Learning path

From tally marks and place value, through fact fluency and binary, to logic gates, how a computer adds, and the jump to universality. See the same ideas as a map in the knowledge graph.

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Counting and quantity
Counting objects, number words and the idea that a number is the same however it is shown.
- One-to-one counting
  Skill
  Give exactly one count word or mark to each object.
  Checking…
  - Counting Creatures 3 · introduces, teaches (Tally Trail)
- Number-word sequence
  Skill
  Say the counting words in order.
  Checking…
  - Counting Creatures 3 · practice (Hand Hills)
- Cardinality
  Idea
  The last word counted tells how many there are.
  Checking…
  - Counting Creatures 3 · teaches (Hand Hills)
- Counting on
  Skill
  Continue counting from a given number instead of starting at one.
  Checking…
  - Counting Creatures 1 · practice (Count ladder)
- Tally marks
  Skill
  Record a quantity with one mark per thing, bundled in fives.
  Checking…
  - Counting Creatures 3 · teaches, check (Tally Trail)
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
- Number as abstraction
  Idea
  "Three" is the same across apples, claps and marks, and a numeral is a symbol for it.
  Checking…
  - Counting Creatures 3 · introduces, teaches (Sloth Forest)
  - Counting Creatures 1 · teaches (The sloth counts in six)
- Zero
  Idea
  Zero is the number of nothing.
  Checking…
  - Counting Creatures 3 · introduces
- Number line
  Idea
  Numbers sit in order as positions along a line.
  Checking…
  No applet or book covers this yet.
- Roman numerals
  Idea
  Symbols for 1, 5, 10, 50, 100, 500 and 1,000 that add side by side (the take-away rule, IV = 4, came later); the rules do some arithmetic for you, but with no symbol above M, big numbers fall back on tallying thousands (BoI 6).
  Checking…
  - Counting Creatures 3 · introduces, teaches, practice (Tally Trail)
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
Place value and number bases
Bundling into groups, positional notation, and writing numbers in bases other than ten.
- Bundling
  Idea
  Group a fixed number of things into one bundle, such as bags of ten.
  Checking…
  - Counting Creatures 3 · introduces, teaches (Tally Trail, Hand Hills)
  - Counting Creatures 2 · teaches (Human world (base 10))
- Tens and ones
  Skill
  Read and build two-digit numbers as tens and ones, teens included.
  Checking…
  - Counting Creatures 3 · teaches, check (Hand Hills)
  - Counting Creatures 2 · teaches, check (Human world (base 10))
- Positional notation
  Idea
  A digit's value depends on its place, and zero holds a place (hundreds and thousands too).
  Checking…
  - Counting Creatures 3 · introduces, teaches (Tally Trail, Hand Hills)
  - Counting Creatures 1 · teaches (Count ladder)
  - Counting Creatures 2 · teaches (Human world (base 10))
- What a base is
  Idea
  A base is the bundle size: the number of fingers or symbols a counting system uses.
  Checking…
  - Counting Creatures 3 · teaches (Sloth Forest, Chip City)
  - Counting Creatures 1 · teaches (Why ten, Side by side)
  - Counting Creatures 2 · teaches (Sloth world (base 6))
- Digits of a base
  Idea
  Base b uses the digits 0 to b-1, so after 5 comes 10 in base 6, and base 16 adds A to F.
  Checking…
  - Counting Creatures 3 · teaches, practice (Sloth Forest, Creature Lab)
  - Counting Creatures 1 · teaches (The sloth counts in six)
  - Counting Creatures 2 · teaches (Sloth world (base 6), Octopus world (base 8), The lab (base 16 and trade-offs))
- Read another base in base ten
  Skill
  Read a base-b numeral as a base-ten number.
  Checking…
  - Counting Creatures 2 · check (Sloth world (base 6), Octopus world (base 8))
  - Counting Creatures 3 · check (Sloth Forest)
- Write a number in another base
  Skill
  Write a base-ten number in base b.
  Checking…
  - Counting Creatures 1 · practice (The sloth counts in six)
  - Counting Creatures 2 · check (Sloth world (base 6), Octopus world (base 8))
  - Counting Creatures 3 · check (Sloth Forest)
- Symbols versus length
  Idea
  More symbols give shorter numerals; fewer symbols give longer but simpler ones.
  Checking…
  - Counting Creatures 1 · introduces (Side by side)
  - Counting Creatures 3 · teaches, practice (Creature Lab)
  - Counting Creatures 2 · teaches (The lab (base 16 and trade-offs))
- Each place is base times the next
  Idea
  Each place is worth the base times the place to its right: 1, 10, 100 and so on.
  Checking…
  - Counting Creatures 3 · introduces
Addition and subtraction facts
Joining and taking away, the single-digit fact families, and carrying in column addition.
- Addition
  Idea
  Addition is joining groups and counting on (Feynman 22-1).
  Checking…
  - Math Quiz · teaches (Math Quiz: Ten frame and visual practice)
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Turnaround facts
  Idea
  a + b = b + a, so each fact has a turnaround twin.
  Checking…
  No applet or book covers this yet.
- Make ten
  Skill
  Make ten and bridge through ten, as on a ten frame.
  Checking…
  - Math Quiz · teaches, practice (Math Quiz: Ten frame and visual practice)
  - Counting Creatures 3 · practice, check (Hand Hills)
- Add 0 facts
  Skill
  Adding zero leaves a number unchanged.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Add 1 facts
  Skill
  Adding one gives the next number.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Add 2 facts
  Skill
  Adding two is the next number after the next.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Doubles
  Skill
  Doubles from 3 + 3 to 9 + 9.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Tough 21
  Skill
  The 21 non-double facts with both addends from 3 to 9 (the hardest six included).
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Hardest six (Sneaky Six)
  Skill
  The six facts with both addends from 6 to 9 and unequal: 6+7, 6+8, 6+9, 7+8, 7+9, 8+9.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Easy 52 addition facts
  Skill
  The 52 ordered facts with both addends 0 to 5, or a 0 or 1 addend: the Math Quiz Stage 1 to 2 milestone set.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Addition fluency
  Skill
  Answer all 100 single-digit sums quickly and correctly.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Carrying
  Idea
  A full place becomes one bundle in the next place, in any base (the bridge to the binary carry).
  Checking…
  - Counting Creatures 3 · teaches, practice (Hand Hills, Sloth Forest)
  - Logic Gates 2 · teaches, practice (Adders, Binary lights)
- Column addition
  Skill
  Add multi-digit numbers in columns, carrying when a column is full.
  Checking…
  - Math Quiz · practice (Math Quiz: Column worksheets)
- Subtraction
  Idea
  Subtraction is taking away, and it undoes addition.
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Minus 0 facts
  Skill
  Facts of the form m - 0 and m - m.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Minus 1 facts
  Skill
  Taking one away gives the number before.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Minus 2 facts
  Skill
  Taking two away gives the number two before.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Halves
  Skill
  Facts of the form 2n - n, the inverse of the doubles.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Tough rest subtraction
  Skill
  Inverse facts whose subtrahend and difference are both 3 to 9 and unequal.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Sneaky 12
  Skill
  The twelve inverse facts of the hardest six, with subtrahend and difference both 6 to 9.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Easy 52 subtraction facts
  Skill
  The inverses of the 52 easy addition facts: the Math Quiz Stage 1 to 2 milestone set for subtraction.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Subtraction fluency
  Skill
  Answer all 100 inverse facts quickly and correctly.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Negative results
  Idea
  a - b with a smaller than b gives a negative number.
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
Multiplication and division facts
Equal groups, the single-digit product families, and division as sharing and as the inverse of multiplication.
- Multiplication
  Idea
  Multiplication is equal groups, arrays and repeated addition.
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Times 0 facts
  Skill
  Any number times zero is zero.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Times 1 facts
  Skill
  Any number times one is itself.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Times 2 facts
  Skill
  Times two is the same as the doubles.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Squares
  Skill
  Products of a number with itself, n x n.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Tough rest products
  Skill
  The remaining products with both factors from 3 to 9.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Sequence 51, Math Quiz: Split 64, Math Quiz: Split 44)
- Easy 52 multiplication facts
  Skill
  The 52 ordered products with both factors 0 to 5, or a 0 or 1 factor: the Math Quiz Stage 1 to 2 milestone set.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Zero to 5, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Multiplication fluency
  Skill
  Answer all single-digit products quickly and correctly.
  Checking…
  - Math Quiz · practice, check (Math Quiz: Auto, Math Quiz: Adaptive quiz, Math Quiz: Fluency Feast, Math Quiz: Target practice, Math Quiz: Problem list, Math Quiz: Quick quiz, Math Quiz: Dragon game, Math Quiz: Zork adventure, Math Quiz: Fun mode, Math Quiz: Assessment, Math Quiz: Sequence 62, Math Quiz: Split 64, Math Quiz: Split 44)
- Division
  Idea
  Division is equal sharing and equal grouping.
  Checking…
  - Counting Creatures 3 · introduces (Sloth Forest)
- Division undoes multiplication
  Idea
  Division finds the unknown factor, so each product belongs to a fact family.
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Division facts
  Skill
  Division facts within 100 with a single-digit divisor and quotient (the Math Quiz "main div" list).
  Checking…
  - Math Quiz · practice (Math Quiz: Starter operation lists)
- Remainders
  Idea
  When things do not share out equally, what is left over is the remainder.
  Checking…
  - Counting Creatures 3 · introduces, teaches, practice (Sloth Forest)
- Powers
  Idea
  A power is repeated multiplication of the same number.
  Checking…
  - Math Quiz · introduces (Math Quiz: Starter operation lists)
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
Fractions
Equal parts of a whole, unit fractions, and fractions as the numbers that make division always work.
- Equal parts
  Idea
  Partition a whole into equal parts.
  Checking…
  No applet or book covers this yet.
- Unit fractions
  Idea
  A unit fraction 1/b is one of b equal parts.
  Checking…
  No applet or book covers this yet.
- Numerator and denominator
  Idea
  a/b means a copies of 1/b.
  Checking…
  No applet or book covers this yet.
- Fractions as division
  Idea
  a/b = a ÷ b: fractions exist so that every division has an answer (Feynman 22-3).
  Checking…
  - Counting Creatures 3 · introduces (Sloth Forest)
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Equivalent fractions
  Skill
  Find fractions that name the same amount, such as 1/2 = 2/4.
  Checking…
  No applet or book covers this yet.
Binary representation
Two-state switches, base two, binary place values and counting and adding in binary.
- Powers of two
  Skill
  The doubling chain 1, 2, 4, 8, 16 and so on.
  Checking…
  - Logic Gates 2 · introduces
- Two states
  Idea
  A switch has two states: on or off, 1 or 0.
  Checking…
  - Counting Creatures 3 · introduces, teaches (Chip City)
  - Logic Gates 2 · introduces (Switches)
  - Zip and the Thinking Machines · teaches (Inside a chip)
  - Counting Creatures 1 · teaches (The computer counts in two)
  - Counting Creatures 2 · teaches (Computer world (base 2))
- Why two states
  Idea
  Two states are easy to tell apart and to restore after noise; only digital signals can be error-corrected, so universal computers must be digital (BoI 6).
  Checking…
  - Counting Creatures 2 · teaches (Computer world (base 2))
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
  - Counting Creatures 3 · teaches (Chip City)
- Binary is base two
  Idea
  Binary is the base-two number system, with only the digits 0 and 1.
  Checking…
  - Counting Creatures 3 · teaches (Chip City)
  - Counting Creatures 1 · teaches (The computer counts in two)
  - Counting Creatures 2 · teaches (Computer world (base 2))
- Binary place values
  Idea
  Each light is worth double the one to its right: 8, 4, 2, 1.
  Checking…
  - Zip and the Thinking Machines · introduces (Inside a chip)
  - Counting Creatures 3 · teaches (Chip City)
  - Logic Gates 2 · teaches, practice (Binary lights)
  - Counting Creatures 2 · teaches (Computer world (base 2))
- Read binary
  Skill
  Read a binary number as a base-ten number.
  Checking…
  - Counting Creatures 2 · check (Computer world (base 2))
  - Counting Creatures 3 · check (Chip City)
  - Logic Gates 2 · check (Binary lights)
- Make binary
  Skill
  Light up a number in binary.
  Checking…
  - Counting Creatures 1 · practice (The computer counts in two)
  - Counting Creatures 3 · practice, check (Chip City)
  - Counting Creatures 2 · check (Computer world (base 2))
  - Logic Gates 2 · check (Binary lights)
- Count up in binary
  Skill
  Add one in binary; the carry ripples left.
  Checking…
  - Counting Creatures 3 · teaches, practice, check (Chip City)
  - Logic Gates 2 · teaches, check (Binary lights)
  - Counting Creatures 2 · check (Computer world (base 2))
- Add two bits
  Skill
  0 + 0 = 0, 0 + 1 = 1, and 1 + 1 = 10.
  Checking…
  - Counting Creatures 3 · teaches, check (Chip City)
  - Logic Gates 2 · teaches, practice, check (Binary lights, Adders)
  - Logic Gates 1 · practice (Half adder)
- Binary column addition
  Skill
  Add two binary numbers by hand in columns, carrying when 1 + 1 makes 10.
  Checking…
  - Logic Gates 2 · teaches
- Bits count values
  Idea
  n bits hold 2 to the n different values; eight bits make a byte.
  Checking…
  - Logic Gates 2 · introduces
- Encoding
  Idea
  Bits can stand for anything: letters, colors, sounds.
  Checking…
  - Zip and the Thinking Machines · introduces (Inside a chip)
Switches, logic gates and adders
Switch circuits, truth tables and gates, and the half, full and ripple-carry adders that let a computer add.
- Switch circuits
  Idea
  A switch controls whether a light is on.
  Checking…
  - Zip and the Thinking Machines · introduces (Inside a chip)
  - Logic Gates 2 · teaches (Switches)
  - Logic Gates 1 · teaches (A switch and a light)
- Transistors
  Idea
  A transistor is a switch flipped by electricity, made from silicon (sand).
  Checking…
  - Zip and the Thinking Machines · introduces, teaches (Inside a chip, Brain and chip)
  - Counting Creatures 1 · teaches (The computer counts in two)
- Truth tables
  Idea
  A truth table lists the output for every combination of inputs.
  Checking…
  - Logic Gates 2 · teaches (Logic gates)
  - Logic Gates 1 · teaches (NOT)
- NOT gate
  Skill
  Predict the output of NOT: it flips its input.
  Checking…
  - Logic Gates 1 · introduces, teaches, check (A switch and a light, NOT)
  - Logic Gates 2 · introduces, teaches, practice, check (Switches, Logic gates, Building with NAND)
- OR gate
  Skill
  Predict the output of OR: on when either input is on.
  Checking…
  - Logic Gates 2 · introduces, teaches, practice, check (Switches, Logic gates, Building with NAND)
  - Logic Gates 1 · teaches, check (OR, Mystery gates)
- AND gate
  Skill
  Predict the output of AND: on only when both inputs are on.
  Checking…
  - Logic Gates 2 · introduces, teaches, practice, check (Switches, Logic gates, Building with NAND)
  - Logic Gates 1 · teaches, check (AND, Mystery gates)
- XOR gate
  Skill
  Predict the output of XOR: on when exactly one input is on.
  Checking…
  - Logic Gates 2 · teaches, check (Logic gates)
  - Logic Gates 1 · teaches, check (XOR, Mystery gates)
- NAND gate
  Skill
  Predict the output of NAND: off only when both inputs are on.
  Checking…
  - Logic Gates 2 · teaches, practice, check (Logic gates, Building with NAND)
  - Logic Gates 1 · teaches, check (NAND, Mystery gates)
- NAND builds every gate
  Idea
  Every gate can be built from NAND alone: a jump to universality in logic.
  Checking…
  - Logic Gates 1 · introduces (The gate family)
  - Logic Gates 2 · introduces, teaches (How a computer adds, Building with NAND)
- Half adder
  Skill
  XOR gives the sum bit and AND gives the carry bit of two bits.
  Checking…
  - Counting Creatures 3 · introduces (Chip City)
  - Logic Gates 2 · teaches, practice, check (Adders)
  - Logic Gates 1 · teaches, check (Half adder)
- Full adder
  Skill
  Add two bits and a carry-in; predict the sum and the carry-out.
  Checking…
  - Logic Gates 2 · teaches, practice, check (Adders, How a computer adds)
  - Logic Gates 1 · teaches, check (Full adder)
- Ripple-carry adder
  Skill
  Chain full adders so the carry ripples along and multi-bit numbers add.
  Checking…
  - Logic Gates 2 · teaches, practice, check (Adders, How a computer adds)
  - Logic Gates 1 · teaches, practice (Ripple-carry adder)
- Overflow
  Idea
  A fixed number of bits can only hold so much; a carry out of the last place overflows.
  Checking…
  - Logic Gates 2 · introduces
- How a computer adds
  Idea
  Mindless switches wired as adders do arithmetic: the heart of a computer adding two numbers.
  Checking…
  - Logic Gates 2 · teaches (How a computer adds)
  - Logic Gates 1 · teaches (You built a computer's heart)
Computation and universality (BoI 6)
Algorithms, special-purpose versus universal machines, and the jump to universality in numerals, writing and computers.
- Algorithms
  Idea
  An algorithm is rules followed without understanding, like column addition, or Roman numerals, whose rules people carry out the way a computer runs a program (BoI 6).
  Checking…
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
- Special-purpose versus universal machines
  Idea
  A special-purpose machine does one job; a universal one can be set to any job (Difference versus Analytical Engine).
  Checking…
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
- Universal computer
  Idea
  One machine can follow any instructions, because programs are data (Babbage, Lovelace, Turing).
  Checking…
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
- Alphabet as a universal system
  Idea
  Phonetic letters can write any word, even words not yet invented.
  Checking…
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
- Positional numerals are universal
  Idea
  Positional digits in any base of two or more can write any number; tallies and Roman numerals cannot in practice.
  Checking…
  - Counting Creatures 3 · teaches, practice (Creature Lab)
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
- The jump to universality
  Idea
  One small improvement can give a system reach over every case of its kind (BoI 6).
  Checking…
  - Logic Gates 2 · introduces (How a computer adds)
  - Counting Creatures 3 · introduces (Creature Lab)
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - Beginning of Infinity episode 6: The Jump to Universality · teaches
- People as universal explainers
  Idea
  People can understand anything that can be understood (BoI 6 and 7).
  Checking…
  - Zip and the Thinking Machines · introduces (You can learn)
  - The Beginning of Infinity, chapter 6: The Jump to Universality · teaches
  - The Beginning of Infinity, chapter 7: Artificial Creativity · teaches
Algebra: inverses and extending numbers (Feynman 22)
Every operation and its inverse, logarithms, and extending the numbers so the rules keep working (Feynman vol. 1 ch. 22).
- Laws of arithmetic
  Idea
  The commutative, associative and distributive laws: the rules we keep.
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Inverse operations
  Idea
  Every operation has an inverse: subtraction, division, roots and logarithms.
  Checking…
  - Math Quiz · introduces (Math Quiz: Starter operation lists)
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Logarithms
  Idea
  A logarithm counts how many times to multiply; log base 2 of how many values you need is how many bits you need (256 values → 8 bits).
  Checking…
  - Math Quiz · introduces (Math Quiz: Starter operation lists)
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Extending the numbers
  Idea
  Keep the rules and extend the numbers: zero, negatives, fractions, irrationals (Feynman 22-3).
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches
- Logs from successive roots
  Idea
  Compute logarithms by taking successive square roots of 10 (Feynman 22-4, enrichment).
  Checking…
  - The Feynman Lectures on Physics, volume I, chapter 22: Algebra · teaches

Side branches

These grow out of the main path and can be explored at any time.

AI and learning machines (Zip)
Neurons, perceptrons and neural nets, training by correction, and AI as a helper that people direct.
- What AI is
  Idea
  AI is a computer program that can learn.
  Checking…
  - Zip and the Thinking Machines · introduces, teaches (Meet Zip)
  - The Beginning of Infinity, chapter 7: Artificial Creativity · introduces
- Neurons
  Idea
  Brains are made of neurons that talk to each other with zaps.
  Checking…
  - Zip and the Thinking Machines · introduces, teaches (Your brain, Brain and chip)
- Practice strengthens paths
  Idea
  Practice makes the paths between neurons stronger.
  Checking…
  - Zip and the Thinking Machines · teaches (Your brain)
- Perceptron
  Idea
  A pretend neuron adds up weighted inputs and fires past a threshold.
  Checking…
  - Zip and the Thinking Machines · teaches (Pretend neurons and nets)
- Neural nets
  Idea
  Perceptrons are arranged in rows; many rows make a deep net.
  Checking…
  - Zip and the Thinking Machines · teaches (Pretend neurons and nets, Teaching a net)
- Weights
  Idea
  Each connection has a strength, its weight, that sets how much gets through.
  Checking…
  - Zip and the Thinking Machines · teaches (Pretend neurons and nets)
- Training by correction
  Idea
  Guess, compare with the right answer, nudge the weights, and repeat.
  Checking…
  - Zip and the Thinking Machines · introduces, teaches (Teaching a net)
- Next-word models
  Idea
  Language models learn to guess the next word.
  Checking…
  - Zip and the Thinking Machines · teaches (Word guessers)
- AI helps, people decide
  Idea
  AI helps doctors, scientists, drivers and farmers, and people make the choices.
  Checking…
  - Zip and the Thinking Machines · introduces, teaches (AI helpers)
Building software with AI agents (Pipa)
Working with a coding agent: commits, repos, branches, worktrees, GitHub, going live, secrets and the builder loop.
- You are the boss of the agent
  Idea
  Code is instructions; the agent writes it and you decide.
  Checking…
  - Pipa's Code Quest · teaches, practice (1. You Are the Boss)
- Commits
  Idea
  Commits are save points with notes, so you can travel back in time.
  Checking…
  - Pipa's Code Quest · teaches, practice (2. Save Points)
- Repos and git
  Idea
  The repo is a treasure chest kept by git, which tracks what changed and what is unsaved.
  Checking…
  - Pipa's Code Quest · teaches, practice (3. The Treasure Chest)
- Branches
  Idea
  Grow a branch, check where you are, keep or toss it, and name it well.
  Checking…
  - Pipa's Code Quest · teaches, practice (4. Branches)
- Worktrees
  Idea
  Worktrees are many desks, so several agents can work at once.
  Checking…
  - Pipa's Code Quest · teaches, practice (5. Many Desks)
- GitHub
  Idea
  Push, pull, and review before merging.
  Checking…
  - Pipa's Code Quest · teaches, practice (6. The Cloud Home)
- Local versus live
  Idea
  Test on your own computer first, then launch; use a practice copy first.
  Checking…
  - Pipa's Code Quest · teaches, practice (7. My Computer or the Whole World)
- Secrets and keys
  Idea
  Know public from private, keep keys secret, and set rules for the agent.
  Checking…
  - Pipa's Code Quest · teaches, practice (8. Locks and Keys)
- The builder loop
  Skill
  Plan, build, check and ship, and recover when the agent gets it wrong.
  Checking…
  - Pipa's Code Quest · teaches, check (9. The Builder Loop)

Learning path

One-to-one counting

Number-word sequence

Cardinality

Counting on

Tally marks

Number as abstraction

Zero

Number line

Roman numerals

Bundling

Tens and ones

Positional notation

What a base is

Digits of a base

Read another base in base ten

Write a number in another base

Symbols versus length

Each place is base times the next

Addition

Turnaround facts

Make ten

Add 0 facts

Add 1 facts

Add 2 facts

Doubles

Tough 21

Hardest six (Sneaky Six)

Easy 52 addition facts

Addition fluency

Carrying

Column addition

Subtraction

Minus 0 facts

Minus 1 facts

Minus 2 facts

Halves

Tough rest subtraction

Sneaky 12

Easy 52 subtraction facts

Subtraction fluency

Negative results

Multiplication

Times 0 facts

Times 1 facts

Times 2 facts

Squares

Tough rest products

Easy 52 multiplication facts

Multiplication fluency

Division

Division undoes multiplication

Division facts

Remainders

Powers

Equal parts

Unit fractions

Numerator and denominator

Fractions as division

Equivalent fractions

Powers of two

Two states

Why two states

Binary is base two

Binary place values

Read binary

Make binary

Count up in binary

Add two bits

Binary column addition

Bits count values

Encoding

Switch circuits

Transistors

Truth tables

NOT gate

OR gate

AND gate

XOR gate

NAND gate