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PERSONAL MATHEMATICS ENVIRONMENT

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Your adaptive study command center.
0day streak
0%overall mastery
0reviews due
LEARN

Build understanding before you drill.

Use lessons, the prerequisite-aware Learning Map, the full curriculum, and your deep study plan from one organized starting point.

4learning destinations
STUDY

Learn a concept

Move from explanation to worked reasoning without leaving the existing lesson system.

ORIENT

See where the math connects

Choose a destination or inspect the complete prerequisite structure before starting.

Evidence boundary: opening a Learn destination never awards mastery. Learning evidence still comes from the existing lesson, practice, review, diagnostic, checkpoint, and exam systems.
PRACTICE

Choose the right kind of evidence.

Build a study session, target one skill, retrieve due material, recalibrate with diagnostics, or run a no-hint exam simulation.

5practice destinations
FOCUSED WORK

Practice and review

MEASURE

Diagnostics and exams

LABS

Explore mathematics by manipulating it.

Nineteen released lab destinations are grouped by mathematical role instead of competing for permanent space in the primary navigation.

19lab destinations
FOUNDATIONS

Number, algebra, and geometry

DATA & UNCERTAINTY

Statistics and probability

FUNCTIONS & CHANGE

Precalculus through advanced calculus

APPLIED & COMPUTATIONAL

Modeling, finance, linear algebra, numerical methods, and ML math

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PROGRESS

Inspect evidence without inventing a story.

Use recorded-evidence analytics for the current learner state and attempt history for the underlying events.

2progress destinations
ANALYZE

Progress intelligence

TRACE

Attempt history

No synthetic trends: missing evidence remains missing evidence. These surfaces do not reconstruct historical mastery states that were never stored.
TOOLS

Use support when the mathematics calls for it.

Tutor, Workbench, Notebook, and the Calculator Suite are grouped here as utilities rather than mixed with progress or curriculum navigation.

4core tools
EXPLAIN & RECORD

Tutor and Notebook

COMPUTE

Calculator Suite

Open the existing scientific, finance, graphing, and time calculators without creating another calculator product.

TODAY'S OBJECTIVE

Build mathematical fluency through deliberate practice.

Take the placement diagnostic so MathPro can establish your starting skill map.

FOCUS AREA

No focus area yet

--

Your highest-priority skill will appear here as evidence accumulates.

Retention model awaiting evidence.
ADAPTIVE COMMAND CENTER

Your next moves, ranked from current evidence.

MathPro reconciles mastery, confidence, retention, prerequisites, mistakes, checkpoints, and today’s goals before choosing what deserves attention.

ADAPTIVE PLAN
Recommendations use your latest learning evidence.
PRIORITY STACK

Now / Next / Later

DOMAIN PULSE

Where attention is concentrating

SUPPORTING EVIDENCE

Detailed progress, queues, and goals

The command center above is the primary decision surface. These panels expose the underlying learner record in more detail.

TODAY'S PLAN

Today's plan

adaptive
LEARNING SUMMARY

Current progress

DAILY TARGET

Session goals

0 / 20questions today
0 / 30active minutes today
MISTAKE PATTERN

Most common mistake

No misconception pattern yet. Wrong answers will be classified and accumulated here.
RETENTION

What needs retrieval

CHECKPOINTS

Skills ready to prove

5-question tests
MASTERY MAP

Highest-priority skills

BEGINNER PLACEMENT 2.0

Find the right starting point without over-testing.

MathPro samples the Beginner Foundations skills adaptively. Correct answers move the boundary upward, misses move it downward, and the diagnostic stops once it has enough evidence.

12–16questions
—foundational skills
0hints
CURRICULUM MAP

Your mathematics path, from foundations to advanced work.

MathPro organizes canonical skills into prerequisite-aware domains so you can see what is ready, what is developing, and what should come next.

ROADMAP

Domains and prerequisites

10 domains
RECOMMENDED LESSON

No lesson recommendation yet

next

DOMAIN READINESS

Curriculum readiness map

Current readiness across the MathPro curriculum. These are learning signals, not official exam scores.

CURRICULUM

Lessons and mastery checkpoints

Your path

Start practice

Choose today’s plan, adaptive practice, a specific skill, or a word-problem session.

Word-problem sessions translate real situations into equations before solving.
SPACED RETRIEVAL

Your review queue is current.

RETENTION MODEL

Review signals

PRIORITY QUEUE

What needs retrieval next

Priority model
MISTAKE INTELLIGENCE

Recurring error patterns

INTELLIGENT SESSION BUILDER

Choose review depth

Each plan uses the same evidence model with a different time budget. Review activity changes scheduling only through real question outcomes.

REVIEW SCHEDULE

Skill-by-skill retention

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SKILLS

Mastery, retention, confidence, and prerequisites

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LEARNING MAP 2.0

Prerequisites, bottlenecks, deep progress, and unlock paths

Navigate the prerequisite map. Map signals are read-only views of existing learner evidence.

Prerequisite graph
NUMBER SENSE 2.0

See how numbers are built, related, regrouped, and rounded.

Explore place value, signed magnitude, factors, equivalent ratios, percentages, and powers of ten without leaving the core mastery system.

NUMBER SENSE MASTERY

Current signals

7 skills
PLACE VALUE

Build the number from positional units

SIGNED NUMBER LINE

Magnitude and direction

Drag directly on the number line, or use the value control above.

FACTOR STRUCTURE

GCF, LCM, and primes

EQUIVALENT REPRESENTATIONS

Fraction → decimal → percent

SCIENTIFIC NOTATION

Powers-of-ten scale

NUMBER SENSE 2.0 · BASE TEN

Compose, decompose, and regroup

NUMBER SENSE 2.0 · ROUNDING

Use midpoint benchmarks

ALGEBRA

Transform structure without breaking equivalence.

Build expressions, solve equations and inequalities, compare systems, factor quadratics, and connect standard form with vertex form.

ALGEBRA MASTERY

Current signals

7 linked skills
EXPRESSION BUILDER

Distribute, combine, verify

Model k(x + b) + mx.

EQUATION SOLVER

Legal transformations

Solve ax + b = c.

SYSTEMS EXPLORER

Two equations, one intersection

a₁x + b₁y = c₁ and a₂x + b₂y = c₂.

FACTORING

Monic quadratic structure

Factor x² + bx + c when integer factors exist.

COMPLETING THE SQUARE

Standard form → vertex form

INEQUALITY EXPLORER

Track when the comparison reverses

VISUAL FRACTIONS

See the numerator over the denominator

interactive

Use the sliders to visualize a fraction as equal parts of a whole.

RATIO SCALING

Equivalent ratios visually

interactive

See how a ratio scales while staying equivalent.

Original

Scaled

EQUATION BALANCE

Visualize why solving equations works

guided

Choose values for a two-step equation of the form ax + b = c. MathPro will build a balanced equation and show the solving path.

Left side
=
Right side
FUNCTION GRAPHER

Explore linear and quadratic functions

interactive graph
Move over or tap the graph to inspect coordinates.
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SYNCHRONIZED REPRESENTATION

Fraction · decimal · percent · number line

ALGEBRA TILES

Distribution as signed area and tile counts

GEOMETRY & COORDINATE MATHEMATICS

Connect formulas to pictures and coordinates.

Use the labs for visual understanding, then send the same topics into the mastery engine for practice and spaced review.

GEOMETRY MASTERY

Current signals

5 skills
COORDINATE PLANE

Distance, midpoint, and slope

interactive
Point A
Point B
PYTHAGOREAN LAB

Right triangles

a² + b² = c²
MEASUREMENT LAB

Area, perimeter, and volume

formula explorer
ANGLE RELATIONSHIPS

Complementary, supplementary, and triangles

interactive
TRANSFORMATION EXPLORER

Translate, reflect, and rotate a point

coordinate geometry
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DYNAMIC TRIANGLE

Coordinates → sides → angles → area

Drag triangle vertices directly. The coordinate fields remain available for exact keyboard entry.

CIRCLE EXPLORER

Radius controls the family

STATISTICS · PROBABILITY · DATA

Turn raw numbers into evidence.

Build statistical intuition with real calculations, visual distributions, probability experiments, and adaptive mastery practice.

STATISTICS MASTERY

Current signals

5 skills
DATA LAB

Describe a dataset

up to 500 values

Enter numbers separated by commas, spaces, or new lines. MathPro calculates center, spread, and a histogram from the same data.

DISTRIBUTION
FREQUENCY TABLE
PROBABILITY LAB

Theory versus simulation

Monte Carlo
Z-SCORE EXPLORER

Standardize an observation

normal model

A z-score tells you how many standard deviations an observation is above or below the mean.

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REGRESSION & RESIDUALS

Fit the data, then inspect the misses

SAMPLING DISTRIBUTION

Repeated samples reveal the behavior of sample means

seeded

Define a population, draw many samples of equal size, and compare the observed distribution of sample means with its theoretical center and standard error.

Run repeated samples to compare observed variability with the theoretical standard error.
STATISTICAL INFERENCE & RESAMPLING

Move from descriptive samples to uncertainty-aware inference.

These modules build on the existing Sampling Distribution Explorer, Distribution Engine, and seeded Simulation Engine. They separate a statistic, its sampling variability, an interval procedure, and a hypothesis-test model.

Interpretation boundary: a confidence level describes the long-run behavior of an interval procedure. A p-value measures compatibility with a null model; it is not the probability that the null hypothesis is true.
CENTRAL LIMIT THEOREM

Sampling mean and standard error

SE = σ / √n
Key distinction: σ describes individual observations; σ/√n describes the spread of repeated sample means. The bell curve shown is exact for normal populations and a CLT approximation when its conditions are appropriate.
MEAN CONFIDENCE INTERVAL

Known-σ z interval and margin of error

estimate ± z*SE
ONE-SAMPLE z TEST

Distance from a null model in standard-error units

z · p-value
p-value: under the null model, the probability of a test statistic at least as extreme in the direction(s) defined by the alternative. It is not P(H₀ is true).
PROPORTION INFERENCE

Wilson confidence interval + null-model z test

p̂
Approximation check: the z test relies on a sufficiently large null-model expected-success and expected-failure count; the Wilson interval is used for interval estimation because it behaves better than the simple Wald interval near boundaries.
POWER & SAMPLE SIZE

Detectable effects under a specified alternative

1 − β
Tradeoff: for fixed α, effect size, and population variability, larger n generally raises power by reducing standard error.
SEEDED BOOTSTRAP

Resample the observed data with replacement

percentile interval
CONFIDENCE-INTERVAL COVERAGE

Repeat the procedure, not the interpretation of one interval

long-run coverage
Long-run meaning: before sampling, a 95% procedure is designed to cover the fixed population parameter in about 95% of repeated samples under the model assumptions.
ADVANCED STATISTICAL INFERENCE

Extend inference beyond known-σ one-sample models.

This wave adds Student t methods, independent and paired comparisons, two-proportion inference, chi-square, ANOVA, and regression slope inference while preserving the existing Statistics Lab and shared engines.

Model discipline: every method has assumptions. The interface reports the mathematics, but learners must still check independence, sampling design, distributional shape, expected counts, and residual conditions where applicable.
UNKNOWN σ

One-sample Student t inference

df = n−1
Why t? replacing unknown σ with sample s adds uncertainty, producing heavier tails and a critical value that depends on degrees of freedom.
INDEPENDENT MEANS

Welch two-sample t inference

unequal variances allowed
PAIRED DESIGNS

Analyze within-pair differences

one-sample t on Δ
Design matters: paired inference is about the distribution of within-pair differences, not two independent groups.
TWO PROPORTIONS

Difference in population proportions

p̂₁ − p̂₂
Two standard errors: the interval uses unpooled sample proportions; the null test for equal proportions uses a pooled estimate.
CATEGORICAL COUNTS

Chi-square goodness of fit

Σ(O−E)²/E
Approximation condition: this implementation requires every expected count to be at least 5.
MULTIPLE MEANS

One-way ANOVA

F = MSbetween / MSwithin
Interpretation: a small ANOVA p-value indicates that the equal-means model is inconsistent with the observed between-group variation relative to within-group variation; it does not identify which group means differ.
REGRESSION INFERENCE

Slope uncertainty and t test

β₁
Inference conditions: slope inference relies on an appropriate linear model, independent observations, stable residual variance, and residual behavior suitable for t-based inference.
PROBABILITY · COMBINATORICS

Count possible worlds, then measure uncertainty.

Build from counting rules into conditional probability, Bayes’ theorem, expected value, and binomial models.

PROBABILITY MASTERY

Current signals

7 skills
COUNTING EXPLORER

Permutations versus combinations

nPr · nCr

Change n and r to compare ordered selections with unordered groups.

CONDITIONAL PROBABILITY

Restrict the sample space

P(A|B)

Use counts to compare conditional probabilities in both directions and check independence.

EXPECTED VALUE

Weight outcomes by probability

E[X]
BINOMIAL DISTRIBUTION

Exactly k successes in n trials

independent trials
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SEEDED BERNOULLI QUICK CHECK

Theoretical probability vs long-run frequency

Run a reproducible Bernoulli experiment through the same reproducible simulation model.
MONTE CARLO · COIN & DICE

Watch experimental probability converge toward theory

reproducible
Choose an experiment and compare a seeded simulation with its exact theoretical probability.
MONTE CARLO π

Approximate π through random area sampling

area ratio
The circle occupies π/4 of its surrounding square. Random points estimate that area ratio.
DISTRIBUTION EXPLORER

Link parameters, theory, and sampled outcomes

shared engine
Compare theoretical mean and spread with a reproducible simulated sample.
RANDOM WALK

From local step probability to a distribution of paths

stochastic paths
For p = 0.5 the theoretical terminal mean remains at the starting value; changing p creates drift.
FINANCIAL MATHEMATICS

Model the value of money across time, cash flows, and portfolios.

Connect time value of money to compound growth, annuities, loans, bonds, investment returns, and weighted portfolio returns.

FINANCE MASTERY

Current signals

7 skills
TIME VALUE OF MONEY

Present value → future value

PV · FV

Change the compounding assumptions and see how time and frequency change a lump-sum investment.

ANNUITY LAB

Repeated contributions

ordinary annuity

Model equal end-of-period deposits and separate contributions from investment growth.

LOAN AMORTIZATION

Payment and interest cost

PMT
BOND PRICING

Discount coupons and face value

price · yield
RETURN LAB

Holding-period return and CAGR

HPR · CAGR
PORTFOLIO MATH

Weighted expected return

Σ wᵢrᵢ

Two-asset portfolio weights are normalized automatically so the relationship stays mathematically valid.

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GROWTH COMPARISON

Simple · compound · continuous

RECURRING CONTRIBUTIONS

Contribution timing matters

INFLATION

Nominal vs real purchasing power

FEE DRAG

Gross vs net compounding

AMORTIZATION DETAIL

Payment schedule

CASH-FLOW MODEL

Move every cash flow to a common point in time.

These modules share one validated financial core. A cash-flow timeline, annuity, loan, bond, or project valuation is a different curriculum experience built on the same time-value mathematics.

Model: cash flows are valued using the configured rate and timing convention. Interpretation: the result is a mathematical equivalence under those assumptions. Limitation: the calculation does not determine whether a real financial decision is appropriate.
CASH-FLOW TIMELINE

Choose a focal date and move every flow there

PV ↔ FV
ANNUITY TIMING

Ordinary, due, and growing cash flows

PV · FV
Invariant: beginning-of-period payments have one additional period to compound. Growing annuities also change the payment stream itself, so payment growth and investment growth are distinct assumptions.
AMORTIZATION + EXTRA PAYMENT

Measure payoff acceleration, not just payment size

shared loan core

Uses the principal, APR, term, and frequency from the Loan Amortization panel above.

VARIABLE-RETURN GROWTH

Arithmetic average is not compound growth

path dependent
Concept: arithmetic mean describes the average single-period return. Geometric mean / CAGR describes the constant compound rate that reproduces the same beginning-to-ending growth path.
CAPITAL BUDGETING

NPV, IRR, payback, and discounted payback

generic project cash flows
Calculation vs judgment: NPV and IRR summarize cash flows under a stated discount-rate model. They do not capture every operational, strategic, financing, or uncertainty consideration in a real project.
PORTFOLIO RISK & FINANCIAL MONTE CARLO

Connect return, volatility, correlation, diversification, and uncertainty.

These experiences use reproducible seeded simulation and portfolio-risk mathematics. Results are mathematical models under stated assumptions, not investment recommendations.

Model boundary: expected return, volatility, VaR, and Monte Carlo outcomes summarize assumptions or observed samples. They do not forecast a security, guarantee outcomes, or decide whether an investment is appropriate.
DIVERSIFICATION EXPLORER

Two assets: expected return, covariance, and portfolio volatility

σₚ² = wᵀΣw
Invariant: expected return is linear in normalized weights. Portfolio variance is not linear because covariance terms connect the assets. Lower correlation can reduce volatility without changing either asset's standalone volatility.
HISTORICAL TAIL RISK

VaR and Expected Shortfall from a return sample

sample model
Interpretation: historical VaR is a sample quantile, while Expected Shortfall averages the observations in the selected loss tail. Both inherit the limitations of the supplied sample.
SEQUENCE-OF-RETURNS RISK

Same returns, different order, different cash-flow outcome

path dependence
Concept: with no intermediate cash flows, reversing multiplicative returns leaves the final product unchanged. Contributions or withdrawals make the path matter because different dollar balances experience each return.
MONTE CARLO WEALTH SIMULATION

Seeded paths, percentile fan, and goal probability

deterministic replay
Simulation assumption: annual growth factors are sampled from a lognormal model calibrated to the entered arithmetic expected return and volatility. Reusing the same seed reproduces the same pseudo-random experiment exactly.
PRECALCULUS · TRIGONOMETRY

Understand functions before calculus asks you to change them.

Train transformations, polynomial behavior, exponential and logarithmic functions, right-triangle trigonometry, the unit circle, and sinusoidal graphs.

PRECALCULUS MASTERY

Current signals

6 skills
FUNCTION TRANSFORMATIONS

Move, stretch, and reflect a parent function

y = a·f(x − h) + k

Drag the highlighted anchor to change h and k. The exact a, h, and k controls remain available for keyboard entry.

PROPERTIES
TABLE
WHAT CHANGED?
UNIT CIRCLE

Connect angle, coordinates, sine, and cosine

interactive
EXPONENTIAL & LOGARITHMIC

Functions and their inverses

bˣ ↔ logᵦ(x)
POLYNOMIAL BEHAVIOR

Degree, leading coefficient, and end behavior

model explorer
TRIG GRAPH LAB

Amplitude, period, and vertical shift

y = A sin(Bx) + D
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FUNCTION MACHINE

Rule → input → table → output

TRIGONOMETRY

Connect triangle ratios, circle coordinates, and periodic functions.

Move between SOH-CAH-TOA, exact standard-angle values, the unit circle, and the full transformed sinusoid.

TRIG MASTERY

Current signals

3 skills
RIGHT TRIANGLE

SOH · CAH · TOA

UNIT CIRCLE & EXACT VALUES

Coordinates and signs

Drag around the unit circle to change θ, or use the angle control above.

SINUSOID TRANSFORMATIONS

Amplitude, period, phase shift, and midline

CALCULUS FOUNDATIONS

Study change and accumulation as visual ideas before memorizing rules.

Build intuition for limits, average and instantaneous rate of change, derivatives, tangent lines, and accumulated area.

CALCULUS MASTERY

Current signals

6 skills
LIMIT EXPLORER

Approach a value from both sides

x → a
SECANT → TANGENT

Watch average rate become instantaneous rate

Δx → 0
DERIVATIVE RULES

Power rule with a live slope

d/dx
ACCUMULATION LAB

Approximate area with Riemann rectangles

Σ → ∫
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CONVERGENCE TABLE

Watch approximation error shrink

ACCUMULATION / FTC

Accumulated amount and current rate

LINEAR ALGEBRA + VECTORS

See equations, vectors, and matrices as transformations of space.

Build intuition for vector magnitude and operations, dot products, matrices, systems of equations, determinants, and linear transformations.

LINEAR ALGEBRA MASTERY

Current signals

7 skills
VECTOR LAB

Add vectors, measure length, and compare direction

u · v
MATRIX LAB

Operate on 2×2 matrices

A · B
A
B
SYSTEMS SOLVER

Connect two equations to one intersection

Ax = b
x +y =
x +y =
LINEAR TRANSFORMATION

Watch a matrix reshape the plane

T(x) = Ax
Transformation matrix
Vector p
LINEAR ALGEBRA

Connect vector geometry, matrix structure, transformations, eigendirections, and approximation.

This release deepens the Linear Algebra view already present in MathPro. It reuses the seven existing canonical Linear Algebra skills and adds a shared mathematical engine rather than creating a second lab architecture.

Core idea: vectors describe quantities in coordinate form; matrices act on those vectors; row reduction solves constraints; determinants describe area scaling and invertibility; eigenvectors expose invariant directions; least squares handles systems that cannot be solved exactly.
DOT PRODUCT + PROJECTION

Decompose one vector relative to another

projᵥ(u)
Orthogonal decomposition: u = projᵥ(u) + residual, and the residual is perpendicular to v.
DETERMINANT + INVERSE

When does a matrix undo itself?

A⁻¹
Invertibility: det(A)=0 means the transformation collapses dimension, so no inverse can restore every input uniquely.
GAUSSIAN ELIMINATION

Row-reduce an augmented system to RREF

Ax=b
Classification: RREF can expose one unique solution, free variables with infinitely many solutions, or a contradictory row that makes the system inconsistent.
SPAN + BASIS

Do two vectors create the whole plane?

linear independence
2D shortcut: two vectors form a basis for R² exactly when their determinant is nonzero.
EIGENDIRECTIONS

Find directions a matrix only stretches or flips

Av = λv
Meaning: an eigenvector keeps its line of direction under the transformation. The eigenvalue tells how strongly that direction is scaled.
LEAST SQUARES

Find the best linear approximation when Ax=b has no exact solution

AᵀAx=Aᵀb
Bridge to statistics: ordinary least-squares regression is also a linear-algebra problem. The best-fit coefficients solve the normal equations when the design matrix has full column rank.
MATHEMATICAL MODELING LAB

Turn assumptions into models you can manipulate.

Connect optimization, discrete change, complex geometry, and dynamical systems through synchronized equations, graphs, numerical experiments, and model comparisons.

PredictManipulateObserveExplainChallengeTransfer
MODELING WORKFLOW

One experiment loop across four mathematical systems

4 modules

Use the Lab shell for guided missions, snapshots, Tutor/Notebook handoff, challenge verification, and Practice transfer. The canvases below remain model controls—not mastery evidence by themselves.

OPTIMIZATION

Linear Programming Studio

max cᵀx

Build a feasible polygon from two half-plane constraints and inspect the objective at every feasible vertex. Drag either constraint line to change its bound, or drag the dotted objective vector to change its coefficients.

DISCRETE CHANGE

Sequences & Partial Sums

aₙ · Sₙ
SERIES APPROXIMATION

Taylor / Maclaurin Error Lab

Pₙ(x)

Increase polynomial degree and compare the local approximation with the exact transcendental function.

COMPLEX GEOMETRY

Complex Plane Studio

z = a + bi

Drag z directly on the plane; exact real/imaginary controls remain available.

DYNAMICAL SYSTEMS

ODE Solver Comparison

y′ = f(x,y)

This module reuses the released Advanced Calculus ODE engine and compares Euler, midpoint, and RK4 over the same slope field.

EulerMidpointRK4
MULTIVARIABLE + DIFFERENTIAL EQUATIONS

Extend calculus from one variable into fields, gradients, and evolving systems.

Build intuition for multivariable functions, partial derivatives, gradients, first-order differential equations, slope fields, growth/decay models, and Euler's method.

ADVANCED CALCULUS MASTERY

Current signals

7 skills
MULTIVARIABLE FIELD

Explore f(x,y), partial derivatives, and the gradient

∇f
SLOPE FIELD

Read a differential equation as local direction

dy/dx
GROWTH & DECAY MODEL

Study y′ = ky as continuous change

y = y₀eᵏᵗ
EULER METHOD

Approximate a solution one tangent step at a time

yₙ₊₁ = yₙ + hf
MULTIVARIABLE CALCULUS, OPTIMIZATION & ODEs

Deepen the Advanced Calculus view with gradients, Hessians, optimization, double integrals, and higher-accuracy ODE solvers.

Explore the seven canonical Advanced Calculus skills in one lab workspace, with linear-algebra tools reused where the mathematics requires them.

Concept bridge: gradients are vectors, Hessians are matrices, optimization uses both, and numerical ODE methods approximate continuous change by controlled finite steps.
GRADIENT + DIRECTIONAL DERIVATIVE

How fast does f change in a chosen direction?

Dᵤf = ∇f·u
Tangent plane: the two partial derivatives are the local x- and y-slopes, so the gradient determines the first-order plane approximation.
HESSIAN + OPTIMIZATION

Classify a critical point and follow gradient descent

Hf
Second derivative test: the Hessian describes local curvature. For a nondegenerate 2D critical point, its determinant and fₓₓ distinguish a local minimum, maximum, or saddle.
DOUBLE INTEGRALS

Accumulate a scalar field across a rectangle

∬R f dA
Iterated accumulation: on rectangular regions, polynomial terms can be integrated exactly by integrating powers independently in x and y.
LOGISTIC DIFFERENTIAL EQUATION

Growth slows as the state approaches carrying capacity

y′ = ky(1−y/K)
Autonomous model: the slope depends on the current state y. Equilibria occur where y′=0, including y=0 and y=K.
NUMERICAL ODE METHODS

Compare Euler, midpoint, and RK4 on y′=y

local approximation → global error
Accuracy hierarchy: for a smooth problem and the same moderate step size, midpoint generally improves on Euler, while classical RK4 is dramatically more accurate. Reducing h provides a convergence experiment rather than a guarantee for every ODE.
BEGINNER FOUNDATIONS

Operations Lab

Build addition, subtraction, multiplication, and division with concrete models before compressing them into symbols.

4 linked skills
ADD / SUBTRACT

Place-value operation model

MULTIPLICATION ARRAY

Equal rows × equal columns

DIVISION SHARING

Share a total into equal groups

BEGINNER FOUNDATIONS

Fractions, Decimals & Ratios Lab

Keep fraction bars, decimals, percents, hundred-grids, and ratios synchronized.

4 linked skills
SYNCHRONIZED REPRESENTATIONS

Fraction → decimal → percent

RATIO TABLE

Scale both quantities together

BEGINNER FOUNDATIONS

Everyday Math Lab

Practice money, time, and measurement with manipulatives tied to real quantities and units.

3 linked skills
MONEY

Make change

TIME

Elapsed time

MEASUREMENT

Convert length units

BEGINNER FOUNDATIONS

Pre-Algebra Foundations Lab

Bridge patterns and unknowns into expressions and equations, with a balance-scale model and simple geometry tiles.

4 linked skills
PATTERNS

Growing sequence builder

VARIABLES

Expression substitution

BALANCE SCALE

Solve ax + b = c

GEOMETRY TILES

Perimeter vs area

NUMERICAL METHODS & SCIENTIFIC COMPUTING

Approximate difficult mathematical problems while measuring error, stability, and convergence.

This lab connects Algebra, Calculus, Linear Algebra, Probability, and Scientific Computing. Every algorithm exposes its assumptions and an error signal rather than treating a numerical answer as automatically exact.

Core discipline: a numerical method is not just an answer generator. It is an algorithm with a stopping rule, an error model, convergence behavior, and sometimes stability or conditioning limits.
ROOT FINDING

Bisection · Newton · Secant

f(x)=0
Tradeoff: bisection is slow but robust with a valid sign-changing bracket; Newton can be very fast but depends on derivative behavior and the starting point; secant avoids an explicit derivative but can still fail.
NUMERICAL DIFFERENTIATION

Forward vs central differences

h → 0
Observed order: central difference should show approximately second-order truncation error before floating-point roundoff dominates at extremely small h.
NUMERICAL INTEGRATION

Trapezoid vs Simpson

quadrature
Exact reference: supported presets have analytic antiderivatives, so the lab can measure quadrature error rather than only displaying an approximation.
INTERPOLATION

Lagrange / barycentric polynomial interpolation

fit through known points
Interpolation vs regression: interpolation passes through every supplied point; least squares instead minimizes residual error when exact interpolation is not the goal.
ITERATIVE LINEAR SOLVER

Jacobi iteration + conditioning

Ax=b
Conditioning: κ₂(A) measures sensitivity of the linear system to relative perturbations. Convergence of Jacobi is a separate algorithmic question and is not guaranteed for every invertible matrix.
MONTE CARLO INTEGRATION

Seeded stochastic quadrature

random sampling
Stochastic error: Monte Carlo standard error typically shrinks like 1/√n. Reusing a seed reproduces the pseudo-random experiment exactly; it does not remove sampling uncertainty.
MACHINE LEARNING MATHEMATICS FOUNDATIONS

See machine-learning models as compositions of linear algebra, calculus, probability, and statistics.

This lab teaches the mathematics underneath common model families rather than turning MathPro into a model-training service. Every probability, loss, gradient, and projection is exposed as a mathematical object.

Core structure: feature vectors enter linear combinations; losses measure mismatch; gradients describe how parameters change the loss; optimization updates the parameters; probability transforms support classification; eigendirections support dimensionality reduction.
FEATURE VECTORS

Cosine similarity + standardization

x · y / ||x||||y||
Similarity is geometry: cosine similarity compares direction rather than magnitude. Standardization instead changes feature scale by centering and measuring values in sample-standard-deviation units.
LINEAR MODEL

MSE loss + gradient descent

ŷ = wx+b
Optimization: the MSE gradient points uphill in parameter space, so gradient descent subtracts a learning-rate-scaled gradient. The learning rate controls step size, not model complexity.
LOGISTIC CLASSIFICATION

Sigmoid probability + binary cross-entropy

σ(wx+b)
Probability vs decision: logistic regression produces a probability under its model. A classification threshold is an additional decision rule, not part of the sigmoid itself.
MULTICLASS PROBABILITY

Softmax + categorical cross-entropy

exp(zᵢ)/Σexp(zⱼ)
Numerical stability: subtracting the largest logit before exponentiation leaves softmax probabilities unchanged while avoiding overflow.
REGULARIZATION

L1 / L2 parameter penalties

loss + λΩ(w)
What regularization changes: the penalty changes the optimization objective. L2 grows quadratically with weight magnitude; L1 grows linearly and is nondifferentiable at zero, where a subgradient convention is required.
PCA

Covariance eigendirections + explained variance

principal components
PCA is variance geometry: after centering the data, the covariance matrix’s leading eigenvector points along the direction of greatest sample variance. PCA is not a supervised predictor.
PERSONAL NOTEBOOK

Capture what you want to remember

new note
NOTEBOOK SIGNALS

Your knowledge base

local knowledge index
SAVED NOTES

Personal reference

0 notes
KNOWLEDGE BACKLINKS

Where this skill appears in your record

actual references only
WORKED-EXAMPLE COMPARISON

Your recorded work vs reference

MISTAKE REPLAY

Replay the latest recorded divergence

MATH REFERENCE ATLAS

Topics, concepts & formula pathways

reference · no synthetic mastery

Explore mathematics beyond a single lesson. Reference topics connect into the existing formula library and only expose a linked skill when MathPro already has validated practice and evidence for it.

PROMOTED CURRICULUM

Assessable learning units

lessons · checkpoints · evidence

Selected reference topics are promoted into structured units with prerequisites, objectives, worked examples, deterministic checkpoints, and recorded evidence.

FORMULA LIBRARY

Core formulas worth knowing

save any formula
L
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PRIVATE LEARNER PROFILE

Learner

Core Mathematics

Stored locally in this browser
LEARNING SNAPSHOT

Your MathPro record

private
IDENTITY & GOAL

Personalize the tutor

STUDY DEFAULTS

Set your normal workflow

Profile changes stay on this device unless you export your MathPro progress.
KeyboardTab moves through controls. Enter submits answers. Escape closes Calculator or confirmation dialogs. Alt+C opens Calculator.
CURRICULUM PATH

Your progress by domain

Curriculum skills
PRIVATE TUTOR

Ask, explain, diagnose

local engine
This standalone tutor is rule-driven and uses your local MathPro data. It does not send your messages to an external model.
WORKED-EXAMPLE TRAINER

Study a complete solution, one step at a time

MISCONCEPTION REMEDIATION

Patterns worth fixing

PERT-STYLE DRILL

12-question mixed drill

Mixed arithmetic, fractions, proportional reasoning, and algebra. No hints. Designed for fast test-prep calibration.

EXAM-STYLE SET

20-question test

A no-hint assessment generated from your current curriculum evidence with a deliberate medium-to-hard difficulty mix.

FOUNDATION SIMULATION

30-question exam

A broader no-hint simulation using the current MathPro foundation graph. It is not an official PERT exam or score predictor.

EXTENDED FOUNDATION TEST

40-question test

A longer no-hint test balanced across core foundation and college-readiness domains, with higher-need skills receiving priority within each domain.

FULL-LENGTH FOUNDATION EXAM

50-question exam

MathPro's longest foundation assessment: 50 no-hint questions spanning beginner math through statistics and probability with balanced domain coverage.

LESSON QUIZ

6-question focused quiz

Tests the currently selected lesson skill when available, otherwise MathPro targets your highest-need skill. No hints.

UNIT CHECKPOINT

10-question focused checkpoint

Checks the selected or highest-need skill with its same-domain prerequisite neighborhood. No hints.

CUMULATIVE REVIEW

12-question retrieval set

Due material is prioritized, then MathPro interleaves high-need skills across the curriculum. No hints.

TIMED PRACTICE

15-question timed set

A 20-minute no-hint set with harder independent questions selected from your current curriculum evidence.

EXAM HISTORY

Recent simulations

Learning evidence loads when you open Analytics.

CURRENT LEARNER STATE

Authoritative evidence model

Current evidence
Recorded evidence window30 days
PROGRESS SNAPSHOT

What the platform actually recorded

PERIOD COMPARISON

Current window vs preceding window

EVIDENCE SOURCES

Where recorded work came from

EVIDENCE TREND

Recorded attempt volume over time

DOMAIN EVIDENCE

Current state plus selected-window performance

SKILL DRILLDOWN

Timestamped evidence without reconstructed mastery history

EVIDENCE COVERAGE

What is represented in this view

SESSION OUTCOMES

Completed logs matching these filters

ERROR PROFILE

Most common misconception signals

DAILY GOALS

Configure training targets

ATTEMPT LOG

Recent work and reasoning signals

0 attempts