How mathematical statements are justified. Includes axioms and postulates accepted without proof, definitions, theorems and what proving one requires, conjectures and why examples never suffice, disproving a general statement with a single counterexample, and indirect proof by assuming the opposite and reaching a contradiction.
Subject: Algebra 1 · 65 slides · symbolic lesson
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Title
Algebra 1 · Chapter 12 — Radicals and More Connections to Geometry
Logical Reasoning: Proof
Objectives
Five outcomes, each one you can test yourself on with a pencil and no answer key.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-745 — the lesson these objectives are drawn from
Warm-up
This whole book has used rules such as the distributive property. This lesson asks where those rules came from and how anything is ever established.
Discussion prompt
Why is it true that a times the quantity b plus c equals a b plus a c? Can you prove it, or is it assumed?
Hint: Try to derive it from something more basic.
Answer:
It cannot be derived from anything simpler within algebra — it is an axiom, one of a small number of properties assumed at the outset. Everything else in the book was built on such assumptions.
That is not a weakness. A system of reasoning has to start somewhere, and being explicit about where is what distinguishes mathematics from a collection of rules that happen to work.
Concept
A postulate or axiom is a property accepted without proof. A definition names a new idea in terms of existing ones. A theorem is a statement that must be proved before it may be used.
theorem — A statement that can be proven to be true. Once proved, a theorem may be used as a reason in the proofs of other theorems.
Every step of a proof must cite an axiom, a definition, given information, or a theorem already proved.
Figure (svg): How the parts of a mathematical system rest on each other
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-741
Section
Section 1
Concept
Axioms are assumed, definitions introduce vocabulary, and theorems are earned. Knowing which is which tells you what may be used freely and what must be established first.
The basic axioms of algebra are the rules of Chapter 2.
Figure (svg): How the parts of a mathematical system rest on each other
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-741 — the sections on axioms, definitions and theorems, and the Basic Axioms of Algebra summary
Picture it
Assumptions at the bottom.
Figure (svg): How the parts of a mathematical system rest on each other
The arrangement is not arbitrary. A proof may cite anything below it in the picture and nothing above, which is what keeps the reasoning from becoming circular.
Worked example
Placing Chapter 2's properties in the structure.
\[ \text{Are the commutative property, the definition of an integer, and } c(-b) = -cb \text{ axioms, definitions or theorems?} \]
Take the commutative property
Why: Assumed at the outset.
Take the definition of an integer
Why: Names a set using existing terms.
Take the third
Why: Provable from the axioms.
Note the consequence
Why: Only the last needed proving.
Figure (svg): How the parts of a mathematical system rest on each other
\[ \text{axiom}, \; \text{definition}, \; \text{theorem} \]
Verify: ask what would happen without each
Why: Removing the axiom would leave nothing to reason from; removing the definition would leave a word undefined; removing the theorem would only mean it had to be reproved when needed. That difference in consequence is what the three categories capture.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-741
Sorting
Three categories.
Sort into buckets
Sort each statement by its status.
Two of each, and the distinction is not about difficulty. The commutative property is simpler than most theorems and is nonetheless assumed rather than proved.
Worked example
A question worth asking once.
\[ \text{Why is the distributive property not proved?} \]
Ask what it would be proved from
Why: Something more basic.
Ask about that statement
Why: It would need proving too.
Note the difficulty
Why: The chain cannot go on forever.
Resolve it
Why: Some statements are assumed.
Figure (svg): How the parts of a mathematical system rest on each other
\[ \text{axioms end the regress} \]
Verify: consider the alternative
Why: If every statement required proof from an earlier one, no proof could ever be completed, since there would always be one more step behind. Choosing a small set of starting assumptions is what makes the whole structure possible.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-740
Trap
The statement is clearly true, so it can be used without proof.
Skip the proof for something that seems evident
Why: Obviousness felt like sufficient justification.
Many false statements seem evident and many true ones do not. Whether something needs proving is settled by its logical status, not by how convincing it looks.
Prove it from the axioms, then use it freely afterwards.
Establish it once, then cite it
Why: That is what theorems are for.
Goldbach's conjecture has seemed evident for nearly three hundred years and remains unproved.
Faded example
Four kinds of reason.
Fill in the blanks
Every step of a proof must be justified by an axiom, a definition, given information, or a previously proved theorem.
Why: Those four sources are exhaustive, which is what makes a proof checkable by someone else. Anything cited that is not one of them is an unjustified assumption.
Elimination
By its logical status.
Eliminate the wrong options
Which kind of statement is accepted without proof?
Survives elimination: A
Why: Axioms are assumed deliberately and definitions need no proof because they only assign names. Conjectures are the interesting case: unproved and, unlike axioms, not accepted either.
Socratic
The rules work either way.
Discussion prompt
Say what is gained by listing the axioms of algebra explicitly. Then say what the Greek geometers changed about mathematics.
Hint: Think about what a reader can check.
Answer:
Listing them makes every later claim checkable: a reader can follow a proof and verify that each step rests on something agreed. Without an explicit starting point there is no way to distinguish a proof from a persuasive argument, and no way to settle a disagreement about whether something has been established.
What the Greek geometers changed, about two and a half thousand years ago, was to stop treating mathematics as a collection of practical rules that worked and start demanding that each result be derived from a small set of stated assumptions. That shift is why mathematical results stay true rather than merely holding up so far, and it is the reason this lesson closes the book.
Section
Section 2
Concept
A proof is a chain of statements in which each one is justified by an axiom, a definition, given information, or a theorem already proved. The justifications are as much a part of it as the algebra.
\[ c(a - b) = ca - cb \]
Once proved, a theorem may be cited in later proofs.
Figure (svg): A short proof with every step justified
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 741-741 — Example 1, Prove a Theorem, and its Study Tip on justifying every step
Picture it
Two columns, both required.
Figure (svg): A short proof with every step justified
Without the right-hand column this is only a calculation. The reasons are what turn a sequence of true statements into an argument that establishes something.
Worked example
This is Example 1 from the textbook.
\[ \text{Using } a - b = a + (-b), \text{ prove that } c(a - b) = ca - cb. \]
Rewrite the subtraction
Why: By the subtraction property.
\[ c [a + (-b)] \]
Distribute
Why: By the distributive axiom.
\[ c a + c(-b) \]
Use the known theorem
Why: That c times negative b is negative c b.
\[ c a + (-c b) \]
Rewrite as a subtraction
Why: By the subtraction property again.
\[ c a - c b \]
Figure (svg): A short proof with every step justified
\[ c(a - b) = ca - cb \]
Verify: check every reason is legitimate
Why: Two steps cite the subtraction property, one cites the distributive axiom, and one cites a theorem proved earlier. All four are among the permitted kinds of justification, so the chain holds.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 741-741
Matching
Every line needs one.
Match the pairs
Why: The same property is cited twice, in opposite directions. That is entirely legitimate: a property may be used to rewrite in either direction, since it asserts an equality.
Worked example
Guided Practice 1.
\[ \text{Prove that } (a + b) + c = (b + c) + a. \]
Swap the first two
Why: Commutative property.
\[ (b + a) + c \]
Regroup
Why: Associative property.
\[ b + (a + c) \]
Swap inside the bracket
Why: Commutative property.
\[ b + (c + a) \]
Regroup again
Why: Associative property.
\[ (b + c) + a \]
Figure (svg): A short proof with every step justified
\[ (a + b) + c = (b + c) + a \]
Verify: check with numbers first
Why: Taking a, b and c as one, two and three gives six on both sides — which supports the claim without proving it. The proof covers every possible triple at once, which no amount of numerical checking could.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 741-741
Error analysis
The student proved a theorem about subtraction.
Annotate
On: \( \begin{aligned} c(a - b) &= ca - cb \\ &\text{because that is how the distributive property works} \end{aligned} \)
The conclusion is correct and the justification is not, which makes this a failed proof rather than a wrong answer. Citing an axiom in a form it was not stated in is the commonest way a proof goes wrong.
Faded example
The axiom is stated for a sum.
Fill in the blanks
c(a - b) = c[a + (-b)] \;\to\; ca + c(-b)
Why: The subtraction property converts the difference into a sum, which is the form the distributive axiom is stated for. Citing an axiom requires the expression to match its statement.
Elimination
In a proof.
Eliminate the wrong options
Which reason may not be cited?
Survives elimination: A
Why: Obviousness is not a logical status, so it justifies nothing. The other three are exactly the permitted kinds of reason, along with information given in the problem.
Socratic
The algebra is the same either way.
Discussion prompt
Say what the justification column adds to a proof. Then say what happens to a proof whose reasons are omitted.
Hint: Who is the proof for?
Answer:
The reasons let someone else check the argument line by line without having to reconstruct your thinking. They also force the writer to confirm that each step really is licensed, which is where errors are caught — the student above would have noticed that the distributive property does not mention subtraction.
Without them the proof becomes a sequence of assertions that a reader must take on trust or verify independently, which defeats the purpose. A proof is a public object, meant to persuade anyone who follows it, and the reasons are what make that possible.
Section
Section 3
Concept
A conjecture is a statement thought to be true but not yet proved. No number of confirming examples establishes a claim about infinitely many cases.
Goldbach's conjecture is still unproved after nearly three centuries.
Figure (svg): Even numbers written as sums of two primes
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 741-741 — the section on conjectures and Example 2 on Goldbach's Conjecture
Picture it
And still not settled.
Figure (svg): Even numbers written as sums of two primes
Every even number ever tested has been a sum of two primes, into the quintillions. That is overwhelming evidence and no proof at all, which is a distinction worth taking seriously.
Worked example
This is Example 2 from the textbook.
\[ \text{Every even integer except } 2 \text{ is a sum of two primes. Does the list } 4 \text{ to } 26 \text{ prove it?} \]
Check the examples
Why: Every one works.
\[ 4 = 2 + 2, \; 26 = 3 + 23 \]
Ask what is claimed
Why: Every even integer.
Compare
Why: Twelve against infinitely many.
Conclude
Why: Evidence, not proof.
Figure (svg): Even numbers written as sums of two primes
\[ \text{unproved, and still open} \]
Verify: consider how much has been checked
Why: Modern computers have verified the conjecture for every even number into the quintillions without finding an exception, and it remains unproved. If that much evidence is not proof, twelve cases plainly are not either.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 741-741
Faded example
Two different things.
Fill in the blanks
Checking many cases provides evidence, but establishing a claim about infinitely many cases requires a proof.
Why: Evidence makes a statement worth believing and worth investigating. Proof makes it certain, and only an argument that reaches every case can do that.
Worked example
Why the caution is not merely pedantic.
\[ \text{Consider the claim that } n^2 + n + 41 \text{ is prime for every whole number } n. \]
Try small values
Why: Nought, one, two.
\[ 41, 43, 47 \text{, all prime} \]
Keep going
Why: Up to thirty-nine.
Try forty
Why: Forty squared plus forty plus forty-one.
\[ 1681 = 41 ^{2} \]
Conclude
Why: The pattern breaks.
Figure (svg): Evidence and proof distinguished
\[ n = 40: \; 1681 = 41 \times 41 \]
Verify: notice how convincing forty cases would feel
Why: Anyone testing the first ten or twenty values would be confident the claim was true, and they would be wrong. That is precisely why mathematics demands an argument rather than a tally, however long the tally is.
Trap
It works for every case I have tried, so it is true.
Generalise from a run of confirming cases
Why: The pattern was consistent and there was no reason to doubt it.
The expression above stays prime for forty consecutive values and then fails. A pattern's track record says nothing certain about the case you have not tried.
It works for every case I have tried, so it is worth trying to prove.
Treat evidence as a reason to investigate, not as a conclusion
Why: That is what a conjecture is.
Most theorems began as conjectures supported by exactly this kind of evidence.
Sorting
Proving or disproving a general claim.
Sort into buckets
Sort each by whether it settles a general statement.
Only two of the six settle anything, and one of those settles it in the negative. Confirming instances, however many, never close the question.
Hypothesis
Even a billion of them.
Predict first
What is the essential obstacle?
Correct: The claim covers cases no finite list can include.
Goldbach's conjecture has been checked past four quintillion and remains open.
Why: A statement about every even integer concerns infinitely many numbers, and any list you can write is finite — so infinitely many cases remain untested no matter how long you continue. Honesty and computational accuracy are beside the point: the gap is one of logic rather than of diligence. That is why an argument covering all cases at once is the only thing that closes it.
Socratic
It is not established.
Discussion prompt
Say what use an unproved conjecture has. Then say what makes one worth attention rather than dismissal.
Hint: Think about what mathematicians do with them.
Answer:
A conjecture directs effort: it says where a theorem is likely to be found and gives something specific to attempt. Whole areas of mathematics have been developed by people trying to prove one, and the tools built along the way often outlast the question.
What makes one worth attention is a combination of strong evidence, connections to other established results, and a statement simple enough to be worth settling. Goldbach's has all three, which is why it has occupied people for nearly three hundred years despite having no practical consequence anyone can name.
Section
Section 4
Concept
To show that a general statement is false, a single counterexample suffices. Proving and disproving are strikingly unequal tasks.
Disproving needs an instance; proving needs an argument.
Figure (svg): Two columns comparing what it takes to prove and to disprove a general claim
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 742-742 — the section on counterexamples and Example 3, Find a Counterexample
Picture it
One instance against all cases.
Figure (svg): Two columns comparing what it takes to prove and to disprove a general claim
The imbalance is the whole point: a claim about everything is fragile in one direction and hard to secure in the other. That is what makes general statements worth stating carefully.
Worked example
The method of Example 3 from the textbook.
\[ \text{Disprove: for all real } x, \; \sqrt{x^2} = x. \]
Read what is claimed
Why: For every real x.
Look for a likely failure
Why: Try a negative value.
\[ x = -3 \]
Test it
Why: The root of nine is three.
\[ \sqrt{9} = 3 \]
Compare
Why: Three is not negative three.
Figure (svg): Two columns comparing what it takes to prove and to disprove a general claim
\[ x = -3: \; \sqrt{(-3)^2} = 3 \ne -3 \]
Verify: check that one instance really is enough
Why: The claim said every real x, so exhibiting one that fails makes it false — no further cases need testing. Note also that the statement is true for every non-negative x, which is why it looked plausible.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 742-742
Elimination
To disprove a general statement.
Eliminate the wrong options
How many cases must fail?
Survives elimination: A
Why: A statement about every case is contradicted by one case where it does not hold. That asymmetry with proof is one of the most useful facts in this lesson.
Worked example
What a counterexample teaches.
\[ \text{How should the claim } \sqrt{x^2} = x \text{ be corrected?} \]
Note where it fails
Why: Only for negative x.
Restrict the claim
Why: To non-negative values.
\[ \text{for } x \ge 0 \]
Or state it generally
Why: Using absolute value.
\[ \sqrt{x^2} = |x| \]
Check the repaired version
Why: At negative three.
\[ |-3| = 3 \;\checkmark \]
Figure (svg): Two columns comparing what it takes to prove and to disprove a general claim
\[ \sqrt{x^2} = |x| \text{ for all real } x \]
Verify: test the repaired claim on both signs
Why: At three the root of nine is three and the absolute value of three is three; at negative three the root is still three and the absolute value is also three. The corrected statement holds everywhere the original was tested.
Trap
Here are five values of x where the claim fails, which shows it is false.
Give several instances for extra certainty
Why: More evidence seemed stronger.
One suffices, and completely. A general claim is false the moment a single case fails, so four of the five add nothing — which is worth knowing, since finding even one can take effort.
At x equal to negative three the claim fails, so it is false.
Exhibit one counterexample and stop
Why: The claim is already refuted.
The asymmetry runs the other way for proving, where no number of instances is ever enough.
Faded example
Negative values often break claims about roots.
Fill in the blanks
\sqrt3 = \sqrt-3 = ___, \text___ ___
Why: The radical gives the positive root, so it cannot return a negative value of x. Testing a negative number is usually the first thing to try against a claim involving square roots.
Sorting
Some values break claims more often.
Sort into buckets
Sort each candidate by whether it is a promising first test.
Negatives, nought, one and fractions below one are where general claims break most often, because each behaves differently from a typical positive whole number. Trying those first is the efficient search.
Socratic
The two tasks look symmetric.
Discussion prompt
Explain the imbalance between proving and disproving a general statement. Then say what it implies about how carefully general claims should be worded.
Hint: What does each task have to cover?
Answer:
Proving requires an argument reaching every case at once, while disproving requires exhibiting a single case — so one task is about all instances and the other about one. That is not a symmetry at all, which is why a claim that took centuries to prove can be refuted in a line if it happens to be false.
It implies that general claims should be stated with their conditions attached, since an unstated restriction is exactly where a counterexample will be found. The claim about square roots was true for every non-negative value and false only for negatives, so wording it carefully would have made it a theorem instead of a mistake.
Section
Section 5
Concept
In an indirect proof, the statement is assumed false and reasoning proceeds until something impossible follows. That impossibility shows the assumption was wrong, so the statement is true.
It is how the root of two is shown to be irrational.
Figure (svg): The shape of an indirect proof
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 742-745 — the key word indirect proof and Example 4, on proving a client's innocence
Picture it
Assume, reason, break.
Figure (svg): The shape of an indirect proof
Nothing about the statement itself is demonstrated directly. What is demonstrated is that its denial cannot stand, which for a statement that is either true or false amounts to the same thing.
Worked example
The proof promised back in Lesson 9.1.
\[ \text{Prove that } \sqrt{2} \text{ cannot be written as a fraction.} \]
Assume the opposite
Why: A fraction in lowest terms.
\[ \sqrt{2} = \tfrac{a}{b} \]
Square and rearrange
Why: Both sides.
\[ 2 b ^{2} = a ^{2} \]
Deduce a is even
Why: Its square is even.
\[ a = 2 k \]
Deduce b is even too
Why: Substituting gives 2b^2 = 4k^2.
Figure (svg): The indirect proof that the square root of two is irrational
\[ \sqrt{2} \text{ is irrational} \]
Verify: identify exactly what was contradicted
Why: The fraction was assumed to be in lowest terms and the reasoning showed both parts are even, so it was not. Since every fraction can be written in lowest terms, no fraction at all can equal the root of two.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 742-745
Faded example
Three moves.
Fill in the blanks
Assume the statement is false, reason from that until something impossible follows, and conclude that the statement is true.
Why: The structure works because a statement is either true or false: ruling out one leaves the other. The whole argument rests on the contradiction being genuine.
Worked example
The method of Example 4, in its everyday form.
\[ \text{How does an alibi establish innocence?} \]
Assume the opposite
Why: Suppose the client is guilty.
Follow the consequence
Why: They were at the scene.
Bring in the evidence
Why: They were elsewhere then.
Conclude
Why: Impossible, so not guilty.
Figure (svg): An indirect argument used to establish innocence
\[ \text{indirect proof} \]
Verify: compare the two arguments
Why: The root of two proof assumes a fraction and breaks lowest terms; the alibi assumes guilt and breaks being in one place. Both reject their assumption because it forced something that cannot be, which is exactly the same logical move.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 745-745
Trap
Assuming the opposite led to a very strange conclusion, so the statement must be true.
Treat an unexpected result as a contradiction
Why: It seemed unlikely enough to count.
Strange is not impossible. An indirect proof needs something that genuinely cannot be — a quantity both even and odd, a fraction both in lowest terms and not — rather than merely a result nobody expected.
Assuming the opposite forced a and b to be both coprime and both even, which cannot happen.
Identify precisely what has been contradicted
Why: Name the two incompatible facts.
Many true statements are surprising, so surprise carries no logical weight at all.
Elimination
After assuming the opposite.
Eliminate the wrong options
What ends the argument?
Survives elimination: A
Why: Only an impossibility rules out the assumption. Being unexpected, hard or counter-intuitive says something about the reader rather than about the mathematics.
Hypothesis
It never computes anything about root two directly.
Predict first
What does the contradiction actually establish?
Correct: That no fraction in lowest terms can equal root 2, and hence no fraction can.
The same argument works for the root of three, and for any prime.
Why: The final step is the important one: every fraction can be reduced to lowest terms, so ruling out those in lowest terms rules out all of them. Without that observation the proof would only have shown something about a particular kind of fraction. Choosing a and b badly is not possible either, since the argument never used anything about them beyond the assumption.
Socratic
A direct argument would seem preferable.
Discussion prompt
Say when an indirect proof is the natural approach. Then say what makes irrationality a case for it.
Hint: What does the statement deny?
Answer:
It suits statements that assert something cannot happen, because a direct proof would have to examine every possible way it might and rule each out. Assuming it does happen gives you something concrete to work with, which a negative statement otherwise fails to provide.
Irrationality is exactly such a statement: it says no fraction whatever equals the root of two, and there are infinitely many fractions to consider. Assuming one does gives an equation to manipulate, and that single equation is enough to break the assumption for all of them at once.
Comparison
Fill the blanks from memory before you scroll back.
Comparison matrix
| Task | What is needed | How much work |
|---|---|---|
| Prove a general statement | an argument covering every case | no number of examples suffices |
| Disprove a general statement | one counterexample | a single case is enough |
| Prove something cannot happen | often an indirect proof | assume it does, and reach a contradiction |
The middle row is the cheapest task in mathematics and the top row is among the most expensive. Knowing which you face determines how to spend your effort.
Pattern
To establish or refute a mathematical claim, these five moves cover it.
Step two before step three is deliberate: a few minutes hunting for a counterexample can save hours attempting to prove something false.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-745
Check
Statuses differ.
Check your understanding
Which kind of statement is accepted without proof?
Answer: A
Why: Axioms, also called postulates, are the properties assumed at the outset so that reasoning has somewhere to start.
Check
One is enough.
Check your understanding
How many counterexamples disprove a general statement?
Answer: A
Why: A statement claiming something about every case is false as soon as one case fails.
Check
Assume the opposite.
Check your understanding
What must an indirect proof reach to succeed?
Answer: A
Why: Something genuinely impossible must follow from the assumption, which shows the assumption cannot hold.
Real world
This is the courtroom question from the lesson opener. A lawyer defending a client often argues indirectly rather than demonstrating innocence directly.
Discussion prompt
Describe how an alibi establishes innocence, name the logical form of the argument, and say what mathematical proof in this lesson has the same shape.
Hint: What is assumed at the start?
Answer:
The lawyer supposes the client is guilty, which would place them at the scene at a particular time, and then produces evidence that they were somewhere else at that time. Being in two places at once is impossible, so the supposition fails and the client is not guilty.
That is an indirect proof: assume the opposite of what you want to establish, reason to something impossible, and reject the assumption.
The proof that the square root of two is irrational has exactly this shape. It supposes the root is a fraction in lowest terms, shows that both its parts must then be even, and observes that a fraction cannot be in lowest terms while both parts share a factor of two. The contradiction is different but the logical move is identical, which is why the courtroom is a fair illustration of a technique that looks purely mathematical.
Commit first
Answer, then rate your confidence honestly. Confident and wrong is the combination worth finding.
Predict first
A claim about all even integers holds for the first billion cases tested. What follows?
Correct: It remains a conjecture.
\[ n = 40: \; 40^2 + 40 + 41 = 1681 = 41^2 \]
Why: The claim concerns infinitely many integers and a billion is a finite list, so infinitely many cases remain untested — no amount of checking closes that gap, which is a matter of logic rather than of diligence. Goldbach's conjecture has been verified past four quintillion and is still open after nearly three hundred years, which shows how little even extraordinary evidence settles. The expression n squared plus n plus forty-one is a sharper warning: it produces primes for forty consecutive values and then fails at forty, so a run of confirmations genuinely can end. An axiom, meanwhile, is something chosen as a starting assumption rather than something a claim graduates into by accumulating evidence.
Explain it
They checked a pattern on ten examples and concluded it is always true.
Discussion prompt
In no more than four sentences, explain what their examples have and have not shown. Then give them the example that makes the point.
Hint: How many cases does the claim cover?
Answer:
A usable answer: ten examples give good evidence that the pattern is worth investigating, but the claim covers infinitely many cases and ten of them leaves infinitely many untested. Only an argument that reaches every case at once can prove it.
The example worth showing them is n squared plus n plus forty-one, which gives a prime for every whole number from nought to thirty-nine and then fails at forty. Forty confirmations in a row and the pattern still breaks.
Exit ticket
Name the weakest spot before you close the deck. That is the one worth ten minutes tonight.
Predict first
Which of these would you least want to be handed cold on a quiz tomorrow?
Correct: Whichever you picked is the right answer — and each one has a specific fix.
Why: Axioms and theorems are separated by asking whether the statement is assumed or derived. Justification is fixed by writing a reason beside every line and checking it is one of the four permitted kinds. The examples point is fixed by remembering that a general claim covers cases no list can reach. Indirect proof is fixed by writing the assumed opposite explicitly at the top, so you know what you are trying to contradict. Pick yours and do five of that kind tonight rather than twenty mixed ones.
Connect it up
Do this on paper. It is worth more than rereading the slides.
Draw it
At the top of a page draw the three levels — axioms, definitions, theorems — and write two examples of each from this book, noting beside each level whether proof is required. Underneath, write out a short proof in two columns, statements on the left and reasons on the right, and check that every reason is an axiom, a definition, given information or an earlier theorem. In the middle, write a general claim, list several confirming examples, and then write one sentence saying exactly what those examples have established and what they have not. Beneath that, take a false general claim, find one counterexample, and then repair the claim by adding the condition its counterexample violated. In the lower half, set out an indirect proof in three labelled parts — the assumed opposite, the reasoning, and the contradiction — using either the root of two or an argument of your own, and name precisely which two statements are incompatible. Finally, in the margin, write the asymmetry between proving and disproving in one line.
Your indirect proof should name the contradiction explicitly rather than leaving it implied. If you cannot state which two things cannot both be true, the argument has reached a surprise rather than a contradiction.
Recap
Five things, and together they are what makes the rest of this book more than a list of rules.
| If the question says | Your first move is |
|---|---|
| Prove this theorem | Find a chain of justified steps from the axioms |
| Is this always true? | Look for a counterexample before attempting a proof |
| Show this statement is false | Exhibit one case where it fails |
| Show this cannot happen | Assume it does and seek a contradiction |
| Does this list prove it? | Ask how many cases the claim covers |
That completes the course. Every technique in this book — solving, factoring, graphing, the quadratic formula, the distance formula — rests on the small set of axioms set out in this lesson, and each was established by exactly the kind of reasoning practised here. What began as rules to follow is, properly seen, a single connected argument.
McDougal Littell Algebra 1: Concepts and Skills, Ch. 12 Radicals and More Connections to Geometry — Lesson 12.9 Logical Reasoning: Proof §12.9, pp. 740-745 — everything on these slides traces back here
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