Gourmet Lesson Plan:
Logic and Proof Writing
Addie Andromeda Evans with Tol Lau
INTRODUCTION:
The goal of this series of lessons is to use the traditional column proof in high school geometry as a context to learn
formal logic, and deductive and inductive reasoning. Logic and reason are important life skills that students can strengthen
in their mathematics classes, if emphasized properly. All too often, I have seen students who get through math classes
by copying procedures they have seen and getting answers that “seem right” to them. With these lessons, we hope to
further instill in students that each conclusion should follow from some true statements and a series of logical implications.
Whether these conclusions pertain to mathematics, politics, or life, the thought process should be the same. The goal is to
empower students to have the confidence to assert their ideas, because they know their ideas are based on logic and sound
reasoning. We can take this opportunity to strengthen the student’s general math habits, by emphasizing justification of
each step. We will see examples from algebra, examples from life and newspapers, abstract examples of symbolic logic,
and two-column proofs.
Learning Objectives:
• To understand deductive and inductive reasoning.
• To be able to write two column proofs.
• To be able to analyze arguments for their validity.
• To be able to shift from symbolic, to geometric, to real life logic.
• To instill habits of a successful mathematician.
• To have the confidence to assert ideas.
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Materials Required:
• The logic game. (See Appendix C.)
• The geometry game (See Appendix E.)
• Pencil and paper.
• Lots of logic examples!
PART 1: Introduction to Logical Thinking
Summary: This lesson is designed to fill a 100 minute period. It covers if-then statements, an introduction to deductive and inductive thinking, and analyzing real world arguments in newspaper articles.
Lecture: The clearest way to introduce the if-then statement is to use a non-mathematical example, so that students can
absorb the idea without “panic-inducing math” looming over their heads. We can ask them,“If it’s raining, does that mean
the streets are wet?” Or, “If it’s raining, does that imply there are cupcakes covering the street?” The difference between
these two statements is that one (should under normal terrestrial conditions) be true, and the other is not always true (in
fact, probably never has happened). Is the converse of an if-then statement true? To lead the students through understanding that the converse is not always true, the teacher can ask, “If it’s wet on the streets, does that mean it is raining?” The
answer is no, city employees could have sprayed down the streets. Here we are emphasizing the directionality of the if-then
statement, and that the implication is not necessarily true in the other direction. Explain to the students that these kinds of
statements are used in deductive reasoning; a logical process that draws conclusions. For example, if all cats have yellow
eyes, and there is a cat sleeping in the room, then I can deduce that this cat must have yellow eyes, even if I can’t see its eyes.
The teacher can check for understanding by asking the students to think up similar true and false statements. Have them
come up with some true statements, and then consider the truth of the converse statement. Next, have them think up false
statements. This is a great time to bring some laughter into the classroom, since it should be quite fun for them to think
up the false statements like, “If the Earth goes around the Sun, then Robert Pattinson1 is a green alien from Mars. This
sentence is false2 , but what about it’s converse? “If Robert Pattinson is a little green alien, then the Earth goes round the
Sun.” This statement is actually true, since when the premise is false, the statement as a whole is always true. This is a
difficult concept for anyone to accept, even for advanced mathematics students. As one colleague put it, “I think even
1 Popular
actor from the movie series Twilight.
trust scientists are accurate about the Earth being the celestial body that travels around the Sun, and Paulos makes a convincing argument in
Innumeracy against the possibility of there being aliens among us.
2I
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in common language this seems counter intuitive. How can a statement be true when you clearly said something false?
Moreover, a statement such as “If Robert Pattinson is a little green alien, then it will rain cupcakes seems twice as false.” 3
So why is this the case? Have the students think about making a bet. Suppose they have a friend who claims they can run
for a week straight without ever stopping to eat, drink, sleep or do anything else. You know that it is impossible for them
to do this, in other words, you know this is a false claim. Therefore, you may be willing to bet anything against their claim.
For example:
• “If you run for a week straight without ever stopping to eat, drink, sleep or do anything else, then I will give you a
million dollars.”
• “If you run for a week straight without ever stopping to eat, drink, sleep or do anything else, then I will give you my
first child.”
• “If you run for a week straight without ever stopping to eat, drink, sleep or do anything else, then I will clean your
room for ten years.”
You probably don’t have a million dollars, you won’t give your friend your first child and you won’t clean your friend’s
room, ever. Are you a liar? The answer is no: these are all true statements since you will never have to do any of them.
The statement “you run for a week straight without ever stopping to eat, drink sleep or anything else” is clearly impossible
and thus a false statement. Since it will never happen, you will never have to pay up, so you are not a liar.
Next, we introduce the idea of inductive reasoning. This means that one thing is likely to follow from another, rather
following absolutely. With inductive reasoning, if a proportion or cats have yellow eyes, and I there is a cat sleeping in the
room, then there is a certain percent chance or probability that this cat has yellow eyes. Or, if all cats that have ever been
observed have yellow eyes, then all cats should have yellow eyes. Another example is, if we observe that a certain majority
proportion of a sample of people will vote for one presidential candidate, then it follows inductively that this candidate
is likely to win.
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This discussion can be drawn out much more, depending on how much the teacher wants to get into
inductive reasoning. The teacher will then introduce an excerpt from a newspaper, and will show the class how to analyze
it. For example:
The following appeared in a letter to the editor of the Clearview newspaper:
”In the next mayoral election, residents of Clearview should vote for Ann Green, who is a member of the Good Earth
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4 For
Yaggie.
now will ignore problems in the representativeness of sampling.
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Coalition, rather than for Frank Braunm a member of the Clearview town council, because the current members are not
protecting the environment. For example, during the past year the number of factories in Clearview has doubled, air pollution levels have increased, and the local hospital has treated 25 percent more patients with respiratory illnesses. If we elect
Ann Green, the environmental problems in Clearview will certainly be solved.”
First, the teacher will point out that the statement that needs to be proved is the last one: ”If we elect Ann Green, the
environmental problems in Clearview will certainly be solved.” The letter seems to imply that because she is a member of
the Good Earth Coalition, she will solve environmental problems. This may or may not be true however. In order to be
more convincing that she would certainly solve environmental problems in the future, she would need to have solved many
environmental problems in the past. There are several other arguments that could be made to show that the logic of this
letter is weak.
The process of deconstructing this argument will be greatly facilitated by drawing diagram showing the existence or lack
of existence of implications. The first question should be, “what is it that we want to prove?” This is a very important habit
of successful mathematicians; if you don’t identify what you are trying to prove, then how can you prove it? Once that is
established, then the teacher can walk the students through finding the implications in the text. What do we know about
Ann Green from this letter? Do the facts provided really imply that she will solve all the environmental problems? Do
they strongly support the conclusion? Or only a little bit? Based on the facts provided, would the students feel comfortable
telling their family and friends to vote for Ann Green? Drawing the diagram should show that you don’t have enough
information to reach the desired conclusion. Given that this letter does not provide enough evidence to support its claim,
does that mean that the claim is false? This is a great discussion on the difference between the lack of proof and a the proof
that something is false.
Activity: Students are given a task card with other excerpts from newspapers (see Appendix A). The task card give
pointers on how to think about the activity:
• What is the claim that the author is making?
• What evidence does the author give to justify the claim?
• Are there any gaps in the the authors logic?
They will analyze the arguments as a group, discussing whether the argument makes sense and the author justifies the claim.
Challenges: Students will likely have a great deal of difficulty working through the analysis of these excerpts. The
first one is similar to the example in that the statement to be proved is already written in if-then form at the end of the
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excerpt. In the second excerpt, there is an if-then statement, but it is less obvious since it is written in backwards order. In
the third excerpt it will be even harder for the students to identify what they are trying to prove, and write it in the if-then
form. Some discussion at the board on rewriting statements in the if-then form may be in order. After the claim is found
and written in if-then form, the next goal is to get the students to find all the facts in the excerpt that could possibly support
the claim. Another thing they can look for is facts that could even possibly negate the claim. Some ideas are presented
as support, but when viewed from another angle, can actually give evidence against the claim. Here, the emphasis should
be on working towards the confidence to assert an idea that was logically deduced based on some known facts. This may
seem non-mathematical but it is none the less the ultimate goal: using mathematics education as a way to teach students to
think clearly and logically about all aspects of their lives.
Assessment: The ideal group discussions should involve students saying things such as, “I don’t see why they think
the politician will do solve the environmental problems.” Or, ”What if there is some other reason why pollution has doubled?” In fact, “what if” statements are the key to attacking logical problems and should be encouraged profusely. The first
step is for students to critically analyze the excerpts by asking questions about what doesn’t make sense or isn’t obvious to
them. The next thing we want to observe is that the students can take the lack of convincing evidence and turn it around
into an assertion that the author is not making a strong case for their claim. The third thing we would like to see them
discuss is what evidence they would need to see in order to be convinced that the claim is true.
PART 2: The Two Column Proof Game
Summary: This lesson was designed to fill a 100 minute period. The students see deductive reasoning used to prove the
zero product principle, play the “proof game,” practice logical flow and backwards reasoning, and analyze incorrect proofs.
Lecture: Now, we dive into using logic with mathematical statements. Students should have been exposed to the property
of real numbers that tells us that if ab = 0, then a = 0 or b = 0. This is a good beginning example of using an If-Then statement in a mathematical context, as well as an opportunity to define the mathematical meaning of ”or” and ”and.” First the
teacher reminds the students of the zero product principle. Next, the teacher can ask the students for examples of numbers
where this is true: 3 · 0 = 0, 0 · −2 = 0 and 0 · 0 = 0 are some examples. This brings to light the mathematical meaning
of the word ”or”. In math, ”or” tells that that one, or the other, or both could have the stated property. Can we think of
an example where this doesn’t work? Could we have that ab = 0 and have it be true that a = 3 and b = −1 ? The answer of course is no for the real numbers. But how do we prove this? The teacher can lead the following proof on the board.
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PROOF: To prove this, we need to split it up into three cases. The first when a 6= 0, second when b 6= 0, and third,
when they are both equal to zero. First we assume that a 6= 0. We want to show that it logically follows that b must be
equal to zero. If ab = 0, then we can divide by a, since a is not equal to zero. Then we get b =
0
a
which means that b = 0.
Similarly, if we assume b 6= 0, then if ab = 0, then we can divide by b since now it is not equal to zero. The third case is
works because 0 × 0 = 0. Thus we get that a = b0 , and so a = 0. Therefore, we have proved that if ab = 0, then a = 0,
b = 0, or they both equal zero.
One difficult thing to understand if the reason for using cases. We do this because of the word “or.” The mathematical
meaning of “a or b” is “one of the following is true: only a, only b, a and b.” In everyday language we often don’t include
the last part, we might say, I’ll do my homework or sleep. But clearly we aren’t going to do homework and sleep at the
same time. But in mathematics, this third case of both statements being true is always a possibility.
Next we can give an example of a statement that is not true, such as ”If ab = 1 then a = 1 or b = 1.” Some students
may immediately know that this is wrong. Here the teacher can encourage them to come up with a counter example. There
are infinitely many counter examples since if ab = 1 then it could be that a = n and b =
1
n
since n × 1n = 1 for all the
infinitely many natural numbers. There is only one example that satisfies this statement: when a = 1 and b = 1. But since
there is at least one case where neither a nor b equal 1, but ab = 1, then the statement cannot be considered true.
The teacher can check for understanding of the if-then statements by asking students to come up with If-Then statements
that come from previous geometry they have learned. One example is the triangle sum conjecture, ”If x, y, and z are the
three different angles of a triangle, then x + y + z = 180.” Another example comes from Pythagorean triples, ”If I have a
right triangle and the two sides adjacent to the right angle have lengths 3 and 4 units, then the hypotenuse has a length of
5 units.” Students can work in groups, and the teacher can visit the groups and discuss whether the statements make sense
or not to the whole group. The statement may be almost correct, like ”If I have a right triangle and two of the sides have
lengths 3 and 4 units, then the other side has a length of 5 units.” This is missing the information about which sides have to
be of which length, and is not true until that information is added in. This exercise should be used to get students to apply
detail and rigor to their understanding of the use of conjectures and geometric properties. This part should not be skimmed
over, but should be the point where developing habits of a successful mathematician is emphasized: i.e. rigor and attention
to detail.
This last part should be brief. Following on the deductive reasoning the students learned the day before, the teacher explains
that if a implies b, and b implies c, this means that a implies c. So if we want to prove “If a then c.” we could write the
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following two-column proof:
Statements
Reasons
a
Given.
b
If a, then b.
c
If b, then c.
Since we started with a, and ended with c, we just proved that a implies c, or rather, that if we have a, then we get c.
Activity: The students will be given a task card (see Appendix C) which instructs them how to play the proof game and
what the task is for the day. Students are given cards that ask them to prove things like ”If A, then D.” They are given a
number of other cards that have different letters, as well as cards that have ”reasons” such as ”Given” or ”If A, Then B.”
(see Appendix B) Here is another example:
Statements
Reasons
A
Given.
B
If A, then B.
C
If A, then C.
D
If B & C, then D.
For the next part of the task card, in order to convince students of the importance of writing clear, logically ordered proofs,
the students are given symbolic logic proofs that have errors, and are asked to find them. Here they will see that reading a
proof is confusing if it is not written in the correct order. The students must find the errors and explain why the proof does
not make sense with these errors. An example of an incorrect proofs is:
Statements
Reasons
A
Given.
C
If B then C.
B
If A then B.
D
If B & C, then D.
In this proof, the only error is that the order of two implications is switched. You can’t get C until you have B, so you have
to prove you have B first. The task card includes three incorrect proofs for students to analyze. The most common error is
incorrect order; the proofs include all the supportive statements needed, but have the implications in the wrong order.
Challenges: While writing proofs, students may have a hard time knowing whether certain steps are “legal moves.” For
example, a student may wonder if they can use A twice to get B and C. To convince them that you can use A twice you can
share the following example. Suppose Chris and Breanna will only go ice skating with you if Monica will come. So you
call Monica, and she agrees to go ice skating. You used the fact that Monica will go twice: once to convince Chris to go,
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and once to convince Breanna to go. In analyzing proofs, students may have a hard time verbalizing what is wrong with
the proofs.
Assessment: An important skill of successful mathematicians is to be able to work backwards. After identifying what
you need to prove, you can look for what you ways to do it. For example, you may be asked to prove that given lines A
and B, both intersected by a line C, with angles x and y on the interior of A and B and opposite sides of C, if you know
x=y, then you know A and B are parallel. How do you show two lines are parallel? If you can show that line D which you
can construct to be perpendicular to line A, is also perpendicular to line B, then you know A and B are parallel. Then you
start from the beginning and try to show that line B is perpendicular to line D. Students may discover on their own that it
can be easier to work backwards, as well as forwards. If a student does this, the teacher can stop the class and showcase
that particular student or group of students had a great idea to work backwards. For low-performing students, this is the
perfect opportunity to boost their confidence.
PART 3: The Geometry Game
Summary: This lesson is meant to fill a 100 minute period. The students will construct proofs of congruency using a
game just like the day before, but this time with pieces that have geometric relationships.
Background knowledge: Students should already have been exposed to the concepts of similarity and congruency, as
well as the congruency shortcuts.
Lecture: The teacher can review the proof game from the day before, emphasizing that we always have the statements on
the left and reasons on right. As well, the teacher should emphasize the proper order, and being clear on what the starting
point is, and what you are trying to prove.
Activity: The students will be given a task card (see Appendix D) which instructs them on how to play the geometry
game and what the task is for the day. Students are given cards with diagrams, asking them to prove that two triangles
are congruent (Appendix E). They are given a number of other cards that have relationships related to the diagrams. One
example5 is:
5 From
the geometry curriculum at Mission High School, San Francisco, California.
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Challenges: First, students have a hard time knowing what to do with midpoints and bisectors. They may not make the
leap that this means that each side of the midpoint and bisector is equal to the other. Once they realize they are equal and
mark them as such, students may think that they do not need to write down that step in their proof. For example, if a point is
a bisector of line, they need to say that each half is congruent to the other. But they actually need to say that the two halves
of the angle are congruent because of the bisector. Make sure students don’t confuse parallel tick marks with congruence
tick marks. Students have a hard time seeing alternate interior angles in parallelograms and hourglass figures. We have to
debunk these misconceptions by giving strong reasoning for why what we are saying makes logical sense.
Assessment: “B is the midpoint of line AD” is not a congruency statement, but can be used to make a statement about
congruency. Students may not say this exactly, but it needs to be clear that they understand this. While it may seem obvious
that to use a congruency shortcut, you need to show three congruencies, the students may not actually be trying to think
logically. They may just want to get through the problem, possibly by guessing. Try not to tell them if they have the correct
answer or not, and insist that the give the justifications needed for the conclusion.
CONCLUSION: What’s next?
We began this lesson with just the proof game. We wanted to extend on it in order to teach students more about logic
and to transition more smoothly into geometry proofs. The strength of the proof game is that it provides scaffolding for
the students to learn how to write two-column proofs. WE extended the proof game, by creating the geometry game, to
transition even further with the scaffolding. The next step is for them to write two-column proofs on their own, without
the aid of pre-written statements and reasons. With the the proof game and geometry game, it was a low-stakes task. Removing the game, and having them write on their own means they have to think about what they are going to write. They
aren’t going to want to write just anything if it means they might have to erase it later. After writing their own congruency
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proofs, we can have them move on to harder proofs, such as ones that integrate different concepts related to circles, area an
the pythagorean theorem. The most important thing is that, although we are using these games to make our lessons more
effective, the efficacy of the lesson is dependent on the teacher’s ability to facilitate learning. The teacher needs to make
sure that each of the challenges we stated is addressed by some sort of teacher move or probing question. The teacher
should not just passively monitor the activities, but should facilitate and draw out the understanding my active engagement
and probing.
Resources:
• Fendel, Dan; Resek, Diane; Alper, Lynne and Fraser, Sherry. Interactive Mathematics Program Year 2. Key Curriculum Press, Emeryville; 1998.
• Johnston, William and McAllister, Alex M. A Transition to Advanced Mathematics. Oxford University Press, New
York; 2009.
• Paulos, John Allen. A Mathematician Reads the Newspaper. Anchor Books, New York; 1995.
• Paulos, John Allen. Innumeracy. Hill and Wang, New York; 1988, 2001.
• Serra, Michael. Discovering Geometry: An Investigative Approach, Fourth Edition. Key Curriculum Press, Emeryville;
2008.
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