Organize parity arguments and understand what an if-and-only-if claim requires.
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Try
A universal integer statement can be split into even and odd cases. A biconditional makes two promises: one condition implies the other, and the reverse implication also holds. Cases and directions solve different organizational problems.
Predict
Predict whether checking positive and negative integers covers the domain. It omits zero. Even/odd cases cover zero, negative integers and positive integers together. Each case needs a formula with an integer witness.
Experiment
For n=−2,−1,0,1,2, mark parity and inspect consecutive factors n and n+1. Record which factor is even. These examples suggest a rule; they do not justify it for every integer. Turn the pattern into the two cases below.
Understand
Worked argument: n(n+1) is even for every integer n. If n=2k, the product is 2[k(2k+1)]. If n=2k+1, the product is 2[(2k+1)(k+1)]. Each bracket is an integer, and every integer falls into one of these two cases. This establishes the claim for the entire domain.
An invalid proof says n is either 2k or 2k+2. Both forms are even, so odd integers have been omitted. Another invalid proof of an equivalence establishes only its easier direction. A complete case split cannot compensate for a missing logical direction.
Apply
Use headings such as “even case,” “odd case,” “first implication,” and “reverse implication” in the practice. At the end, explicitly check that the cases or directions cover what the claim requires. This organization makes gaps visible to a reviewer.
Continue
Depth: prove n² is divisible by 4 exactly when n is even. In the odd case, write n²=4(k²+k)+1, which leaves remainder 1. Compare this stronger residue information with parity alone; do not infer divisibility by 4 merely from evenness.
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Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
At n=−3, compute n²−n.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
To establish P if and only if Q, what is required?
Prove n²−n is even for every integer n using parity cases.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that n² is odd if and only if integer n is odd.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
