The order to teach fractions in, and why that order matters
Teach what a fraction *is* before any operation on one, then comparing, then adding and subtracting, then multiplying and dividing, then decimals, percents and ratios. Most fraction trouble comes from starting at step three.
Updated
The usual sequence starts with adding fractions, because that is what a worksheet can ask for on day one. It is also the reason so many people end up adding denominators: they were given an operation before they were given the object.
A fraction is one number, written as a division that has not been carried out yet. Everything below follows from that sentence, and none of it works without it.
The sequence
Each step needs the one before it. The order is not a preference — it is the dependency graph, and skipping a step reliably produces a specific, predictable mistake.
What a fraction is
Equal parts of one whole, and the same quantity written many ways. Until 2/4 and 1/2 are obviously the same number rather than two facts to memorise, nothing later is stable. Skip this and you get students who can follow a procedure but cannot tell whether the answer is sensible.
Comparing and ordering
Which is bigger, and why the one with the larger denominator is often smaller. This is where the object becomes a number with a place on the line. Skip it and you get answers nobody sanity-checks.
Adding and subtracting
Only now, and only after "you can only add things of the same kind" has been said about something concrete. The common denominator is not a rule to remember — it is the step that makes the pieces the same size so they can be counted. Skip the reason and you get 1/2 + 1/3 = 2/5.
Multiplying and dividing
Deliberately after addition, even though multiplying is procedurally easier. Doing it first teaches "operate on top and bottom separately", which is exactly the habit that ruins addition. Division by a fraction needs "how many of these fit into that" before the flip-and-multiply shortcut.
Decimals and percents
The same numbers in different notation — a decimal is a fraction with a denominator of ten or a hundred, a percent is a fraction out of a hundred. Taught here, they reinforce fractions. Taught as separate topics, they become three unconnected sets of rules.
Ratios and proportions
The last step, because a proportion is an equation between two fractions and needs both fraction fluency and the beginnings of algebra. This is where fractions stop being a topic and become a tool.
The lessons, in that order
Each has worked examples, practice with full solutions, and its own list of mistakes.
- Equivalent & Simplifying FractionsRead a fraction as a division and as a part of a whole, generate equivalent fractions, and reduce to lowest terms.
- Comparing FractionsDecide which of two fractions is larger using common denominators, benchmarks, or cross-multiplication, and order a set of fractions.
- Adding & Subtracting FractionsAdd and subtract fractions and mixed numbers by rewriting them with a common denominator.
- Multiplying & Dividing FractionsMultiply fractions directly and divide by multiplying by the reciprocal, understanding why the rules work.
- DecimalsRead, compare and compute with decimals, and convert between decimals and fractions.
- PercentsTreat a percent as a fraction out of 100 and solve percent-of, percent-change and discount problems.
- Ratios & ProportionsCompare quantities with ratios, find unit rates, and solve proportions that scale a relationship up or down.
What each skipped step looks like later
These are the specific mistakes that show up when the sequence is taken out of order — pulled from the lessons themselves.
Equivalent & Simplifying Fractions
- Adding the same number to the top and the bottom instead of multiplying: writing 3/4 = 4/5.
- Changing only one part of the fraction, so 3/4 becomes 3/12 instead of 9/12.
- Stopping half way when simplifying, leaving 9/12 when 3/4 was asked for.
- Swapping the whole number and the remainder when converting, writing 17/5 as 2 3/5 instead of 3 2/5.
- Using the leftovers as the denominator: calling 18 fletched out of 24 arrows "18/6".
Comparing Fractions
- Comparing numerators alone and concluding that 3/10 is bigger than 1/2.
- Assuming a bigger denominator means a bigger fraction, so claiming 1/8 > 1/3.
- Cross-multiplying but attaching each product to the wrong fraction.
- Changing the denominator to the common one without scaling the numerator to match.
- Giving the answer as the rewritten fractions (9/24, 12/24, ...) instead of the originals the question asked about.
Adding & Subtracting Fractions
- Adding the denominators as well as the numerators: writing 1/2 + 1/3 = 2/5.
- Changing the denominator to the common one but carrying the old numerator across, so 1/2 becomes 1/6.
- Flipping a fraction subtraction round to avoid a negative instead of regrouping a whole.
- Adding only the whole numbers of two mixed numbers and forgetting the fraction parts.
- Leaving an answer unsimplified, or leaving an improper fraction where a mixed number was wanted.
Multiplying & Dividing Fractions
- Multiplying straight across when the question is a division, instead of flipping the divisor first.
- Flipping the first fraction rather than the divisor, which gives the reciprocal of the right answer.
- Looking for a common denominator before multiplying, which is needed for addition but not here.
- Multiplying a fraction by a whole number and multiplying the denominator too, so the fraction never changes.
- Multiplying mixed numbers part by part instead of converting them to improper fractions first.