept that
rotations, reflections, and translations
preserve congruency of segment lengths
1. Make sense of problems and persevere in
solving them;
3. Construct viable arguments and critique
reasoning of others;
5. Use appropriate tools strategically.
Blackline masters for all activity pages are
included at the end of this article. Examples
of student work and suggestions on how to
deliver the material can be found throughout
the text.
Discovering and developing an
understanding of quadrilateral properties
As mathematics teachers, we know
that simply providing students with a list of
definitions and properties of quadrilaterals
to memorize is an ineffective practice. This
approach leads to minimal understanding of
the quadrilateral properties and how they are
derived. In addition, little or no connections
are made between the properties and how
the quadrilaterals relate to each other.
Transformations
give
students
the
opportunity to develop the definitions and
properties
of
quadrilaterals
through
discovery and to build connections among
the family of quadrilaterals. In the activity
we describe, students create quadrilaterals
Illinois Mathematics Teacher – Fall 2012 .....................................................................................37
as weell as angle measures in
i an earlierr
lesson
n.
5. The students determine how thee
quadrrilaterals reelate to eacch other by
y
creatiing a faamily tree for thee
quadrrilaterals (ssee the lasst blacklinee
masteer). In the process off creating a
familly tree, students
s
fo
ormulate a
defin
nition of eacch shape, making
m
suree
each definition
n specifies minimum
m
propeerties whilee at the same timee
exclu
uding any un
nwanted casees.
An
A alternativ
ve to numberr 5 above iss
the use of
o expert an
nd jigsaw grroups. Using
g
this meth
hod, studentts are first grouped
g
into
o
expert groups
g
wherre they creaate a singlee
quadrilatteral and reccord its prop
perties. Then
n
new jigsaaw groups arre formed which contain
n
at least one expert for each quadrilateral.
q
.
Every meember sharees with the jiigsaw group
p
the propeerties of theiir particular shape. Then
n
each jigsaw group creates a quadrilateral
q
l
family treee.
Justiifying definitions and properties
using tran
nsformationss
Discussion
D
provides
p
an opportunity
y
for the teeacher to guiide students in the use off
deductivee reasoning to prove quadrilateral
q
l
propertiees. We desccribe two instances
i
off
student reasoning wiith transform
mations as an
n
t
type off
example of how to facilitate this
on. The firsst group co
onstructed a
discussio
parallelogram (see Fiigure 1).
Figure 1. Student drawing from
f
group
p
discussio
on of parallellograms
Wheen asked w
what they nooticed abouut the
verteex angles of the parrallelogram they
creaated, the sttudents respponded thatt the
oppoosite angles were congruuent. The teacher
prom
mpted the students to providde a
justiification forr their reasooning (CCSS
S MP
3). IIn response, the studennts simply sstated
that they meaasured the angles off the
paraallelogram aand found tthat one paair of
oppoosite angles measures 440 degrees w
while
the other pair oof opposite angles measured
140 degrees (ssee Figure 1). The teacher
askeed the studeents if they w
would be abble to
draw
w the samee conclusionn if the oriiginal
trianngle they staarted with haad different angle
meaasures.
At this point, students were
strugggling to m
move beyoond the speecific
paraallelogram thhey had creaated and think in
term
ms of any aarbitrary parrallelogram. The
teaccher remindeed the studdents that siimply
meaasuring the angles did not constituute a
justiification thaat the propeerty would hold
true for all paarallelogram
ms. The stuudents
weree instructedd to thinkk back too the
propperties of transformattions they had
disccussed in ann earlier lessson. One oof the
studdents in the ggroup now sstated (CCSS
S MP
1) thhat when thhey started w
with the oriiginal
trianngle, the vvertex anggle (markedd 40
degrrees on theeir paper) w
would alwayys be
conggruent to the oppositte vertex angle
becaause they rootated the orriginal trianggle to
form
m the paraallelogram. Since rotaations
alwaays preservee congruenccy, the oppposite
verteex angles w
would be ccongruent. U
Using
singgle and double notation m
marks for thee two
anglles, which foormed the oother vertex angle
of thhe paralleloggram, the sttudents weree able
to j ustify that the other ppair of oppposite
verteex angles aalso had to be congruennt by
the ffact that rotaations preserrved congruuency.
The group sum
mmarized theeir knowledgge of
Figure
the pproperties oof paralleloggrams (see F
2).
Illinois Mathematics
M
Teacher – Fall
F 2012 ..........................................................................................38
Figure 2. Studentt work
discussio
on of parallellograms
from
fr
group
p
The
T second group in an
nother classs
started th
heir construcction of a recctangle with
h
right ACB.
A
They found the midpoint
m
M
on
, drew the median
m
, and then
n
rotated the
t
trianglee 180 degrrees around
d
midpointt M (see Figu
ure 3).
Figure 3.
3 Student drawing of
o rectanglee
Measurin
ng the sidees and the angles, thee
students marked con
ngruent sidess and angless
in the fig
gure. One off the question
ns that arosee
in thhis group waas whether
would allways
be ccongruent too
. The students feltt that
this would alwaays be true, but were unnsure
whyy. When prom
mpted by thee teacher to think
backk to the rellationship beetween the sides
and angles of trriangles, the students reaalized
that if they could show thaat the base aangles
of
were ccongruent, aand then the sides
and
opposite thoose angles w
would
be ccongruent.
The queestion thus became hoow to
show
w that
. First, stuudents
noticced that
is coomplementarry to
, because these are thhe base anglles of
origginal right
. Realiizing that
corrresponds to
undder the rotaation,
theyy concludeed that thhe angles are
conggruent. Thee students concluded that
is a rigght angle. T
Then they arrgued
that
and
because the
lenggth of a seegment is preserved uunder
rotattion. This aallowed the students too say
that
. Hencee, they knew
w that
because thhe two anglees are
corrresponding pparts of conngruent trianngles,
whicch proved tthat
. To deeepen
studdents’ undersstanding, thee teacher proodded
the sstudents to m
make expliccit connectioons to
the side-angle-side trianngle congruuence
posttulate.
Using ttheir prior knowledge of
transsformations
and
applying
this
know
wledge to a series oof investigattions,
studdents gainedd a deeper uunderstandinng of
the pproperties of the quadrilaterals and were
ablee to justify tthe properties. Although the
studdents used transform
mation in their
reas oning, theirr final justifi
fication is noo less
rigo rous than a traditionnal two-coolumn
prooof. This activvity was highly engagingg and
had the benefit of students asking quesstions
abouut the properrties conjectuured.
In one pparticular claassroom, stuudents
dem
monstrated a deeper understanding oof the
quaddrilateral pproperties by creatinng a
quaddrilateral faamily tree. Although more
Illinois Mathematics
M
Teacher – Fall
F 2012 ..........................................................................................39
time waas spent disscovering an
nd deriving
g
propertiees than iss customary
y, studentss
successfu
ully
demonstrated
d
d
theirr
understan
nding by constructing
c
the family
y
tree forr the quad
drilateral sh
hapes. Thee
variation
ns in studen
nt representations weree
evident in
n the family tree constru
uctions.
Some grou
ups organ
nized theirr
quadrilatteral shapes in a bottom
m-up fashion
n
starting with the most
m
specificc shape, thee
square, and
a working
g back up to
t the moree
general
quadrilateeral.
Otheer
groupss
organized
d their familly tree using
g a top-down
n
method with
w the morre general paarallelogram
m
figure att the top and
d working down
d
to thee
more speecific figuress (see Figuree 4).
Figure 4.
4 A top-dow
wn quadrilaateral family
y
tree
Most
M
groupss had an easier timee
placing the
t kite and the trapeezoid in thee
family trree diagram
m than the rest of thee
figures. At
A the centerr of the disccussions wass
the relaationship beetween thee rectangle,,
rhombus, and squaare. After the teacherr
directed students’ atttention to th
he propertiess
a
the figures, thee
that werre similar across
students correctly concluded that every
y
h a rhombuss
square haas the propeerties of both
and a recctangle.
In
n summary
y, the lesso
on can bee
delivered
d as follows. At the begiinning of thee
lessoon, students investigate the propertiies of
a paarticular quuadrilateral in groups. In a
largeer class, yoou may neeed to assignn the
sam
me quadrilateeral to differrent groups;; in a
smaaller class, yyou could asssign each ggroup
morre than one quadrilateraal, or you ccould
assiggn only a suubset. In the first stage oof the
inveestigation, sttudents mosst likely focuus on
disccovering thee properties. In the seecond
stagge, direct thee attention oof the studennts to
expllaining and justifying these propeerties.
Afteer studentss have ccompleted their
inveestigation, yyou may waant to bringg the
classs together too discuss thee similarities and
diffeerences in the properrties discovvered.
Thiss also providdes students an opportunnity to
use mathematiccal argumentts to justify their
concclusions, annd to judgee the validitty of
arguuments madde by theirr peers. Finnally,
studdents make connectionns betweenn the
diffeerent quadrrilaterals byy constructiing a
famiily tree. Whhen using thhis lesson in your
classsroom, youu may wannt to keepp the
folloowing suggeestions in miind:
IIf a group of students initially faiils to
ccome up wiith any propperties, you may
nneed to use a question to point theem in
tthe right dirrection. For instance, “W
What
ddo you nottice when yyou measuree the
vvertex anglees?”
O
Once a grouup of studentts has discovvered
a property, guide thhe studentts in
cconstructingg a justtification uusing
ttransformatiions.
IIf a group of students fiinds it difficult to
ggeneralize from their drawing of a
sspecific parallelogram
m (or annother
qquadrilaterall) to the geeneral case,, you
m
may want tto suggest drawing annother
pparallelogram
m.
Illinois Mathematics
M
Teacher – Fall
F 2012 ..........................................................................................40
Advancing geometric thought through
the use of transformations
The van Hiele levels of geometric
thought (1986) provide a sound justification
of why transformations are a good bridge
between initial student thinking and the
abstract nature of high school geometry. The
model includes five levels of geometric
reasoning: visual, descriptive, informal
deductive, (formal) deductive, and rigor.
Although a student progresses through the
levels linearly, there is no strict dependence
on age. For example, a fourth grader and a
high school geometry student could be at the
same level. The levels overlap as a student
transitions from one level to the next. Most
students entering a high school geometry
course operate at or below the van Hiele
visual level of understanding (Shaughnessy
and Burger 1985). These students are able to
identify different shapes, but they may find
it hard to recognize and reason with specific
characteristics of shapes. For example,
doing transformations with everyday
language as described in the introduction is
at the visual level. Students at this level are
often not yet ready for the abstract reasoning
required in high school geometry. Students
using the mathematical terminology for
transformations are at the overlap of the
visual and descriptive levels. Students
discovering and deriving quadrilateral
properties are at the informal deductive
level.
Transformations provide a bridge
between the initial visual intuition of the
students and the more formal reasoning of
the higher van Hiele levels. The
quadrilateral activities provided entry to
students at the descriptive level. Students
drew conclusions about the sides and angles
of their original triangle and then rotated the
triangles to create the specified quadrilateral.
This provided students the opportunity to
apply prior experiences of transformations
and the properties of triangles. With the
discussions, the teacher guided students to
the informal deductive level by having them
discover the properties of quadrilaterals and
create informal arguments deriving the
quadrilateral properties. Finally, students
unify their understanding of quadrilaterals
by creating the family tree.
Conclusion
Incorporating transformations to
develop understanding in high school
geometry helps students move from the
visual and descriptive van Hiele levels to the
informal deductive and formal deductive
levels. All students, regardless of their van
Hiele level, stand to benefit from an
integration of transformation geometry in a
high school geometry course (Usiskin 1972;
Okolica and Macrina 1992).
In the
quadrilateral activities described above, the
teacher used this approach to guide the
students in building their understanding of
quadrilateral properties. When students can
connect geometric concepts to the prior
knowledge and experiences that they bring
to class, they can formulate deductive
arguments. The study of transformations in
the early grades and the continuing
development of these concepts in middle
and high school can provide the bridge
geometry students need to reach the
informal and formal deductive levels of
geometric reasoning.
References
Coxford, Arthur F., and Zalman P. Usiskin.
Geometry:
A
Transformation
Approach. River Forest, IL: Laidlaw,
1971, 1975.
National
Council
of
Teachers
of
Mathematics (NCTM). Principles and
Standards for School Mathematics.
Reston, VA: NCTM, 2000.
Illinois Mathematics Teacher – Fall 2012 .....................................................................................41
National Governors Association for Best
Practices & Council of Chief State
School Officers (2010) Common Core
State Standards for Mathematics.
http://www.corestandards.org/thestandards
Okolica, Steve, and Georgette Macrina.
“Integrating
Transformational
Geometry Into Traditional High
School Geometry.” Mathematics
Teacher 85, no. 9 (December 1992):
716–19.
Shaughnessy, J. Michael, and William F.
Burger.
“Spadework
Prior
to
Deduction in Geometry.” Mathematics
Teacher 78, no. 6 (September 1985):
419–28.
Usiskin, Zalman P., and Arthur F. Coxford.
“A Transformation Approach to
Tenth-Grade Geometry.” Mathematics
Teacher 65, no. 1 (January 1972): 21–
30.
Usiskin, Zalman P. “The Effects of
Teaching Euclidean Geometry via
Transformations
on
Student
Achievement and Attitudes in TenthGrade
Geometry.”
Journal
of
Research in Mathematics Education 3,
no. 4 (November 1972): 249–59.
Van Hiele, Pierre M. Structure and Insight:
A Theory of Mathematics Education.
Orlando, FL: Academic Press, 1986.
Blackline masters for the activity pages are included below and on subsequent pages.
Building Quadrilaterals: Parallelogram
1. On a sheet of graph paper, draw obtuse
.
Draw one of the sides of
along one of the grid lines.
Be sure all vertices are placed at the intersection of grid lines.
2. Locate the midpoint,
, of
3. Draw the median to side
.
.
4. Label the figure you have drawn by indicating congruent sides, angles, and measures using
appropriate markings.
5. Rotate
by
around point
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures.
6. Discuss with your group the properties of the parallelogram that you created using the
transformations above. Pay attention to the properties of the sides, the vertex angles, the
diagonals, and the symmetry of the figure.
Summarize your findings under the headings on the chart.
Illinois Mathematics Teacher – Fall 2012 .....................................................................................42
Building Quadrilaterals: Rectangle
1. On a sheet of graph paper, draw scalene right
.
Draw both legs of
along grid lines.
Draw the right angle at vertex .
Be sure all vertices are placed at the intersection of grid lines.
2. Locate the midpoint,
, of
.
3. Draw the median to hypotenuse
.
4. Label the figure you have drawn by indicating congruent sides, angles, and measures using
appropriate markings.
5. Rotate
by
around point
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures.
6. Discuss with your group the properties of the parallelogram that you created using the
transformations above. Pay attention to the properties of the sides, the vertex angles, the
diagonals, and the symmetry of the figure.
Summarize your findings under the headings on the chart.
Building Quadrilaterals: Rhombus
1. On a sheet of graph paper, draw scalene right
.
Draw both legs of
along grid lines.
Draw the right angle at vertex .
Be sure all vertices are placed at the intersection of grid lines.
2. Label the figure you have drawn by indicating congruent sides, angles, and measures using
appropriate markings.
3. Reflect
across the line containing
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use
prime marks for the image vertices.
What kind of figure do you have now? Justify your answer.
4. Reflect
across the line containing
. Be sure to also reflect
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use
prime marks for the image vertices.
5. Discuss with your group the properties of the parallelogram
that you created using
the transformations above. Pay attention to the properties of the sides, the vertex angles, the
diagonals, and the symmetry of the figure.
Summarize your findings under the headings on the chart.
Illinois Mathematics Teacher – Fall 2012 .....................................................................................43
Building Quadrilaterals: Square
1. On a sheet of graph paper, draw isosceles right
.
Draw both legs of
along grid lines.
Draw the right angle at vertex .
Be sure all vertices are placed at the intersection of grid lines.
2. Label the figure you have drawn by indicating congruent sides, angles, and measures using
appropriate markings.
3. Reflect
across the line containing
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use
prime marks for the image vertices.
What kind of figure do you have now? Justify your answer.
4. Reflect
across the line containing
. Be sure to also reflect
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use
prime marks for the image vertices.
5. Discuss with your group the properties of the square
that you created using the
transformations above. Pay attention to the properties of the sides, the vertex angles, the
diagonals, and the symmetry of the figure.
Summarize your findings under the headings on the chart.
Building Quadrilaterals: Kite
1. On a sheet of graph paper, draw scalene acute
.
Draw side
of
along a grid line.
Be sure all vertices are placed at the intersection of grid lines.
2. Draw an altitude,
, of
from point
to
.
3. Label the figure you have drawn by indicating congruent sides, angles, and measures.
4. Reflect
across the line containing
.
Label the new image with the correct markings to indicate congruent sides, angles, and measures. Use
prime marks for the image vertices.
5. On a separate sheet of graph paper, repeat steps 1-4 with a scalene obtuse
6. Discuss with your group the properties of the kites
transformations above.
.
that you created using the
Summarize your findings under the headings on the chart.
Illinois Mathematics Teacher – Fall 2012 .....................................................................................44
Building Quadrilaterals: Trapezoid 1
1. On a sheet of graph paper, draw a large scalene
.
2. Label the figure you have drawn by indicating congruent sides, angles, and measures using
appropriate markings.
3. Locate a point
anywhere on
.
Construct a line parallel to
through point .
Construct the intersection of this line with
at point .
4. Discuss with your group the properties of the trapezoid
transformations above.
that you created using the
Summarize your findings under the headings on the chart.
Hint: What connections can be made between the measures of the angles in this figure and your prior
experiences with parallel lines cut by a transversal?
Building Quadrilaterals: Trapezoid 2
1. Now draw a large isosceles
.
Draw the vertex angle at .
2. Locate a point
anywhere on
.
Construct a line parallel to
through point .
Construct the intersection of this line with
at point .
3. Discuss within your group the properties of the trapezoid
transformations above.
that you created using the
Summarize your findings under the headings on the chart.
Hint: What connections can be made between the measures of the angles in this figure and your prior
experiences with parallel lines cut by a transversal?
Illinois Mathematics Teacher – Fall 2012 .....................................................................................45
Names _______________________________________________________________________
Properties of ____________________________________
SIDES:
VERTEX ANGLES:
DIAGONALS:
SYMMETRY:
Illinois Mathematics Teacher – Fall 2012 .....................................................................................46
Quadrilateral Familyy Tree
Your gro
oup will be building
b
a fam
mily tree forr all of the quuadrilateral shapes that yyou have
explored so far in thiis lesson.
Your fam
mily tree willl be a diagraam that connects all the ddifferent quaadrilateral shhapes togetheer
based on their shared
d properties. When desig
gning your ffamily tree itt would be bbest to start w
with
the more generic figu
ures at the to
op of the diag
gram. Thesee would be tthe figures w
whose properrties
were a paart of all the other shapes. You will use the shappes that weree cut out for the “Wheree Do I
Belong Activity.”
A
As you come to moree shapes with
h more speccific and uniqque propertiees, you will need to placce
these und
derneath the first set of figures.
f
Be sure
s
to show
w connectionns between fiigures using line
segmentss that branch
h between different quad
drilaterals.
Your fam
mily trees sho
ould branch off from thee most generric name for a 4-sided figgure:
quadrilatteral. Once you
y have deccided for surre what yourr family treee should lookk like, glue oor
tape the shapes
s
in theeir appropriaate place.
Qu
uadrilateralls Family S
Scrapbook
Using paaper, pencil, and picturess or photos, your
y
group nneeds to desiign a scrapboook for the
family off quadrilateral shapes. Your
Y
entire group
g
will bee responsiblee for creatingg one scrapbbook.
Each perrson in the grroup must crreate at leastt one page off the scrapboook.
Each shaape must hav
ve at least on
ne page in the scrapbookk. Each pagee should conntain at least the
following
g:
A picture of the quadrrilateral shap
pe seen in thhe real worldd
A list of the
t characterristic propertties of that pparticular shaape
In additio
on, the scrap
pbook needs to have a co
opy of the quuadrilateral ffamily tree thhat was creaated
in class.
If your grroup choosees, you may also
a complette the scrapbbook in Pow
werPoint on thhe computerr.
Illinois Mathematics
M
Teacher – Fall
F 2012 ..........................................................................................47