# Real Order, Average Rates, Coordinate Geometry, and Data Summaries: Seven-Round Workflow Complete seven traceable knowledge objects in dependency order on top of the existing number-system and function foundations. This is a suggested learning sequence, not a universal logical dependency or a formal proof. - Stage 1 · required ## [Real number](/en/mko/mko-real-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-real-number) [Concept guides](/en/derivations/?q=mko-real-number) - Stage 1 · required ## [Rational number](/en/mko/mko-rational-number/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-rational-number) [Concept guides](/en/derivations/?q=mko-rational-number) - Stage 1 · required ## [Function mapping](/en/mko/mko-function-mapping/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-mapping) [Concept guides](/en/derivations/?q=mko-function-mapping) - Stage 1 · required ## [Quadratic equations over the reals](/en/mko/mko-quadratic-equation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-quadratic-equation) [Concept guides](/en/derivations/?q=mko-quadratic-equation) - Stage 1 · required ## [Ratio and proportion](/en/mko/mko-ratio-proportion/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-ratio-proportion) [Concept guides](/en/derivations/?q=mko-ratio-proportion) - Stage 1 · required ## [Graphs of real-valued functions](/en/mko/mko-function-graph/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-function-graph) [Concept guides](/en/derivations/?q=mko-function-graph) - Stage 1 · required ## [Affine functions and degree-one functions](/en/mko/mko-linear-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-linear-function) [Concept guides](/en/derivations/?q=mko-linear-function) - Stage 1 · required ## [Coordinate plane](/en/mko/mko-coordinate-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-coordinate-plane) [Concept guides](/en/derivations/?q=mko-coordinate-plane) - Stage 1 · required ## [Natural-number powers and principal square roots](/en/mko/mko-power-root/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-power-root) [Concept guides](/en/derivations/?q=mko-power-root) - Stage 1 · required ## [Euclidean length](/en/mko/mko-euclidean-length/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-euclidean-length) [Concept guides](/en/derivations/?q=mko-euclidean-length) - Stage 1 · required ## [Points, lines, and planes](/en/mko/mko-point-line-plane/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-point-line-plane) [Concept guides](/en/derivations/?q=mko-point-line-plane) - Stage 1 · required ## [Arithmetic operations](/en/mko/mko-arithmetic-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-arithmetic-operations) [Concept guides](/en/derivations/?q=mko-arithmetic-operations) - Stage 1 · required ## [Set operations](/en/mko/mko-set-operations/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Existing canonical prerequisite for the new objects in this pulse. [Related questions](/en/explore/?mko=mko-set-operations) [Concept guides](/en/derivations/?q=mko-set-operations) - Stage 2 · optional ## [Usual order on the real numbers](/en/mko/mko-order-relation/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) The usual real order compares real values and supports finite nonempty extrema, closed intervals, and translation by a common addend. [Related questions](/en/explore/?mko=mko-order-relation) [Concept guides](/en/derivations/?q=mko-order-relation) - Stage 3 · optional ## [Quadratic functions in vertex form](/en/mko/mko-quadratic-function/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) A nonzero-coefficient quadratic on all of R has a vertex, an axis of symmetry, and a global minimum or maximum determined by the coefficient sign. [Related questions](/en/explore/?mko=mko-quadratic-function) [Concept guides](/en/derivations/?q=mko-quadratic-function) - Stage 4 · optional ## [Average rate of change between distinct inputs](/en/mko/mko-rate-of-change/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Average rate of change is an endpoint difference quotient with a nonzero input difference, not an instantaneous derivative. [Related questions](/en/explore/?mko=mko-rate-of-change) [Concept guides](/en/derivations/?q=mko-rate-of-change) - Stage 5 · optional ## [Euclidean distance in an equal-unit orthonormal frame](/en/mko/mko-distance-formula/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) The nonnegative L2 distance in one equal-unit orthonormal coordinate frame is the principal square root of the sum of squared coordinate differences. [Related questions](/en/explore/?mko=mko-distance-formula) [Concept guides](/en/derivations/?q=mko-distance-formula) - Stage 6 · optional ## [Circle boundaries and zero-radius degeneracy](/en/mko/mko-circle/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) A circle here is the boundary point set at a nonnegative radius from its center, not the filled disk; radius zero yields a singleton. [Related questions](/en/explore/?mko=mko-circle) [Concept guides](/en/derivations/?q=mko-circle) - Stage 7 · optional ## [Exact area of half-open rectangles and unit-cell unions](/en/mko/mko-area/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) This entry computes exact area for half-open integer rectangles and finite unit-cell unions, including empty and overlapping cases. [Related questions](/en/explore/?mko=mko-area) [Concept guides](/en/derivations/?q=mko-area) - Stage 8 · optional ## [Descriptive summaries of nonempty observation lists](/en/mko/mko-descriptive-statistics/?from=%2Fen%2Fpaths%2Fpath-pulse-foundations-20261006%2F) Mean, median, range, and the explicitly defined maximal-frequency set summarize a nonempty finite observation list while retaining multiplicities. [Related questions](/en/explore/?mko=mko-descriptive-statistics) [Concept guides](/en/derivations/?q=mko-descriptive-statistics) Path source & statuspath-pulse-foundations-20261006 · candidate [Original paths JSON](/data/architecture/learning-paths.json)