Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 90 minutes, as an unvalidated planning estimate. This collection contains 2 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
State the input convention and all conditions needed for a halving step.
Track coefficients against the declared base inputs; keep the external common power of two separate.
Prove the result with divisibility and a linear-combination identity, not only agreement with a worked example.
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The original inputs are 84 and 140. Remove their greatest common power of two to obtain base inputs a=21,b=35 and scale 4. A changed initialization uses u=a+b=56 with coefficients (1,1), and v=a=21 with coefficients (1,0). Prove that this initialization preserves the base gcd. Repeatedly halve u until it is odd, preserving integer coefficients. Then subtract u from v and halve the new v until odd. At equality, identify the gcd and give a certificate in the original inputs. Explain why directly halving the first coefficient pair fails the integer-coefficient requirement, even though it preserves the numerical equality over rational numbers.
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 verifier for nonnegative integers a,b, not both zero, is offered three claims for a=84,b=132: d=24 with coefficients (5,-3); d=6 because it divides both inputs; and d=12 with coefficients (-3,2). Determine which claims actually certify the gcd. Design a sufficient verifier using only a positive candidate d, two divisibility checks and an integer linear-combination identity. Prove sufficiency without rerunning either Euclidean algorithm. State how it handles (0,75), why it rejects d=0, and why allowing arbitrary rational coefficients would make it unsound.
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.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
The original inputs are 84 and 140. Remove their greatest common power of two to obtain base inputs a=21,b=35 and scale 4. A changed initialization uses u=a+b=56 with coefficients (1,1), and v=a=21 with coefficients (1,0). Prove that this initialization preserves the base gcd. Repeatedly halve u until it is odd, preserving integer coefficients. Then subtract u from v and halve the new v until odd. At equality, identify the gcd and give a certificate in the original inputs. Explain why directly halving the first coefficient pair fails the integer-coefficient requirement, even though it preserves the numerical equality over rational numbers.
A verifier for nonnegative integers a,b, not both zero, is offered three claims for a=84,b=132: d=24 with coefficients (5,-3); d=6 because it divides both inputs; and d=12 with coefficients (-3,2). Determine which claims actually certify the gcd. Design a sufficient verifier using only a positive candidate d, two divisibility checks and an integer linear-combination identity. Prove sufficiency without rerunning either Euclidean algorithm. State how it handles (0,75), why it rejects d=0, and why allowing arbitrary rational coefficients would make it unsound.
