 # Math representations in language

The single biggest problem with communication is the illusion that it has taken place.

George Bernard Shaw

## Overview

The following table summarizes different representations in the language of mathematics!

The mathematical objects (nouns) being operated upon by a mathematical action which can be represented as a manipulation (verb) in a manner in which we believe is appropriate for elementary students and therefore a necessary part of an elementary curriculum. See additional resources

Mathematical Objects (nouns) Mathematical Actions (verbs)
Types of objects Properties of objects Practicle use Operations on objects

Number/quantity

• integers
• rational numbers
• real numbers
• measures of:
• length
• area
• volume
• time
• weight
• order
• between-ness
• part/whole relationships
• units
• dimensions
• counting
• measuring
Arithmetic operations
• subtraction
• multiplication
• division
• expotential

Shape/space

• metric - value between two points
• lines/segments
• polygons
• circles
• conic sections
• spherical shapes
• topological properties - geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.
• distance
• location
• symmetry
• similarity
• connectedness
• enclosure
• mapping and traveling
• designing and building objects
• translation
• rotation
• reflection
• covering, packing, & tessellating
• scaling
• projection
• inversion
• conformal mapping
• homotopies/deformations

Pattern/function

• linear
• power
• rational
• periodic
• transcendental
• input/output
• domain/range
• continuity
• boundedness
• rate of change, curvature, etc.
• maxima and minima
• rate of accumulation
• linear-root, slope/intercept
• power-roots, asymptotic behavior
• rational-roots, singularities, asymptotic behavior
• periodic-frequency, phase
• transcendental-"growth" constant
• expressing how something depends on another - one or more other things
• resolving constraints
• (solving equations and inequalities)
• identifying and describing repetitive relationships
• arithmetic operations (functions on Rn)
• comparison
• equations
• inequalities
• identities
• composition
• translation
• reflection
• dilation/contraction

Arrangements

• combinations
• permutations
• graphs
• networks
• trees
• enumeration
• vertices and edges of graphs/networks
• organizing discrete information
• tends to be closely connected to the idea of repetitive pattern at this level

Chance/data

• discrete
• continuous
• determinism
• randomness
• relative frequency
• distribution
• moments
• dealing with uncertainty
• dealing with lack of precision
• sampling (by counts and/or measures)
• composing
• representing

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