Tutor Sheet: Grades 9–12 Math (US)

Targets: Functions & models · Systems · Quadratics · Exp/Log · Trig · Proof/similarity · Stats/Probability

Quick diagnostics → Fast fixes

[FunctionSense] What is a function?

• Check: Is x2+y2=1x^2+y^2=1x2+y2=1 a function y=f(x)y=f(x)y=f(x)? Why/why not?
• Fix: Vertical-line test; input → one output; compare relation vs function.

[TransformSigns] Graph transformations

• Check: From y=f(x)y=f(x)y=f(x) to y=f(x−3)+2y=f(x-3)+2y=f(x−3)+2: shift which way?
• Fix: Inside = x: opposite sign (→ +3); outside = y: same sign (↑ +2).

[PiecewiseEndpoints] Domain breaks

• Check: Evaluate at boundary; which piece applies?
• Fix: Circle/closed-dot convention; test intervals carefully.

[SysMethodChoice] Solving systems

• Check: Pick method for: clean coefficients vs messy decimals.
• Fix: Elimination when coefficients align; substitution for isolated variable; matrix/graph for models.

[QuadDisc] Roots & discriminant

• Check: Predict root nature without solving.
• Fix: D=b2−4acD=b^2-4acD=b2−4ac: D>0D>0D>0 two real, D=0D=0D=0 double, D<0D<0D<0 complex; connect to graph intercepts.

[CompleteSquareSlip] Completing square errors

• Check: Convert to vertex form.
• Fix: Add/subtract (b/2a)2(b/2a)^2(b/2a)2 inside; factor a first; verify by expansion.

[ExpLogRules] Laws confusion

• Check: Simplify ln⁡(a2b)\ln(a^2b)ln(a2b); solve 3x=73^x=73x=7.
• Fix: ln⁡a2+ln⁡b\ln a^2+\ln blna2+lnb; take logs both sides; never split ln⁡(a+b)\ln(a+b)ln(a+b).

[TrigAngle] Radian/degree & quadrant

• Check: sin⁡(210∘)\sin(210^\circ)sin(210∘), cos⁡(−π/3)\cos(-\pi/3)cos(−π/3).
• Fix: Reference angle + signs by quadrant; unit check; unit circle anchors.

[SOHCAHTOA-Label] Right-triangle labeling

• Check: Given angle θ, identify opp/adj/hyp before using ratios.
• Fix: Mark triangle; write ratio before numbers.

[ProofFrame] Geometry proofs

• Check: Student proof missing reasons.
• Fix: Two-column or flow proof scaffold: Given → To Prove → Statements/Reasons; name the theorem (CPCTC, similarity).

[SimilarityScale] Similar figures

• Check: Find missing side/area scale factor.
• Fix: Ratio of sides kkk ⇒ area k2k^2k2 ⇒ volume k3k^3k3.

[SamplingBias] Stats designs

• Check: Convenience sample vs SRS.
• Fix: Identify bias source; propose fix (randomization/stratify).

[Indep≠Cond] Independence vs conditional

• Check: Are A and B independent if P(A∣B)=P(A)P(A|B)=P(A)P(A∣B)=P(A)?
• Fix: Define independence test; use tree/2-way table; compute P(A∩B)=P(A)P(B)P(A\cap B)=P(A)P(B)P(A∩B)=P(A)P(B).