Subject entry4 topics16 concepts2 starter tracks1 bridge376 min

Math

Enter the current math slice through graph transformations, rational-function asymptotes, exponential change, vectors, complex-plane geometry, trig identities, inverse-angle reasoning, polar coordinates, and parametric motion without leaving the same live-bench product language used elsewhere on the site.

Math is still intentionally compact, but it now has more than one durable branch. Start here when you want the graph-first launch path through transformations, rational asymptote behavior, and exponential change, then widen into plane-based complex numbers, unit-circle and polar geometry, trig identities, inverse-angle recovery from ratios, parametric motion from equations, or the vectors bridge back into mechanics.

Start Functions and Change Open Functions New here? Start here Search Math Browse Math concepts

Starter tracks

Start with a bounded path before branching wider.

Open concept library

Starter track6 concepts139 min5 checkpoints

函數與變化

Track not started

Keep the first math path compact: read parent-curve moves first, then rational asymptotes and domain breaks, then exponential growth and decay, local slope, visible limit behavior, and finally accumulation so change stays graph-first all the way through.

Parent-curve movesShifts and reflectionsRational asymptotes and holesExponential growth and decayTarget time from logsTangent slopeOne-sided limits and continuityLocal rate of changeSigned area and accumulation

Track progress

0 of 11 moments complete

0 of 6 concepts complete and 0 of 5 checkpoints cleared.

1圖像變換

Start here

2有理函數／漸近線與行為

Ahead

3指數變化／增長、衰減與對數

Ahead

4Derivative as Slope / Local Rate of Change

Ahead

5極限與連續／逼近某個值

Ahead

6積分作為累積／面積

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Graph Transformations opens this track and sets up the rest of the path.

Start track Jump to Graph transforms

Starter track6 concepts140 min4 checkpoints

複數與參數運動

Track not started

Start with complex numbers as points on one plane, turn that plane into unit-circle and polar-coordinate geometry, deepen that same bench into trig identities and inverse-angle reasoning, then carry the coordinate language into motion traced from x(t) and y(t).

Complex points on a planeUnit-circle and polar geometryTrig identities from one pointQuadrant-aware inverse trigPath vs traversal

Track progress

0 of 10 moments complete

0 of 6 concepts complete and 0 of 4 checkpoints cleared.

1複平面上的複數

Start here

2單位圓／由旋轉理解正弦與餘弦

Ahead

3極座標／半徑與角度

Ahead

4由單位圓幾何理解三角恆等式

Ahead

5反三角函數／由比值求角

Ahead

6參數曲線／由方程描述運動

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Complex Numbers on the Plane opens this track and sets up the rest of the path.

Start track Jump to Complex numbers

Cross-subject bridge

Keep the bridge visible when it genuinely connects subjects.

Starter track2 concepts50 min

向量與運動橋接

Start with vectors as geometric objects on a 2D plane, then carry the same component language into the existing motion-facing vectors bench.

Starts with 二維向量 across 2 concepts.

Start track

Topics

Use topics when you want the subject shaped into a clearer map.

TopicMath3 concepts65 min1 starter tracks

函數

Use parent-curve moves, a shifted reciprocal family, and one exponential bench so graph moves, asymptotes, domain breaks, growth versus decay, and target-time questions stay tied to the same visual branch before the math path widens into local and accumulated change.

Parent-function movesRational functions and asymptotesExponential growth and decay

Best first concepts

圖像變換有理函數指數變化

Open topic Start 圖像變換

TopicMath4 concepts98 min1 starter tracks

微積分

Start from slope on the graph itself, use one constrained rectangle bench to make a real maximum visible, keep limit and continuity behavior available on one target point, and then widen into signed area and accumulation so rate and total change stay connected on one visual branch.

Tangent and local changeLimits and visible continuityConstrained maxima and minima

Best first concepts

Derivative as slope Limits and continuity Optimization and constraints

Open topic Start Derivative as slope

TopicMath6 concepts140 min1 starter tracks

複數與參數運動

Use one bounded math branch where the complex plane, unit-circle rotation, polar coordinates, trig identities, inverse-angle reasoning, and motion traced from equations all stay tied to the same coordinate language.

Complex plane geometryRotation and projectionsRadius-angle coordinates

Best first concepts

複數 Unit circle rotation Polar coordinates

Open topic Start 複數

TopicMath3 concepts73 min1 starter tracks

向量

Use one 2D plane to read vectors as arrows, ordered pairs, matrix actions, alignment measures, and projections before the same language bridges into motion.

Combination and scalingLinear actions on the planeAlignment and projection

Best first concepts

2D vectors Matrix transforms Dot product

Open topic Start 2D vectors

Best first concepts

Start with one strong concept when you do not need the full path yet.

數學函數

圖像變換

用誠實的控制項移動同一條母函數曲線，讓平移、垂直伸縮與反射都綁在同一張疊圖與標誌點上。

Open 圖像變換

數學函數

有理函數／漸近線與行為

改變同一個平移後的倒數函數族，讓定義域缺口、垂直與水平漸近線、截距與可去間斷都綁在同一張圖上。

Open 有理函數

數學函數

指數變化／增長、衰減與對數

改變同一條曲線的初值、變化率與目標值，讓增長、衰減、倍增／半衰期與對數求時都留在同一個即時圖上。

Open 指數變化

數學複數與參數運動

複平面上的複數

把複數視為平面上的點與向量，再讓加法與乘法保持幾何直覺，而不只停留在符號操作。

Open 複數

數學複數與參數運動

單位圓／由旋轉理解正弦與餘弦

把旋轉點、x/y 投影與正弦餘弦曲線連在一起，讓單位圓成為這兩個函數的即時來源。

Open Unit circle rotation

數學複數與參數運動

極座標／半徑與角度

讓同一個點同時出現在極座標與直角座標中，使半徑與角度如何變成 x、y 直接可見。

Open Polar coordinates

數學複數與參數運動

由單位圓幾何理解三角恆等式

讓旋轉點與其投影保持可見，使核心三角恆等式仍然連著幾何，而不是脫離圖像的規則。

Open Trig identities

數學複數與參數運動

反三角函數／由比值求角

讓一個極座標點與其座標正負保持可見，使反三角函數變成帶象限檢查的由比值求角，而不只是計算器輸出。

Open Inverse trig

數學複數與參數運動

參數曲線／由方程描述運動

把 x(t)、y(t)、描出的路徑與移動點同時顯示，清楚分開形狀本身與沿著曲線前進的方式。

Open Parametric curves

數學微積分

Derivative as Slope / Local Rate of Change

Slide a point along one curve, tighten a secant into a tangent, and connect local steepness to the derivative graph without leaving the same live bench.

Open Derivative as slope

數學向量

二維向量

在同一平面上組合、相減與縮放向量，讓大小、方向與分量都綁在同一個即時物件上。

Open 2D vectors

數學向量

矩陣變換／伸縮、剪切、反射

讓同一個 2×2 矩陣同時作用在網格、基底向量與範例圖形上，使平面變換保持可視化而不只是符號。

Open Matrix transforms

數學向量

內積／夾角與投影

把兩個向量、它們的夾角、其中一個投影到另一個的有號分量，以及內積一起顯示，讓對齊程度從幾何上變得清楚。

Open Dot product