This page is the toolbox. Before you can follow the parent topic, every squiggle it uses must mean something you can see. So we start from a blank slate and earn each symbol one at a time.
Before any fraction, there is the whole — one complete thing. One pizza. One chocolate bar. One length of string. We draw it as a full bar or a full circle.
Figure s01 — What to see: the same "one whole" drawn three ways (an orange bar, a magenta pizza, and the gap from 0 to 1 on the number line). Notice all three are one complete unit before any cutting happens. Every fraction on this page lives inside one of these wholes.
Why we need it: a fraction is always a fraction of something. If you forget the whole, "43" is meaningless — three-quarters of what? The whole is the silent partner in every fraction.
The picture: take the whole bar and slice it into b equal strips. The denominator b is the number of slices in the whole.
Why the topic needs it: you can only add pieces that are the same size, and "same size" means "same denominator." Every "common denominator" step is just making the slice-sizes agree. This idea is the heart of Fractions - meaning and equivalent fractions.
The picture: the bar is cut into b strips (denominator). Shade a of them. That shaded amount is the fraction.
Figure s02 — What to see: one bar cut into 5 equal slices (denominator 5) with 2 shaded, and a thin bar with 1 shaded. The denominator fixes the slice-size; the numerator just counts shaded slices. Because both bars use fifths, combining them is pure counting: 2 slices +1 slice =3 slices =53.
Why the topic needs it: when denominators already match, adding fractions is just counting shaded pieces together — you add numerators and the denominator (piece-size) stays fixed.
Why the topic needs it: in Section 3 of the parent (division), the "complex fraction" dcba only makes sense if you already know the bar means divide.
The picture: the denominator is "how many pieces make one whole." Zero pieces can never make a whole — the instruction "cut the pizza into 0 equal slices" is impossible. So 0a has no meaning.
Why the topic needs it: when you flip a fraction in division (Section 3), you must be sure you never flip d0 into 0d. Guarding b=0 keeps every step legal. (Note b0=0 is fine — it is the bottom that must never be 0.)
The picture: on the number line, all three land at the same point to the left of 0. Because we agreed b>0 (Section 1), the tidy standard form keeps the denominator positive and puts the sign out front: −ba.
Why the topic needs it: when the parent computes things like bdad−bc, the top can come out negative. Knowing b−a=−ba lets you read that answer without panic. Negative fractions live inside the Rational numbers.
The picture: take each shaded piece and cut it into k smaller pieces. You now have k times as many shaded pieces (a⋅k), but the whole also has k times as many pieces (b⋅k). Same shaded area — just finer cuts. Reading the picture backwards (gluing k small pieces back into one) is the cancelling direction.
Figure s03 — What to see: the violet half-bar on the left (21) has each piece re-cut into 3, giving 63 on the right — exactly the same shaded area. Read left-to-right you multiplied top and bottom by 3; read right-to-left you cancelled the common factor 3. Both directions leave the amount of pie untouched.
Why the topic needs it: this is the ONE fact behind all four operations. Common denominators (multiply direction), cancelling before multiplying (divide direction), and "flip and multiply" are all this rule in disguise. It is developed fully in Fractions - meaning and equivalent fractions.
The picture:31 and 41 have mismatched slices. Re-cut both bars into 12 slices (using the golden rule). Now 31=124 and 41=123 — same size, ready to combine.
Why 12? Because 12 is a multiple of both 3 and 4. The smallest such number is the Least Common Multiple, and finding it is the whole job of LCM and HCF. Using the LCM instead of b×d keeps the numbers small.
The picture: if dc shrinks something, its reciprocal cd grows it back to full size — they undo each other, landing on the whole (1).
Why the topic needs it: division "flip and multiply" works because the reciprocal turns the divisor into 1, cancelling it out. This is the subject of Reciprocals and multiplicative inverse. (Note: only nonzero fractions have reciprocals — b0=0 has none, since nothing times 0 gives 1.)
Why the topic needs it: the parent's summary ba±dc=bdad±bc packs addition and subtraction into one line using ±. Knowing the signs (and having the multiply/divide rules named) lets you read every formula on the page.
