Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 95 minutes, as an unvalidated planning estimate. This collection contains 8 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 the unfamiliar cases before revealing solutions. Write reasons and domain checks, not only final numbers.
Labeled coordinate-and-segment descriptions may replace drawings if they identify every requested feature. Use the matrix tool when requested and independently verify its output.
Public practice, saved completion and reviewed understanding remain separate.
Loading saved assessment responses…
0 of 8 tasks have recorded responses. No overall mastery score is assigned.
A cost is C=24+5n(1+s), with n items and dimensionless surcharge s. Treat a compound part as one entity and explain the effect of doubling n. Separately prove (u+v)²−(u−v)²=4uv and use it to calculate 53²−47².
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.
Let P=x²+2x−3 and Q=2x²−x+4. Compute P−Q. Then simplify [(x²−16)/(x²−4x)] ÷ [(x+4)/(x−4)], preserving the original domain. Explain why the simplified result alone is insufficient.
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 (k+2)x<2(k+2) for every real k. Then solve 3x²+6x−2=0 by completing the square and explain how the same operation leads to the general quadratic formula.
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.
Two workers can divide a synthetic job freely and maintain independent constant rates. The slower takes 6 hours longer alone; together they take 4 hours. Define an unknown, derive a rational equation, find both solo times and verify the result. Identify an operational condition that could invalidate the model.
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.
Describe the full solution set of (x+1)(y−2)=0 and justify both directions. Separately draw the feasible region x≥0,y≥0,x+2y≤10,2x+y<8. Identify boundary inclusion, limiting vertices and whether (2,4) and (1,4) belong.
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 shared scaled graphs of y=(7/10)x+11/10 and y=−(2/5)x+37/10 using labeled points. Estimate their crossing, then give exact coordinates and a table bracket for its x-coordinate between 2.36 and 2.37. Explain the relation between the crossing and output equality.
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.
Compare solving 2^x=5−x with solving 1/(x−2)=0. For the first use 2^1.70≈3.24901 and 2^1.80≈3.48220 to bracket a solution and justify uniqueness. For the second explain why opposite signs at x=1 and x=3 do not certify a zero.
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.
Encode 2x+y=6, y+z=5, 2z=6 as Ax=b in order (x,y,z). Use the available inverse calculator or independently documented exact elimination to obtain A inverse and the solution. Verify the inverse by a full matrix product and the solution in all original equations. Then explain what changes if the last coefficient row becomes zero while its right side remains 6.
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 cost is C=24+5n(1+s), with n items and dimensionless surcharge s. Treat a compound part as one entity and explain the effect of doubling n. Separately prove (u+v)²−(u−v)²=4uv and use it to calculate 53²−47².
Let P=x²+2x−3 and Q=2x²−x+4. Compute P−Q. Then simplify [(x²−16)/(x²−4x)] ÷ [(x+4)/(x−4)], preserving the original domain. Explain why the simplified result alone is insufficient.
Classify (k+2)x<2(k+2) for every real k. Then solve 3x²+6x−2=0 by completing the square and explain how the same operation leads to the general quadratic formula.
Two workers can divide a synthetic job freely and maintain independent constant rates. The slower takes 6 hours longer alone; together they take 4 hours. Define an unknown, derive a rational equation, find both solo times and verify the result. Identify an operational condition that could invalidate the model.
Describe the full solution set of (x+1)(y−2)=0 and justify both directions. Separately draw the feasible region x≥0,y≥0,x+2y≤10,2x+y<8. Identify boundary inclusion, limiting vertices and whether (2,4) and (1,4) belong.
Construct shared scaled graphs of y=(7/10)x+11/10 and y=−(2/5)x+37/10 using labeled points. Estimate their crossing, then give exact coordinates and a table bracket for its x-coordinate between 2.36 and 2.37. Explain the relation between the crossing and output equality.
Compare solving 2^x=5−x with solving 1/(x−2)=0. For the first use 2^1.70≈3.24901 and 2^1.80≈3.48220 to bracket a solution and justify uniqueness. For the second explain why opposite signs at x=1 and x=3 do not certify a zero.
Encode 2x+y=6, y+z=5, 2z=6 as Ax=b in order (x,y,z). Use the available inverse calculator or independently documented exact elimination to obtain A inverse and the solution. Verify the inverse by a full matrix product and the solution in all original equations. Then explain what changes if the last coefficient row becomes zero while its right side remains 6.
