level 2 geometry spring 2012 ms. katz. day 1: january 30 th objective: form and meet study teams....

Level 2 Geometry

Spring 2012Ms. 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. When did you take Algebra 1?

2. Who was your Algebra 1 teacher?

3. What grade do you think you earned in Algebra 1?

4. 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

http://www.cpm.org/

http://www.hotmath.com/

http://www.haverford.k12.pa.us/hhs/site/default.asp

Quilts

1-1: First Resource Page

1-1: Second Resource Page

Cut along dotted line

Write sentence and names around the gap.

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

2. Write a rule relating x and y for the table below.

x -3 -1 0 2 4 7

y

x 1 2 3 4 5 6

y 7 10 13 16 19 22

x -3 -1 0 2 4 7

y 11 7 5 1 -3 -9

y = 3x+4

A Complete Graph

y = -2x+5• Create a table of x-values

• Use the equation to find y-values

• Complete the graph by scaling and labeling the axes

• Graph and connect the points from your table. Then label the line.

x -4 -3 -2 -1 0 1 2 3 4

y 13 11 9 7 5 3 1 -1 -3

x

y

-5-10 105

-5

-10

5

10

y = -2x+5

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 2x + 6

A B C

Transformation (pg 34)

Reflection: Mirror image over a line

Rotation: Turning about a point clockwise or counter clockwise

Translation: Slide in a direction

Transformation: A movement that preserves size and shape

Everyday Life Situations

Here are some situations that occur in everyday life. Each oneinvolves one or more of the basic transformations: reflection,rotation, or translation.State the transformation(s) involved in each case.

a. You look in a mirror as you comb your hair.b. While repairing your bicycle, you turn it upside down and spin

the front tire to make sure it isn’t rubbing against the frame.c. You move a small statue from one end of a shelf to the other.d. You flip your scrumptious buckwheat pancakes as you cook

them on the griddle.e. The bus tire spins as the bus moves down the road.f. 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 ___________ to the line of reflection.

2. The line of reflection ______ each of the segments connecting a point and its image.

perpendicular

bisects

1-72

A

B

A’

Isosceles Triangle

Sides: AT LEAST two sides of equal length

Height: Perpendicular to the base AND splits the base in half

Base Angles: Have the same measure

1-72 Isosceles Triangles

1. Two sides are _____ .

2. The ____ angles are equal.

3. The line of reflection ______ the base.

equal

base

bisects

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: Growth or rate of change.

y-intercept: Starting point on the y-axis. (0,b)

m yx

Slope y-intercept

Slope-Intercept Form

33

2y x

First plot the y-intercept on

the y-axis

Next, use rise over run to plot new points

Now connect the points with a line!

You can go backwards if you need!

Parallel Lines (pg 47)

Parallel lines do not intersect.

Parallel lines have the same slope.

For example:

and

y 5

2x 4

y 5

2x 1

Perpendicular Lines (pg 47)

Perpendicular lines intersect at a right angle.

Slopes of perpendicular lines are opposite reciprocals (opposite signs and flipped).

For example:

and

35

2y x

21

3y x

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)

A B

D

C

Condition #1 Condition #2

Satisfies condition 1

only

Satisfies condition 2

only

Satisfies neither

conditionSatisfies both

conditions

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

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:

14 7

1 4x

2 4

4 8

x 1. 2.

Probability (pg 60)

Probability: a measure of the likelihood that an event will occur at random.

Example: What is the probability of selecting a heart from a deck of cards?

Number of Desired Outcomesevent

Total Possible OutcomesP

Number of Hearts 13 1heart 0.25 25%

Total Number of Cards 52 4P

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;


Shape Bucket

2-2

a.

b.

c.

BB’

C’

C

A

m A m B m C

6

or m CAC m C AC

Notation for Angles

Name

or

If there is only one angle at the vertex, you can also name the angle using the vertex:

Incorrect:

F

DE

FEDDEF

E

Measure

Correct:

Incorrect:

45m DEF m A m B

45DEF A B

W X

Y

ZX ?

?

Angle Relationships (pg 76)

Complementary Angles: Two angles that have measures that add up to 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

30°

60°x°

y°

x° + y° = 90°

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


Distributive PropertyThe 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 )

x +3

+5

x x2

( 3x – 2 )( 2x + 7)

+ -4x

= 6x2 + 17x – 14

+ 21x + -14 6x2

• Firsts• Outers• Inners • Lasts• Simplify

( x + 5 )( x + 3 )

+3x

+5x +15

x2 + 8x + 15

Box Method FOIL

Marcos’ Tile Pattern

How can you create a tile pattern with a single parallelogram?

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.



b. What does this mean in terms of the angles in our pattern? Color all angles that must be equal the same color.


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.


Use the following diagram to help answer question 2-15.

L M

J

N P

K

a b

dc

w x

zy

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


Chapter 1 Individual Test TOMORROW

Why Parallel Lines?

x

53°

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!


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


2-23 (a)

a

b

a

More Angles formed by Transversals

>

>48°

48°

48°

48°132°

132°

132°132°

a. Alternate Interiorb. (1) Same Side Interior (2) (3)

Angles formed by Parallel Lines and a Transversal

Corresponding - Congruent

Alternate Interior - Congruent

Same-Side Interior - Supplementary

>

>b

a a = b>

>100°

100°

>

>ba a = b

>

>22°

22°

>

>b

aa + 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°.

Example:

mA mB mC 180

A

B

C

45°

65°70°


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. b and d

2. a and x

3. d and w

4. c and wPossible Choices:

5. x and yVertical Angles

Straight Angle

Alternate-Interior Angles

Corresponding Angles

Same-side Interior Angles

w

dc

b a

z y

x

u

s

v

g

r

q

k f

m p

81°

h57°

57°57°123°

57°

123°

123° 81°

81°99°

99°

42°

2-37: Challenge!f

g

h

k

mp

q

rsuv

2-43 and 2-44

>

>

y

x

2-43 and 2-44

80°

100°B

A

D

C E

2-43 and 2-44

68°

112°

>

>

If Same-Side Interior angles are supplementary, then the lines must be parallel.

If Corresponding angles are congruent,

If Alternate Interior angles are congruent,

then the lines must be parallel.

then the lines must be parallel.

2-45

80°

80°

80°

80°100° 100°

>

>

>

>


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


Area of a Right Triangle

What is the area of the right triangle below? Why?

What about non-right triangles?10 cm

4 cm

http://hotmath.com/util/hm_flash_movie_full.html?movie=/hotmath_help/gizmos/triangleArea.swf

Where is the Height & BaseH

eigh

t

Hei

ght

Hei

ght

BaseBase

Base

Obtuse Triangle

Base

Hei

ght

Extra

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.

1

2A bh

Base

Hei

ght

Base

Hei

ght

Base

Hei

ght


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


Optional E.C: Do Problem 2-49 on a separate sheet of paper and hand it in on Monday. It must be

neat and well- explained 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

Rectangle!

Height

Base

h

h

h

h

Area = b.h

b


Area = b.h

h

b

The area of a parallelogram is the base times the height.

Ex:


Area = b.hh

b

513

20

20

13 A = 20.5 = 100

Area of a Trapezoid

Height

Base Two

Base One

b2

b1

Parallelogram!

h b2

b1

h

b2

b1

b1

b2

h

DuplicateReflectTranslateArea = (b1 + b2) h

Area of a Trapezoid

b2

b1

Area =

h

1 2

1

2b b h

The area of a trapezoid is half of the sum of the bases times the height.

Ex:

Area of a Trapezoid

b2

b1

Area = h

9

5

15

4 5 A = ½ (9+15) 4 = ½ . 24 . 4 = 48

1 2

1

2b b h

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


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 well- explained to be considered for credit.

Warm Up! February 17th

Solve for x in both diagrams

2x + 9°

-(x – 36°)

>

>

7 units

7 + 4x units

The area of the polygon above is 357 un2.

Note card = Height Locator

“Weight”

Base

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


Chapter 2 Team Test Tomorrow

[Review transformations and angle relationship vocabulary]

http://www.cpm.org/flash/technology/triangleInequality.swf

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[D] 5

2. x2 + 9 = 130

[A] 11.79 [B] 11 [C] 8.40[D] 121

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


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

a

b

c

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.

a – b < c < a + b

a – c < b < a + c

b – c < a < b + c

Longest Side: Slightly less than the sum of the two shorter sides

Shortest Side: Slightly more than the difference of the two shorter sides

Triangle Inequality

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



The Pythagorean Theorem

a

b

c

a

b

c

a

b c

a

bc

c2

a

b

ca

b

c

a

b c

a

bcb2

a2

a2 + b2 = c2

Pythagorean Theorem

a2 + b2 = c2a c

A

B

C b

Leg

Leg

Hypotenuse

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 inx

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

9 m

x

5 m

Do you need to

solve for a side or

angle?

Do you have

two sides or

a side and an angle?

Pythagorean

Theorem


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



Dilation

A transformation that shrinks or stretches a

shape proportionally in all directions.

Enlarging


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 TopicsAngles:• Acute, Obtuse, Right, Straight, Circular – p. 24• Complementary, Supplementary, Congruent – p. 76• Vertical, Corresponding, Same-Side Interior, Alternate Interior

– Toolkit and p. 91Lines:• Slopes of parallel and perpendicular lines – p. 47Transformations:• Reflection, Rotation, Translation, and Prime Notation – p.81Shapes:• Name/Define shapes – ToolkitProbability:• Use proper notation…Ex: P(choosing a King) = 4/52 = 1/13

– Page 60

Chapter 1-2 TopicsTriangles:• Triangle Angle Sum Theorem – p.100• Area• Triangle Inequality TheoremArea:• Triangle, Parallelogram, Rectangle, Trapezoid, Square

– Page 112 and Learning Log/ToolkitPythagorean Theorem & Square Roots – p. 115 and 123

Similar Figures

Exactly same shape but not necessarily same size

• Corresponding Angles are congruent• The ratios between corresponding sides

are equal

90°

90°127°

53°90°

90°127°

53°

7

10

45

21

30

1215

Zoom Factor

The number each side is multiplied by to enlarge or reduce the figure

Example:

Zoom Factor = 2

12

3

9

246

18

x2

x2x2

Notation

ABC XY

m ABC XY

Angle ABC Line Segment 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




• 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



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


Notation

ABC XY

m ABC XY

Angle ABC Line Segment 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

60 ft? ft

? ft? ft720 in

? in? in? 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

A

B

C

X

Y

Z

Δ Δ A ZZBC XYXY~

Example: ΔDEF~ΔRST

The order of the letters determines which sides and angles correspond.

ABC

Writing a Proportion

A

B C X

D

W

YZ

13s

10

25

=

ABCD ~ WXYZ

=

BCABAB

AB WX

WXBC

BC XY

XY13s

WX

25XY

10

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:

x

86

12

y3

2.Figure ABCD is similar to WXYZ. Find the length of z:

z

1815

10

A B

D C

W Z

X 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 6

y 9

Similarity and Sides

The following is not acceptable notation:

OR

Acceptable:

AB CD

~AB CD

AB CD

3-54

What Conjecture will we use:

124

3

SSS~ Facts

164

4

84

2

2

3

4

8

16

12C

D

F

Q

T

R

Conclusion

ΔCDF ~ ΔRTQ SSS~

Another Example

What Conjecture will we use:

m A m Z

AA~ Facts

m B m Y

A

B

C

Y

ZX

Conclusion

ΔABC ~ ΔZYX AA~

60°

100°

60°

100°

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


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.

A

8BT

1220

O

45

S

N

2718

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



Conditions for Triangle SimilarityIf you are testing for similarity, you can use the

following conjectures:

SSS~All three corresponding side lengths have

the same zoom factor

AA~Two pairs of corresponding angles have

equal measures.

SAS~Two pairs of corresponding lengths have

the same zoom factor and the angles between the sides have equal measure.

NO CONJECTURE FOR ASS~

3

5

7

6

10

14

55°

40°

55°

40°

70°

40

30

70°

20

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


Chapter 3 Individual Test Thursday

You’re Getting Sleepy…

200 cmx cm

Eye Height Eye

Height

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


Lessons from Abroad

12 + 930 = 942

x

12

6 – 2 = 4

316 ft

Chapter 1-2 TopicsAngles:• Acute, Obtuse, Right, Straight, Circular – p. 24• Complementary, Supplementary, Congruent – p. 76• Vertical, Corresponding, Same-Side Interior, Alternate

Interior – Toolkit and p. 91Lines:• Slopes of parallel and perpendicular lines – p. 47Transformations:• Reflection, Rotation, Translation, and Prime Notation – p.81Shapes:• Name/Define shapes – ToolkitProbability:• Use proper notation…Ex: P(choosing a King) = 4/52 = 1/13

– Page 60

Chapter 1-2 TopicsTriangles:• Triangle Angle Sum Theorem – p.100• Area• Triangle Inequality TheoremArea:• Triangle, Parallelogram, Rectangle, Trapezoid, Square

– Page 112 and Learning Log/ToolkitPythagorean Theorem & Square Roots – p. 115 and 123

Chapter 3 TopicsDilations• Zoom Factor – p. 142Similarity• 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



Warm Up! March 13th

1. Make a table in order to graph the following equation: 2 2 3y x x

[Perhaps use x-values from -5 to 5?]

2. Factor the following equation in order to solve for x:

2 2 3 0x x

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



• 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


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

25 cm

x

68°

hypo

tenu

se

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


Trigonometry

Hypotenuse

(across from the 90° angle)

Adjacent

(forms the known angle)

Opposite

(across from the known angle)

Theta ( ) is always an acute angle

Δy h

Δx

Trigonometry

Hypotenuse

(across from the 90° angle)

Adjacent

(forms the known angle)

Opposite

(across from the known angle)


oh

a

Trigonometry (LL)


h

Opp

osite

Adjacent

Trigonometry (LL)


h

Opposite

Adj

acen

t

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


Chapter 4 Team Test Friday

1. Find the length of x:

2. If a bag contains 6 yellow, 10 red, and 8 green marbles. What is the probability of selecting a red marble at random.

Warm Up! March 20th

x 4

7 9

A 1

4 B 5

12 C 1

3 D 7

12

Warm Up! March 21st

Multiply the following expressions using an area diagram:

1.

2.

7 2x x

2 2 3 1x x

2 5 14x x

26 8 2x x

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

#41

#28

#55

ListenSTART

#81

ReadWrite

Listen

#41

ReadWrite

Listen

One Possibility:

Take Bus #41 and Listen to an MP3

player

ReadWrite

Listen

ReadWrite

Listen

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



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.

2.

27x

23 2x

2 14 49x x

29 12 4x x

Warm Up! March 22nd

Solve for x and y:

60°x

y

tan(60)6

x · 66 ·

6 tan(60) x

10.39 x

6

2 2 26 10.39 y 236 107.9521 y 2143.9521 y

12 y

2 2 2a b c

4-60: Tree Diagram

$100

$300

Keep

DoubleSTART $1500

Keep

Double

Keep

Double

$100

$200

$300

$600

$1500

$3000

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


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

12

Spinner #1

Spi

nner

#2

IT UT AT

IF UF AF

13 1

6

14

34

18

112

124

38

312

324

I U A

T

F

Warm-Up! March 23rd

Make an area diagram to model the game where both spinners below are used. Then find the probabilities below:

1. P(A, X) = 2. P(C, Y) = 3. P(not A, Y) =

A

C

BX Y

B

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


[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. If each spinner is spun once, what is the probability that both spinners show blue?

2. If each spinner is spun once, what is the probability that both spinners show the same color?

3. 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




test.]

Warm-Up! March 27th

Solve for the length of x and y:

10 cm

x71°

yoh

a

Step 1:

Step 2:

Chapter 4 TopicsSlope Angles/Ratios:

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

y

x

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




test.]

Find the area of the triangle:

30°

6 cm

Warm-Up! March 28th

o

h

a

or

Trigonometry

SohCahToa

oh

a

sin( )

cos( )

tan( )

opposite o

hypotenuse h

adjacent a

hypotenuse h

opposite o

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


Chapter 4 Individual Test Tomorrow – STUDY!

Are the following triangles similar? If so, make a flowchart. If not, explain why they are not similar and/or what information is missing.

Warm-Up! March 29th

1. 2.

Day 43: March 30th



• 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


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 Roots1. 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:

6 236 272 36 2

3 39 327 9 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


Warm-Up! April 12th

Solve for the measures of x and y:

18 in

x

10 in

y

o

h

ao

h

a

30° – 60° – 90°

A 30° – 60° – 90° is half of an equilateral (three equal sides) triangle.

s

s

.5s

60°

30°

You can use this

whenever a problem has an

equilateral triangle!

30° – 60° – 90°

Hypotenuse

60°

30°

Short Leg (SL)

Long

Leg

(LL

)

30° – 60° – 90°

Remember √3 because there are 3 different angles

2

60°

30°

1

√3

You MUST know SL first!

SL Hyp÷√3

x2

LL

÷2

x√3

Isosceles Right Triangle 45° – 45° – 90°

45°

45°

1

1

√2

Remember √2 because 2 angles are the same

Leg(s) Hypotenuse

÷√2

x√2

Isosceles Right Triangle 45° – 45° – 90°

A 45° – 45° – 90° triangle is half of a square.

s

s45°

45°

You can use this

whenever a problem

has a square with its diagonal!

d

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.



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 = c2

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


Ch. 5 Team Test Wednesday


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


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:

2.2 mi

55° 72°

h

Solve:

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


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 TopicsTrigonometry:• Tangent, Sine, and Cosine Ratios – p. 241• Inverse Trigonometry – p. 248Special Right Triangles:• 45-45-90 –p. 260 and LL• 30-60-90 –p. 260 and LL• Pythagorean Triples – p. 260Non-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

2x + 4° x + 24°

2x – 13°

A B

C

2 13 2 4 24 180x x x 5 15 180

5 165

33

x

x

x

2 33 13

53

m C

m C

Extra Practice

http://cpm.org/pdfs/skillBuilders/GC/GC_Extra_Practice_Section2.pdf

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 IV. 15, 21, 36III. 7, 8, 14

No

5+9<20

20–9>5

Yes

6+10>13

13–10<6

No

15+21=36

36–21=15

Yes

7+8>14

14–8<7

Extra Practice

http://www.haverford.k12.pa.us/1923209819181327/lib/1923209819181327/_files/Triangle_Inequality.doc

Area of a Triangle

What is the area of the shaded region?

8 units

6 un

its 2

1

21

4 62

12 u

A bh

A

A

Extra Practice


Similarity Based on Statements

Given BAC DEF, write an equation that

could be used to solve for .x

=

=

DFEF AC

BC

x25

811

A

B

C

D

F

E7

8

11

x

25

Extra Practice


Probability

Greg is going to flip a coin twice. What is the probability heads will not come up?

12

12

1 1 1

2 2 4

1 1 1

2 2 4

12

12

12

12

H

H

HT

T

T

1 1 1

2 2 4

1 1 1

2 2 4

TT¼

TH¼

HT¼

HH¼

START

First FlipT ( ½ ) H ( ½ )

Sec

ond

Flip

H (

½ )

T

( ½

)

TA

Probability Extra Practice


Rotation in a Coordinate Grid

Rotate the point (-4,5) either 90° or 180°

(-4,5)

(-5,-4)(4,-5)

(5,4)

Extra Practice


Angle Measures in Right Triangles

Find the measure of angle A to the nearest degree:

A

C

B

24

26

24sin

26A

1 24sin

26A

67m A

Transformations: Rotation

Rotate ΔABC counter-clockwise around the origin. What are the coordinates of A’?

C

BA

C BA

Angle Relationships: Equations

Solve for x:

5x 3 5x

5 3 5 180x x 4 180x

45x

Area: Trapezoid

Find the area of the trapezoid:

1 2

1

21

10 15 202

A h b b

A

110 35

2A

2175 A in

15 in.

10 in.

20 in.

12 in.

Area and Lengths: Triangle

The area of ΔABC is 60 sq. inches. What is the length of segment KC?

1

2A bh

60 5bA

B

CK8 in.

10 in. 160 10

2b

12 b AC 8KC AC 12 8KC 4 .KC in

Algebraic Areas: Square

Find the perimeter and area of the square below:

2 2x

4 2 2P x 8 8 un.P x

2 2 2 2A x x 24 4 4 4A x x x 2 24 8 4 unA x x

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


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


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.



Ch. 6 Team Test Monday

Ch. 6 Individual Test FridayIs 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

BC

D

Example 2

Determine if the triangles below are congruent. If the triangles are congruent, make a flowchart to justify your answer.

A

B

C

E

D

>

>

Conditions for Triangle SimilarityIf you are testing for similarity, you can use the

following conjectures:

SSS~All three corresponding side lengths have

the same zoom factor

AA~Two pairs of corresponding angles have

equal measures.

SAS~Two pairs of corresponding lengths have

the same zoom factor and the angles between the sides have equal measure.

NO CONJECTURE FOR ASS~

3

5

7

6

10

14

55°

40°

55°

40°

70°

40

30

70°

20

15

Conditions for Triangle Congruence

SSSAll three pairs of corresponding side

lengths have equal length.

ASATwo angles and the side between them

are congruent to the corresponding angles and side lengths.

SASTwo pairs of corresponding sides have

equal lengths and the angles between the sides have equal measure.

3

5

7

55°

40°

10

70°

20

15

3

5

7

55°40°

10

If you are testing for congruence, you can use the following conjectures:

70°

20

15

Conditions for Triangle Congruence

AASTwo pairs of corresponding angles and

one pair of corresponding sides that are not between them have equal measure.

HLThe 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

44°

42°

51

23

19

If you are testing for congruence, you can use the following conjectures:

23

19

44°

42°51

Problem 6-12Complete 6-12 on page 295:

Use your triangle congruence conjectures to determine if the following pairs of triangles must be congruent.

SAS SSS

SAS ASSAASASA

Problem 6-12 ContinuedComplete 6-12 on page 295:

Use your triangle congruence conjectures to determine if the following pairs of triangles must be congruent.

AASSSS

AAAASS

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

Calculate :m C

2 2 2 2 cosc a b ab C

2 2 211 10 17 2 10 17cos C

121 100 289 340cos C 121 389 340cos C

268 340cos C -389 -389

340cos268

340 340

C

268cos

340C

1 268cos

340m C

38m C

Practice with Congruent TrianglesDetermine whether or not the two triangles in each

pair are congruent. If they are congruent, show your reasoning in a flowchart.

(1) (2) (3)

1816

18

16

41°22°

41°22°

A

B CD

F

E

G

H

J K

M

L

N

Q

P

SR

5 4

4 5

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)

level 2 geometry spring 2012 ms. katz. day 1: january 30 th objective: form and meet study teams....

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