Lattice Multiplication vs Surface Explorer

Lattice Multiplication helps students learn and practice the lattice multiplication method step by step. The app shows each part of the lattice multiplication process visually. Students first complete the multiplication steps and then add along the diagonals. Animated numbers move into the correct places, helping students understand how the method works. Lattice multiplication is a compact visual method for multiplying larger numbers. It is algorithmically equivalent to long multiplication and is also excellent times tables practice. Students can solve randomly generated problems or enter their own multiplication problems. Wired.com: “This was one form of multiplication that I’d never done before, but the app taught me very clearly. It makes multiplying large numbers easier, and allows you to see it all visually.” Teacher endorsement: “The math apps by Esa Helttula allow students to practice and reinforce the specific computation strategies taught in class, with as much or as little scaffolding as necessary. They also provide individualized feedback to each student, with an immediacy that is not always possible through pencil and paper practice.” — Rita Zeni, Ecole Sandy Hill Elementary, Abbotsford, BC, Canada Features: - Step-by-step lattice multiplication practice - Multiplicand can have up to 5 digits - Multiplier can have up to 3 digits - Randomly generated or manually entered problems - Current operation for each step can be hidden - Operands for the current operation can be highlighted - Three visual themes: black, gray, and gold - Adjustable animation speed - Different interfaces for iPhone and iPad The Lattice Multiplication Method Lattice multiplication is algorithmically equivalent to long multiplication. A lattice (a grid) guides the calculation. All the multiplications are done first and then the additions. The method was introduced to Europe in 1202 in Fibonacci's Liber Abaci. In this groundbreaking book Fibonacci presented many algorithms for working with Arabic numerals. Ancient Indians and Chinese originally invented some of the algorithms. Fibonacci presented both the current standard long multiplication and also an originally Indian method called lattice multiplication, which is faster and more compact for working with larger numbers. Other iDevBooks Math Apps: Other iDevBooks math apps include Column Addition, Long Multiplication, Long Division, Column Subtraction, Visual Multiplication Table, Fraction Math, Partial Differences Subtraction, Partial Quotients Division, Partial Sums Addition, Decimal Rounding, and more. Feedback: New ideas to make this app better are welcome. Please visit idevbooks.com to give feedback. Privacy: This app has no ads, no in-app purchases, and does not transmit user data during operation. The app also does not contain links to other apps or to the web.

Surface Explorer is a high-performance visualizer for intricate mathematical surfaces — 2D manifolds embedded in three- or four-dimensional space, and 3D slices of four-dimensional hypersurfaces — bringing abstract geometry to life with real-time GPU rendering. KEY FEATURES • Cross Section — slice four-dimensional hypersurfaces and watch their 3D sections change shape as you rotate them in 4D or slide the cutting plane along the fourth axis, even along a 4D camera path. • Ray Marching Engine — real-time rendering of implicit surfaces and signed-distance fields, with procedural displacement, transparency, adjustable shells, and a precise mode for high-degree equations. • Multi-Mesh Surfaces — build figures from several independent grids, set with a single field or a script, each with its own color, transparency, texture, image and domain. • Solid Curve — write a curve in 3D or 4D and see the tube drawn around it, with adjustable thickness; several tubes from a script. • Surface Borders — add a tube along the edges of a parametric surface, with its own color, texture and thickness. • Geodesic Flow — explore intrinsic geometry on curved manifolds (S³, H²×R, SL(2,R) and more), integrating geodesics from initial conditions. • 4D Spatial Exploration — dedicated controls for hyperspatial rotation (Omega, Phi, Psi), projection, and 3D and 4D camera paths. • 4D Lighting Models — Directional, Observer, and Slice lighting designed for hyperspace. • Dynamic GLSL Scripting — write custom shaders for procedural textures and surface logic in the built-in editor. • Backgrounds — images and procedural effects, fixed on screen or wrapped around the scene as a sphere, cylinder or cube. • Audio-Visual Synthesis — synchronize geometric parameters with a mathematical sound synthesizer. • Video Recording — capture your explorations directly in the app and save them to your device. A library of ready-made presets ships with the app, from classic parametric surfaces to black-hole ergospheres, curved-space tori and slices of the 3-torus, together with a searchable built-in manual. Built with Qt and Metal for smooth, native performance on iPhone and iPad.