Number structures: a mixed construction and reasoning portfolio
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 180 minutes, as an unvalidated planning estimate. This collection contains 7 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 public mixed problems before revealing their worked guidance. Keep the original domains, units and coordinate conventions.
For every drawing, either provide a labeled equal-scale sketch or an exact coordinate and directed-segment description. Written evidence needs human mathematical review; matching a numeric entry does not assess a drawing or proof.
These are public transfer practice, not a secured examination. State any theorem used without proof.
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0 of 7 tasks have recorded responses. No overall mastery score is assigned.
Evaluate 243^(−2/5) by explaining the root definition. For real x≠0, simplify √(25x²)/x and explain its domain and both sign cases. Diagnose the claim ((−16)²)^(1/2)=(−16)^1.
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 t be irrational and r a nonzero rational. Prove r(t+2) is irrational. Is the product of two irrational numbers necessarily irrational? Supply a verified counterexample, then an irrational-result example using only the known irrationality of √3.
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.
Compute q=(4−7i)/(3+i), verify by multiplication, and compute |q|². Derive the denominator-conjugate method for a general nonzero denominator, including why its new real denominator is positive.
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 z=−4+3i and w=2−5i. Construct w−z, the midpoint and the reflection of z across the real axis. Find the distance. Then verify the quadratic with leading coefficient one and roots −2±3i; state what this and the general quadratic argument establish about the Fundamental Theorem of Algebra.
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 constant velocity vectors have magnitudes 10 and 6 m/s and directions 0° and 90°. Construct their sum by end-to-end arrows and a parallelogram; find speed and direction. Starting at (−1,4) m, find the position after 2 seconds. Explain the frame assumptions.
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 v=(−4,3), construct (−2/3)v and 0v. Find their magnitudes and describe their directions. Prove the magnitude-scaling rule for an arbitrary planar vector and real scalar.
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.
Rows are workshops A,B and columns screws,clips. Stock S=[[8,11],[14,6]], delivery D=[[5,3],[2,9]] and use U=[[7,8],[9,4]] are item counts. Compute remaining stock and interpret it. Separately, for P=[[2,1],[1,1]], find its inverse and check both products. Explain why this invertible example does not authorize cancellation by every nonzero matrix. Include a dimension-correct distributive argument.
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
Evaluate 243^(−2/5) by explaining the root definition. For real x≠0, simplify √(25x²)/x and explain its domain and both sign cases. Diagnose the claim ((−16)²)^(1/2)=(−16)^1.
Let t be irrational and r a nonzero rational. Prove r(t+2) is irrational. Is the product of two irrational numbers necessarily irrational? Supply a verified counterexample, then an irrational-result example using only the known irrationality of √3.
Compute q=(4−7i)/(3+i), verify by multiplication, and compute |q|². Derive the denominator-conjugate method for a general nonzero denominator, including why its new real denominator is positive.
Let z=−4+3i and w=2−5i. Construct w−z, the midpoint and the reflection of z across the real axis. Find the distance. Then verify the quadratic with leading coefficient one and roots −2±3i; state what this and the general quadratic argument establish about the Fundamental Theorem of Algebra.
Two constant velocity vectors have magnitudes 10 and 6 m/s and directions 0° and 90°. Construct their sum by end-to-end arrows and a parallelogram; find speed and direction. Starting at (−1,4) m, find the position after 2 seconds. Explain the frame assumptions.
For v=(−4,3), construct (−2/3)v and 0v. Find their magnitudes and describe their directions. Prove the magnitude-scaling rule for an arbitrary planar vector and real scalar.
Rows are workshops A,B and columns screws,clips. Stock S=[[8,11],[14,6]], delivery D=[[5,3],[2,9]] and use U=[[7,8],[9,4]] are item counts. Compute remaining stock and interpret it. Separately, for P=[[2,1],[1,1]], find its inverse and check both products. Explain why this invertible example does not authorize cancellation by every nonzero matrix. Include a dimension-correct distributive argument.
