The School of Numbers & Logic · mathematics, pure and applied
Arithmetic & Everyday Math
Whole numbers, fractions, percentages, and estimation — the arithmetic a person uses daily, done confidently with or without a calculator.
The four operations on whole numbers, built from place value so the algorithms make sense instead of merely working.
Syllabus · 4 units · ~18 hours
Unit I — Place Value
Reading and writing numbers to the millions · Comparing and ordering whole numbers · Rounding to a chosen place · Why our number system uses columns at all
Unit II — Addition & Subtraction
Regrouping across columns · Estimating a sum before computing it · Subtraction as distance and as take-away · Checking answers by the inverse operation
Unit III — Multiplication
Multiplication as repeated addition and as area · The times table, learned by structure not chant · The standard algorithm, one partial product at a time · Multiplying by tens, hundreds, and thousands
Unit IV — Division
Sharing and grouping: two pictures of division · Long division, step by honest step · Remainders and what the problem wants done with them · Estimating quotients to catch mistakes
Part-of-a-whole in its three costumes, with the conversions and operations that let you move freely among them.
Syllabus · 5 units · ~22 hours
Unit I — What a Fraction Names
Fractions as parts of a whole and points on a line · Equivalent fractions and why they exist · Comparing fractions with common denominators and benchmarks
Unit II — Adding & Subtracting Fractions
Same-denominator sums as counting pieces · Finding a common denominator without superstition · Mixed numbers and improper fractions
Unit III — Multiplying & Dividing Fractions
A fraction of a fraction, drawn before computed · Why invert-and-multiply is legal · Fraction word problems from cooking and carpentry
Unit IV — Decimals
Decimal place value as fractions of ten · Operations with decimals and where the point goes · Converting between fractions and decimals · Repeating decimals and what they conceal
Unit V — Percent
Percent as a fraction of one hundred · Finding the part, the whole, or the rate · Percent change, markups, and discounts · Percentage points versus percent — a common trap
Techniques for computing in your head and knowing roughly what an answer must be before anyone reaches for a device.
Syllabus · 3 units · ~10 hours
Unit I — Rearranging the Work
Compatible numbers and friendly pairs · Left-to-right addition · Compensation: adjust, compute, correct
Unit II — Multiplication Shortcuts
Doubling and halving · Multiplying by 5, 9, 11, and 25 without effort · Squaring numbers ending in five · Breaking a factor apart with the distributive law
Unit III — Estimation as a Habit
Rounding to one significant figure · Bracketing an answer between two easy bounds · Sanity checks: units, magnitude, parity · Fermi problems: estimating what nobody has counted
Comparing quantities by division — the reasoning behind unit prices, speed, scale drawings, and doubled recipes.
Syllabus · 4 units · ~14 hours
Unit I — Ratios
Ratio as a comparison by division · Equivalent ratios and ratio tables · Part-to-part versus part-to-whole
Unit II — Rates
Unit rates and unit prices · Speed, flow, and other everyday rates · Comparing deals with a unit rate
Unit III — Proportional Relationships
Testing whether two quantities are proportional · The constant of proportionality in tables and graphs · Solving proportions — and why cross-multiplication works · When proportion fails: fixed costs and other spoilers
Unit IV — Scale
Scale drawings and maps · Scaling recipes up and down · Similar figures as proportion in geometry
Length, area, volume, and mass in both major systems, with dimensional analysis as the one conversion method worth learning.
Syllabus · 3 units · ~8 hours
Unit I — Measuring at All
What a unit is and why standards matter · Reading rulers, scales, and measuring cups · Precision: how close is close enough
Unit II — The Metric System
The design of the metric system: powers of ten · Milli, centi, kilo — moving the decimal point · Liters, grams, meters and how they interlock
Unit III — Converting Between Systems
Dimensional analysis: multiplying by clever forms of one · Chained conversions without memorized shortcuts · Area and volume conversions — where squaring bites · Temperature scales as a different kind of conversion