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.
Define every edge family before reasoning from a picture.
Give a proper coloring for an upper bound and a separate impossibility proof for a lower bound.
Track which hypotheses survive each changed construction; a failed proof is not automatically a false conclusion.
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0 of 2 tasks have recorded responses. No overall mastery score is assigned.
Start from the seven-cycle with vertices v0 through v6. Its shadow lift adds independent shadows u0 through u6, joins each ui to the original neighbors of vi, and joins an apex w to all shadows. Prove the resulting graph is triangle-free and has chromatic number 4. Supply an explicit coloring of every vertex, compute vertex and edge counts and all vertex degrees, then repeat the construction once and compute the new counts and chromatic number. Explain why a search over colorings of the first graph alone cannot prove the general repeated-construction theorem.
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.
In the shadow lift of the five-cycle, delete just the apex edge w-u0 and leave all other edges unchanged. A classmate applies the original recoloring proof and concludes the changed graph still needs four colors. Find the precise unsupported step. Determine the changed graph’s chromatic number, proving both bounds. Use original vertex colors (3,1,2,1,2) as a starting point, and assign every shadow and the apex explicitly. Does this example say that every possible deletion from every lifted graph lowers its chromatic number?
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
Start from the seven-cycle with vertices v0 through v6. Its shadow lift adds independent shadows u0 through u6, joins each ui to the original neighbors of vi, and joins an apex w to all shadows. Prove the resulting graph is triangle-free and has chromatic number 4. Supply an explicit coloring of every vertex, compute vertex and edge counts and all vertex degrees, then repeat the construction once and compute the new counts and chromatic number. Explain why a search over colorings of the first graph alone cannot prove the general repeated-construction theorem.
In the shadow lift of the five-cycle, delete just the apex edge w-u0 and leave all other edges unchanged. A classmate applies the original recoloring proof and concludes the changed graph still needs four colors. Find the precise unsupported step. Determine the changed graph’s chromatic number, proving both bounds. Use original vertex colors (3,1,2,1,2) as a starting point, and assign every shadow and the apex explicitly. Does this example say that every possible deletion from every lifted graph lowers its chromatic number?
