VisuApps
Interactive visualizations for the lower-secondary curriculum — matching the example course Mathematics Grade 9 (Baden-Württemberg).
Apps
Tree diagram as a flow tree: product rule along paths, sum rule in the P(E) bar, reverse the tree for Bayes. Parts, urn, coin, plant, test; fractions, decimals or frequencies, two-way table.
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Split and rearrange squares and rectangles to explore the three binomial identities and make the often-missed middle term visible.
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Approximate π: inscribed/circumscribed polygons clamp π between two bounds, or Monte Carlo with random points (fraction ×4 ≈ π). Estimate vs. true π live.
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Drag a point on the unit circle → cos α = x, sin α = y; on "unrolling" the sine curve emerges on the right (period 2π, range [−1;1]). Degrees/radians, sin/cos/tan switchable, play animation.
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N(t)=N₀·qᵗ with sliders for N₀ and q (q>1 growth, q<1 decay). Comparison with linear growth, scenarios bacteria/interest/decay, doubling time/half-life marked.
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Model Ulm’s city library as a glass pyramid on a stepped base. Explore glass surfaces and calculate area and cleaning costs step by step.
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Apply the same operation to both sides of a linear equation to preserve balance and isolate x. Compare an incorrect one-sided operation.
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An atom grid decays in sync with the curve N(t)=N₀·(½)^(t/T). Half-life T constant: at T/2T/3T still 50/25/12,5% remain. Switchable to growth with doubling time.
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Explode composite solids into simpler parts, highlight surfaces and build volume and area formulas. Explore hollow solids and missing lengths using Pythagoras.
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Prism, cylinder, pyramid, cone, sphere rotatable in 3D; volume & surface area live with formula. Plausibility: cone = ⅓ cylinder, pyramid = ⅓ prism; Cavalieri.
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Adjust slope and intercept, or drag two points. See how a slope triangle connects horizontal steps with vertical change.
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Fold nets into cubes, pyramids, cylinders, cones and triangular prisms. Match faces and inspect why an invalid net cannot form the intended solid.
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A Ferris wheel + a height-time diagram for the general sine function f(t)=a·sin(b(t−c))+d. Amplitude, period, phase, midline via slider; scenarios Ferris wheel / day length / tides.
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The graph of f(x)=a·xⁿ with sliders for n and a. Even n → axis-symmetric, odd → point-symmetric; n<0 → hyperbola. Function family, symmetry highlight, notable points.
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Connect percentages, fractions, decimals and sectors of a circle. Explore how the base amount changes the result and practise all three percentage problems.
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Pythagoras in space on a cuboid: face diagonal d=√(a²+b²), space diagonal e=√(a²+b²+c²). Two right triangles highlighted, rotatable (3D). Sliders a, b, c; cube preset.
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Rotate plane shapes around an axis to create cylinders, cones, spheres and composite solids. Connect generating profiles, radius, height and axial sections.
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A right triangle with squares on all sides; a playable rearrangement proof of a²+b²=c². Sliders for the legs a, b; live a², b², c² and c=√(a²+b²).
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Two rays from the apex Z, cut by two parallels: 1st intercept theorem ZA/ZA′ = ZB/ZB′, 2nd intercept theorem AB/A′B′ = ZA/ZA′. Measure the segments live; application mode "measure height" (forester's triangle/shadow).
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sin/cos/tan on the right triangle — as a side ratio and as a value. Drag the angle α, exact values 30/45/60°, the unit-circle bridge (hypotenuse = 1 → sin α = y, cos α = x).
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√x as the inverse/reflection of x² across y=x. Root index n and parameters a, c, d; domain and range highlighted.
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Dilate a figure from the center Z with factor k: rays through every vertex, ZP′ = k·ZP. k=1 → congruent (congruence), k<0 → point reflection. Live: lengths ×|k|, area ×k². Z and the vertices freely draggable.
School mathematics (Gymnasium)Try itWant to use these apps in your courses? Let’s find out what works for you.
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