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Mr. Wolf
Thursday 9/25/08
Geometry
Grades 10-12
Unit 2: Introduction to Proofs and Logic
Review of Proofs
Materials and Resources:
 Why Proofs? Warm-up (1 per student)
 Proof Toolbox sheet (1 per student)
 Flatland Section 3 (1 per student)
 Proofs Game Show PowerPoint
 Proof By Numbers HW sheet A (1 per student)
 Proof By Numbers HW sheet B (1 per student)
 Proof By Numbers HW sheet C (1 per student)
 Proof By Numbers HW sheet D (1 per student)
 Proof By Numbers HW sheet E (1 per student)
 Proof By Numbers HW sheet F (1 per student)
 Exit Ticket (1 per student)
PA Standards Addressed:
2.4.11 A. Use direct proofs, indirect proofs or proof by contradiction to validate
conjectures.
2.4.11 B. Construct valid arguments from stated facts.
Instructional Objectives:
 Students will be able to complete two-column geometric proofs by working in
groups to complete a game show activity.
Time
10 min
1 min
10 min
10 min
Activity
Warm-up
Agenda
Homework Check
Review Homework
20 min
Complete the Proof!
Game Show
Description
Pass out the Warm-up and discuss.
Review the goals for the day.
Instruct students to copy HW problems pg. 59 #26.
Review the HW solutions and answer any
questions.
Modeling: Pass out the Proofs Game Show Answer
sheet and put students into groups. Explain the
rules of the game (similar to Review Jeopardy!).
Guiding: Present the PowerPoint and “host” the
game show.
Independent Practice: Students will have an
opportunity to complete five proofs.
Assessment: Informally assess understanding as
students complete the five proofs. Formally asses
by collecting and grading the Exit Ticket at the
completion of class.
Modifications:
Students with special needs will be placed in
groups that can offer help and support in
10 min
Commercial Break:
Journey Into the
World of Flatland
20 min
Complete the Proof!
Gameshow
Agenda
Conclusion
1 min
5 min
Homework:
Proof By Number sheets
Lesson Reflection:
completing the proofs. Additionally, these students
may receive handouts with printed proofs to ease in
completion.
Advanced students will be group leaders who help
their groups arrive at the correct answers.
Modeling: Pass out the Flatland Section and lead a
class reading.
Guiding: Call on students to read paragraphs aloud.
Independent Practice: Students will complete the
Critical Thinking Questions at the completion of
the reading.
Assessment: Review solutions to the Critical
Thinking Questions.
Modifications:
Students with special needs will be given handouts
with scaffolds in place such as definitions of
difficult words and explanations of the reading.
Advanced students will be asked to draw the
figures described in the section.
Complete the Game Show.
Revisit goals and identify whether they were met.
Pass out the Exit Ticket and collect at the bell.
Geometry Fall 2008
Name: ________________________
Why Proofs? Warm-up
Why do we make an effort to prove statements and reasons about geometric figures?
It is important to train ourselves to reason in a logical and sequential manner so that we
can solve difficult problems and apply critical thinking skills in new contexts and
situations.
Below is an example of a proof applied in a context outside of math class. The
statements and reasons are provided – you must choose where to fill them in.
Statements:
Jimmy will get in trouble with his teacher at school.
Jimmy will get caught cutting class.
Jimmy will be grounded.
Cutting class is against school rules.
Jimmy will get in trouble at home.
Reasons:
Jimmy cut class.
Attendance records will show Jimmy cut class.
Teacher calls Jimmy’s parents to tell them he cut class.
Given.
Jimmy’s teacher is not pleased about the cut.
Given: Cutting class is against school rules.
Prove: If Jimmy chooses to cut class, Jimmy will be grounded by his parents.
Statements
Reasons
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2)
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5)
5)
Abbott, E.A. Flatland
Section 3
Concerning the Inhabitants of Flatland
THE GREATEST length or breadth of a full grown inhabitant of Flatland may be
estimated at about eleven of your inches. Twelve inches may be regarded as a maximum.
Our Women are Straight Lines.
Our Soldiers and Lowest Class of Workmen are Triangles with two equal sides, each
about eleven inches long, and a base or third side so short (often not exceeding half an
inch) that they form at their vertices a very sharp and formidable angle. Indeed when their
bases are of the most degraded type (not more than the eighth part of an inch in size),
they can hardly be distinguished from Straight lines or Women; so extremely pointed are
their vertices. With us, as with you, these Triangles are distinguished from others by
being called Isosceles; and by this name I shall refer to them in the following pages.
Our Middle Class consists of Equilateral or Equal-Sided Triangles.
Our Professional Men and Gentlemen are Squares (to which class I myself belong) and
Five-Sided Figures or Pentagons.
Next above these come the Nobility, of whom there are several degrees, beginning at SixSided Figures, or Hexagons, and from thence rising in the number of their sides till they
receive the honourable title of Polygonal, or many-Sided. Finally when the number of the
sides becomes so numerous, and the sides themselve so small, that the figure cannot be
distinguished from a circle, he is included in the Circular or Priestly order; and this is the
highest class of all.
It is a Law of Nature with us that a male child shall have one more side than his father, so
that each generation shall rise (as a rule) one step in the scale of development and
nobility. Thus the son of a Square is a Pentagon; the son of a Pentagon, a Hexagon; and
so on.
But this rule applies not always to the Tradesman, and still less often to the Soldiers, and
to the Workmen; who indeed can hardly be said to deserve the name of human Figures,
since they have not all their sides equal. With them therefore the Law of Nature does not
hold; and the son of an Isosceles (i.e. a Triangle with two sides equal) remains Isosceles
still. Nevertheless, all hope is not such out, even from the Isosceles, that his posterity may
ultimately rise above his degraded condition. For, after a long series of military
successes, or diligent and skillful labours, it is generally found that the more intelligent
among the Artisan and Soldier classes manifest a slight increase of their third side or
base, and a shrinkage of the two other sides. Intermarriages (arranged by the Priests)
between the sons and daughters of these more intellectual members of the lower classes
generally result in an offspring approximating still more to the type of the Equal-Sided
Triangle.
Rarely--in proportion to the vast numbers of Isosceles births--is a genuine and certifiable
Equal-Sided Triangle produced from Isosceles parents [footnote 1]. Such a birth requires,
as its antecedents, not only a series of carefully arranged intermarriages, but also a longcontinued exercise of frugality and self-control on the part of the would-be ancestors of
the coming Equilateral, and a patient, systematic, and continuous development of the
Isosceles intellect through many generations.
The birth of a True Equilateral Triangle from Isosceles parents is the subject of rejoicing
in our country for many furlongs round. After a strict examination conducted by the
Sanitary and Social Board, the infant, if certified as Regular, is with solemn ceremonial
admitted into the class of Equilaterals. He is then immediately taken from his proud yet
sorrowing parents and adopted by some childless Equilateral, who is bound by oath never
to permit the child henceforth to enter his former home or so much as to look upon his
relations again, for fear lest the freshly developed organism may, by force of unconscious
imitation, fall back again into his hereditary level.
Critical Thinking Questions:
1) A triangle with two equal sides is an ________________ triangle.
2) A triangle with three equal sides is an _________________ triangle.
3) What is the best definition for the term “Polygonal?”
___________________________________
4) According to the Law of Nature, the son of a Heptagon (7-sided figure) would be an
__________.
5) Using context clues, what do you think the word “Regular” means in the last paragraph?
Geometry Fall 2008
Name: ________________________
Proof Toolbox
Below is a list of all of the properties, postulates, and theorems that you have at your
disposal when completing proofs.
Properties of Equality
Addition Property:
If a = b and c = d then a + c = b + d.
Multiplication Property:
If a = b then ca = cb.
Substitution Property:
If a = b then either a or b may be
substituted for the other in any equation.
Reflexive Property:
a=a
Transitive Property:
If a = b and b = c then a = c.
Subtraction Property:
If a = b and c = d then a - c = b – d.
Division Property:
a b
If a = b and c  0 then  .
c c
Distributive Property:
a(b  c)  ab  ac
Symmetric Property:
If a = b then b = a.
Segment Addition Postulate
If B is between A and C, then AB + BC = AC.
Angle Addition Postulate
If point B lies in the interior of AOC , then mAOB  mBOC  mAOC .
If AOC is a straight  and B is any point not on AC then mAOB  mBOC  180
Midpoint Theorem
1
1
If M is the midpoint of AB , then AM  AB and MB  AB
2
2
Angle Bisector Theorem
1
1
If BX is the bisector of ABC , then mABX  mABC and mXBC  mABC
2
2
Vertical Angle Theorem
Vertical angles are congruent.
Perpendicular Lines Theorems
If two lines are perpendicular, then they form congruent adjacent angles.
If two lines form congruent adjacent angles, then the lines are perpendicular.
If the exterior sides of two adjacent acute angles are perpendicular, then the angles are
complementary.
Supplementary Angles Theorem
If two angles are supplementary to congruent angles (or the same angle) then the two
angles are congruent.
Complementary Angles Theorem
If two angles are complementary to congruent angles (or the same angle) then the two
angles are congruent.
Geometry Fall 2008
Name: ________________________
Proofs! The Game Show
Answer Sheet
Use this sheet to record your answers to the proofs presented in the PowerPoint.
Note – you may not fill up each Statement or Reason section.
Proof #1:
Statements
Reasons
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5)
Proof #2:
Statements
Reasons
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5)
Proof #3:
Statements
Reasons
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Proof #4:
Statements
Reasons
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Proof #5:
Statements
Reasons
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5)
Geometry Fall 2008
Name: ________________________
Proof by Numbers HW
Geometry Fall 2008
Name: ________________________
Proof by Numbers HW
Geometry Fall 2008
Name: ________________________
Proof by Numbers HW
Geometry Fall 2008
Name: ________________________
Proof by Numbers HW
Geometry Fall 2008
Name: ________________________
Proof by Numbers HW
Geometry Fall 2008
Name: ________________________
Proof by Numbers HW
Geometry Fall 2008
Name: ________________________
Exit Ticket
Below is Proof #3 from the game show today. Please complete and submit.