Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 240 minutes, as an unvalidated planning estimate. This collection contains 11 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.
Attempt these mixed tasks before opening worked guidance. State assumptions and justify all steps; numerical agreement alone is not a proof.
These public problems require human mathematical review. Saved responses and revealed solutions do not establish secure independent examination performance.
Work is untimed. Use the suggested duration for planning, then review errors, return to linked lessons and revise.
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0 of 11 tasks have recorded responses. No overall mastery score is assigned.
A point in H has cosh distances3/2,9/8,9/4 from anchors3+2i,3+4i,5+2i respectively. Recover it by a known half-plane isometry and justify uniqueness. If only the first two measurements are retained, give the other candidate. Classify every hyperbolic isometry fixing the first two anchors pointwise, proving the exhaustive claim rather than listing examples. Which one exchanges5+2i and1+2i?
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.
Construct the affine plane over F₅ using coordinate points and lines. Count its points and lines, verify unique joining lines and unique parallels, and explain why it is not a model of ordered real-plane geometry. State what a model establishes about consistency.
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 the Euclidean distance triangle inequality from the dot product and state its equality case. Explain which parts of an isosceles-triangle SAS argument are neutral, and why using a triangle angle sum of π to prove Playfair would be circular in a neutral-geometry development.
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.
Find the intersection coordinates of x²+y²=4 and (x−2)²+y²=4. Explain why rational-coordinate geometry is not closed under these circle intersections. Prove that a cube of volume seven cannot have its side constructed from a unit segment by straightedge and compass, and contrast algebraic closure under square roots with Dedekind completeness.
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.
Classify F(x,y)=(−y+4,x+2), find its fixed point and derive its distance preservation. Compose reflections across y=1 and y=−2 in that order. Finally explain how a similarity of ratio 5/3 changes lengths, areas and angles, and why an arbitrary affine map need not preserve angles.
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.
For A=(1,2), B=(7,2), C=(1,8), find the centroid and show affine maps preserve it. Find the projective intersection of y=−3x+2 and y=−3x−5. Compute CR(0,2;3,7) using ((c−a)(d−b))/((c−b)(d−a)), and explain why cross-ratio survives projective coordinate changes but equal spacing need not.
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.
For the projective conic X²+2Y²−Z²=0, derive the tangent at [1:0:1]. Explain why selecting a different line at infinity can make the same nonsingular real projective conic appear as an ellipse, parabola or hyperbola. Why does this not contradict Euclidean preservation of eccentricity under similarities?
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.
For z=1+2i and w=5+2i, compute cosh d and the center/radius of their hyperbolic geodesic in the half-plane model. Verify that T(z)=2z+3 preserves the metric |dz|/Im z, and classify its determinant-one matrix by trace squared.
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.
Take a Poincaré disk point on the positive x-axis with radius 2/5. Find its Klein coordinate, its hyperboloid coordinates using ((1+r²)/(1−r²),2r/(1−r²),0), and cosh of its distance from the origin. Explain why straight Klein chords do not imply Euclidean angle measurement.
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 curvature −1 right hyperbolic triangle has leg cosines cosh a=5/3 and cosh b=13/5. Find cosh c and explain why the Euclidean Pythagorean equation is not exact. On a sphere of radius 3, compute the area of a simple disk-like geodesic triangle with angles 2π/3, π/2 and π/2. For unit sphere points with dot product −√3/2, compare their spherical distance and their elliptic-quotient distance.
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 geodesic triangle on a surface of constant curvature −9 has angle sum 2π/3. Find its area. Explain what Gauss–Bonnet states when curvature varies, and formulate a valid relative-independence argument for the parallel postulate using Euclidean and hyperbolic models.
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
A point in H has cosh distances3/2,9/8,9/4 from anchors3+2i,3+4i,5+2i respectively. Recover it by a known half-plane isometry and justify uniqueness. If only the first two measurements are retained, give the other candidate. Classify every hyperbolic isometry fixing the first two anchors pointwise, proving the exhaustive claim rather than listing examples. Which one exchanges5+2i and1+2i?
Construct the affine plane over F₅ using coordinate points and lines. Count its points and lines, verify unique joining lines and unique parallels, and explain why it is not a model of ordered real-plane geometry. State what a model establishes about consistency.
Prove the Euclidean distance triangle inequality from the dot product and state its equality case. Explain which parts of an isosceles-triangle SAS argument are neutral, and why using a triangle angle sum of π to prove Playfair would be circular in a neutral-geometry development.
Find the intersection coordinates of x²+y²=4 and (x−2)²+y²=4. Explain why rational-coordinate geometry is not closed under these circle intersections. Prove that a cube of volume seven cannot have its side constructed from a unit segment by straightedge and compass, and contrast algebraic closure under square roots with Dedekind completeness.
Classify F(x,y)=(−y+4,x+2), find its fixed point and derive its distance preservation. Compose reflections across y=1 and y=−2 in that order. Finally explain how a similarity of ratio 5/3 changes lengths, areas and angles, and why an arbitrary affine map need not preserve angles.
For A=(1,2), B=(7,2), C=(1,8), find the centroid and show affine maps preserve it. Find the projective intersection of y=−3x+2 and y=−3x−5. Compute CR(0,2;3,7) using ((c−a)(d−b))/((c−b)(d−a)), and explain why cross-ratio survives projective coordinate changes but equal spacing need not.
For the projective conic X²+2Y²−Z²=0, derive the tangent at [1:0:1]. Explain why selecting a different line at infinity can make the same nonsingular real projective conic appear as an ellipse, parabola or hyperbola. Why does this not contradict Euclidean preservation of eccentricity under similarities?
For z=1+2i and w=5+2i, compute cosh d and the center/radius of their hyperbolic geodesic in the half-plane model. Verify that T(z)=2z+3 preserves the metric |dz|/Im z, and classify its determinant-one matrix by trace squared.
Take a Poincaré disk point on the positive x-axis with radius 2/5. Find its Klein coordinate, its hyperboloid coordinates using ((1+r²)/(1−r²),2r/(1−r²),0), and cosh of its distance from the origin. Explain why straight Klein chords do not imply Euclidean angle measurement.
A curvature −1 right hyperbolic triangle has leg cosines cosh a=5/3 and cosh b=13/5. Find cosh c and explain why the Euclidean Pythagorean equation is not exact. On a sphere of radius 3, compute the area of a simple disk-like geodesic triangle with angles 2π/3, π/2 and π/2. For unit sphere points with dot product −√3/2, compare their spherical distance and their elliptic-quotient distance.
A geodesic triangle on a surface of constant curvature −9 has angle sum 2π/3. Find its area. Explain what Gauss–Bonnet states when curvature varies, and formulate a valid relative-independence argument for the parallel postulate using Euclidean and hyperbolic models.
