KinderCalculus
KinderCalculus is an experimental home curriculum that teaches typical K6 children Calculus by relieving them of the burden of large number arithmetic.
Freed from this burden, children in my family reached Calculus in their K6 years with 2 hours/week of effort. A wider test is needed but that is difficult to do because families will rarely commit a child to a 7year curriculum throughout K6. Though our coverage is not as thorough as the high school materials, we've been able to manipulate equations at midlevel difficulty by using: (1) unified algebraic concepts (2) a simplified vocabulary (3) shapes to declutter equations (4) infinitesimal numbers. These methods are included here.
With these basic skills, we've even rederived some of humanity's crowning intellectual achievements on reality (Einstein), infinity (Cantor), and creation (fractals). I think it's worth rethinking the burden of big arithmetic if we can replace it with these these wonders of the world.
The "kinder" part of the name KinderCalculus refers to the German term for young children and not Kindergarten. Email inquiries can be sent to numiri@yahoo.com with the subject heading of "KinderCalculus".
Vocabulary
Because of its unconventional goal, KinderCalculus uses an unconventional approach. I propose to replace much of the arcane vocabulary with simple relatable kidfriendly terms. It is my personal belief that the choice of vocabulary plays a substantive role in a subject's difficulty  any subject:
 in language. Ask yourself: would I start teaching my child to read with a sentence like "observe the blotchly decorated canine as it executes a rapid gait", or would I start with "See Spot run"?
 in business. Should we teach kindergardeners to practice "collaborative consumption", or just teach them to "share"?
 in physics. Most children cannot grasp what an "inertial reference frame" in Special Relativity is, but they certainly know what being "adrift" is.
Let's face it, if we force cumbersome language on children, we place a substantive cognitive load on a child and distract the child from the concept at hand. KinderCalculus replaces such polysyllabic abuses with significantly shorter words like "verbs" for "operators", "nouns" for "operands", "speed" for "derivative", and "peers" for "commutative and associative operands". Terms are chosen based on its mnemonic value, brevity, descriptiveness, relatability, and their appeal to a child's sense of subversion. For example, with K2 children, "potty humor" is an endless source of amusement so I use the term "blunderwear" to describe certain algebraic associative pitfalls similar to the inner and outer layers of clothing. With children, the silly becomes memorable and the subversive offers an illusion of empowerment. See the Glossary for more terms.
Remarks
Thoroughness: at this stage, this content is only an outline of a teacher's guide  it is NOT meant to be read by children and it is NOT a complete textbook. As such, the focus is on alternate ways to explain advanced concepts to children, so I don't cover all the concepts in the K12 curriculum. I only cover areas where this approach diverge from the standard exposition for Algebra & Calculus. I skip nonalgebra & noncalculus areas such as number sense, probabilities, proportions, reading data tables, etc.
Exercises: rather than do a wide breadth of exercises for Calculus homework, I take the lazy teacher approach of doing hard exercises over and over again, such as repeatedly doing proofs of the Fundamental Theorem, Chain Rule, Product Rule, and time dilation in Special Relativity. This stems from a lack of teaching preparation time more than anything else.
Prerequisite for teachers: the reader is assumed to have mastered high school algebra & calculus.
Prerequisite for children: fractions & 2digit arithmetic.
Table of contents
Grade 12  Grade 23  Grade 34  Grade 45  Grade 56  Appendix 

Grammar & Syntax

Algebra 
Applications

Calculus

Calculus 
For more information, email numiri@yahoo.com w/ the subject "KinderCalculus"