THE INTERACTIVE LAB · 44 TOPICSMake the mathematics visible.
Predict what will happen. Change a value, test a boundary and explain the evidence. Every investigation is inside its corresponding Notes lesson.
1 · Predict→2 · Explore→3 · Explain
44 investigations
- You control
- filling pattern · elapsed time
- You observe
- Exact volume, equal-time gains and synchronized tank/graph
- You explain
- Name both the direction of change and how the gains change.
Test the boundary
- increasing with decreasing gains is not decreasing
- patterns are specified formulas, not inferred from a few dots
- You control
- first interval start · interval width
- You observe
- Two secants on the same quadratic and exact change quotients
- You explain
- State the rate, interval and output units per input unit.
Test the boundary
- positive net change does not establish increasing throughout
- no zero-width secant
- You control
- model family · input spacing · quadratic coefficient
- You observe
- Input/output, first-difference, second-difference and AROC tables
- You explain
- Link constant differences to equal input increments and the stated model family.
Test the boundary
- a=0 is linear
- unequal steps invalidate the usual second-difference shortcut
- finite tables do not prove a global model
- You control
- cubic shape · leading sign · secant center
- You observe
- Exact turning/flat points, monotonic intervals and short secants
- You explain
- Justify an extremum with a change in direction, not with a flat tangent alone.
Test the boundary
- x cubed has no local extremum at zero
- short secant is not exact instantaneous rate
- local and global extrema differ
- You control
- real-zero multiplicity · conjugate-pair factor · symmetry test
- You observe
- Graph at a repeated zero, factor degree and complex-root inventory
- You explain
- Explain crossing/touching and count zeros with multiplicity.
Test the boundary
- nonreal zeros do not appear as x-intercepts
- conjugate pairing assumes real coefficients
- odd/even degree is not odd/even function symmetry
- You control
- degree · leading coefficient · tail probe
- You observe
- Complete polynomial, leading term and exact output ratio
- You explain
- State two directional limit statements and explain lower-order terms.
Test the boundary
- tail probe is finite evidence, not infinity
- nonzero leading coefficient is required
- You control
- degree case · leading coefficient · tail probe
- You observe
- Separate rational branches, horizontal/slant comparison and remainder table
- You explain
- Compare the whole function with its quotient and show the remainder tends to zero.
Test the boundary
- horizontal asymptote may be crossed
- slant is not a horizontal asymptote
- domain x not equal to -1
- You control
- numerator root · excluded denominator root · sign-test input
- You observe
- Original domain, allowed zero or hole, and interval signs
- You explain
- Check numerator equals zero AND denominator is nonzero.
Test the boundary
- coincident roots give a hole at height one and no zero
- undefined is neither positive nor negative
- You control
- pole location · remaining order · coefficient sign
- You observe
- Split branches, exact pole sides and near-pole table
- You explain
- Write separate left-hand and right-hand limit statements.
Test the boundary
- even pole order has matching sides
- coefficient sign flips both sides
- a denominator root alone is insufficient before cancellation
- You control
- excluded input · numerator multiplicity · denominator multiplicity · extension overlay
- You observe
- Original/reduced rules, remaining order and open versus filled point
- You explain
- Classify the discontinuity after cancellation while retaining the original domain.
Test the boundary
- partial cancellation can leave a pole
- 0/0 is not enough to conclude a hole
- equal multiplicities give hole height one
- You control
- representation task · excluded root or shift · expansion power
- You observe
- Equivalent forms, remainder check and binomial coefficient row
- You explain
- Select a form that reveals zeros, a quotient, or expansion coefficients, and retain domains.
Test the boundary
- division remainder is essential
- simplified rational rule alone extends the original domain
- binomial coefficients require powers as well as coefficients
- You control
- vertical scale · horizontal scale · horizontal shift · vertical shift · parent point
- You observe
- Mapped coordinates, parent and transformed graphs, vertex or constant case
- You explain
- Use the input equation to explain why the horizontal scale is reciprocal.
Test the boundary
- a=0 has no unique vertex
- b cannot be zero in the point-map formula
- negative b reflects the input
- You control
- data scenario · candidate model · prediction input
- You observe
- Observed versus modeled outputs, residuals and held-out prediction
- You explain
- Support a model choice and state an assumption plus a domain limitation.
Test the boundary
- finite exact fit does not prove truth outside the data
- extrapolation requires caution
- inverse model excludes zero
- You control
- context · base/fixed cost · additional/variable rate · query input
- You observe
- Contextual graph, exact cost, boundary agreement or asymptotic cost
- You explain
- Define variables/units, write the rule and interpret the requested value within its domain.
Test the boundary
- shipping boundary belongs to one branch only
- shared count must be a positive integer
- a context can restrict an algebraic domain
- You control
- initial value · common difference · common ratio · inspect term · continuous extension
- You observe
- Discrete points and consecutive differences or ratios update together.
- You explain
- State the rule and its whole-number domain.
Test the boundary
- zero initial value
- negative ratio
- ratio 1
- You control
- model · interval start · interval width
- You observe
- A fixed difference or a fixed output ratio over equal input steps.
- You explain
- Include the equal-length input condition.
Test the boundary
- linear output zero
- negative linear output
- non-unit step
- You control
- base · vertical multiplier · horizontal shift · vertical shift
- You observe
- The transformed curve, its range and direction.
- You explain
- Use the multiplier and base together.
Test the boundary
- zero multiplier
- negative multiplier
- decay base
- You control
- period factor · period length · elapsed time
- You observe
- Three equivalent formulas evaluate to the same output.
- You explain
- The new base is a root of the period factor.
Test the boundary
- decay factor
- fractional elapsed time
- one-unit period
- You control
- initial population · later population · elapsed days · prediction day
- You observe
- The model passes through both observations; prediction type is labelled.
- You explain
- Distinguish a factor from a percent rate.
Test the boundary
- constant population
- decay
- extrapolation
- You control
- data pattern · measurement noise · candidate model · validation reveal
- You observe
- Least-squares fit, signed residuals and validation error.
- You explain
- Use residual pattern and validation error.
Test the boundary
- in-sample fit versus extrapolation
- noise
- zero residual
- You control
- composition order · square-root shift · input
- You observe
- Input, inner output, radicand and final output.
- You explain
- Derive the domain from the radicand.
Test the boundary
- boundary input
- outside domain
- order-dependent domain
- You control
- domain branch · test height · input
- You observe
- Horizontal-line test and reflected inverse branch.
- You explain
- State the domain for each inverse formula.
Test the boundary
- unrestricted square
- vertex
- negative branch
- You control
- base · exponent · reveal exponent
- You observe
- Base-power-output machine with the exponent unknown.
- You explain
- Use the same base and positive argument.
Test the boundary
- fractional exponent
- negative exponent
- base below one
- You control
- base · point input · show inverse · show reflection
- You observe
- A point and its reversed coordinates.
- You explain
- Exchange domain and range; verify composition.
Test the boundary
- negative log output
- decay base
- zero input
- You control
- base · multiplier · boundary shift · vertical shift · probe input
- You observe
- The domain boundary and a probe that rejects forbidden inputs.
- You explain
- Require the argument, not output, to be positive.
Test the boundary
- zero multiplier retains domain
- negative output
- base below one
- You control
- proposed law · first argument · second argument
- You observe
- Both expressions report their own domains and values.
- You explain
- A numerical coincidence is not an identity.
Test the boundary
- negative arguments
- zero argument
- square of negative input
- sum trap
- You control
- problem type · target · candidate · growth/decay base · inverse branch and input
- You observe
- Candidates tested in original equation; inequality direction and inverse domains checked.
- You explain
- Show domain and verify accepted solutions.
Test the boundary
- extraneous root
- zero argument
- inequality reversal
- You control
- horizontal shift · start argument · argument multiplier · vertical multiplier · base
- You observe
- Inputs, arguments and exact output difference.
- You explain
- Compare argument ratios for a shifted log.
Test the boundary
- nonzero shift
- negative multiplier
- zero multiplier
- You control
- data set · initial value · growth factor · raw/log display
- You observe
- Raw points transform only when all outputs are positive.
- You explain
- Translate slope and intercept back to the original model.
Test the boundary
- negative logarithms valid
- zero or negative outputs invalid
- constant case
- You control
- Waveform · Period · Candidate shift · Observation time
- You observe
- A shifted graph and paired output table.
- You explain
- A repeat shift preserves every output; a single matching pair is insufficient.
Test the boundary
- A triangle wave is periodic but not sinusoidal.
- Half a period is not a repeat interval for these models.
- You control
- Angle in degrees
- You observe
- Circle coordinates, projections and the tangent denominator.
- You explain
- Sine is the vertical coordinate; tangent is undefined when cosine is zero.
Test the boundary
- Axes are not quadrants.
- Non-special angles use approximate values.
- You control
- Special angle · Radius
- You observe
- Signed exact coordinates and the reference angle.
- You explain
- Multiply both unit-circle coordinates by the radius.
Test the boundary
- Reference-angle magnitudes do not determine signs.
- Axis points are identified separately.
- You control
- Rotation angle · Visible traces
- You observe
- The same angle on a circle, two graphs and a quarter-cycle table.
- You explain
- Cosine starts at 1 and sine at 0; both repeat after one revolution.
Test the boundary
- At 2π the full circle is visibly complete.
- A second revolution repeats the outputs.
- You control
- Sine or cosine · Signed vertical scale · Midline
- You observe
- Extrema, midline and a complete-cycle graph.
- You explain
- Amplitude is nonnegative and equals half the range width.
Test the boundary
- Zero scale produces a constant with no fundamental period.
- A negative scale reflects the graph.
- You control
- Amplitude scale · Frequency factor · Phase shift · Vertical shift
- You observe
- A transformed sine graph with its five quarter-cycle anchors.
- You explain
- Use 2π/|B| for the period of A sin(B(x−h))+k.
Test the boundary
- A=0 makes a constant.
- The horizontal shift is h in the factored argument.
- You control
- Lowest height · Wheel diameter · Revolution time · Starting position · Elapsed time
- You observe
- A moving wheel, physical height and quarter-cycle observations.
- You explain
- The model assumes constant angular speed and uses height units and time units.
Test the boundary
- Starting at the bottom requires a negative cosine term.
- A periodic process need not be sinusoidal without this assumption.
- You control
- Horizontal factor · Horizontal shift · Vertical scale
- You observe
- Separate branches, asymptotes and the central branch interval.
- You explain
- The period is π/|B|, and tangent is never defined at its poles.
Test the boundary
- The graph never connects across a vertical asymptote.
- The central branch endpoints are excluded.
- You control
- Inverse function · Input value
- You observe
- A restricted original curve, its inverse and a reflected point pair.
- You explain
- An inverse requires a one-to-one restriction and returns only its principal value.
Test the boundary
- arcsin/arccos inputs are in [−1,1].
- arctan has range (−π/2,π/2), with no finite-input endpoints.
- You control
- Sine or cosine · Target · Equation or inequality
- You observe
- Intersections and shaded satisfying angle intervals.
- You explain
- Check both branches, endpoint inclusion and the original interval.
Test the boundary
- 0 is included when valid; 2π is excluded.
- At a maximum or minimum, duplicate roots merge.
- You control
- Reciprocal function · Probe angle
- You observe
- Reciprocal graphs, denominator values and valid or excluded points.
- You explain
- sec and csc have no zeros; cot has zeros where cosine is zero and sine is nonzero.
Test the boundary
- A reciprocal never equals zero.
- Undefined is different from zero.
- You control
- Identity family · Angle
- You observe
- Two expressions evaluated side by side and their common domain.
- You explain
- Equal simplified values do not restore an input excluded by the original denominator.
Test the boundary
- Cancellation requires sin θ≠0.
- A double-angle identity remains valid for all real angles.
- You control
- Signed radius · Angle · Equivalent representation
- You observe
- The reference ray, actual point, distance and equivalent coordinates.
- You explain
- Negative radius places the point opposite the reference ray; distance is |r|.
Test the boundary
- At r=0 every angle represents the origin.
- Adding π and reversing radius preserves the point.
- You control
- Curve family · Angle
- You observe
- A complete polar graph, radius table and moving point.
- You explain
- Graph (r cos θ,r sin θ), including negative values of r.
Test the boundary
- A negative radius is not a negative distance.
- r=0 places the point at the pole.
- You control
- Polar model · Initial angle · Interval width
- You observe
- A radius-versus-angle secant and its two corresponding polar points.
- You explain
- Divide Δr or Δ|r| by Δθ and state the units and interval.
Test the boundary
- Equal endpoint radii can hide motion between them.
- A finite secant is not an instantaneous rate.