Live canvas — every figure is drawn, not drawn once
Mathematics you can drag, plot and watch behave.
MorphMath Academy turns limits, derivatives and polygons into instruments. Pull a vertex, push a slider, and the measurements recompute in front of you — then follow the same argument line by line until it clicks.
18
interactive labs
150+
worked solutions
3
topic tracks
Morph loop — move your pointer over the canvas to probe r(θ), perimeter and area.
01 / Topic library
Three tracks, one habit of mind
Every topic is taught the same way: build the object, measure it, then write the argument that explains the measurement.
Track 02 — flagship
Calculus: limits
Epsilon and delta without the hand-waving. Shrink h, watch the secant slope converge, then squeeze the error below any tolerance you choose.
Track 01
Algebra and functions
Factor, transform and compare functions. Meet slope as a rate of change in the function lab before it ever becomes a formula.
Track 03
Geometry and measurement
Drag a vertex and watch angles, perimeter and area recompute live. Convexity becomes a number you can read, not a shape you guess at.
Every track
Worked solutions
150+ problems solved one line at a time. Amber hints arrive before the answer, never after the deadline.
02 / Interactive labs
Three tracks, one habit of mind
No downloads, no app. Each lab is a small program that draws itself in your browser and answers back the moment you change something.
Lab A — morph loop
Morphing geometry
A closed curve interpolates between a circle, a square, a triangle, a star and a blob while a probe ray reports r(theta), perimeter and area.
Lab B — five sliders
Function lab
Move a, b, c, x-zero and h. The curve, the secant through x-zero plus or minus h and the tangent at x-zero all redraw as you drag.
Lab C — drag the vertices
Polygon lab
Every vertex is draggable. Side lengths, interior angles, the angle sum, area and convexity update on every frame while you pull.
03 / Step-by-step solutions
Limits, line by line
A worked solution should read like an argument, not a magic trick. Click once and the next line appears — with the reason attached to it.
Step 01
Read the structure. Substituting x = 2 gives 0/0, an indeterminate form. That tells you the quotient is undefined at the point — it says nothing yet about what happens nearby.
Step 02
Factor the numerator. It is a difference of squares, so it splits into two linear factors.
x² − 4 = (x − 2)(x + 2)
Step 03
Cancel the common factor. The cancellation is legal everywhere except the single point you excluded.
(x² − 4) / (x − 2) = x + 2, x ≠ 2
Step 04
Take the limit of the simplified expression. Polynomials are continuous, so substitution — which failed a moment ago — now works.
lim (x → 2) of (x + 2) = 4
Step 05
State the conclusion. The limit is 4 even though f(2) is undefined: a limit is a claim about nearby values, never about the value at the point itself.
lim (x → 2) of (x² − 4) / (x − 2) = 4
Stuck at step 2? Ask for the amber hint: it names the algebraic identity to use without handing over the answer.
How the reveal works
- Each step is a claim plus its justification, never a bare line of algebra.
- Hints are amber and name the next move, not the result.
- Nothing collapses behind you — the whole argument stays on screen to read back.
- Reveal state is local to your browser. Refresh and the walkthrough starts clean.
04 / Practice
Answer, pulse, understand
Wrong answers flash pink and stay open so you can try again. Right answers pulse green and open the worked solution. Nothing is scored until you ask for a score.
Which statement about lim (x → 2) of (x² − 4) / (x − 2) is exactly right?
The discontinuity at x = 2 is removable. Because (x² − 4) / (x − 2) = x + 2 for every x other than 2, the two expressions agree on a punctured neighbourhood of 2 — and a limit only ever looks at that neighbourhood. The limit is therefore the value of the simplified expression at x = 2, namely 4.
05 / Classrooms and kitchen tables
Teaching limits this term?
Put a lab on the projector and improvise with sliders instead of hunting for the right slide. Lesson plans, answer keys and a scope-and-sequence map are ready to print — for teachers and for parents running a study table at home.
Slide-free lesson plans
Forty-five minute sequences that assume only a browser and a whiteboard marker.
Answer keys and rubrics
Every practice set ships with a marking guide that rewards justification, not just the final number.
Parent study guide
A one-page explanation of what a limit is, so homework help does not turn into a second lecture.