Block 28 - Math GR. 6-8

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MA.7.A.1.3
Solve problems involving
similar figures
Block 28
Similarity is the basis of all
measurement. It reveals the
secret of map making and scale
drawings. Similarity helps explain
why a hummingbird's heart beats
so much faster than a human
heart, and why it is impossible for
a small creature such as a
praying mantis to become as
large as a horse.
Vocabulary
•
•
•
•
•
Similar figures
Corresponding sides
Corresponding angles
Scale factor
Double, twice, triple, half
Do we really understand
the definition of similarity?
Activity 1 & 2
Testing for Similarity
Completing the enlargement
Similarity?
• What is it?
• When are two triangles similar?
– Squares/circles/rectangles?
– Line segments?
– Arbitrary shapes?
• How to test for similarity?
Checking for similarity
• Distribute a copy of the worksheet
“Complete Enlargement” to each
participant
• Participants are asked to complete the
enlargement of a heptagon
Discussion after activity
• What were you trying to accomplish as you
made the enlargement?
• What was your method to finish the
enlargement?
• What changed in the enlarged figured? What
stayed the same?
• What was the scale factor?
• How could this activity be used in the
classroom?
• Do students need to have a formal
understanding of similarity before engaging in
this activity?
Proportional and Nonproportional relationships
Activity 2
Similarity Problems
Proportional relationships
• Are linear, but not all linear
relationships are proportional
• Are multiplicative
• Y-intercept is zero
• Have a constant of proportionality (k)
y=k*x
Linear (or directly) Proportional
Relationships
• Major topic in elementary mathematics
education
• Basic model for advance problems in
pure and applied mathematics
• Students tend to believe that every
numerical relationship is linear (known in
math education as linear misconception
or linear obstacle)
• Students over-generalize the linear model
Research on properties of similar figures
• Gaining insight into the relationships
between lengths, areas and volumes of
similar figures is usually a slow and difficult
process. (Cognitive difficulty: high)
• “… most students in grades 5–8
incorrectly believe that if the sides of a
figure are doubled to produce a similar
figure, the area and volume also will be
doubled” (NCTM, 1989, pp. 114–115).
Why are proportions difficult?
A mathematical analogy
 Analogies allow the mind to transfer
information
 Analogies are the interstate freeway of
cognition
 Analogies were key to the old SAT
SAT analogies
• Follow form  A : B :: C : D
• nurse : hospital :: scholar : school
• analogy : thought :: core : apple
DRIP : GUSH
A. Cry : Laugh
B. Curl : Roll
C. Stream : Tributary
D. Dent : Destroy
E. Bend : Angle
What is a Proportion?
A mathematical analogy
• A statement of equality between two
ratios
6 12
30

60
• Each number is related to two others
 to:
• As opposed
•WALK : LEGS :: CHEW : MOUTH
GeoGebra:
Perimeter_area_similar
Open the GeoGebra file
1. As you change the scale factor,
notice the changes in Fig B
2. The point “Perimeter” moves in the
coordinate system as the scale
factor changes, what path does it
describe?
3. Check the box perimeter to verify
your conjecture as you change the
scale factor
4. Is the relationship linear
(proportional)? If yes, what is the
equation of the line? What does the
slope represent?
5. Do the same for the point “Area”
• How can we help students understand
that not all problems are linear?
• Does the shape of the figure used in
the problem matter?
• Does the presence (or absence) of a
drawing help students?
With the following activity we will explore
some of these answers.
Length of a Segment
Distance between the end points in
Cartesian coordinates
2
2
L'
L  d  x1  x 2   y1  y 2 
s
L
Magnify by a factor s
L'
sx1  sx 2   sy1  sy 2 
2
2

 s 2 x1  x 2   s 2 y1  y 2 
2
s
 sL
2
x1  x 2   y1  y 2 
2
2
Area of simple shapes
• Rectangle A  b  h
• Triangle
A  12 b  h
2
A


r
• Circle

Contains the product of 2 length A'  s2

A

Magnify by factor s
Each length in each formula times s
A bh
A' sb  sh  s
2
2

(b  h)  s A
Volume of simple shapes
V' 3
• Cube
s
V
• Sphere V  4 r 3
3
Is the product of three lengths

Magnify by factor s

V  L3
V
 LW  H
V ' sL  sW  sH
 s 3  L  W  H  s 3V
In General
Length 1-d measure
L ∝ s1
Area 2-d measure
A ∝ s2
Volume 3-d measure V ∝ s3
General
P ∝ sD
D scaling exponent (dimension)
not necessarily an integer
• ∝ = proportional to
Activity
• Will use applied mathematics problems
dealing with lengths and areas of two
similar geometric figures
• Divide participants in 3 groups
• Give a worksheet to each participants
• Participants should work individually
• Group 1 is not suppose to do any drawings, only
calculations in order to solve the problems
• Group 2 must make a sketch or drawing before
solving the problem
• Group 3 already has a correct drawing in each
problem.
Discussion after problems were done
• Did all problems follow the linear model?
• What problems were easier?
• Do you think that problems were easier to
solve for a particular group? Why?
• Do you think that a sketch or drawing has a
beneficial effect on the students’
performance?
• Does it help more if students do the drawing
or if it is provided in the problem?
• Are problems more difficult for different types
of plane figures?
• Does the shape of the figure matter?
Students make their own Drawings
• They must construct a proper (mental)
representation of the essential elements
and relations involved in the problem.
Can all students do that?
• Especially for the non-linear problems, this
representational activity should help
students to detect the inappropriateness
of a stereotyped linear proportional
reasoning, and to determine the nature
of the non-linear relationship connecting
the known and the unknown elements in
this problem.
Giving Students the Correct Drawing
• Students may not succeed in making a
correct, usable drawing themselves.
• Giving students a correct ready-made
drawing, could be more effective than
instructing them to generate such a drawing
on their own, especially for students with
learning disabilities
• Is giving a correct drawing more effective for
all students?
• Does making your own drawing reinforces
comprehension of the problem? Does it help
understand what they are reading?
Does the shape of the figure matter
in the difficulty of the problem?
• Does it matter for linear and/or nonlinear problems?
• What strategies can students use to
find the solution to these non-linear
problems?
• How can the drawings help?
In the problem with squares, students
could choose among three
appropriate solution strategies:
(1) ‘paving’ the big square with little ones
(2) calculating and comparing the areas of
both figures by means of the area formula
(3) applying the general principle ‘if length
is increased by r, then area is increased
by r2’.
Strategies for the problems with circles
• The first solution strategy becomes
impossible
• The second strategy is more errorprone (because of the greater
complexity of the formula for finding
the area of a circle)
• The last strategy is the best
• In problems with circles, does a
drawing help?
Strategies for problems with
irregular shapes
• Can only be solved by applying the
general principle (the third solution
strategy).
• How can we help students understand
the general principle?
A study using 12-13 year old
students concluded that:
• Most students were able to solve the
proportional items correctly, whereas the nonproportional items were seldom solved
correctly.
• Drawings, either student made or given, does
not increase students’ performance.
• The type of figure used in the problem has a
significant effect on the percentage of
correct responses.
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