Foundations — Equivalent fractions, simplifying fractions

Before you can trust a single line of the parent note, you need to own every symbol it throws at you. Below is every mark, word and idea it uses — built from nothing, each one leaning on the one before it.

1. The counting numbers — the raw material

The picture: dots in a row. Three dots = the number $3$. There are no half-dots yet; everything is a full, unbroken thing.

Why the topic needs it: a fraction is built from two whole numbers stacked on each other. If you can't count equal pieces, you can't say "how many I took" or "into how many I cut". Whole numbers are the alphabet; fractions are the words.

2. "A whole" and cutting it into equal parts

Look at the figure. On the left the bar is split into 2 equal parts — the red part is a true half. On the right the bar is split into 2 unequal parts — the red part looks like "1 out of 2" but it is clearly not half the chocolate.

Why the topic needs it: the whole definition of a fraction says "cut into $b$ equal parts". Drop the word "equal" and every rule on the parent page collapses.

3. The fraction bar and the two numbers — $\frac{a}{b}$

The picture: the denominator sets how finely the whole is sliced (bigger $b$ = thinner slices), and the numerator (in red) says how many of those slices you're holding.

Why "top over bottom" and not the reverse? You must know the total number of pieces before "how many I took" means anything. The bottom builds the stage; the top acts on it. Swapping them changes the amount: $\frac{2}{3}$ (two of three parts) is a very different amount from $\frac{3}{2}$ (three of two parts).

4. Why $b\ne 0$ — the forbidden denominator

The picture: try to cut a bar into $0$ equal pieces. There is no picture — you cannot draw it. "How big is each of zero pieces?" is a question with no answer.

Why the topic needs it: a fraction stands for the number $x$ that fills the sentence "$b\times x=a$". If $b=0$ that says "$0\times x=a$", which no number $x$ can satisfy (unless $a=0$, where every number works). Either way there's no single answer, so we forbid $b=0$. This is what "division by zero is undefined" means.

5. Multiplication and division as picture-actions

The picture: $\times k$ re-cuts each slice into $k$ thinner slices; $÷k$ glues $k$ slices back into one fatter slice. Crucially, doing this to both numerator and denominator at once does not change how much chocolate you're describing — you only re-labelled the slices.

Why the topic needs it: this is the engine of equivalent fractions. Adding does not have this "amount-preserving" property (that's the classic mistake on the parent page); only $\times$ and $÷$ do.

6. The number $1$ in disguise — $\frac{k}{k}$

The picture: $\frac{4}{4}$ = four quarters = one full bar.

Why the topic needs it: multiplying by $1$ never changes a value. Since $\frac{k}{k}=1$, multiplying a fraction by $\frac{k}{k}$ leaves the amount untouched while it changes the numbers. That single fact is the whole proof that $\frac{a}{b}=\frac{ak}{bk}$.

7. Factor, common factor, and the GCD

The picture: the factors of $18$ and $24$ are two circles; where they overlap sit the common factors $1,2,3,6$. The largest in the overlap — in red — is $6=\gcd (18,24)$.

Why the topic needs it: simplifying means removing all shared factors. Divide top and bottom by the GCD and you strip out the biggest chunk in one move, landing straight in lowest terms. See Factors, multiples and GCD/HCF and Prime factorisation for how to find the GCD reliably.

8. "Lowest terms" and the symbol $\gcd (a,b)=1$

The picture: the two overlap-circles from figure 3 now touch at only the number $1$. Nothing left to strip out; the slices are as big as they can be while still naming the same amount.

Why the topic needs it: it's the finish line for simplifying. Stopping before $\gcd =1$ (like leaving $\frac{9}{12}$) is the "stopped too early" mistake.

9. The double-arrow $⟺$ and cross-multiplication

Why the topic needs it: it's a fast, division-free way to check equivalence, and it's the exact same tool used in Ratios and proportion.

How these feed the topic

Equipment checklist

Test yourself — reveal only after you've answered out loud.

Connections

• Parent: Equivalent & simplifying fractions — everything here is the toolkit for that page.
• Factors, multiples and GCD/HCF — where factors and the GCD come from.
• Prime factorisation — the reliable way to find a GCD.
• Adding and subtracting fractions — uses equivalent fractions to match denominators.
• Ratios and proportion — cross-multiplication reappears there.
• Decimals and percentages — other costumes for the same amount.
• Number Systems – rational numbers — a fraction in lowest terms is the canonical rational number.