Learning Paths
The learning paths below use English high-school lessons. Early mathematics and Grades 1–9 resources are currently in Vietnamese. High-school lessons are available in Vietnamese and English; not every lesson has both versions.
For early mathematics and Grades 1–9, see the free library (in Vietnamese) or the learning paths for all levels (in Vietnamese).
Choose a focused sequence. Check the prerequisites, work through the lessons in order and try the practice before reading the solutions. These routes cover selected goals, rather than the whole curriculum.
Choose a learning path
9 lessons
From 3D coordinates to orthogonal projection
Use coordinates, dot products and the normal direction to project a point onto a plane and check the result.
View this learning path9 lessons
From counting to probability
Distinguish ordered and unordered counting models, then move from sample spaces to addition, conditional probability and multiplication.
View this learning path10 lessons
From limits and derivatives to graph sketching
Combine domains, limits, derivative signs and local extrema to read and sketch the function families studied.
View this learning pathFrom 3D coordinates to orthogonal projection
Goal: Use coordinates, dot products and the normal direction to project a point onto a plane and check the result.
Before you start: Know basic vector operations, perpendicularity, midpoints and centroids, algebraic manipulation and two-by-two determinants. Start at step 1 to review coordinates; this route does not teach vectors from the beginning.
Review midpoint and centroid coordinates through the linked lessons if needed. In the centroid lesson, some angle and area checks use the dot product; return to those checks after step 3.
- Three-dimensional Cartesian coordinates: points and vectors
Distinguish point coordinates, vector coordinates and distances.
- Vector coordinates and length from two endpoints
Keep the initial and terminal points in the correct order when finding vector coordinates and length.
- The dot product in 3D: coordinate formula and applications
Use the dot product to express perpendicularity and check the conditions for an angle formula.
- Cross products and scalar triple products: formulas and applications
Learn the cross product before constructing a normal from two nonparallel directions.
- Normal vectors to a plane: definition and methods
Recognize nonzero normal vectors and the methods for finding them.
- Equations of planes in 3D: general and intercept forms
Write a plane equation from a point and normal vector, then check its constant term.
- Direction vectors of a line in space
Distinguish a line’s direction vector from a plane’s normal vector.
- Equations of lines in space: parametric and symmetric forms
Use parametric equations, including direction vectors with a zero component.
- Orthogonal projection of a point onto a plane in 3D
Follow the normal direction and check that the projection lies in the plane, including a point already in the plane.
From counting to probability
Goal: Distinguish ordered and unordered counting models, then move from sample spaces to addition, conditional probability and multiplication.
Before you start: Know finite sets, elements, unions and intersections, natural numbers, fractions and parity. Powers and polynomial multiplication are needed for the binomial theorem.
If counting principles, ordered selections and combinations are already secure, start the probability section at step 5. The binomial theorem at step 4 is an application of combinations, not a prerequisite for probability.
- The addition and multiplication principles
Specify distinct outcomes, then add disjoint cases or multiply stage counts.
- Permutations and circular arrangements
Distinguish row arrangements, circular order and labeled seats.
- Permutations and combinations: counting with and without order
Decide whether order changes an outcome before choosing a formula.
- The binomial theorem and Pascal’s triangle
Optional counting extension; if studying probability only, continue at step 5.
- Random experiments, sample spaces, and events
Separate an outcome from an event, and represent unions, intersections and complements.
- Probability in a finite equally likely model
Use a ratio of counts only in a finite equally likely model, with a consistent sampling order.
- The addition rule of probability
Check overlap before adding probabilities; use a complement where appropriate.
- Conditional probability and independence
Retain a positive conditioning probability and distinguish independence from mutual exclusivity.
- The multiplication rule of probability
Use conditional-probability chains; replace factors by marginal probabilities only when independence permits.
From limits and derivatives to graph sketching
Goal: Combine domains, limits, derivative signs and local extrema to read and sketch the function families studied.
Before you start: Know functions and domains, sequence limits, polynomial factorization, quadratic/product signs and even functions. Trigonometric examples use radians. This route does not teach algebra or sequence limits from the beginning.
Review sequence limits and trigonometry through the linked lessons if needed; each review has its own prerequisites. Absolute extrema and counting solutions with graphs are follow-on branches, not prerequisites for this route.
- Limits of functions: domains, signs, and indeterminate forms
Track the domain, approach direction and signs when transforming limits.
- Continuity at a point, one-sided continuity, and closed intervals
Compare a limit with the function value, distinguishing interior and endpoint conditions.
- Derivatives: definition, rules, and tangent lines
Apply the derivative definition and rules where their conditions hold, and identify the tangent interpretation.
- The chain rule: both the outer and inner derivatives
Identify the function layers and retain both the outer and inner derivatives.
- Monotone functions and derivative-sign conditions
Analyze derivative signs on each domain interval without crossing an undefined point.
- Local extrema: definitions, derivative signs, and the domain
Check interior points and sign changes; a derivative root alone does not establish a local extremum.
- Analyzing cubic functions: extrema, inflection, and graphs
Use derivative signs, the inflection point and end behavior to sketch a cubic.
- Analyzing even quartic functions: one or three local extrema
Combine even symmetry with derivative signs to distinguish one or three local extrema.
- Linear fractional functions: asymptotes, monotonicity, and symmetry
Check the denominator and degenerate cases before identifying asymptotes, branches and the center of symmetry.
- Graphs of y = |f(x)|: reflection, corners, and local extrema
Starting with a known graph, reflect its part below the horizontal axis and check corners and local extrema.