Download L33 Clarifying Examples and Activities2 - mi-math

HSCE:
L3.3.1 Know the basic structure for the proof of an “If…, then…”
statement (assuming the hypothesis and ending with the
conclusion) and know that proving the contrapositive is
equivalent.
Clarification statements:
Conditional Statements: If A (hypothesis), then B (conclustion)
Statement
Converse
Inverse
Contrapositive



A B
B A
A : B
B : A
If A then B
If B then A
Negate A and B
Reverse and negate
A and B
A statement and its converse need not be true.
A statement and its contrapositive always have the same truth value.
They are either both true or both false. Thus, B : A also proves
A  B . This is sometimes helpful when a clear path from the original
hypothesis and conclusion is not readily apparent.
The inverse and converse always have the same truth value. They are
either both true or both false.
Given the opportunity to describe how one can go about proving
an “If …, then…” statement, begin by assuming the hypothesis is
true. This information is the “given” of the situation. Students then
develop a chain of logical reasoning, ending with showing that the
conclusion must be true. This chain of logic can take several forms:
a paragraph proof, a flow proof, a two-column proof, a proof using
figures on the coordinate plane, and so on.
Example 1:
Prove that corresponding angles formed by parallel lines cut by a
transversal are congruent.
Direct Proof:
Rewrite the statement; “If parallel lines are cut by a transversal, then
the corresponding angles formed are congruent.” Next, identify the
hypothesis to determine what information we are given to use. In this
case, we are working with “parallel lines cut by a transversal”. We
could draw a diagram at this point and label the angles:
Since the conclusion says “the
corresponding angles formed
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are congruent”, at some point in
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the proof one would need to
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identify some corresponding
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m
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angles and state why they are identified as such. One would also need
to probably work with pairs of angles and other angle relationships
which are already known (through postulates, theorems, definitions).
By following a step-by-step chain of logical reasoning, show that “the
corresponding angles formed are congruent.
Proof by using the contrapositive:
First write the contrapositive: “If two lines are cut by a transversal and
the corresponding angles formed are NOT congruent, then the lines
are NOT parallel”
We then would assume we have two lines and a transversal, but with
the angles which form corresponding pairs of angles NOT being
congruent. We would follow a similar chain of reasoning, which would
eventually lead us to the conclusion that, in fact, the two lines cannot
be parallel.
VOCABULARY
Proof, “If…, then…” statement, hypothesis, conclusion, assumptions,
givens, contrapositive, logically equivalent.
Web Resource:
Finding Proof, Annenberg Video Library #9
Students are introduced to the historical context of proofs and to Greek
mathematician Thales, who sought proof for mathematical c
http://www.learner.org/resources/series34.html?pop=yes&vodid#onclusions.
Activity 5: Different Methods of Proof. This PDF highlights proof, both
in algebra and geometry.
http://mdk12.org/instruction/curriculum/hsa/geometry/clg.html
HSCE:
L3.3.2 Construct proofs by contradiction; use counterexamples,
when appropriate, to disprove a statement.
Clarification statements:
Proof by contradiction:
The "Proof by Contradiction" is also known as reductio ad absurdum, which
is probably Latin for "reduce it to something absurd". Here's the idea:
1. Assume that a given proposition is untrue.
2. Based on that assumption, reach two conclusions that contradict
each other.
This is based on a classical formal logic construction known as Modus Tollens:
If P implies Q ( P  Q ) and Q is false ( Q ), then P is false ( P ).
http://www.programmingforfun.com/Programs/Math_Topics/proof_by_contra
diction.htm
Students will likely use direct reasoning/direct proof strategies
when constructing proofs. However, there are times when a direct
reasoning path is hard to see, or when using indirect reasoning is
more elegant. Students should see several proofs using indirect
methods (proof by contradiction) over their high school mathematics
experience. They should also be able to construct fairly simple proofs
by contradiction.
This expectation highlights the importance of counterexamples.
One good counterexample can spoil what looked like a very good
generalization. Students also need to realize that just because
numerous examples supporting a generalization can be provided, they
themselves do not constitute a proof. They may suggest a
generalization, and even point to how the generalized statement
might be proved. However, if a counterexample is provided, the
generalization will need to be rethought.
VOCABULARY:
Proof, proof by contradiction, contradiction, counterexample, prove,
disprove, statement, “If..., then…” statement, hypothesis, conclusion,
paragraph proof, flow proof, two-column proof, coordinate proof, direct
reasoning, indirect reasoning.
Helpful Web Sites:
“This lesson focuses on using Venn diagrams to explore direct, indirect, and
transitive reasoning.” – from the NCTM Illuminations webpages.
http://illuminations.nctm.org/LessonDetail.aspx?id=L384
Indirect Reasoning
http://postpositivism.wordpress.com/2006/04/27/an-exercise-inindirect-reasoning/
Quick and easy class activity involving indirect reasoning.
http://regentsprep.org/Regents/mathb/1e/indirectteacher.htm
Intro to Logic
http://people.hofstra.edu/faculty/stefan_Waner/RealWorld/logic/logic4
.html
From NCTM – short student sheet on indirect reasoning
http://cpscia.k12.ar.us/Curriculum/Math/Math%20912/Geometry/LG/LG.1.G.1/INDIRECT%20REASONING%20%20SHEET%202.htm
HSCE:
L3.3.3 Explain the difference between a necessary and a sufficient
condition within the statement of a theorem; determine the
correct conclusions based on interpreting a theorem in which
necessary or sufficient conditions in the theorem or hypothesis
are satisfied.
Clarification statements:
Given a statement of a theorem, the student will explain the
difference between necessary and sufficient conditions. The student
will also identify conditions from given theorems as “necessary”,
“sufficient”, “necessary and sufficient”. The student will also draw
correct conclusions when presented with a theorem and one or more
necessary or sufficient conditions, which have been fulfilled
(satisfied).
Condition A is a “necessary condition” for condition B to occur
as long as if A is false, B is then false. Or, put another way, you
can’t have B without also having A.
Condition A is a “sufficient condition” for condition B to occur as
long as whenever A is true, B must also be true. Or, put
another way, if A happens, then B must happen.
When working with “If…, then…” statements, the conclusion (phrase
following the “then”) is “condition B”, and the hypothesis (phrase
following the “if”) is “condition A”.
The concepts of necessary and sufficient can be effectively explored in
class discussion or small group discussion using a number of everyday
language examples and picking them apart. This can be a humorous
activity, as students explore the variety of necessary conditions for a
given situation.
Example 1:
If a figure is a right triangle, then the figure has exactly one right
angle. A  B
Condition A: the figure is a right triangle
Condition B: the figure has exactly one right angle
The condition “the figure has exactly one right angle” is a
necessary condition for a “figure [to be] a right triangle”. It’s
“necessary” since if a figure did not have a right angle, it certainly
would not be a right triangle.
Now the question is whether “the figure has exactly one right
angle” is a sufficient condition for a “figure [to be] a right triangle”.
In other words, if we are told someone has drawn a figure with
exactly one right angle, are we certain we can say, “Ah, the figure
must be a right triangle”? Of course, the answer is no, we can’t be
certain. The person could have drawn a pentagon (or any n-gon, or
even a non-polygon) with one right angle. So “the figure has exactly
one right angle” is a necessary, but not sufficient condition for a
“figure [to be] a right triangle”.
Example 2:
List the necessary and sufficient conditions for a person using a cell
phone to have a conversation with a friend.
Sample necessary conditions:
 The person must be initiating the conversation must be in
possession of a cell phone.
 The cell phone must be in good working order.
 The batteries must have sufficient charge to supply power for
the duration of the conversation.
 The phone must be within a cell phone service area.
 The friend must be available to take the call.
 The person is alive to place the call and for the duration of the
conversation.
 …etc.
As far as sufficient conditions…no single ones jump to mind. If an
exhaustive listing of necessary conditions could be compiled, they
would then constitute a set of sufficient conditions.
VOCABULARY:
Necessary condition, sufficient condition, theorem, hypothesis,
conclusion, to satisfy a condition; necessary and sufficient; necessary
but not sufficient; sufficient, but not necessary; neither necessary nor
sufficient.
Web Sites:
A very nice explanation of these two concepts, with examples.
http://www.sfu.ca/philosophy/swartz/conditions1.htm
http://en.wikipedia.org/wiki/Necessary_and_sufficient_conditions
http://www.msu.edu/~susse/Necessary&Sufficient.htm
http://faculty.uncfsu.edu/jyoung/necessary_and_sufficient_conditions.htm
Nice site for Logic in general.
http://philosophy.hku.hk/think/meaning/nsc.php