I.

Triangle
Inequality


II.

Grade
Level:
Geometry
8


by
Kim
Dinwiddie
 
 


III.

Length
of
Lesson:
2‐3
days



IV.

Overview


V.

Context
of
the
Lesson


VI.

Connections
to
State
and
National
Standards



 
 


Common
Core
State
Standards:






 In
this
inquiry
lesson,
students
will
investigate
the
question,
What
is
the
relationship
between
the
 longest
side
and
the
sum
of
the
remaining
sides
of
a
triangle?
Students
will
use
materials
to
create
 models
of
triangles
and
non‐triangles,
and
they
will
measure
the
sides
and
angles
of
a
scalene
triangle.
 They
will
investigate
the
relationship
between
the
longest
side
of
a
triangle
and
the
sum
of
its
other
two
 sides,
as
well
as
the
relationship
between
the
largest
angle
and
the
longest
side
of
a
triangle.

 
 
 This
unit
should
be
taught
during
a
unit
on
triangles.
Before
this
lesson,
students
should
already
have
 explored
how
to
classify
triangles
by
angles
and/or
sides
into
acute,
obtuse,
right,
scalene,
equilateral,
 and
isosceles
triangles.
This
lesson
serves
as
a
good
transition
into
congruence
theorems.
During
this
 guided
inquiry
lesson,
some
students
will
also
investigate
the
relationship
between
an
angle
of
a
 triangle
and
its
opposite
side:
this
is
a
great
segue
into
a
lesson
(that
should
immediately
follow
this
 one)
on
ordering
the
triangle
angles
based
on
the
lengths
of
the
sides
and
vice
versa.

 
 Before
the
lesson,
give
a
pre‐assessment
that
addresses
students’
measuring
skills
with
a
ruler
and
a
 protractor
and
their
understanding
of
the
relationship
between
sides
of
a
triangle.
Use
the
pre‐ assessment
results
to
group
students
of
similar
levels
of
skill
and
understanding
together.
 
 This
lesson
uses
a
variety
of
adaptations
for
diverse
learners
(differentiation
strategies):
kinesthetic
 learners
can
use
materials
to
create
triangles
(or
non‐triangles);
visual
learners
can
present
data
in
a
 visual
way;
and
auditory
learners
can
listen
to
class
and
group
discussion.
 
 
 National
Mathematics
Standards
for
Grades
6‐8
Geometry:
In
grades
6‐8
all
students
should
 
 • understand
relationships
among
the
angles,
side
lengths,
perimeters,
areas,
and
volumes
of
similar
 objects,
 • create
and
critique
inductive
and
deductive
arguments
concerning
geometric
ideas
and
 relationships,
such
as
congruence,
similarity,
and
the
Pythagorean
relationship.
 • use
coordinate
geometry
to
represent
and
examine
the
properties
of
geometric
shapes.
 • draw
geometric
objects
with
specified
properties,
such
as
side
lengths
or
angle
measures.


•

7.G.A.2

Draw
(freehand,
with
ruler
and
protractor,
and
with
technology)
geometric
shapes
with
 given
conditions.
Focus
on
constructing
triangles
from
three
measures
of
angles
or
sides,
noticing
 when
the
conditions
determine
a
unique
triangle,
more
than
one
triangle,
or
no
triangle.





 Virginia
Standards
of
Learning
(SOLs)
for
Mathematics:
 
 Triangle
Inequality
 


1


•


 •

SOL
8.6
The
student
will
 a)
verify
by
measuring
and
describe
the
relationships
among
vertical
angles,
adjacent
angles,
 supplementary
angles,
and
complementary
angles;
and
 b)
measure
angles
of
less
than
360°
 SOL
G.5
The
student,
given
information
concerning
the
lengths
of
sides
and/or
measures
of
angles
 in
triangles,
will
 c)
determine
whether
a
triangle
exists;
and
 d)
determine
the
range
in
which
the
length
of
the
third
side
must
lie.



 VII.

Unit
Goals
and
Lesson
Objectives
 





a. Know
(facts) o Angle
types
(acute,
obtuse,
right)
 o Degree
 o Triangle
parts
 b. Understand
(big
idea)



 
 


Patterns
help
us
predict.
 c. Do
(skills)








VIII.

o o o o o o

Use
the
triangle
inequality
to
solve
problems
involving
triangles.
 Measure
triangle
side
lengths
with
rulers.
 Measure
triangle
angles
with
protractors.
 Given
the
lengths
of
three
sides,
determine
whether
a
triangle
is
formed.
 Given
the
lengths
of
two
sides
of
a
triangle,
determine
the
range
of
the
third
side’s
length.
 Use
mathematical
terminology/vocabulary
for
angles:
acute,
obtuse,
right.


Pre‐assessment
of
students’
prior
knowledge
and/or
skills



 Before
this
lesson,
students
should
be
comfortable
with
and
skilled
at
using
a
ruler
and
a
protractor
to
 measure
lengths
and
angles.
Use
the
pre‐assessment
(see
Resources)
to
assess
this
skill
in
order
to
see
 which
students
may
need
some
extra
guidance
during
the
lesson.
The
pre‐assessment
will
also
give
 you
an
idea
of
how
well
students
already
understand
the
relationship
between
the
longest
side
and
 the
sum
of
the
other
two
sides
of
a
triangle.
Given
three
side
lengths,
students
will
answer
whether
a
 triangle
can
be
constructed:
while
they
may
be
able
to
answer
this
correctly,
they
will
likely
have
 difficulty
justifying
their
answers,
which
indicates
real
comprehension.
 
 Give
the
pre‐assessment
the
day
before
the
inquiry
lesson,
then
use
the
results
to
group
students
of
 similar
skill
and
understanding
levels
together,
in
order
to
encourage
collaboration,
enable
more
 advanced
students
to
further
challenge
themselves
and
each
other,
and
give
you
the
opportunity
to
 give
more
focused
guidance
to
those
students
who
are
struggling.





IX.

Materials




 Pre‐assessment:
 • Pre‐assessment
paper/pencil
worksheet
 • Pencils
 


Triangle
Inequality





2





Guided
Inquiry
Lesson:
 • Supplies
to
construct
triangles:
uncooked
spaghetti
noodles;
straws;
etc
(Place
15
or
so
of
these
in
 plastic
baggies
to
cut
down
on
material‐gathering
time;
keep
some
in
reserve
in
case
students
 need
more
than
is
in
the
baggie.)
 • Scissors
 • Rulers
 • Protractors
 • Pencils
 • Paper
 • Formative
Assessment
Data
Notes
Taking
Chart
(for
you)
 • Day
1
Homework
handout
 • Day
2
Exit
Card
 Post‐assessment:
 • Post‐assessment
paper/pencil
worksheet
 • Pencils
 


X.

Level
of
Inquiry:
Guided



 This
is
a
guided
inquiry
lesson.
You
will
present
the
question,
but
students
will
work
together
in
groups
 to
plan
and
carry
out
their
own
investigations,
collect
and
analyze
data,
and
present
data
to
the
class.
 You
should
act
as
a
facilitator
rather
than
an
instructor
and
ensure
that
your
students
are
actively
 engaged.





XI.

Teaching
Strategies




 Pre‐Assessment
 Give
this
the
day
before
beginning
the
lesson
in
order
to
form
groups
of
similar
skill
and
understanding
 levels.
 
 Day
1
 Divide
students
into
the
groups
you
have
determined
based
on
pre‐assessment
results.
Pass
out
two
 sets
of
three
sticks
to
each
group,
one
set
that
will
form
a
triangle
and
one
that
won’t.
After
giving
 them
time
to
manipulate
the
sticks,
explain
that
not
all
sets
of
three
sticks
(line
segments)
can
form
a
 triangle.
WHY?
Tell
them
that
this
is
what
they
will
be
exploring
in
class.
 
 Write
the
investigation
question
on
the
board:
What
is
the
relationship
between
the
longest
side
and
 the
sum
of
the
remaining
sides
of
a
triangle?

Give
students
10‐15
minutes
to
discuss
hypotheses
and
 procedures
for
finding
the
answer.
As
they
discuss,
circulate
through
the
room,
jotting
down
notes
and
 offering
help
and
guiding
questions
as
needed.
In
their
procedures,
students
may
include
only
sides
of
 an
already‐complete
triangle.
If
this
happens,
have
a
brief
class
discussion
of
the
experimental
benefits
 of
using
lengths
that
do
not
create
triangles
as
well.
 
 Remind
students
that
they
will
present
their
methods,
results,
and
conclusions
to
the
class
tomorrow.
 Do
not
tell
them
how
to
present
their
data:
leaving
this
up
to
them
will
likely
result
in
a
range
of
data
 recording
and
presenting
methods
that
should
yield
a
rich
class
discussion
of
data
collection
and
 presentation
methods.

 
 Have
students
work
in
their
groups
for
the
remainder
of
class
to
conduct
their
experiments.
Monitor
 their
progress
by
observing
their
experiments
and
jotting
down
notes,
using
the
Formative
Assessment
 Data
Note
Taking
Chart
(see
Resources).
If
they
need
help,
do
not
tell
them
what
to
do
or
what
they


Triangle
Inequality





3


need
to
know,
but
ask
probing
questions
to
guide
them
through.
Make
sure
that
they
are
using
more
 than
one
trial
(more
than
one
triangle)
to
answer
the
question.
 
 Some
of
the
more
advanced
groups
may
finish
early;
ask
these
groups
to
use
a
procedure
similar
to
the
 one
they
have
been
using
to
investigate
the
relationship
between
the
length
of
a
side
and
the
size
of
 the
angle
opposite
it.
 
 Pass
out
the
homework
assignment
for
Day
1
(see
Resources).
 
 Day
2
 Have
each
group
present
their
findings
to
the
class.
Ideally,
they
should
all
have
reached
the
 conclusion
that
regardless
of
type
of
triangle,
the
sum
of
the
two
smaller
sides
must
be
greater
than
 the
longest
side
of
the
triangle.
Take
notes
on
the
presentations
as
part
of
your
formative
assessment.
 
 Discuss
as
a
class
the
different
procedures
groups
followed.
Some
groups
may
have
measured
 different
types
of
triangle
than
others
did;
discuss
this
and
how
it
relates
to
the
investigation
question.
 
 Discuss
as
a
class
the
findings
of
the
groups
that
investigated
the
relationship
between
angle
and
 opposite
side.
If
students
who
did
not
investigate
this
question
challenge
the
conclusion
(that
the
 largest
angle
is
opposite
the
longest
side
and
the
smallest
opposite
the
shortest
side),
invite
everyone
 to
measure
their
own
triangles
with
rulers
and
protractors
to
discover
this
relationship
for
themselves.
 
 At
the
end
of
the
day,
give
students
the
Day
2
Exit
Card
(see
Resources).
 
 Post‐assessment
 Give
the
post‐assessment
a
day
after
the
lesson.
 


XII.

Assessment
Plan



XIII.

Resources




 As
stated
in
section
VII,
the
goals
for
this
guided
inquiry
lesson
include
learning
about
the
relationship
 between
the
longest
side
of
a
triangle
and
the
other
two
sides,
as
well
as
the
relationship
between
size
 of
angle
and
length
of
opposite
side.

 
 Summative
assessments
include
the
pre‐
and
post‐assessment.
These
test
student
understanding
of
 these
relationships
in
various
ways,
including
justifying
their
answers,
which
will
give
you
a
good
sense
 of
how
successfully
the
lesson
reached
the
learning
objectives.

 
 Formative
assessments
include
your
informal
notes
on
the
note‐taking
chart;
your
notes
on
the
 student
presentations;
the
homework
assignment
from
Day
1,
and
the
Exit
Card
from
Day
2.
These
will
 all
give
you
a
sense
not
only
of
how
well
the
students
learned
the
relationships
that
were
the
learning
 objectives,
but
also
of
how
well
they
take
to
and
like
inquiry‐based
learning.
 
 
 Books:
 Llewellyn,
D.
(2007).
Inquire
Within:
Implementing
Inquiry‐Based
Science
Standards
in
Grades
3‐8,
2nd
 Edition.
Corwin
Press.


Websites:
 Johnson,
H.
L.
(n.d.).
Illuminations:
Inequalities
in
Triangles.
Retrieved
August
20,
2014
from
 http://illuminations.nctm.org/Lesson.aspx?id=2339 Triangle
Inequality
 


4


Pre‐assessment
 


1.
 Measure
the
length
of
the
line
segment
to
the
nearest
centimeter.
 
 
 
 
 
 
 
 
 2.
 Measure
the
angle
to
the
nearest
degree.
 
 
 
 
 
 
 
 
 
 
 
 
 For
questions
3
–
5,
can
you
construct
a
triangle
with
the
given
lengths?

Why
or
why
not?
 
 3.
 3cm,
8
cm,
14
cm
 4.

 12
cm,
12
cm,
12
cm
 
 5.

 8
cm,
10
cm,
14
cm


Triangle
Inequality





5


Formative
Assessment
Data
Notes
Chart
 Student
Name


What’s
the
 relationship?


Organize
 Data


Largest
Side
 Opposite
Largest
 Angle



Notes


























































































































































































































































































































































































 Put
check
mark
if
observed
~
include
some
details
if
necessary
 Triangle
Inequality





6






 
 


Homework,
Day
1:


 
 
 
 In
the
box
below,
write
3
inequalities
that
are
always
true
for
a
triangle
with
side
lengths
a,
b,
 and
c.
 
 
 
 
 
 ________
+
________
>
_________
 
 
 
 
 
 
 ________
+
________
>
_________
 
 
 
 
 
 
 
 ________
+
________
>
__________


Triangle
Inequality





7



 Exit
Card,
Day
2



 
 1.
 
 
 
 
 
 
 2.
 
 
 
 
 
 
 3.
 
 
 
 
 
 
 4.

 
 
 
 
 
 
 
 5.
 
 

 
 
 
 
 6.
 
 


Can
you
construct
a
triangle
with
side
lengths
3
cm,
8
cm,
and
14
cm?

Why
or
why
not?


Can
you
construct
a
triangle
with
side
lengths
12
cm,
12
cm,
and
12
cm?

Why
or
why
not?


Can
you
construct
a
triangle
with
side
lengths
8
cm,
10
cm,
and
14
cm?

Why
or
why
not?


Two
sides
of
a
triangle
are
6
cm
and
10
cm.

Determine
a
range
of
possible
measures
for
the
 third
side.


Two
sides
of
an
isosceles
triangle
measure
3
cm
and
7
cm.

Which
of
the
following
could
be
 the
measure
of
the
third
side?
 


a.


9





b.


7





c.


3


In
triangle
ABC,
the
measure
of
angle
A
is
30
degrees
and
angle
B
is
50
degrees.

Which
is
the
 longest
side
of
the
triangle?
 


a.


Triangle
Inequality


AB





b.


AC








c.


BC


8





Post‐assessment
 
 1) Hannah
is
putting
a
border
around
her
triangular
garden.
 Two
sides
of
the
garden
have
lengths
of
12
feet
and
17
feet.

 
 What
is
the
range
in
which
the
third
side
of
the
garden
must
fall?
 
 
 
 
 
 
 2) In
ΔABC
,
with
BC
>
AC
,
which
of
the
following
statements
must
be
true?
 
 A. 
m
∠C
is
greater
than
m
∠B
 B. 
m
∠B
is
greater
than
m
∠A
 C. 
m
∠
A
is
greater
than
m
∠B
 D. 
m
∠C
is
greater
than
m
∠A
 
 
 3) In
ΔDEF,
side
DE
=
8
inches,
FE
=
6
inches,
and
FD
=
10
inches.
Which
lists
the
 angles
in
order
from
smallest
to
largest?
 
 A. 
∠E,
∠F,
∠D
 B. 
∠F,
∠D,
∠E
 C. 
∠D,
∠E,
∠F
 D. ∠D,
∠F,
∠E
 



 






 
 


4) John
wants
to
make
a
triangular
garden.
Which
of
the
following
are
possible
dimensions?
 
 A. 4
ft
by
5
ft
by
10
ft
 B. 6
ft
by
8
ft
by
10
ft
 C. 8
ft
by
12
ft
by
20
ft
 D. 6
ft
by
6
ft
by
12
ft
 
 5) Two
sides
of
a
triangle
measure
14
inches
and
8
inches.
Which
cannot
be
the
length
of
the
r emaining
side?
 
 A. 21
in.
 
 
 B.
6
in.

 
 C.

14
in.
 
 D.
8
in.
 


Triangle
Inequality





9



 
 


6) Jennifer
made
these
measurements
on
ΔABC
.
BC
must
be
—

 
 A. between
12
and
22
inches
 B. between
10
and
12
inches
 X
 C. greater
than
22
inches
 D. less
than
10
inches
 50 
 °°
 B
 
 


C


60


10


12


A


7) Josh
is
planning
a
trip
and
the
path
of
his
top
three
destinations
forms
triangle
ABC.



Order
the
angles
 from
smallest
to
largest.
 
 
 
 
 
 
 
 
 
 
 8) Kari
needs
to
create
a
triangular
structure
for
her
science
project.
Her
teacher
has

 allowed
her
to
pick
three
pieces
of
wood
from
a
large
pile
to
use
for
her
project.
 Which
three
pieces
of
wood
could
create
a
triangle?
 
 A. 2.7
in,
4.3
in,
6.8
in
 B. 6.7
in,
4.2
in,
2.4
in
 C. 1.3
in,
8.1
in,
6.4
in
 D. 2.3
in,
5.1
in,
7.4
in
 


Triangle
Inequality





10