Mathematics — Grade 4

In fourth grade, children determine the extent of a flat surface using the small square as a unit of measure. They learn to establish the area of rectangles and of figures made up of several rectangles by counting the units that cover them, linking the arrangement in rows and columns to multiplication, or working out the length of one side starting from a known area.

The most common error consists in confusing area with perimeter: instead of counting the inner squares that fill the figure, the pupil counts the segments that form the outer boundary. A similar difficulty emerges when the unit of measure changes value, as in cases where one tile corresponds to two squares: there is often a tendency to count the single squares absentmindedly without applying the required proportion.

Practical activities involve visual and concrete tasks based on squared grids:

• calculating how many squares cover a rectangle or finding out how many tiles are still missing to complete it;
• determining the area of a figure made up of two rectangles or comparing two shapes to understand by how much one surface is larger than the other;
• solving true or false questions, identifying and correcting an incorrect area, or drawing a rectangle with a given area.

In quarta primaria i bambini imparano a riconoscere la regola matematica che guida una successione di numeri. Non si limitano a proseguire un semplice schema intuitivo: osservano con precisione di quanto aumenta o diminuisce ogni passaggio, calcolano il passo della sequenza e applicano questa logica per continuare la serie, procedere a ritroso o calcolare un valore posizionato molto più avanti.

L'errore tipico consiste nell'osservare soltanto i primi due termini della serie e tirare a indovinare la regola, senza verificare se il passo resta coerente nei passaggi successivi. Un'altra difficoltà frequente emerge quando bisogna trovare un numero mancante nel mezzo: spesso si applica una somma o una sottrazione a partire da un estremo senza controllare che il risultato si raccordi correttamente con il numero che segue.

Nelle attività pratiche i bambini affrontano schede con compiti mirati, tra cui:

• scoprire di quanto cresce ogni volta una sequenza o continuare un ritmo decrescente;
• completare i termini mancanti, indicando quale numero manca nel mezzo o risalendo a qual era il primo numero;
• identificare gli errori in serie completate male e confrontare più ritmi per capire quale sequenza cresce più in fretta.

In fourth grade, children begin to interpret negative whole numbers through a real situation: the graduated scale of the thermometer. They learn to use zero as a reference point, to read both positive temperatures and those below zero, and to compare two values to determine which measurement is warmer.

The most common mistake appears when temperatures drop below zero: accustomed to counting from the bottom up, many children count the marks in the wrong direction, or think that -5 °C is a warmer temperature than -2 °C, because they automatically associate the number 5 with a greater quantity.

In the worksheets, children work with illustrations of graduated thermometers. The exercises ask them to read the temperature from the thermometer, choose between true or false statements, correct wrong readings, or calculate changes, for example by finding by how much the temperature has risen or how many degrees are missing to reach a certain value.

In fourth grade, the child learns to determine the perimeter of a square using the standard formula: multiplying the length of a side by 4, taking advantage of the fact that all sides have the same length. They work both by counting the units or squares on a square grid, and by performing direct calculation with the given measurements. In addition, they learn to use the inverse procedure: finding the length of a side by dividing the perimeter by 4.

The most frequent difficulty concerns confusion about the geometric meaning of the formula: the child often tends to multiply the side by itself instead of by 4. In inverse problems, a common error is trying to subtract a quantity instead of performing the division into four equal parts.

In practical activities, the exercises propose:

• calculating the perimeter from a drawing on a square grid or from the numerical value of the side;
• finding the side of the square from the perimeter through division;
• multiple-choice questions, statements to evaluate as true or false, and worksheets in which calculation errors must be corrected;
• the comparison of figures to determine by how much the perimeters of two different squares differ.

In fourth grade, children explore a notation system different from ours through Roman numerals from I to XII, which is also commonly used when reading a clock. In this unit they associate the fundamental symbols (I, V, X) with their respective quantities and understand how they combine through addition (such as VII for 7) and subtraction (such as IV for 4 or IX for 9).

The most common mistake arises from confusing the modern positional value with the additive or subtractive rules of Roman numerals. Pupils often mix up the order of the symbols (for example, swapping IV with VI) or tend to repeat the same sign four times (writing IIII instead of IV), forgetting that a symbol placed to the left of a symbol of greater value must be subtracted.

In practical activities, children approach this topic through several types of exercises:

• arranging the numbers from I to XII in ascending or descending order;
• choosing the correct notation or identifying the error in reading a Roman numeral;
• solving simple true or false questions and calculating sums expressed directly with these symbols.

In grade four, children learn to use tables and bar graphs to extract information and make targeted comparisons. At this level, pupils do not simply identify the highest or lowest value; they use subtraction to determine how many more items appear in one row compared to another, and they check how many categories exceed a specific numerical threshold.

The most frequent mistake occurs precisely in comparative questions: when asked how many more items there are, many children simply copy the absolute value of the highest column, leaving out the calculation of the difference. Another common difficulty is reading the table's rows in the wrong order or confusing the name of a category with the numerical value associated with it.

In worksheets, the tasks present data tables or bar graphs alongside structured questions in which the child must:

• identify which category has the greatest quantity;
• calculate the difference between two rows or between the largest and smallest value, giving the correct answer or checking whether a statement is true or false;
• count how many groups exceed the number indicated in the instructions.

In fourth grade, students learn to determine the perimeter of a rectangle through different procedures: by adding the lengths of all the sides or by applying direct formulas such as (base + height) × 2. They work both on figures drawn on grid paper, counting the units of measure along the outline, and on figures with assigned numerical measurements, arriving at the measure of a missing dimension when the perimeter is already known.

The most common error consists of adding only two consecutive sides (the base and the height), forgetting to double the sum or to add the other two sides to complete the entire outline. In inverse tasks, when looking for a missing side, it often happens that the known side is subtracted directly from the total perimeter instead of from half of the perimeter.

Practical assignments propose a variety of activities: determining or choosing the correct perimeter among several options, checking whether a given calculation is correct, finding the length of the missing side, drawing a rectangle starting from an assigned perimeter, and comparing several figures to identify which one has the greatest perimeter.

In fourth grade, children approach fractions starting with the most immediate divisions: the half and the quarters. They work both with a continuous whole (a geometric figure divided into 2 or 4 equal parts) and with discrete sets of objects, learning to calculate the fraction of a numerical quantity — such as finding a quarter of 24 elements by division — and to do the reverse process, that is, to go back to the total quantity knowing the value of just one part.

The most common error appears when moving from the geometric figure to the group of objects: many pupils tend to count the leftover elements instead of comparing them with the whole, or they forget that the parts must be made up of a strictly identical number of elements. Confusion also often arises between the parts considered and the remaining ones, for example by mixing up the coloured fraction with the uncoloured one.

In practical activities, children practise with targeted tasks:

• calculating how many elements make up a fraction of a set, or identifying how many uncoloured parts remain in a figure through true or false questions;
• solving inverse situations to find the total, as in the case of how many are there in all if one part is...;
• analysing given representations to correct division errors, or inventing sets that can be shared equally into two or four groups.

In fourth grade, starting from practical and visual contexts, children learn to intuit and explain which of two events is more likely, or to recognize whether they are equally likely. In the simplest cases, they reach a first numerical quantification based on direct counting: for example, they compare the number of favorable outcomes (such as 6 red marbles against 2 blue ones) to argue why one choice is more advantageous than the other.

The most common error at this age is relying on subjective beliefs or "feelings of luck" instead of counting the elements present. Children often focus only on the one outcome they would like to see come up, or they believe that a result that has already come up cannot be repeated right away, ignoring the actual composition of the group of objects.

In the practical activities, the exercises present very clear illustrated situations:

• Exercises such as A bag of marbles or What will you draw most easily, in which a group of colored objects is observed and the most likely outcome is indicated;
• Worksheets of the type Fill the bag, where the pupil must draw or distribute the elements to obtain a greater, smaller or equal probability;
• Questions in which the reasoning of a character is analyzed, to establish whether the comparison was correct and, if necessary, to correct the error.

In fourth grade, children learn to determine the perimeter of a triangle by adding the lengths of its three sides. Beyond direct calculation, they apply inverse procedures: knowing the total perimeter, they use subtractions to find the length of an unknown side, or they exploit the properties of figures (such as the equality of the two equal sides in an isosceles triangle) to determine the missing measurements.

The typical difficulty arises precisely with triangles that have equal sides: pupils often subtract the base from the total perimeter but forget to divide the remaining difference in half, confusing the sum of the two equal sides with the measure of a single side.

In the worksheets, the tasks vary between multiple-choice exercises and true or false questions about the perimeter. Practical exercises ask students to calculate the total measure of the outline, to find the length of the third side or of a equal side of an isosceles triangle, or to compare two different figures to discover by how much the perimeter of one triangle is greater than that of the other.

In fourth grade, children learn to use numerical approximation to find their way through everyday situations. In this unit they work mainly with rounding to the nearest ten or rounding up, applying it to sums, subtractions and concrete data such as prices in euros or measurements of quantities and space.

The typical mistake is wanting to calculate the exact result first and only round it at the very end: many pupils struggle to trust a calculation done "by eye". They also often confuse rounding to the nearest ten with rounding up, which is instead used to make sure a sum of money is enough before going to the checkout.

In the practical worksheets, children tackle contextualised tasks through targeted exercise models:

• Roughly how much will we pay in total and Estimate a bill by rounding up, to evaluate a shopping basket;
• Estimate the sum: round to the nearest ten and Estimate the difference, to get used to simplifying numbers before operating on them;
• Will it fit? and Find prices that match the estimate, to check the plausibility of measurements, dimensions and receipts.

In fourth grade, children analyze concrete situations to determine with confidence whether an event will certainly happen, might happen, or can never happen. Working with game contexts and visual models, they learn to justify their answers and to distinguish between what is certain, possible, or impossible.

The most common mistake is confusing a very likely event with a certain one, or a very rare event with an impossible one. For example, if a group contains ten tokens and only one is yellow, many pupils tend to call drawing the yellow token "impossible" simply because it is unlikely, or call drawing one of the other colors "certain," forgetting that a single exception is enough to make an event merely possible.

In practical activities, pupils tackle multiple-choice questions ("Certain, possible, or impossible? Choose") and true-or-false statements, where they must also identify and correct errors in reasoning. In hands-on exercises such as "Fill the bag," they are instead asked to draw or distribute the elements with the right colors so that a given draw is certain, possible, or completely impossible.

In fourth grade, children learn to extract useful information from numerical tables and pictograms. At this level they can calculate the overall total by adding the data in the different rows, compare quantities, and interpret pictograms in which each drawing or icon represents a multiplied value (for example, a drawing worth 2, 5, or 10 units).

The most common error occurs precisely with pictograms: many children tend to simply count the number of symbols drawn as if each were worth one, ignoring the key that assigns a larger quantity to each figure. In tables, another frequent difficulty appears when the total is known and an missing quantity must be found: the pupil tries to add the visible numbers instead of subtracting them from the overall total.

In practical activities, children work on structured worksheets focused on specific tasks:

• calculating the sum of the data with How many in total in the table or by choosing the correct option among several answers;
• checking whether a calculation is correct through questions such as Total from the table: true or false;
• multiplying the number of icons by their value with the exercise Pictogram: one drawing stands for several things;
• using subtraction to find the missing data in the activity The missing row when the total is known.

In fourth grade, children learn to extend written addition and subtraction to decimal numbers. The main goal is to correctly handle the place value of digits, aligning units with units, decimal points with each other, and the decimal parts (tenths, hundredths and thousandths) so that they can carry out the calculation with confidence.

The most frequent error concerns column alignment: many pupils tend to line up numbers on the right, as they did with whole numbers, placing digits of different value on top of each other (for example, adding tenths to hundredths). In subtraction, moreover, the typical mistake is not adding the placeholder zeros needed to make the decimal digits match, which makes the carrying and borrowing incorrect.

In the worksheets, the exercises appear as pure calculations and practical problems based on specific models:

• Add decimal numbers and Subtract decimal numbers, in which students are asked to align the addends, or the minuend and subtrahend, correctly in order to calculate the result;
• Add masses written with decimals, where numbers with decimal points are applied to real weight measurements expressed in the same unit of measurement.

In fourth grade, children learn to find their way among the most common units of length (from the millimetre to the kilometre), moving from one measurement to another. Relying on multiplication and division by 10, 100 and 1000 and on the use of decimal numbers, they can rewrite a measurement while keeping its overall value intact.

The most frequent stumbling block concerns the direction of the conversion: when moving to a smaller unit, pupils tend to divide instead of multiply (or vice versa), confused by the fact that a measurement expressed with a larger number corresponds to a physically shorter unit. Difficulties often also arise in correctly repositioning the decimal point or in adding the missing zeros.

In practical activities, the exercises come in three main formats:

• Convert into a smaller unit of length, by multiplying the starting measurement;
• Convert into a larger unit of length, by dividing the measurement to obtain larger multiples;
• Write the length in a single unit, recombining measurements expressed in different units into one whole or decimal number.

In fourth grade, children learn to determine the area of a rectangle by applying the geometric formula with multiplication (base times height). They also learn to work with the inverse formula: if they know the total area and the measure of one side, they perform a division to find the length of the unknown side.

The most frequent stumbling block is confusing perimeter with area. Students often add the sides instead of multiplying them, or they look for the missing side by subtracting the known measure from the area or by dividing it in half, forgetting that the inverse operation of multiplying the dimensions is dividing the area by the known side.

In practical activities, this work is presented through three graded worksheets:

• Area of the rectangle: exercises with geometric figures in which the given measures of base and height are multiplied;
• Missing side of the rectangle: backward tasks in which division is applied to determine the height or the base;
• The area of the rectangle in everyday life: simple problems set in real contexts, such as measuring the floor of a room or a notebook.

In fourth grade, children use units of time to calculate durations and make simple conversions between hours, minutes and seconds. Unlike other units of measurement based on the decimal system, here they work with the sexagesimal system: they know that 60 seconds make a minute and 60 minutes make an hour, and they use multiplications and additions to convert from one unit to another or to calculate differences between times.

The typical difficulty lies in treating time as if it were decimal. When faced with a time interval, for example between 14:50 and 15:10, students often apply a traditional column subtraction, forgetting that the next hour begins at 60 minutes rather than at 100.

In the teaching worksheets, this work is proposed through targeted activities:

• Minutes and seconds into seconds: calculations in which minutes are multiplied by 60 and the remaining seconds are added to find the total value;
• How many minutes have passed?: questions starting from given times to find the time elapsed until the next moment;
• How long did it last?: problems in which the start time and end time are compared to determine the total duration of an action.

In fourth grade, children learn to recognize and describe the regularities that link a series of numbers. At this level, pupils generate and extend sequences based on two fundamental operations: adding a constant step or multiplying each term by the same number.

The most common mistake occurs when the numbers grow quickly: the child often assumes right away an additive rule, simply calculating the difference between the first two terms without checking whether multiplication describes the whole sequence better. Another typical difficulty is applying the rule to the first step without checking that it remains valid for all the numbers already given in the exercise.

In worksheets, exercises appear in three typical formats:

• Sequence: add the same step, where the chain is built by repeatedly adding a given value;
• Sequence: multiply by the same number, in which each following term is obtained by multiplying by the given factor;
• Find the rule and continue the sequence, where the child must first analyze the steps between the written numbers to discover the correct operation and then complete the empty boxes.

In fourth grade, children face situations that require several chained operations to reach the solution. At this level they learn to identify the structure of a text and organise the data so that calculations are carried out in the correct order: they combine quantities using additions and multiplications, or use division with remainder to determine how many portions or complete groups can be formed with the materials available.

The most frequent obstacle is the rush to calculate: many children take all the numbers in the text and immediately add or multiply them, without planning the intermediate steps. In questions involving stock or supplies, the typical error concerns the handling of the remainder of the division, which is sometimes ignored or added inappropriately, without considering that the leftover quantities are not enough to complete one more group.

In the worksheets, exercises come in two main forms: the word problem with multiple steps (combining), where one must find an intermediate value before answering the final question, and practical tasks such as how many sets at most can be made from the supplies, in which the maximum number of packs that can be made is calculated by dividing the available supplies and correctly interpreting the remainder obtained.

In fourth grade, children explore an ancient system of numerical notation, learning to recognize and compose Roman numerals up to 3000. They use the fundamental symbols (I, V, X, L, C, D, M) and apply the additive and subtractive rules that make it possible to move from our positional notation to the Roman one.

The most common mistake concerns the position of the symbols and the subtraction rule. Children often confuse the value when a smaller digit precedes a larger one (as in IX or CD), or they tend to repeat the same symbol four times in a row (writing, for example, IIII instead of IV), ignoring the constraint of a maximum of three repetitions.

In the practical activities, the work focuses on two types of exercise:

• Read a Roman numeral: starting from a writing with Roman letters up to 3000, translate it into the corresponding number in digits;
• Write a number in Roman numerals: a decimal number is given and students must compose it correctly using Roman symbols.

In fourth grade, children begin calculating with fractions. In this unit, they learn to perform simple additions and subtractions limited to fractions that have the same denominator, operating only on the numerators to find the resulting part of the whole.

The typical mistake is to add or subtract the denominators as well, writing for example that 2/5 + 1/5 = 3/10 instead of 3/5. This happens because children tend to apply the operation to all the numbers on the line, forgetting that the denominator only indicates how many parts the whole is divided into, and not a quantity to be counted.

The worksheets present activities based on the models Add fractions with the same denominator and Subtract fractions with the same denominator. These are horizontal operations in which the student must keep the denominator unchanged and calculate the sum or difference of the numerators alone.

In fourth grade, children work with large numbers and carry out written calculations with multi-digit whole numbers. This step requires being able to line up the addends or the terms of a subtraction accurately, aligning ones, tens, hundreds and thousands correctly so they can proceed confidently with the calculations.

The typical difficulty arises when the two numbers have different lengths: it often happens that digits are aligned on the left instead of the right, thus confusing different orders of magnitude. In subtractions, moreover, the presence of intermediate zeros creates uncertainty in handling borrowing, leading to errors when exchanging between different place values.

In practical activities, the work develops through worksheets based on the models of Column Addition and Column Subtraction. The exercises present the operations to be performed on squared grid paper, helping the pupil keep the alignment correct and clearly write down carries and borrowings.

In fourth grade, when tackling large numbers, pupils learn to simplify values in order to judge the order of magnitude of calculations. Specifically, they develop the ability to determine whether a number is closer to the lower or higher ten or hundred, by carefully observing the position of the digit that determines the change (the units for the tens, the tens for the hundreds).

The most common mistake occurs with the middle digit 5: faced with numbers such as 45 or 350, children often round down instead of up. Another typical confusion consists in looking at the wrong digit, for example checking the units instead of the tens when the goal is to find the nearest hundred.

In worksheet activities, the work is presented through targeted exercises entitled Round to the nearest ten and Round to the nearest hundred. Children are asked to rewrite series of numbers by transforming them into their rounded value, training their eye to recognise quantities and making it easier to estimate the result in mental or written calculations.

In fourth grade, children learn to determine the perimeter of polygons by adding the lengths of all their sides. They also know how to do the reverse: starting from the total perimeter and the measures of the known sides, they use addition and subtraction (or division, if the polygon is regular) to work out the length of the missing side.

The typical difficulty appears when looking for the unknown side: students often subtract only one known side from the perimeter instead of subtracting the sum of all the other sides, or they forget to count a side in the drawing, which alters the final result.

In the worksheets, this learning is practised through two types of activities:

• Perimeter of the polygon, where the total measure is calculated by adding the values assigned to each side of the figure;
• Missing side from the perimeter, in which the total perimeter is given together with some sides, and students are asked to calculate the length of the segment left blank.

In fourth grade, children apply the study of measures and decimal numbers to the monetary system, working with practical conversions between euros and cents. Pupils learn to manage a real budget: they add up the cost of several items, calculate the difference from the initial amount to find the change, and make numerical comparisons between different prices.

The most common error appears in writing and aligning cents. Children often confuse coins under ten cents by writing, for example, 0.5 € instead of 0.05 €. Another typical difficulty is calculating change starting from whole amounts (such as a 10 or 20 euro banknote), where they forget to place the decimal point and the zeros for the cents in order to perform the subtraction in columns.

In practical activities, this work takes the form of concrete problems based on two exercise models:

• How much money is left?: starting from an initial sum and from the receipt or the price of one or more purchases, the remaining money is calculated.
• Which offer is better value?: comparing two different prices or packages of the same product, the most economically advantageous choice is determined.

In fourth grade, children tackle the study of large multi-digit numbers. At this level they learn to precisely recognize the value of each digit according to its position (ones, tens, hundreds, thousands) and to understand how each move to a higher place value makes a digit ten, one hundred, or one thousand times larger.

A typical difficulty arises when numbers contain intermediate zeros or have different lengths: children often get misled by the size of the individual leading digits instead of first considering the total number of digits, or they forget to consider the actual value of zero as a placeholder.

The proposed activities guide pupils step by step through specific worksheets:

• with the template Compare multi-digit numbers, pairs of numbers are compared by inserting the symbols for greater than, less than, or equal to;
• with How many times larger? Place value, students reason about the multiplicative relationships between digits shifted to different positions within the number.

In fourth grade, children work with large numbers and learn to round them to the nearest thousands or ten thousands. This skill makes it possible to find their way among numbers with four or five digits and is essential for learning to estimate at a glance the likely result of an operation, checking whether the final calculation makes sense.

The typical difficulty lies in identifying the right digit to look at in order to decide whether to round up or down. Children often look at the digit in the required position instead of the one immediately to its right, or they round the indicated digit correctly but forget to replace with zero all the following positions, writing for example 35.720 instead of 35.000.

In the worksheets, the exercises focus on targeted tasks such as Round to the nearest thousand or Round to the nearest ten thousand. Faced with numbers such as 4.820 or 37.150, the pupil identifies the reference value and rewrites the approximate number, consolidating the mechanism on concrete quantities.

In fourth grade, pupils learn to describe and name plane figures with three sides by applying two distinct classification criteria: according to their sides (equilateral with three equal sides, isosceles with two equal sides, scalene with all sides different) and according to their angles (right with one right angle, acute with three acute angles, obtuse with one obtuse angle).

The most common mistake is confusing the two criteria with each other, for example using the term isosceles to describe an angle, or being misled by the orientation of the drawing on the page, struggling to recognise a right triangle when the right angle does not sit on the horizontal base.

In the practical activities, the work focuses on the worksheets Type of triangle according to angles and Type of triangle according to sides. The children observe geometric figures drawn in various positions, compare the length of the sides or the opening of the angles, and match each figure with its correct name.

In fourth grade, children learn to round numbers in order to estimate the result of an addition before calculating it exactly. This skill helps develop number sense with increasingly large numbers, making it possible to predict the order of magnitude of a total and check whether the result obtained makes sense.

The typical mistake is to carry out the exact column calculation anyway and only round at the very end. Children often confuse estimation with precise calculation out of uncertainty, or they round the individual addends incorrectly, for example by changing the order of magnitude and obtaining a value that is completely far from the real one.

In worksheets based on the Estimate a sum model, the exercise presents numerical additions in which students are explicitly asked not to do the column calculation: the learner must mentally replace the addends with more convenient numbers and indicate the approximate result among several alternatives or write a plausible estimate.

In fourth grade, children learn that the same quantity can be expressed with different numbers. To expand a fraction, they multiply both the numerator and the denominator by the same whole number (for example, doubling both terms of 2/3 to get 4/6), concretely checking that the overall value of the fraction does not change.

The most common error consists of multiplying only one of the two terms — often only the denominator — or of adding a quantity instead of multiplying (adding the same number to the top and bottom). When this happens, the pupil loses sight of the concept of proportion and changes the real value of the fraction.

In worksheets with the exercise Find the equivalent fraction, the task requires completing guided equalities: given a starting fraction and an assigned new denominator, the pupil identifies the multiplicative factor and calculates the missing numerator to maintain equivalence.

In fourth grade, children learn to compare the size of angles using the right angle as the main reference point. Pupils look at figures and decide whether the opening between two rays corresponds to a right angle, whether it is smaller (acute angle) or larger (obtuse angle), until they recognize straight angles and full turns.

A very common mistake is confusing the length of the drawn segments with the actual size of the angle. Children often judge an angle to be "bigger" only because its sides are drawn longer, or they struggle when the angle is rotated or oriented in an unusual way relative to the page.

In the worksheets with the exercise Identify the type of angle, children practise on geometric figures and single angles presented in various positions: they compare the given opening with the right angle and write the correct classification.

In fourth grade, children learn to find a fractional part of a given whole number in order to describe and solve concrete situations. To determine, for example, 3/4 of 24, they apply two precise steps: they divide the whole number by the denominator to calculate the unit fraction (24 : 4 = 6), and multiply the value obtained by the numerator (6 × 3 = 18).

The most common error consists in reversing the steps or confusing the roles of the two terms of the fraction: some pupils divide by the numerator instead of the denominator, losing the connection with the practical meaning of dividing a quantity into equal parts and then considering a certain number of them.

In activities of the type Calculate the fraction of a number, the exercises require direct calculations starting from whole numbers (such as calculating 2/5 of 35) or brief contexts linked to everyday life in which to apply the sequence of division and multiplication.

In fourth grade, children learn to represent the same rational value in different ways, converting decimal numbers into the corresponding fractions with denominator 10, 100, or 1000. The pupil learns to connect the place value of the decimal digits (tenths, hundredths, thousandths) to the structure of the fraction, placing the digits in the numerator and the correct power of ten in the denominator.

The most common error concerns identifying the correct denominator when a zero is present: for example, writing 0.04 as 4/10 instead of 4/100, confusing hundredths with tenths. Another frequent difficulty arises with numbers greater than one, such as 1.5, where the child tends to isolate only the decimal part by writing 5/10 and forgetting the whole unit, instead of recording the fraction 15/10.

In the worksheets with the template Write the decimal number as a fraction, the task requires observing a number with a decimal point (such as 0.7 or 1.25) and directly writing the equivalent fraction. This practice makes the concrete link between decimal notation and the division into decimal parts visible.

In quarta primaria, i bambini imparano ad associare la scrittura frazionaria a quella decimale, comprendendo che una frazione rappresenta una parte dell'unità esprimibile anche attraverso un numero decimale finito. Gli alunni collegano così il valore del denominatore (come decimi o centesimi) alla corretta posizione delle cifre dopo la virgola.

L'errore più frequente riguarda la gestione dello zero e del valore posizionale: molti bambini tendono a scrivere semplicemente numeratore e denominatore separati dalla virgola (trasformando ad esempio 3/10 in 3,10), oppure confondono decimi e centesimi scrivendo 3/100 come 0,3 anziché 0,03.

Nelle attività pratiche del modello Scrivi la frazione come numero decimale, ai bambini viene presentata una frazione da trasformare nel rispettivo numero con la virgola, abituandoli a passare con sicurezza da una rappresentazione numerica all'altra.

In fourth grade of primary school, children learn to compare pairs of decimal numbers. To do this, they first examine the whole-number part and, if it is the same, they move on to comparing the decimal part digit by digit, starting from the tenths and then considering hundredths and thousandths.

The most common mistake consists in treating the digits after the decimal comma as if they were an ordinary whole number. Often the pupil thinks that 4,18 is greater than 4,5 simply because 18 is bigger than 5, forgetting to first compare the place value of the tenths.

In the worksheets for the activity Compare decimal numbers, the task typically requires observing two numbers with a decimal comma and placing between them the correct symbol chosen from >, < or =.

In fourth grade, children learn to break down a natural number by identifying all of its divisors. To do this, they apply division with zero remainder and the relationships between multiplication and division learned through the times tables, working in a systematic way so as not to miss any possibility.

The most common mistake is forgetting the number 1 or the number itself, which are always starting divisors. Confusion between multiples and divisors often arises as well, or the search is stopped too early because pairs are not considered (for example, for 20: 1 and 20, 2 and 10, 4 and 5).

In practical activities, the typical task requires: "Write all the divisors of a number". Given an assigned number, the pupil calculates and lists in ascending order all the exact corresponding divisors.

In this fourth-grade primary school unit, children practice carrying out equivalences by moving from one unit of mass to a smaller one (for example, from kilograms to grams). To make this conversion, they apply multiplication by 10, 100, or 1,000, drawing on their knowledge of numbers and the place value of digits.

The most common error consists of confusing the operation to be performed or miscounting the steps on the scale of measures. Children often become disoriented when they see the unit become smaller and apply division instead of multiplication, or they forget a factor of ten, for example converting from kilograms to grams by adding only two zeros instead of three.

In the worksheets with the exercise Convert to a smaller unit of mass, a starting measure expressed in a given unit is presented, and the student is asked to calculate and write the corresponding value in the indicated smaller unit.

In this stage of the fractions journey, the pupil learns to compare fractions with the same denominator. Knowing that the whole is divided into the same number of equal parts, the child understands that the larger fraction is the one that takes more parts, that is, the one with the higher numerator.

The most common mistake occurs when children get confused in using the symbols for greater than and less than, or when they look carelessly at the digits without connecting them to their concrete meaning, forgetting that, when the total number of parts is the same, only the value indicated by the numerator matters.

In the practice worksheets dedicated to this topic, the typical exercise offers pairs of fractions to compare (such as 3/8 and 5/8), asking to insert the correct sign to establish the order of magnitude.

In fourth grade, the child begins to use the fraction as a concrete operator to calculate a part of a known quantity. They learn to find the fraction of a group of elements (such as a/b of the total): they understand that the denominator indicates into how many equal parts to divide the whole to obtain the unit fraction (1/b), while the numerator specifies how many of these parts are to be taken into consideration.

The typical error consists in confusing the roles of the two terms of the fraction, inverting the operations: for example, multiplying by the denominator and dividing by the numerator, or adding the numbers together. Children often struggle to visualize the intermediate step, forgetting to first calculate the value of a single part and then multiply it.

In the worksheets on the model Fraction of a set: a/b of the whole, the pupil tackles numerical and contextualized problems: for example, calculating 3/5 of a group of 25 elements or 2/3 of 18 objects. To solve them, they first calculate the division (25 : 5 = 5) and then multiply the result obtained by the numerator (5 × 3 = 15).

In fourth grade, children strengthen their ability to order and compare natural numbers on the number line, working with graduated scales with intervals wider than a single unit. At this level, the focus is on reading the number line through jumps of ten units, consolidating mastery of the place value of digits.

The most common error occurs when the student automatically treats every tick mark as a single unit rather than as a ten, or when they lose track of the count while crossing the next hundred (for example, moving from 190 to 200).

In practical activities based on the model of jumping by tens on the number line, the child observes a partially graduated line and must work out the value of the empty tick marks, moving forward or backward to add or subtract whole tens.

In fourth grade, children consolidate written calculation by tackling multiplications between multi-digit numbers and a single-digit multiplier. The operation is set up by placing the single-digit factor under the units of the multiplicand, then multiplying in order: units, tens, hundreds and thousands.

The most frequent obstacle concerns handling the carry. Often the child adds the carry to the next digit before multiplying instead of after, or completely forgets to add it to the partial product just calculated.

In worksheets based on the Multiply in column by a single digit model, the exercise requires arranging the numbers in the calculation grid and carrying out the steps in column, recording the carries in the designated spaces until the final result is found.

In fourth grade, children learn to handle the main units of capacity by focusing on converting to a smaller unit of measure. Working with the most commonly used units — from the liter to its submultiples such as deciliters, centiliters and milliliters — pupils apply multiplications by 10, 100 or 1,000 to express the same quantity with a different unit.

The typical mistake is confusing the value of the number with the size of the unit: knowing that the target unit is smaller, some children are tempted to divide instead of multiply. Another frequent difficulty is miscounting the "steps" between non-consecutive units, such as forgetting that between liters and centiliters two places must be considered and one must multiply by 100.

In activities of the type Convert to a smaller unit of capacity, the child works with equalities to complete: starting from a given quantity, such as 5 l or 12 dl, they must calculate and write the corresponding value in the requested unit, such as centiliters or milliliters.

In fourth grade, as part of the work on large numbers and written calculation, children learn to compute column multiplications with multi-digit factors. At this level they tackle multiplication by two-digit numbers, working out the individual partial products and then adding them together to obtain the final result.

The most common error occurs when multiplying by the tens digit: children often forget to place the placeholder zero (or empty space) on the right, misaligning the second partial product. Another typical difficulty is confusing the carry-overs from the multiplication with those from the final addition, especially when they are written down in a disorderly way next to the calculations.

Worksheets based on the model Multiply multi-digit numbers in columns offer operations set up on grid paper or to be aligned independently, guiding the pupil to complete each step of the calculation in order up to the final sum.

In fourth grade, children tackle long division of natural numbers using a one-digit divisor. By applying the written procedure step by step, they learn to calculate the partial divisions, carry out the necessary subtractions, and clearly identify the quotient and the final remainder, if any.

The most frequent stumbling block concerns managing the intermediate steps: children often forget to bring down the next digit or make mistakes calculating the partial remainder before continuing. Another typical difficulty arises when an intermediate digit is smaller than the divisor and requires writing a zero in the quotient before proceeding.

In the worksheets of the Long division by a one-digit number template, the task is presented with operations to be set up in columns or already laid out with the designated spaces for the dividend, divisor, and intermediate calculations, asking the student to develop the entire algorithm until stating the result and the remainder obtained.

In fourth grade, children learn to use graphical representations to obtain concrete information. Faced with a bar chart, they can observe the different columns, compare them with each other, and quickly identify the most frequent value, the least frequent one, or the order relationship among the various elements analyzed.

The typical difficulty lies in confusing the visual height of the columns with the actual value when the numerical scale on the axis does not advance by one at a time. Students often simply count the tick marks or grid squares, forgetting to check whether each interval on the scale represents 2, 5, 10 or more units.

In the activities of the worksheet Compare the bars of the chart, the child is shown a bar chart alongside guided questions. The exercises ask them to directly compare two or more bars to determine which quantity is greater, identify which category corresponds to a given value, or calculate the numerical difference between two columns.

In fourth grade, children extend their knowledge of the number system beyond whole units, learning to read and write decimal numbers. At this level they connect the presence of the comma to the splitting of the whole number and recognize the place value of the digits that make up the decimal part, in particular tenths and hundredths.

The most common difficulty lies in confusing the place order after the comma: many children tend to swap tenths with hundredths, thinking that places develop from right to left as with whole numbers, or they ignore the comma and read the decimal part as if it were a separate number.

In practical activities, the work focuses on visual and positional recognition of values. With exercises such as Find the hundredths digit, the pupil is faced with a number containing a comma and must precisely identify the digit that occupies the second decimal place.

In fourth grade, children learn to identify and interpret decimal numbers on a number line, observing how the space between two whole numbers is divided into equal parts, such as tenths.

The most common mistake is counting the intermediate marks as if they were separate whole numbers, without considering the value of the interval. Often, after the value 1.9, pupils are tempted to read the next mark as 1.10 instead of recognising the step up to the whole number 2.

In exercises of the type Read the decimal number on the line, a number line is shown with some endpoints marked and a slider or arrow pointing to an intermediate mark: the child must count the correct steps starting from the reference whole number and write the number with the corresponding decimal point.

In fourth grade, children consolidate their use of the number line by learning to represent and read fractions as well. At this level, the pupil learns to interpret the space between whole numbers as a whole divided into equal parts: the total number of these parts corresponds to the denominator, while the steps taken from the start of the interval determine the numerator.

The most common difficulty consists in counting the drawn vertical tick marks rather than the actual spaces between them. Pupils often also count the zero tick, or simply count the graphic marks, thus assigning an incorrect denominator compared to the real parts into which the unit is divided.

In activities of the type Read the fraction on the number line, a graduated line is shown on which an arrow or a point highlights a precise position. The child must observe the division of the unit, count the segments, and write the fraction corresponding to the indicated point.

In fourth grade, children learn to confidently distinguish the main geometric relationships on the plane. Using the square grid as a visual guide, they analyze pairs of segments to see whether they keep the same distance from each other without touching (parallelism) or whether they meet forming right angles (perpendicularity).

The typical difficulty arises when the segments are drawn diagonally with respect to the grid. Many pupils tend to think that parallel lines must be exclusively horizontal or vertical, or they confuse any crossing of two lines with perpendicularity, neglecting the actual check for the presence of a right angle.

In the exercises of the worksheet Parallel or perpendicular?, the child observes several pairs of segments drawn on the grid. For each pair, they use the squares to verify their inclination and their mutual distance, finally determining whether the segments are parallel, perpendicular, or generic intersecting segments.

In fourth grade, children learn to measure the size of angles using a specific tool: the protractor. The student learns to place the center of the instrument on the vertex of the angle and align one of the sides with the zero line, then read the correct measure in degrees along the graduated arc.

The most frequent mistake is linked to the presence of a double scale of numbers on the protractor (running from 0° to 180° both from right and from left). Children often read the value on the opposite scale, confusing for example an acute angle of 50° with an obtuse angle of 130°, because they do not check which zero lies on the first side of the angle.

In the practice activities of the model Read the angle on the protractor, the child observes figures in which an angle is already positioned on a graduated instrument. The task requires identifying the correct starting scale, following the second side of the angle, and writing down the exact measure obtained.

In fourth grade, children develop an initial capacity for geometric visualization in space and in the plane. At this level they are able to observe graphic models or materials and correctly identify the main solids: prisms, pyramids, cylinders, cones and spheres.

The most common error concerns confusing plane figures with solid figures, such as calling a sphere a "circle" or a cube a "square", or mixing up solids with similar bases, for example taking a cone for a pyramid or a cylinder for a prism because they do not pay attention to the curvature of the surfaces.

In the practical activities of the model What solid is this?, the pupil is shown the depiction of a solid and asked to identify its correct type, matching the image to the corresponding geometric name.