It is mid-September, and in Year 1 of an Italian art high school (liceo artistico) maths has already been on the timetable for two weeks. The entrance test is done, and the teacher is just starting natural numbers and integers, powers, GCD and LCM, fractions and signed numbers. With 3 hours a week — against 5 at the science high school (liceo scientifico) — the syllabus is not lighter, only more compressed: the first test arrives in five to eight weeks, so between the end of October and mid-November, and it asks for exactly these topics.
Many students arrive at an art school thinking maths will be a pass-through subject, good for a 6 without much studying, because they chose the school for drawing, painting and design. But the syllabus is the same as at any other high school, and the first test shows it at once: whoever comes up from middle school convinced they can get by on intuition with fractions and signed numbers usually gets it wrong.
Why maths surprises people most here
In the first two years of the art high school, maths has 3 hours a week out of a total of 34: fewer than the 5 of the science high school and the 4 of the technical institute. Around those 3 hours there are graphic and pictorial disciplines (4 h), geometric disciplines (3 h), plastic and sculptural disciplines (3 h) and the art workshop (3 h): 13 hours of subjects with drawings and projects to hand in by a deadline, which in many weeks end up swallowing the time that at home would go to maths. The syllabus is not lighter: it competes with a load of deadlines that other schools don’t have, and whoever puts off studying numbers “because it’s only a little” arrives at the test with the same syllabus as a science student but with far less practice behind them.
What the first test really asks
The typical format is 6-10 exercises in an hour, almost all pure calculation: no theory questions to recite from memory, just expressions to work through to a numerical result. The topics, in the order you are doing them in class:
- natural numbers and integers, a review of the four operations and their properties;
- powers, including those with a negative exponent;
- GCD and LCM, both on their own and inside fractions;
- fractions, with the four operations and multi-step expressions;
- signed numbers, that is numbers with a sign, and the rules for addition, subtraction and multiplication.
The critical point flagged for these weeks, in schools that follow this programme, is precisely fractions and signed numbers: these are the topics where the gaps carried over from middle school concentrate, even before you get to monomials and special products, which come later in the syllabus here. If your teacher has already mentioned monomials, don’t worry: in most art high schools full factorisation and algebraic fractions slip to the start of Year 2, and the first Year 1 test stays on numbers.
The mistakes that cost marks
They are always the same, and you can spot them at a glance in a marked test.
1. The minus sign in front of a power does not go inside the power. Without brackets, raising to a power comes before the sign:
Whoever writes has done the right calculation on the wrong number: they have raised the sign to the power as well, when in fact the sign stays outside until a bracket includes it.
2. A negative exponent changes the number, not the sign of the result. is not : it is the reciprocal of .
The result of a power with a positive base always stays positive, whatever the sign of the exponent.
3. Adding fractions without bringing them to the same denominator. The most common mistake is to add “across”, numerators together and denominators together: becomes , which is wrong. The correct result, using the LCM of the denominators:
4. Subtracting a negative number and getting the sign wrong. is not : the two consecutive minus signs become a plus.
5. Ignoring the order of operations. In the multiplication comes before the addition: the result is , not . This holds inside longer expressions with powers and fractions too: first powers, then multiplications and divisions, finally additions and subtractions.
6. Dividing two fractions without flipping the second. Dividing by a fraction is the same as multiplying by its reciprocal: whoever divides “straight across”, numerator by numerator, gets a different result.
The sign rule, in a table
With signed numbers, multiplication and division follow a fixed rule that is worth keeping in mind at all times, because it comes back in every exercise from here on:
| Operation | Same signs | Different signs |
|---|---|---|
| Product | positive result | negative result |
| Division | positive result | negative result |
It also works in a chain: , because three negative signs give a negative (the third sign “flips” the result again). With addition and subtraction, on the other hand, there is no rule of this kind: you have to look at the values and the signs case by case, as in examples 3 and 4 above.
The method, step by step
Faced with a long expression, the order in which to proceed is always the same, and following it removes most of the mistakes above:
- Work out the innermost brackets first.
- Calculate the powers, turning any with a negative exponent into a fraction straight away.
- Do multiplications and divisions, from left to right.
- Bring the fractions to the same denominator (the LCM) before adding or subtracting.
- Do additions and subtractions, checking the signs one last time.
Never skip a step to do it “in your head”: that is where the mark is lost, not in the calculation itself.
Fully worked exercises
Exercise 1. Calculate:
Powers first: (the sign stays outside); . Then the multiplication: . What remains is:
If you want to check yourself with other numbers, you’ll find 15 questions of the same kind, with instant marking, in the Year 1 maths test — numbers, fractions and powers: the correct answer is free, and the guided step-by-step solution unlocks with your first lesson.
Exercise 2. Calculate:
, because the exponent is odd. The second term becomes positive: . Rewriting with the LCM of 3 and 6, which is 6:
The most frequent mistake in this exercise is not the calculation: it is forgetting that a power with an odd exponent and a negative base stays negative, while with an even exponent it becomes positive.
An exercise from the entrance test: GCD and LCM
Two shuttle buses leave the school gate together: one every 12 minutes, the other every 18. After how many minutes will they leave together again? The answer is the LCM of 12 and 18:
They will leave together again after 36 minutes. It is the same calculation pattern as the September entrance test: factorise, then take each prime factor with the highest exponent found in either number.
How we work in lessons
With 3 hours a week of maths, there is no time in class to go back over a gap from middle school: if a block — usually fractions or signed numbers — stays shaky, it is carried forward to the test and shows up there. In lessons I start with a five-minute diagnostic mini-test to find out which of the two blocks (numbers with powers, or fractions and signs) is the one that is really shaky, and then I work only on that: targeted exercises on the specific weak point, not a generic review of the whole syllabus, with immediate correction at every step and not at the end of the week. For those with project deadlines piling up, this counts double: one well-aimed hour recovers what would otherwise take three hours of disorganised studying the evening before.
If we haven’t yet had a lesson together, the best place to start is a free 30-minute introductory call, no strings attached, to work out where to begin. After that, people who carry on regularly usually move from single lessons to the 10-hour package — you’ll find current rates and packages on the pricing page.
If physics comes up in class too, I have written about it for the same school in physics at art high school: measurements and significant figures — it is a Year 3 article, but the same “few hours, compressed syllabus” pattern applies there. If you are at a science high school or a technical institute instead, the October syllabus is similar but not identical: you’ll find it in the first maths test in Year 1 of high school. For the full syllabus of the year there is the maths page. Ratios and percentages, which follow fractions in the first weeks, are in ratios and percentages in Year 1.
If you need a hand with these topics before the test arrives, write to me: catching up now, on one specific block, takes far less time than chasing a failed subject in January.
