Part I
Reentry
- 014 demonstrations
You Already Do Mathematics
Sorting becomes mathematics the moment the rule is public enough for someone else to test.
- 024 demonstrations
Why Your Mind Goes Blank
The arithmetic stays the same. What changes is where the middle of the work can live.
- 034 demonstrations
The Most Useful Wrong Answer
A counterexample points at the sentence that promised too much, never at the person who wrote it.
- 044 demonstrations
Patterns Are the Point
Name the change, name what counts as the same, then check every feature that standard needs.
Part II
Quantity
- 054 demonstrations
How Big Is It?
Read size from place first, then state what an estimate changed before you check it exactly.
- 064 demonstrations
Addition Moves, Subtraction Measures
A sign belongs to a number, a move has a direction, and every subtraction can be checked by addition.
- 074 demonstrations
Multiplication Scales, Division Asks
Every product has two roles in it, and every division asks which role is missing.
- 084 demonstrations
Whole Numbers Have Architecture
Ask what a number is made of, and its factors, powers and square roots answer.
Part III
Units and fractions
- 094 demonstrations
Fractions Are Places
A fraction names a place on the number line: count equal steps from zero.
- 104 demonstrations
Common Units Explain the Rules
Add counts only when they count the same size of part.
- 114 demonstrations
When Multiplication Shrinks
A factor below 1 keeps only part of the amount, and dividing by a small group counts many groups.
- 124 demonstrations
Ratios Run the World
Name what is compared, what counts as one, and which base a percent is of.
Part IV
Relationships
- 134 demonstrations
A Variable Is a Name
A letter saves a place for any value the context allows, and the grouping tells you what to do with it.
- 144 demonstrations
Equality Is a Promise
An equals sign says two sides have the same value. Solving keeps that promise one equal change at a time.
- 154 demonstrations
A Graph Is a Story
A graph reports two quantities at once, and its labels, domain and scale are part of the story.
- 164 demonstrations
Functions Are Relationships
A function is a dependable assignment: each allowed input gets exactly one output.
Part V
Shape and motion
- 174 demonstrations
Measurement Is a Model
A number without a unit is not wrong. It is unfinished.
- 184 demonstrations
Triangles Know the Distance
Find the right angle first, then let the squares of the legs tell you the diagonal.
- 194 demonstrations
Why Circles Keep Saying Pi
Every ideal circle answers the same question: how many diameters fit around the boundary?
- 204 demonstrations
How a Circle Becomes a Wave
A wave can begin as a point going around in circles.
Part VI
Uncertainty and data
- 214 demonstrations
Random Is Not the Same as Equal
Random does not mean every pattern takes turns.
- 224 demonstrations
The Base Rate You Forgot
A percentage is only half a sentence until you say which group it is a percentage of.
- 234 demonstrations
Build an Unfair Game
A game can smile at you while its expected value does the accounting.
- 244 demonstrations
The Average Is Hiding Something
An average is a useful witness. It is not the whole story.
Part VII
Nearness and infinity
- 254 demonstrations
Patterns That Grow
A pattern is not a prophecy until someone says how it continues.
- 264 demonstrations
A Hotel With No Vacancy
A full hotel can still make room, provided its rooms are labels rather than walls.
- 274 demonstrations
Getting Close Counts
Getting close is not the same as giving up before you arrive.
Part VIII
Change and accumulation
- 284 demonstrations
What Is Happening Right Now?
An average can tell what happened over an interval. It cannot tell what was happening at one moment.
- 294 demonstrations
Rules of Change, Unmasked
A derivative rule is a receipt for work someone has already checked.
- 304 demonstrations
Where Is the Best Point?
The best point is never best until the question says best for what, and over which inputs.
- 314 demonstrations
How Much Has Accumulated?
The gauge reports the rate. The running total keeps the receipts.
- 324 demonstrations
The Speedometer Meets the Odometer
One accumulated quantity keeps direction. The other records every bit of movement.
Notebooks on GitHub
Every chapter has a Jupyter notebook that reproduces its worked examples and figures, so you can run the calculations yourself.
github.com/KarpelesPublishing/math-class-didnt-show-its-work