Separate examples, universal claims and counterexamples by identifying the allowed objects.
Loading your saved responses…
Try
Choose a number, square it, and compare the result with the original. Negative integers and integers above one seem to support the claim that squaring never makes a number smaller. Try a fraction between zero and one before accepting that claim.
Predict
Predict whether changing the domain from integers to real numbers changes the answer. A domain is the collection of permitted inputs. A predicate is a statement whose truth may depend on those inputs. “For every integer x, x²≥x” includes a domain, a quantifier and a predicate; “squaring makes things bigger” leaves important details unstated.
Experiment
Use the counterexample investigation to compare finite tests with a general argument. Record one successful integer case and one failing fractional case. Every counterexample must satisfy the assumptions and make the conclusion false. An input outside the domain cannot refute the statement.
Understand
Consider the related claim x²≥4 for every integer x with |x|≥2. If x≥2, multiplying the nonnegative inequality by x gives x²≥2x≥4. If x≤−2, put y=−x≥2, so x²=y²≥4 by the first case. These cases cover the stated domain. This is a proof because its reasoning applies to an arbitrary allowed integer; the same proof actually works for real x with |x|≥2.
An invalid proof checks only −2 and 2 and says the result follows for everything between or beyond them. That step supplies no rule connecting unchecked values to the examples. In particular, removing the assumption |x|≥2 would make x=0 a counterexample.
Apply
Write the exact domain before beginning the practice. For a proposed counterexample, record its domain membership, the left side and the right side. When repairing a false statement, identify whether you changed its assumptions or its conclusion; a repaired claim does not retrospectively prove the original.
Continue
Depth: solve x(x−1)≥0 over the real line by locating the two roots and checking the signs on each interval. Explain why an integer domain avoids the interval where the product is negative. Continue to quantifiers to distinguish checking one input from handling every input.
Apply · explain · revise
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.
For x=1/2, calculate x²−x.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Where is x²≥x true for every allowed x?
Prove x²≥x for every integer x without checking a finite list.
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.
A learner checks x=−2,−1,0,1,2 and concludes x²≥x for every real x. Diagnose and repair the claim.
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.
