Question:
### SubjectName: Linear Sequences (nth term) ### ConstructName: Match linear sequences to other representations, such as tables and graphs ### Question: ![The first three terms of a sequence represented using boxes. The first term is made of 3 boxes arranged with 2 on the base and one box above the furthers left box. The second term is the same pattern as the first but with two additional boxes added to the left hand side of the shape, one on top of the other. The third pattern is the second pattern with an additional two boxes added to the left hand side of the shape, one above the other again.]() \begin{tabular}{|l|c|c|c|}
\hline Position & \( 1 \) & \( 2 \) & \( 3 \) \\
\hline Term & \(  \) & \( \bigstar \) & \(  \) \\
\hline
\end{tabular} What value should replace the star?

Thinks the term in a sequence is double the position value

Believes dividing two positives will give a negative answer

Believes dividing a negative by a negative gives a negative answer

Thinks a divide and a negative sign next to each other makes a minus

Believes a negative power results in a negative answer

Believes multiplying two negatives gives a negative answer

Correctly finds the reciprocal of a number, but believes it should be opposite in sign to the original number.

Thinks the subtraction sign means divide

Believes dividing a positive by a negative gives a positive answer

Believes dividing a negative by a positive gives a positive answer

When solving an equation where there is a negative sign in front of the variable and its equivalent value, assumes the variable has a negative value

Believes that reciprocals are undefined for negative numbers

Has used a subtraction sign to represent division

Believes multiplying two positives gives a negative answer

Thinks the fraction bar means subtract rather than divide

When substiting into an algebraic term, raises the coefficient to the power of the variable rather than multiplying the coefficient and the variable..

Heads towards zero when subtracting from a negative

When substituting into an algebraic term, replaces variables with numbers without multiplying.

When there's a negative sign in the question, thinks the answer must be negative

when subtracting a fraction from an integer, just adds their absolute value and places a negative sign in front

When subtracting a negative thinks the answer must be negative

Believes that a fraction means dividing the denominator by the numerator

When subtracting a negative number, subtracts the absolute value rather than adding it

Thinks the division sign means to subtract

Assumes the denominator is divided by the numerator

Does not recognise when one part of a fraction is the negative of the other

Believes division is sometimes commutative

Does not consider the variable when evaluating an expression that requires substitution.

Thinks a divide and a negative sign next to each other makes a plus

Believes dividing by a unit fraction is equivalent to dividing by its reciprocal

When substituting into an algebraic term, believes they can disregard the coefficient

When dividing a fraction by an integer, divides the denominator by the integer

Makes the number negative when asked to find the reciprocal

Thinks the subtraction sign means multiply

Believes the square of a negative will also be negative

When subtracting a negative number, adds the absolute values and then writes the answer with a negative sign

When subtracting past zero, adds the absolute values and then writes the answer with a negative sign

When dividing fractions, does the reciprocal of the first fraction instead of the second

Does not realise that subtracting a negative is the same as adding it's additive inverse

When solving an equation, uses the same operation rather than the inverse.

When dividing a fraction by an integer, subtracts the integer from the denominator

When dividing a fraction by an integer, divides both the numerator and denominator by the integer

Assumes the negative sign in a power has no meaning

Does not recognise that subtracting a larger number from a smaller number will give a negative answer

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Weight Units ### ConstructName: Convert between g and kg ### Question: \( 4000 \mathrm{~g} \) is the same as \( k g \)

Thinks the reciprocal of a fraction is what number adds to that fraction to make one whole.

Thinks the inverse of multiplication is addition

Thinks if you multiply a fraction by itself you get a whole

When solving an equation thinks they add when in fact they need to multiply

Believes equivalent fractions are formed with addition not multiplication

When solving a problem that requires an inverse operation (e.g. missing number problems), does the original operation

Believes the fraction line represents addition

Adds or subtracts from numerator first when solving an equation with a fraction

When adding an integer to a fraction, thinks you add the integer to the numerator

When solving an equation, adds instead of dividing

When solving a problem that requires an inverse operation (e.g. missing number problems), does subtraction when division is required

Thinks the inverse of subtraction is multiplication

Thinks the inverse of addition is multiplication

When solving an equation, multiplies instead of adding

When solving an equation thinks they divide when in fact they need to multiply

Thinks the fraction bar means subtract rather than divide

Thinks a variable next to a number means addition rather than multiplication

When solving an equation,  subtracts instead of multiplying

Simplifies a fraction by adding or subtracting the same amount from the numerator  and denominator

When asked to multiply fractions, uses the method for adding instead

When converting from an improper fraction to a mixed number, adds on the part left to make the whole, rather than subtracting it

Believes that a number divided by itself equals the original number

Divides by the numerator and multiplies by the denominator when finding a fraction of an amount

Does not understand how to maintain equivalence in fractions

Believes that the reciprocal of a number is equal to that number (1/x = x)

when multiplying fractions multiplies by the reciprocal of the 2nd fraction instead

Forgets to change the numerators when finding equivalent fractions

When multiplying fractions, multiplies the numerator and adds the denominator

When solving an equation, uses the same operation rather than the inverse.

Believes they should multiply each numerator by its original denominator when adding/subtracting fractions

Multiplies by the denominator instead of dividing when finding a fraction of an amount

Uses numerator as whole part of a fraction

Thinks the inverse of addition is division

When multiplying a fraction by an integer, multiplies the denominator instead of the numerator

Thinks the inverse of subtraction is division

When finding a missing part in a ratio, divides rather than multiplies

Can identify the multiplier used to form an equivalent fraction but does not apply to the numerator

When multiplying a vector by a fraction, adds the numerator to the top number of the vector, and the denominator to the bottom number of the vector.

Thinks the multiplication sign means to add

Believes 0 multiplied by a number gives the number

Does not know how to solve linear equations involving fractions

Thinks two variables next to each other represent addition

When multiplying a fraction by an integer, multiplies both numerator and denominator by the integer

Believes to complete a part whole model you select a value already given in the model

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Expanding Double Brackets ### ConstructName: Carry out multiplication problems involving one negative integer ### Question: John is expanding these two brackets.

What should he get when he multiplies the two terms indicated by the arrows? ![The two brackets are (x+5)(x-3). The arrows are pointing at the +5 in the first bracket and the -3 in the second bracket.]()

Adds instead of multiplying when expanding bracket

Confuses collecting like terms with multiplication

Does not realize we can omit the multiplication symbol in algebraic expressions

Writes the variable before the coefficient in an algebraic term

Multiplies all terms together when simplifying an expression

Does not realise that two letters together, or a number and letter together, are multiplying

Does not realize we can simplify a term that involves powers

Does not understand how to multiply algebraic terms

When substituting into an algebraic term, believes they can disregard the coefficient

Does not realise that two letters together are multiplying

Does not think that a variable with no coefficient has a coefficient of 1.

Adds instead of multiplies when multiplying algebraic terms

Assumes you cannot use the fact that multiplication is commutative when simplifying an algebraic term.

Does not think a factorised expression is equivalent to its multiplied out form

Does not include brackets when attempting to multiply an expression

In algebra, thinks it's ok to write (for example) 0x rather than just 0

Writes repeated letters instead of using index form of a variable

When collecting like terms, only simplifies when there is an addition

Only multiplies the numerical terms when expanding a bracket

When collecting like terms, assigns positivity or negativity incorrectly.

When substituting into an algebraic term, replaces variables with numbers without multiplying.

Thinks index notation represents repeated addition rather than repeated multiplication

Only multiplies the first term in the expansion of a bracket

Does not know how to simplify algebraic fractions

Thinks the subtraction sign means multiply

When simplifying an algebraic fraction, only looks for number factors

Multiplies coefficients and letters separately and writes as an addition

Does not understand when like terms can be collected

Ignores coefficients when forming an expression from a question

Thinks a variable next to a number means addition rather than multiplication

When multiplying just writes the numbers next to each other

Thinks any 2 values multiplied together can be denoted with a squared symbol

When collecting like terms, treats subtractions as if they are additions

Misunderstands order of operations in algebraic expressions

When substiting into an algebraic term, raises the coefficient to the power of the variable rather than multiplying the coefficient and the variable..

Thinks you can ignore the variable when simplifying algebraic fractions

Does not understand the interpretation of the fx column in a frequency table

Cannot spot a coefficient within a written scenario

Writes a variable to the power of n as nx

Mixes up squaring and multiplying by 2 or doubling

Has multiplied the value of the first term by the the term being asked

Does not account for grammar when writing algebra e.g. ,then making a group

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Percentages of an Amount ### ConstructName: Calculate percentage of an amount mentally where the percent is greater than 100 ### Question: \( 130 \% \) of \( 40= \)

Found the percentage but forgot to add to 100%

When subtracting fractions, subtracts the numerators and denominators

Incorrectly cancels what they believe is a factor in algebraic fractions

Does not recognise when one part of a fraction is the negative of the other

Does not understand how to subtract algebraic fractions

When adding fractions, adds the numerators and denominators

When subtracting fractions, divides the numerators

Does not find a common denominator when adding/subtracting fractions

Believes that a fraction means dividing the denominator by the numerator

Removes the common denominator when making equivalent algebraic fractions

Does not understand equivalent fractions

When subtracting fractions, subtracts the numerators and multiplies the denominators

Forgets to change the numerators when finding equivalent fractions

Assumes the denominator is divided by the numerator

When adding fractions, changes to common denominator but leaves numerators the same

When adding fractions with identical numerators, leaves the numerator and adds the denominators

Uses the numerator of an improper fraction as the denominator of the equivalent mixed number

Does not consider the denominator when adding/ subtracting fractions. Adds/subtracts the numerator and makes the denominator the one with the highest value.

When rationalising the denominator, confuses the numerator with the denominator

When adding/subtracting fractions, adds/subtracts the denominators and multiplies the numerators

Simplifies a fraction by adding or subtracting the same amount from the numerator  and denominator

Mixes up the numerator and denominator when writing a division as a fraction

Subtracts numerators when dividing fractions with a common denominator

Thinks that when you cancel identical terms from the numerator and denominator, they just disappear

Mixes up the terms for the parts of a fraction

Does not understand how to maintain equivalence in fractions

Uses the total as the numerator of the fraction rather than the denominator.

Believes that if the numerators of 2 fractions are equal then the fractions are equal, regardless of the denominator

When asked to subtract a fraction from a unit, removes the unit value from the denominator

Believes that fractions with the same denominator are equal

Thinks the fraction bar means subtract rather than divide

Subtracts from left to right, ignoring the word ‘from’

Believes the subtraction button writes a fraction

Does the reciprocal of both fractions when dividing fractions

Thinks can combine the numerator and denominator after simplifying an algebraic fraction

When adding fractions, adds the numerators and multiplies the denominators

Believes they can change the subject of a formula just by swapping two letters

When asked to subtract a fraction from a unit, removes the unit value from the numerator.

Ordered fractions by comparing denominators rather than using equivalent fractions.

Is not familiar with the names of the different parts of a fraction

Has not noticed the difference between 'subtract' and 'subtract from'

When converting a fraction to a decimal, divides the denominator by the numerator

Adds or subtracts from numerator first when solving an equation with a fraction

Thinks you can ignore the variable when simplifying algebraic fractions

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Mental Addition and Subtraction ### ConstructName: Given the solution to one problem, deduce the answer to a related problem using the same operation ### Question: Tom and Katie are discussing related calculations
Tom says if you know that \( 4122+426=4548 \), then \( 4122+428 \) must equal \( \mathbf{4 5 5 0} \)

Katie says if you know that \( 4122-426=3696 \), then \( 4122-428 \) must equal \( \mathbf{3 6 9 4} \)
Who is correct?

Does not realise that adding a larger number gives a larger answer

When multiplying numbers with different bases, multiplies the powers and multiplies the base numbers.

Thinks we have to add the base numbers as well as the powers when multiplying indices

When multiplying numbers with the same base, multiplies the powers

Has multiplied the base number and the power

When multiplying algebraic terms written in the same base, adds the base variables and multiplies the powers.

Believes you multiply matching bases when carrying out a multiplication

Has added the powers rather than multiplying them

Believes that when multiplying a number with a power, the multiplier is added onto the power

When dividing numbers with different bases, divides the powers and divides the base numbers.

Thinks powers with different bases cannot be equal

Multiplied the powers rather than added them

Thinks multiplying by a number is the same as raising to the power of that number

When dividing numbers with different bases subtracts the powers but divides the base numbers.

Does not realise that multiplication can be broken down into factors

When dividing numbers with different bases, subtracts the powers and subtracts the base numbers.

Believes the columns in a base are multiples not powers of the base

Believes multiplying together all prime factors gives the highest common factor

When dividing numbers with the same base divides the base numbers and the powers

When dividing numbers with the same base correctly subtracts the powers but divides the base numbers.

Mixes up squaring and multiplying by 2 or doubling

Repeats the digits twice when squaring a number

Believes a power of 2 has no impact on its value

Believes the lowest common multiple is found my multiplying the numbers together

Believes when dividing numbers with the same base you divide the powers, rather than subtracting

When squaring a variable with a power, thinks that the power is squared rather than doubled.

Believes multiply all numbers in a prime factor venn diagram gives the value of one sets number

When multiplying large numbers, splits numbers into addends rather than factors

Thinks any 2 values multiplied together can be denoted with a squared symbol

Believes all the multiples of a number will also be multiples of its multiples

When multiplying by a multiple of 10, gives an answer 10 times bigger than it should be

When converting between non standard form and standard form assumes changes to the base will not affect the power

Thinks that when raising a variable with a coefficient to a power, only the coefficient is affected by the power

When squaring a variable, believes they also need to square the coefficient

Thinks multiplying by a variable is the same as raising to the power of that variable

When multiplying surds in the form ✓a × b✓c, believes the answer is  ✓(abc)

Thinks that doubling and then doubling again is not the same as multiplying by 4

Has added on the power of 1/4 rather than raise to the power of 1/4

Thinks multiplying a variable by a number is the same as raising that variable to the power of that number

Believes that if more than one number in a multiplication is divided by the same power of ten, the answer will be divided by the same power of ten.

Thinks that dividing both numbers in a multiplication sum by the same power of 10 results in equal answers

Thinks a power raised to a power leads to addition of the powers

Counts repeated factors as two separate factors

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Volume and Capacity Units ### ConstructName: Convert between cl and l ### Question: \[
650 \text { centilitres }=\bigstar \text { litres }
\]
What should replace the star?

When multiplying decimals, divides by the wrong power of 10 when reinserting the decimal

When multiplying decimals, ignores place value and just multiplies the digits

Does not understand place value within a number

When multiplying decimals, does not realize that the order and position of the digits (relative to each other) has to remain constant.

When multiplying decimals, multiplies rather than divides when reinserting the decimal.

When multiplying a decimal less than 1 by an integer, gives an answer 10 times smaller than it should be

When multiplying a decimal less than 1 by an integer, gives an answer 100 times smaller than it should be

When multiplying a decimal less than 1 by an integer, gives an answer 10 times bigger than it should be

When multiplying a decimal by a power of 10, just adds a zero to the end of the number, not changing the place value of the digits

When multiplying just writes the numbers next to each other

Does not understand the value of zeros as placeholders

When dividing decimals, does not realize that the order and position of the digits (relative to each other) has to remain constant.

Believes multiplying by 0.1 is the same as multiplying by 10

When adding decimals, just adds the digits and ignores place value

When multiplying a decimal by an integer, ignores decimal point and just multiplies the digits

When multiplying, treats digits in other columns as if they are units

Believes you write the carried value in the number rather than adding it to the next place value column

Divides instead of multiplies when multiplying by a decimal

Does not understand place value after the decimal point

Does not write the place holder zero when doing column multiplication

When subtracting decimals with a different number of decimals, subtracts one digit from more than one column

When dividing decimals, divides rather than multiplies when reinserting the decimal.

Identifies wrong place value column after decimal point

When subtracting decimals with a different number of decimal places, lines up the digits incorrectly

Thinks that when multiplying decimals (less than 1), the number will get bigger

Changes wrong place value column when rounding

Believes multiplying two negatives gives a negative answer

When adding decimals with a different number of decimal places, lines up the digits incorrectly

When multiplying involving a multiple of 10, gives an answer 10 times smaller than it should be

When doing operations with decimals, ignores significant zeros after the decimal point

When dividing a decimal by an integer, divides by the wrong power of 10 when reinserting the decimal.

When asked for the value of a digit, gives an answer 10 times too small

Believes that when multiplying by a power of 10 you add zeroes after the decimal point

When adding decimals, does not keep place value the same as the original digits

When two digits multiply to 10 or more during a multiplication problem, does not add carried value to the preceding digit

Believes dividing by 0.1 is the same as dividing by 10

When finding the square root of a decimal, ignores place value

When dividing a decimal by a power of 10 adds zeroes to start of the number ignoring the placement of the initial decimal point

When multiplying multiples of ten and the answer requires an extra digit, leaves off that extra digit

Adds 3 decimal places rather than moving 3 places

Hasn't lined up the place value columns when writing out the calculation

When adding, just writes the digit to the right of the number without considering place value

When two digits sum to 10 or more during addition of decimals, does not carry across decimal point to preceding digit.

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Averages and Range from Frequency Table ### ConstructName: Identify the mode from a frequency table ### Question: The frequency table shows the grade achieved by some students in a maths exam.
What was the modal grade? ![\begin{tabular}{|c|c|}
\hline Grade & Frequency \\
\hline \( 1 \) & \( 3 \) \\
\hline \( 2 \) & \( 7 \) \\
\hline \( 3 \) & \( 3 \) \\
\hline \( 4 \) & \( 9 \) \\
\hline \( 5 \) & \( 1 \) \\
\hline
\end{tabular}]()

Believes the mode is the most common frequency rather than the highest frequency

Multiplies the number of whole units by the numerator when converting from a mixed number to an improper fraction

Forgets to add the numerator when converting from a mixed number to an improper fraction

Believes the number of wholes in a mixed number multiplies by the fraction part

Thinks you add the number of wholes to the numerator when converting a mixed number to an improper fraction.

Thinks you multiply by the numerator and add the denominator when converting a mixed number to an improper fraction

Confused mixed numbers with improper fractions

Finds an equivalent improper fraction when asked for a mixed number

Writes the numerator of an improper fraction as the number of wholes when converting to a mixed number

Believes a mixed number can contain an improper fraction

thinks when writing a mixed number fraction as an improper fraction, you need to multiply the denominator by the whole

When converting from an improper fraction to a mixed number, just writes the tens digit as the number of wholes

Thinks the number of wholes in a mixed number represents the tens from the numerator of an improper fraction

Uses numerator as whole part of a fraction

Believes you add the whole to the numerators, ignoring denominators, when adding a mixed number to a proper fraction with the same denominator.

Believes the whole and remainder are the other way when changing an improper fraction to a mixed number

When subtracting a fraction from a mixed number, assumes they just subtract the numerator from the integer of the mixed number

When multiplying mixed numbers, forgets to multiply the fraction

Uses the numerator of an improper fraction as the denominator of the equivalent mixed number

When subtracting a fraction from a mixed number, assumes they subtract the numerator from the integer and the numerator of the mixed number

When converting from an improper fraction to a mixed number, adds on the part left to make the whole, rather than subtracting it

Simplifies integers when simplifying a mixed number

Thinks converting an integer to a fraction means both the numerator and denominator are equal

When the dividend is a mixed number and the divisor is an integer, thinks they only perform the calculation with the integers.

When the dividend is a mixed number and the divisor is an integer, thinks they only divide the fraction by the integer.

Multiplies the numerators when adding fractions with a common denominator

When multiplying mixed numbers thinks they add the integers

Believes when changing an improper fraction to a mixed number you divide the numerator by 10, regardless of the denominator

Mixes up the terms for the parts of a fraction

When subtracting mixed numbers, does not realise that a negative fractional answer needs to be taken away from their integer answer

When given a non-unit fraction of an amount, treats it like a unit fraction

Assumes they multiply by the numerator of fractional index

Finds unit amount but doesn't multiply up

Thinks numbers greater than 1 cannot be converted into fractions

Does not know how to write a fraction - just given an integer answer

Believes they should multiply each numerator by its original denominator when adding/subtracting fractions

Does not know how to interpret a fraction worded problem

Identifies the wrong part of a fraction model as the whole

Forgets to change the numerators when finding equivalent fractions

Denominator used for other parts rather than total parts

Uses the total as the numerator of the fraction rather than the denominator.

Multiplies up without finding unit amount

Is not familiar with the names of the different parts of a fraction

Finds the unit fraction when asked to find a non-unit fraction of an amount

Believes mixed numbers cannot be simplified

Believes the amount given is for the whole rather than a part

Mixes up the numerator and denominator when writing a division as a fraction

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: BIDMAS ### ConstructName: Use the order of operations to carry out calculations involving addition, subtraction, multiplication, and/or division ### Question: What does the following equal?
\[
3-5+4 \times 2
\]

Ignores the negative signs when adding positive and negative numbers

Believes that adding a negative will increase the value of the original number

Believes adding two negatives gives a positive answer

Thinks signs are interchangeable when adding or subtracting positive and negative numbers

Performs addition ahead of any other operation

Believes a negative number can be replaced with its positive equivalent when answering an addition problem from a numberline

Believes that adding a positive to a negative makes your answer more negative

Thinks two variables next to each other represent addition

When two digits sum to 10 or more during an addition problem, does not add one to the preceding digit

Thinks positive or negative sign affects the term before it

When adding, just writes the digit to the right of the number without considering place value

Believes we move down a number line when we add

Believes multiplying a positive by a negative gives a positive answer

Believes the range is largest add smallest

Doesn't recognise commutativity of addition with negative numbers

Performs addition ahead of multiplication

Multiplied the powers rather than added them

When adding an integer to a fraction, thinks you add the integer to the numerator

Thinks multiplication and addition are the same

Lines up the digits incorrectly when doing an addition involving two numbers with a different amount of digits

When subtracting a negative number from a positive number, uses a method which assumes one of the negative signs can be ignored

Thinks they find the length of a line between 2 points by adding coordinates, rather than subtracting

Carries out operations from left to right regardless of priority order

When collecting like terms, treats subtractions as if they are additions

To find the range adds the biggest and smallest number rather than subtract

When two digits sum to 10 or more during addition of decimals, does not carry across decimal point to preceding digit.

Believes adding b to any integer gives a multiple of b

Has increased by the value of the first value, rather than the difference

Does not realise that adding a larger number gives a larger answer

When adding decimals with a different number of decimal places, lines up the digits incorrectly

Thinks the difference between a positive number and a negative number is equal to the positive number

When subtracting a negative number, subtracts the absolute value rather than adding it

Confuses the direction of vectors when adding or subtracting

Does not realise addition is commutative

Thinks a divide and a negative sign next to each other makes a plus

Multiplies the numerators when adding fractions with a common denominator

When adding fractions, changes to common denominator but leaves numerators the same

When solving an equation, multiplies instead of adding

When adding variables with powers, thinks they can add the powers

Does not divide by 2 when finding midpoint

When adding fractions, adds the numerators and denominators

Ignores negative sign when expanding bracket

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Direct Proportion ### ConstructName: Use direct proportion to solve non-unitary missing amounts in problems (e.g. recipes) ### Question: Twelve rulers cost \( 96 p \).
How much do five rulers cost?

Multiplies up without finding unit price in proportion question

Does not know that sample size affects reliability

Confuses relative frequency and expected frequency

Gives frequency when asked for relative frequency

Believes any sample gives an equal chance of selection

Believes 'data' and 'sample' are interchangeable

Believes the frequency represented by a pie chart sector can be judged visually, without considering the total it represents

Confuses theoretical and experimental probability

Believes biased selection methods can be random

Cannot evaluate if an estimate is an underestimate or overestimate

Believes data implies equal chance of selection

Thinks a higher average means more consistent data

Thinks only one variable can introduce bias

Misunderstands how to adjust proportions for different sample sizes in stratified sampling

Thinks that probabilities of an event occurring or not occurring must be equal

Thinks a greater range means more consistent data

Believes the mean is always representative

Thinks a lower average means the data is more consistent

Thinks that we cannot calculate the range if values are repeated

Confuses random and stratified sampling methods

Uses the other part instead of total frequency as denominator for relative frequency

Misunderstands the capture-recapture method

Does not understand equivalent fractions

Believes you can make assumptions about size of pie chart sectors without enough information

Does not understand that a probability of 1 represents something that is certain

Believes all possible outcomes must have equal probability

Believes stratified samples should be evenly split

Confuses the method of selection with the group selected in statistics

Believes that any percentage of a larger number will be greater than any percentage of a smaller number

Believes we can categorise data as primary or secondary without knowing the source

Does not know how to read information off a pie chart

Uses numerators to convert fractions to ratio despite different denominators

Believes it is not possible to calculate the frequency for a pie chart sector despite being given enough information to do so

Mixes up quantitative and qualitative data types

Ignores a more than question and just states one of the values

Does not think 'other' is an acceptable category for data

Does not understand that a probability of 0 represents something that is impossible

Believes that fractions with the same denominator are equal

Believes if you changed all values by the same proportion the range would not change

Does not know the meaning of attribute data

Does not find a common denominator when comparing fractions

Doesn't take into account cost when calculating expected return/profit

Believes you do not include an "other" category when finding a total frequency

Does not recognise that probabilities of all possible outcomes will sum to 1

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Adding and Subtracting Negative Numbers ### ConstructName: Carry out missing number subtraction problems involving negative integers ### Question: ![Number line with -7 and -5 marked on it. From the -7 there is an arrow to the right labelled +5. Then there is an arrow pointing to the left labelled b and ending at -5.]() What is the value of \( b \) ?

Counts the starting number when counting backwards

Thinks that only the sides need to be equal for a polygon to be regular

Thinks that only the angles need to be the same for a polygon to be regular

Assumes regular shape refers only to a square

Does not know the meaning of the word regular

Does know know equilateral means the same as regular

Does not understand that all angles and sides must be equal for regular shapes

Does not realise that all regular shapes of a certain type will be similar

Does not remember that a rhombus has equal sides

Thinks rule for calculating interior angles only applies to regular polygons

Does not know the properties of a rectangle

Believes a rectangle has four equal sides

Thinks the term polygons only applies to shapes with more than 3 sides

Does not know the properties of a rhombus

Does not know the properties of an equilateral triangle

Does not know the meaning of equilateral

Does not recall rectangle side length property

Believes a rectangle has one pair of equal sides

Believes a rhombus has one pair of equal sides

Confuses isosceles and equilateral triangles

Does not understand the naming structure of polygons

Does not understand that shapes are congruent if they have the same size and shape

Does not recognise the properties of a kite

Does not know the meaning of uniform cross section

Does not know a rhombus is a type of parallelogram

Does not know a rectangle is a type of parallelogram

Does not believe a shape can have two correct names e.g. rectangle and parallelogram

Believes a polygon can have curved sides

Thinks the interior angle of a regular polygon is 10n

Does not know the properties of an isosceles triangle

Does not understand to find each interior angle of a regular polygon you divide the sum by the number of sides

Believes an arrowhead has four equal sides

Does not know the properties of a parallelogram

Thinks rectangles have rotational symmetry or order 4

Does not realise that all circles are similar

Does not realise similar has a specific mathematical meaning

Does not understand that base angles in an isosceles are equal

Does not know what dashes on the sides of shapes mean

Thinks congruent shapes are the same shape but can be different sizes

Believes a parallelogram has one pair of equal sides

Believes only straight lines count as edges on 3D shapes

Does not know the meaning of the word equidistant

Believes there is only one orientation for the isosceles triangle base angles

Does not know how to find order of rotational symmetry

Cannot identify a line of symmetry in a simple shape

Does not know that a single letter labels a vertex

When given the perimeter of a regular polygon, subtracts instead of divides to find each side length

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Converting between Decimals and Percentages ### ConstructName: Convert decimals less than 1 with 2 decimal place to percentages ### Question: Convert this decimal to a percentage
\(
0.07
\)

Thinks you need to just add a % sign to a decimal to convert to a percentage

Believes an outlier is a value that does not fit the same characteristics as the other values

Believes an outlier is the value in a data set with the biggest denominator, even if this value is close to the mean

Believes an outlier is the smallest value in a data set, even if this value is close to the mean

Believes an outlier is not determined by distance from the mean

Believes an outlier has to be a whole number

Believes extreme values can affect the mode

Believes extreme values can affect the median

Does not understand that removing an outlier will impact the range

Thinks a higher average means more consistent data

Believes an outlier is determined by the number of decimal places a number has

Identifies the biggest value for the range

Thinks a greater range means more consistent data

Thinks only one variable can introduce bias

Believes that fractions with larger denominators are greater in size

Chosen the longest value rather than the greatest

Does not understand equivalent fractions

Thinks a lower average means the data is more consistent

Believes when identifying the mode you look at all values, not just frequencies, for the highest number

Does not think 'other' is an acceptable category for data

Ordered fractions by comparing denominators rather than using equivalent fractions.

Believes that fractions with the same denominator are equal

Includes a data point with the value, when asked to find the number of data points above a value

When comparing unit fractions believes that fractions with larger denominators are greater in size.

When asked for the mean of a list of data, gives the mode

Does not find a common denominator when comparing fractions

Thinks that we cannot calculate the range if values are repeated

believes that a fraction with the largest denominator will be smallest regardless of the numerator

Believes the lower quartile is the lowest value in a list

Believes the mode is the most common frequency rather than the highest frequency

When asked for the mean of a list of data, gives the median

Believes even numbers can have odd multiples

Mixes up smallest and largest when ordering fractions

Does not understand the meaning of "value"

Does not know that 7 and -7 are different

Has increased by the value of the first value, rather than the difference

Thinks the difference between the numerator and denominator determines how close the fraction is to 1

Believes you don't need to numerically order the data before calculating the median

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Written Multiplication ### ConstructName: Multiply 3 digit integers by 2 digit integers using long multiplication ### Question: What is a correct way to set up this problem to solve it using long multiplication?
\[
128 \times 26
\]

Hasn't lined up the place value columns when writing out the calculation

Mixes up sides of a ratio, when working with vectors for line segments.

Assumes a ratio is part to part instead of part to whole

Does not know how to find the length of a line segment from coordinates

In ratio where total is given, treats that as one part without dividing

In ratio question where one side of ratio is given, treats that as the total

In ratio question where one side of ratio is given, uses that as one part in ratio without dividing

When finding a point on a line, assumes the point is always the midpoint

Divides total amount by each side of the ratio instead of dividing by the sum of the parts

Shares equally instead of in given ratio

Believes angles on a straight line add to 360 degrees

Thinks the number in the ratio is the total

When converting ratio to fractions, puts the other side of the ratio as the denominator rather than the total

Finds one part of a ratio but doesn't multiply that by the number of parts needed

When drawing bar diagram for ratio, shows the total amount as the total of one side

When finding a missing part in a ratio, divides rather than multiplies

Forgets to add starting value when calculating new coordinates

Divides length of original by length of image to find scale factor, rather than image by original.

Thinks that all angles anywhere along a straight line add to 180

In ratio question where the difference between the sides is given, uses that as one part without dividing

When reading a dash on a number line does not take into account the number at the start or the width of each division

When drawing bar diagram for ratio, shows the total as the difference between parts

When describing the relationship between the parts in a ratio, replaces the : with a decimal point to say how many times bigger a part is than the other

Does not divide by 2 when finding midpoint

Believes adding coordinates of endpoint and midpoint of a line will give the other endpoint

Believes the amount given is for the whole rather than a part

Beileves you can use the full length when finding an area of part of a composite shape, when part of the length is needed

Finds the circumference of the whole circle instead of arc length

Believes the total of a part whole model is always 1 (or equivalent) instead of the given total

Makes an assumption about line segments being equal within a shape

Does not understand that in the ratio 1:n the total number of parts would be 1+n

When calculating a scale factor, thinks the division is the other way around.

When finding the distance between coordinates, includes the starting number

Believes a proportion of a whole is whole/part not part/whole

When writing ratio, writes total as one side

Believes the gradient of a line is given by the x intercept

Believes ratios can be combined without equalizing common terms

Does not realise that the order is important in a ratio

Believes quartiles are at 1/4 and 3/4 through the range of the data

Thinks they find the length of a line between 2 points by adding coordinates, rather than subtracting

When enlarging a shape, thinks the scale factor divides the movement from the centre of enlargement to a vertex rather than multiplies

Believes that 1 part of a bar model will always have a value of 1

Divides length of original by length of image rather finding length of image divided by length of original to find an enlargement scale factor.

Confuses section and whole when using a bar model

Thinks the gradient of a line is found from 2 points by finding the length of that line between those points

When converting fractions to ratios, puts the other side of the ratio as the total of the numerator and denominator

Thinks co-interior angles sum to 360 degrees

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Time Series and Line Graphs ### ConstructName: Interpret continuous data using a line graph ### Question: This graph shows the amount of potatoes consumed by a restaurant in a week.

How many more potatoes were consumed on Sunday compared to Saturday? ![Chart of potatoes consumed with kilos against day of week]()

Reads one point on a time series graph instead of all the values from a given time period

When converting from an improper fraction to a mixed number, adds on the part left to make the whole, rather than subtracting it

When asked to subtract a fraction from a unit, removes the unit value from the numerator.

Confused mixed numbers with improper fractions

Writes the numerator of an improper fraction as the number of wholes when converting to a mixed number

When asked to subtract a fraction from a unit, removes the unit value from the denominator

When converting from an improper fraction to a mixed number, just writes the tens digit as the number of wholes

Believes the whole and remainder are the other way when changing an improper fraction to a mixed number

Uses numerator as whole part of a fraction

Uses the numerator of an improper fraction as the denominator of the equivalent mixed number

When expressing one quantity (x) as a fraction of another (y), believes the answer is 1/x.

When subtracting a fraction from a mixed number, assumes they subtract the numerator from the integer and the numerator of the mixed number

When subtracting a fraction from a mixed number, assumes they just subtract the numerator from the integer of the mixed number

Finds an equivalent improper fraction when asked for a mixed number

Thinks the fraction 1/n can express sharing any number of items between n people

Does not know how to interpret a fraction worded problem

Thinks numbers greater than 1 cannot be converted into fractions

Believes a mixed number can contain an improper fraction

Thinks that a unit fraction can be simplified further

Simplifies integers when simplifying a mixed number

Forgets that a number divided by itself is 1

When asked to take off a fraction of an amount, subtracts the fraction itself

Multiplies the number of whole units by the numerator when converting from a mixed number to an improper fraction

Thinks converting an integer to a fraction means both the numerator and denominator are equal

Does not know that you can leave a fraction in improper form

Thinks you add the number of wholes to the numerator when converting a mixed number to an improper fraction.

Forgets to add the numerator when converting from a mixed number to an improper fraction

When given a non-unit fraction of an amount, treats it like a unit fraction

When subtracting fractions, subtracts the numerators and denominators

Thinks the fraction 1/n can express sharing n items between any number of people

Thinks you multiply by the numerator and add the denominator when converting a mixed number to an improper fraction

Only applies a division to one of multiple terms in a numerator when simplifying an algebraic fraction

Thinks if you multiply a fraction by itself you get a whole

Does not understand that a remainder can be written as a fraction

When adding fractions, adds the numerators and denominators

Does not understand equivalent fractions

Identifies the wrong part of a fraction model as the whole

Uses the total as the numerator of the fraction rather than the denominator.

Simplifies a fraction by adding or subtracting the same amount from the numerator  and denominator

Thinks that fractions involving a fraction as the denominator can just be read as mixed numbers

Does not consider the denominator when adding/ subtracting fractions. Adds/subtracts the numerator and makes the denominator the one with the highest value.

when subtracting a fraction from an integer, just adds their absolute value and places a negative sign in front

Mixes up the terms for the parts of a fraction

Believes that a fraction means dividing the denominator by the numerator

Writes fraction rather than fraction of amount

Denominator used for other parts rather than total parts

Believes when changing an improper fraction to a mixed number you divide the numerator by 10, regardless of the denominator

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Converting between Fractions and Decimals ### ConstructName: Convert fractions less than 1 to terminating decimals of 2 decimal places ### Question: \( \frac{6}{25}=? \)

Converts a fraction to a decimal by placing a decimal point between the numerator and denominator

Believes multiplying a positive by a negative gives a positive answer

Believes multiplying two negatives gives a negative answer

Believes dividing a negative by a positive gives a positive answer

Believes dividing a positive by a negative gives a positive answer

Believes adding two negatives gives a positive answer

Believes dividing a negative by a negative gives a negative answer

Thinks signs are interchangeable when adding or subtracting positive and negative numbers

Believes multiplying two positives gives a negative answer

Thinks multiplying two or more negative numbers together always gives a positive answer, regardless of how many negative numbers there are in the sum

When multiplying with negative numbers, assumes any negative sign can be ignored

Believes the square of a negative will also be negative

Ignores negative sign when expanding bracket

Ignores the negative signs when adding positive and negative numbers

When factorising into double brackets, believes the product of the constants in the brackets is of the opposite sign to the constant in the expanded equation.

Believes raising a negative number to an odd power will give a positive answer

Believes that the sign changes when factorising an expression into single brackets

When subtracting a negative number, subtracts the absolute value rather than adding it

Does not realise that subtracting a negative is the same as adding it's additive inverse

Confuses negative and positive correlation

Believes a number raised to a power will always give a positive answer

Believes two odd numbers multiply to give an even number

Believes that the directions of positive and negative vectors are the opposite way around.

When solving simultaneous equations, believes the terms must have the same sign in order to be eliminated.

Believes dividing two positives will give a negative answer

Does not understand the link between the sign of the root and the sign in the bracket

When calculating the power of a negative number, thinks they can ignore the sign when multiplying and then put a negative sign back on at the end, regardless of the power

Factorises difference of two squares with the same sign in both brackets

Only multiplies the first term in the expansion of a bracket

Adds instead of multiplying when expanding bracket

Only multiplies second term in the expansion of a bracket

Does not think a square root can be negative

Does not believe that a negative multiple of a vector is in the opposite direction

When subtracting a negative number from a positive number, uses a method which assumes one of the negative signs can be ignored

When multiplying out 2 or more brackets, does not multiply all the combinations of terms

When subtracting from a negative thinks the answer must be positive

When subtracting a negative thinks the answer must be negative

Believes reflection in y = -x changes sign of x coordinate only

Believes they only need to multiply the first and last pairs of terms when expanding double brackets

Believes multiplication is not commutative

Thinks the subtraction sign means multiply

Believes the centre of rotation is always the origin

Does not understand that the negative outside the bracket means answer will be negative

Thinks the difference between a positive number and a negative number is equal to the positive number

Does not know that 7 and -7 are different

Believes reflection in y = -x changes sign of y coordinate only

Does not realise that when you multiply by 0 the answer will always be 0

Thinks positive or negative sign affects the term before it

Believes they can multiply/divide through by a negative number without changing the direction of the inequality

Believes squaring a negative number just changes the sign

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Pie Chart ### ConstructName: Calculate the sector angle on a pie chart for a given group ### Question: A diver wants to make a pie chart of his trip. \begin{tabular}{|c|c|c|}
\hline & Frequency & Degrees \\
\hline Manta Ray & \( 16 \) & \( 180 \) \\
\hline Hammerhead & \( 4 \) & \( 45 \) \\
\hline Conger Eel & \( 12 \) & \( \mathrm{~F} \) \\
\hline
\end{tabular} Which one of these is a correct method for finding F?

Finds the degrees for 1 person rather than the required number of people for a pie chart sector

When completing the square, believes the constant in the bracket is the coefficient of x

When completing the square, believes the constant in the bracket is double the coefficient of x

When completing the square in the form (x - a)^2 + b, believes b is the difference between a and the constant of the original quadratic

When completing the square in the form (x - a)^2 + b, believes b = the original constant

When completing the square, believes a coefficient of x^2 does not affect the value of the constant inside the bracket

When completing the square, believes the constant in the bracket is the square root of the coefficient of x

When completing the square, believes that a is always half the x-coefficient.

When completing the square in the form (x - a)^2 + b, believes b = a^2 + the original constant

When completing the square in the form (x - a)^2 + b, believes b = -a^2 regardless of the original constant

In completed square form, confuses the x and y coordinates of the turning point

Believes that when factorising a quadratic with a coefficient of x squared, the two constants in the brackets still add to give the coefficient of x

When factorising into double brackets, believes the sum of the two values in the brackets equals bx rather than b

Believes that in completed square form, the x co-ordinate of the turning point has the same sign as the constant in the brackets.

When factorising into double brackets, finds the two values which sum to give the constant rather than the coefficient of x

Believes the second term in a 2 term quadratic equation is always a coefficient and the variable

Believes the coefficent of x in an expanded quadratic comes from multiplying the two numbers in the brackets

Believes that in completed square form, they need to change the sign of the constant to get the turning point

Mixes up squaring and multiplying by 2 or doubling

Believes that in completed square form, they need to change the signs of both values to get the turning point

Believes the constant in an expanded quadratic comes from adding the two numbers in the brackets

Thinks factorising a quadratic means factorising the variable terms into a single bracket, and leaving out the constant

Only multiplies the first term in the expansion of a bracket

When factorising a quadratic with a non-unit coefficient of x squared, believes that coefficient will be in front of both x terms in the factorised form

Believes the numbers in factorising should multiply to give the coefficient of x and add to give the constant

When multiplying out 2 or more brackets, does not multiply all the combinations of terms

Adds instead of multiplying when expanding bracket

Only multiplies second term in the expansion of a bracket

When solving an equation thinks they add when in fact they need to multiply

When factorising into single brackets, factorises the first term correctly but subtracts the number outside the bracket with the second term in the expression to find the second term in the bracket

Believes the inverse of square rooting is doubling

Believes that when factorising a quadratic, they can choose any two numbers that multiply to give the constant

Thinks that the square root of an expression square roots each term in the expression, rather than square rooting the whole expression

When squaring a variable, believes they also need to square the coefficient

Does not include brackets when attempting to multiply an expression

Believes they can factorise a difference of two squares by placing the constant in both brackets without square rooting

Believes we can only take the square root of a perfect square

Writes a repeated root once, rather than in a squared bracket

When factorising, finds a factor that goes into only the first term of the expression

Squares when asked to find the square root

Thinks converting an integer to a fraction means both the numerator and denominator are equal

Believes that when factorising a quadratic, they can choose any two numbers that add to give the coefficient of x

Believes a power of 2 has no impact on its value

When solving an equation, uses the same operation rather than the inverse.

Believes expanding a squared expression means writing it as a set of double brackets

When solving an equation forgets to eliminate the coefficient in front of the variable in the last step

Does not think that a variable with no coefficient has a coefficient of 1.

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Mental Multiplication and Division ### ConstructName: Recognise and use efficient methods for mental multiplication ### Question: Tom and Katie are discussing ways to calculate \( 49 \times 9 \) mentally
Tom multiplies \( 49 \) by \( 10 \) and then subtracts \( 49 \) from the answer.
Katie does \( 50 \times 9 \) and then subtracts \( 9 \) from the answer
Who would get the correct answer?

Believes that the compensation strategy for the multiplication of 2 numbers can only be used one way round.

When multiplying decimals, divides by the wrong power of 10 when reinserting the decimal

When multiplying a decimal less than 1 by an integer, gives an answer 10 times bigger than it should be

When multiplying decimals, ignores place value and just multiplies the digits

When multiplying decimals, multiplies rather than divides when reinserting the decimal.

When multiplying a decimal by an integer, ignores decimal point and just multiplies the digits

When multiplying decimals, does not realize that the order and position of the digits (relative to each other) has to remain constant.

When multiplying a decimal less than 1 by an integer, gives an answer 100 times smaller than it should be

When multiplying a decimal less than 1 by an integer, gives an answer 10 times smaller than it should be

Does not understand place value within a number

When multiplying a decimal by a power of 10, just adds a zero to the end of the number, not changing the place value of the digits

When multiplying just writes the numbers next to each other

Thinks that when multiplying decimals (less than 1), the number will get bigger

When adding decimals, just adds the digits and ignores place value

Mixes up squaring and multiplying by 2 or doubling

When multiplying, treats digits in other columns as if they are units

Believes multiplying by 0.1 is the same as multiplying by 10

Believes you write the carried value in the number rather than adding it to the next place value column

When multiplying by 1 gives an answer of 1 (for a number other than 1)

Ignores place value when adding a whole number and a decimal

Divides instead of multiplies when multiplying by a decimal

Has multiplied the base number and the power

When dividing a decimal by a whole number, ignores decimal point and just divides the digits

When dividing decimals, does not realize that the order and position of the digits (relative to each other) has to remain constant.

When finding the square root of a decimal, ignores place value

When doing operations with decimals, ignores significant zeros after the decimal point

Believes a power of 2 has no impact on its value

Believes multiplying two negatives gives a negative answer

Has written something to the power of 1 as 0.1

When dividing decimals, divides rather than multiplies when reinserting the decimal.

Believes that when multiplying an integer by a decimal greater than 1, you multiply by the integer component of the decimal but add on the decimal component multiplied by 10.

When multiplying with negative numbers, assumes any negative sign can be ignored

Thinks any 2 values multiplied together can be denoted with a squared symbol

When dividing a decimal by an integer, divides by the wrong power of 10 when reinserting the decimal.

When dividing a decimal by a power of 10 adds zeroes to start of the number ignoring the placement of the initial decimal point

When adding, just writes the digit to the right of the number without considering place value

Does not understand place value after the decimal point

Believes that when multiplying by a power of 10 you add zeroes after the decimal point

Does not realise that in standard form the decimal we use must be between 1 and 10

When multiplying out 2 or more brackets, does not multiply all the combinations of terms

Does not understand the value of zeros as placeholders

Added rather than multiplied by a scale factor

When adding decimals, does not keep place value the same as the original digits

Believes dividing by 0.1 is the same as dividing by 10

When two digits sum to 10 or more during addition of decimals, does not carry across decimal point to preceding digit.

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Ordering Decimals ### ConstructName: Read a decimal on a scale where the required measurement is marked by a dash between two numbers ### Question: ![A number line from 0 to 1 split into 5 parts with a red arrow pointing to the marker for the value 0.4]() What number is the arrow pointing to?

When reading decimals on a number line, assumes each dash is 0.1

Thinks the number in the ratio is the total

Shares equally instead of in given ratio

Thinks a difference of one part in a ratio means the quantities will differ by one unit

In ratio question where one side of ratio is given, treats that as the total

Finds one part of a ratio but doesn't multiply that by the number of parts needed

Does not believe you can scale up a proportional statement

Estimates shares of a ratio instead of calculating

When answering questions regarding adding items to get a new ratio, looks at the difference between the numbers in the original and new ratio rather than finding the size of a part

In ratio question where the difference between the sides is given, uses that as the total for one side instead

Thinks adding same amount to each side of a ratio gives an equivalent ratio

In ratio question where the difference between the sides is given, uses that as one part without dividing

In ratio question where one side of ratio is given, uses that as one part in ratio without dividing

When writing ratio from diagram, writes total as one side

Does not scale up a ratio to find an equivalent value

Finds unit amount but doesn't multiply up

When converting fractions to ratios, puts the other side of the ratio as the total of the numerator and denominator

When writing ratio, writes total as one side

Multiplies up without finding unit amount

In ratio where total is given, treats that as one part without dividing

Believes they can change only one side of a ratio and it will remain equivalent

Thinks you need to include the total as well as the individual parts when asked to write a ratio

Difficulty with translating a mathematical answer into a real world context

Assumes a ratio is part to part instead of part to whole

Does not understand that in the ratio 1:n the total number of parts would be 1+n

Does not realise that the order is important in a ratio

Does not round up when an answer is non-sensical in a real-life context

Ignores numbers written as words in worded questions, and just calculates using those written in digits

Mixes up sides of a ratio, when working with vectors for line segments.

Divides total amount by each side of the ratio instead of dividing by the sum of the parts

May have estimated when using ratios with geometry

When converting ratio to fractions, puts the other side of the ratio as the denominator rather than the total

Believes that when converting ratio to fractions, the denominator is the difference between the parts of the ratio

Forgets to include values outside of the Venn diagram when finding the total

Attempts to simplify ratio by subtracting from both sides

Finds total instead of difference when asked 'How many more'

Does not realise ratio in worded problem can be simplified

Adds rather than multiplies when scaling up in direct proportion problems

Believes ratios can be combined without equalizing common terms

Uses the denominator when converting from fractions to ratio, rather than numerator.

Believes you do not include an "other" category when finding a total frequency

Does not know the meaning of consecutive

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Volume and Capacity Units ### ConstructName: Convert between l and  cm³ ### Question: Complete this statement:
\( 5 \) litres \( = \) ________\( \mathrm{cm}^{3} \)

Thinks the interior angles of any polygon add up to 180 degrees

Thinks the interior angle of a regular polygon is 10n

Thinks the interior angles of any polygon add up to 360

Thinks the interior angles of a polygon add up to 100n

Does not know how to calculate the sum of interior angles

Does not understand to find each interior angle of a regular polygon you divide the sum by the number of sides

Believes an interior angle in a regular polygon can be found using 180(n+2)/n

Believes an interior angle in a regular polygon can be found using 360 subtract (180 over number of sides)

Believes interior angles are outside a shape

Does not subtract 2 from the number of sides when finding the sum of interior angles

Finds interior angle rather than given angle

Confuses interior angles with exterior angles

Believes that you are unable to calculate the sum of the interior angles of an irregular polygon given the number of sides

Believes you add 2 instead of subtracting 2 to the numbers of sides when finding total interior angles

Believes the only way to find the interior angle of a regular polygon is to subtract the exterior angle from 180.

Identified the interior not exterior angle at a point

Divides by (n-2) rather than n when finding an interior angle

Subtracts 1 rather than 2 from the number of sides when finding the sum of interior angles

Thinks rule for calculating interior angles only applies to regular polygons

Does not understand what an interior angle is

Does not recognise that interior and exterior angles are supplementary

When given the perimeter of a regular polygon, subtracts instead of divides to find each side length

When given the perimeter of a regular polygon, multiplies instead of divides to find each side length

Assumes an angle has been split into equal parts

Thinks that only the sides need to be equal for a polygon to be regular

Does not understand that all angles and sides must be equal for regular shapes

Thinks that only the angles need to be the same for a polygon to be regular

Believes lines inside the shape are included when finding perimeter

Does not know the meaning of the word regular

Thinks co-interior angles sum to 90 degrees

Believes angles around a point sum to 180

Believes a polygon can have curved sides

Thinks the value of an exterior angle is 360 degrees

Calculates perimeter when asked for area

Ignores unlabelled angles when applying angle sum rules

Believes angles in a quadrilateral sum to 450 degrees

Finds the degrees for 1 person rather than the required number of people for a pie chart sector

Believes if a circle is divided into n equal sectors, the angles at the centre will each be n degrees

Does not understand the naming structure of polygons

Calculates area when asked for perimeter

Believes alternate angles sum to 180 degrees

Mixes up squaring and multiplying by 2 or doubling

Divides 360 degrees by sum of sides rather than given amount of angles

Believes that an exterior angle can have a value of 360 degrees

Thinks there are 180 degrees in a full turn

Does not recognise parallel lines angle rules applied inside a shape

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Solving Linear Inequalities ### ConstructName: Solve one-step linear inequalities in one variable where the variable appears on one side of the equation ### Question: Solve the inequality:
\(-\frac{1}{3} y \leq-6\)

Believes they can multiply/divide through by a negative number without changing the direction of the inequality

When describing a translation, counts the corners of squares (including the starting one) rather than the number of jumps between squares.

When describing a translation, miscounts when crossing an axis

When describing a translation, identifies the correct number of jumps but mixes up the directions .

When describing a translation, goes from the image to the original

Struggles to visualise the new position of an object after transformations

Rotates by the wrong center of rotation (not including the origin)

Reflects horizontally or vertically instead of across a diagonal line of symmetry

Counts the squares rather than the diagonals when reflecting in y=x or y=-x

Confuses negative and positive coordinates when working with graphs in four quadrants

Translates rather than reflects across a line of symmetry

When finding the distance between coordinates, includes the starting number

Does not recognise that a shape translated would not change orientation

Thinks a positive x value in a column vector means to move left rather than right

Counts the starting number when counting backwards

Counts squares instead of the correct scale when plotting coordinates

Counts on by 1, when asked to count forward/backwards in a step other than 1

Forgets to add starting value when calculating new coordinates

When asked to rotate by 180 degrees rotates by 90 degrees

Mixes up squaring and multiplying by 2 or doubling

When asked for a specific term in a sequence gives the term after

Has completed only one of the two operations.

Believes f(x - a) translates the function down a units

Believes displacement continues to increase even when returning to starting point

Believes the centre of rotation is always the origin

Believes a coordinate cannot be placed beyond the bounds of the grid shown

Does not notice all changes in a sequence of visual patterns

Reflects in an axis, instead of the given mirror line

Chooses an incorrect point in between the shapes when describing a 90 or 270 degree rotation

Chooses an incorrect point on the object when describing a 90 or 270 degree rotation

Counts the starting number when counting on

Does not recognise the origin as the starting point for coordinates

Does not know the meaning of square sign for right angle

Has used the wrong data point on the graph

Does not know how to find area by counting squares on a grid

Thinks positive values in a column vector denote movement left and down (and vice-versa for negative values).

Does not follow the arrows through a function machine, changes the order of the operations asked.

Does not count from the centre of enlargement to find the image

Makes an assumption about the squares on a scale representing either 1 or 10

Counts a side length more than once when counting perimeter due to not marking a starting point

Believes the square of a negative will also be negative

Carries out operations from left to right regardless of priority order

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Combined Events ### ConstructName: Calculate the probability of two independent events occurring without drawing a tree diagram ### Question: ![Two spinners shown. The first spinner has the numbers 1-4 and the second spinner has the number 1-5.]() You spin the above fair spinners
What is the probability of getting a \( 1 \) on both spinners?

When multiplying fractions, multiplies the numerator and adds the denominator

Does not know how to identify common factors from algebraic terms

Identifies a number as a common factor but doesn't identify a letter as a common factor

Misunderstands the concept of common factors in algebra

Does not know the meaning of common, eg. common factor

Identifies a letter as a common factor but doesn't identify a number as a common factor

Cannot identify a common factor when simplifying algebraic fractions

Believes factors of algebraic terms/expressions will only be numbers

When asked for factors of an algebraic expression, thinks a term will be a factor

When asked for factors of an algebraic expression, thinks any part of a term will be a factor

Does not recognise difference of two squares

Thinks a factor of the final digit is also a factor of the whole number

Does not understand how to divide algebraic terms

Does not understand how to multiply algebraic terms

Does not know how to simplify algebraic fractions

Identifies a common factor but not the highest common factor

Does not recognise when to find a factor from a worded question

Does not understand when like terms can be collected

Thinks they can simplify algebraic fractions by cancelling unlike variables

When factorising into single brackets, only factors out the numerical values and forgets to factorise possible variables

Does not recognise factorising a quadratic

Does not understand how to subtract algebraic fractions

Does not realise that two letters together, or a number and letter together, are multiplying

Identifies a multiple of a number but not a common multiple of a group of numbers

Believes only the first term needs to be divided when factorising an expression

Does not use the associative property of multiplication to find other factors of a number

Thinks only like terms can be multiplied

Incorrectly cancels what they believe is a factor in algebraic fractions

Thinks you can divide terms by different factors when simplifying an algebraic fraction

Does not know the meaning of the identity symbol

Does not understand that a number is divisible by the factors of its factors

When simplifying an algebraic fraction, only looks for number factors

Does not realise that two letters together are multiplying

Can not recognise all factor pairs of numbers

When factorising, finds a factor that goes into only the first term of the expression

Thinks you can ignore the variable when simplifying algebraic fractions

Does not know the meaning of the word identity

Believes a factor pair doesn't have to be 2 integers

Identifies a common multiple of two numbers when asked to consider a common multiple of three numbers

Does not realize we can omit the multiplication symbol in algebraic expressions

Believes that they can fully factorise an expression by taking out any common factor, not just the HCF

Does not think a number can be a factor of itself

Believes multiplication is not commutative

Does not realise similar has a specific mathematical meaning

Believes you need to know the value of all algebraic variables before any general conclusions can be made

When factorising, thinks the smallest coefficient will be the numerical factor, regardless of whether it divides into all terms

Believes sets are the same if the elements within them have a shared property

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Length Scale Factors in Similar Shapes ### ConstructName: Solve length problems involving similar shapes where the missing length is the longer one ### Question: What is the ratio of \( x \) to \( y \) ? ![A triangle. The bottom width is marked '15cm'. One side is split into two parts, one part is called 'x', the other is marked 'y'. A second line goes across the triangle from where the side is split into two parts. It is marked '5cm'. ]()

Does not understand that in the ratio 1:n the total number of parts would be 1+n

When writing a number as a percentage of another number gets them the wrong way around

When the change in value is greater than the original amount, thinks percentage change = original/change x 100

Does not understand that you can have percentages greater than 100%

Found the percentage but added it to 100%

Multiplied by 100 rather than divided by 100

Thinks when finding a percentage you divide by the percentage being asked

Finds 10% of an amount rather than the percentage being asked

Believes that a greater percentage represents a greater quantity

Believes a proportion of a whole is whole/part not part/whole

Does not understand a percentage is out of 100

Thinks they had to give the new amount rather than the total percentage that the original has been increased by.

Has considered the percentage rather than the percentage of the amount

Increases by the given percentage rather than finding the percentage of an amount

Found the percentage but forgot to add to 100%

Thinks when finding a percentage you multiply by the percentage being asked

Has found the total as a percentage of the amount being asked

Gives the change in value when asked for the percentage change

Thinks they double the numerator to turn a fraction into a percentage

Converts a fraction to a percentage by writing the numerator followed by the denominator

Believes a pie chart makes 360% not 360 degrees

Added the values together instead of finding the percentage

Finds a percentage when asked for a fraction

Error finding percentage value after finding 1%

When calculating a scale factor, thinks the division is the other way around.

Converts fraction to percentage by putting % sign after the denominator

Thinks you just add a % sign to the numerator to make a fraction into a percentage

Multiplies instead of divides by 100 when converting a percentage to a decimal

Multiplies by 1000 instead of 100 when converting a decimal to a percentage

Believes a fraction out of a number other than 100 represents a percentage

Thinks you need to just add a % sign to a number to make it a percentage

Believes that any percentage of a larger number will be greater than any percentage of a smaller number

Believes that a fraction means dividing the denominator by the numerator

Does not know that you can have a decimal percentage

Believes you can divide the change by 100 to find the percentage change

When finding a percentage of an amount, assumes they divide the percentage by 10, but not the amount, before multiplying

Does not understand how to find percentages of an amount

Does not understand a percentage can be applied to something not out of 100

Multiplied by the scale factor rather than divided in direct proportion problems

Believes a percentage of a number will be a multiple of the percentage

Divides length of original by length of image rather finding length of image divided by length of original to find an enlargement scale factor.

Divided by 100 rather than multiplied by 100

Multiplies rather than divides when answering worded problems

Multiplies by the denominator instead of dividing when finding a fraction of an amount

Thinks that to decrease by a percentage you multiply by 1+ the decimal equivalent to that percentage

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question: ### SubjectName: Converting between Decimals and Percentages ### ConstructName: Convert integer percentages with 1 digit to decimals  ### Question: How do you write \( 2 \% \) as a decimal?

Does not understand a percentage is out of 100

Forgets to change the numerators when finding equivalent fractions

Does not find a common denominator when adding/subtracting fractions

Does not understand how to maintain equivalence in fractions

When subtracting fractions, subtracts the numerators and denominators

Does not understand equivalent fractions

Uses the numerator of an improper fraction as the denominator of the equivalent mixed number

When adding fractions, changes to common denominator but leaves numerators the same

Does not find a common denominator when comparing fractions

When subtracting a fraction from a mixed number, assumes they subtract the numerator from the integer and the numerator of the mixed number

Removes the common denominator when making equivalent algebraic fractions

Believes they should multiply each numerator by its original denominator when adding/subtracting fractions

Confused mixed numbers with improper fractions

Believes that if the numerators of 2 fractions are equal then the fractions are equal, regardless of the denominator

Believes a mixed number can contain an improper fraction

Finds an equivalent improper fraction when asked for a mixed number

When subtracting a fraction from a mixed number, assumes they just subtract the numerator from the integer of the mixed number

When subtracting fractions, subtracts the numerators and multiplies the denominators

Believes that fractions with the same denominator are equal

Does not consider the denominator when adding/ subtracting fractions. Adds/subtracts the numerator and makes the denominator the one with the highest value.

When subtracting fractions, divides the numerators

When adding fractions, adds the numerators and denominators

Does not understand how to subtract algebraic fractions

Adds or subtracts from numerator first when solving an equation with a fraction

When adding fractions with identical numerators, leaves the numerator and adds the denominators

Subtracts numerators when dividing fractions with a common denominator

Can identify the multiplier used to form an equivalent fraction but does not apply to the numerator

After multiplying to get a common denominator, simplifies back to the original fractions

Believes equivalent fractions are formed with addition not multiplication

Incorrectly cancels what they believe is a factor in algebraic fractions

Mixes up the terms for the parts of a fraction

Uses numerators to convert fractions to ratio despite different denominators

When converting from an improper fraction to a mixed number, adds on the part left to make the whole, rather than subtracting it

Thinks you multiply by the numerator and add the denominator when converting a mixed number to an improper fraction

Believes you add the whole to the numerators, ignoring denominators, when adding a mixed number to a proper fraction with the same denominator.

Is not familiar with the names of the different parts of a fraction

When adding/subtracting fractions, adds/subtracts the denominators and multiplies the numerators

Believes when changing an improper fraction to a mixed number you divide the numerator by 10, regardless of the denominator

When subtracting mixed numbers, does not realise that a negative fractional answer needs to be taken away from their integer answer

Believes a common denominator must always be the two denominators multiplied together

When solving a problem that requires an inverse operation (e.g. missing number problems), does the original operation

Cannot identify a fraction on a number line

Does not see fraction diagrams as equivalent if the shading is in a different pattern

Believes a common denominator can only be the lowest common multiple

Believes the whole and remainder are the other way when changing an improper fraction to a mixed number

When asked to subtract a fraction from a unit, removes the unit value from the denominator

Does not know how to solve linear equations involving fractions

Uses numerator as whole part of a fraction

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Writing Expressions ### ConstructName: Use brackets to write function machines as calculations ### Question: Tom and Katie are arguing about the result of this Function Machine:
Tom says the output is: \( 3 n-12 \)
Katie says the output is: \( 3(n-4) \)
Who is correct? ![A function machine with input n and operations subtract 4, multiply by 3]()

Does not think a factorised expression is equivalent to its multiplied out form

When solving an equation, subtracts instead of dividing

When solving an equation, uses the same operation rather than the inverse.

When solving an equation, adds instead of dividing

When solving an equation,  subtracts instead of multiplying

When solving an equation where there is a negative sign in front of the variable and its equivalent value, assumes the variable has a negative value

When solving an equation, multiplies instead of dividing

When solving an equation thinks they divide when in fact they need to multiply

Subtracts instead of adds when using inverse operations to solve an equation

When solving an equation, thinks the divisor and dividend are the opposite way around

When solving an equation forgets to eliminate the coefficient in front of the variable in the last step

When solving an equation thinks they add when in fact they need to multiply

When solving an equation, multiplies instead of adding

Confusion over whether multiplication or division is the correct way to solve the problem.

Does not reverse the order of operations when solving an equation

Thinks the inverse of subtraction is division

Swaps the dividend and divisor in order to get an integer answer

Thinks the inverse of multiplication is addition

Thinks they solve mx + c = a, by substituting in the value of a for x and therefore calculating ma + c

Adds or subtracts from numerator first when solving an equation with a fraction

When solving a problem that requires an inverse operation (e.g. missing number problems), does subtraction when division is required

Thinks the inverse of subtraction is multiplication

Misunderstands the process of making a variable the subject of an equation

When solving a problem that requires an inverse operation (e.g. missing number problems), does the original operation

Thinks the inverse of addition is division

Believes that a number divided by itself equals 0

Believes dividing by 0.1 is the same as dividing by 10

Believes that the solution to ax + b = 0 is x = -a/b

Assumes the denominator is divided by the numerator

Stops iteration too early when solving equations

Believes dividing by a unit fraction is equivalent to dividing by its reciprocal

Thinks the inverse of addition is multiplication

Believes you can subtract from inside brackets without expanding when solving an equation

Believes they must change the direction of the inequality when dividing through by a positive number.

Believes you can add or subtract from inside brackets without expanding when solving an equation

Believes the solution of mx + c = 0 is x = m

Thinks the subtraction sign means divide

Believes they can multiply/divide through by a negative number without changing the direction of the inequality

Believes 0 multiplied by a number gives the number

Incorrectly clears fractions from equations

Believes you keep the divisor unchanged when dividing by a fraction

Believes the solution to mx + c = a is the x intercept of y = mx +c

Believes that dividing by a decimal is the same as dividing by its reciprocal

Does not understand that when multiplying both sides of an equation by an amount every term must be multiplied by the same amount

Believes that a number divided by itself equals the original number

Believes dividing two positives will give a negative answer

Heads towards zero when subtracting from a negative

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Place Value ### ConstructName: Write 3 digit numbers as words  ### Question: How do you write this number in words?
\( 107 \)

The student does not understand the place value of 3-digit numbers

Finds volume when asked for surface area

When calculating surface area, only includes visible faces

Finds surface area when asked for volume

Multiplies all given dimensions when calculating an area

Cannot identify the correct side lengths to use when asked to find the area of a face

Believes you find the surface area of a prism by finding the total area of the unique faces

Calculates perimeter when asked for area

Thinks all dimensions given must be used when calculating the area

Believes volume of a cuboid is the area of the three unique faces multiplied together

adds all given dimensions when calculating area

Includes all faces when finding surface area of a compound shape, including those which are stuck together

Calculates area when asked for perimeter

Adds when calculating volume instead of multiplying

Calculates circumference when asked for area

Finds area of one face when asked for volume

When calculating perimeter, only adds two sides

Adds when calculating area instead of multiplying

Does not know how to find missing lengths in a composite shape

Does not know how to calculate the volume of a prism

When calculating perimeter, multiplies instead of adds

Has used slant height and base to find area rather than perpendicular height and base

Finds the circumference instead of the area

Uses an incorrect side length when splitting a composite shape into parts

When calculating the area of a rectilinear compound shape, calculates the are of the surrounding rectangle

Has multiplied base by slant height and perpendicular height to find area of a triangle

Does not recognise that all lengths on a cube are equal

Multiplies all terms together when simplifying an expression

Does not convert measurements to have the same units before calculating area or volume

Does not divide by 2 when calculating the area of a triangle

Believes the area of a rectangle is calculated by length add width

When calculating volume expressed in cubes, counts the faces of all the cubes

Believes the number of edges on a 3D shape match those of the 2D cross section

Thinks measures of area must end in squared

Thinks only labelled dimensions need to be included when calculating the perimeter

Multiplies all given dimensions when calculating the length of the hypotenuse in a right angled triangle

Mixes up squaring and multiplying by 2 or doubling

When calculating volume expressed in cubes, counts only the squares on one face

Does not have when calculating the area of a triangle

When multiplying out 2 or more brackets, does not multiply all the combinations of terms

Struggles to visualise nets of 3D shapes

Believes you multiply all of the values given when using the capture-recapture sampling method

Adds instead of subtracts when using an area model for multiplication

When calculating perimeter, only adds the marked lengths

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.


Question:
### SubjectName: Averages and Range from Frequency Table ### ConstructName: Calculate the mean from a frequency table ### Question: This table shows the ages of people on a university running team.
To calculate the mean age, what is the most sensible heading for the third column? \begin{tabular}{|c|c|l|}
\hline Age & Frequency & \\
\hline \( 18 \) & \( 7 \) & \\
\hline \( 19 \) & \( 3 \) & \\
\hline \( 20 \) & \( 1 \) & \\
\hline \( 21 \) & \( 2 \) & \\
\hline
\end{tabular}

Added rather than multiplied by a scale factor

Subtracted rather than divided by a scale factor

Thinks a variable next to a number means addition rather than multiplication

When multiplying just writes the numbers next to each other

Thinks the multiplication sign means to add

Believes they add the conversion factor when converting between units

Thinks two variables next to each other represent addition

Adds when calculating area instead of multiplying

Thinks a variable next to a number means write the digits together

Writes the index as a digit on the end of a number

Adds when calculating volume instead of multiplying

Adds instead of multiplying when expanding bracket

Believes the area of a rectangle is calculated by length add width

When adding, just writes the digit to the right of the number without considering place value

When calculating perimeter, multiplies instead of adds

Forgets to add starting value when calculating new coordinates

When multiplying fractions, multiplies the numerator and adds the denominator

Adds rather than multiplies for combined probabilities

Multiplies rather than adds when given the command word 'more'

Adds rather than multiplies when asked for a product

Added the values together instead of finding the percentage

Adds rather than multiplies when scaling up in direct proportion problems

Calculates perimeter when asked for area

Confuses additive and multiplicative relationships

Multiplied by the scale factor rather than divided in direct proportion problems

Thinks multiplication and addition are the same

Adds instead of multiplies when using grid method

When calculating a scale factor, multiplies instead of dividing.

Believes the fx column for calculating mean is value + frequency

Adds instead of multiplies when multiplying algebraic terms

When adding column vectors, multiplies rather than adds

Thinks the subtraction sign means multiply

When solving an equation thinks they add when in fact they need to multiply

Adds instead of multiplies when calculating total frequency

Believes they write the digits next to each other instead of multiplying, when multiplying a vector by an integer.

When asked to multiply fractions, uses the method for adding instead

Adds rather than multiplies when asked for a total

When solving an equation, multiplies instead of adding

When scaling the perimeter multiplies the number of sides by the scale factor

Prompt: A math problem, its correct answer, and an incorrect answer are given. Retrieve the reasoning behind how the incorrect answer was derived.