High-school Number and Quantity objective evidence

Why fractional powers mean roots — Real numbers, complex coordinates, vectors and matrices

• For a positive real a and positive integer n, explain why extending (a^r)^n=a^(rn) to r=1/n forces a^(1/n) to be the positive nth root. Why is positivity part of this convention?Written reasoning: review required
• Explain why 64^(2/6) and 64^(1/3) agree. Then justify a^(2/6)=a^(1/3) for every positive a without relying only on this example.Written reasoning: review required

Rewrite radicals without losing the domain — Real numbers, complex coordinates, vectors and matrices

• For a>0, rewrite √a·∛(a²) as a single power, then as a sixth root. Justify both representations.Written reasoning: review required
• Prove √(x²)=|x| for every real x. Use it to simplify √(16x²y⁴) for real x,y.Written reasoning: review required
• Simplify √(x²)/x for real x where it is defined. Give the domain and a piecewise description; explain why it is not always 1.Written reasoning: review required

Which number types survive arithmetic? — Real numbers, complex coordinates, vectors and matrices

• Prove the sum and product of two arbitrary rational numbers are rational. State the denominator checks.Written reasoning: review required
• Let r be rational and u irrational. Prove r+u is irrational. Explain why showing one example is insufficient.Written reasoning: review required
• Prove ru is irrational when r is nonzero rational and u is irrational. Explain exactly what changes if r=0.Written reasoning: review required
• Give irrational pairs whose sum is rational and irrational, and irrational pairs whose product is rational and irrational. Justify every assertion; you may use the known irrationality of √2.Written reasoning: review required

Choose quantities, units and honest axes — Functions in graphs, measurements and growth decisions

• A rectangular display is 150 cm by 80 cm. What is its area in square meters?Automatically checked response; reasoning still matters
• For the supplied temperature observations at 0,2,4,6 hours: 18,19,21,20 °C, propose axes for showing small changes. Defend an origin and a scale, and explain how a truncated vertical scale should be labeled.Written reasoning: review required
• A paint covers 8 m² per liter. A wall is 3 m by 240 cm and needs two coats, with no openings. Compute required liters and distinguish model assumptions from a purchasing decision.Written reasoning: review required

Choose quantities, units and honest axes — Functions in graphs, measurements and growth decisions

• School A recycles 90 kg for 300 students; school B recycles 70 kg for 140 students. Choose a useful quantity for comparing participation scale, compute it and state a limitation.Written reasoning: review required
• A class compares two bus services with different trip lengths and passenger counts. Define a quantity that answers a specific efficiency question and explain the data needed.Written reasoning: review required

Report what measurements support — Functions in graphs, measurements and growth decisions

• A display rounded to the nearest 0.2 kg shows 18.6 kg. State a compatible interval and defend a report that does not imply milligram accuracy.Written reasoning: review required
• A board length is 12.0±0.1 cm and width 5.0±0.1 cm, interpreted as inclusive guaranteed bounds. Give area bounds and choose an honest summary.Written reasoning: review required
• A ruler has 1 mm markings but may be miscalibrated by 3 mm. Explain why ±0.5 mm is not a justified total error bound.Written reasoning: review required

Calculate with a real and an imaginary coordinate — Real numbers, complex coordinates, vectors and matrices

• For real a,b,c,d, derive (a+bi)(c+di)=(ac−bd)+(ad+bc)i. Diagnose the answer (ac+bd)+(ad+bc)i.Written reasoning: review required
• A learner claims (3+2i)²=9−4=5. Repair the calculation and explain when the missing imaginary term would vanish for (a+bi)².Written reasoning: review required

Calculate with a real and an imaginary coordinate — Real numbers, complex coordinates, vectors and matrices

• Compute (1−2i)+(−4+3i)−(2+i), and verify by a coordinate table.Written reasoning: review required
• For real a,b,c,d, derive (a+bi)(c+di)=(ac−bd)+(ad+bc)i. Diagnose the answer (ac+bd)+(ad+bc)i.Written reasoning: review required
• Find the real part of (−3+5i)−(4−2i).Automatically checked response; reasoning still matters

Calculate with a real and an imaginary coordinate — Real numbers, complex coordinates, vectors and matrices

• Define the conjugate of z=a+bi and prove z times its conjugate equals a²+b². Check z=−2+5i.Written reasoning: review required

Use conjugates to divide and measure — Real numbers, complex coordinates, vectors and matrices

• Compute (3−4i)/(1+2i) and verify your result by multiplying by the original denominator.Written reasoning: review required
• Prove that z=a+bi≠0 has inverse (a−bi)/(a²+b²). Explain why this does not define an inverse of zero.Written reasoning: review required

Draw complex arithmetic and midpoints — Real numbers, complex coordinates, vectors and matrices

• Plot −√3+i and give a polar description with positive modulus and angle in [0°,360°). Explain why the two descriptions represent the same point; compare multiplication by 2i.Written reasoning: review required

Draw complex arithmetic and midpoints — Real numbers, complex coordinates, vectors and matrices

• Draw z=2+3i and w=−2+i. Construct z−w using an arrow between their tips and by adding −w at the tip of z. A labeled coordinate/segment description is an equally valid response.Written reasoning: review required
• For z=−1+4i, construct its conjugate and iz. Describe coordinates and explain the geometric operations.Written reasoning: review required
• Plot −√3+i and give a polar description with positive modulus and angle in [0°,360°). Explain why the two descriptions represent the same point; compare multiplication by 2i.Written reasoning: review required

Multiply complex numbers by scaling and rotating — Trigonometry, polar geometry and precalculus representations

• Derive the product rule for r(cos a+i sin a) and s(cos b+i sin b), and use it to explain the integer-power formula for nonnegative integers n.Written reasoning: review required

Draw complex arithmetic and midpoints — Real numbers, complex coordinates, vectors and matrices

• For arbitrary complex z=a+bi and w=c+di, prove their average is the midpoint of the joining segment and their distance is |w−z|.Written reasoning: review required
• Find the distance between 1−2i and 7+6i.Automatically checked response; reasoning still matters

Why every quadratic has two complex roots — Real numbers, complex coordinates, vectors and matrices

• For x²+6x+13=0, find the real part of either root.Automatically checked response; reasoning still matters
• For real a,b,c with a≠0, prove ax²+bx+c has exactly two complex roots counting multiplicity. Separate positive, zero and negative discriminants.Written reasoning: review required

Why every quadratic has two complex roots — Real numbers, complex coordinates, vectors and matrices

• Verify x⁴+16=(x²−2√2 x+4)(x²+2√2 x+4), and factor each quadratic over the complex numbers.Written reasoning: review required

Why every quadratic has two complex roots — Real numbers, complex coordinates, vectors and matrices

• For real a,b,c with a≠0, prove ax²+bx+c has exactly two complex roots counting multiplicity. Separate positive, zero and negative discriminants.Written reasoning: review required
• The preceding proof assumed real coefficients. Show that every complex number d=u+vi has a complex square root, and use that fact to extend the quadratic argument to complex a,b,c with a≠0. You may use principal square roots of nonnegative reals.Written reasoning: review required
• A real quartic has no real-axis crossings. Does that contradict the Fundamental Theorem of Algebra? State the theorem accurately and illustrate with x⁴+1.Written reasoning: review required

Turn endpoints into a directed quantity — Real numbers, complex coordinates, vectors and matrices

• Draw the arrows from A=(−2,1) to B=(1,5) and C=(3,−3) to D=(6,1). Explain why they represent the same vector although their endpoints differ. Coordinate/segment descriptions are accepted.Written reasoning: review required
• A displacement has magnitude 10 m and direction 120°. Determine exact components, sketch it and check the magnitude.Written reasoning: review required

Turn endpoints into a directed quantity — Real numbers, complex coordinates, vectors and matrices

• A displacement runs from (−3,2) m to (4,−2) m. What is its horizontal component in metres?Automatically checked response; reasoning still matters
• Prove reversing endpoints negates a displacement but preserves its magnitude. What direction can be assigned when the endpoints coincide?Written reasoning: review required

Turn endpoints into a directed quantity — Real numbers, complex coordinates, vectors and matrices

• A walker moves with constant velocity (3,−4) m/s for 6 seconds from (2,1) m. Find displacement, final position and speed; explain why these are different quantities.Written reasoning: review required

Add arrows and find the actual resultant — Real numbers, complex coordinates, vectors and matrices

• An aircraft has air-relative velocity (−12,0) m/s and wind velocity (0,5) m/s. Find ground-relative velocity, speed and direction, and explain why this is a model assumption.Written reasoning: review required

Add arrows and find the actual resultant — Real numbers, complex coordinates, vectors and matrices

• For u=(3,1) and v=(−1,3), construct u+v by end-to-end arrows and a parallelogram. Give exact coordinates for every vertex.Written reasoning: review required
• Two vectors have magnitudes 8 and 8 and directions 30° and 150°. Find the magnitude and direction of their sum, then explain why adding their magnitudes would fail.Written reasoning: review required

Add arrows and find the actual resultant — Real numbers, complex coordinates, vectors and matrices

• Two vectors have magnitudes 8 and 8 and directions 30° and 150°. Find the magnitude and direction of their sum, then explain why adding their magnitudes would fail.Written reasoning: review required
• An aircraft has air-relative velocity (−12,0) m/s and wind velocity (0,5) m/s. Find ground-relative velocity, speed and direction, and explain why this is a model assumption.Written reasoning: review required

Add arrows and find the actual resultant — Real numbers, complex coordinates, vectors and matrices

• Draw u=(2,−1) and v=(−3,2) from the origin. Which tip-to-tip arrow represents u−v? Verify by adding −v at the tip of u.Written reasoning: review required

Turn endpoints into a directed quantity — Real numbers, complex coordinates, vectors and matrices

• Prove reversing endpoints negates a displacement but preserves its magnitude. What direction can be assigned when the endpoints coincide?Written reasoning: review required

Scale, reverse, and handle the zero vector — Real numbers, complex coordinates, vectors and matrices

• Draw v=(2,−3), 2v, −v and 0v from the same origin, or give labeled endpoints and directions. Explain the zero case.Written reasoning: review required
• For v=(−6,8), find the horizontal component of (−1/2)v.Automatically checked response; reasoning still matters

Scale, reverse, and handle the zero vector — Real numbers, complex coordinates, vectors and matrices

• For v=(x,y) and real c, prove ||cv||=|c| ||v||. State exactly when a direction comparison is meaningful.Written reasoning: review required
• A velocity is v=(4,−1) m/s. Compare the displacement after 3 seconds with the dimensionless scalar multiple −3v, stating units for both. Separately interpret (−3 seconds) times v. Does that formal time offset describe waiting for a negative elapsed duration?Written reasoning: review required

Keep the labels when operating on matrices — Real numbers, complex coordinates, vectors and matrices

• Using rows A,B and columns pens,pads, compute [[12,7],[5,9]]+[[3,4],[6,2]]−[[8,5],[4,6]]. Interpret every resulting entry and check physical feasibility.Written reasoning: review required
• A table A has columns [pens,pads] and values [[4,9],[6,2]]. Table B for the same rows has columns [pads,pens] and values [[1,3],[5,7]]. Find the combined stock correctly and explain why matching dimensions alone was insufficient.Written reasoning: review required
• Which operations are defined and meaningful: adding a 2×3 count matrix to a 3×2 count matrix; adding two equally labeled 2×3 count matrices; subtracting a 2×3 dollar table from a 2×3 count table? Justify rather than only naming the shapes.Written reasoning: review required

Keep the labels when operating on matrices — Real numbers, complex coordinates, vectors and matrices

• A payoff table P=[[2,−1],[0,3]] gives dollars per play, with rows your choices and columns the other player’s choices. Compute 3P and explain the negative and zero entries. When would this scaling model fail?Written reasoning: review required
• A cost matrix has an entry 7/4 dollars per kit. All costs increase by 20%. What is the new entry in dollars per kit?Automatically checked response; reasoning still matters

Keep the labels when operating on matrices — Real numbers, complex coordinates, vectors and matrices

• Using rows A,B and columns pens,pads, compute [[12,7],[5,9]]+[[3,4],[6,2]]−[[8,5],[4,6]]. Interpret every resulting entry and check physical feasibility.Written reasoning: review required

Which number laws survive for matrices? — Real numbers, complex coordinates, vectors and matrices

• For A=[[1,2],[0,1]] and B=[[2,0],[3,1]], find entry (1,1) of AB.Automatically checked response; reasoning still matters
• Let A be m×n and B,C be n×p. Prove A(B+C)=AB+AC entry by entry. Also state the compatible shapes and argument for (D+E)F.Written reasoning: review required

Which number laws survive for matrices? — Real numbers, complex coordinates, vectors and matrices

• Compute AB and BA for A=[[1,2],[0,1]], B=[[2,0],[3,1]]. Why does this disprove a universal commutative law rather than merely showing a rounding discrepancy?Written reasoning: review required
• For A of size m×n, B of size n×p and C of size p×q, justify (AB)C=A(BC). Explain why this allows regrouping but not reordering.Written reasoning: review required
• Let A be m×n and B,C be n×p. Prove A(B+C)=AB+AC entry by entry. Also state the compatible shapes and argument for (D+E)F.Written reasoning: review required

Which number laws survive for matrices? — Real numbers, complex coordinates, vectors and matrices

• Explain A+0=A and AI=IA=A for a 2×2 matrix. Prove a 2×2 matrix [[a,b],[c,d]] is invertible exactly when ad−bc≠0. Include a nonzero singular example and show why cancellation can fail.Written reasoning: review required

Undo a square linear system when possible — Linear Algebra I: systems, spaces and linear maps

• For A=[[2,1],[0,3]], enter the (1,2) entry of A⁻¹.Automatically checked response; reasoning still matters

Use determinants to detect collapse and track signed volume — Linear Algebra I: systems, spaces and linear maps

• A square matrix has determinant zero exactly when:Automatically checked response; reasoning still matters

Undo a square linear system when possible — Linear Algebra I: systems, spaces and linear maps

• Find and verify the inverse of [[1,1],[1,2]].Written reasoning: review required
• If B and C are both two-sided inverses of square A, prove B=C.Written reasoning: review required

Multiply matrices in the order the maps act — Linear Algebra I: systems, spaces and linear maps

• For A=[[2,−1],[3,0]] and x=(4,5), enter the second coordinate of Ax.Automatically checked response; reasoning still matters
• If x first passes through B and then A, the composite output is:Automatically checked response; reasoning still matters

Use determinants to detect collapse and track signed volume — Linear Algebra I: systems, spaces and linear maps

• Explain why det [[1,2],[0,3]]=3 means area scale 3, and what a negative determinant would change.Written reasoning: review required