arithmetic

• top - numerator
• bottom - denominator
• top heavy - improper fraction
• bottom heavy - proper fraction, may be unreduced
• cops - mnemonic for Covering Prime String, which is the Least Common Denominator

algebra

• layers - order of operation, the parse tree of an expression
• evert - generalized factorization or distribution applicable to a(b+c) = ab + ac, (f+g)' = f' + g', etc. switch inner and outer operators.
• nevadab - initialism for "combining like terms". "nev" = neighboring verb criteria, "ab" = alike base criteria, ad = add depth
• peers - operands that are both commutative and associative. Commutivity is symmetry of a verb, and associativity is symmetry of a verb chain, eg. (ab)c = a(bc).
• blead - the base-leaf-depth relationship of binary operators
• left & right divide - the inverse-exponential operators root & log

• noun - operands
• verbs - operators
• adjective - the negative (-) and reciprocal (1/*) noun modifiers
• shape - (verb) to assign a shape to a term. Shaping is useful for substitutions.

• invisible - implicit verbs & layers such as parentheses or multiplication in xy. the exponential operator is also invisible
• neighboring verbs - the relationship between a seed verb (+ & *) and its iterative counterpart (* & ^)
• forward verbs: +, *, ^ (exponential)
• reverse verbs: -, /, o--, --o
• forq: the forward-reverse-sequence identity, eg. (a*b)/c = a/c * b/c, log(a^b) = blog(a), or root(a^b) = root(a)*root(b)
• seed & repeater verbs: in the context of self-plus and self-times, the primitive operation is called the seed verb and the resulting operation from the repetition is the repeater verb. For instance, in 3x = x + x + x, the seed verb is "+" and the repeater verb is times. Likewise, x^3 = x * x * x has the seed verb "*" and repeater verb as "^".
• self-ish verbs: repeater verb
• center: of a verb is the group-theoretic Identity element, either 0 or 1, for verbs + or - respectively

• blunderwear - the algebraic pitfall where an inner layer is peeled off before an outer layer, or a layer is put onto an expression inside of the outer most layer
• snowflake - the pictorial representation of exponentiation, logarithms, and roots

• colo: slightly generalized factorization. This is an abbreviation for "common-leftover", emphasizing those the "common" part that is extracted and the remaining parts that are leftover in a factorization, eg. xy + xz has x as the common factor, and y & z as the leftovers. generalizing a bit, (y^x)*(z^x) = (yz)^x. colo & spread are opposites.
• spread: to distribute, as in x( y + z ) = xy + xz. spread & colo are opposites.
• sprolo: spread-colo identities, aka factoring & distributive identities. eg. a(b+c) = ab + ac, and many more.
• common leaves: common factors in reducing a fraction. The leaves stem from the factor tree.
• nudify: to solve for a variable, leaving it bare and naked. Used for comedic value and may be dismissed by conservative minded parents.
• fit: to satisfy an equation by substituting in values. For instance, x = 2 fits the equation x + 2 = 4

trigonometry

• tick - coordinate
• grid - coordinate system
• round ticks - polar coordinates
• circle grid - polar coordinate system
• square ticks - cartesian coordinates
• square grid - cartesian coordinate system
• bionic number - complex number

calculus

• mirrors = the diametric opposites in the product rule. In the integration-by-parts formula, the product g'f has the conjugate f'g. I use conjugate in this way, not to be confused with the complex conjugate of a+bi as a-bi. If you prefer another term altogether, "mirror" or "reflection" are good alternatives.
• forever big, forever small - infinity & infinitessimal respectively
• leak = the speed identity (cf)' = c(f'), where c is a constant.
• line = (verb) to assign a solid or dashed line to a shape to indicate speeded-function vs. a plain-function. This term is useful for exercises in integration by substitution or by parts.
• nudge = change in input or output. For example, x2-x1 is an input nudge, where f(x2)-f(x1) is the output nudge.
• nudge ratio = input nudge / output nudge
• segment = the interval of integration in a definite integral.
• sliver = a skinny rectangle in a Riemann sum
• speed = derivative
• steepness = the graphical interpretation of derivative as the slope of the tangent
• studr = (pronounced "studder") portmanteau that abbreviates the non-standard calculus definition of speed as the STandard of the nUDge Ratio. This is a mnemonic to help students remember the definition.
• sweep = Riemann Sum, also anti-derivative as appropriate
• tess = infinitesimal, forever small
• graphing terms:
  • cutting x-axis = in graphing, extreme points & inflection points divides the x-axis into regions of positive/negative y' & y"
  • dump water = concave down
  • falling = decreasing function in graphing
  • flat point = solutions of the equation y'(x)=0. we don't have a term for solutions of y"(x)=0. "Inflection" does not reflect all of the solutions
  • hold water = concave up
  • rising = increasing function in graphing
  • trap = solving for x in the equation y'(x)=0 or y"(x)=0. Trapping is an imprecise act, as opposed to "finding" the critical point because trapping critical points gets too much and we have to take the additional step to exclude the inflections.