Level 2 Geometry
Spring 2012
Ms. Katz
Day 1: January 30th
Objective: Form and meet study teams. Then work together
to build symmetrical designs using the same basic
shapes.
• Seats and Fill out Index Card (questions on next slide)
• Introduction: Ms. Katz, Books, Syllabus,
Homework Record, Expectations
• Problems 1-1 and 1-2
• Möbius Strip Demonstration
• Conclusion
Homework: Have parent/guardian fill out last page of
syllabus and sign; Problems 1-3 to 1-7 AND 1-17 to 1-18;
Extra credit tissues or hand sanitizer (1)
Respond on Index Card:
1.
2.
3.
4.
When did you take Algebra 1?
Who was your Algebra 1 teacher?
What grade do you think you earned in Algebra 1?
What is one concept/topic from Algebra 1 that Ms. Katz
could help you learn better?
5. What grade would you like to earn in Geometry?
(Be realistic)
6. What sports/clubs are you involved in this Spring?
7. My e-mail address (for teacher purposes only) is:
Support
• www.cpm.org
–
–
–
–
Resources (including worksheets from class)
Extra support/practice
Parent Guide
Homework Help
• www.hotmath.com
– All the problems from the book
– Homework help and answers
• My Webpage on the HHS website
– Classwork and Homework Assignments
– Worksheets
– Extra Resources
Quilts
1-1: First Resource Page
1-1: Second Resource Page
Write
sentence
and
names
around
the gap.
Cut
along
dotted
line
Glue sticks are rewarded when 4 unique symmetrical designs
are shown to the teacher.
Day 2: January 31st
Objective: Use your spatial visualization skills to investigate reflection.
THEN Understand the three rigid transformations (translations,
reflections, and rotations) and learn some connections between them.
Also, introduce notation for corresponding parts.
•
•
•
•
•
•
•
Homework Check and Correct (in red) – Collect last page of syllabus
“Try This!” Algebra Review (x2)
LL – “Graphing an Equation”
Problems 1-48 to 1-51, 1-53
Problems 1-59 to 1-61
LL – “Rigid Transformations”
Conclusion
Homework:
Problems 1-54 to 1-58 AND 1-63 to 1-67; GET
SUPPLIES; Extra credit tissues or hand sanitizer (1)
Try This! Algebra Review
1. Complete the table below for y = -2x+5
x
-3
-1
0
2
4
7
y
11
7
5
1
-3
-9
1. Write a rule relating x and y for the table
below.
y = 3x+4
x
1
2
3
4
5
6
y
7
10
13
16
19
22
A Complete Graph
•
y = -2x+5
Create a table of x-values
x -4 -3 -2 -1 0 1 2 3 4
y 13 11 9 7 5 3 1 -1 -3
•
•
•
y
10
5
y = -2x+5
x
-10
-5
5
-5
Use the equation to find
-10
y-values
Complete the graph by scaling and
labeling the axes
Graph and connect the points from your
table. Then label the line.
10
Try This! Algebra Review
Solve the following Equation for x and check
your answer:
6x + 3 – 10 = x + 47 + 2x
Solving Linear Equations (pg 19)
• Simplify each side: Combine like terms
• Keep the equation balanced: Anything added or taken
away from one side, must also be added or taken
away from the other
• Move the x-terms to one side of the equations: Isolate
the letters on one side
• Undo operations: Remember that addition and
subtraction are opposites AND division and
multiplication are opposites
Day 3: February 1st
Objective: Understand the three rigid transformations (translations, reflections,
and rotations) and learn some connections between them. Also, introduce
notation for corresponding parts. THEN Begin to develop an understanding
of reflection symmetry. Also, learn how to translate a geometric figure on a
coordinate grid. Learn that reflection and reflection symmetry can help
unlock relationships within a shape (isosceles triangle).
•
•
•
•
•
•
Homework Check and Correct (in red)
Try This!
Problems 1-59 to 1-61
LL – “Rigid Transformations”
Problems 1-68 to 1-72
Start Problems 1-87 to 1-89 (Notes if time)
Homework:
Problems 1-73 to 1-77 AND 1-82, 85, 86; GET SUPPLIES;
Extra credit tissues or hand sanitizer
Try This! February 1st
The distance along a straight road is
measured as shown in the diagram
below. If the distance between towns A
and C is 67 miles, find the following:
1. The value of x.
2. The distance between A and B.
5x – 2
A
2x + 6
B
C
Transformation (pg 34)
Transformation: A movement that preserves size and shape
Reflection: Mirror
image over a line
Translation: Slide in a direction
Rotation: Turning about
a point clockwise or
counter clockwise
Everyday Life Situations
Here are some situations that occur in everyday life. Each one
involves one or more of the basic transformations: reflection,
rotation, or translation.
State the transformation(s) involved in each case.
a.
b.
c.
d.
e.
f.
You look in a mirror as you comb your hair.
While repairing your bicycle, you turn it upside down and spin
the front tire to make sure it isn’t rubbing against the frame.
You move a small statue from one end of a shelf to the other.
You flip your scrumptious buckwheat pancakes as you cook
them on the griddle.
The bus tire spins as the bus moves down the road.
You examine footprints made in the sand as you walked on the
beach.
Day 4: February 2nd
Objective: Begin to develop an understanding of reflection symmetry. Also,
learn how to translate a geometric figure on a coordinate grid. Learn that
reflection and reflection symmetry can help unlock relationships within a
shape (isosceles triangle). THEN Learn about reflection, rotation, and
translation symmetry. Identify which common shapes have each type of
symmetry.
•
•
•
•
•
•
Homework Check and Correct (in red)
Finish Problems 1-70 to 1-72
LL – Notes
Problems 1-87 to 1-89
LL – Notes
Start Problem 1-97 if time
Homework:
Problems 1-92 to 1-96 AND 1-100; GET SUPPLIES;
Extra credit tissues or hand sanitizer
1-71 Reflections
1. Lines that connect corresponding points
are perpendicular
___________ to the line of reflection.
2. The line of reflection bisects
______ each of the
segments connecting a point and its
image.
1-72
B
A
A’
Isosceles Triangle
Sides: AT LEAST
two sides of
equal length
Base Angles:
Have the same
measure
Height: Perpendicular
to the base AND splits
the base in half
1-72 Isosceles Triangles
1. Two sides are _____
equal .
2. The ____
base angles are equal.
3. The line of reflection bisects
______ the base.
Reflection across a Side
The two shapes MUST meet at a side that
has the same length.
Polygons (pg 42)
Polygon: A closed figure made up of straight
segments.
Regular Polygon: The sides are all the same
length and its angles have equal measure.
Line: Slope-Intercept Form (pg 47)
y = mx + b
Slope
Slope: Growth or rate of change.
y-intercept
y
m
x
y-intercept: Starting point on the y-axis. (0,b)
Slope-Intercept Form
3
y  x 3
2
You
Next,
use
rise
Firstcan
plotgo
the
Now
connect
backwards
if
over
run towith
plot
y-intercept
on
the
points
new
points
theneed!
y-axis
ayou
line!
Parallel Lines (pg 47)
Parallel lines do not intersect.
Parallel lines have the same slope.
For example:
5
y  x 4
2
and
5
y   x 1
2
Perpendicular Lines (pg 47)
Perpendicular lines intersect at a right angle.
Slopes of perpendicular lines are opposite
reciprocals (opposite signs and flipped).
For example:
2
3
y   x  5 and y  x  1
2
3
Day 5: February 3rd
Objective: Begin to develop an understanding of reflection symmetry. Also,
learn how to translate a geometric figure on a coordinate grid. Learn that
reflection and reflection symmetry can help unlock relationships within a
shape (isosceles triangle). THEN Learn about reflection, rotation, and
translation symmetry. Identify which common shapes have each type of
symmetry.
•
•
•
•
•
Homework Check and Correct (in red)
Wrap-Up Problem 1-89
LL – Notes
Problem 1-98
Problems 1-104 to 1-107
Homework:
Problems 1-101 to 1-103 AND 1-110 to 1-114; SUPPLIES;
Chapter 1 Team Test Monday
Symmetry
Symmetry: Refers to the ability to perform a transformation without
changing the orientation or position of an object
Reflection Symmetry: If a shape has reflection symmetry, then it
remains unchanged when it is reflected across a line of
symmetry. (i.e. “M” or “Y” with a vertical line of reflection)
Rotation Symmetry: If a shape has rotation symmetry, then it can
be rotated a certain number of degrees (less than 360°) about
a point and remain unchanged.
Translation Symmetry: If a shape has translation symmetry, then it
can be translated and remain unchanged. (i.e. a line)
Venn Diagram
#1: Has two or
more siblings
#2: Speaks at least
two languages
Venn Diagrams (pg 42)
Condition #1
Condition #2
Satisfies
condition 2
only
Satisfies
condition 1
only
A
B
C
Satisfies
neither
condition
Satisfies both
conditions
D
Problem 1-98(a)
#1: Has at least one
pair of parallel sides
#2: Has at least two
sides of equal length
Problem 1-98(a)
Has at least one pair
of parallel sides
Both
Has at least two
sides of equal length
Neither
Problem 1-98(b)
Has only three sides
Both
Has a right angle
Neither
Problem 1-98(c)
Has reflection
symmetry
Both
Has 180° rotation
symmetry
Neither
Describing a Shape
Shape Toolkit
Shape Toolkit
Day 6: February 6th
Objective: Assess Chapter 1 in a team setting. THEN Develop
an intuitive understanding of probability, and apply simple
probability using the shapes in the Shape Bucket.
•
•
•
•
Homework Check and Correct (in red)
Try This! Algebra Review
Chapter 1 Team Test
Problems 1-115, 116, 119
Homework:
Problems 1-121 to 1-125 AND CL1-126 to 1-129;
Chapter 1 Individual Test Friday
Try This! February 6th
Solve the following equations for x:
1.
x2 4

4
8
2.
14
7

x 1 4
Probability (pg 60)
Probability: a measure of the likelihood that an event will
occur at random.
Number of Desired Outcomes
P  event  
Total Possible Outcomes
Example: What is the probability of selecting a heart
from a deck of cards?
Number of Hearts
13 1
P  heart  

  0.25  25%
Total Number of Cards 52 4
Day 7: February 7th
Objective: Develop an intuitive understanding of probability, and
apply simple probability using the shapes in the Shape Bucket.
THEN Learn how to name angles, and learn the three main
relationships for angle measures, namely supplementary,
complementary, and congruent. Also, discover a property of
vertical angles.
•
•
•
•
Homework Check and Correct (in red)
Try This! Algebra Review
Problems 1-116, 119
Problems 2-1 to 2-7
Homework:
Problems CL1-130 to 1-134 AND 2-8 to 2-11;
Chapter 1 Individual Test Friday
Shape Bucket
2-2
A
C’
B
B’
C
a. mA  mB  mC
b.
6
c. mCAC or mCAC
Notation for Angles
F
E
D
Name
DEF
or
FED
If there is only one angle at the
vertex, you can also name the
angle using the vertex: E
Y
W
X
Z
?
Incorrect:
X
?
Measure
mDEF  45
Correct:
mA  mB
Incorrect:
DEF  45
A  B
Angle Relationships (pg 76)
Complementary Angles: Two
angles that have measures that
add up to 90°.
30°
x°
60°
y°
x° + y° = 90°
Supplementary Angles: Two angles
that have measures that add up
to 180°.
Example: Straight angle
Congruent Angles: Two angles that
have measures that are equal.
Example: Vertical angles
70°
110°
x° y°
x° + y° = 180°
85°
85°
x°
y°
x° = y°
Day 8: February 8th
Objective: Use our understanding of translation to determine that when a
transversal intersects parallel lines, a relationship exists between
corresponding angles. Also, continue to practice using angle
relationships to solve for unknown angles. THEN Practice naming
angles and stating angle relationships.
•
•
•
•
•
Homework Check and Correct (in red)
Distributive Property: Algebra Review
Finish Problems 2-5 to 2-6
Problems 2-13 to 2-17
“Naming Angles 2” Worksheet
Homework:
Problems 2-18 to 2-22
Chapter 1 Individual Test Friday
Distributive Property
The two methods below multiply two expressions and
rewrite a product into a sum.
Note: There must be two sets of parentheses:
( x – 3 )2 = ( x – 3) ( x – 3 )
FOIL
Box Method
( x + 5 )( x + 3 )
+5 +5x
+15
x
+3x
x2
x
+3
x2 + 8x + 15
•
•
•
•
•
Firsts
( 3x – 2 )( 2x + 7)
Outers
Inners
6x2 + 21x + -4x + -14
Lasts
2 + 17x – 14
=
6x
Simplify
Marcos’ Tile Pattern
How can you create a tile pattern with a
single parallelogram?
Marcos’ Tile Pattern
a. Are opposite angles of a parallelogram
congruent?
Pick one parallelogram on your paper. Use color
to show which angles have equal measure. If
two measures are not equal, make sure they are
different colors.
Marcos’ Tile Pattern
b. What does this mean in terms of the
angles in our pattern? Color all angles that
must be equal the same color.
Marcos’ Tile Pattern
c. Are any lines parallel in the pattern? Mark
all lines on your diagram with the same
number of arrows to show which lines are
parallel.
Marcos’ Tile Pattern
J
a
L
c
w
N
y
b
M
d
x
P
z
K
Use the following diagram to help answer
question 2-15.
Day 9: February 9th
Objective: Use our understanding of translation to determine that when a
transversal intersects parallel lines, a relationship exists between
corresponding angles. Also, continue to practice using angle
relationships to solve for unknown angles. THEN Practice naming
angles and stating angle relationships.
•
•
•
•
•
•
Homework Check and Correct (in red)
Finish Problems 2-16 to 2-17
“Naming Angles 2” Worksheet
Review Chapter 1 Team Test and Algebra Concepts
Problems 2-23 to 2-25
More Chapter 1 Review if time
Homework:
Problems 2-29 to 2-33
Chapter 1 Individual Test TOMORROW
Why Parallel Lines?
53°
x
2-16
X
X
Day 10: February 10th
Objective: Assess Chapter 1 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Take the test
• Second: Check your work
• Third: Hand the test to Ms. Katz when you’re done
• Fourth: Correct last night’s homework
Homework:
Worksheet: “Angles and Parallel Lines”
WILL BE COLLECTED AND GRADED ON
CORRECTNESS ON MONDAY – SHOW
WORK WHEN POSSIBLE!
Day 11: February 13th
Objective: Apply knowledge of corresponding angles, and develop
conjectures about alternate interior and same-side interior angles. Also,
learn that when a light beam reflects off a mirror, the angle of the light
hitting the mirror equals the angle of the light leaving the mirror. THEN
Discover the triangle angle sum theorem, and practice finding angles in
complex diagrams that use multiple relationships.
•
•
•
•
Homework Check and Correct (in red)
Problems 2-23 to 2-28
Problems 2-34 to 2-37
Conclusion
Homework:
Problems 2-38 to 2-42
2-23 (a)
a
b
More Angles formed by Transversals
132° 48°
48° 132°
132° 48°
48° 132°
>
>
a. Alternateb.Interior(1) Same Side
(2) Interior
(3)
Angles formed by Parallel Lines and a
Transversal
Corresponding - Congruent
b
>
a=b
a
>
100°
100° >
>
Alternate Interior - Congruent
b
a
>
a=b
>
22°
22°
>
>
Same-Side Interior - Supplementary
b
a
>
a + b = 180°
>
60°
120°
>
>
Hands-On Activity
1. Draw a large triangle (about 4 to 5 inches
wide) using a ruler.
2. Make sure that your triangle looks
different than the other triangles in your
group.
3. Use scissors to cut out your triangle.
4. Tear-off the angles of your triangle.
5. Connect the three vertices of the torn
angles
Triangle Angle Sum Theorem
The measures of the angles in a triangle add up to
180°.
mA

mB

mC
180
Example:
B
45°
65°
A
70°
C
Day 12: February 14th
Objective: Discover the triangle angle sum theorem, and practice finding
angles in complex diagrams that use multiple relationships. THEN
Learn the converses of some of the angle conjectures and see
arguments for them. Also, apply knowledge of angle relationships to
analyze the hinged mirror trick seen in Lesson 2.1.1. THEN Learn
how to find the area of a triangle and develop multiple methods to find
the area of composite shapes formed by rectangles and triangles.
•
•
•
•
•
Homework Check and Correct (in red) & Quick Warm-Up
Finish Problems 2-35 to 2-37
Problems 2-43 to 2-48
Start Problems 2-66 to 2-69
Conclusion
Homework:
Problems 2-51 to 2-54 AND 2-62 to 2-65
Warm Up! February 14th
Name the relationship between these pairs of angles:
1.
2.
3.
4.
5.
b and d
a and x
d and w
c and w
x and y
b
c
x
w
z
y
a
d
Possible Choices:
Vertical Angles
Straight Angle
Alternate-Interior Angles
Corresponding Angles
Same-side Interior Angles
2-37: Challenge!
f
g
h
k
m
p
m
57° 123°
h k57°
123°
99°
p 81°
q
g 99°
81°
f
q
r
s
u
v
42°
s
r
81°
57°
v 57°
u
123°
2-43 and 2-44
>
x
y
>
2-43 and 2-44
A
100°
C
B
E
80°
D
2-43 and 2-44
>
112°
68°
>
2-45
80°
>
100°
80°
>
80°100°
80°
>
>
If Same-Side Interior angles are supplementary, then
the lines must be parallel.
If Corresponding angles are congruent, then the lines
must be parallel.
If Alternate Interior angles are congruent, then the
lines must be parallel.
Day 13: February 15th
Objective: Learn how to find the area of a triangle and develop multiple
methods to find the area of composite shapes formed by rectangles
and triangles. THEN Use rectangles and triangles to develop
algorithms to find the area of new shapes, including parallelograms
and trapezoids.
•
•
•
•
Homework Check and Correct (in red) & Quick Warm-Up
Problems 2-66 to 2-69
Problems 2-75 to 2-79
Conclusion
Homework:
Problems 2-70 to 2-74 AND 2-81 to 2-85
Area of a Right Triangle
What is the area of the right triangle below?
Why?
4 cm
10 cm
What about non-right triangles?
Height
Height
Where is the Height & Base
Base
Height
Base
Base
Height
Obtuse Triangle
Extra
Base
Area of Obtuse Triangle = Area of Right Triangle
= ½ (Base)(Height)
Area of a Triangle
The area of a triangle is one half the base
times the height.
Base
Base
Height
Height
Height
1
A  bh
2
Base
Day 14: February 16th
Objective: Use rectangles and triangles to develop algorithms to find the
area of new shapes, including parallelograms and trapezoids. THEN
Explore how to find the height of a triangle given that one side has
been specified as the base. Additionally, find the areas of composite
shapes using the areas of triangles, parallelograms, and trapezoids.
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Problems 2-75 to 2-79
Problems 2-86 to 2-89
Conclusion
Homework:
Problems 2-90 to 2-94
Optional E.C: Do Problem 2-49 on a separate sheet of
paper and hand it in on Monday. It must be neat and wellexplained to be considered for credit.
Warm-Up! February 16th
Answer the following questions:
1. The area of a triangle is 40 in2 and the base is 8
inches. What is the length of the height?
2. Find the value of x in the figure below if the area
of the triangle is 60 in2.
2x + 1
8 in
Can We find the Area?
YES!
YES!
YES!
YES!
YES!
YES!
YES!
YES!
Area of a Parallelogram
Height h
h
h
Base
b
h
Area Rectangle!
= b.h
Area of a Parallelogram
h
b
.
Area = b h
Area of a Parallelogram
The area of a parallelogram is the
base times the height.
.
Area = b h
h
b
20
Ex:
13
5
20
13
A = 20.5 = 100
Area of a Trapezoid
b21
b1
h
Base One
h
Height
b2
h
Base Two
b2
b21
b1
Duplicate
Parallelogram!
Translate
Reflect
Area
= (b1 + b2) h
Area of a Trapezoid
b1
h
b2
1
Area =  b1  b2  h
2
Area of a Trapezoid
The area of a trapezoid is half of the sum of
the bases times the height.
b1
1
Area =  b1  b2  h
2
h
b2
Ex:
9
5
5
4
15
A = ½ (9+15) 4 = ½ . 24 . 4 = 48
Answers to 2-79
a. 0.5(16)9 = 72 sq. un
b. 26(14) = 364 sq. un
c. 11(11) = 121 sq. un
d. 0.5(6+21)8 = 108 sq. un
Day 15: February 17th
Objective: Explore how to find the height of a triangle given that one side
has been specified as the base. Additionally, find the areas of
composite shapes using the areas of triangles, parallelograms, and
trapezoids. THEN Review the meaning of square root. Also,
recognize how a square can help find the length of a hypotenuse of a
right triangle.
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Problems 2-86 to 2-89
Problems 2-95 to 2-97
Conclusion
Homework:
Problems 2-100 to 2-104 (Skip 101)
Optional E.C: Do Problem 2-49 on a separate sheet of
paper and hand it in on Monday. It must be neat and wellexplained to be considered for credit.
Warm Up! February 17th
Solve for x in both diagrams
-(x – 36°)
>
7 + 4x units
2x + 9°
>
7 units
The area of the polygon
above is 357 un2.
Note card = Height Locator
Base
“Weight”
Day 16: February 21st
Objective: Review the meaning of square root. Also, recognize how a
square can help find the length of a hypotenuse of a right triangle. THEN
Learn how to determine whether or not three given lengths can make a
triangle. Also, understand how to find the maximum and minimum
lengths of a third side given the lengths of the other two sides.
•
•
•
•
•
Homework Check and Correct (in red) & Collect Optional E.C.
Quick Warm-Up!
Problems 2-96 to 2-97
Problems 2-105, 2-106 to 2-108
Conclusion
Homework:
Problems 2-109 to 2-113
Chapter 2 Team Test Tomorrow
[Review transformations and angle relationship vocabulary]
Warm Up! February 21st
Solve the 2 equations for x. Are there more
solutions not listed?
1. x2 = 15 + 13
[A] 784
[B] 5.29
[C] 28
2. x2 + 9 = 130
[A] 11.79
[B] 11
[C] 8.40 [D] 121
[D] 5
Day 17: February 22nd
Objective: Assess Chapter 2 in a team setting. THEN Learn how
to determine whether or not three given lengths can make a
triangle. Also, understand how to find the maximum and
minimum lengths of a third side given the lengths of the other
two sides.
•
•
•
•
Homework Check and Correct (in red)
Chapter 2 Team Test
Problems 2-105, 2-106 to 2-108
Conclusion
Homework:
Problems 2-118 to 2-122
Chapter 2 Individual Test Tuesday
Pink Slip
Can these three side lengths form a triangle? Why?
a) 12, 4, 8
b) 13, 10, 5
c) 11, 9, 30
Triangle Inequality
Each side must be shorter than the sum of
the lengths of the other two sides and longer
than the difference of the other two sides.
b
a–b<c<a+b
a
a–c<b<a+c
c
b–c<a<b+c
Triangle Inequality
Longest Side: Slightly less than the sum of the two
shorter sides
Shortest Side: Slightly more than the difference of the
two shorter sides
Day 18: February 23rd
Objective: Learn how to determine whether or not three given
lengths can make a triangle. Also, understand how to find the
maximum and minimum lengths of a third side given the lengths
of the other two sides. THEN Develop and prove the
Pythagorean Theorem.
•
•
•
•
Homework Check and Correct (in red)
Finish Problems 2-105, 2-106 to 2-108
Problems 2-114 to 2-117
Conclusion
Homework:
Problems CL2-123 to 2-131
Chapter 2 Individual Test Tuesday
The Pythagorean Theorem
a
a
c
b
a
c
c
a
b
2
a +b =c
2
2
b
c
b
c
a
b
2
c
a
c
b
b
c
2
a
a
b
c
a
b
2
Pythagorean Theorem
Leg
B
a
C
2
b
2
2
a +b =c
c
A
Leg
When to use it:
•
If you have a right triangle
•
You need to solve for a side length
•
If two sides lengths are known
Practice Problem
Solve for x
6 in
x
7 in
Do you need to
solve for a side or
angle?
Do you have
two sides or
a side and an angle?
Pythagorean
Theorem
Practice Problem
Solve for x
9m
5m
x
Do you need to
solve for a side or
angle?
Do you have
two sides or
a side and an angle?
Pythagorean
Theorem
Day 19: February 24th
Objective: Learn the concept of similarity and investigate the
characteristics that figures share if they have the same shape.
Determine that two geometric figures must have equal angles to
have the same shape. Additionally, introduce the idea that similar
shapes have proportional corresponding side lengths.
•
•
•
•
•
Homework Check and Correct (in red)
Review Chapter 2 Team Test
Problems 3-1 to 3-5
Time? More Chapter 2 Review Time
Conclusion
Homework:
Problems 3-5 to 3-10
Chapter 2 Individual Test Tuesday
Dilation
A transformation that
shrinks or stretches a
shape proportionally in all
directions.
Enlarging
Day 20: February 27th
Objective: Determine that multiplying (and dividing) lengths of shapes by
a common number (zoom factor) produces a similar shape. Use the
equivalent ratios to find missing lengths in similar figures and learn
about congruent shapes. THEN Examine the ratio of the perimeters of
similar figures, and practice setting up and solving equations to solve
proportional problems.
•
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Review Problems 3-5 and 3-10 and Terms: “Dilation” and “Similar”
Problems 3-11 to 3-15
Start Problems 3-22 to 3-25
Conclusion
Homework:
Problems 3-17 to 3-21 AND STUDY
Chapter 2 Individual Test Tomorrow
Do this in your graph notebook:
A triangle has the following coordinates:
(-3,4), (2,4), and (2,-1)
1. Plot and connect the points on a graph that
goes from -10 to 10 on both axes.
2. Find the area of the triangle.
3. Find the length of the hypotenuse.
4. Find the perimeter.
Chapter 1-2 Topics
Angles:
• Acute, Obtuse, Right, Straight, Circular – p. 24
• Complementary, Supplementary, Congruent – p. 76
• Vertical, Corresponding, Same-Side Interior, Alternate
Interior – Toolkit and p. 91
Lines:
• Slopes of parallel and perpendicular lines – p. 47
Transformations:
• Reflection, Rotation, Translation, and Prime Notation – p.81
Shapes:
• Name/Define shapes – Toolkit
Probability:
• Use proper notation…Ex: P(choosing a King) = 4/52 = 1/13
– Page 60
Chapter 1-2 Topics
Triangles:
• Triangle Angle Sum Theorem – p.100
• Area
• Triangle Inequality Theorem
Area:
• Triangle, Parallelogram, Rectangle, Trapezoid, Square
– Page 112 and Learning Log/Toolkit
Pythagorean Theorem & Square Roots – p. 115 and 123
3-10
Similar Figures
Exactly same shape but not
necessarily same size
• Corresponding Angles are congruent
• The ratios between corresponding sides
are equal
21
127°
7
5
127°
90°
15
90°
12
4
53°
90°
10
53°
90°
30
Zoom Factor
The number each side is multiplied
by to enlarge or reduce the figure
x2
x2
Example:
18
3
9
12
x2
24
Zoom Factor = 2
6
Notation
Angle ABC
Line Segment XY
ABC
XY
mABC
XY
The Measure of
Angle ABC
The Length of
line segment XY
Notation
Acceptable
Not Acceptable
mR  mT
R   T
KT  GB
KT  GB
Day 21: February 28th
Objective: Assess Chapter 2 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Take the test
• Second: Check your work
• Third: Hand the test to Ms. Katz when you’re done
• Fourth: Correct last night’s homework
Homework:
Problems 3-27 to 3-31
Day 22: February 29th
Objective: Examine the ratio of the perimeters of similar figures, and
practice setting up and solving equations to solve proportional
problems. THEN Apply proportional reasoning and learn how to write
similarity statements.
•
•
•
•
•
Homework Check and Correct (in red)
Review Problem 3-29 as a class
Finish Problems 3-22 to 3-25
Problems 3-32 to 3-37
Conclusion
Homework:
Problems 3-38 to 3-42
Notation
Angle ABC
Line Segment XY
ABC
XY
mABC
XY
The Measure of
Angle ABC
The Length of
line segment XY
Notation
Acceptable
Not Acceptable
mR  mT
R   T
KT  GB
KT  GB
George Washington’s Nose
720 in
60 ft
in
? ft
? ft
in
ft
? in
Day 23: March 1st
Objective: Apply proportional reasoning and learn how to write similarity
statements. THEN Learn the SSS~ and AA~ conjectures for
determining triangle similarity.
•
•
•
•
Homework Check and Correct (in red)
Finish Problems 3-32 to 3-37
Problem 3-43
Conclusion
Homework:
Problems 3-48 to 3-52 (SKIP 3-49)
Writing a Similarity Statement
Example: ΔDEF~ΔRST
The order of the letters determines which
sides and angles correspond.
B
Z
C
Y
A
ΔABC
ABC ~ ΔZXY
X
Writing a Proportion
B
s
C
W
13
A
25
X
10
D
Z
AB
ABCD
WXYZ
BC ~ WX
XY
WX
AB
=
XY
BC
25
13
=
s
10
Y
Day 24: March 5th
Objective: Apply proportional reasoning and learn how to write similarity
statements. THEN Learn the SSS~ and AA~ conjectures for
determining triangle similarity.
•
•
•
•
•
•
Homework Check and Correct (in red)
Review of Classroom Expectations
Finish Problems 3-35 to 3-37
Problems 3-43 to 3-47
Review Chapter 2 Individual Test
Conclusion
Homework:
Problem 3-49, AND Worksheet #2,3,6,7,8 – Show work!
[Worksheet will be collected and graded on accuracy.]
Warm Up! March 5th
1.The figures are
drawn to scale and
are similar, find the
length of x and y:
2.Figure ABCD is
similar to WXYZ.
Find the length of z:
A 15 B
8
6
10
z
12
x
3
W 18 Z
X
y
D
C
Y
First Two Similarity Conjectures
SSS Triangle Similarity (SSS~)
If all three corresponding side lengths share
a common ratio, then the triangles are
similar.
AA Triangle Similarity (AA~)
If two pairs of angles have equal measure,
then the triangles are similar.
Day 25: March 6th
Objective: Learn the SSS~ and AA~ conjectures for determining triangle
similarity. THEN Learn how to use flowcharts to organize arguments
for triangle similarity, and continue to practice applying the AA~ and
SSS~ conjectures.
•
•
•
•
•
Homework Check and Correct (in red)
Finish Problems 3-46 to 3-47
Problems 3-53 to 3-58
Start Problems 3-64 to 3-67
Conclusion
Homework:
Problems 3-59 to 3-63 (Can skip 3-62)
Warm-Up! The Triangles are Similar
1.Find PT and PR:
2.Find the length of y:
4
y
6
9
Similarity and Sides
The following is not acceptable notation:
AB ~ CD
OR
AB  CD
Acceptable:
AB  CD
3-54
T
D
3
C
16
4
2
F
12
Q
8
R
What Conjecture will we use: SSS~
Facts
12
4
3
8
4
2
16
4
4
Conclusion
ΔCDF ~ ΔRTQ SSS~
Another Example
Y
B
100°
100°
A
60°
C
60°
X
Z
What Conjecture will we use: AA~
Facts
mA  mZ
mB  mY
Conclusion
ΔABC ~ ΔZYX AA~
Day 26: March 7th
Objective: Practice making and using flowcharts in more challenging
reasoning contexts. Also, determine the relationship between two
triangles if the common ratio between the lengths of their
corresponding sides is 1. THEN Complete the list of triangle similarity
conjectures involving sides and angles, learning about the SAS~
Conjecture in the process.
•
•
•
•
•
•
Homework Check and Correct (in red)
Warm-Up!
Wrap-Up Problem 3-58 (LL Entry and Math Notes)
Problems 3-64 to 3-67
Start Problems 3-73 to 3-77
Conclusion
Homework:
Problems 3-68 to 3-72
Chapter 3 Team Test Friday
Warm Up! March 7th
Decide if the triangles (not drawn to scale) below
are similar. Use a flowchart to organize your
facts and conclusion.
O
A
18
20
B
8
27
S
12
T
45
N
Day 27: March 8th
Objective: Complete the list of triangle similarity conjectures involving
sides and angles, learning about the SAS~ Conjecture in the process.
THEN Practice using the three triangle similarity conjectures and
organizing reasoning in a flowchart.
•
•
•
•
•
•
Homework Check and Correct (in red)
Warm-Up! Finish Problem 3-66
Finish Problems 3-66 to 3-67
Problems 3-73 to 3-77
Problem 3-83
Conclusion
Homework:
Problems 3-78 to 3-82
Chapter 3 Team Test Tomorrow
Conditions for Triangle Similarity
If you are testing for similarity, you can use the
following conjectures:
SSS~
All three corresponding side lengths have
the same zoom factor
7
3
10
5
AA~
Two pairs of corresponding angles have
equal measures.
14
6
55°
40°
40°
55°
SAS~
Two pairs of corresponding lengths have
the same zoom factor and the angles
between the sides have equal measure.
40
20
70°
30
NO CONJECTURE FOR ASS~
70°
15
Day 28: March 9th
Objective: Practice using the three triangle similarity conjectures and
organizing reasoning in a flowchart. THEN Assess Chapter 3 in a
team setting.
•
•
•
•
•
•
Homework Check and Correct (in red)
Warm-Up! Start Problem 3-85
Problems 3-85 to 3-86
Chapter 3 Team Test
Time? Problem 3-93 (Interesting mirror activity)
Conclusion
Homework:
Problems 3-88 to 3-92
Chapter 3 Individual Test Thursday
You’re Getting Sleepy…
Eye
Height
Eye
Height
x cm
200 cm
Day 29: March 12th
Objective: Practice using the three triangle similarity conjectures and
organizing reasoning in a flowchart. THEN Review Chapters 1-3.
•
•
•
•
•
Homework Check and Correct (in red)
Problem 3-94
Chapter 1-3 Topics
Problems CL3-101 to CL3-105
Conclusion
Homework:
Problems 3-96 to 3-100 AND CL3-107 to CL3-110
Chapter 3 Individual Test Thursday
Lessons from Abroad
x
316 ft
12 + 930 = 942
6–2=4
12
Chapter 1-2 Topics
Angles:
• Acute, Obtuse, Right, Straight, Circular – p. 24
• Complementary, Supplementary, Congruent – p. 76
• Vertical, Corresponding, Same-Side Interior, Alternate
Interior – Toolkit and p. 91
Lines:
• Slopes of parallel and perpendicular lines – p. 47
Transformations:
• Reflection, Rotation, Translation, and Prime Notation – p.81
Shapes:
• Name/Define shapes – Toolkit
Probability:
• Use proper notation…Ex: P(choosing a King) = 4/52 = 1/13
– Page 60
Chapter 1-2 Topics
Triangles:
• Triangle Angle Sum Theorem – p.100
• Area
• Triangle Inequality Theorem
Area:
• Triangle, Parallelogram, Rectangle, Trapezoid, Square
– Page 112 and Learning Log/Toolkit
Pythagorean Theorem & Square Roots – p. 115 and 123
Chapter 3 Topics
Dilations
• Zoom Factor – p. 142
Similarity
• Writing similarity statements – p.150
• Triangle Similarity Statements: AA~, SSS~, SAS~
– Page 155 and 171
• Flowcharts
• Congruent Shapes – p. 159
Solving Quadratic Equations – p. 163
Day 30: March 13th
Objective: Recognize that all slope triangles on a given line are similar to
each other, and begin to connect a specific slope to a specific angle
measurement and ratio.
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Quick Look @ Team Tests
Start Problems 4-1 to 4-5
Conclusion
Homework:
Problems 4-6 to 4-10
Chapter 3 Individual Test Thursday
Warm Up! March 13th
1. Make a table in order to graph the following equation:
y  x  2x  3
2
[Perhaps use x-values from -5 to 5?]
2. Factor the following equation in order to solve for x:
x  2x  3  0
2
Day 31: March 14th
Objective: Recognize that all slope triangles on a given line are similar to
each other, and begin to connect a specific slope to a specific angle
measurement and ratio.
• Homework Check and Correct (in red)
• Problems 4-2 to 4-5
• Conclusion
Homework:
Angles Puzzle Worksheet
Chapter 3 Individual Test Tomorrow
Day 32: March 15th
Objective: Assess Chapter 3 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Take the test
• Second: Check your work!
• Third: Hand the test to Ms. Katz when you’re done
• Fourth: Correct last night’s homework & sit quietly
Homework:
Enjoy one free night from math homework!
Day 33: March 16th
Objective: Connect specific slope ratios to their related angles and use
this information to find missing sides or angles of right triangles with
11°, 22°, 18°, or 45° angles (and their complements). THEN Use
technology to generate slope ratios for new angles in order to solve for
missing side lengths on triangles.
•
•
•
•
Homework Check and Correct (in red)
Problems 4-11 to 4-15
Start Problems 4-21 to 4-24
Conclusion
Homework:
Problems 4-16 to 4-20
Day 34: March 19th
Objective: Use technology to generate slope ratios for new angles in
order to solve for missing side lengths on triangles. THEN Practice
using slope ratios to find the length of a leg of a right triangle and learn
that this ratio is called tangent. Also, practice re-orienting a triangle
and learn new ways to identify which leg is Δx and which is Δy.
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Finish Problems 4-22 to 4-24
Problems 4-30 to 4-35
Conclusion
Homework:
Problems 4-25 to 4-29 (Skip 28) AND
Problems 4-36 to 4-40 (Skip 39)
Warm-Up! March 19th
Solve for x
x
68°
25 cm
Day 35: March 20th
Objective: Practice using slope ratios to find the length of a leg of
a right triangle and learn that this ratio is called tangent. Also,
practice re-orienting a triangle and learn new ways to identify
which leg is Δx and which is Δy. THEN Apply knowledge of
tangent ratios to find measurements about the classroom.
•
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Finish Problems 4-34 to 4-35
Problems 4-41 to 4-42
Review Chapter 3 Individual Test
Conclusion
Homework:
Problems 4-43 to 4-47
Trigonometry
Theta ( ) is always an acute angle
Opposite
(across from the
known angle)
Δy
h
Hypotenuse
(across from the
90° angle)
Δx
Adjacent
(forms the known angle)
Trigonometry
Theta ( ) is always an acute angle
Opposite
(across from the o
known angle)
h
Hypotenuse
(across from the
90° angle)
a
Adjacent
(forms the known angle)
Trigonometry (LL)
Opposite
Theta ( ) is always an acute angle
h
Adjacent
Trigonometry (LL)
Adjacent
Theta ( ) is always an acute angle
h
Opposite
Day 36: March 21st
Objective: Review the tangent ratio. THEN Learn how to list
outcomes systematically and organize outcomes in a tree
diagram. THEN Continue to use tree diagrams and introduce a
table to analyze probability problems. Also, investigate the
difference between theoretical and experimental probability.
•
•
•
•
•
Homework Check and Correct (in red) & Warm-Ups!
Review Tangent (Practice Problems)
Problems 4-49 to 4-53
Problem 4-59
Conclusion
Homework:
Problems 4-54 to 4-58
Chapter 4 Team Test Friday
Warm Up! March 20th
1. Find the length of x:
9
7
x
4
1. If a bag contains 6 yellow, 10 red, and 8
green marbles. What is the probability of
selecting a red marble at random.
1
A
4
5
B
12
1
C
3
7
D
12
Warm Up! March 21st
Multiply the following expressions using
an area diagram:
1.  x  7  x  2 
x  5 x  14
2
2.  2 x  2 3x  1 
6 x  8x  2
2
When to use Trigonometry
1. You have a right triangle and…
2. You need to solve for a side and…
3. A side and an angle are known
Use Trigonometry
My Tree Diagram
Read
Write
S
T
A
R
T
#41
#28
#55
#81
Listen
Read
Write
Listen
Read
Write
Listen
Read
Write
Listen
One Possibility:
Take Bus #41 and
Listen to an MP3
player
Day 37: March 22nd
Objective: Continue to use tree diagrams and introduce a table to analyze
probability problems. Also, investigate the difference between
theoretical and experimental probability. THEN Learn how to use an
area model (and a generic area model) to represent a situation of
chance.
•
•
•
•
•
Homework Check and Correct (in red) & Warm-Ups!
Problem 4-60
Problems 4-68 to 4-70
Problems 4-77 to 4-78
Conclusion
Homework:
Problems 4-63 to 4-67 AND 4-72 to 4-74
Chapter 4 Team Test Tomorrow
Problem 4-71 can be counted for E.C. – see Ms. Katz for a worksheet.
(Complete individually – if I think you shared/copied, no points will be
awarded.) Due Monday
Warm Up! March 22nd
Multiply the following expressions using
an area diagram:
1.  x  7  
2
2.
 3x  2 
2
x  14 x  49
2
 9 x  12 x  4
2
Warm Up! March 22nd
Solve for x and y:
6
a b  c
2
x
60°
x
6 · tan(60)  · 6
6
6  tan(60)  x
10.39  x
2
6  10.39  y
36  107.9521  y 2
2
y
2
2
2
143.9521  y
12  y
2
4-60: Tree Diagram
S
T
A
R
T
$100
$300
Keep
$100
Double
$200
Keep
$300
Double
$600
Keep
$1500
Double
$3000
$1500
Day 38: March 23rd
Objective: Learn how to use an area model (and a generic area model)
to represent a situation of chance. THEN Assess Chapter 4 in a team
setting.
•
•
•
•
•
•
Homework Check and Correct (in red) & Warm-Up!
Finish Problem 4-77
Chapter 4 Team Test
Problems 4-78 to 4-80
Math Notes Box – Notes in LL
Conclusion
Homework:
Problems 4-75 to 4-76 AND 4-82 to 4-86
Chapter 4 Individual Test Next Friday
Problem 4-71 can be counted for E.C. – see Ms. Katz for a worksheet.
(Complete individually – if I think you shared/copied, no points will be
awarded.) Due Monday
4-77: Area Diagram
Spinner #1
Spinner #2
I
T
F
U
A
1
 2
1
 6
 3
 4
IT
UT
AT
1
1
24
1
12
 4
IF
UF
AF
3
3
3
1
3
8
8
24
1
12
Warm-Up! March 23rd
Make an area diagram to model the game
where both spinners below are used. Then find
the probabilities below:
A B
B C
1. P(A, X) =
2. P(C, Y) =
3. P(not A, Y) =
X Y
Day 39: March 26th
Objective: Develop more complex tree diagrams to model biased
probability situations. THEN Review Chapter 4 by working through
closure problems.
***NEW SEATS***
• Homework Check and Correct (in red) & Collect Optional E.C.
• Warm-Up!
• Problems 4-78 to 4-80 and Math Notes Box – Notes in LL
• Review Chapter 4 Team Test
• Problems CL4-96 to CL4-105
• Conclusion
Homework:
Problems 4-91 to 4-95 AND CL4-100 to CL4-105
Chapter 4 Individual Test Friday
[If you know you’re not going to be here due to extenuating
circumstances, you must see me ahead of time to take the test.]
Warm-Up! March 26th
Use an area model or tree diagram to answer these
questions based on the spinners below:
1.
2.
3.
If each spinner is spun once, what is the
probability that both spinners show blue?
If each spinner is spun once, what is the
probability that both spinners show the same color?
If each spinner is spun once, what is the
probability of getting a red-blue combination?
Day 40: March 27th
Objective: Review Chapter 4 by working through closure problems.
THEN Learn about the sine and cosine ratios, and start a Triangle
Toolkit.
•
•
•
•
Homework Check and Correct (in red)
Warm-Up! Slide and Do Problems CL4-96 to 4-99
Problems 5-1 to 5-6
Conclusion
Homework:
Problems 5-7 to 5-11
Chapter 4 Individual Test Friday
[If you know you’re not going to be here due to extenuating
circumstances, you must see me ahead of time to take the test.]
Warm-Up! March 27th
Solve for the length of x and y:
Step 1:
h
y
o 10 cm
71°
x
a
Step 2:
Chapter 4 Topics
Slope Angles/Ratios:y
x
Trigonometry:
• Tangent Ratio – p. 200
• Use tangent to solve for a missing side of a slope triangle
• As the slope angle increases, does the slope ratio increase or
decrease? (Look at yellow Trig Table)
• Problems like the Leaning Tower of Pisa, Statue of Liberty, etc
(Clinometer activities)
Probability:
• Tree Diagrams
• Area Models
• Equally likely events (like the bus problem)
• Biased events (like Problem 4-69 and 4-77)
• Math Notes on Page 219
Day 41: March 28th
Objective: Learn about the sine and cosine ratios, and start a Triangle
Toolkit. THEN Develop strategies to recognize which trigonometric
ratio to use based on the relative position of the reference angle and
the given sides involved.
•
•
•
•
•
Homework Check and Correct (in red)
Warm-Up!
Problems 5-5 to 5-6
Problems 5-12 to 5-15
Conclusion
Homework:
Problems 5-16 to 5-20
Chapter 4 Individual Test Friday
[If you know you’re not going to be here due to extenuating
circumstances, you must see me ahead of time to take the test.]
Warm-Up! March 28th
Find the area of the triangle:
h
o 6 cm
30°
a
or
Trigonometry
h
o
a
SohCahToa
opposite
o
sin( ) 

hypotenuse h
adjacent
a
cos( ) 

hypotenuse h
opposite o
tan( ) 

adjacent a
Day 42: March 29th
Objective: Develop strategies to recognize which trigonometric ratio to
use based on the relative position of the reference angle and the given
sides involved. THEN Use sine, cosine, and tangent ratios to solve
real world application problems.
•
•
•
•
•
Homework Check and Correct (in red)
Warm-Up!
Finish Problems 5-12 to 5-15
Problems 5-31 to 5-33
Conclusion
Homework:
Problems 5-36 to 5-40
Chapter 4 Individual Test Tomorrow – STUDY!
Warm-Up! March 29th
Are the following triangles similar? If so, make a
flowchart. If not, explain why they are not similar
and/or what information is missing.
1.
2.
Day 43: March 30th
Objective: Assess Chapter 4 in an individual setting.
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Take the test
• Second: Check your work
• Third: Give test & formula sheet to Ms. Katz when you’re done
• Fourth: Correct last night’s homework
Homework:
Problems 5-26 to 5-30
Enjoy your week away from school!
Day 44: April 10th
Objective: Review previous material. THEN Understand
how to use trig ratios to find the unknown angle measures
of a right triangle. Also, introduce the concept of “inverse.”
*Beginning of Quarter 4*
• Homework Check and Correct (in red)
• Review Chapter 4 Test in detail
• Trig Practice WS - #1, 2, 3, 4, 9, 11, 13
• Problems 5-21 to 5-25
• Conclusion
Homework: Problems #5, 6, 7, 8, 10, 12, 14 on WS
Day 45: April 11th
Objective: Understand how to use trig ratios to find the unknown
angle measures of a right triangle. Also, introduce the concept
of “inverse.” THEN Recognize the similarity ratios in 30°-60°90° and 45°-45°-90° triangles and begin to apply those ratios
as a shortcut to finding missing side lengths.
•
•
•
•
•
Homework Check and Correct (in red)
Finish Problems 5-21 to 5-25
Trig/Inverse Trig Practice Worksheet
Problems 5-41 to 5-45
Conclusion
Homework:
Problems 5-46 to 5-50 (skip 49)
When to use Inverse Trig
1. You have a right triangle and…
2. You need to solve for a angle and…
3. Only two sides are known
Use Inverse Trigonometry
Perfect Squares
The square of whole
numbers.
1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 ,
121, 144 , 169 , 196 , 225, etc
Simplifying Square Roots
1. Check if the square root is a whole number
2. Find the biggest perfect square (4, 9, 16, 25,
36, 49, 64) that divides the number
3. Rewrite the number as a product
4. Simplify by taking the square root of the
number from (2) and putting it outside
5. CHECK!
Note: A square root can not be simplified if there
is no perfect square that divides it.
ex: √15 , √21, and √17
Just leave it alone.
Simplifying Square Roots
Write the following as a radical (square root)
in simplest form:
72  36  2  36 2  6 2
27  9  3  9 3  3 3
Day 46: April 12th
Objective: Recognize the similarity ratios in 30°-60°-90° and 45°45°-90° triangles and begin to apply those ratios as a shortcut
to finding missing side lengths. THEN Learn to recognize 3:4:5
and 5:12:13 triangles and find other examples of Pythagorean
triples. Additionally, practice recognizing and applying all three
of the new triangle shortcuts.
•
•
•
•
•
Homework Check and Correct (in red)
Problems 5-43 to 5-45
Review HW Problem 5-46
Problems 5-51 to 5-55
Conclusion
Homework:
Problems 5-56 to 5-60
Warm-Up! April 12th
Solve for the measures of x and y:
o
a
x
10 in a
o
18 in
h
y
30° – 60° – 90°
A 30° – 60° – 90° is half of an equilateral
(three equal sides) triangle.
30°
s
60°
.5s
s
You can
use this
whenever a
problem
has an
equilateral
triangle!
Long Leg (LL)
30° – 60° – 90°
30°
60°
Short Leg (SL)
30° – 60° – 90°
Remember
√3 because
there are 3
different
angles
You MUST
know SL first!
√3
30°
2
÷2
60°
1
÷√3
SL
LL
x√3
x2
Hyp
Isosceles Right Triangle
45° – 45° – 90°
Remember
√2 because 2
angles are
the same
45°
√2
1
45°
÷√2
Leg(s)
1
Hypotenuse
x√2
Isosceles Right Triangle
45° – 45° – 90°
A 45° – 45° – 90° triangle is half of a square.
45°
s
d
45°
s
You can
use this
whenever a
problem
has a
square with
its diagonal!
Day 47: April 13th
Objective: Learn to recognize 3:4:5 and 5:12:13 triangles and find
other examples of Pythagorean triples. Additionally, practice
recognizing and applying all three of the new triangle shortcuts.
THEN Review tools for finding missing sides and angles of
triangles, and develop a method to solve for missing sides and
angles for a non-right triangle.
•
•
•
•
Homework Check and Correct (in red)
Problems 5-51 to 5-55
Problems 5-61 to 5-65
Conclusion
Homework:
Problems 5-67 to 5-72
Ch. 5 Team Test Wednesday
Midterm Exam Friday (?)
Pythagorean Triple
A Pythagorean triple consists of three positive
integers a, b, and c (where c is the greatest)
such that:
a2 + b2 = c 2
Common examples are:
3, 4, 5 ; 5, 12, 13 ; and 7, 24, 25
Multiples of those examples work too:
3, 4, 5 ; 6, 8, 10 ; and 9, 12, 15
Day 48: April 16th
Objective: Recognize the relationship between a side and the angle
opposite that side in a triangle. Also, develop the Law of Sines,
and use it to find missing side lengths and angles of non-right
triangles.
•
•
•
•
Homework Check and Correct (in red)
Review Math Notes prior to Problem 5-67
Problems 5-73 to 5-76
Conclusion
Homework:
Problems 5-79 to 5-84
Ch. 5 Team Test Wednesday
Midterm Exam Friday (?)
Day 49: April 17th
Objective: Complete the Triangle Toolkit by developing the Law of
Cosines. THEN Review tools for solving for missing sides and
angles of triangles.
•
•
•
•
•
Homework Check and Correct (in red)
Warm-Up!
Problems 5-85 to 5-87
Problem 5-98
Conclusion
Homework:
Problems 5-89 to 5-94 (Skip 5-91)
Ch. 5 Team Test Tomorrow
Midterm Exam Friday (?)
Warm-Up! April 17th
The angles of elevation to an airplane from two people on
level ground are 55° and 72°, respectively. The people
are facing the same direction and are 2.2 miles apart.
Find the altitude (height) of the plane.
Diagram:
Solve:
h
55°
72°
2.2 mi
Solution/Answer:
The airplane is about 5.87 miles above the ground.
Day 50: April 18th
Objective: Review tools for solving for missing sides and angles of
triangles. THEN Assess Chapter 5 in a team setting.
•
•
•
•
Homework Check and Correct (in red)
Warm-Up! Do Problems 5-98 and 5-122
Chapter 5 Team Test
Conclusion
Homework:
Problems 5-100 to 5-105 & Work on Triangle Review WS
Midterm Exam Friday (?)
Day 51: April 19th
Objective: Review and practice Chapter 1-5 topics.
•
•
•
•
Homework Check and Correct (in red)
Problems 5-98(a), 5-106 to 5-109
Problems CL5-126 to 5-130, 5-133, 5-134 and Check
Conclusion
Homework:
Worksheets:
“Special Right” – Left side on the front (4 problems)
“Law of Sines and Cosines” – ODDS (front & back)
Midterm Exam Friday
Day 52: April 20th
Objective: Review and practice Chapter 1-5 topics.
•
•
•
•
•
•
Homework Check
Review Homework Worksheets
Problem 5-114
Practice Problems
Triangle Review Worksheet (some of you already have it)
Conclusion
Homework:
Problems 5-117 to 5-121 AND 5-124
Midterm Exam Wed. and Thurs.
Chapter 5 Topics
Trigonometry:
• Tangent, Sine, and Cosine Ratios – p. 241
• Inverse Trigonometry – p. 248
Special Right Triangles:
• 45-45-90 –p. 260 and LL
• 30-60-90 –p. 260 and LL
• Pythagorean Triples – p. 260
Non-Right Triangle Tools:
• Law of Sines – p. 264 and LL
• Law of Cosines – p. 267 and LL
Algebraic Triangle Angle Sum
Find the measure of C
C
2x – 13°
mC  2  33  13
mC  53
2x + 4°
x + 24°
B
A
2x 13  2x  4  x  24  180
5 x  15  180
5 x  165
x  33
Extra Practice
The Triangle Inequality
1. Which of the following lengths can form
a triangle?
2. Which of the following lengths cannot
form a triangle?
I. 5, 9, 20
II. 6, 10, 13 III. 7, 8, 14
IV. 15, 21, 36
No
Yes
Yes
No
5+9<20
6+10>13
7+8>14
15+21=36
20–9>5
13–10<6
14–8<7
36–21=15
Extra Practice
Area of a Triangle
What is the area of the shaded region?
6 units
8 units
1
A  bh
2
1
A   4  6 
2
2
A  12 u
Extra Practice
Similarity Based on Statements
Given BAC DEF, write an equation that
could be used to solve for x.
B
7
E
11
EF
AC
=
BC
DF
8
x
= 11
25
A
8
x
C
D
25
F
Extra Practice
Probability
A
T
Greg is going to flip a coin twice. What is
the probability heads will not come up?
Second Flip
H(½) T(½)
First Flip
T(½) H(½)
TT
¼
HT
¼
TH
¼
HH
¼
S 1
T T 2 
A
R H
T  12 
Probability
1 

T 2
H 1
2

1 

T 2
H 1
2
1 1 1
 
2 2 4
1 1 1
 
2 2 4
1 1 1
 
2 2 4

1 1 1
 
2 2 4
Extra Practice
Rotation in a Coordinate Grid
Rotate the point (-4,5) either 90° or 180°
(-4,5)
(5,4)
(-5,-4)
(4,-5)
Extra Practice
Angle Measures in Right Triangles
Find the measure of angle A to the nearest
degree:
24
sin  A  
A
26
1  24 
A  sin  
B
26 

26
24
C
m A  67
Transformations: Rotation
C
B
A
C
Rotate ΔABC counter-clockwise around the
origin. What are the coordinates of A’?
A
B
Angle Relationships: Equations
Solve for x:
x5
x  5  3x  5  180
4x  180
3x  5
x  45
Area: Trapezoid
Find the area of the trapezoid:
15 in.
10 in.
12 in.
20 in.
1
A  h  b1  b2 
2
1
A  10 15  20 
2
1
A  10  35 
2
A  175 in 2
Area and Lengths: Triangle
A
The area of ΔABC is 60 sq. inches. What is the
length of segment KC?
1
B
A  bh
2
1
10 in.
60  b 10
2
8 in.
C
K
KC  AC  8
KC  12  8
KC  4 in.
60  5b
12  b  AC
Algebraic Areas: Square
Find the perimeter and area of the square below:
P  4  2 x  2
P  8x  8 un.
2x  2
A   2 x  2 2 x  2
2
A  4x  4x  4x  4
A  4 x 2  8 x  4 un 2
Day 53: April 23rd
Objective: Practice identifying congruent triangles by first
determining that the triangles are similar and that the ratio of
corresponding sides is 1.
• Homework Check
• Problems 6-1 to 6-3
• Conclusion
Homework:
Problems 6-4 to 6-9 (Skip 6-8) AND STUDY!
(Remember:
You need to know Laws of Sines & Cosines!)
Midterm Exam Wed. and Thurs.
Day 54: April 25th
Objective: Assess Chapters 1-5 in an individual setting.
*MULTIPLE CHOICE #1-18 ONLY TODAY*
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Take the test
• Second: Check your work
• Third: Give test & formula sheet to Ms. Katz when you’re done
• Fourth: Correct last night’s homework
• Work on Problem 6-2 with Ms. Katz
Homework:
Problems 6-13 to 6-18
Day 55: April 26th
Objective: Assess Chapters 1-5 in an individual setting.
*MULTIPLE CHOICE #19-25 AND OPEN-ENDED*
• Silence your cell phone and put it in your school bag (not your
pocket)
• Get a ruler, pencil/eraser, and calculator out
• First: Take the test
• Second: Check your work
• Third: Give test & formula sheet to Ms. Katz when you’re done
• Fourth: Correct last night’s homework
• Work on Problem 6-3 with Ms. Katz
• Start Problems 6-10 to 6-12
Homework:
Problems 6-24 to 6-29
Day 56: April 27th
Objective: Use understanding of similarity and congruence to develop triangle
congruence shortcuts. THEN Extend the use of flowcharts to document
triangle congruence facts. Practice identifying pairs of congruent triangles
and contrast congruence arguments with similarity arguments.
•
•
•
•
Homework Check and Correct (in red)
Problems 6-11 to 6-12
Problems 6-19 to 6-23
Conclusion
Homework:
Problems 6-43 to 6-48
Ch. 6 Team Test Monday
Ch. 6 Individual Test Friday
Is your book damaged? Torn/missing pieces of book cover means that your
book needs to be replaced. Bring cash or check for $19 ASAP. If you
think it can be repaired, see Ms. Katz – do NOT make a mess of it with
tape! You will be getting Volume 2 of the textbook on Tuesday.
Example 1
Determine if the triangles below are congruent. If the
triangles are congruent, make a flowchart to justify
your answer.
A
B
C
D
Example 2
Determine if the triangles below are congruent. If the
triangles are congruent, make a flowchart to justify
your answer.
A
C
>
>
E
B
D
Conditions for Triangle Similarity
If you are testing for similarity, you can use the
following conjectures:
SSS~
All three corresponding side lengths have
the same zoom factor
7
3
10
5
AA~
Two pairs of corresponding angles have
equal measures.
55°
55°
40
20
70°
NO CONJECTURE FOR ASS~
40°
40°
SAS~
Two pairs of corresponding lengths have
the same zoom factor and the angles
between the sides have equal measure.
14
6
30
70°
15
Conditions for Triangle Congruence
If you are testing for congruence, you can use the following conjectures:
5
SSS 
All three pairs of corresponding side
lengths have equal length.
3
SAS 
Two pairs of corresponding sides have
equal lengths and the angles between
the sides have equal measure.
7
3
5
ASA 
Two angles and the side between them
are congruent to the corresponding
angles and side lengths.
7
40°
10
55°
10
55°
40°
20
20
70°
70°
15
15
Conditions for Triangle Congruence
If you are testing for congruence, you can use the following conjectures:
AAS 
Two pairs of corresponding angles and
one pair of corresponding sides that
are not between them have equal
measure.
51
51 42°
42°
44°
44°
HL 
The hypotenuse and a leg of one right
triangle have the same lengths as the
hypotenuse and a leg of another right
triangle.
NO CONJECTURE FOR ASS 
19
23
23
19
Problem 6-12
Complete 6-12 on page 295:
Use your triangle congruence conjectures to
determine if the following pairs of triangles
must be congruent.
SAS
SAS
SSS
ASS
ASA
AAS
Problem 6-12 Continued
Complete 6-12 on page 295:
Use your triangle congruence conjectures to
determine if the following pairs of triangles
must be congruent.
SSS
AAS
ASS
AAA
Day 57: April 30th
Objective: Extend the use of flowcharts to document triangle congruence facts.
Practice identifying pairs of congruent triangles and contrast congruence
arguments with similarity arguments. THEN Recognize the converse relationship
between conditional statements.
•
•
•
•
•
Homework Check and Correct (in red) & Warm-Ups!
Finish Problems 6-22 to 6-23
Problems 6-30 to 6-31
Chapter 6 Review Sheet
Conclusion
Homework:
Problems 6-35 to 6-40 & BRING TEXTBOOK FROM HOME
Ch. 6 Team Test TOMORROW & Ch. 6 Individual Test Friday
Is your book damaged? Torn/missing pieces of book cover means that your book
needs to be replaced. Bring $19 TOMORROW. If you think it can be
repaired, see Ms. Katz – do NOT make a mess of it with tape! You will be
getting Volume 2 of the textbook TOMORROW.
Warm-Up! April 30th
11

10

17

2

10

17cos
C


Calculate m C :
121  100  289  340cos  C 
121  389  340cos  C 
2
2
2
-389 -389
c2  a2  b2  2ab cos  C 
268  340cos  C 
268

340
340
268
 cos  C 
340
1  268 
m C  cos 

340 

m C  38
Practice with Congruent Triangles
Determine whether or not the two triangles in each
pair are congruent. If they are congruent,
show your reasoning in a flowchart.
A
(1)
(2)
(3)
G
18
16
D
41°
H 22°
C
18
B
4
Q
5
J
16
F
E
L
P
N
K
41°
22°
M
R
5
4
S
Day 58: May 1st
Objective: Assess Chapter 6 in a team setting. THEN Review
Chapters 5 and 6.
• Homework Check and Correct (in red)
• Chapter 6 Team Test
• Work on Chapter 6 Review Sheet
Homework: Problems CL6-87 to 6-94 (and check solutions)