Fractions Explained Simply: Examples and a Quiz

Priya was ten, in my second year of teaching fifth-grade math in Chicago, and she told me she'd take 1/8 of the pizza over 1/4 "because eight is more than four." She was right about the numbers and wrong about the pizza. It remains the most reasonable wrong answer I have ever heard, and it contains most of what a child needs to sort out about fractions.

Fractions explained simply: a fraction is a fair share. The denominator (bottom number) says how many equal parts the whole is cut into, and the numerator (top number) says how many of those parts you have. In 3/4, a whole is cut into four equal parts and you hold three. Bigger denominator, smaller parts.

Fractions explained simply: the bottom number is the name

Before any writing, cut something. A chocolate bar, or a strip of paper. Cut it into four equal parts and each one is a quarter. Take three and you have three quarters: 3/4.

The detail that matters most is that the denominator works like a name, the way "apples" does in "three apples". The numerator counts. Suddenly the rules stop looking arbitrary. Three quarters plus two quarters make five quarters, exactly as three apples plus two apples make five apples. More cuts mean thinner pieces, so 1/5 is smaller than 1/3: the same bar is shared among five people instead of three. And a fraction can go past 1, because five quarters is a whole bar and a quarter left over.

One rule has no exceptions: the parts must be equal. Four uneven chunks are not quarters, however many lines you drew. Children who sketch quickly skip this, and I stop them every time.

To remember which number is which, the oldest trick still works. The denominator sits Down below. The numerator is the Number you count with.

Why 1/5 looks bigger than 1/3

Since kindergarten, a child has relied on one rule that never failed: bigger digits, bigger number. 7 beats 3, 50 beats 5. Then fractions arrive, two whole numbers stacked on top of each other, and the rule gets applied to each one separately. Xiaoyan Ni and Xinlin Zhou (2005, Educational Psychologist) called this the whole number bias: doi.org/10.1207/s15326985ep4001_3. It tells you nothing about how smart the child is. A rule worked for three years, so the child keeps using it.

The stakes reach well past next week's quiz. Robert Siegler and colleagues (2012, Psychological Science) analysed large longitudinal samples from the US and the UK. Children who knew more about fractions and division at age 10 did better in algebra and in math overall in high school five or six years later, even after the researchers accounted for other math skills, IQ, working memory, and family income and education: doi.org/10.1177/0956797612440101. The US National Mathematics Advisory Panel (2008) had already singled out fractions as a critical foundation for algebra in its final report.

That is correlation. It does not prove that extra fraction practice at ten will lift a GCSE or SAT grade at sixteen. Still, I think of fractions as a toll booth. A child who gets through without understanding pays later, in algebra, with interest.

Where fractions land in school (England and the US)

The idea is the same on both sides of the Atlantic, but the calendar differs. In England the national curriculum starts with halves and quarters in Year 1, brings in equivalent fractions and adding same-denominator fractions across Years 3 and 4, and reaches different denominators in Year 6 (DfE programme of study). In the US, Common Core spreads it over three grades: unit fractions and the number line in 3rd grade, equivalent fractions and like denominators in 4th, unlike denominators and multiplying fractions in 5th (standards 3.NF to 5.NF).

So a Year 4 pupil and a 3rd grader, both around 8 or 9, are working on similar ground. Our guide to 4th grade Common Core math goes through the fraction standards one by one, and the Year 6 SATs revision plan shows where fractions turn up in the reasoning papers.

Three examples, from easiest to most useful

Example 1: comparing two fractions

Which is bigger, 2/3 or 2/5? You take two pieces either way, but thirds are bigger than fifths. So 2/3 wins.

Trickier: 3/4 or 5/8? The parts have different names, so give them the same one. Cut each quarter in half and a quarter becomes two eighths, which means 3/4 = 6/8. Six eighths beat five eighths, so 3/4 is bigger. Two strips of paper of the same length, one folded in four and one in eight, show it in ten seconds.

Example 2: a fraction of a quantity

Maya has 18 trading cards and gives 2/3 of them to her brother. How many does she give away?

The denominator says how many equal groups: 18 ÷ 3 = 6 cards in each group. The numerator says how many groups she hands over: 2 × 6 = 12. She gives away 12 and keeps 6, which is one third of 18, as it should be. Divide first, multiply second. The other order works too, but the numbers grow for no reason.

If a child is stuck, draw a bar for the 18 cards, split it into three, and shade two parts. English mastery classrooms call this a bar model, and it is the most useful picture I know for these problems.

Example 3: adding in a recipe

A recipe asks for 3/4 pint of milk for the batter and 1/2 pint for the sauce. How much milk in total?

The parts don't share a name yet, so give them one: a half is two quarters, so 1/2 = 2/4. Now 3/4 + 2/4 = 5/4 pints, which is a pint and a quarter. Different denominators belong to Year 6 or 5th grade, so a younger child won't meet this one yet. I kept it because a kitchen is where most people first run into it.

The mistake that comes back every year

1/2 + 1/3 = 2/5. I have seen that line on more papers than I could count. The student adds the tops and adds the bottoms as if these were whole numbers, and the bias strikes again.

A sanity check rescues almost everybody: try 1/2 + 1/2. Add the bottoms too and you get 2/4, which is a half. Two halves making one half? No. Two halves make a whole. Whenever an answer contradicts the picture of the share, something went wrong.

Its cousin is 3/8 + 2/8 = 5/16. "Eighths" is only a name, so it stays put.

A dad in Bristol messaged me through EduBoost last January. His Year 5 daughter could recite every rule and kept losing marks on "find 3/5 of 40", because she divided 40 by 3, the top number. I asked him to stop explaining rules and to draw: a bar for 40, cut into five equal pieces, three of them shaded. A couple of weeks later he wrote back to say she was now sketching the bar on her own, even when the teacher hadn't asked for it. I can't promise it works for every child. When a child has the rule and lacks the picture, though, I would put the picture first.

My position, and I will defend it: don't teach cross-multiplying, the "butterfly" or any other shortcut for fractions until the child can draw the share. A shortcut with nothing under it collapses the first time the question looks different.

Five-question quiz

1. A pizza is cut into 8 equal slices and you eat 3. What fraction did you eat?
2. Which is bigger, 1/4 or 1/6?
3. What is 3/4 of 24?
4. Calculate 2/7 + 3/7.
5. A student writes 1/2 + 1/4 = 2/6. Where is the mistake, and what is the right answer?

Answers: 1) 3/8. 2) 1/4, because quarters are bigger than sixths. 3) 18 (24 ÷ 4 = 6, then 3 × 6 = 18). 4) 5/7, since the denominator doesn't change. 5) They added the tops and the bottoms. Write 1/2 as 2/4, then 2/4 + 1/4 = 3/4. The answer 2/6 equals one third, which is smaller than 1/2, and adding something can't shrink a number.

If two answers wobble, get the paper strips out again and redo the examples by hand before moving on to exercises. A good test is to have your child explain, out loud, to a younger sibling or a stuffed toy, why you can't add the bottoms. The Feynman technique makes the gap show up in about a minute. To make fractions stick, spaced repetition beats a big cram: ten minutes every few days does more than an hour the night before, and our guide to spacing your revision has the schedules.

For practice at your own pace, the EduBoost AI tutor builds harder variations and goes back over each mistake. Families in England can start with Year 5 maths tutoring, and families in the US with 4th grade math tutoring. If you'd rather work with a person, take a look at our online tutoring options. And if homework time is the real battleground, our practical guide to helping with homework was written for exactly that.

Back to Priya

Priya got there once we cut a paper circle into quarters and eighths and laid the pieces side by side. "Oh," she said. "I want the quarter." Fractions explained simply always come back to that picture: one whole, cut into fair pieces. So next time your child gets a fraction wrong, what do you reach for first, the rulebook or the scissors?