Unit 2
Triangle similarity
October 1, 2012
DO NOW
• NO D.E.A.R TODAY for 3rd block
• Triangle A’B’C’ is a translation image of
triangle ABC. What is the rule of the
translation?
A’B’C’ ordered points are: A’ (-2, 5)
B’ ( 1, 2) and C’ ( 3, 4)
ABC ordered points are:
A( 3, - 1) B (6, -4) and C ( 8, -2)
what are polynomials.htm
Activating/launch
Balance – what is balance? What can we balance? Why balance?
What motive is there for understanding balance, especially in
a math class?
What is a polynomial?
made up of terms that are only added,
subtracted or multiplied.
Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:
How do you remember the names? Think cycles!
We do:
Matching Expressions and Words
4(n +2)
Multiply n by four, then add two
2(n + 4)
Add four to n, then multiply by two.
4n + 2
Add two to n, then multiply by four.
P-4
YOU DO
• Each student will receive a copy of the
assessment task Interpreting Expressions and
a mini-whiteboard, pen, and eraser.
• Each pair of students will need a copy of Card
set A: Expressions, Card Set B: Words, a glue
stick, a felt-tipped pen, and a large sheet of
paper or card for making a poster.
Wrap up
• Summary: How are theorems/rules useful in everyday
school subjects and life outside of school?
• Where are ratios found in everyday life?
• Homework: Assigned 10/1 due 10/5
• Devise an area for your pet, to play basketball, soccer,
football, to dance, or act. Come up with a blueprint, label
the coordinates of your area, (create on graph paper) how
would you come up with the scale factor for the actual
area, how would you calculate perimeter of the area if the
units are in feet? In addition, you want to put a wall/fence
around your area, the fencing or wall is $3 per yard, how
much will it cost to fence the entire area?
• 15 minutes D.E.A.R
10 – 2 - 12
1. DO NOW – what is a ratio? What is scale
factor? And what is the image of P (3, - 4)
reflected across line x = - 1
2. Launch – video
3. Discuss rotations, theorems, reflections, glide
reflections
4. Symmetry
5. Summary: How are theorems/rules useful in
everyday school subjects and life outside of
school?
rotations
• Pg. 603
• Reflections
• Pg. 612
• Glide reflections
What is the image of
triangle TEX for glide
translation where
translate is (x, y – 5) and
line reflection is x = 0
Stations – ratios of rectangles
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Small groups of 3 or 4
Answer question on the graph paper
Turn in, this is an assignment
9-4 symmetry (form K)
similar polygons preview
practice
• Assessment
• Skill based task
– worksheet
• Practice
– Problem task
Each set of partners/threes has to
create a poster and an example for
their specific rule
• Theorems 6 – 9 Proofs using coordinate
geometry
• Properties of Parallelograms pg. 369A
• Special Parallelograms pg. 396 “special
parallelograms” 6-13, 6-14
• Coordinate geometry pg. 369B
• Why should we use variables as coordinates
when writing a coordinate proof?
• Polygon Angle sum theorem pg. 371
• Corollary to the polygon angle sum thm pg. 372
• Polygon exterior angle sum thm pg. 373
• Theorem 6-3, 6-4, 6-5, 6-6 pg. 378 – 384
parallelogram
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Theorem 6-7 pg. 385
Theorem 6-8, 6-9, 6-10 pg. 388
Theorem 6-11, 6-12 pg. 390
Theorem 6 -16, 6 -17, 6- 18 pg. 404
Theorem 6 – 19, 6-20, 6-21 pg. 410
• Theorem 6-23 pg. 415
10-2-12
• Academic Vocabulary: center of dilation, line
segment
Skills to master
– Given a center & scale factor, experiment to
visually see that when performing dilations the
line segment, the pre-image, the segment which
becomes the image is longer or shorter based on
the ratio given by the scale factor
Additional skills to master
• Given a line segment, a point not on the line
segment, and a dilation factor, construct a
dilation of the original segment
• Recognize that the length of the resulting
image is the length of the original segment
multiplied by the scale factor & that the
original & dilated image are parallel to each
other
Activities
• We do: 9-5 enrichment questions and regular
9-5 questions
• Homework: 9 -6 compositions of reflections
due Wednesday Oct 3rd
assessment
• Skill based
– Create a dilation of
segment AB through C,
with a scale factor of 2
to create segment EF.
Find the lengths of EF,
AC, BC, CE, and CF
• Problem Task
– Locate the center of
dilation and scale factor
in the following pair of
triangles
– Need ordered pairs for
triangles TRS, T’R’S’
Wrap up
• How is scale factor calculated? How is it
applied to create images from
transformations?
10-3-12
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Do Now
Launch
Instructions & Review
Quiz
Think about a Plan
• Similar polygons 7-2
Teacher Created Argumentation Tasks (W1-MP3&6)
•Your classmate provides the following solutions to the problems below.
In complete sentences, identify and explain the error in each explanation,
and tell me how you would help your classmate reach an accurate conclusion.
(From www.pearsonsuccessnet.com –
Chapter 7: Find the Errors! for sections 7.2-7.3)
Skill-based task
•Solve the following proportion
• Prentice Hall Geometry Textbook
1.Pg. 437 # 27 – 32
The triangles below are
H.O.T ?
Wrap up
G-SRT.2 Ratios, Proportions, Similar
Polygons
• What you will discover:
1. Determine if 2 figures are similar using properties of
transformations
2. Determine if 2 triangles are similar, given their angle
measures and side lengths
3. Calculate scale factor
4. Given 2 similar triangles determine angle measures
and side lengths
Ratio & proportions worksheet
Congruence
• 2 triangles are congruent if and only if
corresponding pairs of sides & corresponding
pairs of angles are congruent
• We will be using what we know about proving
angles congruent
AP: Skill based task
• Complete the following statements:
– Given triangle QXR congruent to triangle NYC
– A) line segment QX congruent to line segment ___
– B) Angle Y congruent to angle ____
– Explain your answer/rationale, use your notes
from class
Problem Task
• If each angle in one triangle is congruent to its
corresponding angle in another triangle, are
the two triangles congruent? Explain
Coordinates to prove simple geometric
theorems algebraically
• Given a triangle, use slopes to verify that
length and height are perpendicular
• Explore perimeter of a variety of polygons
• Textbook 1.8 Perimeter, Circumference, Area
• Google earth
• Unit 2 G.GPE.7 Worksheet 1
Essential questions
1) How do you factor a trinomial?
2) How is scale factor calculated? How is it
applied to create images from transformations?
3) What are the characteristics of similar
polygons? Similar triangles? What is AA
Similarity Theorem?
4) How do you prove the Pythagorean Theorem?
How can the Pythagorean theorem prove
distance formula?
Essential Question
• 5) How can coordinates be used to calculate
area of triangles? What about the perimeter
of other polygons?
Perform Arithmetic Operations on
Polynomials
• Interpreting algebraic expressions
• Create algebraic operations with polynomials
Skill-based task
Solve the following equations:
1.Solve the following equations
a. x2 + 9x = 36
b. 3x2 + 8x + 2 = 0
c. 4x2 – 25 = 0
1.A square has a side length of (x – 2). Write the equations that represent the
perimeter and area of the square.
2.A rectangle has an area of 8 square yards and is represented by the equation
(10x^2 + 5x).
a. Find the equations that represent the dimensions of the rug.
b. Solve for x.
3.The width of a rectangle is six less than two times its length.
a. Write the equation that represents the width.
b. What equation represents the perimeter of this rectangle?
c. What equation represents the area of this rectangle?
d. What is the smallest possible length of the rectangle?
Problem Task
1)Jake determines that the area of his square is represented by the
equation (x^2 + 81). Is he correct? If so, explain why. If not, how
would you explain to him why he is incorrect?
2) The area of Jane’s rectangular dining room table is represented by
the equation (x^2 + 8x + 15). If the width is represented by the
equation (x + 3), write an equation to represent the perimeter of the
table.
3) The perimeter of a rectangular playground is represented by the
equation (6x + 6).
If the length is (x + 4), what equation represents the width?
Using your answer from part a, what is the area of this
playground?
4) The sides of a triangle are represented by the equations 2x+1, 4x,
and 5x – 5. If this triangle is dilated by a scale factor of 2x, what will
be the perimeter?
Problem Task (H.O.T. ?)
http://illustrativemathematics
.org/illustrations/603 – “Are
they similar” activity
Jan uses an overhead
projector to enlarge a picture
5 in. high and 7 in. wide. She
projects the picture on a
blackboard 4 ft 2 in. high and
12 ft wide. What are the
dimensions of the largest
picture that can be projected
on the blackboard?
Pythagorean Theorem
10-5-12
Understand and Apply Pythagorean
Theorem
• Know that in a right triangle
• a^2 + b^2 = c^2 (Pythagorean Thm)
• Explore various proofs of the Pythagorean
Thm
• Students find examples of right triangles in
your own personal environment
Resources
Textbook Correlation: 8.1 The Pythagorean Theorem
and its Converse
“Proofs of the Pythagorean Theorem” Activity
http://map.mathshell.org/materials/download.php?fileid=804
MARS Tasks (HS):
E04: Proofs Of The Pythagorean Theorem
E08: Pythagorean Triples
MARS Problem Solving Lesson (HS): Proofs of the
Pythagorean Theorem
Texas Instrument 8.G.6 Lessons
Texas Instrument 8.G.7 Lessons
CMP2 Resources
Skill-based Task
Solve for x in each problem below
5. If the height of a cone is 10 meters and the radius is 6 meters,
what is the slant height?
Problem Task
Prove the Pythagorean Theorem: a² + b² = c².
Explain the logical reasoning behind a proof of the Pythagorean
Theorem. (Why did it make sense to prove the theorem using this
method?)
Investigate the historical context of one of the proofs of the
Pythagorean Theorem and present the proof in context to the class.
TVs are measured along their diagonal to find their dimension. How
does a 52-inch HD (wide-screen) TV compare to a traditional 52-inch
(full screen) TV?
A 65 ft. ladder is propped against a building and reaches a point 33 ft.
high. If the base of the ladder slides 7ft away from the building, what is
the new height reached by the ladder?
Theorems
• Derive distance formula using Pythagorean
Thm
• Derive midpoint formula using Pythagorean
Thm
• Overlap a map with coordinate grid and use
the Pythagorean Thm to find the distance
between two locations
Skill Based
• Using the Pythagorean Theorem, find the
distance between (4, 2) and (7, 10).
• Worksheet from Pennsylvania Department of
Education Website – “FOG – Unit 2 – 8.G.8
Worksheet 1”
Problem Task
• Using the Pythagorean Theorem to prove the distance
formula.
• (Teachers: For assistance, review pg. 52 of the textbook
and the following website. Select the link that says
“The distance between any two points” -http://www.themathpage.com/alg/pythagoreandistance.htm )
•
• List 3 coordinate pairs that are 5 units away from the
origin in the first quadrant. Describe how to find the
points and justify your reasoning. (Note: Points on the
axes are not in the quadrant.)
• Skill-based task
• Calculate the area of triangle ABC with
altitude CD, given
• A(-4, -2), B(8, 7), C(1, 8) and D(4, 4)
• Problem Task
• Find the perimeter of a real-world shape using a
coordinate grid and Google Earth.
• Jack is building a play area for his dog. On the
blueprint, the coordinates are located at (1, 6) (4,
2) (6, 10) (8, 2) and (11, 6).
– What is the perimeter of the play area if the units are
in feet?
• If fencing is $3 per yard, how much will it cost to
fence the entire area?
Problem Task
Locate the center of dilation and scale factor in the following pair of triangles.