Download Similarity Summary

Title:
Grade: 6
BIG Idea:
How Do They Compare?
Author(s): Gayle
Herrington
Similarity
Prior Knowledge Needed:
Students understand the meaning of ratios and how to set up proportions
Students know how to measure angles and lengths with the appropriate
tools
GPS Standards:
Objectives:
M6G1c. Use the concepts of ratio,
1. Students will measure angles and
proportion and scale factor to
line segments using appropriate tools
demonstrate the relationships between
similar plane figures.
2. Students will be able to identify the
corresponding parts (angles and sides)
M6A1. Students will understand the
of similar figures
concept of ratio and use it to represent
quantitative relationships.
3. Students will be able to approximate
(or estimate) the ratio of corresponding
M6A2. Students will consider
sides in two similar figures.
relationships between varying
quantities.
4. Students will be able to informally
describe the meaning of similarity based
M6P2. Students will reason and
on the comparisons made between
evaluate mathematical arguments.
similar and non-similar figures.
M6P3. Students will communicate
mathematically.
M6P4. Students will make connections
among mathematical ideas and to
other disciplines.
Materials:
Students: How Do They Compare? handouts – guided and open versions ( pg 1
is optional); How Do They Compare? Test Your Knowledge – guided and open
versions; 2 sheets of Patty Paper (or wax paper)
Teacher: How Do They Compare? transparencies, Blank Transparency,
Overhead Pen
Task:
Investigating Similarity.
Pentagon PQRST is similar to pentagon ABCDE (figures provided). List the pairs
of corresponding angles in PQRST for each angle in ABCDE. What is the
relationship of the measures of each pair of corresponding angles in the
pentagons?
List the pairs of corresponding sides in PQRST for each side in ABCDE. Find
the ratio of the lengths of each pair of corresponding sides in the pentagons
(PQRST: ABCDE). What is the relationship of the ratios of each pair of
corresponding sides in the pentagons?
Complete the statement based on your investigation: If two figures are similar,
then___.
Test Your Knowledge
Hexagon GLKJIH is similar to two of the hexagons shown. Use ratios of
corresponding sides and the measures of corresponding angles to determine
which hexagon is not similar to GLKJIH. Use diagrams, words, and symbols to
explain how your measurements support your answer.
Description and Teacher Directions:
Initial goal is that students recognize that similar
figures look alike. Then, direct attention to the
relationship between corresponding sides and
angles. The big ideas that you are gradually
building with this series of tasks are:
Similar figures are two figures that are the same
shape but not necessarily the same size
Tests for similarity: (1) the measures of each pair
of corresponding angles are equal & (2) the
lengths of the corresponding sides increase by
the same factor, called the scale factor.
HOW DO THEY COMPARE?
Display T1 on the overhead projector / interactive
whiteboard and distribute the first handout. Ask
students to identify enlargements/reductions for
Figures 1-3. Have students work individually or with
a partner for a few minutes before engaging in a
class discussion.
Figure 1 I,G Figure 2 C, F Figure 3 J,E
At this time, students should note that enlargements
and reductions are not congruent to the original
figure. Students will likely describe the figures
based on what they see:
Same shape but larger/smaller
Stretched or pushed in all directions
Stretched or pushed in one direction
Distorted
Narrow and tall
Short and wide
Remind students about the general definition of
similarity within the context of the enlargements and
reductions.
What does it mean for two figures to have the same
shape? We are going to investigate further…
Teacher
Commentary:
Alternates:: (1) each pair
of corresponding angles
are congruent & (2) the
lengths of every pair of
corresponding sides are
proportional
If students struggle, ask
them about enlarging and
reducing on the copy
machine or digital photos;
pick 2 images and ask
them to tell you how they
are alike and/or different;
etc.
Review the meaning of CONGRUENCE and
COMPARE with the class.
When you compare the sizes of figures like
length or area, your task is to determine
whether the measures are equal or not
equal. (AB=CD or mAB = mCD )
When you compare the shapes of figures,
your task is to determine whether the
figures are congruent or not congruent.
( AB ≅ CD )
Model for them how to show that 2 figures are
congruent using the overhead projector. Show T2
and use a blank transparency as your patty paper
(or wax paper). Trace a figure on your
transparency then, place it on top of another to
compare.
Note that if you place one figure (polygon,
angle, segment, etc) on top of the other, they
will match exactly if congruent.
INVESTIGATING SIMILARITY?
Have students practice at their seats or on the
board as needed. When you feel that the students
understand how to compare figures using patty
paper, distribute Investigating Similarity sheets and
patty paper.
**Use Teacher Checklist to informally assess
students as they work on the investigation**
Students can work individually or with a partner on
questions 1-4. Move around the room to monitor
student progress.
What two figures are you comparing?
How would you describe corresponding
angles? Sides?
How did you identify the corresponding
parts?
How do you name an angle? Segment?
What patterns do you notice as you
compare____?
Have full group come together to discuss their
comparisons of the two figures (use T4 as needed).
Have pairs report their findings. The goal is for
students to note which measures are congruent and
which are not.
Students should note that the corresponding
angle pairs are congruent or that they have
equal measures. You can verify this
Remind students that
angles show the amount
of turn not the length of its
sides.
Monitor notation as
students list the segments
and angles. You may
want to verify the lists of
corresponding parts as a
class or individually before
students make
comparisons.
If you see students
struggle with the angle
measurements, you might
suggest that they trace
and compare one angle at
a time. This might also be
helpful with the segments.
conjecture as a class.
Students should note that the corresponding
segment pairs are not congruent or that they
do not have equal measures. You can verify
this conjecture as a class.
Individual line segments can also be used as a
‘ruler’ to measure lengths. Show T3 to model how
patty paper can be used to compare lengths. If two
segments are not congruent, then one must be
longer than the other. How much longer?
How many AB segments will it take to
measure___?
Students should also be
able to approximate a
fraction of a segment (i.e.,
half, third, etc.)
Good opportunity to
remind students that
ratios can be written in
many ways—fraction,
decimal, percent, a:b, a to
b.
AB= 1 unit
DE= 2 units
(or segment DE is twice as long as segment
AB.)
FG = 3 ½ units
(or segment DE is three and a half times as
long as segment AB.)
Remind students that all
comparisons are based on
AB= 1 unit.
If needed, model for students how to measure the
edge of their desk (or worksheet) using AB .
Then have students resume work with question 5.
Move around the room to monitor student progress.
What patterns do you notice as you
compare____?
Is that measure reasonable?
How did you find the length of ____?
What two figures are you comparing?
How many AB segments
will it take to measure the
edge of the desk?
Have full group come together to discuss their
comparisons of the two figures (use T4 as needed).
Have pairs report their findings.
Students should note that the
corresponding segment pairs have the
same ratio or are multiplied by the same
factor.
In this case, there is a 1:2 ratio. The segments in
PQRST are half the length of the segments in
ABCDE.
If students describe the
ratio as 2:1, note the order
that they have listed the
ratios
ABCDE
2 units
=
= 2.0
PQRST
1 unit
PQRST
1 unit
=
= 0.5
ABCDE 2 units
You can verify this conjecture as a class.
Corre spon ding Angle Pairs are Congruent
m∠AED m∠EDC m∠BAE m∠CBA m∠DCB
113°
92°
107°
134°
94°
m∠RQP m∠TPQ m∠QRS m∠RST m∠STP
134°
107°
94°
92°
113°
Q
The segments in ABCDE
are twice the length of
the segments in PQRST.
R
C
B
P
S
T
A
D
Students can later use a
ruler to measure the
segments to the nearest
0.1 cm for further
exploration.
The RATIO of Correspondi ng Side Pai rs is Constant
OR Corresponding Si des are Pro portion al
SR
QR
ST
PQ
TP
DC
BC
DE
AB
EA
0.50 0.50
EA
AB
0.50 0.50 0.50
DE
BC
DC
TP
PQ
ST
QR
2.00 2.00 2.00 2.00
SR
2.00
Continue the full group discussion to summarize the
findings. Discuss the meaning of similarity.
When you compare the two figures, which
features changed? Which remained the
same?
What do you think it means for two
figures to be similar?
What measurements did you record?
How do the lengths of the sides compare
in similar figures?
How do the angles compare in similar
figures?
How can you decide if two figures are
similar?
Also mention
enlargements and
reductions to revisit the
intro task and emphasize
that order does matter
when verbally describing
the action.
Ask students to construct a definition for similar
figures based on the activity. Record the class
definition on the board for students to add to their
notes.
As students work through the unit, we will refine the
definition of similarity. Choose the phrasing that
best captures what your students noted while
exploring. Also consider the vocabulary previously
used for ratios and proportions.
After selecting a working definition for similar figures Alternates:: (1) each pair
as a class, have students work on Test Your
of corresponding angles
Knowledge.
are congruent & (2) the
lengths of every pair of
corresponding sides are
Key ideas to note:
Similar figures are two figures that are the
proportional
same shape but not necessarily the same
size
Tests for similarity: (1) the measures of
each pair of corresponding angles are
equal & (2) the lengths of the
corresponding sides increase by the same
factor, called the scale factor
TEST YOUR KNOWLEDGE
Distribute the Test Your Knowledge handoutStudents may work alone or with their partner to
synthesize the information that they have learned
about similarity. Move around the room to monitor
student progress.
Ask questions, as needed, to redirect student
thinking. Try not to tell them what to do.
Explanations will vary. Focus on how students
address the corresponding angles and sides.
Hexagon FABCDE is not similar to
Hexagon GLKJIH. ( corresponding
angle pairs are not congruent &
shapes are different)
Hexagon TUVXYZ is similar to
Hexagon GLKJIH. (TUVXYZ is 1.5
times larger than GLKJIH. Students
will most likely see scale factor of 1.5
rather than 3:2)
Hexagon MNOPQR is similar to
Hexagon GLKJIH. (MNOPQR is
congruent to GLKJIH. This is a special
case where the scale factor is 1 or
1:1)
Note that three angles in
FABCDE are
approximately one degree
different than those in
GLKJIH. Students may
think that they are
congruent with the given
tools. The two angles
shown are clearly different
in size.
Students may be puzzled
about the figures being
congruent. Have them
revisit the definition
regarding corresponding
angles and the ratio of
corresponding sides. (1:1
Scale factors and non-congruent angles are also
shown below with the images.
A
MR
GH
= 1.00
P
F
m ∠IHG
m ∠ DEF
91 °
100 °
Q
B
C
O
N
M
R
E
D
L
G
m ∠ HIJ
m ∠ EDC
119 °
114 °
U
T
K
H
TZ
GH
Z
I
X
V
= 1.50
Y
J
is the ratio)
Modifications/Extensions:
Modification:
Investigating Similarity-Trace the segments and angles in advance on a blank
transparency for students to use for the comparisons.
Extension:
MORE COMPARISONS
How do the perimeters of similar figures compare? The perimeter
increases/decreases by the same scale factor as the corresponding sides. (Perimeter
measures the sum of the side lengths)
How do the areas of similar figures compare? The area increases/decreases by the
square of the scale factor or (scale factor)2. Students can estimate the area of the
figures using a grid outlined on patty paper or a transparency if they do not have the
formula for finding the area.
D
D
F
B
C
A
E
D
AC = 1 cm
ED = 2 cm
AB = 1 cm
DF = 2 cm
BD = 1 cm
FD = 2 cm
DC = 1 cm
DE = 2 cm
ED
AC
=2
(Perimeter FDED)
(Perimeter CDBA)
(Area FDED)
(Area CDBA )
=4
=2
AC ⋅2+AB ⋅2+BD ⋅2+DC⋅2 = 8 cm
2⋅(AC+AB+BD+DC) = 8 cm
AC ⋅2⋅AB ⋅2 = 4 cm 2
4⋅AC ⋅AB = 4 cm 2
22⋅AC ⋅AB = 4 cm 2
(Perimeter NMRQPO)
= 1.00
(Perimeter LGHIJK)
(Area NMRQPO )
(Area LGHIJK)
(Area UTZYXV)
(Area LGHIJK)
= 1.00
A
= 2.25
P
F
(Perimeter FABCDE)
(Perimeter LGHIJK)
(Perimeter UTZYXV)
(Perimeter LGHIJK)
= 1.21
Q
= 1.50
(Area FABCDE)
B
C
O
(Area LGHIJK)
= 1.37
N
M
R
E
D
G
U
T
K
H
J
I
V
Z
Y
Resources:
Discovering Geometry
Geometry Activities for Middle School Students with the Geometer’s Sketchpad