Mathematics — Grade 8

In 8th grade, students determine the probability of a specific event in simple random experiments. They systematically record all possible outcomes and relate the number of favorable outcomes to the total number of all possible cases in order to express the result as a fraction, decimal, or percentage.

Favorable and unfavorable outcomes are frequently confused with each other: instead of dividing favorable cases by all possible cases (for example, 2 out of 6), learners mistakenly relate favorable outcomes to unfavorable ones (2 to 4). Incomplete counting of the total set also repeatedly leads to errors.

In classroom practice, students encounter concrete experimental setups:

• Rolling a die: Calculating the probability of events such as an even number or a number less than 3.
• Drawing a ball from a box: Determining the probability of a specific color given a certain distribution of colors.
• Drawing a numbered card: Determining the chance of drawing, for example, a prime number or a number divisible by 5 from a set of cards.

In 8th grade, students learn to reliably identify the two defining characteristics of a linear function in the coordinate system. They read the y-intercept as the intersection of the line with the vertical axis and determine the slope either visually from the function graph or algebraically using the coordinates of two points.

A typical mistake occurs when setting up the slope ratio: coordinate differences are often swapped, so that the difference in x-values is mistakenly divided by the difference in y-values. On the graph itself, the y-intercept is also occasionally confused with the x-intercept on the x-axis, or the negative sign of the slope is overlooked for falling lines.

In practice, students work with drawings in the coordinate system from which they directly read off the y-intercept or the slope using a slope triangle. In purely algebraic exercises, two points are given to determine the slope of a line through two points.

In 8th grade, students apply percentage calculations appropriately to changes. Starting from a base value, they determine the changed final value after an increase or decrease. To visualize parts and the total value, they use conceptual models such as percentage bars.

A typical error is failing to combine the calculated percentage amount with the initial value: learners calculate the absolute increase or discount correctly, but then forget to add it to or subtract it from the original value. Occasionally, the percentage itself is mistakenly subtracted directly from the initial value as if it were an absolute amount.

In practice, students work on targeted exercises from the templates Increase by percent and Decrease by percent. In each case, an initial quantity is given that increases or decreases by a fixed percentage to calculate the new total step by step.

In 8th grade, students apply the distributive property in both directions to transform algebraic expressions. They expand products of a number or variable and parentheses or, conversely, break down sums and differences by factoring out common factors.

Common mistakes occur primarily when dealing with negative signs, such as when a minus sign in front of the parentheses or within the factor is overlooked. Likewise, students sometimes forget to multiply the outer factor by every single term inside the parentheses.

In the exercises, learners work on tasks from two templates: In Expanding parentheses, given expressions with parentheses are converted into a sum. In Factoring out, students determine the greatest common factor of several terms of an expression in order to factor it out and represent the expression as a product.

In 8th grade, students grasp proportional relationships and calculate unknown quantities systematically. They relate ratios to one another and use calculation methods such as the rule of three or rearranging proportion equations to solve real-world problems mathematically.

Typical errors usually arise when proportional relationships are confused with additive patterns: students then add the same difference to both sides instead of multiplying or dividing by a constant factor. Carelessly swapping reference quantities when setting up fractions also regularly leads to incorrect intermediate steps.

In the exercises, learners encounter two main types of problems:

• Completing proportion equations: An equation consisting of two fractions or ratio pairs contains a missing value that must be calculated.
• Rule of three: from one quantity to another: In word problems, starting from a known pair of values, an intermediate step is used to determine the desired target quantity.

In 8th grade, students use the fixed relationship of interior angle sums: in every triangle, the sum of all interior angles is 180°, and in every quadrilateral, it is 360°. They add up the known angle measures and subtract this sum from the respective total to determine the missing angle size by calculation.

The two sum values are often confused, leading to the accidental assumption of 180° for quadrilaterals as well. Another typical error occurs during the step-by-step subtraction of multiple values from 180° or 360°, when subtotals are formed inaccurately or regrouping errors are made.

In practice, students work on exercise formats such as The third angle in a triangle, where two angle measures are given and the remaining angle is calculated. Correspondingly, the template The fourth angle in a quadrilateral requires determining the fourth angle from three given degree measurements.

In 8th grade, students work confidently with rational numbers in the form of decimals, just as they encounter them in everyday life. They perform additions and subtractions with positive and negative decimal numbers and use these calculations both mentally and as a check for more complex problems.

Common mistakes arise from confusing operation signs and positive/negative signs—especially when subtracting a negative number, such as in -2.4 - (-1.5). Many learners apply sign rules uncertainly or lose track of the sign of the final result when calculating decimal places.

In the task formats Adding signed decimals and Subtracting signed decimals, students solve specific problems with two or more signed decimal numbers, determining the correct sign and exact value step by step.

In Grade 8, students perform arithmetic operations with integers — that is, with positive numbers, negative numbers, and zero. They master multiplication and division both mentally and for checking calculations, and they understand the significance of the sign rules in these arithmetic operations.

Typical errors arise primarily from confusing the sign rules between multiplication/division and addition/subtraction. The mnemonic "minus times minus equals plus" is often mistakenly applied by learners to addition problems, or the negative sign is completely overlooked in calculations like -4 · 6 because only the absolute values of the numbers are calculated.

Typical exercises from the templates Multiplying Integers and Dividing Integers require quickly determining the sign and absolute value in calculations like (-7) · 8 or (-36) : (-4).

In 8th grade, students systematically explore two-stage random experiments. They determine all possible combinations of two consecutive steps and calculate the probabilities of specific overall outcomes using favorable and possible outcomes.

A typical mistake is assuming that all overall outcomes are equally likely. For instance, when rolling two dice simultaneously or consecutively, it is often assumed that each sum of the dice occurs with equal frequency. This overlooks the fact that a sum like 7 can result from six different pairs (such as 1 and 6, 2 and 5), whereas a sum like 2 can only result from a single pair (1 and 1).

In the exercises, students work on specific task formats:

• Sum of two dice: All 36 possible individual outcomes are recorded in a table to calculate probabilities for specific sums or ranges of values.
• Drawing twice with replacement: Balls are drawn consecutively from a total number, with the ball being returned to the urn after the first step so that the initial probabilities remain unchanged in the second step.

In 8th grade, students use the probability of an event to determine the expected absolute frequency for a fixed number of trials. They multiply the relative frequency or probability by the total number of repetitions to determine the expected value as a benchmark.

A typical mistake lies in a misunderstanding of randomness: many learners expect the calculated result to occur exactly in reality. They confuse theoretical expectation over many trials with a fixed guarantee for a specific experiment, or inappropriately round non-integer counts of successes.

In task formats such as How often is this to be expected? or How often can a red marble be expected?, urn or drawing experiments are usually used. Students extract the proportion of favorable outcomes from the problem description (for example, red marbles in an urn) and calculate the expected number of hits for a series of approximately 50, 100, or more draws.

In 8th grade, students learn how to systematically expand products of two sums or differences. They apply the distributive law by multiplying each term in the first set of parentheses by each term in the second set of parentheses, and then simplifying the resulting terms as much as possible.

A typical mistake is forgetting the middle terms: often, only the first and last terms are multiplied together. In addition, minus signs within the parentheses regularly lead to sign errors, as the sign must remain closely attached to the subsequent number or variable.

In practice, students encounter exercises based on templates such as expanding (x + a)(x + b), as well as more general practice problems focusing on expanding two sets of parentheses, where expressions are converted step by step into a sum without parentheses.

In 8th grade, students learn to systematically simplify algebraic expressions. They recognize which terms of an expression share the same variables and combine their numerical values through addition and subtraction. They apply this both to expressions with a single variable and to expressions with two different variables.

A common mistake occurs when different variables are merged together—for example, when 3a + 2b is incorrectly combined into 5ab. Frequently, the sign preceding a term is also overlooked, leading to plus and minus signs being swapped when rearranging terms.

Typical exercises follow formats such as Combining like terms and Combining like terms with two variables. Students are given unordered expressions such as 5x + 3y - 2x + 4y, identify or sort the corresponding parts, and write the final result in simplified form.

In grade 8, students learn to apply percentage calculations appropriately and conceptually to determine the base value (i.e., 100 %) from a known percentage value and the corresponding percentage rate. Visual models such as the percentage bar help students understand the relationship between the part and the total amount.

A typical mistake is confusing the given value with the base value. Many learners then incorrectly multiply the percentage value directly by the percentage rate, instead of recognizing that they must first divide to find 1 % and then calculate up to the full 100 %.

Typical exercises from the template Base Value from the Percentage Value provide a specific share (for example: “12 euros correspond to 20 %”). The task is to mathematically determine the desired original quantity of 100 %.

In 8th grade, students confidently apply the basic arithmetic operations to rational numbers. When dividing fractions, they use the rule of converting division into multiplication by the reciprocal. They perform these calculations on paper and, for simple numbers, also mentally to quickly check intermediate results or expressions.

A typical error occurs when the first fraction is accidentally inverted instead of the second, or when the numerator is mistakenly divided by the numerator and the denominator by the denominator. The latter leads to complicated complex fractions if the numbers are not divisible without a remainder.

Typical exercises from the template Dividing a fraction by a fraction require calculating expressions such as 3/4 : 2/5. Students form the reciprocal of the divisor (here 5/2), multiply the numerators and denominators together, and simplify the final result completely.

In 8th grade, students consolidate their skills in calculating with rational numbers. They multiply fractions with confidence by multiplying numerator by numerator and denominator by denominator. For simpler values, this is done mentally, or in writing to check their own calculation steps.

A typical mistake occurs due to confusion with the addition of fractions: some learners mistakenly first look for a common denominator or cross-multiply numerators and denominators instead of simply multiplying directly across.

In the worksheets on the topic of multiplying fractions, students work through concrete multiplication problems involving two or more fractions, simplify intermediate results skillfully, and determine the fully reduced result.

In this area, students multiply fractions by a natural number. They multiply the number by the numerator and leave the denominator unchanged. Before calculating or at the end, they cancel common factors to express the result as a fully simplified fraction or as a mixed number.

A typical mistake is multiplying both the numerator and the denominator by the natural number. Many confuse this operation with expanding fractions. As a result, the value of the fraction remains the same instead of being multiplied.

Practice exercises from the template Multiply a fraction by a natural number present direct calculation problems, such as four times two-sevenths. Learners perform the multiplication, identify opportunities to simplify, and write the final result in simplest form.

In this skill, students divide simple unit fractions with a numerator of 1 by whole numbers, as well as whole numbers by such unit fractions. In doing so, they mathematically grasp two fundamental processes of division: either a fractional part is broken down even further (as with one-third divided by 2), or they determine how many equal fractional parts fit into one or more whole units.

Confusion often arises with problems in which a whole number is divided by a fraction, such as 3 ÷ 1/4. Many learners have internalized the idea that division must make the result smaller, and mistakenly calculate 3 ÷ 4 = 3/4. They overlook that the problem is asking how many one-fourth pieces fit into three wholes, which leads to a larger result (12).

Typical worksheets titled How many times does the unit fraction fit? focus precisely on this concept. Learners calculate step by step how many times a specific unit fraction fits into a given number of wholes, thereby solidifying their understanding of division before moving on to more complex fraction operations.

In 8th grade, students use sound conceptual understandings of arithmetic operations with rational numbers to understand multiplication as scaling. They recognize that multiplying by a fraction increases, decreases, or leaves an initial number unchanged, depending on whether the fractional factor is greater than, less than, or equal to 1.

A typical stumbling block is the rule of thumb internalized in the early school years that multiplication always means "more" or "bigger." If this assumption is applied to rational numbers such as proper fractions (like 1/2 or 3/4), it leads to errors, because here the product is smaller than the initial number.

In exercises using the task format Comparing products without calculating, students directly compare expressions such as 24 · 3/5 with the initial number 24. Instead of calculating the value by hand, they justify the relationship using the fractional part and insert the appropriate comparison symbol.

In 8th grade, students calculate the size of a missing component angle. When a total angle is divided into two or more sections, they subtract the known angle measures in degrees from the total value to determine the exact remaining size.

A common mistake occurs when fixed angle measures, such as a right angle (90°) or a straight angle (180°), are assumed without checking, even though the problem specifies a different total angle. In addition, calculation errors can creep in when several component angles must be subtracted one after another.

In tasks on calculating the missing part of an angle, a geometric diagram with labeled angles is usually shown. For example, a total angle of 125° is given, consisting of a known part of 48° and an unknown angle. Students set up the calculation 125° - 48° = 77° and enter the result.

In 8th grade, students determine what percentage corresponds to a specific proportion. They appropriately relate a part to its corresponding base value and convert this fraction into a percentage. To develop a solid understanding of orders of magnitude, they work using conceptual models and visual aids such as percentage strips, from which the part and the whole can be read directly.

A typical error occurs when the base value and the percentage value are confused: frequently, the smaller number is hastily chosen as the numerator and the larger as the denominator—or vice versa—without considering the actual meaning of the information. This leads to incorrectly calculated proportions or results over 100% not being recognized as implausible.

In practical exercises such as the worksheet “What percentage is that?”, students determine the desired percentage from two given quantities by first writing the proportion as a fraction and then converting it into a percentage.

In the 8th grade, students solve realistic problems involving inversely proportional relationships, in which two quantities are inversely related (“The more, the less”). They use calculation methods such as the rule of three or the constant product of pairs of values to reliably determine unknown quantities in applied situations.

The typical mistake is to apply the pattern of a directly proportional relationship out of habit. For example, when more workers are deployed, some learners mistakenly calculate a longer instead of a shorter duration because they multiply by the same factor on both sides instead of dividing on the opposite side.

Typical tasks in the Inverse Proportionality template present classic real-world contexts: for instance, calculating how the required time changes when more excavators dig a pit, how long supplies last when the number of people changes, or how filling times decrease when using multiple inlet pipes.

In 8th grade, students determine the percentage value for a given base value and percentage rate. They apply percentage calculations appropriately, using visual aids such as percentage bars to determine proportions not just schematically, but with a genuine understanding of magnitude.

Typical mistakes occur primarily when the percentage value and the percentage rate are confused, or when converting the percentage rate fails—such as when the percentage rate is treated directly as a fixed quantity instead of calculating it as a hundredth part of the base value.

Typical tasks in the area of calculating the percentage value present concrete real-world or calculation situations: Students are given a base value and a percentage rate and calculate the required value from them, such as the savings from a 15% discount on a purchase value of €240.

In grade 8, students apply this fundamental step of the rule of three to solve real-world problems involving proportional relationships. From a known total quantity and a total value, they calculate the value of exactly one piece, one kilogram, or another basic unit by performing the appropriate division.

A common mistake occurs when it is unclear which value should be divided by which: for example, the number of items is often divided by the total price instead of the total price by the number of items. While this results in a number, it is not the desired price for a single unit.

In tasks such as How much is it for one?, learners encounter real-life scenarios, such as the total price of a bulk pack or the weight of several identical objects. Through division, students determine the unit price or individual weight as an intermediate step in order to then calculate any other quantities.

In 8th grade, students determine the area of a parallelogram by multiplying a base by the perpendicular height. They apply the formula A = b · h and learn to read the required dimensions directly from geometric drawings or real-world contexts.

A typical mistake occurs when the height is confused with the slanted side edge. Since students are used to simply multiplying two adjacent sides in a rectangle, they often use the slanted side length instead of the actual perpendicular height.

In the exercises of the template Area of a Parallelogram, learners mostly work with labeled figures. They identify the corresponding pairs of base and height, substitute the numerical values into the formula, and calculate the correct area.

In 8th grade, students learn how to determine the area of trapezoids. To do this, they identify the two parallel sides as well as the perpendicular distance between them, the height, and calculate the result using the formula A = (a + c) : 2 · h.

A typical mistake occurs when a slanted side is mistakenly used as the height. However, the height must always be at a right angle to the parallel bases. Likewise, when substituting into the expression, students sometimes forget to add the two bases first before dividing or multiplying.

In the exercises for the task template Area of a Trapezoid, students either work with geometric drawings from which they read off the required measurements, or they calculate the area directly from given lengths for the bases and height.

In 8th grade, students learn to reliably solve linear equations where the unknown variable appears on both sides. They use targeted operations on both sides of the equals sign to collect all terms with x on one side and all constant numbers on the other side, until the value of x is clearly determined.

Typical difficulties often arise with signs when terms with a minus sign need to be moved to the other side. In addition, students sometimes accidentally combine terms with x with pure numbers instead of keeping both types of terms cleanly separated.

In exercises such as the template Equation with x on both sides, learners encounter problems like 5x + 3 = 2x + 12. Step by step, they subtract terms like 2x and numbers like 3 on both sides to simplify the equation and calculate the value of x.

In Grade 8, students test whether a given numerical value satisfies an equation by using substitution. They replace the variable with the test number and calculate both the left and right sides of the equation step by step to check if both sides have the same value.

Typical errors occur especially when substituting negative numbers. If the number is not properly enclosed in parentheses, sign errors frequently arise in multiplications or when applying the order of operations. Additionally, some students mistakenly try to rearrange the equation instead of simply substituting the number and evaluating both sides separately.

In exercises such as Is the number a solution?, an equation is given along with a number. Students evaluate both expressions, compare the results, and finally decide with “Yes” or “No” whether it is a valid solution.

In 8th grade, students learn to systematically solve equations with one variable for x. To do this, they apply two transformation steps in succession: first, they isolate the variable term using addition or subtraction, before dividing by the coefficient in the second step.

A typical mistake is reversing the order of the calculation steps: learners often try to divide by the coefficient in front of the x first, instead of moving the constant term to the other side first. Calculation errors involving negative signs also occur regularly.

In the practice exercises, students work on problems of the form ax + b = c, such as 3x + 5 = 20 or 4x - 7 = 17. They perform the transformation steps line by line and write down the respective inverse operation on both sides of the equation.

In 8th grade, students learn to substitute specific numerical values for variables in an algebraic expression and calculate the overall value step by step. In doing so, they reliably apply fundamental arithmetic rules, such as the order of operations and handling parentheses.

Typical mistakes occur especially when substituting negative numbers. If a negative number is not carefully enclosed in parentheses, sign errors easily happen – such as when (-3)² is mistakenly calculated as -9 instead of 9. Working from left to right while ignoring the order of operations also frequently leads to incorrect intermediate results.

In the exercises for the template Calculating the value of an expression, students are given an expression with variables such as x or y (for example, 2x² - 4x + 3) and are asked to evaluate it precisely for specific values such as x = 3 or x = -2.

In 8th grade, students work with systems of linear equations consisting of two equations with two variables. They use algebraic transformations to systematically isolate or eliminate a variable, calculating step by step the exact values for both unknowns.

Typical errors usually occur with sign changes during transformations—such as when an equation is subtracted or brackets preceded by a minus sign are expanded. Another common source of error is stopping prematurely: after calculating the first variable, students often forget to substitute the value back into one of the original equations to determine the second unknown as well.

Tasks from the Linear System of Equations template provide specific pairs of equations. Students determine the appropriate pair of numbers that satisfies both equations simultaneously and can then check their result by substituting it into the original equations.

In 8th grade, students determine the function value y or f(x) for a given linear relationship. To do this, they substitute a given number for the variable x into the function equation and evaluate the expression step by step by multiplying by the slope and then adding the y-intercept.

Typical calculation errors occur primarily when dealing with negative numbers. If a negative x-value is substituted, the sign is often overlooked or calculated incorrectly during multiplication by the slope. Another common stumbling block is ignoring the order of operations ("multiplication before addition"), such as prematurely combining the y-intercept with the slope before multiplying by x.

In the exercises for the template Function value of a linear function, an equation such as f(x) = 3x - 5 is usually given. Students use it to calculate, for example, the value for f(-2) or complete a table of values by systematically substituting individual numbers and recording the final result.

In 8th grade, students substitute the coordinates of a point into the equation of a linear function. They calculate the value for the x-coordinate and check, through computational comparison with the y-coordinate, whether a true statement is obtained and the point is therefore on the line.

Typical mistakes often arise from confusing the x- and y-values when substituting them into the function equation. In addition, sign errors, especially with a negative slope or negative coordinates, often lead to an incorrect result.

In tasks from the topic Does the point lie on the graph?, a linear function equation is given along with the coordinates of a point. Students perform the point test in writing and give a reasoned decision as to whether the point lies on the line.

In 8th grade, students determine the zero of a linear function of the form y = mx + b. They set the function value to zero (0 = mx + b) and rearrange the equation step by step to calculate the value of x at which the graph of the function intersects the horizontal axis.

Typically, the zero is confused with the y-intercept: many mistakenly substitute x = 0 instead of setting the equation equal to zero, or simply read off the value b. Calculation errors also frequently occur when solving, especially sign errors when subtracting the intercept or during the subsequent division by a negative slope m.

In the exercises of the template Zero of a Linear Function, a function equation is usually given. Learners calculate the required x-value and record the result either as a number or as a complete coordinate point on the axis.

In 8th grade, students appropriately apply the fundamentals of percentage calculation to banking and financial topics. Using visual representations—such as percentage strips—they determine how much interest a specific savings balance (the principal) generates over the course of a year at a fixed interest rate.

Learners often confuse the interest rate (the percentage) with the actual interest amount (the monetary amount in euros). In addition, many find it difficult to imagine the scale when calculating purely with formulas, rather than first visualizing the balance and the shares using a strip model.

Typical problems in the area of interest on savings provide an amount of money and an annual interest rate. Students use this to determine the annual interest earned or calculate the new total balance at the end of one year.