Lesson 04 · Wu, §13.1
2/3, 4/6, 6/9 … these aren’t different fractions that happen to be “equal.” They are one point on the number line, written with finer and finer subdivisions. Once you see that, there is only one idea to learn here — not two rules to memorize.
Drag the slider to set c. Watch two things at once: the subdivisions of the unit segment get finer as c climbs, and the dot marking 2/3 never moves. Its position was fixed once, at the start — the slider only changes how the line is cut, never where the point sits.
Wu’s own chain (p. 204) — click any term to jump the slider there:
Wu states a single theorem. Reading it left to right builds bigger, finer names for the same fraction. Reading the identical equation right to left is what students call “reducing.” It is not a second procedure — it is the same equality, run backwards.
Theorem 13.1 · H. Wu
“For any fraction m/n and any nonzero whole number c,”
mn = cmcn
— Understanding Numbers in Elementary School Mathematics, §13.1, p. 204
Corollary 13.1.1 · H. Wu
“For any fraction k/l and any nonzero whole number c that divides both k and l,”
kl = k÷cl÷c
— §13.1, p. 205 — the Cancellation Law
Building up only asks you to pick a c. Reducing asks you to find one — the common divisor hiding inside both numerator and denominator. You can eyeball that 2 and 3 share no factor, or that 4 and 6 share a 2. You cannot eyeball it for Wu’s harder example, 253/161 (p. 205) — same law, but the reverse direction hides its input. That gap — knowing the law but not being able to find c by eye — is exactly the problem the Euclidean algorithm exists to solve, and it’s where this series goes next.
On relative size · note
Two fractions are equal exactly when they name the same point on the number line — no matter how the numerator and denominator are written. That is the whole content of Theorem 13.1: it doesn’t just say the two fractions are “the same size,” it says why, by pointing at the one dot both names refer to.
The usual approach teaches “simplifying” and “finding common denominators” as two separate skills, each with its own steps to memorize. Here there is one theorem, and the slider makes both directions the same motion: drag right, you’re building up; drag left, you’re reducing. What a student keeps isn’t a procedure — it’s the image of a dot that refuses to move no matter how finely the line gets cut.