Download Integrated 1

Integrated 2
Unit 3: Triangle Properties and Proof
Lesson Plan
5 weeks
2011-12
Use deductive reasoning to prove that a valid geometric statement is true. Valid proofs may be presented in paragraph, two-column, or flow-chart formats. Proof by
contradiction is a form of deductive reasoning. Embed
M2.3.B
Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid statements about geometric relationships.
M2.3.D
Distinguish between definitions and undefined geometric terms and explain the role of definitions, undefined terms, postulates (axioms), and theorems.
M2.3.E
Know, explain, and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle.
M2.3.F
Determine and prove triangle congruence (and similarity) and other properties of triangles.
M2.3.G
Know, prove, and apply the Pythagorean Theorem and its converse.
M2.6.D
Generalize a solution strategy for a single problem to a class of related problems, and apply a strategy for a class of
related problems to solve specific problems.
M2.6.E
Read and interpret diagrams, graphs, and text containing the symbols, language, and conventions of mathematics.
M2.6.G
Synthesize information to draw conclusions and evaluate the arguments and conclusions of others.
Sect and Std
M2.3.A
3.1
T4 L1
M2.3.F
Key ideas and
Vocabulary
parallel lines
transversal
vertical angles
corresponding
angles
alternate interior
congruent
sum of angles of
triangle
Supplies for this unit:
Patty Paper can be purchased
at Cash and Carry or on line at
www.eaieducation.com
Resources
T: Teacher Page
S: 1 copy per student
G: 1 copy per group
Classwork
Assessments
C: 1 class set
EX: Extensions
I: Internet Resource
Create a graphic organizer or reference page for student to use during the unit.
What do you remember about Parallel Lines?
G: 3.1 Memory
(from Integrated 1 Unit 6) This is needed for the Triangle
Sum Proof.
There are 2 choices.
1. Game: Memory Angle Vocabulary
Cards must be cut in advance.
Have students lay all the cards facedown in an array.
Take turns. Turn over two cards. If the picture
matches with the term, they have a match.
If the cards do not match, turn them back over.
Summary – post a list of vocabulary. Have students
write definitions for words they did not remember.
2. Put it All Together. Review assignment from Unit 6.
Angle Vocabulary
© Evergreen Public Schools 2010
9/27/11
doc pdf
S: 3.1 Put it All
Together
doc pdf
3.1 Parallel Lines
TICKET
More Practice
3.1 Practice U6 L 1
doc pdf
Note: U3
Foundations
Assessment
3.1 Practice U6 L 2
should be done in
doc pdf
advance so you
know what review Need Keys
your students
need.
Page 1 of 8
Integrated 2
3.2
Unit 3: Triangle Properties and Proof
Angle Sum of
Triangle
T4 L1
Why is the sum of the angles of a
triangle 180?
Conjecture: Students tear the angles of a
triangle and put them together to show the angle
sum is 180.
Proof: As a class do a proof. Focus on reason
and not form.
Note: Students need these notes from section 6.
Add problems solved algebraically.
M2.3.A
M2.3.E
M2.6.E
Lesson Plan
T: 3.2 Teacher Notes
C: 3.2 Triangle Sum
Conjecture TASK CARD
doc pdf
S: 3.2 Triangle Sum
Conjecture Cut Triangles
(2 on a page) doc pdf
5 weeks
3.2 Triangle Sum
TICKET
2011-12
3.2 Practice Angles
in Triangles KUTA
TPR Target
Check 1
doc pdf
3.2 KEY
3.3 Exterior
Angle TICKET
3.3 Practice 1
doc pdf
3.3 KEY 1
S: 3.2 Triangle Sum
Conjecture STUDENT
NOTES
doc pdf
3.3
T3 L3
M2.3.E
M2.6.E
angles
complementary
supplementary
vertical
interior
remote interior
exterior
perpendicular
What is an exterior angle of a
triangle?
Triangle Investigation 1
T: 3.3Teacher Notes
Triangle Exterior Angle Conjecture
The lesson starts with students reviewing
vertical, complementary and supplementary
angles. Then use inductive reasoning to find the
conjecture “The exterior angle of a triangle is
equal to the sum of the two remote interior
angles”.
Note: It remains a conjecture until we prove it.
Add problems solved algebraically.
S: 3.3 Triangle
Investigation
doc pdf
3.3 Practice 2
doc pdf
3.3 KEY 2
3.3 Exterior Angle
Theorem KUTA
3.3 KEY Exterior
angle
© Evergreen Public Schools 2010
9/27/11
Page 2 of 8
Unit 3: Triangle Properties and Proof
Integrated 2
3.4
T3 L3
M2.3.E
M2.6.E
altitude
congruent
construct
endpoint
isosceles
triangle
perpendicular
bisector
equilateral
triangle
equiangular
A. What is special about the angles
of an isosceles triangle?
Isosceles Triangle Conjecture
1) copy a segment
2) construct an isosceles triangle
3) make conjecture about angles of isosceles
triangle.
4) identify altitude and perpendicular bisector
in an isosceles triangle
5) apply conjecture
6) construct an equilateral triangle.
B. Continue. . .
And add special case of Equilateral Triangle.
Lesson Plan
Triangle Investigation 2
T: 3.4ATeacher Notes
C: 3.4A Task Card
doc pdf
5 weeks
2011-12
3.4A Isosceles
3.4 Isosceles and
Triangle TICKET Equilateral Triangles
KUTA
TPR Target
Check 2
3.4 KEY
doc pdf
I: Constructions:
3.4A Copy a segment pdf
3.4A Construct an
Isosceles Triangle
How to construct with a
bullseye compass
doc pdf
3.4A Construct Isosceles
Triangle with standard
compass
pdf
How to construct with
bullseye compass
3.4B Construct an
Equilateral Triangle
doc pdf
3.4B Isosceles
Triangle
TICKET.
How to construct with a
standard compass
3.4B Construct Equilateral
Triangle
pdf
3.5
T3 L3
T4 L1
M2.3.F
M2.6.E
C: 3.5 Triangle Properties
TEAM PRACTICE
Team Practice
Problem Solving: Triangle Properties
doc
doc
pdf
Use problems from
KUTA software
worksheets
3.5 KEY
M2.3.A
M2.3.E
© Evergreen Public Schools 2010
pdf
Concept Check T3
L3
9/27/11
Page 3 of 8
Unit 3: Triangle Properties and Proof
Integrated 2
3.6
T4 L3
theorem
flowchart
M2.3.A
M2.3.D
M2.3.F
M2.6.G
3.7
T4 L3
Lesson Plan
T: 3.6 ppt pps
What is a theorem?
Introduce Flowchart proofs.
They write a flowchart to make a peanut butter
and jelly sandwich. You might want PB &J and
knife for demonstration from one student’s
flowchart. Do exactly what the flowchart says.
T: 3.6 Triangle Sum
TEACHER NOTES
S: 3.6 Triangle Sum
STUDENT NOTES
doc
exterior triangle
theorem
C: 3.7 Ext Angle Proof Parts
Prove Exterior Angle Conjecture
M2.3.A
M2.3.D
M2.3.F
M2.6.G
doc
3.8
T4 L3
M2.3.F
M2.6.E
congruent
figures
congruent
triangles
corresponding
sides
corresponding
angles
A. How do we know if two figures
are congruent?
Congruent Figures – students are asked to
identify congruent figures. . Do not define
congruent figures. Have patty paper available.
TPF Target Check
1
doc
pdf
Gradebook Quiz
T3 L3 3.7
doc
pdf
Concept Check T3
L3 3.7
continued
doc
3.7 KEY
pdf
T: 3.8A ppt
T: 3.8A Teacher Notes
This should be
review, but model a
few before you send
them home.
3.8A Information in
Geometric Diagrams
KUTA
S: 3.8A Congruent Figures
STUDENT
doc pdf
Where do we see congruent
triangles?
3.8A KEY
T: 3.8B Teacher Notes
B. What makes a triangle
congruent?
Congruent Triangles – use protractors and rulers
to measure sides and find congruent
corresponding parts.
© Evergreen Public Schools 2010
3.7 Exterior Angle
Theorem KUTA
(same as section 3.3)
pdf
Building the “Why?” -- see for photos
3.8
3.6 Practice Flowchart
pdf
S: 3.7 Ext Angle Thm
STUDENT NOTES
doc
3.6 Triangel Sum
TICKET
2011-12
pdf
T: 3.7 Teacher Notes
How can we prove the Exterior
Angle Conjecture?
5 weeks
9/27/11
3.8 Congruent
Triangles TICKET
3.8B Congruence and
Triangles KUTA
S: 3.8B Congruent Triangles
doc
pdf
3.8BKEY
Page 4 of 8
Integrated 2
3.9
T4 L3
M2.3.E
M2.3.F
M2.6.E
Unit 3: Triangle Properties and Proof
AA Similarity
SSS
SAS
ASA
AAS
Lesson Plan
Triangle Investigation 3
G: 3.9A Triangle
Investigation TASK CARD
A. How can we guarantee two
triangles are congruent?
AA, SSS, SAS Explorations
doc
5 weeks
3.9A Congruent
Triangles TICKET
3.9B Congruent
Triangles TICKET
3.9B ASA and AAS
Congruence KUTA
Gradebook Quiz
T3 L3
3.9B ASA and AAS
Congruence KEY
doc
G: 3.9C Congruent Triangle
TEAM PRACTICE
C. Congruent Triangle TEAM
PRACTICE
doc
KEY
Look at edits in MTV groupshare
pdf
3.9A SSS, and SAS
Congruence KUTA
3.9A KEY
pdf
B. ASA, AAS, SSA Explorations
2011-12
pdf
TPF Target
Check 2
doc pdf
T: 3.9C KEY
3.10
A. Extension: Level 4 OPTIONAL
T4 L3
How can we prove SSS, etc?
SSS and SAS are postulates
Prove SAA and ASA using the postulates.
This is best to do with direct instruction.
Our book uses AA~ Postulate and the fact that
congruent triangles have a scale factor of 1. But
only proves AAS.
M2.3.A
M2.3.E
M2.3.F
M2.6.E
M2.6.G
3.10
continued
prove congruent
triangles
I: Done with proof by
contradition
3.10 SSS, SAS, ASA
KEY
B. How can we prove two triangles
are congruent?
Proof Manipulatives
T: 3.10B Teacher Notes
In team, practice applying SSS, SAS, ASA and
AAS.
C: 3.10B Proof Manipulatives
C. Continue…
S: 3.10B Student Notes
© Evergreen Public Schools 2010
9/27/11
3.10 SSS, SAS, ASA,
and AAS Congruence
KUTA
doc
Concept Check T4
L3
pdf
Page 5 of 8
Unit 3: Triangle Properties and Proof
Integrated 2
Lesson Plan
doc
doc
3.11
T4L3
M2.3.F
M2.6.D
parts of
congruent
triangles
Corresponding parts of congruent
triangles are congruent. (CPCTC)
Launch: Read letter to Dr. Math.
There are 13 proofs. DO NOT ASSIGN THEM
ALL!!! This s a differentiation opportunity.
2011-12
pdf
S: 3.10D Prove Congruent
Triangle TEAM PRACTICE
D. TEAM PRACTICE
5 weeks
3.10 Ticket
pdf
T: 3.11 CPCTC Teacher
Model
3.11 Practice
C: 3.11 Letter to Dr. Math
3.11 KEY
doc
doc
pdf
pdf
C: 3.11 CPCTC Proofs
doc
pdf
3.11 KEY
3.12
T3L3
M2.3.E
M2.6.E
T3L4
M2.3.F
construct
perpendicular
through point
angle bisector
perpendicular
bisector
median
A. How can I draw accurately?
S: 3.12A angle bisector task
Perform basic constructions with compass and
straight edge. These skills are needed to find
the segments in triangles (next lesson).
S: 3.12A perpendicular task
doc
doc
pdf
3.12A Practice
Constructions
pdf
S: 3.12A perp thru point
(altitude) task
doc
pdf
S: 3.12A median task
doc
pdf
C: How to sheets from Math
Open Reference
3.12A Construct angle bisect
3.12A Construct perp bisect
3.12A Construct
perpendicular thru point
3.12A Construct medians
I: www.mathopenref
© Evergreen Public Schools 2010
9/27/11
Page 6 of 8
Integrated 2
Unit 3: Triangle Properties and Proof
altitude
angle bisector
median
perpendicular
bisector
B. Which triangle segments make 2
new congruent triangles?
Students draw the 4 segments in scalene,
obtuse, isosceles, equilateral and right triangles.
They must prove the 2 new triangles are
congruent.
Lesson Plan
T: 3.12B Teacher Notes
S: 3.12B Altitude Invest
doc
pdf
S: 3.12B Angle Bisector
Invest
doc
pdf
5 weeks
Gradebook Quiz
T4 L3
doc
2011-12
3.12B Angle Bisectors
of Triangles KUTA
pdf
3.12B KEY Angle
Bisect
3.12B Medians KUTA
S: 3.12B Median Invest
doc
pdf
3.12B KEY Median
S: 3.12B Perpendicular
Bisector Invest
Section C
is on the next page
doc
pdf
I: www.mathopenref.com
I: www.mathopenref.com
I Need keys!
3.12
continued
orthocenter
incenter
centroid
circumcenter
C. Extension (Level 4):
What is special about these
segments?
S: 3.12C Altitude Invest L4
Make conjectures about the intersections of all 3
segments (from yesterday).
S: 3.12C Median Invest L4
doc
pdf
S: 3.12C Angle Bisector
Invest L4
doc
doc
3.12C Practice Centers
of Triangles JMAP
doc
pdf
pdf
pdf
S: 3.12C Perpendicular
Bisector Invest L4
doc
pdf
C: Instructions from Math
Open Reference
3.12C Construct Orthocenter
3.12C Construct Incenter
3.12C Construct
Circumcenter
3.12C Construct Centroid
Need keys! They can be done
by hand and I will scan
them.
© Evergreen Public Schools 2010
9/27/11
Page 7 of 8
Integrated 2
Unit 3: Triangle Properties and Proof
3.13
T4L4
M2.3.F
M2.6.G
Why are the base angles of an
isosceles triangle congruent?
There are 3 proofs
Level 3 with median or angle bisector
Level 4 with altitude
Hint cards are available for all 3 proofs.
You can have half to one-third of the pairs
doing one of the proof methods. Have pairs
show their method to another pair.
Lesson Plan
T: 3.13 Isosceles Triangle
Thm Proof TEACHER
NOTES
Can I Do This?
15
Targeted Review
16
Assessment
2011-12
3.13 Angles in
Triangles KUTA
3.13 KEY
T: 3.13 clue card organizer
S: 3.13 Isosceles Triangle
Thm Proof STUDENT
NOTES
-each student gets one of the
2 (or 3) proof pages.
doc
14
5 weeks
pdf
Unit 4 Foundations
Int2 U3 T3 2010 Assmt Form A
doc
pdf
Int2 U3 T3 2010 Assmt Form B
doc
pdf
Int2 U3 T4 2010 Assmt Form A
doc
pdf
Int2 U3 T4 2010 Assmt Form B
doc
pdf
Consider letting students use a list of theorems and
properties
© Evergreen Public Schools 2010
9/27/11
Page 8 of 8