First maths test in Year 1: how to prepare

Every year, between mid-October and the first days of November, I get the same message from parents of children who have just started Year 1 of high school: “in middle school he had 8 in maths, he got 4 on the first test, we don’t understand what happened”. Nothing mysterious happened. What happened is that high school maths asks for something different from middle school maths, and the first test is the exact point where the difference turns into a grade.

The good news is that this test is the most predictable of the year. The September-October syllabus is almost identical in all schools, the mistakes that lose marks are five or six, always the same, and you are still in time to step in before the hole widens. Here you’ll find what really comes up, where the marks are lost, three worked exercises and a four-week plan.

When it comes and what it looks like

The first maths test almost always falls between mid-October and the start of November, after the September entrance test. Whether that test counts towards your average depends on the school and the individual teacher — in many classes it stays out of the assessment, in others it is recorded as a first grade: ask, don’t take it for granted. In any case it is mainly for the teacher to understand the starting level. Don’t confuse the two events: the entrance test looks back, to middle school; the test looks ahead, to the new syllabus.

The typical format of maths tests in the first two years of the science track is 6-10 graded exercises in an hour, from the most mechanical to the most reasoned. In the first test, though, expect calculation exercises and possibly a theory question on definitions: the “mixed” test with the geometric proof comes later in the year, when geometry has reached the congruence criteria for triangles (usually between November and December). At a technical institute there are generally 4-6 exercises, normally within the lesson hour — but the duration is decided by your teacher, so check in class.

The content depends on the type of school:

Science high school (liceo scientifico; maths 5 hours a week, with computer science in the first two years): from September to October the work is on the number sets $N$ , $Z$ , $Q$ , on powers, GCD and LCM, fractions, percentages and scientific notation; then sets and logic, and in geometry primitive concepts, segments and angles. Monomials, polynomials and special products arrive from November onwards, so in the first test they appear at most as a taste.
Technical institutes (maths 4 hours): same start — number sets, powers, ratios and percentages — with sets and logic and basic Euclidean geometry between November and December. Here monomials and polynomials come later, typically between January and February.

A note for anyone in Year 1 at a technical institute this year: the Year 1 classes of 2026/27 start with the new curriculum (Ministerial Decree no. 29 of 19 February 2026), which merged Physics and Chemistry into a single subject, “Experimental sciences”. Maths, however, stays at 4 hours with the same Year 1 objectives: nothing changes for the October test.

If you don’t know for sure what your teacher has covered, the reference is their register, not the book’s contents page: ask in class which sections are in the test.

Why the grade collapses even if you got 8 in middle school

In middle school most exercises are solved in two or three steps, and the calculation is almost always with “friendly” numbers. In high school three things change at once.

Exercises become long. An expression with fractions and powers can take ten steps. A single wrong sign at the third step makes everything after it wrong, even if the seven following steps are impeccable. In middle school a mistake cost a mark; here it costs the exercise.

Knowing the rule isn’t enough, it has to be automatic. If you need half a minute to remember that $a^{- n} = \frac{1}{a ^{n}}$ , you won’t reach the end in an hour-long test. The rule has to be known like your times tables: without thinking.

You move from numbers to letters. This is the real jump of Year 1. As long as you write $3 + 5$ you have a mental image; when you write $a + b$ you have to trust the structure, not the result. Many students who are brilliant at arithmetic get stuck here, and it is not a question of intelligence: it is a question of habit, which is built by doing exercises.

Then there is a fourth factor, less technical but heavy: many students in middle school got 8 without studying at home, just following well in class. In high school that method stops working around October, and it is the first test that says so. I wrote about this in how to study maths without stress.

The mistakes that cost the most marks

Marking Year 1 papers, the lost marks concentrate almost entirely on five points. They are not conceptual difficulties: they are missing automatisms.

1. The minus sign in front of a bracket. Changing the sign of only the first term is the number one mistake of all:

5 - (3 - 7 + 2) = 5 - 3 + 7 - 2 = 7

Not $5 - 3 - 7 + 2$ . The minus is distributed over all the terms inside the bracket.

2. The order of operations. In $10 - 2 \cdot 3$ the result is $4$ , not $24$ : multiplication is done before subtraction. The same goes for powers, which come even before that: $- 3^{2} = - 9$ , while $(- 3)^{2} = 9$ . The bracket completely changes the meaning, and teachers always put in at least one exercise on this.

3. Powers with a negative exponent. A negative exponent does not make the result negative: you take the reciprocal. So

2^{- 3} = \frac{1}{2 ^{3}} = \frac{1}{8}

and for a fraction you just flip it:

(\frac{2}{3})^{- 2} = (\frac{3}{2})^{2} = \frac{9}{4}

Remember also that $a^{0} = 1$ for every $a \neq = 0$ , and that $0^{0}$ is not defined.

4. Adding fractions by adding the denominators. $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$ : you need the common denominator, and the result is $\frac{5}{6}$ . It seems trivial, but under pressure it resurfaces — especially when the fractions are inside a long expression.

5. Confusing GCD and LCM. The greatest common divisor is smaller than or equal to the starting numbers (it is used to simplify, to divide into equal parts); the least common multiple is greater than or equal to them (it is used for the common denominator, for phenomena that repeat and meet). If you get a GCD bigger than the given numbers, you have made a mistake: it is a check that costs two seconds.

To these I add a mistake that appears as soon as letters begin: $(a + b)^{2}$ is not $a^{2} + b^{2}$ . The square of a binomial has three terms, $a^{2} + 2 ab + b^{2}$ , and that double product is where most marks are lost from November on.

Three worked exercises

1. Expression with fractions and powers

Calculate:

(\frac{3}{2})^{- 2} \cdot (\frac{3}{4})^{- 1} : (\frac{2}{3})^{3} + \frac{1}{2}

First thing: sort out all the negative exponents, before doing any calculation. I flip the fractions:

(\frac{3}{2})^{- 2} = (\frac{2}{3})^{2} = \frac{4}{9} (\frac{3}{4})^{- 1} = \frac{4}{3}

Now the multiplication:

\frac{4}{9} \cdot \frac{4}{3} = \frac{16}{27}

Then the division. Dividing by a fraction means multiplying by its reciprocal, and $(\frac{2}{3})^{3} = \frac{8}{27}$ :

\frac{16}{27} : \frac{8}{27} = \frac{16}{27} \cdot \frac{27}{8} = 2

Only at the end the addition, which has lower priority:

2 + \frac{1}{2} = \frac{5}{2}

The point to take home: the order in which you worked is not random. Negative exponents, then multiplications and divisions from left to right, then sums. Whoever adds the $\frac{1}{2}$ halfway through gets a different number and doesn’t notice.

2. GCD and LCM in a problem

Two lighthouses flash at regular intervals: the first every 84 seconds, the second every 120. If they are flashing together now, when will it happen again?

You always start from the prime factorisation:

84 = 2^{2} \cdot 3 \cdot 7120 = 2^{3} \cdot 3 \cdot 5

The LCM takes all the factors, common and not common, with the largest exponent:

LCM (84, 120) = 2^{3} \cdot 3 \cdot 5 \cdot 7 = 840

So the two lighthouses flash together again after 840 seconds, that is 14 minutes.

The GCD, which is what you need when you have to divide something into equal parts as large as possible, takes only the common factors with the smallest exponent:

GCD (84, 120) = 2^{2} \cdot 3 = 12

A quick check that I always recommend: the product of GCD and LCM must give the product of the two starting numbers. Here $12 \cdot 840 = 10, 080$ and $84 \cdot 120 = 10, 080$ . It matches, so the factorisation was right.

3. The first exercise with letters

Simplify:

(2 x - 3 y)^{2} - (2 x + 3 y) (2 x - 3 y)

The first piece is a square of a binomial: square of the first, double product (with its sign), square of the second.

(2 x - 3 y)^{2} = 4 x^{2} - 12 x y + 9 y^{2}

The second piece is a sum times a difference, which gives the difference of squares:

(2 x + 3 y) (2 x - 3 y) = 4 x^{2} - 9 y^{2}

Now the subtraction, with the bracket written out explicitly — this is where mistakes happen:

4 x^{2} - 12 x y + 9 y^{2} - (4 x^{2} - 9 y^{2}) = 4 x^{2} - 12 x y + 9 y^{2} - 4 x^{2} + 9 y^{2}

The $4 x^{2}$ terms cancel and what remains is

- 12 x y + 18 y^{2}

which can be factorised as $6 y (3 y - 2 x)$ . If you didn’t write the bracket at the subtraction step, the $- 9 y^{2}$ stays negative and the result is wrong: writing the bracket even when it is “obvious” is the single habit that gains the most marks in Year 1.

Sets: the part that looks easy

Sets and logic look like a gift — no calculations — and precisely for that reason they are studied little. The three things people get wrong:

Membership versus inclusion. An element belongs to a set ( $3 \in A$ ), a set is contained in another ( $B \subset A$ ). Swapping $\in$ and $\subset$ is a red-pencil mistake even when the reasoning is correct.
The empty set. $\emptyset$ is a subset of every set, but $\emptyset$ and ${\emptyset}$ are not the same thing: the second has one element.
The representations. Tabular, by characteristic property and with Euler-Venn diagrams all describe the same set. If the exercise asks for a specific form, that is the one to use.

It is twenty minutes of study that are worth one or two sure marks: don’t skip it because “it’s only theory”.

How to prepare in four weeks

Four weeks are plenty, as long as you work a little but every day. Thirty minutes a day for five days beats three hours on Saturday: maths needs close repetition, not marathons.

Week 1 — arithmetic under control. Fractions (sum, product, division, simplification), order of operations, brackets and signs. Ten numerical expressions, all with the book closed and no calculator. Goal: no sign errors.

Week 2 — powers. Properties (same base, power of a power), zero exponent, negative exponent, powers of fractions, scientific notation. This is where the automatism has to be built: when you see a negative exponent, your hand must flip the fraction on its own.

Week 3 — GCD, LCM, percentages and ratios. Prime factorisation of three-digit numbers, LCM and GCD with the product check, applied problems. If you are at a science high school and the teacher has already started monomials and special products, add the square of a binomial and the sum times difference here.

Week 4 — mock test and targeted review. A full paper timed at one hour, with the alarm, without looking at the book. Then the part that matters most: classify every mistake into three categories — I didn’t know the rule, I knew the rule but made a calculation mistake, I didn’t understand what the question asked. They are three different problems and they are cured in three different ways. In the last days you work only on the biggest category.

The mistake notebook is the most underrated tool of the first two years. One page, two columns: on the left the mistake as you made it, on the right the correct step and the reason. Before the test, reread that, not the book chapter.

Using a practice test as a self-assessment

The paper you need right now is the practice test on numbers, fractions and powers: 15 questions in 50 minutes on the October block, that is exactly what your teacher is about to ask. When the algebra test arrives there is its test on monomials, polynomials and special products, and at the end of the term the mixed paper that puts them back together. The full list, block by block, is on the page of Year 1 high school tests.

There is also a second paper, and it is useful in a different way: the practice paper for the Year 1 maths catch-up exam covers the whole syllabus of the year — algebra, factorisation, algebraic fractions with the conditions of existence, first-degree equations, sets and logic, Euclidean geometry, statistics. In October you haven’t yet done half of those topics: don’t be frightened if you get them wrong, use it as a map of where the year is going and which topics weigh the most.

On the answers I play with my cards face up: the correct answer to every question is free and needs no registration; the guided step-by-step solution unlocks with your first lesson. The mixed paper that puts everything together is here; if instead you are reading because you already have a failed subject to make up, the dedicated paper is Year 1 maths catch-up and the guide is how to prepare for the maths catch-up exam.

On the day of the test

Three habits worth half a grade, regardless of how much you have studied.

Read all the exercises before writing. Two minutes spent working out which are the three easy ones and which is the long one let you manage the hour, instead of suffering it.
Start with the exercises you can do. The test isn’t marked in order and there is no prize for sequence. If you are stuck for more than four or five minutes, move on and come back later.
Write out the steps. Without steps the teacher can’t give you partial marks and you can’t check your work. One line per step, always.

If the problem is not preparation but the block in front of the page — empty mind, everything you knew gone — that is a separate thing, tackled with specific techniques: I wrote about it in managing test anxiety in maths and physics.

A 4 on the first Year 1 test decides nothing. What you do in the three weeks after decides a lot: the October syllabus is the foundation on which the whole first two years rest, and recovering it now costs a fraction of what it will cost in June.

If you want to understand where to start in your case, the free introductory call is for that: you send me the marked test or tell me what you have done in class, and I tell you honestly whether you need a hand or whether it is enough to fix the method on your own. Lessons are online and one-to-one, single lesson or 10-hour package: you’ll find current rates and packages on the pricing page. The full programme of maths tutoring is here, year by year.

First maths test in Year 1 of high school: how to prepare