Mathematics: quantities, relationships, and models
Start with what numbers mean, then build toward arithmetic, fractions, symbols, and relationships.
GUIDE9 published lessons · Updated 2026-09-27
Before you begin
Start at lesson 1. No previous mathematics study is assumed. Read one idea at a time, work through the example on paper, then try the short exercise. Use a calculator when useful. Move on when you can explain the answer, not just repeat it. Fractions and symbols are introduced before equations.
See the pathway from novice to doctoral study
Working toward: Translate a simple quantitative question into an equation, solve it, and check what the result means.
Read each explanation, work through the example, and attempt the exercise before opening its answer. Published lessons are ready to study; unfinished roadmap topics remain planned.
Validation: Original calculation-based examples with explicit assumptions. These lessons do not report physical experiments.
1. Numbers describe quantities
Before this lesson: No prior subject knowledge. Start here.
A number answers a question such as “how many?” A quantity combines a number with what is being counted or measured. Three notebooks and three metres both use 3, but they describe different things. A unit is an agreed amount used for measurement, such as a metre for length. Start by saying what each number means before doing anything with it.
Worked example
There are 4 notebooks on a shelf and 2 more on a desk. Counting all the notebooks gives 6 notebooks. A notebook that is 20 cm tall has a measured height; 20 is the number and cm is the unit.
Your turn
In “a ribbon is 8 centimetres long,” identify the number, the unit, and what is measured.
Show answer and reasoning
Number: 8. Unit: centimetres. Measured property: the ribbon’s length. “8” alone leaves out what the measurement describes.
Watch for: Numbers with different meanings cannot always be combined: adding 3 metres to 2 seconds does not produce a meaningful length or time.
Link to this lesson2. Adding, subtracting, multiplying, dividing
Before this lesson: Read 1. Numbers describe quantities first; return to its example if the new terms are unfamiliar.
Addition combines quantities of the same kind. Subtraction finds a difference or what remains. Multiplication can count equal groups. Division can share a total into equal groups or ask how many groups fit. The symbols +, −, ×, and ÷ name those operations. The equals sign = says that the expressions on either side have the same value.
Worked example
Three trays each contain 4 counters. Repeated addition is 4 + 4 + 4 = 12; multiplication writes this as 3 × 4 = 12. Sharing 12 counters equally among 3 trays gives 12 ÷ 3 = 4 per tray.
Your turn
Four boxes each hold 3 pencils. How many pencils are there? If 5 are removed, how many remain?
Show answer and reasoning
4 × 3 = 12 pencils. Then 12 − 5 = 7 pencils remain. Multiplication counts the equal groups; subtraction removes the five pencils.
Watch for: Division by zero is undefined. Equal-group division also requires knowing whether you are finding the group size or the number of groups.
Link to this lesson3. Parts of a whole
Before this lesson: Read 2. Adding, subtracting, multiplying, dividing first; return to its example if the new terms are unfamiliar.
A fraction describes a number using division. In 3/4, the bottom number 4 says the whole is divided into four equal parts; the top number 3 counts three of those parts. A decimal uses place values: 0.1 is one tenth and 0.01 is one hundredth. Fractions and decimals can name the same number: 1/2 = 0.5.
Worked example
Divide one metre into four equal lengths. Each is 1/4 m = 0.25 m. Three such lengths total 3/4 m = 0.75 m. The whole and the size of its equal parts stay fixed.
Your turn
A strip is divided into 8 equal parts. You use 4 parts. What fraction of the strip did you use? Express it as a decimal.
Show answer and reasoning
4/8 = 1/2 = 0.5. Four of eight equal parts is half the strip. Dividing both numerator and denominator by 4 gives 1/2.
Watch for: Parts must be equal. Half of a small strip and half of a large strip need not have the same length.
Link to this lesson4. Zero, negatives, and the number line
Before this lesson: Read 3. Parts of a whole first; return to its example if the new terms are unfamiliar.
A number line orders numbers from smaller on the left to larger on the right. Zero is a reference point. Positive numbers lie to its right and negative numbers to its left. A negative sign can be part of a number, such as −3; a subtraction sign describes an operation, such as 5 − 3. The same symbol has these two related jobs.
Worked example
Start at −2. Adding 5 moves five steps right: −1, 0, 1, 2, 3. Therefore −2 + 5 = 3. Subtracting 4 from 3 moves four steps left and gives −1.
Your turn
Start at −3 on a number line and add 7. Where do you finish?
Show answer and reasoning
At 4. Three steps reach zero; four more steps reach 4, so −3 + 7 = 4.
Watch for: The number −8 is smaller than −2, even though 8 is larger than 2. Position on the number line determines order.
Link to this lesson5. Reading brackets, powers, and letters
Before this lesson: Read 4. Zero, negatives, and the number line first; return to its example if the new terms are unfamiliar.
Brackets group an expression to be evaluated together. Multiplication and division come before addition and subtraction; operations at the same priority are read left to right. A power such as 3² means 3 × 3. A letter can stand for a number: if x = 4, then x + 2 = 6. Juxtaposition such as 3x means multiplication, 3 × x.
Worked example
2 + 3 × 4 = 14 because multiplication comes first. (2 + 3) × 4 = 20 because the brackets are evaluated first. With x = 5, 2x + 1 = 2 × 5 + 1 = 11.
Your turn
If x = 3, evaluate 2(x + 4).
Show answer and reasoning
First substitute 3: 2 × (3 + 4). Evaluate the brackets: 2 × 7. The result is 14.
Watch for: x² means x × x, not 2 × x. Powers and coefficients do different jobs.
Link to this lesson6. Ratios and unit conversions
Before this lesson: Read 5. Reading brackets, powers, and letters first; return to its example if the new terms are unfamiliar.
A ratio compares two quantities by division. A conversion factor expresses the same quantity in two units, so multiplying by it changes the unit without changing the physical amount. Write units in the calculation and cancel matching units before evaluating the numbers. A percentage is a ratio expressed per hundred.
Worked example
A model uses 1 cm for every 2 m. A 7.5 cm line represents 15 m. Separately, 90 km/h × 1000 m/km ÷ 3600 s/h = 25 m/s.
Your turn
Convert 72 km/h to m/s and explain the unit cancellation.
Show answer and reasoning
72 × 1000 ÷ 3600 = 20 m/s. Kilometres cancel against kilometres in the denominator; hours cancel against hours in the numerator. Metres per second remain.
Watch for: A scale ratio written 1:200 uses the same units on both sides. Do not confuse 1 cm : 2 m with 1:2.
Link to this lesson7. Equations preserve a balance
Before this lesson: Read 6. Ratios and unit conversions first; return to its example if the new terms are unfamiliar.
An equation states that two expressions have the same value. Applying the same reversible operation to both sides preserves its solutions. Undo addition or subtraction, then multiplication or division, to isolate a variable. Substitution into the original equation checks the result; a plausible-looking number is not enough.
Worked example
A fictional hire charge is C = 12 + 4h. For a total of 32 units: 32 − 12 = 4h, so h = 5 hours. Check: 12 + 4 × 5 = 32.
Your turn
Solve 3x + 7 = 28, then check the result.
Show answer and reasoning
Subtract 7 from both sides: 3x = 21. Divide by 3: x = 7. Substitution gives 3 × 7 + 7 = 28.
Watch for: Dividing by an expression that might be zero can discard solutions or create an invalid step.
Link to this lesson8. Functions, slope, and intercept
Before this lesson: Read 7. Equations preserve a balance first; return to its example if the new terms are unfamiliar.
A function assigns exactly one output to each allowed input. In a linear function y = mx + b, m is the change in output per unit change in input, and b is the output at input zero. A model also needs a domain: a formula may produce numbers outside the range where its assumptions are useful.
Worked example
For a model reservoir volume V = 80 − 5t in litres, with t in minutes, the slope is −5 L/min and the intercept is 80 L. The constant-drain model is valid only until the reservoir empties at 16 minutes.
Your turn
For V = 60 − 3t litres, find V at t = 8 minutes and the time when the reservoir empties.
Show answer and reasoning
V(8) = 60 − 24 = 36 L. Set V = 0: 3t = 60, so t = 20 minutes. A negative volume after 20 minutes would indicate use outside the model’s domain.
Watch for: A straight line fitted to observations does not by itself establish cause or justify unlimited extrapolation.
Link to this lesson9. Probability and independent events
Before this lesson: Read 8. Functions, slope, and intercept first; return to its example if the new terms are unfamiliar.
For a finite set of equally likely outcomes, an event’s probability is its number of favourable outcomes divided by the total number. Independent events do not change one another’s probabilities. For two independent events, multiply their probabilities to find the chance that both occur. Independence is an assumption to justify, not a default for every pair of events.
Worked example
For two independent tosses of a fair coin, HH, HT, TH, and TT are equally likely. The probability of two heads is 1/2 × 1/2 = 1/4; exactly one head has probability 2/4 = 1/2.
Your turn
A fair six-sided die is rolled twice independently. What is the probability that both rolls are six?
Show answer and reasoning
Each roll has probability 1/6 of six. Independence gives (1/6) × (1/6) = 1/36. There is one favourable ordered pair among 36 equally likely pairs.
Watch for: A short run of outcomes need not match the theoretical proportions. Previous fair independent tosses do not make the next toss due for a particular result.
Link to this lessonFrom first steps to doctoral study
Start slowly. Each stage builds on the previous one; later branches have their own prerequisites. The foundation lessons are available now. The remaining curriculum is a plan for future lessons, not a complete course.
1. First steps · FOUNDATION LESSONS AVAILABLE
Before you start: No previous mathematics study.
Quantities, arithmetic, fractions, decimals, signed numbers, notation, ratios, and introductory equations.
Ready to continue when you can: Explain a calculation in words and check its answer using a second method.
Start the foundation lessons →2. School foundations · PLANNED
Before you start: Comfort with first-step arithmetic and symbols.
Percentages, powers and roots, algebra, inequalities, geometry, coordinate graphs, trigonometry, and introductory statistics.
Ready to continue when you can: Solve unfamiliar multistep problems and state units, assumptions, and domain restrictions.
3. Undergraduate core · PLANNED
Before you start: Algebra, functions, geometry, and trigonometry.
Single- and multivariable calculus, linear algebra, discrete mathematics, proof methods, differential equations, and probability.
Ready to continue when you can: Write elementary proofs and connect symbolic, geometric, and numerical solutions.
4. Advanced undergraduate study · PLANNED
Before you start: Proof methods, calculus, and linear algebra.
Real analysis, abstract algebra, complex analysis, topology, numerical analysis, and mathematical statistics.
Ready to continue when you can: Explain why a theorem’s hypotheses matter and construct counterexamples when they fail.
5. Graduate foundations · PLANNED
Before you start: Advanced undergraduate study in the chosen branch.
Measure theory, functional analysis, graduate algebra and topology, advanced probability, and partial differential equations as branch-specific options.
Ready to continue when you can: Solve sustained proof problems and present a rigorous reading of an advanced text.
6. Research specialization · PLANNED
Before you start: Relevant graduate prerequisites; no requirement to study every branch.
Choose a branch such as analysis, algebra, geometry, topology, probability, or applied mathematics; follow its literature and methods.
Ready to continue when you can: Reproduce a published argument or computational result, identify assumptions, and compare competing approaches.
7. Doctoral research practice · PLANNED
Before you start: A focused specialization and substantial proof or modelling experience.
Research questions, literature synthesis, conjectures, original proofs or models, seminars, reproducible computation, and thesis writing.
Ready to continue when you can: Formulate a tractable question, defend a method, and communicate an original contribution under expert critique.
Doctoral-level work involves specialization, original research, and expert critique. This pathway supports that preparation; completing site lessons does not confer a degree or establish research competence.
References
Original AUWEN lessons, with upstream documentation for further study and version checks.
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